Chemical Reactions and Their Control on the Femtosecond Time Scale: 20th Solvay Conference on Chemistry

ADVANCES IN CHEMICAL PHYSICS VOLUME 101

EDITORIAL BOARD BRUCE,J. BERNE,Department of Chemistry, Columbia University, New York, New York, U.S.A. KURTBINDER,Institut fur Physik, Johannes Gutenberg-Universit Mainz, Mainz, Germany A. WELFORD CASTLEMAN, JR., Department of Chemistry, The Pennsylvania State University, University Park, Pennsylvania, U.S.A. DAVID CHANDLER, Department of Chemistry, University of California, Berkeley, California, U.S.A. M. S. CHILD, Department of Theoretical Chemistry, University of Oxford, Oxford, U.K. WILLIAM T. COFFEY,Department of Microelectronics and Electrical Engineering, Trinity College, University of Dublin, Dublin, Ireland F. FLEMING CRIM, Department of Chemistry, University of Wisconsin, Madison, Wisconsin, U.S.A. ERNEST R. DAVIDSON, Department of Chemistry, Indiana University, Bloomington, Indiana, U.S.A. GRAHAMR. FLEMING,Department of Chemistry, The University of Chicago, Chicago, Illinois, U.S.A. KARLF. FREED, The James Franck Institute, The University of Chicago, Chicago, Illinois, U.S.A. PIERRE GASPARD, Center for Nonlinear Phenomena and Complex Systems, Brussels, Belgium ERICJ. HELLER,Institute for Theoretical Atomic and Molecular Physics, HarvardSmithsonian Center for Astrophysics, Cambridge, Massachusetts, U.S.A. ROBIN M. HOCHSTRASSER, Department of Chemistry, The University of Pennsylvania, Philadelphia, Pennsylvania, U.S.A. R. KOSLOFF,The Fritz Haber Research Center for Molecular Dynamics and Department of Physical Chemistry, The Hebrew University of Jerusalem, Jerusalem, Israel RUDOLPH A. MARCUS, Department of Chemistry, California Institute of Technology, Pasadena, California, U.S.A. G. Nrco~is,Center for Nonlinear Phenomena and Complex Systems, Universith Libre de Bruxelles, Brussels, Belgium THOMAS P. RUSSELL,Department of Polymer Science, University of Massachusetts, Amherst, Massachusetts DONALD G. TRUHLAR, Department of Chemistry, university of Minnesota, Minneapolis, Minnesota, U.S.A. JOHND. WEEKS,Institute for Physical Science and Technology and Department of Chemistry, University of Maryland, College Park, Maryland, U.S.A. PETERG. WOLYNES, Department of Chemistry, School of Chemical Sciences, University of Illinois, Urbana, Illinois, U.S.A.

Advances in CHEMICAL PHYSICS Chemical Reactions and Their Control on the Femtosecond Time Scale XXth Solvay Conference on Chemistry Edited by

PIERRE GASPARD Center for Nonlinear Phenomena and Complex Systems Universiti Libre de Bruxelles Brussels, Belgium

and

IRENE BURGHARDT Institut fur Physikalische und Theoretische Chemie der Universitiit Bonn Bonn, Germany

Series Editors

I. PRIGOGINE

STUART A. RICE

Center for Studies in Statistical Mechanics and Complex Systems The University of Texas Austin, Texas and International Solvay Institutes Universitk Libre de Bruxelles Brussels, Belgium

Department of Chemistry and The James Franck Institute The University of Chicago Chicago, Illinois

VOLUME 101

AN INTERSCIENCE@ PUBLICATION

NEW YORK

JOHN WILEY & SONS, INC.

CHICHESTER

WEINHEIM

BRISBANE

SINGAPORE TORONTO

A NOTE TO THE READER This book has been electronically reproduced from digital information stored at John Wiley & Sons, Inc. We are pleased that the use of this new technology will enable us to keep works of enduring scholarly value in print as long as there is a reasonable demand for them. The content of this book is identical to previous printings.

This text is printed on acid-free paper. An InterscienceB Publication Copyright 0 1997 by John Wiley Sc Sons, Inc. All rights reserved. Published simultaneously in Canada. Reproduction or translation of any part of this work beyond that permitted by Section 107 or 108 of the 1976 United States Copyright Act without the permission of the copyright owner is unlawful. Requests for permission or further information should be addressed to the Permissions Department, John Wiley & Sons, Inc. Library of Congress Catalog Number: 58-9935 ISBN 0-471-18048-3 Printed in the United States of America 10 9 8 7 6 5 4 3 2

ADMINISTRATIVE COUNCIL OF THE INSTITUTS INTERNATIONAUX DE PHYSIQUE ET DE CHIMIE founded by E. SOLVAY J. SOLVAY, Prksident

F. BINGEN,Vice Prksident & Trksorier G. NICOLIS, Secrituire D. JANSSEN A. JAUMOTTE

F. LAMBERT J.-M. RRET 3.

REISSE

L.WIJNS

I. PRIGOGINE, Directeur I. ANTONIOU, Directeur-adjoint

SCIENTIFIC COMMITTEE IN CHEMISTRY OF THE INSTITUTS INTERNATIONAUX DE PHYSIQUE ET DE CHIMIE founded by E. SOLVAY S . A. EWE,Prisident, The University of Chicago, U.S.A. M.EIGEN,Max Planck Institute, Gottingen, Germany K. FUKUI,Institute for Fundamental Chemistry, Kyoto, Japan B. HESS,Max Planck Institute, Dortmund, Germany V

vi

ADMINISTRATIVE COUNCIL AND SCIENTIFIC COMMITEE

E. KATCHALSKI-KATZIR, The Weizmann Institute of Science, Rehovot, Israel J.-M. LEHN,Universit6 de Strasbourg, France W. N. LIPSCOMB, Harvard University, Cambridge, U.S.A. V. PRELOG, ETH Zurich, Switzerland Lord TODD,University of Cambridge, United Kingdom A. BELLEMANS, Secrc%uire, Universit6 Libre de Bruxelles, Belgium

PARTICIPANTS H. R. H.the PRINCE LAURENT of Belgium, President of the Royal Institute for the Sustainable Management of Natural Resources and the Promotion of Clean Technologies P. W. BRUMER, University of Toronto, Ontario, Canada I. BURGHARDT, Universith Libre de Bruxelles, Belgium G. CASATI,University of Milan, Italy L. S. CEDERBAUM, Universitit Heidelberg, Germany M. CHERGUI, Universitk de Lausanne, Switzerland M. S. CHILD,University of Oxford, United Kingdom M. DESOUTER-LECOMTE, UniversitC de Likge, Belgium V. ENGEL,Universitat Wurzburg, Germany U. EVEN,Tel-Aviv University, Israel R. W. FIELD,Massachusetts Institute of Technology, Cambridge, U.S.A. The University of Chicago, U.S.A. G. R. FLEMING, P.GASPARD, Universitk Libre de Bruxelles, Belgium D. GAUYACQ, UniversitC de Paris-Sud, Orsay, France G. GERBER, Universitit Wurzburg, Germany M. GODEFROID, Universith Libre de Bruxelles, Belgium H. HAMAGUCHI, The University of Tokyo, Japan P.-H. HEENEN, UniversitC Libre de Bruxelles, Belgium M. HERMAN, Universith Libre de Bruxelles, Belgium Benno HESS,Max Planck Institute, Heidelberg, Germany Bemd. A. HESS,Universit5t Bonn, Germany J. JEENER, Universitk Libre de Bruxelles, Belgium R. JOST,Service National des Champs Intenses, Grenoble, France Ch. JUNGEN, UniversitC de Paris-Sud, Orsay, France M. E. KELLMAN,University of Oregon, Eugene, U.S.A vii

...

Vlll

PARTICIPANTS

A. KIRSCH-DE MESMAEKER, Universitk Libre de Bruxelles, Belgium T. KOBAYASHI, The University of Tokyo, Japan B. KOHLER, The Ohio State University, Columbus, U.S.A. F. KONG,Chinese Academy of Sciences, Beijing, China V. S. LETOKHOV, Russian Academy of Sciences, Moscow, Russia R. D. LEVINE,The Hebrew University, Jerusalem, Israel J. LIEVIN,Universiti Libre de Bruxelles, Belgium M. LOMBARD[,tJniversit6 Joseph Fourier, Grenoble, France J.-C. LORQUET, Universitk de Likge, Belgium P. MANDEL, Universiti Libre de Bruxelles, Belgium J. MANZ,Freie Universit2t Berlin, Germany R. A. MARCUS, California Institute of Technology, Pasadena, U.S.A. J.-L. MARTIN,Ecole Polytechnique, Palaiseau, France W. H. MILLER,University of California, Berkeley, U.S.A. S. MUKAMEL, University of Rochester, U.S.A. D. M. NEUMARK, University of California, Berkeley, U.S.A. H. J. NEUSSER, Technische Universitiit Munchen, Garching, Germany G. NICOLIS,Universitk Libre de Bruxelles, Belgium T. OKADA,Osaka University, Japan J.-P. PIQUE,Universitk Joseph Fourier, Grenoble, France E. POLLAK, The Weizmann Institute of Science, Rehovot, Israel I. FRIGOGINE, Universitk Libre de Bruxelles, Belgium M. QUACK,ETH Zurich, Switzerland H. RABITZ,Princeton University, U.S.A. J. REISSE,Universiti Libre de Bruxelles, Belgium F. REMACLE, Universiti de Likge, Belgium S. A. RICE,The University of Chicago, U.S.A. J.-P. RYCKAERT, Universitk Libre de Bruxelles, Belgium K. SCHAFFNER, Max Planck Institute, Mulheim an der Ruhr, Germany R. ScHiNKE, Max Planck Institute, Gottingen, Germany E. W. SCHLAG, Technische Universitiit Munchen, Garching, Germany

PARTICIPANTS

M.SHAPIRO, The Weizmann Institute of Science, Rehovot, Israel

University of Oxford, United Kingdom T. P. SOFTLEY, D. J. TANNOR, The Weizmann Institute of Science, Rehovot, Israel J. TROE,UniversiGt Gottingen, Germany M.VANDER AUWERAER, Katholieke Universiteit Leuven, Belgium L. Wosm, Freie UniversiGt Berlin, Germany K. YAMANOUCHI, The University of Tokyo, Japan K. YAMASHITA, The University of Tokyo, Japan A. H.ZEWAIL, California Institute of Technology, Pasadena, U S A . J.-Y. ZHOU,Zhongshan University, Canton, China

ix

SCIENTIFIC SECRETARY OF THE CONFERENCE P. GASPARD, Conference C h a i m n I. BURGHARDT D. DAEMS B. GREMAUD

S. R. JAIN

KARLSON D. MEI A.

A. SUAREZ

P.VAN EDEVAN DER PALS

ADMINISTRATIVE SECRETARY OF THE CONFERENCE M.ADAM

A X . BLONDLET N. GALLAND M.KIES P. KINET

M.n E t N

M. KUNEBEN I. SAVERINO J. TACHELET S. WELLENS xi

GROUP PHOTOGRAPH OF THE PARTICIPANTS IN THE XXTH SOLVAY CONFERENCE ON CHEMISTRY 1. I. Prigogine 2. S. A. Rice 3. R. A. Marcus 4. E. W. Schlag 5. 1. Troe 6. M. Quack 7. L. Woste 8. P. Gaspard 9. H. Rabitz 10. Benno Hess 11. G. R. Fleming 12. A. H.Zewail 13. A. Karlson 14. 1. Burghardt 15. D. Mei 16. D. Daems 17. E. Pollak 18. T. Okada 19. T. Kobayashi 20. M. S. Child 21. D. M.Neumark

22. K. Yamanouchi 23. S. Mukamel 24. G. Casati 25. R. D. Levine 26. G. Gerber 27. P. W. Brumer 28. P. van Ede van der Pals 29. S. R. Jain 30. B. Grimaud 31. A. Sukez 32. M. Shapiro 33. J. Manz 34. R. W. Field 35. M. Herman 36. D. Gauyacq 37. M. Lombardi 38. M. E. Kellman 39. H. J. Neusser 40. B. Kohler 4 1. R. Jost 42. H.Hamaguchi

M. Godefroid D. J. Tannor L. S. Cederbaum M. Chergui V. Engel Ch. Jungen J.-L. Martin 50. F. Remacle 51. F. Kong 52. U. Even 53. M.Van der Auweraer 54. J. LiCvin 55. J.-P. Pique 56. R. Schinke 57. T. Softley 58. Bemd A. Hess 59. K. Yamashita 60.J.-Y. Zhou 6 I. K. Schaffner 62. L. Michaille

43. 44. 45. 46. 47. 48. 49.

...

Xlll

CONTRIBUTORS TO VOLUME 101 A. BARTANA, Department of Physical Chemistry and The Fritz Haber

Research Center, The Hebrew University, Jerusalem, Israel

T. BAUMERT, Physikalisches Institut, Universitit Wurzburg, Wurzburg, Germany C. BECK,Max-Planck-Institut fiir Stromungsforschung, Gottingen, Germany R. S. BERRY,Department of Chemistry, The James Franck Institute, The University of Chicago, Chicago, Illinois V. BONACIC-KOUTECKY, Walter Nernst-Institut Humbold-Universitiit zu Berlin, Berlin, Germany P. BRUMER, Chemical Physics Theory Group and The Ontario Laser and Lightwave Research Centre, University of Toronto, Toronto, Canada

I. BURGHARDT, Service de Chimie Physique and Center for Nonlinear Phenomena and Complex Systems, Universitk Libre de Bruxelles, Brussels, Belgium Z. CHEN,Chemical Physics Theory Group and The Ontario Laser and Lightwave Research Centre, University of Toronto, Toronto, Canada M. CHERGUI, Institut de Physique Exp6rimentale Facult; des Sciences, BSP Universite de Lausanne, Lausanne-Dorigny, Switzerland M. CHO,Department of Chemistry, Massachusetts Institute of Technology, Cambridge, Massachusetts H. CHOI,Department of Chemistry, University of California Berkeley, California, and Chemical Sciences Division, Lawrence Berkeley Laboratory, Berkeley, California C. CIORDAS-CIURDARIU, Department of Chemistry, University of Rochester, Rochester, New York A. J. DOBBYN,Theory, Computational Science and Computing Division, Daresbury, Warrington, Cheshire, England R. W. FIELD,Department of Chemistry, Massachusetts Institute of Technology, Cambridge, Massachusetts G. R. FLEMING, Department of Chemistry, James Franck Institute, University of Chicago, Chicago, Illinois xv

CONTRIBUTORS TO VOLUME 101

xvi

H. FLOTHMANN, Max-Planck-Institut fur Stromungsforschung, Gottingen, Germany P. GASPARD, Service de Chimie Physique and Center for Nonlinear Phenomena and Complex Systems, Universit6 Libre de Bruxelles, Brussels, Belgium J. GAUS,Walter Nemst-Institut, Humbold-Universitat zu Berlin, Berlin, Germany

G. GERBER,Physikalisches Institut, UniversiGt Wurzburg, Wurzburg, Germany

I. GOLUB,Department of Chemical Physics, The Weizmann Institute of Science, Rehovot, Israel J. HELBING, Physikalisches Institut, Universitit Wurzburg, Wurzburg, Germany A. HISHIKAWA, Department of Pure and Applied Sciences, The University of Tokyo, Tokyo, Japan H. ISHIKAWA, Department of Chemistry, Massachusetts Institute of Technology, Cambridge, Massachusetts M. P. JACOBSON, Department of Chemistry, Massachusetts Institute of Technology, Cambridge, Massachusetts C. JEANNIN, Institut de Physique Expirimentale, BSP Universit6 de Lausanne, Lausanne-Dorigny, Switzerland T. JOO, Department of Chemistry, James Franck Institute, University of Chicago, Chicago, Illinois

CH.JUNGEN, Laboratoire Aim6 Cotton du CNRS, Universit6 de Paris-Sud, Orsay, France

H.-M. KELLER,Max-Planck-Institut fur Stromungsforschung, Gottingen, Germany V. KHIDEKEL, Department of Chemistry, University of Rochester, Rochester, New York

M.V. KOROLKOV, Belarus Academy of Sciences, Institute of Physics, Republic of Belarus

R. KOSLOFF,Department of Physical Chemistry and The Fritz Haber Research Center, The Hebrew University, Jerusalem, Israel TH. LEISNER,Institut fur Experimentalphysik, Freie Universitit Berlin, Berlin, Germany

CONTRIBUTORS TO VOLUME 101

xvii

V. S. LETOKHOV, Institute of Spectroscopy, Russian Academy of Sciences, Troitzk, Moscow Region, Russia R. D. LEVINE,The Fritz Haber Research Center for Molecular Dynamics, The Hebrew University, Jerusalem, Israel, and Department of Chemistry and Biochemistry, University of California Los Angeles, Los Angeles, California S. R. MACKENZIE, Physical and Theoretical Chemistry Laboratory, Oxford, United Kingdom J. MANZ,Institut fur Physikalische und Theoretische Chemie, Freie Universitiit Berlin, Berlin, Germany R. A. MARCUS, Noyes Laboratory of Chemical Physics, California Institute of Technology, Pasadena, California F. MERKT,Laboratorium fur Physikalische Chemie, Zurich, Switzerland W. H. MILLER, Department of Chemistry, University of California, Berkeley, California, and Chemical Sciences Division, Lawrence Berkeley Laboratory, Berkeley, California Department of Chemistry, University of California, BerkeD. H. MORDAUNT, ley, California S. MUKAMEL, Department of Chemistry, University of Rochester, Rochester, New York R. NEUHAUSER, Institut fur Physikalische und Theoretische Chemie, Technische Universitit Miinchen, Garching, Germany D. M. NEUMARK, Department of Chemistry, University of California, Berkeley, California, and Chemical Sciences Division, Lawrence Berkeley Laboratory, Berkeley, California

H. J. NEUSSER, Institute fur Physikalische und Theoretische Chemie, Tech-

nische Universitit Munchen, Garching, Germany J. P. OBRIEN, Department of Chemistry, Massachusetts Institute of Technology, Cambridge, Massachusetts K. OHDE,Department of Pure and Applied Sciences, The University of Tokyo, Tokyo, Japan D. L. OSBORN, Department of Chemistry, University of California, Berkeley, California, and Chemical Sciences Division, Lawrence Berkeley Laboratory, Berkeley, California G. K. PARAMONOV, Belarus Academy of Sciences, Institute of Physics, Minsk, Republic of Belarus

xviii

CONTRIBUTORS TO VOLUME 101

M. T. PORTELLA-OBERLI, Institut de Physique Ex+rimentale, BSP Universitt de Lausanne, Lausanne-Dorigny, Switzerland W. F. POLIK,Department of Chemistry, Massachusetts Institute of Technology, Cambridge, Massachusetts H. RABITZ,Department of Chemistry, Princeton University, Princeton, New Jersey B . RJZISCHL-LENZ, Institut f i r Physikalische und Theoretische Chemie, Freie Universitit Berlin, Berlin, Germany S. A. RICE,Department of Chemistry, The James Franck Institute, The University of Chicago, Chicago, Illinois D. ROLLAND, Physical and Theoretical Chemistry Laboratory, Oxford, United Kingdom

H. RUPPE,Institut fur Experimentalphysik, Freie Universi~tBerlin, Berlin, Germany

S. R u n , Institut fur Experimentalphysik, Freie Universitit Berlin, Berlin, Germany R. SCHINKE, Max-Planck-Institut Fur Stromungsforschung, Gottingen, Germany

E. W. SCHLAG, Institut fur Physikalische und Theoretische Chemie, Technische Universitat Munchen, Munchen, Gemany E. SCHREIBER, Institut fur Experimentalphysik, Freie Universitat Berlin, Berlin, Germany M. SHAPIRO, Department of Chemical Physics, The Weizmann Institute of Science, Rehovot, Israel A. SHNITMAN, Department of Chemical Physics, The Weizmann Institute of Science, Rehovot, Israel I. SOFER, Department of Energy and Environment, The Weizmann Institute of Science, Rehovot, Israel T. P. SOFTLEY,Physical and Theoretical Chemistry Laboratory, Oxford, United Kingdom S. A. B. SOLINA, Department of Chemistry, Massachusetts Institute of Technology, Cambridge, Massachusetts M. STUMPF, Physical and Theoretical Chemistry Laboratory, Oxford, United Kingdom D. J. TANNOR, Department of Chemical Physics, The Weizrnann Institute of Science, Rehovot, Israel

CONTRIBUTORS TO VOLUME 101

xix

J. "ROE,Institut fiir Physikalische Chemie, Universitat Gottingen, Gottingen,

Germany ST.VAJDA,Institut fiir Experimentalphysik, Freie Universitilt Berlin, Berlin, Germany R. DE VIVIE-RIEDLE, Institut fiir Physikalische und Theoretische Chemie, Freie Universitiit Berlin, Berlin, Germany S. WOLF,Institut f i r Experimentalphysik, Freie Universitilt Berlin, Berlin, Germany L. Wosm, Institut fur Experimentalphysik, Freie UniversiGt Berlin, Berlin, Germany K. YAMANOUCHI, Department of Pure and Applied Sciences, The University of Tokyo, Tokyo, Japan A. YOGEV,Department of Energy and Environment, The Weizrnann Institute of Science, Rehovot, Israel Arthur Amos Noyes Laboratory of Chemical Physics, CaliA. H. ZEWAIL, fornia Institute of Technology, Pasadena, California

INTRODUCTION Few of us can any longer keep up with the flood of scientific literature, even in specialized subfields. Any attempt to do more and be broadly educated with respect to a large domain of science has the appearance of tilting at windmills. Yet the synthesis of ideas drawn from different subjects into new, powerful, general concepts is as valuable as ever, and the desire to remain educated persists in all scientists. This series, Advances in Chemical Physics, is devoted to helping the reader obtain general information about a wide variety of topics in chemical physics, a field that we interpret very broadly. Our intent is to have experts present comprehensive analyses of subjects of interest and to encourage the expression of individual points of view. We hope that this approach to the presentation of an overview of a subject will both stimulate new research and serve as a personalized learning text for beginners in a field. I. PRIGNINE STUART A. RICE

xxi

The XXth Solvay Conference on Chemistry was held at the Free University of Brussels from November 28 to December 2, 1995. It gathered 65 participants, including scientists from the Free University of Brussels, and has consisted of 17 reports, 11 communications, and 9 discussion sessions. This volume contains the papers and discussions presented at the Conference. During the past decade, an unprecedented time resolution has been achieved in the study of chemical reactions thanks to the development of femtosecond lasers. This remarkable progress currently allows one to observe the dynamics of molecules on the intrinsic time scale of their vibrations, dissociative motions, and electronic excitations. In addition, this new laser technology has concretized the possibility of the coherent control of chemical reactions as well as of more general quantum systems. It is to these problems of fundamental importance and interest that this XXth Solvay Conference has been devoted. Special emphasis was placed on the new perspectives opened at this challenging and promising new frontier of science, including possibilities of technological applications. We wish to express our thanks to Ilya Prigogine, Director of the Solvay Institutes, to Grigoire Nicolis, Secretary of the Solvay Instibtes, and to Stuart A. Rice, Chairman of the Scientific Committee in Chemistry of the Solvay Institutes, for their support and advice in the preparation of this Conference. Special thanks are due to Mr. Jacques Solvay for the active interest he has taken in the Conference. It is also our great pleasure to thank Nadine Galland, as well as the other members of the Scientific and Administrative Secretaries, for their essential role in the organization and running of the Conference and in the preparation of the Proceedings. The Conference has been financially supported by the International Institutes of Physics and Chemistry, founded by E. Solvay and as an Advanced Research Meeting by a grant of the DG XI1 of the European Commission, which are gratefully thanked.

P. GASPARD I. BURGHARDT

xxiii

CONTENTS xxxi

OPENING REMARKS J. Solvay

FEMTOCHEMISTRY: FROMISOLATED MOLECULES TO CLUSTERS FEMTOCHEMISTRY: CHEMICAL REACTION DYNAMICS AND THEIR CONTROL

3

A. H.Zewail Discussion on the Report by A. H.Zewail

COHERENT CONTROL WITH FEMTOSECOND LASERPULSES

47

T. Baumert, J. Helbing, and G. Gerber Discussion on the Report by G. Gerbet

GENERAL DISCUSSION ON FEMTOCHEMISTRY: FROM ISOLATED MOLECULES TO CLUSTERS FEMTOCHEMISTRY:

83

FROMCLUSTERS TO SOLUTIONS

SIZE-DEPENDENT ULTRAFAST RELAXATION PHENOMENA IN METAL CLUSTERS

101

R. S. Berry, K Bonu&5Kouteck$ J. Gaus, Th. Leisner, H. Ruppe, S. Rutz, E. Schreiber, St. Vajak, S. Wolf;L. Woste, J. Manz, B. Reischl-Lenz, and R. de Vivie-Riedle Discussion on the Report by L Woste FEMTOSECOND CHEMICAL DYNAMICS IN CONDENSED PHASES

141

G. R. Fleming, T Joo, and M. Cho Discussion on the Report by G. R. Fleming

FEMTOSECOND LASERCONTROL ULTRAFAST DIFFRACTION

OF ELECTRON BEAMSFOR

185

V S. Letokhov Discussion on the Communication by K S. Letokhov xxv

xxvi

CONTENTS

GENERAL DISCUSSION ON SOLUTIONS

FEMTOCHEMISTRY:

FROMCLUSTERS TO

193

LASER CONTROL OF CHEMICAL REACTIONS ON THE CONTROL OF QUANTUM MANY-BODY DYNAMICS: APPLICATION TO CHEMICAL REACTIONS

PERSPECTIVES

213

S.A. Rice

Discussion on the Report by S.A. Rice

EXPERIMENTAL OBSERVATION OF LASER CONTROL: ELECTRONIC BRANCHING IN THE PHOTODISSOCIATION

OF N a 2

285

A. Shnitmun, I. Sofer, I. Golub, A. Yogev,M. Shapiro, Z. Chen, and I? Brumer Discussion on the Communication by M. Shupiro

COHERENT CONTROL OF BIMOLECULAR SCATTERING

295

R Brumer and M. Shapiro LASERHEATING, COOLING, AND TRANSPARENCY OF INTERNAL DEGREES OF FREEDOM OF MOLECULES

301

D. J. Tannor, R. Koslofi and A. Bartana Discussion on the Communication by D. J. Tannor RAMIFICATIONS

DYNAMICS

OF

FEEDBACK FOR CONTROL OF QUANTUM

315

H. Rabitz Discussion on the Communication by H.Rabitz

THEORY OF LASERCONTROL OF VIBRATIONAL TRANSITIONS AND PULSES CHEMICAL REACTIONS BY ULTRASHORT INFRARED LASER

327

M. V Korolkov, J. Manz, and G. K. Paramonov Discussion on the Communication by J. Manz

TIME-FREQUENCY AND COORDINATE-MOMENTUM WIGNER WAVEPACKETS IN NONLINEAR SPECTROSCOPY

S. Mukamel, C. Ciordas-Ciurdariu, and V Khidekel GENERAL DISCUSSION ON REACTIONS

LASERCONTROL OF CHEMICAL

345

373

xxvii

CONTENTS INTRAMOLECULAR SOLVENT

DYNAMICS

DYNAMICS AND RRKM THEORY OF CLUSTERS

39 1

R. A. Marcus Discussion on the Report by R. A. Marcus

HIGH-RESOLUTION SPECTROSCOPY DYNAMICS

AND INTRAMOLECULAR

409

H. J. Neusser and R. Neuhauser Discussion on the Report by H. J. Neusser GENERAL DISCUSSION ON

INTRAMOLECULAR

REGULAR AND IRREGULAR FEATURESIN SPECTRA AND DYNAMICS INTRAMOLECULAR

DYNAMICS

449

UNIMOLECULAR

DYNAMICS IN THE FREQUENCY DOMAIN

463

R. W Field, J. I! O’Brien, M. I! Jacobson, S. A. B. Solina, W E Polik, and H. Ishikuwa OF CLASSICAL PERIODIC ORBITS AND CHAOS IN EMERGENCE INTRAMOLECULAR AND DISSOCIATION DYNAMICS

49 1

€? Gaspard and I. Burghardt

GENERAL DISCUSSION ON REGULAR AND IRREGULAR FEATURES IN UNIMOLECULAR SPECTRA AND DYNAMICS

583

MOLECULAR RYDBERG STATESAND ZEKE SPECTROSCOPY 607

ZEKE SPECTROSCOPY E. W Schlag Discussion on the Report by E. W Schlag

SEPARATION OF R m SCALES IN THE DYNAMICS OF HIGH MOLECULAR RYDBERGSTATES

625

R. D. Levine GENERAL DISCUSSION ON MOLECULAR RYDBERG STATES AND

ZEKE SPECTROSCOPY:

PART

I

647

xxviii

CONTENTS

FROM RYDBERG STATE DYNAMICS TO ION-MOLECULE REAmlONS USING ZEKE SPECTROSCOPY

667

T F! Sofley, S. R. Mackenzie, E Merkt, and D. Rolland Discussion on the Report by T. l? Sofley QUANTUM DEFECTTHEORY OF THE DYNAMICS OF MOLECULAR RYDBERG STATES

701

Ch. Jungen Discussion on the Report by Ch. Jungen SUBPICOSECOND STUDY OF BUBBLE FORMATION UPON STATE EXCITATION IN CONDENSED RAREGASES

RYDBERG

711

M.-T Portella-Oberli, C. Jeannin, and M. Chergui Discussion on the Communication by M. Chergui GENERAL DISCUSSION ON MOLECULAR RYDBERG STATES AND ZEKE SPECTROSCOPY: PARTXI

719

TRANSITION-STATE SPECTROSCOPY AND PHOTODISSOCIATION PHOTODISSOCIATION SPECTROSCOPY AND DYNAMICS OF THE VINOXY(CH2CHO) RADICAL

729

D. L. Osbom, H. Choi, and D. M. Neumurk Discussion on the Report by D. M. Neumrk RESONANCES IN UNIMOLECULAR DISSOCIATION: FROM MODE-SPECIFIC TO STATISTICAL BEHAVIOR

745

R. Schinke, H.-M. Keller, H. Flothmann, M. Stumpf; C. Beck, D. H. Mordaunt, and A. J. Dobbyn Discussion on the Report by R. Schinke PHOTODISSOCIATING SMALL POLYATOMIC MOLECULES IN THE VUV REGION: RESONANCES IN THE l ~ + - l C +BANDOF OCS

789

K. Yamanouchi,K. Ohde, and A. Hishikawa Discussion on the Communicationby K. Yamanouchi PHASE AND AMPLITUDE IMAGING OF BY SPECTROSCOPIC MEANS

EVOLVING WAVEPACKETS

M. Shapiro Discussion on the Communication by M. Shapiro

799

CONTENTS

GENERAL DISCUSSION ON TRANSITION-STATE SPECTROSCOPY AND PHOTODISSOCIATION REACTION

xxix 809

RATE THEORES

RECENT ADVANCES IN STATISTICAL ADIABATIC CHANNEL DYNAMICS CALCULATIONS OF STATE-SPECIFIC DISSOCIATION

819

J. Troe Discussion on the Report by J. Tme

QUANTUM AND SEMICLASSICAL THEORIES OF CHEMICAL REACTION RATES

853

M! H. Miller Discussion on the Report by M! H . Miller FEMTOSPECTROCHEMISTRY: NOVEL POSSIBILITIES WITH RESOLUTION THREE-DIMENSIONAL (SPACE-TIME)

873

K S. btokhov Discussion on the Communication by F S. Letokhov TO KING

ACADEMIC SESSION AT THE CASTLE OF LAEKEN: bSENTATI0N ALBERT

889

MODERNPHOTOCHEMISTRY

889

S. A. Rice

FEMTOCHEMISTRY

892

A. H. Zewail

CONCLUDING REMARKS

S.A. Rice and K S. Letokhov

893

AUTHORINDEX

899

SUBJECT INDEX

927

OPENING REMARKS J. Solvay President, Administrative Council, International Institutes for Physics and Chemistry It is a pleasure and an honor to welcome the participants to this XXth Solvay Conference on Chemistry. The subject of your investigations is photochemistry, with particular emphasis on chemical reactions on the femtosecond time scale. When browsing through the abstracts for the conference that we have received, I find that they often refer to quantum mechanics as applied to the understanding of chemical reactions. The theory of quanta has been a central preoccupation of the Solvay Conference since the first Solvay Conference on Physics, which took place in 1911. On that occasion, the fundamental discrepancies between classical theory and experimental data, which appeared at the beginning of the century, were discussed by major figures of modem science like Marie Curie, Albert Einstein, Max Planck, Henri Poincark, and many others. Today, the theory of quanta has led to many applications, among which are the femtosecond lasers that can be used for the control of quantum systems. This modem laser technology allows the detailed observation of different types of molecular motions on ultrashort time scales with unprecedented resolution in energy. Knowing from personal experience the difficulty of control in industrial applications,the control of chemical reactions at the level of the femtosecond inspires my admiration for the range of applications of quantum theory. In turn, these new results appear particularly challenging for the theory of quanta. A major issue is to understand the relationships between the ultrashort and the long-time scales as well as between the microscopic and macroscopic properties of chemical reactions. In this context, the new possibilities of controlling microsystems highlight the fundamental paradoxes of quantum mechanics, where probabilistic features coexist so closely with coherent behaviors. Might the new frontier at ultrashort time scales contribute to understanding at a deeper level the role of quantum mechanics in irreversible processes such as chemical reactions? Such questions were discussed in 1962 at the XIIth Solvay Conference on Chemistry, entitled “Energy Transfer in Gases,” which gathered G. Herzberg, R. Karplus, R. G. W. Nonish, xxxi

xxxii

OPENING REMARKS

J. C. Polanyi, G. Porter, 0. K. Rice, N. N. Semenov, N. B. Slater, A. R. Ubbelohde, and E. P. Wigner, among others. Today, these questions remain as puzzling and important as ever. The recent advances in modem technology continue to open new opportunities for the observation of chemical reactions on shorter and shorter time scales, at higher and higher quantum numbers, in larger and larger molecules, as well as in complex media, in particular, of biological relevance. As an example of open questions, the most rapid reactions of atmospheric molecules like carbon dioxide, ozone, and water, which occur on a time scale of just a few femtoseconds, still remain to be explored. Another example is the photochemistry of the atmospheres of nearby planets like Mars and Venus or of the giant planets and their satellites, which can help us to understand better the climatic evolution of our own planet. We hope that this Conference will contribute to solving these fundamental questions. In creating the Solvay Institutes of Physics and Chemistry, the hope of Ernest Solvay has been that international meetings at the highest level should foster progress through reports and discussions. My wife and I are looking forward to having most of you for a buffet dinner at home this evening. You will be able to tell me if this conference indeed fulfills the expectations of Ernest Solvay. I would like to express my gratitude to the European Commission, which provided substantial support in the financing of this Conference.

ADVANCES IN CHEMICAL PHYSICS VOLUME 101

FEMTOCHEMISTRY: FROM ISOLATED MOLECULES TO CLUSTERS

FEMTOCHEMISTRY: CHEMICAL REACTION DYNAMICS AND THEIR CONTROL A. H.ZEWAIL Arthur Amos Noyes Laboratory of Chemical Physics California Institute of Technology Pasadena, California

CONTENTS 1. Introductory Remarks 11. Concept of Coherence and the Evolution to Ferntochemistry

A. B. C. D.

Coherence and Dephasing Coherence Control by Phase-Coherent Pulses Coherence in the States of Isolated Molecules: IVR Coherence in Orientation: Molecular Structures E. Coherence in Reactions: Wavepackets and Nuclear Motions F. Coherence in Solvation: Clusters and Dense Fluids G. Coherence Control of Wavepackets: Reactive and Nonreactive Systems H. Coherence in Electron Diffraction: Complex Molecular Structures 111. Prototype Systems: Uni- and Bimolecular Reactions A. Resonances in Unimolecular Reactions B. Barrier Reactions: Saddle-Point Transition State C. Bimolecular Reactions: Ground-State Dynamics D. Complex Organic Reactions E. Electron Transfer Reactions F. Tautomerization Reactions of DNA Models IV. Scope of Reactions Studied V. Concluding Remarks Bibliography References

Advances in Chemical Physics, Volume 101: Chemical Reactions and Their Control on the Femtosecond Time Scale, XXth Solvay Conference on Chemistry, Edited by Pierre Gaspard, Advances in Chemical Physics, Volume 101: Chemical Reactions and Their Control on the Femtosecond Time Scale. XXth Solvay Conference on Chemistry, Edited by Pierre Gaspard, Irene Burghardt, I. Prigogine, and Stuart A. Rice. ISBN 0-471-18048-3 0 1997 John Wiley & Sons, Inc.

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A. H. ZEWAIL

I. INTRODUCTORY REMARKS The International Institutes of Physics and Chemistry were founded by Ernest Solvay at the beginning of this century. The Solvay Conferences in Brussels have played an essential role in the history of physics, as remarked by one of the founders of quantum mechanics, Werner Heisenberg. The first Solvay Conference on Physics in 1911 became famous for its discussions on the birth of quantum mechanics, a marked departure from classical concepts, by Marie Curie, Albert Einstein, Max Planck, Henri Poincarb, and many others. The XXth Solvay Conference on Chemistry was devoted to “chemical reactions and their control on the femtosecond time scale.” The conference followed the wonderful tradition of appreciation of scientists and scientific discoveries. We owe our appreciation to Ilya Prigogine, Director of the Solvay Institutes, Stuart Rice, Chairman of the Scientific Committee, and Pierre Gaspard, the Conference Scientific Secretary and Organizer. I had the honor to review the field, as described by the title of this chapter, I would like to take this opportunity here to focus on some concepts that were essential in the development of femtochemistry: reaction dynamics and control on the femtosecond time scale. The following is not an extensive review, as many books and articles have already been published [l-121 on the subject, but instead is a summary of our own involvement with the development of ferntochemistry and the concept of coherence. Most of the original articles are given in a recent two-volume book that overviews the work at Caltech [5], up to 1994. Reaction dynamics on the femtosecond time scale are now studied in all phases of matter, including physical, chemical, and biological systems (see Fig. 1). Perhaps the most important concepts to have emerged from studies over the past 20 years are the five we summarize in Fig. 2. These concepts are fundamental to the elementary processes of chemistry-bond breaking and bond making-and are central to the nature of the dynamics of the chemical bond, specifically intramolecular vibrational-energy redistribution, reaction rates, and transition states. In a classical Bohr orbit, the electron makes a completejourney in 0.15 fs. In reactions, the chemical transformation involves the separation of nuclei at velocities much slower than that of the electron. For a velocity -lo5 cm/s and a distance change of lo-* cm (1 A), the time scale is 100 fs. This is a key concept in the ability of femtochemistry to expose the elementary motions as they actually occur. The classical picture has been verified by quantum calculations. Furthermore, as the deBroglie wavelength is on the atomic scale, we can speak of the coherent motion of a single-moleculetrajectory and not of an ensemble-averaged phenomenon. Unlike kinetics, studies of dynamics require such coherence, a concept we have been involved with for some time.

5

CHEMICAL REACTION DYNAMICS AND THEIR CONTROL Gas phase molecular beam Barrier reactions Dissociation Electron transfer Proton transfer Vander Waals reactions Eimolecular reactions Rydberg reactions Organometallic reactions Exchange reactions lsomerization reactions Cleavageladdition Abstract reactions Norrish reactions Elimination reactions ~

Clusters Acid-base reactions Step-wise solvation Metal clusters

Surfaces Femto-STM dynamics Desorption

Caging reactions Harpoon reactions Excimers Exciplexes Polymer reactions Semiconductor reactions

Femtosecond Resolution in Chemistry

-

&

Dense fluids Dissociation reactions Recombination reactions Dephasing Vibrational relaxation

-

‘I:

Biology

Liquids Coherent dissociation Geminate recombination Dephasing Proton transfer Electron transfer Vibrational relaxation Barrierless reactions Bimolecular reactions Ionic reactions Solvation dynamics Friction dynamics Polarization (kerr) ~

Control

t*

U*(d

I

~

Biological systems Ligand-myoglobin Protein dynamics Bacteriorhodopsin Light harvesting Pigment-protein complexes Photosynthetic reaction centers

Figure 1. Schematic indicating the different phases studied with femtosecond resolution and the area of control studied by spatial (r*), temporal (t*), phase (a*), or potential-energy (v*)manipulation.

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SOME CONCEPTS IN FEMTOCHEMISTRY Concept of Time Scales and Atomic-Scale

Electronmotion Nudear motion DeBroglie wavelength

Concept of Coherence and Single-Molecule

State, orientation, wave packet coherence

UncertaintyprincipIeMdwherence Single-molecule, not ensemble, trajectory Dynamics, not kinetics Complex systems, robustnessof phenomena

Concept of Physical and Chemical Time Scales

Bond breaking & Bond making time scales Spreadingin space & dephasing time scales Analogy with Ti and Tz

Freezing structures at fs resolution Probing with mass spectrometry, LIP, m,

Probing Transition States and Intermediates

PE, SEF' and absorption Reaction mechanisms,bonding, structures

~~

Concept of Controlling Reactivity Figure 2.

Reaction wave packet wntml Time &space control Phase conml PES control

Summary of some key concepts in ferntochemistry discussed in text.

What coherence is, how it can be probed, and why it is robust in molecular systems are some fundamental issues of concern here. In what follows, we discuss these issues and then provide examples for a number of different classes of reactions. 11. CONCEPT OF COHERENCE AND THE

EVOLUTION TO FEMTOCHEMISTRY

Twenty years ago, the concept of coherence in molecular systems was new. In the beginning, and certainly within the chemistry community, the relevance

CHEMICAL REACTION DYNAMICS AND THEIR CONTROL

7

and importance of a notion such as dephasing in molecular and reaction dynamics was suspect! Coherence as a theoretical concept was fully developed in the physics and optics literature, especially following the invention of lasers and the development of nonlinear optics. In chemistry, however, the concept was first appreciated only in the context of nuclear magnetic resonance (NMR) research.

A. Coherence and Dephasing In the early days of studying molecular coherence (Fig. 3), the focus was on the advancement of new techniques and on the observation of the phenomenon of dephasing. In the 1970s, nanosecond lasers were used to form a coherent superposition of ground and excited electronic states, and the resulting polarization was monitored to measure the coherence of the ensemble. At Caltech, we made these pulses by the “switching” of a single-mode laser, with particular emphasis on the generation of laser acoustic-diffraction pulses. Coherent transients were observed in solids and in gases, both in “bulbs” and in molecular beams (Fig. 4).

La

ry=

1

/

*

I

,+I

I

Figure 3. Coherence, induced between two states (lowest and first-harmonic wave functions) and the nature of the hybrid superposition, which evolves with time.

t :ppy - I 8

A.

H.ZEWAIL lntenrity

Emission intensity

900

IOV

&I

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100

1%

20

loo

180

260

350

340

I - 0

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1

2

3

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5

300

6

6aou 1

7

Figure 4. Coherent transients observed in gases and molecular beams. Shown are the photon echo (detected by spontaneous emission), the free induction decay, and T i for different pressures (iodine gas and beam).

The importance of these results is manifested in the ability of separating homogeneous from inhomogeneous dephasings in complex molecular systems. Previously, the true homogeneous dynamics were often masked in the apparent absorption line shapes, occurring on a time scale four orders of magnitude slower than would have been inferred from the absorption spectra. One significant development in this area, which we used later for phase control, was the detection of coherence on the incoherent spontaneous emis-

CHEMICAL REACTION DYNAMICS AND THEIR CONTROL

9

sion of molecules using a pulse train (in phase!) to convert coherence (xy plane of the rotating frame) to population (kz axis).

B. Coherence Control by Phase-Coherent Multiple Pulses

In 1980, with W. Warren, we decided to extend these multiple-pulse experiments on molecular systems to include a prescribed phase of each pulse, opening the way for optical phase control and for the optical analogue of multiple-pulse NMR spectroscopy. The multiple-pulse phase-coherent sequences were generated using the laser acoustic diffraction method mentioned in the above section. This way, we observed the photon echo with a pulse sequence, such as XXX-XXZ (in the rotating-frame notation), and we could use a y pulse to lock the polarization (photon locking), making it possible to control dephasing due to collisions, as demonstrated for iodine (Fig. 5). Other pulse sequences, including composite pulses, were developed, both theoretically and experimentally, to control the fluorescence of inhomogeneous ensemble as well as the coherence (photon echo) in an inhomogeneous system (Fig. 5). These techniques of control using phase-coherent pulses have been revisited and extended to the picosecond (and more recently to the femtosecond) time domain by several groups, and represent an active field of current interest (see Ref. 1).

C. Coherence in the States of Isolated Molec~~ies: IVR The concepts of dephasing and coherence were introduced to the field of isolated molecular dynamics, also in 1980, via the following question: Can a single, isolated large molecule with many degrees of freedom exhibit intramolecular dephasing by developing its own heat bath? Using molecular beams and picosecond pulses, we were (pleasantly) surprised to observe coherence among the enormously large number of vibrational states in the anthracene molecule. We observed not only the coherence among a restricted number of levels in a bath of other states but also the phase of the oscillatory motion (Fig. 6). These studies were extended to different energies and different molecules and were generalized to include the effect of rotations and rotational-vibrational couplings on coherence. The phenomenon of intramolecular vibrational-energy redistribution (IVR) was now on solid ground, as its nature and origin could be related to the concept of phase coherence in the hybridized eigenstutes of the isolated molecule. From these and other studies, we could define different regions of IVR (restricted and dissipative) in molecules, and this same behavior was found in many other molecules studied in different laboratories. Such observation of “coherent motion” in isolated molecular systems triggered our interest in molecular reaction dynamics in the 1980s and the 1990s.

10

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x x x xxx m..-m Tz 941 f 17 as

L 100 200 300 400 500

:rnL I

PE

Aru- 1.00

20 30 Rcuurc. mTorr

10

T,me. nr

h

P,:60,-3~-60,

P,:1 q - 1%

0

PI: 200, 1

2

3

4

11)

05

0

Figure 5. Composite pulse trains for control of fluorescence and echo. Also shown are the echoes (iodine gas) and T I and T2 at different pressures.

CHEMICAL REACTION DYNAMICS AND THEIR CONTROL

11

Restricted (Coherent)

IVR

I

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Dissipative .

IVR

Fast Slow

N

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I

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I

7

Figure 6. Restricted and dissipative IVR observed for an anthracene beam at a rotational temperature of -3 K. Note the in-phase and out-of-phase nature of the coherent oscillations (top).

D. Coberence in Orientation: Molecular Structures In extending the studies of vibrational coherence to rotational coherence in isolated molecules, we formulated the concept of rotational recurrences (echoes!), which led to rotational coherence spectroscopy. A polarized picosecond (and later femtosecond) pulse was used to orient a molecular ensemble (Fig. 7). The molecules then rotate freely with different speeds

12

A. H. ZEWAIL

OrientationalAliQnment & Structural Determination

t=o

Time

b

Classical Motion

tU)

Figure 7. (a) Concept of time-dependent alignment as a method for structural determination. Top: Initial alignment at t = 0, dephasing, and recurrence of alignment at later times. Bottom: Classical motion of a rigid prolate symmetric top. (b) Structures of stilbene and tryptamine-water complex from rotational coherence spectroscopy; transients are shown. [see ref. 131.

(depending on the angular momentum), but despite the difference of their speeds, they realign again at well-defined times. This recurrence, or echo, gives direct information on the molecular structure, namely the moment of inertia along the principal axes. Figure 7 gives two examples of using rotational coherence spectroscopy to determine the molecular structure; there are now more than 100 structures obtained by this method. The approach has also been used to study ground- and excited-state structures. (For a recent review, see Ref. 13.)

w

L

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I

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m r y

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4

(b) Figure 7. (Continued)

I

Structures from Rotatlonal Coherence Spectroscopy

14

A. H. ZEWAIL

E. Coherence in Reactions: Wavepackets and Nuclear Motions The nature of coherence, when created on the femtosecond time scale, opened up the possibility of making localized nuclear wavepackets in molecules and reactions. This was achieved and probed in both nonreacfive and reactive systems. In this field of femtochemistry, one is studying the fundamentals of bond breaking and bond making. Initially, the elementary femtosecond dynamics for bound molecules, for elementary unimolecular reactions, and for bimolecular reactions were the focus used to establish the methodology and the synthesis of wavepackets in molecular systems. The reaction trajectories were also observed, and the spreading (dephasing) of the wavepacket and its recurrence or echo was seen. These femtochemical real-time probings of the elementary dynamics of the nuclear motion have been extended over the last 10 years to different phases (gases, molecular beams, clusters, surfaces, liquids, solids, and biological systems) and to many classes of reactions (elementary, more complex, solvated, etc.). Some examples are given below in Section 111, and these and other studies (see Section IV) are detailed in the collected works on femtochemistry mentioned above [5] and in some recent reviews [7].

F. Coherence in Solvation: Clusters and Dense Fluids The concept of coherent nuclear motion was extended to studies of solvation, as the time scale of solute-solvent collisions could be of the same magnitude or longer as the time scale of the wavepacket motion. Thus, on the femtosecond time scale, the system can be “frozen” in time and its interaction with the solvent can be mapped out by probing the evolution of the dynamics at longer times. The approach we took is that of stepwise solvation in two regimes of the dynamics. Studies of elementary and complex reactions were made in (1) molecular beams (clusters) and in (2) dense Auids (high pressures). Figure 8 shows the wavepacket motion of iodine in different argon solvent densities (up to 1900 atrn); elsewhere (see references below), we discuss this system up to 3000 atm. Of particular interest were the dephasing and predissociation rates as a function of solvent density, the mean vibrational frequency of the packet, and the molecular dynamics in the solvent. Studies in clusters of argon (Fig. 9) were made and compared to those in dense fluids without regular solvent structures. Complex reactions such as acid-base, isomerization, and electron and proton transfers have been examined similarly.

G. Coherence Control of Wavepackets: Reactive and Nonreactive Systems With the concept of molecular coherence well established, we wrote a review making the point that with ultrashort pulses we should be able to control

CHEMICAL REACTION DYNAMICS AND THEIR CONTROL

4

0

0

1

b p

12

2

3

n-4ddW.p

4

5

15

1

100

~~.115

cm-’

Figure 8. Coherence and solvation in dense fluids. The damping of coherence by dephasing is clearly manifested for iodine in helium at different pressures. Fourier transforms are to the right.

molecular dynamics [14]. With the success we had in probing such dynamics and the knowledge of multiple-pulse techniques (see above), we decided to focus some effort on the use of femtosecond multiple pulses to control the dynamics of a chemical reaction. In Fig. 10, we show how two successive pulses with well-defined phase angle can add constructively or destructively

16

A. H. ZEWAIL

-

Time. P

Figure 9. Femtosecond dynamics of an elementary reaction (12 21) in solvent (Ar) cages. The study was made in clusters for two types of excitation: to the dissociative A state and to the predisswiative B state. The potentials in the gas phase govern a much different time scale for bond breakage (femtosecond for A state and picosecond for B state). Based on the experimental transients, three snapshots of the dynamics are shown with the help of molecular dynamics simulations at the top. The bond breakage time, relative to solvent rearrangement, plays a crucial role in the subsequent recombination (caging) dynamics. Experimental transients for the A and B states and molecular dynamics simulations are shown.

-

the population of a wavepacket. Such an approach was exploited to turn on and @the yield of a chemical reaction: Xe + 12 XeI + I, as illustrated in Fig. 10. We extended the same approach to a unimolecular reaction, the dissociation of NaI, where the yield in a given channel can be changed by the control from the transition-state region (Fig. 11). The use of different pulses

CHEMICAL REACTION DYNAMICS AND THEIR CONTROL

17

Figure 10. Top: Multiple-pulse (femtosecond) preparation of wavepackets in molecular iodine. Two pulses were used for excitation, with a well-defined delay time (or phase angle), and one pulse for probing (middle). The phase of the wavepacket motion relative to time zero is shown on the left with the in-phase and out-of-phase feature of the oscillation (note the similarity to the IVR problem). Because of this known phase difference, wavepacket population can be “added or “subtracted” with multiple pulses as shown (at right) experimentally and theoretically. Bottom: Femtosecond control of a reaction yield: Xe + I2 XeI + I. The potential along three coordinates (1-1. I2 ... Xe, and Xe ... I) is shown. The product yield XeI was monitored using the chemiluminescence (right, bottom). The change in the yield of XeI with the delay between the initiating and controlling pulses is shown in the panel at the bottom left. The wavepacket motion of I2 is also shown and clearly tracks the modulation in the product yield. For this bimolecular reaction, studied in a gas cell, the controlling pulse acts as a “switch” to lift the wavepacket to the “harpoon region” of the reaction, where entry to product Xel is made.

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A. H.ZEWAIL

18 Potentid energy. Id cm-’ 30r I

1

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3101620/589/(589)

0 1

blk At variable

-30

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310/620/589/(589) 3101620/-/(589)

x , q : Affucd

0

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10 15 htunuc~earseparation.

Femtosecond pulse sequence: XI x.

A

0

I

2

3

Time delay, ps

x;

Figure 11. Reaction channel control in the unimolecular dissociation of NaI. The potentials (a), the pulse sequence, and the results (b)are shown. The pulse A, controls the relative yield in the two channels: Na + I and Na* + I.

to manipulate the packet on such potentials is at the heart of the Rice-Tannor scheme of control discussed at the conference (see the chapters by these authors). There are other schemes discussed at the conference and for recent reviews see the excellent articles by Wilson’s and Manz’s groups [15, 21.

H. Coherence in Electron Diffraction: Complex Molecular Structures As a closing example of the powerful application of the concept of coherence in structures and dynamics, we point to its importance in obtaining molecular structural changes with time using ultrafast electron diffraction (UED) (Fig. 12). The UED technique has been developed, so far with -1-ps resolution (Fig. 12). We have reported recently that the introduction of rotational orientation (Section D above) to the diffraction in real time can provide a “three-dimensional image” of the structure, instead of the conventional two-

CHEMICAL REACTION DYNAMICS AND THEIR CONTROL

19

& CLCI

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s. inverse A

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s,invtrse A

'c

0

3

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,in

Y

b-8 Molcc~larsample (isotropic)

Detector

Figure 12. Coherence and UED. Shown are the experimental diffraction results (top) and the coherent alignment (bottom).

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A. H.ZEWAIL

dimensional molecular scatterings (Fig. 12) observed in the gas phase. These methods complement the spectroscopic approach outlined above and promise numerous applications, especially in complex systems.

III. PROTOTYPE SYSTEMS: UNI- AND BIMOLECULAR REACTIONS A. Resonances in Unimolecular Reactions One example that illustrates the methodology and the concept of femtosecond transition-state spectroscopy (FTS)is the dissociation reactions of alkali halides. For these systems, the covalent (M+ X,where M denotes the alkali atom and X the halogen) potential and the ionic (M+ + X-)potential cross at an internuclear separation larger than 3 A. The reaction coordinate therefore changes character from being covalent at short distances to being ionic at larger distances. The reaction occurs by a harpoon mechanism and has a large cross section because of the involvement of the ionic potential. In studying this system, the first femtosecond pulse takes the ion pair M+X- to the covalent branch of the MX potential at a separation of 2.7 The activated complexes [MX]*$ ,following their coherent preparation, increase their internuclear separation and ultimately transform into the ionic [M+ . . X-]*form. With a series of pulses delayed in time from the first one the nuclear motion through the transition state and all the way to the final M + X products can be followed. The probe pulse examines the system at an absorption frequency corresponding to either the complex [M - - XI** or the free atom M. Figure 13 gives the observed transients of the NaI reaction for the two detection limits. The localized wavepacket and its motion is shown as calculated from ab initio quantum dynamics. The resonance along the reaction coordinate describes the oscillatory motion of the wavepacket when the activated complexes are monitored at a certain internuclear separation. The steps describe the quantized buildup of free Na, with separations matching the resonance period of the activated complexes. Not all of the complexes dissociate on every outbound pass, since there is a finite probability that the I atom I internuclear separation reaches can be harpooned again when the Na the crossing point at 7 A. The complexes survive for about 10 oscillations before completely dissociating to products. When adjusting the position of the crossing point by changing the difference in the ionization potential of M and the electronegativity of X (e.g., NaBr), the survival of the complex changes (NaBr complexes, e.g., survive for only one period). The results in Fig. 13 illustrate additionally some important features of the dynamics. The oscillatory motion is damped in a quantized fashion due to

A.

e . 1

21

CHEMICAL REACTION DYNAMICS AND THEIR CONTROL

Theoretical: Quantum wavepacket motion

Potential-energysurfaces

I Covalent

. I Ionic

r 0

1 : Internuclearseparation (A)

u 0

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Experimental

g/ -2

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Figure 13. Femtosecond dynamics of dissociation (NaI) reaction. Bottom: Experimental observations of wavepacket motion, made by detection of the activated complexes ”all** or the free Na atoms. Top: Potential energy curves (left) and the “exact” quantum calculations (right) showing the wavepacket as it changes in time and space. The corresponding changes in bond character are also noted: covalent (at 160 fs), covalent/ionic (at 500 fs), ionic (at 700 fs), and back to covalent (at 1.3 ps).

bifurcation of the wavepacket at the crossing point of the covalent and ionic potentials, as shown in the quantum calculation given in the figure. In fact, it is this damping that provides important parts of the dissociation dynamics, namely the reaction time, the probability of dissociation, and the extent of covalent and ionic characters in the bond. These observations and their anal-

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yses have been discussed in more detail elsewhere, and other systems have been examined similarly. Numerous theoretical studies of these systems have been made to test quantum, semiclassical and classical descriptions of the reaction dynamics and to compare theory with experiment.

B. Barrier Reactions: Saddle-Point Transition State

The simplest system for addressing the dynamics of barrier reactions is of the type [ABA]$ -+ AB + A. This system is the half-collision of the A + BA full collision (see Fig. 14). It involves one symmetrical stretch (Qs),one and one bend (4);it defines a barrier along the asymmetrical stretch reaction coordinate. The “lifetime” of the transition state over a saddle point near the top of the barrier is the most probable time for the system to stay near this configuration. It is simply expressed, for a one-dimensional reaction coordinate (frequency w ) near the top of the barrier, as = l / w . For values of tiw from 50 to 500 cm-’ ,T $ ranges from 100 to 10 fs. In addition to this motion, one must consider the transverse motion perpendicular to the reaction coordinate, with possible vibrational resonances, as discussed below. The IHgI system, representing this class of reactions, was one of the first reactions studied in femtochemistry [5]. The activated complexes [IHgI]*$, for which the asymmetric (translational) motion gives rise to vibrationally cold (or hot) nascent HgI, were prepared coherently at the crest of the energy barrier (Fig. 14). The barrier-descent motion was then observed using series of probe pulses. As the bond of the activated complex breaks during the descent, both the vibrational motion (-3oOfs) of the separating diatom and the mtutional motion (-2.3 ps) caused by the torque can be observed. These studies of the dynamics provided the initial geometry of the transition state, which was found to be bent, and the nature of the final torque, which induces rotations in the nascent HgI fragment. Classical and quantum molecular dynamics simulations show the important features of the dynamics and the nature of the force acting during bond breakage. Two snapshots are shown in Fig. 14. The force controls the remarkably persistent coherence in products, a feature that was unexpected, especially in view of the fact that all trajectory calculations are normally averaged (by Monte Carlo methods) without such coherences. Only recently has theory addressed this point and emphasized the importance of the transverse force, that is, the degree of anharmonicity perpendicular to the reaction coordinate. The same type of coherence along the reaction coordinate, first observed in 1987 by our group, was found for reactions in solutions, in clusters, and in solids, offering a new opportunity for examining solvent effects on reaction dynamics in the transition-state region.

tea),

CHEMICAL REACTION DYNAMICS AND THEIR CONTROL

23

Potential energy surfaces

II

I*

HgI + I

I + HgI

Theoretical: Quantum wave packet motion t=Ots

tr600fs

(a1 Figure 14. (a)Potential-energy surfaces, with a trajectory showing the coherent vibrational motion as the diatom separates from the I atom. nkto snapshots of the wavepacket motion (quantum molecular dynamics calculations) are shown for the same reaction at t = 0 and r = 600 fs. (b) Femtosecond dynamics of barrier reactions, IHgI system. Experimental obser-

vations of the vibrational (femtosecond) and rotational (picosecond) motions for the barrier (saddle-point transition state) descent, [IHgI]** --r HgI(vib, rot) + I, are shown. The vibrational coherence in the reaction trajectories (oscillations) is observed in both polarizations of FTS.The rotational orientation can be seen in the decay of FTS spectra (parallel) and buildup of FTS (perpendicular) as the HgI rotates during bond breakage (bottom).

A. H. ZEWAIL

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1000 1500 2000 2500 3000 3500

Figure 14. (Conrinued)

Even more surprising was the fact that this same phenomenon was also found to be robust and common in biological systems, where wavepacket motion was found in the twisting of a bond (e.g., in rhodopsin and bacteriorhodopsin), in the breakage of a bond (e.g., in ligand-myoglobin systems), in photosynthetic reaction centers, and in the light-harvesting antenna of purple bacteria. The implications as to the global motion of the protein are fundamental to the understanding of the mechanism (coherent vs. nonstatistical

CHEMICAL REACTION DYNAMICS AND THEIR CONTROL

25

energy or electron flow) and such new observations are triggering numerous theoretical studies in these biological systems. For the wavepacket motion in dissociation and barrier reactions discussed above, there have been recent studies of the same (or similar) systems in solutions, and the results are striking. Sundstrom and co-workers have observed the wavepacket motion for the twisting process in a barrierless isomerization reaction in solutions. Their findings give a direct view of the motion and examine the nature (coherent vs. diffusive) of the coupling to the solvent during the reaction. This wavepacket-type behavior indicates the persistence of coherence along the reaction coordinate and provides the time scales for intramolecular motion and solvation. Hochstrasser's group has shown for Hg12 in ethanol solutions that the HgI is formed in a coherent state, similar to the observation we made in the gas phase. Their study is rich with information regarding solvated wavepacket dynamics, relaxation in the solvent, and the effective Potential Energy Surface (PES). Of particular interest is the study of solvent-induced relaxation of nascent fragments.

C. Bimolecular Reactions: Ground-State Dynamics

Real-time clocking of abstraction reactions was first performed on the I-H/COz system for the dynamics on the ground-state PES [5]:

H + C02

[HOCO]* --t CO + OH

Two pulses were used, the first to initiate the reaction and the second delayed to probe the OH product. The decay of [HOCO]* was observed in the buildup of the OH final fragment in real time. The two reagents were synthesized in a van der Waals complex. The results established that the reaction involves a collision complex and that the lifetime of [HOCO]* is relatively long, about a picosecond. Wittig's group [13 has recently reported accurate rates with subpicosecond resolution, covering a sufficiently large energy range to test the description of the lifetime of [HOCO]' by an RRKM theory. Recent crossed-molecular beam studies of OH and CO, from the group in Perugia, Italy, have shown that the angular distribution is consistent with a long-lived complex. Vector correlation studies by the Heidelberg group addressed the importance of the lifetime to IVR and to product state distributions. The molecular dynamics calculations (with ab initio potentials) by Clary, Schatz, and Zhang are also consistent with such lifetimes of the complex and provide new insight into the effect of energy, rotations, and resonances on the dynamics of [HOCO]' . This is one of the reactions in which both theory and experiment have been examined in a very critical and detailed manner.

26

A. H. ZEWAIL

For exchange reactions, the femtosecond dynamics of bond breaking and bond making were examined in the following system: Br + I2 -+[BrII]$ -+BrI + I The dynamics of this Br + 12 reaction (Fig. 15) were resolved in time by detecting the BrI with the probe pulses using laser-induced fluorescence. The reaction was found to be going through a sticky collision complex lasting tens of picoseconds. More recently, McDonald’s group has monitored this same reaction using multiphoton ionization mass Spectrometry and found the rise of I (and I;!) to be similar to the rise of BrI (Fig. 15). They proposed a picture of the PES for the dynamics, including the involvement of the Br* + I, surface. With molecular dynamics simulations, comparison [5] with the experimental results showed the trapping of trajectories in the [BrII]* potential well; the complex is a stable molecular species on the picosecond time scale. Gruebele et al. drew a simple analogy between collision (Br + 12) and half-collision (hv + 12) dynamics based on the change in bonding and employed frontier orbitals to describe it (see Ref. 5). The PES may involve avoided crossings. Other bimolecular reactions of complex systems, such as those of benzene and iodine and acid-base reactions, have also been studied. Currently, we are examining the inelastic and reactive collision of halogen atoms with polyatomics (e.g., CH3I). Other groups at the National Institutes of Science and Technology and at the University of Southern California have studied a new class of reactions: 0 + CHq [CH3OH]* CH3 + OH and H + ON;! HO + N2 or HN + NO.

-

-.

-+

D. Complex Organic Reactions One of the most well-studied addition/elimination reactions, both theoretically and experimentaily,is the ring opening of cyclobutane to yield ethylene or the reverse addition of two ethylenes to form cyclobutane (Fig. 16). Such is a classic case study for a Woodward-Hoffmann description of concerted reactions. The reaction, however, may proceed directly through a transition state at the saddle region of an activation barrier or it could proceed with a diradical intermediate, beginning with the breakage of one u bond to produce tetramethylene, which in turn passes through a transition state before yielding final products. The concept, therefore, besides being important to the definition of diradicals as stable species, is crucial to the fundamental nature of the reaction dynamics: a concerted one-step process vs. a two-step process with an intermediate. Experimental and theoretical studies have long focused on the possible existence of diradicals and on the role they play in affecting the processes

CHEMICAL REACTION DYNAMICS AND THEIR CONTROL

27

E? a, 5

a,

Experimental

Theoretical

zz nme delay (ps)

'2

-

3

4

S o

6

7

Br-I d i E t a n e e (A)

Figure 15. Femtosecond dynamics of the Br + 12 BrI + I exchange reaction. Here, the collision complex is long lived, zC = 53 ps. As shown by the molecular dynamics, the [BrIIIS complex is trapped in the transition-state region; the reaction may also involve avoided crossings (see text).

of cleavage, closure, and rotation. The experimental approach is based primarily on studies of the stereochemistry of reactants and products, chemical kinetics, and the effect of different precursors on the generation of diradicals. The time "clock" for rates is internal, inferred from the rotation of a single bond, and is used to account for any retention of stereochemistry from reac-

A. H.ZEWAIL

28

"6'

Diradical intermediate

Parent

diradical

t o 3,O

e

,

3.5

, <, 28 41 5556 I

.

.

I

84

Mass, amu

Figure 16. Femtosecond dynamics of addition/cleavage reaction of the cyclobutaneethylene system. Bottom: Experimental observation of the intermediate diradical by mass spectrometry. Top: The PES showing the nonconcerted nature of the reaction, together with three snapshots of the structures at 10 (initial), fd (diradicai) and ff (final). The parent precursor is also shown.

CHEMICAL REACTION DYNAMICS AND THEIR CONTROL

29

tants to products. Theoretical approaches basically fall into two categories: Those involving thermodynamical analysis of the energetics (enthalpic criterion) and those concerned with semiempirical or ab initio quantum calculations of the PES describing the motion of the nuclei. It appears, therefore, that real-time studies of these reactions should allow one to examine the nature of the transformation and the validity of the diradical hypothesis. We recently reported direct studies of the femtosecond dynamics of the transient diradical structures. The aim was at “freezing” the diradicals in time in the course of the reaction. Various precursors were used to generate the diradicals and to monitor the formation and the decay dynamics of the reaction intermediate(s). The parent (cyclopentanone) or the intermediate species was identified distinctly using time-of-flight mass spectrometry. The concept behind the experiment and some of the results are given in Fig. 16. The mass spectra obtained at different femtosecond time delays show the changes of the precursor and intermediate species. At negative times there is no signal present. At time zero, the parent mass (84 amu) of the precursor cyclopentanone is observed whereas the intermediate mass of 56 amu is not apparent. As the time delay increases, a decrease of the 84 mass signal was observed; for the 56 mass, first an increase and then a decrease of the signal was observed. The 56 mass corresponds to the parent minus the mass of CO, and its dynamics directly reflect the nature of the transition-state region. Considering the dynamics of nuclei at the top of the barrier, it is impossible at these velocities to obtain such time scales if a wavepacket is moving translationally on a “flat,” one-dimensionaf surface. For example, over a distance of 0.5 A, which is significantly large on a bond scale, the time in the transition-state region will be -40 fs. The reported (sub)picosecond times therefore reflect the involvement of other nuclear degrees of freedom. In the original publication (see below) the rates were related to the stability of the diradical species. By varying the total energy and using different substituents, these studies gave evidence that the diradical is a stable species on the global PES.The approach is general for the study of other reactive intermediates in reactions and since then has been extended to cover different classes of reactions. We have completed studies of trimethylene and many of the isotopically substituted species. Without the femtosecond-resolved mass spectra, it would have been impossible to observe the evolution of the parent reagents and the dynamics of the intermediates. Using the same techniques, elimination reactions were clocked in order to address a similar problem: the nature of two-center elimination by either a one-step or two-step process. The reaction of interest is

30

A. H. ZEWAIL

x

F (ethanes)

(ethylenes)

where the 2X (in this case 21) elimination leads to the transformation of ethanes to ethylenes. The questions then are: Do these similar bonds break simultaneously or consecutively with formation of intermediates? What are the time scales? In methyl iodide, the C-I nonbonding to antibonding orbital transition (at -2800 A) is known to fragment and form iodine in both spin-orbit states (I and I*). The CH3 fragment is produced vibrationally excited. In both I and I* channels, the iodine signal rises in a time of less than 0.5 ps. For I-CF2-CFz-I, the situation is entirely different. The bond breakage process leading to elimination is consecutive and nonconcerted. Similar to methyl iodide, the first bond breakage occurs in -200 fs; however, the second step is much slower (32 ps). The dynamics of the prompt breakage can be understood by applying the theoretical techniques mentioned above for coherent wavepacket motion in direct dissociation reactions, but what determines the dynamics of the slower secondary process? The observation of a 30-ps rise for formation of I indicates that after the recoil of the fragments in the primary fragmentation the total internal energy in the intermediate [CF21-CF,Js is sufficient for it to undergo secondary dissociation and produce I in the ground state. From the photon energy used in the experiment (102 kcal/mol), the C-I bond energy (52.5 kcal/mol), and the mean translational energy, the internal energy of the fragment was found to be comparable with the activation energy. The 30 ps-' represents the average rate for this secondary bond-breaking process (barrier 3-5 kcal/mol). When the total energy was decreased in these experiments, the decay became slower than 30 ps. This energy dependence of the process could be understood considering the time scale for IVR and the microcanonical rates, k ( E ) at a given energy, in a statistical RRKM description. Such dynamics of consecutive bond breakage are common to many systems and are also relevant to the mechanism in different classes of reactions discussed in the organic literature. E. Electron 'Ikansfer Reactions An approach that makes it possible to study directly the transition-state dynamics of charge transfer (CT) reactions was recently reported by our group. The entire system is prepared on a reactive potential energy sur-

CHEMICAL REACTION DYNAMICS AND THEIR CONTROL

31

face and in a well-defined impact geometry. To define the zero of time, we start from the van der Wads configuration in a molecular beam, similar to other real-time studies discussed above in Section 1II.C. A femtosecond pulse induces the CT. We then follow the dynamics of the transition state using probe pulses that monitor the transition state or the final products of the reaction using mass spectrometry. The system of interest here is the bimolecular reaction of benzene (Bz) and similar derivatives with iodine: Bz.-*I-I (reactants)

4

[Bz'*.*I-...I]' (transition state)

-+

BZI+I (products)

This system (see Fig. 17) is unique in many aspects of the structure and dynamics and has historic roots for nearly 50 years since the seminal works by Hildebrand and Mulliken. Mixing of benzene and iodine results in a new color, a new absorption spectrum, and a new theory. Mulliken attributed the strong absorption band of the system to the excitation of the ground-state complex to the CT state with the aromatic molecule acting as the electron donor and the iodine as the acceptor, that is, Bz' . I2-. Several spectroscopic and theoretical studies have predicted that the Bz . 12 ground state has a c6"axial structure with the 1-1 bond being perpendicular to the benzene molecular plane. The heat of formation of this complex in the gas phase was determined by spectrometric methods to be on the order of 2-3 kcal/mol and our ab initio calculations support these values. The product we monitor is again the I atom using femtosecond-resolved mass spectrometry (the other product is the Bzi species). We also monitor the initial complex evolution. The initial femtosecond pulse prepares the system in the transition state of the harpoon region, that is, Bz+I2-. The iodine atom is liberated either by continuing on the harpoon PES and/or by electron transfer from iodine (I2 -) to Bz' and dissociation of neutral I2 to iodine atoms. We have studied the femtosecond dynamics of both channels (Fig. 17) by resolving their different kinetic energies and temporal behavior. The mechanism for the elementary steps of this century-old reaction is now clear. The observed femtosecond dynamics of this dissociative CT reaction is related to the nature of bonding. Upon excitation to the CT state, an electron in the highest occupied molecular orbital (HOMO) of benzene*(n) is promoted to the lowest occupied molecular orbital (LUMO) of I2 (a ). Vertical electron attachment of ground state 12 is expected to produce molecular iodine anions in some high vibrational levels below the dissociation limit. In other words, after the electron transfer, the 1-1 bond is weakened but not yet broken. While vibrating, the entire 12 and benzene complex begins an excursion motion within the coulombic field and the system proceeds

A. H.ZEWAIL

32

-A-

[M+..-BCI*

M* + BC

M+B' +

(lonlc Channel)

+

(Neutral Channel)

+

. )

Reaction Coordinate 7

3 v

6

+f

5

k

E N

4

3

2

2

3

4

5

R(Bz+I--..J)/A

6

7

(a) Figure 17. (a)Generic reaction path for charge transfer reactions with both channels of harpooning and electron transfer indicated. Molecular dynamics of the Bz/I2 birnolecular reaction is shown at the bottom. (b) Observed transient for the Bz/12 reaction (I detection) and the associated changes in molecular structure. Note that we observe the two channels of the reaction, shown in (a), with different kinetic energies and rises of the I atom.

w w

I

I

-500

t-

I

I

0

-

trl I

t

I

500

I

t'

I

1500

I

Figure 17. (Continued)

t"

I

2000

Reaction Time (fs)

I

1000

I

2500

I

3000

2=750+50 fs

I

t f

b

34

A. H. ZEWAIL

from the transition-state region to final products. Additionally, an electron may return to the benzene cation, leaving 12 on a dissociative potential. The resulting neutral Bz ' I then loses the I atom. The apparent 750-fs reaction time actually is made of two components, one fast (400 fs), describing the back electron transfer, and one slow (1.4 ps), describing both the ionic channel and the secondary Bz - I dissociation. This is an important conclusion pertinent to dissociative CT reactions in solutions, to CT surface reactions, and to future transition-state studies of surface-aligned, photoinduced reactions. To give more insight into the molecular dynamics in the transition-state region, we performed classical trajectory calculations. The results, which are detailed elsewhere, reveal that the transition state of CT reactions can be studied directly at well-defined impact geometries. The dissociative CT reaction of benzenes with iodine occurs with an elementary harpoon/electron transfer mechanism. The time scales for the CT and for the product (I) formation define the degree of concertedness and, as reported elsewhere, are significant to the recent elegant studies in condensed media by Wiersma and colleagues and by Sension. So far we have studied the electron donors of benzene, toluene, xylene, mesitylene, and cyclohexane and we plan extension to other systems. We have also studied the effect of solvation in clusters and in solutions. The above CT systems represent the case for intermolecular electron transfer. There is some analogy to proton transfer in acid-base reactions [5]. We have also examined intramolecular electron transfer systems and studied the influence of IVR and geometric changes; this work is detailed elsewhere 151. Other reactions involving ultrafast electron transfer are those of harpooning in Xe + 12, NaI, and more recently Xe/C12 (see Ref. 1).

F. Tautomerization Reactions of DNA Models

Multiple hydrogen bonds commonly lend robustness and directionality to molecular recognition processes and supramolecular structures. In particular, the two or three hydrogen bonds in Watson-Crick base pairs bind the double-stranded DNA helix and determine the complementarity and the pairing. Watson and Crick pointed out, however, that the possible tautomers of base pairs, in which hydrogen atoms become attached to the donor atom of the hydrogen bond, might disturb the genetic code, as the tautomer is capable of pairing with different partners. But the dynamics of hydrogen bonds in general, and of this tautomerization process in particular, are not well understood. Recently, we reported observations of the femtosecond dynamics of tautomerization in model base pairs (7-azaindole dimers) containing two hydrogen bonds. Because of the femtosecond resolution of proton motions, we were able to examine the cooperativity of formation of the tautomer (in

CHEMICAL REACTION DYNAMICS AND THEIR CONTROL

35

which the protons on each base are shifted sequentially to the other base) and to determine the characteristic time scales of the motions in a solventfree environment. The first step was found to occur on a time scale of a few hundreds femtoseconds, whereas the second step, to form the full tautomer, is much slower, taking place within several picoseconds; the time scales are changed significantly by replacing hydrogen with deuterium. These results establish the molecular basis of the dynamics and the role of quantum tunneling. The molecular structures and some of the transients are shown in Fig. 18. There are two possible mechanisms of double proton transfer in these model base pairs: a stepwise transfer from the base-pair structure (BPS) to the tautorner structure (TS) through an intermediate (zwitterionic) structure (IS) or a direct cooperative transfer of BPS to TS. We have studied the femtosecond transients for the fully undeuterated (NH, NH, CH) pair at two different vibrational energies E and for the isotopic species. For the base pair, 236 amu mass, the decay was fit to a biexponential function giving decay times of 71 = 650 fs and 7 2 = 3.3 ps when E = 0. On the other hand, when the vibrational energy content became -1.5 kcal/mol, these rates changed significantly, giving 71 = 200 fs and 7 2 = 1.6 ps. This drastic change reflects the presence of a reaction barrier. The observed decays for E = 0 (and at higher energies) give the rates at which the BPS is changing with time due to proton transfer. The fact that the initial tautomerization is on the femtosecond time scale, when the total vibrational energy is zero, indicates that the proton transfer motion is direct and does not involve the entire vibrational phase space of the pair. The implication is that the motion can be described as “localized” in the coordinate of N-H - - . :N. Furthermore, the two decay components indicate the presence of the intermediate structure, which reflects the two-step motion in the transfer. The rate of tautomerization can be related to a simple model describing the transformation of the BPS to IS by considering the motion of the proton in a double-well potential, with the system either in the N-H or the NH’ configuration of the two moieties. In this model, the rate is given by the tunneling expression:

k = Y exp(-?rud2mU~) where v is the reaction coordinate frequency (N-H), uo = hu is the halfwidth of the energy barrier, and UOis its height. Here, m is the effective mass of the particle. Using the measured k and taking v for the N-H of 2800 cm-’ and a0 = 0.27 A, we obtained UO= 1.2 kcal/mol. This picture assumes that

m

W

Intermediate Tautorner

Figure 18. Molecular structures of the DNA base-pair model of 7-azaindole.The femtosecond transients and the mass spectra for protonated and deuterated pairs are shown.

I

Base pair

I

CHEMICAL REACTION DYNAMICS AND THEIR CONTROL

37

the distance between the two nitrogens is fixed. When relaxing this condition and averaging over the stretch motion of the N . .. N centers at E = 0, we again obtained Llo = 1.3 kcal/mol. For self-consistency, we have repeated these two types of calculations and obtained satisfactory agreement for the effect of isotope substitution and also for the excess vibrational energy. We have also made similar studies on the deuterated structures and observed the decrease in rates due to quantum tunneling (E = 0). The phenomenon could be general to biological systems and is relevant to Lowdin’s description of quantum genetics. With the equivalence of the two hydrogen bonds in the static structure of the molecule, it is interesting to ask: What is the nature of the process that leads to the dynamical structures? We proposed the following picture. Because the time scale of the proton motion is observed to be relatively short, compared to the energy redistribution, the “reaction center” involves primarN motions. The time scale of the proton motions, ily the N-H and N however, is longer than or comparable to the changes in the electronic distribution upon excitation and the nuclear vibrational motions of the N-H and N . - .N stretches. This last inequality allows for the asymmetric motion of one of the protons, and because one moiety is excited, the proton ultimately transfers, leading to the IS. A consequence of this transfer is a stability for the second N-H motion and a higher barrier toward TS formation. The N ... N stretch is -120 cm-’ and the N-H is -2800 cm-I, giving 280 and 12 fs, respectively. Therefore, on the time scale of 0.5-10 ps (typical reaction times), the “asymmetric reaction coordinate” for the two particles is established. Very recent ab initio calculations support this proposed model. The process of mutation by tautomerization is similar to the excited-state process described here. If a “misprint” induced by a tautomer takes place during replication, then an error is recorded. Because reaction path calculations of DNA base pairs show similar potential-energy characteristics to those discussed here, we anticipate being able to explore the relevance of tautomerization dynamics to mutagenesis. In this area, we are currently examining these and other systems, also in solutions.

--

IV. SCOPE OF REACTIONS STUDIED Applications to different classes of reactions in different phases are numerous in many laboratoriesaround the world; here, we limited ourselves to some examples studied by the Caltech group. The specific examples given are chosen to span classes of reactions that display different structures and dynamics in the transition-state region. The scope of reactions studied is highlighted in the summary in Fig. 19. Details can be found in the original articles given in Refs. 5 and 7 and the bibliography to different sections and below.

38

A. H. ZEWAIL

Norrish Reactions

Figure 19. Schematic indicating the reactions studied at Caltech (so far).

V. CONCLUDING REMARKS The discussions, illustrations, and references provided here give a summary of the presentation made at the Conference. The one emerging concept is coherence in molecular dynamics, and over the last 20 years, this concept has proven powerful in observing nuclear motion in reactions, examining microscopic dynamics and structures, disentangling dephasing and relaxations in

CHEMICAL REACTION DYNAMICS AND THEIR CONTROL

39

condensed media, and controlling elementary dynamics. More recently, the same phenomena were found in biological systems, defining the importance of global motions in, for example, proteins, and in a variety of other systems. Given these broad and diverse applications, it appears that the initial skepticism raised 20 years ago is no longer justified; coherence is here to stay! The other concepts outlined in Fig. 2 have been detailed elsewhere [5, 7, 11, 141. I wish to conclude by emphasizing that the experimental and theoretical efforts covered in this lecture owe much of their success to the great dedication of my graduate students and postdoctoral fellows at Caltech, whose contributions I have had the pleasure of recalling throughout the lecture.

Acknowledgments This work was supported by grants from the National Science Foundation and the U.S. Air Force Office of Scientific Research. I thank J. L. Herek for helpful comments.

Bibliography The following selected publications give a representation of our contribution in the areas of research highlighted by the section headings; the reader may find them helpful for more details. Section 1I.A Measurementsof Molecular Dephasing and Radiationless Decay by Laser-Acoustic Diffraction Spectroscopy, T. E. Orlowski, K. E. Jones, and A. H. Zewail, Chem. Phys. Lert. 54, 197 ( 1978).

Nonlinear Laser Spectroscopy and Dephasing of Molecules: An Experimental and Theoretical Overview, M. J. Bums, W. K. Liu, and A. H. Zewail, in Spectroscopy and Excitation Dynamics of Condensed Molecular Systems, Series in Modem Problems in Condensed Matter Sciences, Vol. 4, V. M. Agranovich and R. M. Hochstrasser, Eds., North-Holland Publishing, Amsterdam, New York, Oxford, 1983, Chapter 7, p. 301. Optical Molecular Dephasing: Principles of and Probings by Coherent Laser Spectroscopy, A. H. Zewail, Acc. Chem. Res. 13, 360 (1980). Radiationless Relaxation and Optical Dephasing of Molecules Excited by Wide- and NarrowBand Lasers. 11. Pentacene in Low-TemperatureMixed Crystals. T. E. Orlowski and A. H. &wail, J. Chem. Phys. 70, 1390 (1979). Spontaneously Detected Photon Echoes in Excited Molecular Ensembles: A Probe Pulse Laser Technique for the Detection of Optical Coherence of Inhomogeneously Broadened Electronic Transitions, A. H. Zewail. T. E. Orlowski, K. E. Jones, and D.E. Godar, Chem. Phys. Lett. 48, 256 (1977).

Section 1I.B Locking of Dephasing and Energy Redistribution in Molecular Systems by Multiple-Pulse Laser Excitation, E. T. Sleva, M. Glasbeek, and A. H . Zewail, J. Phys. Chem. 90, 1232 (1986).

40

A. H. ZEWAIL

Multiple Phase-Coherent Laser Pulses in Optical Spectroscopy. I. The Technique and Experimental Applications, W. S. Warren and A. H. Zewail, J. Chem. Phys. 78, 2279 (1983). Multiple Phase-Coherent Laser Pulses in Optical Spectroscopy. 11. Applications to Multilevel Systems, W. S. Warren and A. H. Zewail, J. Chem. Phys. 78,2298 (1983). Optical Analogues of NMR Phase-Coherent Multiple-Pulse Spectroscopy, W. S. Warren and A. H. Zewail, J. Chem. Phys. 75,5956 (1981). Optical Multiple Pulse Sequences for Multiphoton Selective Excitation and Enhancement of Forbidden Transitions, W. S. Warren and A. H. &wail, J . Chem. Phys. 78 (II), 3583 ( 1983). Phase- and Energy-Changing Collisions in Iodine Gas: Studies by Optical Multiple-PulseSpectroscopy, E. T. Sleva and A. H. &wail, Chem. Phys. Lett. 110,582 (1984). Phase Coherence in Multiple-Pulse Optical Spectroscopy, W. S. Warren and A. H. Zewail, in Photochemistry and Photobiology: Proceedings of the International Conference, Vols. I and 11, Alexandria, Egypt, A. H. Zewail, Ed., Harwood Academic, Chur, Switzerland, 1983; Laser Chem. 2 (I-fj), 37, (1983). Photon Locking, E. T.Sleva, I. M. Xavier, Jr., and A. H. Zewail, J. Opr, Soc. Am. 3, 483 (1986).

Section ZZ.C Direct Observation of Nonchaotic Multilevel Vibrational Energy Flow in Isolated Polyatomic Molecules, P. M. Felker and A. H. Zewail, Phys. Rev. Lett. 53, 501 (1984). Dynamics of Intramolecular Vibrational-Energy Redistribution (IVR), P. M. Felker and A. H. &wail, J. Chem. Phys. 82,2961,2975, 2994, and 3003 (1985). Energy Redistribution in Isolated Molecules and the Question of Mode-Selective Laser Chemistry Revisited, N. Bloembergen and A. H. Zewail, J. Phys. Chem. 88, 5459 (1984). Picosecond Time-Resolved Dynamics of Vibrational-Energy Redistribution and Coherence in Beam-Isolated Molecules, P. M.Felker and A. H. Zewail, Adv. Chem. Phys. 70, 265 (1988). Quantum Beats and Dephasing in Isolated Large Molecules Cooled by Supersonic Jet Expansion and Excited by Picosecond Pulses: Anthracene, W. R. Lambert, P. M. Felker, and A. H. Zewail, J. Chem. Phys. 75, 5958 (1981). Ultrafast Dynamics of IVR in Molecules and Reactions, P. M. Felker and A. H. Zewail, in Jet Spectroscopy and Molecular Dynamics, M. Hollas and D. Phillips, Eds., Chapman & Hall, Blackie Academic, Oxford, 1995, p. 222. Section Z1.D Determination of Excited-State Rotational Constants and Structures by Doppler-Free Picosecond Spectroscopy, J. S. Baskin and A. H. Zewail, J. Phys. Chem. 93,5701 (1989). Doppler-Free Time-Resolved Polarization Spectroscopy of Large Molecules: Measurement of Excited State Rotational Constants, J. S. Baskin, P. M. Felker, and A. H. Zewail, J. Chem. Phys. 84,4708 (1986). Femtosecond Wave Packet Spectroscopy: Coherences, the Potential, and Structural Determination, M. Gruebele and A. H. Zewail, J. Chem. Phys. 98, 883 (1993). Molecular Structures from Ultrafast Coherence Spectroscopy, P.M. Felker and A. H. Zewail, in Femtosecond Chemistry, J. Manz and L. Woste, Eds., Verlag Chemie, New York, 1994.

CHEMICAL REACTION DYNAMICS AND THEIR CONTROL

41

Purely Rotational Coherence Effect and Time-Resolved Sub-Doppler Spectroscopy of Large Molecules, P. M. Felker, J. S. Baskin, and A. H. Zewail. J. Chem. fhys. 86,2460 and 2483 (1987). Sub-Doppler Measurement of Excited-State Rotational Constants and Rotational Coherence by Picosecond Multiphoton Ionization Mass Spectrometry, N. F. Scherer, L. R. Khundkar, T. S. Rose, and A. H.Zewail, J. fhys. Chem. 91,6478 (1987).

Section 1I.E Direct FemtosecondMapping of the Trajectories in a Chemical Reaction, A. Mokhtari, P. Cong, 3. L. Herek, and A. H. Zewail, Nature 348, 225 (1990). Femtochemistry, A. H. Zewail, J. fhys. Chem. 97, 12427 (1993). Femtochemistry: Concepts and Applications, A. H. Zewail, in Femtosecond Chemistry, J. Manz and L. Woste, Eds., Verlag Chemie, New York. 1994. Femtochemistry:Ultrafast Dynamics of the Chemical Bond, A. H.Zewail, Vols. I and 11, World Scientific, Singapore, 1994. Femtosecond Probing of Persistent Wave Packet Motion in Dissociative Reactions: Up to 40 Picoseconds, !F Cong, A. Mokhtari, and A. H. Zewail, Chem. fhys. Lert. 172, 109 ( 1990). Femtosecond Real-Time Observation of Wave Packet Oscillations (Resonance) in Dissociation Reactions, T. S. Rose, M. J. Rosker, and A. H. Zewail, J. Chem. Phys. 88, 6672 (1988). Real-Time Femtosecond Robing of 'Transition States" in Chemical Reactions, M. Dantus, M. J. Rosker, and A. H. Zewail, J. Chem. Phys. 87,2395 (1987).

Section I1.F Femtochemistry at High Pressures: Solvent Effect in the Gas-to-Liquid Transition Region, C. Lienau and A. H.Zewail, Chem. fhys. Lett. 222,224 (1994). Femtochemistryat High Pressures: The Dynamics of an Elementary Reaction in the Gas-Liquid Transition Region, C. Lienau. J. C. Williamson, and A. H. Zewail, Chem. fhys. Lett. 213, 289 (1993). Femtochemistry:Recent Advances and Extension to High-Pressures, A. H.Zewail, M. Dantus, R. M. Bowman, and A. Mokhtari. J. fhotochem. fhotobiol. A: Chem. 62/3, 301 (1992). Femtosecond Chemical Dynamics in Solution: Wave Packet Evolution and Caging of 12, Y. Yan,R. M. Whitnell, K. R. Wilson, and A. H. Zewail, Chem. Phys. Lerr. 193,402 (1992). Femtosecond Dynamics of Dissociation and Recombination in Solvent Cages, Q. Liu, J.-K. Wang, and A. H. Zewail, Nature 364,427 (1993). Femtosecond Dynamics of Reactions: Elementary Processes of Controlled Solvation, A. H. Zewail, Berichte der Bunsengesellschafi fur Phys. Chem., 99, 474 (1995). Femtosecond, Velocity-Gating of Complex Structures in Solvent Cages, P. Y. Chen, D. Zhong, and A. H. Zewail, J. f h y s . Chem. 99, 15733 (1995). Microscopic Friction and Solvation in Barrier Crossing: Isomerization of Stilbene in SizeSelected Hexane Clusters, A. A. Heikal, s. H. Chong, J. S. Baskin, and A. H. Zewail, Chem. Phys. Lett. 242, 380 (1995). Solvation Ultrafast Dynamics of Reactions: VIII. Acid-Base Reactions in Finite-Size Clusters of Naphthol in Ammonia, Water, and Piperidine, S. K. Kim, J. J. Breen, D. M.Willberg, L. W. Peng, A. Heikal, J. A. Syage, and A. H.Zewail, J. Phys. Chem. 99, 7421 (1995).

42

A. H. ZEWAIL

Solvation Ultrafast Dynamics of Reactions: IX. Femtosecond Studies of Dissociation and Recombination of Iodine in Argon Clusters, J-K. Wang, Q. Liu, and A. H. Zewail, J. Phys. Chem. 99, 11309 (1995). Solvation Ultrafast Dynamics of Reactions: X. Molecular Dynamics Studies of Dissociation, Recombination and Coherence, Q. Liu, J-K. Wang, and A. H. Zewail, J. fhys. Chem. 99, 11321 (1995). Solvation Ultrafast Dynamics of Reactions, C. Lienau and A. H. Zewail, J. Phys. Chem. 100, 18629 (1996); see also ibid p. 18650 and 18666. Section ZZ.G

Femtosecond Control of an Elementary Unimolecular Reaction From the Transition-State Region, J. L. Herek, A. Matemy, and A. H. Zewail, Chem. Phys. Lett. 228, 15 (1994). Femtosecond Laser Control of a Chemical Reaction, E. D. Potter, J. L. Herek, S. Pedersen, Q. Liu, and A. H. Zewail, Nature 355, 66 (1992). Femtosecond Selective Control of Wave Packet Population, J. J. Gerdy, M. Dantus, R. M. Bowman, and A. H. Zewail, Chem. fhys. Lett. 171, 1 (1990). Laser Selective Chemistry: Is It Possible? A. H. Zewail, fhys. Today 33, 27 (1980). Section ZZ.H

Ultrafast Electron Diffraction. IV. Molecular Structures and Coherent Dynamics, J. C. Williamson and A. H. Zewail, J. fhys. Chem. 98,2766 (1994). Ultrafast Electron Diffraction. V.Experimental Time Resolution and Applications, M.Dantus, S. B. Kim, J. C. Williamson, and A. H. Zewail, J. fhys. Chern. 98,2782 (1994). Section IZZ.D

The Validity of the Diradical Hypothesis: Direct Femtosecond Studies of the Transition-State Structures, S. Pedersen, 3. L. Herek, and A. H. Zewail, Science 266, 1359 (1994). Retro-Diels-Alder Femtosecond Reaction Dynamics. B. A. Horn, J. L. Herek and A. H. Zewail, J. Am. Chem. SOC. 118, 8755 (1996). Section IILE

Transition States of Charge-Transfer Reactions: Femtosecond Dynamics and the Concept of Harpooning in the Bimolecular Reaction of Benzene with Iodine, P. Y.Cheng, D. Zhong, and A. H. Zewail, J. Chem. fhys. 103, 5153 (1995). Microscopic Solvation and Ferntochemistry of Charge-Transfer Reactions: The Problem of Benzene(s)-IodineBinary Complexes and Their Solvent Structures, F! Y.Cheng, D. Zhong, and A. H. Zewail, Chem. Phys. Lett. 242, 368 (1995). Femtosecond Real-Time probing of Reactions: XXI Direct Observation of Transition-state Structure and Dynamics in Charge Transfer Reactions, P. Y. Chen, D. Zhong and A. H. Zewail, J. Chem. fhys. 105, 6216 (1996). Section IIZ-F

Femtosecond Molecular Dynamics of Tautomerization in Model Base Pairs, A. Douhal, S. K. Kim, and A. H. Zewail, Nature 378, 260 (1995).

CHEMICAL REACTION DYNAMICS AND THEIR CONTROL

43

References 1. M. Chergui, Ed.. Femtochemistry: Ultrafast Chemical and Physical Processes in Molecular Systems (The Lausanne Conference), World Scientific, Singapore, 1996.

2. 3. Manz and L.Woste, Eds., Femtosecond Chemistry (The Berlin Conference), Vols. I and 11, Verlag Chemie, Weinheim, 1994. 3. J. Manz and A. W. Castleman, Jr., Eds., Femtosecond Chemistry (special issue of Journal of Physical Chemistry), December issue, 97 (1993). 4. V. Letokhov in Femtochemistry: Ultrafast Chemical and Physical Processes in Molecular Systems, M. Chergui, Ed., World Scientific, Singapore, 1996. 5. A. H. Zewail, Femtochemistry: Ultrafast Dynamics of the Chemical Bond, Vols. I and 11, World Scientific, Singapore, 1994. 6. D. A. Wiersma, Ed., Femtosecond Reaction Dynamics, Royal Netherlands Academy of Arts and Sciences, North Holland, Amsterdam, 1994. 7. A. H. Zewail, J. Phys. Chem. 97, 12427 (1993); J. Phys. Chem. 100, 12701 (1996); in Femtosecond Chemistry, J. Manz and L. Woste, Eds., Verlag Chemie, Weinheim, 1994. 8. G . Beddard, Molecular Photophysics, Repts. Prog. Phys. 56, 63 (1993). 9. M. A. El-Sayed, I. Tanaka, and Y. Molin, Eds., Ultrafast Processes in Chemistry and Photobiology, IUPAC, Blackwell Science, Oxford, 1995. 10. J. C. Polanyi and A. H. Zewail, Acc. Chem. Res. 28, 119 (1995). 11. A. H. Zewail and R. B. Bernstein, in The Chemical Bond: Structure and Dynamics, A. H. Zewail, Ed., Academic, Boston, 1992.

12. B. Garraway and K-A. Suominen, Reports on Progress in Physics, 58,365 (1995).

13. P. M. Felker and A. H. Zewail, in Femtosecond Chemistry, J. Manz and L. Woste, Eds., Verlag Chemie, Weinheim, 1994. 14. A. H. Zewail, Phys. Today 33, 27 (1980). 15. B. Kohler, J . L. Krause, F. Raksi, K. R. Wilson, V. V. Yakovlev, R. M. Whitnell and Y. Yan,Acc. Chem. Res. 28, 133 (1995).

DISCUSSION ON THE REPORT BY A. H. ZEWAIL Chairman: S.A. Rice G. Casati: In the first part of your talk you discussed the “longtime” propagation of an initially localized quantum packet. Did you check whether in such situations the corresponding classical motion is in dynamically stable regions or instead is chaotic? Indeed, the maximum time t up to which the quantum packet can propagate before its destruction depends on how far one is in semiclassical regions and scales in different ways depending on the nature of the corresponding classical motion. More precisely, we expect t (l/A)* for dynamically stable systems and t I In A( for classically chaotic systems.

-

-

44

A. H. ZEWAIL

A. H. Zewaik About this problem, Dr. Gaspard has done very interesting calculations to identify and study the nature of resonances above the saddle point and compared with our experiments on HgI, and other systems. I would like to refer to his contribution in this conference. M. Chergui: Prof. &wail, I probably missed the point but it was not clear to me why you see a double peak in the NaI case when you increase the time resolution? A. H. Zewail: By increasing the resolution, the probe opens a spatial window at a given internuclear separation (R).Thus, we can observe the motion to the right and the motion to the left as the wavepacket crosses the region of the probe. The observed splitting gives the trajectory in R and t and hence the spatial resolution AR. D. J. Tannor: I also have a question to Prof. Zewail on the doublepeak structure in NaI. Would the double-peak structure be characteristic of ($01 # I(t)), where # I is the first excited state of the oscillator, as can be seen from Wigner phase-space pictures (cf. Fig. l)?

z

z Figure 1.

A. H. Zewail: The splitting is not due to the reflection. It is actually predictable by classical and quantum calculations and as said to the comment by Prof. Chergui maps the trajectory of the motion. [See A. Mokhtari, P. Cong, J. L. Herek, and A. H. Zewail, Nature 348, 225 (1990); P. Cong et al., J. Phys. Chem. 100,7832 (1996).] T. Kobayashi: I have three questions concerning Prof. Zewail's report: 1. In the experiment of proton (respectively deuteron) transfer it is understandable that both the fast component (7, = 0.2 ps) and the

CHEMICAL REACTION DYNAMICS AND THEIR CONTROL

45

slow component (72 = 1.6 ps) become longer in a fully deuterated base pair (respectively 71 = 3 ps; 72 = 25 ps). Why then in a monodeuterated base pair (71 = 1.5 ps; 72 = 11 ps) is one of the two time constants not the same as that in a nondeuterated system? 2. Why are there two independent time constants corresponding to two successive proton transfer processes in such a system with an equivalent component of

3. Why does the process proceed via thermal fluctuations? What is expected if the experiment is performed at 0 K? May quantum fluctuations work as the ignition?

A. H.Zewail: For the base pair, we did the deuteration of both protons and observed transients for HH, DD, and HD species, indicating the nature of quantum tunneling. Please note that the experiments were performed in a molecular beam and the initial temperature is low. The motion is induced by the finite energy deposited microcanonically. H.Rabitz: Prof. Zewail, could you identify the physical limitations of the time resolution expected from electron diffraction? A. H. Zewail: Reaching the 100-ps time scale with our apparatus was not difficult. For the next step in reaching better resolution with a factor of 100, we had to build the system I described, which is sensitive to single-electron detection. Currently we can obtain -1 ps resolution, and with better electron trajectories we anticipate reaching the 50-fs limit outlined by Prof. Prokhorov’s and Dr. Schelev’s groups. We are collaborating with them to obtain electron sources to install in our apparatus to reach this time resolution.

COHERENT CONTROL WITH FEMTOSECOND LASER PULSES T. BAUMERT, J. HELBING, and G. GERBER*

Physikulisches Institut Universitat Wiinburg Wurzburg, Germany

CONTENTS I. Introduction

11. Experiment 111. Pump-Probe Schemes

IV. Phase-Sensitive Pump-Probe Experiments V. Coherent Control with Phase-Modulated Femtosecond Laser Pulses VI. Influence of Laser Pulse Duration VII. Coherent Control with Intense Laser Pulses VIII. Conclusion References

I. INTRODUCTION Microscopic control of the outcome of a chemical reaction, the well-defined breaking (if not formation) of one or several selected bonds in a molecule, is a long-standing dream in chemistry. Many technological advances, above all the invention and development of lasers, have therefore been welcomed in the past as a possible decisive step toward its realization. At first it was hoped, that the precise frequency of these new light sources could be employed to selectively excite individual bonds, weakening or breaking them, thus enhancing the formation of one product and discrimi*Report presented by G. Gerber Advances in Chemical Physics, Volume 101: Chemical Reactions and Their Control on the Femtosecond Time Scale, XXrh Solvay Conference on Chemistry, Edited by Pierre Gaspard, Irene Burghardt, I. Prigogine, and Stuart A. Rice. ISBN 0-471-18048-3 0 1997 John Wiley & Sons, Inc.

47

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T. BAUMERT, J. HELBING, AND G. GERBER

nating against others. However, it was soon found that the local laser excitation of a single molecular bond is not possible using continuous-wave (CW) lasers, because the deposited energy is rapidly redistributing throughout the molecule on a very short time scale. Only much later it was realized that the excellent coherence of laser light offers another, maybe much more powerful control parameter, which allows us to make use of quantum mechanical interference. This principle forms the basis of what is today generally referred to as coherent control. In a first example, Brumer and Shapiro proposed to simultaneously excite an exit channel via two different excitation pathways. The wave functions corresponding to the two pathways (usually one- and three-photon excitation, Y I = 3 ~ 2 may ) then interfere constructively or destructively depending on the phase relation between the two lasers used [1, 21. The availability of ultrashort laser pulses has opened up further possibilities: Ultrashort, spectrally wide femtosecond lasers provide a wide range of frequencies with a fixed phase relation in a single laser pulse. Besides allowing to deposit large amounts of energy in a specific molecular bond on a time scale too short for significant energy redistribution, femtosecond laser pulses can therefore often coherently excite several rovibrational levels of a molecule simultaneously. Tannor et al. 13, 41 suggested to let the resulting wavepacket evolve until a molecular configuration is reached that is favorable to the excitation of the desired final state by a second probe pulse. These dynamical control schemes were later improved, stimulated by advances in temporal and spectral phase shaping. Shi et al. [5], Peirce et al. [6], Shi and Rabitz [7,83, Warren et al. [9], and Amstrup et al. [lo] proposed to use phase- and amplitude-modulated femtosecond pulses and trains of pulses to control not only the time of propagation but also the specific shape of molecular wavepackets (optimal control theory). In a different approach, Chelkowski et al. [ l l ] suggested to construct laser pulses with a frequency sweep that follows the vibrational energy spacings on a molecular potential, thus making the nuclear wave function “climb up” the vibrational ladder. The latter scheme is especially applicable when intense ultrashort lasers are employed. High laser intensities may bring another important advance in controlling chemical reactions, both by increasing the total yield and by allowing the molecule to modify itself in the intense laser field [12, 131.

Despite the expectations raised with any new scheme for coherent control, their experimental realizations have so far (like the theoretical models) been widely limited to simple molecules or even atoms. Nevertheless, the feasibility of most schemes could be demonstrated in the laboratory. The picture that presents itself is therefore encouraging in principle even though for practical applications a lot of work remains to be done. It is the inten-

COHERENT CONTROL WITH FEMTOSECOND LASER PULSES

49

tion of this chapter to summarize what has been achieved experimentally by the work performed in our own group. Several parameters have been demonstrated to influence the excitation and dissociation of (small) molecules using femtosecond laser pulses and will be treated separately: pumpprobe delay time, phase relation between pump and probe pulse, phase modulation, pulse length, and laser intensity.

II. EXPERIMENT In order to achieve coherent control in a laboratory experiment, three major requirements are to be met. Well-defined final states cannot be reached without the preparation of a well-defined initial state. Ultrashort, spectrally wide and intense laser pulses at different wavelengths must be produced for excitation and a good characterizationof the final product states must be achieved. To prepare well-defined initial states the molecules studied in this contribution (Na2 and Na3) are prepared almost completely (>!lo%) in the lowest vibrational level (u" = 0) of their electronic ground states in a supersonic molecular beam. The apparatus used consists of two differentially pumped torr high-vacuum chambers with a nominal background pressure of 2 x in the interaction chamber. Vibrationally cold Na2 is produced in an oven operated between 500 and 600°C and ejected (adiabatic expansion) through a 200-pm nozzle. The Na3 production is enhanced at higher temperatures or by coexpanding the sodium molecular beam with argon at 4 bars (seeded beam technique) [14]. The required pulses were produced in a recently set up new laser system (Fig. 1) consisting of a Ti-sapphire oscillator, chirped pulse amplification, and an optical parametric generator [travelling-wave optical amplification of superfluorescence in combination with second-harmonic generation and sum frequency mixing (TOPAS), Light Conversion]. The home built Ar' ion laser pumped oscillator provides 30-fs pulses at a repetition rate of 85 MHz and 2 nJ energy that are subsequently amplified at 1 kHz in a modified commercial regenerative amplifier. This delivers 9O-fs, 1.2-d pulses of Gaussian shape at a wavelength of 790 nm and a bandwidth of 22 nm [full width at half maximum (FWHM)]. Due to the large bandwidth of this seed pulse, TOPAS can be used to produce laser pulses over a wide range of the visible spectrum that may be compressed to their bandwidth limit of about 40 fs in a prism compressor. At 618 nm, 20-pJ pulses of 40 fs duration are generated. In a Michelson-type setup the beam can be split into equal parts and realigned with a variable time delay between the two pulses. By weakly focusing with a 300-mmachromatic lens, peak intensities of approximately 10" W/cm2 are reached. Some of the earlier experiments were carried out using our home-built colliding pulse modelocked (CPM) ring dye laser. Equipped with two excimer

0

m

Figure 1. Overview of our new Ti--sapphire laser system, producing amplified bandwidth-limited 40-fs pulses over a wide range of the visible spectrum.

COHERENT CONTROL WITH FEMTOSECOND LASER PULSES

51

laser-pumped bow-tie amplification stages, this laser system produces 80-fs pulses of 40 pJ energy at 620 nm with a repetition rate of 100 Hz. Different wavelengths can be produced by selecting and reamplifying different frequency components from the white-light continuum generated in a cell containing methanol in a grating arrangement. To obtain most complete information about the final states produced in our experiments, we use ion and electron time-of-flight (TOF) detection. The same linear TOF spectrometer is used for both mass- and energy-resolved measurements of ions and ionic fragments and for energy-resolved electron detection under very similar conditions. Fragment energies can be determined to an accuracy of approximately 0.1 eV. The electron spectrometer setup is calibrated by producing electrons of well-known energies via resonance-enhanced multiphoton ionization (REMPI) of atomic sodium with a nanosecond laser using several different wavelengths. Thus an energy resolution of approximately 50 meV is achieved for photoelectrons of 1 eV kinetic energy. A schematic overview of the experimental arrangement is shown in Fig. 2. All TOF spectra are recorded with a 500-MHz digital oscilloscope (LeCroy) and averaged over several thousand laser shots. Boxcar averagers (Stanford Research) are used to integrate the signal of individual ion mass or electron energy peaks in the pumpprobe experiments. The data are later corrected for laser fluctuations that are monitored by a photodiode.

Figure 2. Schematic view of the experimental setup. The femtosecond laser beam is crossed with a cold sodium molecular beam. A linear TOF spectrometer is used for massand energy-resolved ion and electron detection.

52

T. BAUMERT, J. HELBING, AND G . GERBER

111. PUMP-PROBE SCHEMES The original idea to use lasers for the excitation of individual chemical bonds was based on the simple imagination of chemical bonds as individual springs of different strengths and characteristic frequencies. From a quantum mechanical point of view this classical picture is of course too simple; however, the path from (time-independent) quantum mechanics (energy eigenstates smeared out in space) to classical trajectory dynamics was pointed out as early as 1926 by Schdinger 1151. When several quantum mechanical eigenstates are coherently excited with fixed phase relations, the resulting wave function resembles a wavepacket localized in space, which propagates according to the classical equations of motion. Thus in order to stretch or squeeze individual chemical bonds in the classical sense, one must coherently couple vibrational levels to form a wavepacket for which femtosecond laser pulses with their intrinsically large bandwidth are ideally suited. This was first demonstrated experimentally by Dantus et al. [16], who observed the propagation of a vibrational wavepacket on the B state of 12. A beautiful experiment demonstrating coherent control in the sense of the Tannor-Kosloff-Rice scheme was carried out by Baumert et al. [17] using resonant three-photon ionization and fragmentation of Na2. Figure 3 shows the relevant potential energy curves for excitation of Na2 near 620 nm. Ionization is predominantly due to REMPI, whereas nonresonant multiphoton processes play only a minor role. The molecule is excited from the v" = 0 vibrational level of the neutral ground state to a range of vibrational levels in the 2 'IIg state resonantly enhanced by the A 'Z: state 1181. From the 2 'IIg state different photoionization and fragmentation processes are possible. Process 1 is the direct ionization into the ionic ground state 'E;, which yields Na2+. Process 2 is the excitation of a bound doubly excited neutral Na2** state at large internuclear distances followed by autoionization and autoionization-induced fragmentation yielding Na(3s) and Na+ fragments. When an ultrashort, spectrally broad laser pulse is used for excitation, vibrational wavepackets are formed at the inner turning points of the A 'Et;and 2 'II, potentials, which oscillate with time constants of 320 and 380 fs, respectively. A second time-delayed probe laser may then ionize either via process 2 when the 2 'Itgstate wavepacket has propagated to large internuclear distances (i.e., when the Na-Na bond has stretched) or it will transfer population into the bound ionic ground state *E; (process 1). This is shown in the transient Na' and Na2+ signals in Fig. 4, which were obtained using 80-fs pulses at 618 nm from our new Ti-sapphire laser system and reproduce the data published in Ref. [17]. The Na2+ transient has maxima each time the probe pulse is fired when the A 'Xi state wavepacket has returned to its inner turning point but shows only a very small contri-

COHERENT CONTROL WITH FEMTOSECOND LASER PULSES

@ ..... .. . . . . ~................. -____----

53

Na(3s)+Na+

*Cd(Na2+)

I

Na(3s)+Na(3s)

x 'zg+ ~

,

2

T

l

4

.

l

6

,

8

l

,

l

10

,

R14 Figure 3. Potential curve diagram for the excitation, multiphoton ionization, and fragmentation of Na2 using 620-nm photons.

bution from a wavepacket propagating on the 2 lIIg state potential. This is because the Franck-Condon maximum for the 2 'II,-A 'qtransition is found at small internuclear distances. The 2 q - 2 'II, transition on the other hand can occur over the whole range of the nuclear coordinate, thus Na2+ formation is insensitive to wavepacket motion on the 2 'II, potential. Since the doubly excited state Na2" can only be reached, when the 2 'II, state wavepacket is located at large internuclear distances, the Na+ signal is modulated predominantly with the 2 'IIg state frequency and is out of phase with the Na2+ transition. [The contribution from the A state to the Na+ signal arises from fragmentation of Na2+ following direct ionization (process 3)]. The ratio Naz+/Na+can therefore be controlled by varying the time delay

I

Figure 4. The Na+ and N%+ transient signals obtained from pump-probe measurements using 8O-fs pulses at 618 nm. The power spectra (insets) obtained from a fast Fourier transformation (FFT)show the different frequency components of the two transients.

pump-probe delay Ips]

Figure 4. The Na+ and N%+ transient signals obtained from pump-probe measurementsusing 80-fs pulses at 618 nm. The power spectra (insets) obtained from a fast Fourier transformation (FFT)show the different frequency components of the two transients.

pump-probe delay Ips]

4

55

COHERENT CONTROL WITH FEMTOSECOND LASER PULSES

010

0:5

1:o

1:5

pumpprobe delay [ps]

2:o

23 1

Figure 5. Ratio of Na+ and Na2+ ion signals as a function of pump-probe delay using 618-nm pulses. The relative yields of fragments versus molecular ions can be controlled via the delay time.

of the ionizing pulse. Figure 5 clearly shows the oscillations of the ion to fragment signal ratio. Coherent control by varying the time delay between a pump laser pulse that prepares an evolving wavepacket in the system and a probe laser pulse that excites the desired product state has also been demonstrated with twophoton ionization. Figure 6 shows the relevant potential diagram of Na2. The double-minimum state 2 'C: of the sodium dimer is used for the evolution of the molecule to the desired nuclear configuration. Due to the origin of this state from an avoided crossing between two adiabatic potential curves, a vibrational wavepacket on the 2 'qpotential created by excitation from the u" = 0 level of the ground state with sufficient energy to overcome the potential barrier (340 nm) may propagate to very large internuclear distances. At large internuclear distances, the repulsive ionic state can be reached directly using 540-nm light and Na(3s) and Na+ fragments are formed. Ionization at all other internuclear separations yields only Na2+ in its bound ground state (the additional energy is carried away by the photoelectrons). Again the ratio of Na+/Na2+ shows a strong oscillatory behavior, as can be seen in Fig. 7. This time the change in ratio is even greater than observed for three-photon ionization at 620 nm 1191. Potter et al. were the first to apply this pump and control scheme to a

56

T. BAUMERT, J. HELBING, AND G. GERBER

0

0 0 0 (D

0 0 0 0

Na(3s)+Na*

Ln

0 0 0 0 n *

c

I

E O

20 0 w

o

M

0 0

0

0 N

0 0

0 0

Na(3s)+Na(3s)

v-

x 0

2

11;

b

4

6

8

R [A1

10

12

'

Figure 6. Potential energy curves relevant for the pump-probe experiment on the 2 Et double minimum state of Na2. A femtosecond pump pulse forms a wavepacket at the inner turning point above the barrier. A second probe pulse (540 nm) can either only ionize (excitation of the 2Ei ground state of Na2+) or ionize and fragment (excitation of the 2C: repulsive state) the molecule depending on the location of the wavepacket.

bimolecular (atom-molecule) reaction: Xe + 12 -+ XeI + I [20]. A vibrational wavepacket was created in the B state of 12 by a pump pulse. The XeI yield (detected by photoluminescence) was then observed to vary strongly as a function of time delay of a second UV-femtosecond pulse. Formation of the product XeI could therefore be switched on or off depending on the position of the wavepacket when the control pulse was applied.

COHERENT CONTROL WITH FEMTOSECOND LASER PULSES

0 .o 0

2

4 6 pump-probe

a 10 delay [ps]

57

12

Figurr 7. Ratio of Na+ and Na2+ ion signals as a function of pump-probe delay for the two-photon process depicted in Fig. 6.

IV. PHASE-SENSITIVEPUMP-PROBE EXPERIMENTS Not only the time delay between pump and probe pulses but also their phase relation was controlled in an experiment presented by Scherer et al. [21]. Using a piezocontrol, this group adjusted the time delay between two femtosecond pulses very accurately, thus keeping the relative phase constant. Both pulses excite population on the B state of I2 in analogy to the control scheme proposed by Brumer and Shapiro [l] and beautifully demonstrated experimentally by Chen, Yin, et al. [22-241 as well as by Park, Kleinman et al. [25, 261 for C W lasers. The first wavepacket “stores” the phase information of the pump laser. Thus, each time this wavepacket returns to the Franck-Condon region at which excitation from the ground state is possible, interference is observed with the wavepacket excited by the second phaselocked laser pulse. Whether positive or negative interference occurs depends on the relative phase between the two laser pulses and the additional phase acquired by the first excited wavepacket while propagating on the molecular potential. Either relative phase angle or pumpprobe delay can therefore be modified to control the final excited-state population. When the pump-probe delay is varied slowly and continuously (i.e., both parameters are varied simultaneously), the high-frequency oscillations due to the optical phase of the wavepacket can be resolved in the transient signal, as shown by Blanchet et al. [27], who monitored the wavepacket motion and

58

T.BAUMERT, J. HELBING. AND G.GERBER

interference on the B state of Cs2 via two-photon ionization. The amplitude of the high-frequency oscillations is a measure of the interference between the wavepackets excited by the first and second laser pulse and is therefore large each time the first excited wavepacket has returned to its point of formation. We have employed this phase-sensitive pump-probe technique to further investigate the multiphoton ionization of Na2 with 6 18-nm femtosecond pulses as discussed in the previous paragraph and have observed the interference of the A El and 2 'IIgwavepackets created by the first pulse and those created by the second pulse in the Na2+ signal. The amplitude of the high-frequency oscillations in the Na2+ signal was obtained as a function of pump-probe delay by filtering the transient with the laser frequency. It is shown in Fig. 8 (top). Below the "averaged" Na2+ transient of Fig. 4 is

'

$0

1

1do0

I

2000 I

150

I

3600

250

I

4WO

Pump-Probe Delay [fs]

Figure 8. Frequency-filtered Na2+ pumpprobe signal in comparison to the averaged signal of Fig. 4. The filtered signal measures by how much the Na2+ signal is modulated with the laser frequency. Such modulations occur when there is interference between excitation by the probe pulse and the wavepackets formed by the pump laser pulse. This interference effect causes both the A 'Ci and the 2 'IIg state wavepacket motion to be observable in the signal.

COHERENT CONTROL WITH FEMTOSECOND LASER PULSES

59

shown for comparison. As already observed in Ref. 21, information about the phase of the wavepacket (measured by the phase-sensitive or frequency-filtered transients) is lost much more quickly than information about the location of the vibrational wavepacket excited by the pump pulse (measured by the averaged transient). This may be due to the influence of rotation [21] and may limit the applicability of phase-sensitive coherent control. On the other hand, additional spectroscopic information can be obtained by making use of this technique: The Fourier transform of the frequency-filtered transient (inset in Fig. 8) shows that the time-dependent modulations occur with the vibrational frequencies of the A 'Zt and the 2 I l l g state. In the averaged Na2+ transient there was only a vanishingly small contribution from the 2 Illg state, because in the absence of interference at the inner turning point ionization out of the 2 'IIg state is independent of internuclear distance, and this wavepacket motion was more difficult to detect. In addition, by filtering the Na2+ signal obtained for a slowly varying pumpprobe delay with different multiples of the laser frequency, excitation processes of different order may be resolved. This application is, however, outside the scope of this contribution and will be published elsewhere.

V. COHERENT CONTROL WITH PHASE-MODULATED FEMTOSECOND LASER PULSES Shi et al. [4] showed very early theoretically that a given system may be driven very efficiently from a chosen initial state to a selected final state in a specified time interval using optimally shaped laser pulses. Kosloff et al. [28] incorporated this idea in a modified version of the Tannor-Kosloff-Rice scheme. They proposed to utilize modulation of wavepacket evolution by optimally shaped pulses on an excited-state potential energy surface to influence the selectivity of product formation in the ground state. Amstrup et al. then performed calculations involving more potential energy surfaces [101. Only in the past few years, however, has it become possible to generate and characterize (simple) phase-modulated femtosecond laser pulses (see, e.g., Refs. 29-34) and to think about an experimental realization of the schemes proposed. Kohler et al. [35] generated vibrational wavepackets using linearly up and down chirped femtosecond laser pulses on the B state potential curve of 12, probing by excitation of a higher lying state with a second laser at an internuclear distance slightly short of the outer turning point. The laser-induced fluorescence signal that is proportional to the population in the higher lying state was observed to exhibit a maximum after one reflection of the B state wavepacket at its outer turning point when a down chirped laser pulse was used, whereas a decrease of the signal occurred at the same pumpprobe delay time for excitation with an up-chirped laser pulse. The

60

T.BAUMERT, J. HELBING, AND G. GERBER

mechanism by which this behavior can be explained is quite general: Using chirped pulses the dispersion of a wavepacket on a potential surface can be inverted by exciting each vibrational component with the correct phase factor. While a wavepacket excited by a transform-limitedpulse is spreading out in time, a wavepacket formed by a (down-) chirped laser pulse is maximally focused only after a given time delay. The application of this scheme (like all pumpprobe schemes) is however limited to situations where there is no significant energy redistribution on the time scale needed for the refocusing of the wavepacket. It is therefore desirable, especially for larger molecules, to achieve the desired final configuration within the duration of a single ultrashort-phase-shaped laser pulse. Bardeen et al. managed to selectively excite vibrational wavepackets on the ground-state potential of the laser dye LD690 using strongly chirped pulses of 70 nm bandwidth (transform limit 12 fs!) [36]. The effect demonstrated in their experiment is due to the very common fact that the transition frequency between two electronic states depends on the vibrational coordinate (Franck-Condon principle). In this special case the optimal frequency for transition between the ground state of LD690 and a ring-breathing mode is decreasing with increasing vibrational coordinate. As the dye ring expands during the interaction with a laser pulse, that is, the wavepacket formed by the leading edge of the pulse on the excitedstate potential propagates to larger internuclear distances, population may be transferred back down into the ground state (resonant Raman process). This is favored when the frequency within the laser pulse is decreasing in time (down chirp) and suppressed for an oppositely (up-) chirped pulse. We can report on the observation of a strong chirp dependence of the three-photon ionization probability of Na2 using single-phase-shaped femtosecond laser pulses [37]. Figure 9a shows photoelectron spectra obtained from ionizing Na, with up-chirped and down-chirped laser pulses at 620 nm. The chuped pulses were produced by increasing or decreasing the optical pathway in a prism sequence (SF10) that is used to compress the pulses coming out of the optical parametric generator (OPG) to their transform limit of 40 fs. The upper and lower spectra in Fig. 9a were obtained with linearly chirped pulses (f3500fs’), which correspond to a pulse duration of approximately 240 fs. The ionization yield is seen to double when the frequency order is switched from blue first to red first in the exciting laser pulse. Note that the up- and down-chirped pulses are identical in all their pulse parameters, except for being reversed in time. This indicates that the change in the electron spectra is indeed due to the phase modulation and not to other effects such as different pulse durations or different intensity distributions. In order to better understand the experimental results, we performed quantum mechanical calculations using the fast Fourier transform (FFT) splitoperator technique, which was previously employed by Meier and Engel [38]

COHERENT CONTROL WITH FEMTOSECOND LASER PULSES

I

0,6

'

I

0,7

'

I

0,8

' '

I

0,9

'

61

Down Chirp I

1,0

'

1

1

1,l

-

1,2

Down Chirp

0:s

0:s

017

019

IlO

111

112

e--Kinetic Energy [ey Figure 9. (a) Electron spectra measured with single up-chirped (+35Wfs2)down-chirped ( - 3 5 0 0 - f ~ ~and ) unchirped laser pulses. The transfonn-limited pulses of 40 fs duration are centred at a wavelength of 618 nm. The chirped pulses are of 240 fs duration. (b) Calculated spectra using the same parameters as (a).

to study the interaction of Na2 with unchirped femtosecond laser pulses at 618 nm. Since the electric field of a linearly chirped Gaussian laser pulse can be written in analytical form (see, e.g., Ref. 39), the methods of Meier and Engel [38] could be easily extended to incorporate chirp effects. In accordance with the experimental conditions (the laser beam was attenuated appropriately), calculations were performed in the weak-field limit, taking A 'E:, and 2 'IIg neutral electronic states and couinto account the X pling the 2 I l l g state to the discretized continuum of the ionic ground state 'C; (Na2+).The R-independent dipole matrix elements were assumed for all

'q,

62

T.BAUMERT, J. HELBING, AND G. GERBER

transitions. The calculated electron spectra depicted in Fig. 9b qualitatively reproduce the measured results. In addition, the population in the 2 Hgstate as a function of time was calculated for both chirp directions, which is shown in Fig. 10. X Cd state Since the Franck-Condon maximum for both the A transition as well as for the 2 Illg +- A 'qstate transition is shifted toward the red of the central laser wavelength, the up-chirped laser pulse transfers population to the excited states earlier, whereas a down-chirped laser pulse can efficiently excite the intermediate states only with its trailing edge. For the subsequent ionization process this temporal behavior is essential. In order to achieve a high ionization yield, the population in the 2 lugmust be high at the maximum laser intensity, which is achieved with up-chirped but not with down-chirped laser pulses. However, Fig. 10 also yields a very surprising result. The total population transferred to the 2 'I& state after the end of the pulse is much larger for a down-chirped laser although the ionization yield with this pulse is smaller. This is not due to population transfer to the ionic ground state since in the weak-field limit ionization does not (significantly) decrease the neutral state population. Rather a mechanism very similar to that in the experiments performed by Bardeen et al. [36] is responsible for this effect. To understand

'

'

I

.- '

Down Chirp

time [fs] Figure 10. Temporal development of the population in the 2 'IIg state during interaction with up- and down-chirped laser pulses (f3500 fs2). The chirped pulse profile is shown as a dotted line.

COHERENT CONTROL WITH FEMTOSECOND LASER PULSES

63

it, a semiclassical argument based on a difference potential analysis [40,41] is very illuminating. By the Franck-Condon principle the nuclear kinetic energy must be conserved during an electronic transition. Thus the equation for overall energy conservation,

reduces to

-

A 'Ci transition is thus The classically allowed region for the 2 I l l g given by the point of intersection of the difference potential on the left-hand side of Eq. ( 2 ) with the laser energy. When a chirped laser pulse is used, the photon energy hv is changing in time. For a down-chirped pulse the decreasing laser frequency follows the decrease in the difference potential VR (2 'TIg) - VR(A'q), as the excited-state wavepacket propagates to larger internuclear distances. Excitation is classically allowed over a wider range of the nuclear coordinate for a down-chirped laser pulse, and the corresponding final population in the 2 Ingstate is higher. With the opposite chirp direction (up chirp) one observes a higher ionization yield while the intermediate state population is kept low. This kind of optimization is desired in coherent control schemes.

VI. INFLUENCE OF LASER PULSE DURATION Besides the strong chirp dependence of the ionization yield observable in the electron spectra of Fig. 9, a very different electron signal is observed for unchirped 40-fs larger pulses compared to the chirped laser pulses of 240 fs duration (3500 fs'). The short laser pulse predominantly yields electrons with kinetic energy of about 0.9 eV; the electron spectra obtained with the longer (chirped) pulses are dominated by electrons around 0.8 eV. This behavior can again be understood by a difference potential analysis, this time for the transition from the 2 state to the ionic ground state. For transitions into the ionic continuum, the emitted electron can carry away additional energy, so Eq. ( 2 ) now reads

Since V R ( ' ~ , '-) vR(2 I l l g ) is increasing with internuclear distance, the elec-

64

T. BAUMERT, J. HELBING, AND G . GERBER

trons released have less kinetic energy when formed at the outer turning point of a wavepacket propagating in the 2 'IIg potential than those formed at the inner turning point [42,43]. The duration of the transform-limited 40-fs pulse is much shorter than the oscillation period (approximately 380 fs) of the excited vibrational levels of the 2 'IIgstate, and the wavepacket has no time to move to large internuclear distances during the laser interaction. The upand down-chirped laser pulses, however, are of much longer duration, which allows the wavepackets to sweep the whole range of allowed internuclear distances while ionization takes place. The resulting electron spectra therefore extend to lower kinetic energies, which correspond to the outer turning point where the wavepackets spend more time. Using pulses of different duration, one can also influence the fragmentation of Naz. As discussed in the context of the pumpprobe schemes in Section 111, the doubly excited state Na***, which may subsequently yield Na(3s) and Na+ ions of low kinetic energy (0.1 eV) via autoionization-induced fragmentation, is reached only at large internuclear distances (process 2 in Fig. 3). Much more energetic fragments (1.3 eV) are formed when an additional photon is absorbed by the Na2+ ion in its ground state, which can be reached independent of internuclear distance (process 3). Figure 11 shows the ion TOF spectra obtained with 240-fs (chirped) pulses as well as an unchirped 40-fs pulse. The broad peak at 4.5 ps is due to the low-energy fragments coming from process 2, whereas the peaks at 4.1 and 5 p s arise from fragmentation of Na2+ via process 3. The earlier peak is due to fragments with kinetic energy emitted in the direction of the detector; the fragments emitted in opposite direction, which are forced to turn around

+

Up Chirp

TOF [ P I Figure 11. The Na+ fragment TOF spectra obtained with single long (chirped) pulses of 240 fs duration and 40-fs transform-limited laser pulses at 61 8 nm.The ratio of low energetic versus high energetic fragments is seen to be influenced by the pulse duration.

COHERENT CONTROL WITH FEMTOSECOND LASER PULSES

65

by the extracting electric field, give rise to the later peak [ 18,441. The narrow structure on top of the low-energy fragment signal are Na' ions from the ionization of atomic sodium. As expected, according to the discussion of the electron signals, the Na+ spectra show larger contributions of low-energy fragments for the longer laser pulses, since their formation involves the Na2** excitation at large internuclear distances. However, the bandwidth-limited 40-fs pulse yields almost equal amounts of fragments from processes 2 and 3.

VII. COHERENT CONTROL WITH INTENSE LASER PULSES From a practical point of view a major disadvantage of the above control schemes is that they have been applied both theoretically and experimentally mostly in the perturbative regime, thus yielding very small amounts of the desired products. However, intense laser fields give high yields and may even be used to modify the excited molecule in such a way as to drive it to the desired final state. Melinger et al. [45, 461 have demonstrated control of the population in 3 2P3p and 3 2P1p states of atomic sodium and of the B state of I2 with intense linearly chirped laser pulses by making use of adiabatic rapid passage (ARF'). They also pointed out the difficulty of achieving selectivity when many (rotational or vibrational) levels lie within the bandwidth of the laser pulses used. Chelkowski et al. [ I l l proposed to employ intense laser pulses with a chirp tailored in accordance with the vibrational energy spacings of a molecular potential to achieve high dissociation yields. Boers, Balling, et al. [47, 481 have demonstrated the feasibility of such an excitation on the three-level model system provided by atomic rubidium. For a demonstration of a control scheme based on Rabi-type cycles, we look again at the Na2 system at 620 nm as a first example. Using 80-fs laser pulses (CPM laser) of three different intensities (10,0.510, and 0.110) above the perturbative regime, the Na2' transients of Fig. 12 are obtained [49]. The modulation frequency of the transient signal is changing as a function of laser intensity, which can best be seen from the corresponding Fourier transforms. While at more moderate intensities mainly the A state frequency (110 cm-') contributes to the transient signal [remember that in the perturbathe regime (Fig. 4), there can be no 2 'TI, state contribution to the Na2+ transient], the 2 I l l g state frequency (90 cm-') is beginning to dominate at higher intensities. For the highest intensity shown wavepacket motion is detected even on the electronic ground state X (157 cm-'). It is created through stimulated emission during the time the ultrashort pump pulse interacts with the molecule and is detected via direct three-photon ionization by the time-delayed probe pulse.

'

T.BAUMERT, J. HELBING, AND G. GERBER

44 Na; S i g n a l

1. our 0.3*1

O.l*I Pump-Probe D e l a y [PSI

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I

I

I

-2

-1

0

1

I

2

3

x /

/

I

/ /

I

/ / /

I

/

I

I

I

I

Na; FFT [cm-'] Figure 12. Transient Na2+ spectra as a function of delay between identical 80-fs, 620nm pumpprobe pulses for three different laser intensities (top) and corresponding Fourier transforms (bottom).

COHERENT CONTROL WITH FEMTOSECOND LASER PULSES

67

Similar transient signals were obtained from time-dependent quantum mechanical calculations performed by Meier and Engel, which well reproduce the observed behavior [49]. They show that for different laser field strengths the electronic states involved in the multiphoton ionization (MPI) are differently populated in Rabi-type processes. In Fig. 13 the population in the neutral electronic states is calculated during interaction of the molecule with 60-fs pulses at 618 nm. For lower intensities the A 'qstate is preferentially populated by the pump pulse, and the A '% state wavepacket dominates the transient Na2+ signal. However, for the higher intensities used in the

0,l

-1bo

..,.. . .., -50

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so

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time [fs]

. ....... -. ." .. . . . . . . rigure 13. ropuration in me eiecuonic stales invoivea In me muiupnoron ionizaiion of 1 .

.*

Na2 during the interaction with ultrashort laser pulses of different intensities. The calculations were performed for 60-fs pulses at 618 nm, l o = 3 x IO'O W/cm2.

68

T. BAUMERT, J. HELBING, AND G. GERBER

experiment the calculations indicate that the situation may well be reversed and consequently the contribution from the 2 'I& state is dominant in the measured transient signal of Fig. 12. This shows that by varying the intensity of an ultrashort laser pulse, the population transferred to the various neutral electronic states of Na2 can be modified. Recording the transient ion signal in a pumpprobe setup can serve to monitor and control the achieved result. The drop in NaZ+ signal for zero pumpprobe delay at the highest intensity of Fig. 12 is due to fragmentation of Na2+ by absorption of another photon after ionization. At higher intensities it is therefore advantageous to record the transient electron spectrum rather than the Na2+ signal to monitor the bound-state population. Figure 14 shows the transients of electrons formed upon direct ionization of Na2 (process 1 in Fig. 3) recorded for three different intensities using intense 40-fs transform-limited pulses. This time the contribution of a 2 'IT, state wavepacket is disappearing at the highest intensities and the ground state is eventually dominating the transient signal. Again the result can be explained by calculations of the population in the bound electronic states after the pump pulse excitation for different laser intensities (Fig. 15). They show that despite the high laser intensities the 2 I l l g state population may have dropped again to zero after interaction with the pump pulse. From the calculations it can also be seen that for extremely short pulses (Fig. 15) Rabi oscillations take place predominantly between the X 'Ei and the A 'El electronic states while for longer pulses in Fig. 13 they are observed between the A 'Ci and 2 Ill, states. This additional pulse length effect (see above) is due to the fact that the A-X transition is enhanced only at small internuclear distances. On the other hand, the A 'El-hv and 2 'll,-2hv dressed-state potentials calculated for a laser intensity of 5 x 10" W/cm2 in Fig. 16 are almost parallel, which indicates that 2 '+-A '2; transitions may become equally likely over a wide range of internuclear distances in intense laser fields. The fact that 2 'II,-A 'Xi transitions can occur over a wide range of internuclear distances due to potential curve deformation may actually be an undesired side effect of intense laser fields, because it may lead to a loss of selectivity in pumpprobe control schemes. The intensity control scheme described above can also be applied to triatomic molecules as we now demonstrate for the case of Na3. This molecule is produced in our molecular beam apparatus at very high oven temperatures or by using argon at 4 bars as a seedgas. The excitation scheme for Na3 using laser pulses near 620 nm can be seen in Fig. 17. The B state extends from 600 to 625 nm [50] and can be reached by one-photon excitation. A further photon can ionize Na3, yielding Na3+ in its ground state. Three-photon absorption leads to fragmentation. In the ionic ground state Na3+ has an equilateral triangle configuration, whereas the neutral ground state and

COHERENT CONTROL WITH FEMTOSECOND LASER PULSES

I

I I

0

I

50

1 I

I

I

2

3 I

pump-probe delay [ps]

160

200 e- FFT [cm-'1 140

I

'

240

69

1

360

Figure 14. Transient signal of the electrons formed upon direct ionization of Na2 (process 1 in Fig. 3) as a function of delay between identical 40-fs, 618-nm pumpprobe pulses for three different laser intensities (top) and corresponding Fourier transforms (bottom).

the B state of N q are slightly deformed due to the Jahn-Teller effect. Due to these different geometric structures, one expects configuration-dependent transition probabilities from the Franck-Condon principle. It should therefore be possible to generate and probe vibrational wavepacket motion. Figure 18 shows the Na3+ transient with the corresponding Fourier transform

70

T. BAUMERT, J. HELBING, AND G. GERBER

... .

.. .

time [fs] 10"

Figure 15. Like Fig. 13 but calculated for shorter (30-fs), 618-nm laser pulses, I0 = 2 x W/cm2.

COHERENT CONTROL WITH FEMTOSECOND LASER PULSES

W

71

0-31

Figure 16. Diabatic (top) and adiabatic (bottom) dressed states relevant for the multiphoton ionization of Na2 with 618-nm hotons. The adiabatic dressed states were calculated for a laser intensity of 5 x 10" w/cm .

z

obtained with strongly attenuated 80-fs pulses at 618 nm. The dominant-frequency component around 105 cm-' maps the symmetric stretch oscillation (Bss)of the molecule excited by the pump pulse in the B state. A smaller contribution near 72 cm-' can be tentatively assigned to the asymmetric stretch (B,) and bending (Bas)normal modes in the same electronic state. The fre-

72

T. BAUMERT, J. HELBING, AND G . GERBER

[1 0 ~ ~ ~ 4 1

50

LO

30

2C

10

R Figure 17. Schematic view of the relevant electronic states for the interaction of Na3 with 620-nm laser pulses.

quencies observed below 40 cm-' agree well with those obtained by Broyer, Delacritaz, Rakowsky, et al. [50-531 from nanosecond laser experiments and an analysis of pseudorotation on the B state. Using high laser intensities the transient shown in Fig. 19 is obtained [54]. The higher intensity is immediately apparent from the drop in the Na3+ signal for zero pump-probe delay, which is due to three-photon fragmentation. Many more frequencies con-

COHERENT CONTROL WITH FEMTOSECOND LASER PULSES

73

I

'

6

'

i

.

!

?

.

9

pump-probe delay [ps]

'

i

I

140

Na3+FFT [cm-'1 Figure 18. Transient Na3' signal for strongly attenuated 80-fs pumpprobe laser pulses of 620 nm. The frequencies observed in the Fourier transform are due to vibrational wavepacket motion on the B state potential.

tribute to the signal in this case: Besides the wavepacket propagation on the B state already observed at low intensities, there are now also contributionsat 50,90, and 140 cm-' . These can be assigned (see again the work performed by Broyer et al. [53]) to the bending mode Xb,the asymmetric stretch mode X, and the symmetric stretch mode X, in the ground state of Na3. This shows that the intense pump laser pulse has coherently transferred population back into the neutral ground state, thus creating wavepacket motion on this potential surface, which is subsequently probed by two-photon ionization. As in the case of Naz, population transfer between electronic states in Rabi-type processes can therefore be controlled by varying the laser intensity and can be monitored in a pumpprobe experiment.

74

T.BAUMERT, J. HELBING, AND G. GERBER

0

2

1

3

pump-probe delay [ps]

,;fi,$

4

'b

40

60

4 140

Na3+FFT [cm-'1

Figure 19. Transient Na3+ signal for intense 80-fs pumpprobe laser pulses of 620 nm. In addition to the B state frequencies observed for low intensities, the Fourier transform now also shows contributions attributable to wavepacket motion on the ground-state potential.

VIII. CONCLUSION In summary, we have chosen Naz and Na3 to demonstrate coherent control of excited-state population, ionization yield, fragmentation, and ionic product formation using a variety of control parameters. For a three-photon and a two-photon process we have shown that vibrational wavepacket propagation excited by an ultrashort laser pulse can be used to drive a molecule to a nuclear configuration where the desired product formation by a second probe pulse is favored (Tannor-Kosloff-Rice scheme). In both cases the relative fragmentation and ionization yield of Na;! was controlled as a function of pump-probe delay. By varying the delay between pump and probe pulses very slowly and therefore controlling the phase relation between the two pulses, additional interference effects could be detected.

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75

Using linearly chirped pulses, we demonstrated in a single-pulse experiment that the ionization yield of Na2+ can be maximized while the intermediate-state population is significantly reduced. With the help of quantum mechanical calculations and a semiclassical difference potential analysis, we elaborated that this effect is partly due to a very common variation of the transition frequency between two electronic states with internuclear distance. Making the frequency ordering of a chirped laser pulse follow this change in transition energy as a wavepacket moves along the nuclear coordinate should be very widely applicable as a technique for coherent control. By varying the duration of a laser pulse (40and 240 fs). the range of the nuclear coordinate swept during the interaction was varied, which is reflected by a drastic change in the electron energy distribution for the three-photon ionization of Na2. It was shown that the pulse length can be used to influence the relative yield of high versus low energetic fragments formed via two different fragmentation processes. Finally, coherent transfer of population between electronic states was demonstrated using intense ultrashort laser pulses of different durations. Aided by calculations, it was shown that the population in various neutral electronic states of both Na2 and Na3 at the end of the interaction with a laser pulse can be controlled by varying the laser intensity. A second (intense) probe laser was used to ionize the molecules. The Fourier transform obtained from the transient ion signal can be used to experimentally monitor the population distribution created by the first laser pulse.

Acknowledgments The authors have performed the experiments presented in this contribution together with M. Strehle, R. Thalweiser, B. Waibel, V. Weiss, and E. Wiedemann. Financial support was given by the Deutsche Forschungsgemeinschaft through SFB 276 “Korrelierte Dynamik hochangeregter atomarer und molekularer Systeme” in Freiburg. A. Assion, D. Schulz, V. Seyfried.

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44. T.Baumert, J. L. Herek, and A. H.a w a i t , J. Chem. Phys. 99,4430 (1993). 45. J. S. Melinger, S. R. Gandhi, A. Hariharan, D. Goswami, and W. S. Warren, J. Chem. Phys. 101, 6439 (1994). 46. J. S. Melinger, A. Hariharan, S. R. Gandhi, and W. S. Warren, J. Chem. Phys. 95, 2210 ( 199 1). 47. B. Boers, H. B. Van Linden van den Heuvell, and L. D. Noordam, Phys. Rev. Lett. 69, 2062 (1992). 48. P. Balling, D. J. Maas, and L. D. Noordam, Phys. Rev. A 50,4276 (1994). 49. T. Baumert, V. Engel, C. Meier. and G. Gerber, Chem. Phys. Lett. 200, 488 (1992). 50. M. Broyer, G. Deiacr6taz. P. Labastie, R. L. Whetten, J. P. Wolf, and L. Woste, Z. Phys. D 3, 131 (1986). 51. G. Delacrktaz, E. R. Grant, R. L. Whetten, L. Woste, and J. F. Zwanziger, Phys. Rev. Lett. 56,2598 (1986). 52. S . Rakowsky, F. W. Herrmann, and W. E. Emst, 2 Phys. D 26 (1993). 53. M.Broyer, G. Delacritaz, G. Q. Ni, R. L. Whetten, J. P. Wolf, and L. Woste, Phys. Rev. Lett. 62, 2100 (1989). 54. T. Baumert, R. Thalweiser, and G. Gerber, Chem. Phys. Lett. 209,29 (1993).

DISCUSSION ON THE REPORT BY G. GERBER Chairman: S. A. Rice

L. Woste: You showed that the above-threshold ionization process always ends at the bottom of the ionic state when exciting the system with femtosecond pulses. So, going to higher laser powers, you observe the consecutive onset of multiphotonic processes. What happens when you cross the double-ionization barrier? Is the same true for doubly charged clusters? G. Gerber: The observed molecular AT1 (above-threshold ionization) in Na2 occurs for laser intensities above 10i2W/cm2.The observation that no additional fragmentation channels (except the *% channel) open up in Na2+ might explain why in femtosecond cluster experiments additional ion fragmentation channels do not show up. Within the femtosecond interaction time even under AT1 conditions the lowest Franck-Condon factor (FCF-allowed) vibrational levels are the only ones observed. At even higher intensities when multiple ionization of clusters occurs, the situation can be different. This has not yet been investigated in detail. A. € Zewail: I. I have two questions for Prof. Gerber: 1. In large clusters do you observe new fragmentation pathways due to “Coulomb explosion” as observed in the nice experiments of Castleman’s group?

78

T. BAUMERT, J. HELBING, AND G. GERBER

2. What about rotationally selected wavepackets in Na3 as reported in the new scheme shown by Leone’s group for Liz?

G. Gerber 1. For high laser intensity we observe multiply charged clusters. Even higher charged clusters undergo Coulomb explosion. As far as we have measured the initial kinetic energy release, we observe different fragment energies for different fragments. 2. At the same time a vibrational wavepacket is prepared also a rotational wavepacket is formed in our experiments. However, we have not explored that yet. It is clear what happens based upon your earlier experiments.

J. Troe: Concerning the lack of dependence of the Na, * lifetime

on cluster size n discussed by Prof. Gerber, is it not possible that the excitation leads into repulsive excited electronic states from which the fragmentation is “direct,” that is, not related to phase-space volumes and densities of states? G. Gerber: We do observe a variation of the lifetimes T depending on the cluster size n and also on the particular intermediate cluster resonance Nan* for a given size n. For these (pump-probe) decay time measurements we always selected a very specific cluster size in the detection channel! However, what is currently not understood is the irregular variation of T for one resonance and the obviously regular behavior (independence of n) for another cluster resonance. However, what is clear is that the decay times need to be related to fragmentation processes. J. Manz 1. Prof. Gerber has demonstrated to us three different strategies for laser control with applications to Naz: (i) Control of different product channels

Naz-Naz

*

-

-+

NaZ++e Na+ Na’

+e

by choosing the selective delay time T between the pump and control laser pulses [l], following the general strategy of Tannor et al. [2]. (ii) Control of different ionization pathways by selecting appropriate laser intensities Z [3]. This strategy exploits the fact that increasing

COHERENT CONTROL WITH FEMTOSECOND LASER PULSES

79

intensities may enhance not only electronic excitation but also deexcitation (i.e., stimulated emission) processes, in particular from excited electronic states back to the ground state, thus increasing the number of photons involved in the overall photoionization process. This strategy may be supported by excitations of coherent vibrations in the electronic ground state, as predicted in Ref. 4. (iii) Control of different ionization pathways by selective choices of the laser pulse duration fp [5]. To the best of my knowledge, this is a new strategy, and I wish to ask Prof. Gerber for a more detailed explanation of the mechanism. I would also like to use the opportunity and point to some of the following strategies of laser control: (i) In Ref. 6, we combine the strategy of Tannor et al. [2] with vibrationally mediated chemistry [73, with applications to photodissociation of the model dimer Na - NH3. (ii) In Ref. 8, we demonstrate the control of multiphoton ionization pathways by laser intensities Z for the model system K2. Specifically, low or moderate intensities induce three-photon ionizations via configurations with long bond lengths r > re of K2, whereas high intensities stimulate five-photon ionizations via excitations of a coherent wavepacket in the electronic ground state, which is then driven to a favorable new Franck-Condon window for ionization at short bond lengths r c re. (iii) In Ref. 9, we show that different choices of laser pulse durations, specifically fp = 120 fs versus 1.5 ps at low intensities, facilitate the monitoring of different vibrations of Na3 excited to the electronic B state, specifically the symmetric stretch v1 versus the angular (p) pseudorotation vV. Further details of this selectivity will be explained below (see the following comment by R. de Vivie-Riedle, J. Manz, B. Reischl, and L. Woste). I . T. Baumert, R. Thalweiser, V. Weiss, and G. Gerber, in Femtosecond Chemistry, J. Manz and L. Woste, Eds., Verlag Chemie, Weinheim, 1995. Chapter 12, p. 397. 2. D. J. Tannor and S. A. Rice, J. Chem. Phys. 83, 5013 (1985); D. J. Tannor, R. Kosloff, and S. A. Rice, J. Chem. Phys. 85,5805 (1986). 3. T.Baumert and G. Gerber, Adv. At. Molec. Opt. Phys. 35, 163 (1995). 4. B. Hartke, R. Kosloff, and S. Ruhman, Chem. Phys. Lett. 158,238 (1989). 5. G. Gerber, private communication. See also A. Assion, T. Baumert, J. Helbing, V. Seyfried, and G. Gerber, Chem. Phys. Leu.,259,488 (1996). 6. C. Daniel, R. de Vivie-Riedle, M.-C. Heitz, J. Manz, and P. Saalfrank, Int. J. Quanr. Chem. 57,595 (1996). 7. V. S. Letokhov, Science 180, 451 (1973); F. F. Crim, Science 249, 1387 (1990); J. E. Combariza, C. Daniel, B. Just, E. Kades, E. Kolba, J. Manz, W. Malisch,

80

T. BAUMERT, J. HELBING, AND G. GERBER

G. K. Paramonov, and B. Warmuth, in Isotope Effects in Gas-Phase Chemistry. J. A. Kaye, Ed., ACS Symp. Ser. 502,310 (1992). 8 . R. de Vivie-Riedle, J. Manz, W. Meyer, B. Reischl, S. Rutz, E. Schreiber, and L. Woste, J . Chem. Phys. 100,7789 (1996). 9. R. de Vivie-Riedle, J. Gaus, V. Bona6f-Kouteck9, J. Manz. B. Reischl, S. Rutz, E. Schreiber, and L. Woste, in Femtosecond Chemistry and Physics of Ultrafast Processes, M.Chergui, Ed., World Scientific, Singapore, 1996; B. Reischl, R. de Vivie-Riedle, S. Rutz, and E. Schreiber, J. Chem. Phys. 104,8857 (1996).

2. My question to Prof. Gerber is the following: Could you please explain the different virtues of femtosecond pump-pulse experiments versus ultrashort zero electron kinetic energy (ZEKE) spectroscopy? Do they yield complementary information on the molecular dynamics or are there specific domains where one of them should be preferred with respect to the other?

G. Gerber 1. The control through variation of the femtosecond pulse duration is probably a very general scheme. In the example I discussed, using a long pulse duration ( 4 5 0 fs) we reach the outer turning point and induce the “two-electron” process. For short pulse duration ( 6 0 fs) we only have the channel that is open at small internuclear distances, namely the “one-electron”direct ionization. Since the (bound-free) ionization process at the inner turning point is much less probable (due to the oscillator strength) compared to the (bound-bound) process of excitation of the second electron at the outer turning point, for longer pulse durations essentially only the “two-electron” process plays a role. 2. The ZEKE detection opens up an additional and sometimes different view (compared to the ion detection channel) because of the different find states involved. This has been discussed in a recent publication [ 11 for the particular case of the B state dynamics of Na3, which we had investigated. 1. A. J. Dobbyn and J. M . Hutson, Chem. Phys. Lett. 236,547 (1995).

T. Kobayashi: I have the following questions about the report by Prof. Gerber: 1. If the variation of the population as a function of delay time features damped oscillations with Rabi frequency, it is expected to see the Rabi splitting. Can this be observed? 2. Why is the two-photon ionization spectrum so broad? Is the spectrum mainly homogeneously broadened or inhomogeneously broad-

COHERENT CONTROL WITH FEMTOSECOND LASER PULSES

81

ened? If the first is the case, then why is the lifetime so long, that is, extending from a 1.1-ps component to a 215-ps component? 3. Is the intervalley scattering time on a GaAs surface faster or slower than that in bulk GaAs?

G. Gerber 1. We coherently couple different electronic states in the molecule during our intense femtosecond pulse. For higher laser intensities we induce Rabi oscillations between the different electronic states, which finally lead to a control of the population put in the different states. With an intense 50-fs pulse exhibiting an intrinsic spectral broadening of -30 meV we did not observe a Rabi splitting in the ion detection channel. However, in the electron detection channel (according to a theoretical paper by Engel [l]), one should observe the splitting. 2. The two-photon ionization spectrum of the Nazo cluster is broadened by many possible vibrational transitions. So the width is not related to the decay time of the resonance. 3. To my knowledge they are very similar, but depend on the specific surface. 1. V. Engel, Phys. Rev. Lerr. 73, 3207 (1994).

V. S. Letokhov: Prof. Gerber, can we exploit the analogy between the asymmetric fragmentation process (Na13 -+ Nalo + Na3) and the fission of nuclei? G. Gerber: By applying two-photon ionization spectroscopy with tunable femtosecond laser pulses we recorded the absorption through intermediate resonances in cluster sizes Nan with n = 3, ... , 21. The fragmentation channels and decay pattern vary not only for different cluster sizes but also for different resonances corresponding to a particular size n. This variation of T and the fragmentation channels cannot be explained by collective type processes (jellium model with surface plasmon excitation) but rather require molecular structure type calculations and considerations. U. Even: Prof. Gerber, could you follow the metal-nonmetal transition in mercury clusters? G. Gerber: At moderate laser intensities we do see in femtosecond pumpprobe experiments a very similar “slow” time and “long” time dynamics in all cluster sizes n > 5 up to n = 50 (largest size investigated up to now) irrespective of the charge state of the particular Hg, cluster. From single-pulse TOF mass spectrometry we infer that the

82

T. BAUMERT, I. HELBING, AND G. GERBER

nonmetal-metal transition takes place for a cluster size with n = 80 Hg atoms. Due to the transition from localized to delocalized excitations occumng in a metal, we no longer observe multiply charged Hg, clusters at moderate laser intensities. This transition at n = 80 is in agreement with recent calculations. M. Chergui: My question to Prof, Gerber relates to the lifetime of the absorption resonance at 510 nm in Na, clusters. Since you dealt with clusters in the n = 5, . ., ’45 range and, therefore, you are in the nonmetal regime (maybe even in the van der Waals regime), could one envisage that the fragmentation process occurs via a “bubble-type” mechanism whereby you are exciting a low-n Rydberg electron of a center in the cluster, which has a repulsive interaction with neighboring atoms? The impulsive blowing up of a “bubble” could give rise to a deformation that propagates to the outskirts of the cluster and leads to a boil-off of atoms. This type of mechanism has been described by Jortner and co-workers for XeAr,(n I50) clusters [ I , 21. My second question is: How does the lifetime of the resonance absorption and the fragmentation process evolve if you increase the size of the cluster (n > 45), especially when it reaches the metal regime? 1. D. Scharf, J. Jortner, and U. Landman, J . Chem. Phys. 88,4273 (1988). 2. A. Goldberg, A. Heidenreich. and J. Jortner, J. Phys. Chern. 99, 2662 (1995).

G. Gerber: I do not think this bubble picture applies to the fragmentation of particular Na, clusters. We observed that the intennediate resonance strongly influences the time and the predominant decay channel. We did not study yet cluster sizes beyond n = 41. It appears that, for larger cluster sizes, no predominant picosecond decay channel exists. D. M. Neumark Because of the short pulses used in Prof. Gerber’s experiments, the overall energy resolution is -30 meV. What then is gained by performing ZEKE versus conventional photoelectron spectroscopy (PES) on Na3? With PES, one could map out the Na3 wavepacket onto the entire Na3+ manifold using a single ionization wavelength at each time delay. G. Gerber: The reported experiment was done with fixed-frequency laser pulses for the pump and the probe laser. I do agree that the observation of ZEKE electrons and the dynamics of the signal using tunable femtosecond laser pulses would be of interest.

Advances In Chemical Physics, Volume101 Edited by I. Prigogine, Stuart A. Rice Copyright © 1997 by John Wiley & Sons, Inc.

GENERAL DISCUSSION ON FEMTOCHEMISTRY: FROM ISOLATED MOLECULES TO CLUSTERS Chairman: S. A. Rice

B. Kohler: I would like to ask two questions to Prof. Zewail. First, in your investigation of the electron transfer reaction in a benzene-I2 complex, the sample trajectory calculations you showed appear to suggest that the charge transfer step may induce vibrationally coherent motion in 12-. Have you tested this possibility experimentally?My second question concerns your intriguing results on a tautomerization reaction in a model base-pair system. In many of the barrierless chemical reactions you have studied, you have been able to show that an initial coherence created in the reactant molecules is often observable in the products. In the case of the 7-azaindole dimer system your measurements indicate that reaction proceeds quite slowly on the time scale of vibrational motions (such as the N-H stretch) that are coupled to the reaction coordinate. What role do you think coherent motion might play in reactions such as this one that have a barrier? A. H. Zewaik The observation of coherent motion in the benzene-iodine system should be related to the Iz- motion and hopefully with better time resolution we should be able to resolve it. As for the base-pair experiment, the key motion is that of the N .. N stretch and N-H asymmetric motions, and our time scale of observation was appropriate for the dynamics to be observed. M. Quack Prof. Zewail and Gerber, when you make an interpretation of your femtosecond observations (detection signal as a function of excitation), would it not be necessary to try a full quantum dynamical simulation of your experiment in order to obtain a match with your molecular, mechanistic picture of the dynamics or the detailed wavepacket evolution? Agreement between experimental observation and theoretical simulation would then support the validity of the underlying interpretation (but it would not prove it). The scheme is of the following kind: 83

84

GENERAL DISCUSSION Input

E*perimerimentallycontrolled parameters

'Ibeoretical model (molecular mechanisms and wavepackets) free parameters to be adjusted Quantum simulation

The question is then, first, how often has such a complete match between experiment and theoretical simulation been achieved? Second, are there good examples where complete simulations have been carried out but lead to two or more equally acceptable models to interpret the experimental results? I refer to this question of ambiguity also in relation to a very similar problem arising in the interpretation of nontime-resolved high-resolution spectroscopy data [1,2], which provided in fact, the first experimental results on nontrivial threedimensional wavepacket motion on the femtosecond time scale [3]. I.

M.Quack, Chapter 27, p. 781. in Femtosecond Chemistry, J. Manz and L. Woste,

Eds., Verlag Chemie, Weinheim, 1995. 2. M.Quack, Jerusalem Symp. 24,47 (1991); in Mode Selective Chemistry, J. Jortner, R. D. Levine, and B. Pullman, Eds., Reidel, Dordrecht. 3. R. Marquardt, M. Quack, J. Stohner, and E. Sutcliff, J. Chem. SOC.Faraday Trans. 2 82, 1173 (1986).

A. H. Zewail: If we solve for the molecular Hamiltonian, we will be theorists! I do, of course, understand the point by Prof. Quack and the answer comes from the nature of the system and the experimental approach. For example, in elementary systems studied by femtosecond transition-state spectroscopy one can actually clock the motion and deduce the potentials. In complex systems we utilize a variety of template-state detection to examine the dynamics, and, like every other approach, you/we use a variety of input to reach the final answer. Solving the structure of a protein by X-ray diffraction may appear impossible, but by using a number of variant diffractions, such as the heavy atom, one obtains the final answer. B. A. Hess: In regard to the point discussed by Profs. Quack and Zewail, let me comment that, in general, inverse problems have a unique solution only under very restrictive circumstances; thus we should expect that we can find cases where the same spectroscopic data are compatible with different molecular structures. H. Hamaguchi: I have a question directed to Prof. Zewail. You

FEMTOCHEMISTRY I

85

briefly mentioned the wavepacket dynamics of the stilbene photoisomerization. What could you tell, based on this wavepacket formalism, about the stilbene ;hotoisomerization in solution? As you know, the rate of photoisomerization increases about 100 times on going from the isolated molecule to solution. How can you account for this acceleration within the framework of the wavepacket formalism? A. H. Zewail: There are many beautiful experiments done in the condensed phase that show such coherent nuclear wavepacket motion. For example, the work of Ruhman on 13- and Hochstrasser on HgI, and cis-stilbene. The latter two have shown direct analogy to the results observed in the isolated gas phase. It appears that the time scale for intermolecular couplings is somewhat longer than those of the intramolecular dynamics, even though solvent-induced dephasing and vibrational relaxation are integral parts of the dynamics. For cisstilbene, the wavepacket motion indicates that the molecule is twisting, utilizing a second motion, the phenyls, in addition to the expected ethylenic torsion. As for the acceleration of the rates in solution for trans-stilbene this is due to the microscopic friction and the lowering of the barrier. [See A. A. Heikal, S. H. Chong, J. S. Baskin, and A. H. Zewail, Chern. Phys. Lett. 242, 380 (1985).] J.-L. Martin: Prof. Zewail, how do you expect your work on photoinduced tautomerization of base pairs to apply to the real world of DNA, where such a reaction would happen on ground-state potential surface in a water environment? A. H. Zewail: The relationship to real-life DNA structures is certainly not known to us. However, as you can see from the paper in Nature [l], the transient structures have never been isolated and there is another important point. The presence of an “ionic intermediate” is possibly significant for recognition by enzymes of mutation by this mechanism, if operative.

F. Goodman, Nature 378,237 (1995);A. Donhal, S. K. Kim, and A. H. Zewail. Nature 378,260 (1995).

1. M .

K. Yamanouchi: I have two comments concerning Prof. Zewail’s report: 1. By using the presented second-generationgas electron diffraction apparatus, it would also be possible to probe vibrational motion in real time. Especially when a molecule is photodissociated, a series of snapshots of a diffraction pattern would facilitate understanding the photodissociation process because it describes how a molecule vibrates in the course of the separation of two frag-

86

GENERAL DISCUSSION

ments flying apart. This vibrational motion during the dissociation process is subject to the so-called intramolecular vibrational energy redistribution (IVR), which plays a central role in a unimolecular dissociation reaction. The new gas electron diffraction experiments presented here by Prof. Zewail could have powerful potential to visualize IVR through the real-time probing. 2. In conventional gas electron diffraction experiments, an effusive beam is used in which vibrational levels of molecules are thermally populated and the width of a peak in a radial distribution curve is determined by thermally averaged mean amplitudes. When a molecular beam or a free jet is used, mean amplitudes could become small, since the contribution from the vibrationally excited levels is reduced significantly. As a consequence, sharper peaks are expected in the radial distribution curve, and the spatial resolution of the snapshot could be improved. However, it seems that the observed peaks in the radial distribution curve are considerably broad even though a molecular beam is used. There could be some reasons to have such broadened peaks in the radial distribution curve.

A. H. Zewail: Prof. Yamanouchi is correct in pointing out the relevance of ultrafast electron diffraction to the studies of vibrational (and rotational) motion. In fact, Chuck Williamson in our group [ 11 has considered precisely this point, and we expect to observe changes in the radial distribution functions as the vibrational amplitude changes and also for different initial temperatures. The broadening in our radial distribution function presented here is limited at the moment by the range of the diffraction sampled. 1. J. C. Williamson and A. H. Zewail, J. Phys. Chem. 98, 2766 (1994).

P. Backhaus, J. Maw, and B. Schmidt:* Prof. A. H. Zewail has demonstrated to us some fascinating new pumpprobe ferntochemistry investigations of bimolecular reactions (e.g., Ref. 1; see also Ref. 2)

H + COz -+HOCO’

OH + CO

+

To the best of our knowledge, however, these types of “bimolecular” studies start from weakly bound precursor systems, for example,

IH. oco -+IH*. *Comment presented by J. Manz.

co2-.I + HOCO$ -+I

+OH + co

(2)

87

FEMTOCHEMISTRY I

From the viewpoint of the van der Waals type or hydrogen-bounded reactant, these reactions may therefore be considered as “unimolecular” even though they exhibit various characteristics of bimolecular processes. Here we wish to point to a new type of femtosecond chemistry investigations of bimolecular reactions, demonstrated first by Dantus et al. [3] for the prototype system

2 Hg -+Hg2*

(3)

Essentially, an ultrashort laser pulse associates two colliding reactants * to a new product. In the present case, a metastable excimer Hg2 is formed from two mercury atoms. This photoassociation process may be considered as the reverse of photodissociation. From a theoretical point of view, the important difference between processes (2) and (3) is that the unimolecular photodissociation (2) starts from bound states, whereas the bimolecular photoassociation (3) starts from continuum states, as demonstrated by model simulations in Fig. 1 (supported by Deutsche Forschungsgemeinschaft). 1. N. F. Scherer, L. R. Khundkar, R.B. Bemstein, and A. H. Zewail, J. Chem. Phys. 87, 1451 (1987); N. F. Scherer, C. Sipes, R. B. Bernstein, and A. H.Zewail, J. Chem.

Phys. 92, 5239 (1990); A. H . Zewail, in Femrosecond Chemistry, J. Manz and L. Woste, Eds., Verlag Chemie, Weinheim, 1995, p. 15; A. H.Zewail, J . Phys. Chem. 100 (1996). in press. 2. S. I. Ionov, G. A. Brucker, C. Jaques, L. Valachovic, and C. Wittig, J. Chem. Phys. 99, 6553 (1993); M. Alagia, N. Balucani, P. Casavecchia, D. Stranges, and G. G. Volpi, J. Chem. Phys. 98, 8341 (1993); G. C. Schatz and M. S.Fitzcharles, in Selectiviry in Chemical Reactions, I. C. Whitehead, Ed., Kluwer, Dordrecht, 1988, p. 353; E. M. Goldfield, S. K. Gray, and G. C. Schatz, J. Chem. Phys. 102, 8807 (1995); D. C. Clary and G. C. Schatz, J. Chem. Phys. 99, 4578 (1993). D. H. Zhang and J. Z. H. Zhang, J. Chem. Phys. 103,6512 (1995); see also L. Krim, P.Qiu, N. Halberstadt, B. Soep, and J. P. Visticot, in Femtosecond Chemistry, J. Manz and L. Woste, Eds., Verlag Chemie, Weinheim, 1995, p. 433. 3. U. Marvet and M. Dantus, Chem. Phys. Lert. 245, 393 (1995). 4. P. Backhaus, M.Dantus, J. Manz, and B. Schmidt, in preparation. 5. J. Koperski, J. B. Atkinson, and L. Krause, Can. J. Phys. 72, 1070 (1994). 6. F. H. Mies, W. J. Stevens, and M. Krauss, J. Mofec. Specrmsc. 72, 303 (1978). 7. E. W. Smith, R. E. Drullinger, M.M. Hessel, and J. Cooper, J. Chem. Phys. 66, 15 (1977).

J. Maw: Moreover, I have another comment on the contributions by A. H. Zewail and G. Gerber and a question to all my colleagues and, in particular, to the Chairman, Prof. S. A. Rice:

C D

v)

c

0

s

m

w m N

m

0

(D

m

10

* E

.

'I

w z m N

P

n m

c

0

* 3

N

88

[r

W

00

-

-

Hg2* by ultrashort femtosecond laser pulses. The figure shows Figure 1. Photoassociation2Hg a sequence of three snapshots of the wavepacket dynamics tailored to the experiment of Marvet and Dantus [3]. At t = 0 fs, the reactants Hg(X0;) are in a continuum state with energy E and rotational quantum numbers J,M.The pump laser pulse A(,, = 312 nm, ,,Z = 2 x l o i 1W/cm2, T = 65 fs) creates a small fraction of dimers Hg2* (lu(63P1)),and the probe pulse (A, = 624 nm, I,, = 2 x 10’3 W/cm*, 7 = 65 fs) monitors the coherent vibration of these Hgz* (1,) excimers by depopulation due to the I, + 1,(61P,) transition. The remaining population of Hg, *(Iu)depends on the delay time of the pump and probe laser pulses (in the present case t, = 480 fs) and is measured by fluorescence. All laser pararneters are adapted to the experimental situation [3]. The present simulation is exemplary for E = 0.529 eV, J = 0, M = 0, and Boltzmann averaging over equivalent calculations for other values of E, J, M accounts for the thermal (T= 433.15 K ) situation of the experiment; see Ref. 4. The potential energies and transition dipoles are adapted from Refs. 5.6, and 7,respectively; for the 1, 1, transition, we assume = leao: This does not affect the relative pump and probe signal (in arbitrary units).

90

GENERAL DISCUSSION

Prof. G. Gerber and A. H. Zewail have presented to us three fascinating experiments on femtosecond laser control of the branching ratio of competing product channels:

--

X ~ . I ~ ~ L X ~ + . . . I - . . .-+ I ~ ~Se*+ 1 2 XeI*+I (Ref. 1) (1) Na2+ + e N a + Na+ + e (Ref. 2) (2) NaI -+NaI* -+ Na + I -+ NaI (Y >>O) (Ref. 3) (3)

Na2 +Na2

*

--t

where NaI (v >> 0) denotes the vibrationally excited NaI molecule in the electronic ground state. Essentially, they employ two femtosecond laser pulses with proper time delay fd. The first pump pulse initiates ( t = 0) a coherent nuclear motion in the electronic excited state. This is represented by a nuclear wavepacket that is driven from the molecular equilibrium configuration of the reactant at time t = 0 toward new configurations close to those of the products at time t = t d . The second control pulse (t = t d ) then induces an electronic transition that serves to stabilize the desired product. Historically, it is amazing and certainly encouraging and gratifying that this type of femtosecond laser control by two femtosecond pump and control laser pulses had been predicted by Tannor et al. already in 1985 [4] (see also Ref. 5), that is, one or two years before the first femtosecond laser chemistry investigation of a chemical reaction, carried out by Zewail et al. in 1987 IS]. I should like to ask whether the experimental (see Refs. 1-3) or theoretical pioneers (see Ref. 4) or any other colleagues could point to any other experimental verifications of the Tannor-Rice-Kosloff strategy [4] beyond processes (1-3). 1. E. P. Potter, J. L. Herek, S. Pedersen, Q. Liu, and A. H. Zewail, Nature 355, 66 (1992).

2. T. Baumert, R. Thalweiser, V. Weiss, and G. Gerber, in Femtosecond Chemistry, Vol. 2, J. Manz and L. Woste, Eds., Verlag Chemie, Weinheim, 1995, p. 397.

3. J. L. Herek, A. Materny, and A. H. Zewail, Chem. Phys.

Leff. 228, 15 (1994).

4. D. 3. Tannor and S . A. Rice, J. Chem. Phys. 83, 5013 (1985); D. 5 . Tannor, Kosloff, and S. A. Rice, J. Chem. Phys. 85,5805 (1986).

R.

5. S . A. Rice, J. Chem. Phys. 90,3063 (1986); S . A. Rice, Perspectives on the control of quantum many body dynamics: Application to chemical reactions, Adv. Chem. Phys., 101, (1997).

6. M. Dantus, M. J. Rosker, and A. H. Zewail, J. Chem. Phys. 87,2395 (1987).

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S. A. Rice: My answer to Prof. Manz is that, as I indicated in my presentation, both the Brumer-Shapiro and the Tannor-Rice control schemes have been verified experimentally. To date, control of the branching ratio in a chemical reaction, or of any other process, by use of temporally and spectrally shaped laser fields has not been experimentally demonstrated. However, since all of the control schemes are based on the fundamental principles of quantum mechanics, it would be very strange (and disturbing) if they were not to be verified. This statement is not intended either to demean the experimental difficulties that must be overcome before any verification can be achieved or to imply that verification is unnecessary. Even though the principles of the several proposed control schemes are not in question, the implementation of the analysis of any particular case involves approximations, for example, the neglect of the influence of some states of the molecule on the reaction. Moreover, for lack of sufficient information, our understanding of the robustness of the proposed control schemes to the inevitable uncertainties introduced by, for example, fluctuations in the laser field, is very limited. Certainly, experimental verification of the various control schemes in a variety of cases will be very valuable. S. Mukamel: In relation to the report by Prof. Gerber, I would like to comment that chemical bonding can also be viewed as electronic coherence. By looking at the relevant single-electron density matrix in the atomic orbital representation, we note that the diagonal elements give the local charges whereas the off-diagonal elements (coherences) represent bond order. Our studies of nonlinear optical spectroscopy of conjugated polyenes have shown that, using this view, one can define electronic normal modes and view the electronic system as a collection of coupled harmonic oscillators representing collective electronic motion [l]. This is a very different picture than using the global electronic eigenstates. Using this picture, it is possible to treat electronic and nuclear degrees of freedom along similar lines. It also furnishes a very powerful means in computing optical response functions, which are size consistent [2]. I. S. Mukamel, A. Takahashi, H. X. Wang, and G . Chen, Science 266, 250 (1994). 2. V. Chernyak and S. Mukamel, J. Chem. Phys. 104,444 (1996).

G. Gerber: In response to Prof. Mukamel, I should remark that the coherence between molecular electronic states induced by our intense ultrashort laser pulse is not restricted to bound states but also includes repulsive electronic surfaces. In that sense chemical bonding is related to electronic coherence.

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Another somewhat different example of coherence of electronic states is a radial electronic Rydberg wavepacket whose dynamics can be studied in pumpprobe experiments. A. H. Zewail: Prof. Mukamel's point is very interesting and we should think of this different language. If I understand, the point is that bonding can be described as an "off-diagonal" density matrix. As with the nuclear motion, we now can picture the process and think of controlling the bonding. S. Mukamel: You are absolutely correct. Although chemical bonding is a complex many-body problem, for most practical purposes, it is sufficient to look at the off-diagonal elements of the reduced singleelectron density matsix. Historically, nuclear molecular motions have been treated using normal modes, whereas electronic properties are calculated using many-electron wave functions. The density matrix and its equation of motion obtained using the time-dependent Hartree-Fock theory provides a normal-mode representation of electron dynamics. The density matrix may also provide a natural extension of density functional theories, resulting in a new way for computing groundstate properties [V. Chernyak and S. Mukamel, Phys. Rev. A 52, 3601 (1995)l. P. W. Brumer: As we know, in quantum mechanics, time evolution and coherence are synonymous. Thus, if I see time evolution, then coherences underlie the observation. Hence, in moving my arm I have created a molecular coherence. We should all be asking why this is so easy to create compared to the complex experiments described in these talks? Is it due to the closely lying energy levels in large systems? If so, then it suggests that experiments on larger molecules would be easier. B. A. Has: The reason that macroscopic motions display coherence is that they are in most cases at the classical limit of quantum dynamics. In this case, a suitable occupation of quantum states ensures that quantum mechanical expectation values equal the classical value of an observable. In particular, the classical state of an electromagnetic field (the coherent state) is one in which the expectation value of the operator of the electromagnetic field equals the classical field strengths. M. Quack: Paul Brumer has asked why it is so easy to generate the coherent state corresponding to his waving hand compared to the difficulties of generating similar coherences in molecules by femtosecond spectroscopy. I shall give a very incomplete answer to this based on Schrodinger's interpretation [l]. For a small atomic and molecular system, the quantum energy spacings are very large, requiring large excitation bandwidths (or short times) to generate the coherences. In

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a heavy-mass (macroscopic) body the quantum-level energy spacings are small, and thus it is easy to generate coherences on very long time scales. In relation to Paul Brumer’s comment that quantum mechanics surely applies also to macroscopic bodies, I would like to turn around the question, however: Why is it so difficult to generate stationary states or more generally superposition states of classically localized states? We have been interested for some time in doing this type of superposition experiment for large, chiral, polyatomic molecules [2] including “chiral” oscillator motion [3-51. When one extends such considerations to a macroscopic classical pendulum, one must admit that we do not know whether the superposition principle of quantum mechanics applies: To date this has never been tested experimentally in such cases. Until we have experimental proof, we must leave the answer to this question open. Although there are certainly many among us who would wish to accept the validity of quantum mechanics for macroscopic bodies (classical mechanics being only a limiting law to quantum mechanics), I might point out that the validity of the superposition principle has been questioned [6] even for the superposition experiment for chiral molecules that I have mentioned above. Again this question will have to be solved by experiment [7], although most workers in the field would certainly assume the validity of the superposition principle here. I think that there are more such open questions around than we usually wish to admit, and in this sense I fully agree with Paul Brumer’s comment. 1. E. Schriidinger, Naturwissenschafren 14,664 (1926).

2. 3. 4. 5.

M. Quack, Chem. Phys. Lett. 132, 147 (1986). R. Marquardt and M. Quack, J. Chem. Phys. 90,6320 (1989). R. Marquardt and M. Quack, J. Phys. Chem. 98, 3486 (1994). R. Marquardt and M. Quack, Z Physik D 36,229 (1996). 6. P. Pfeifer, in Energy Storage and Redistribution in Molecules (proceedings of two workshops, Bielefeld, 1980). J. Hinze, Ed., Plenum, New York, 1983, p. 315; H. Primas, Quantum Mechanics and Reductionism. Springer, Berlin, 198 1. 7. M. Quack, in Energy Storage and Redisfribution in Molecules (proceedings of two workshops, Bielefeld, 1980). J. Hinze, Ed.,Plenum, New York, 1983; in Femtosecond Chemistry, J. Manz and L. Woste, Eds., Verlag Chemie, Weinheim, 1995, Chapter 27, p. 781.

M. S. Child: The comments of Brumer and Quack raise questions about the correspondence between classical and quantum mechanics. In this connection one must first recognize that the classical analogue of a wavepacket is not a single particle but an ensemble. Second, the

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spreading of the wavepacket arises largely from the spreading of this ensemble: Quantum effects come in via interference between different components of the ensemble. Hence the oscillation of a macroscopic object such as Brumer’s arm is seen as classical because the interference effects become smaller and smaller. The coherence in this case is classical in the sense that the components of the ensemble move together. Consequently, in considering what properties of a system favor coherence, the mass or size of the species will be less important than the underlying classical coherence. We know that a harmonic oscillator is coherent classically because the oscillators of each ensemble component have the same frequency and quantum mechanically because all energy spacings are equal. Hence coherence is favored by harmonic behavior regardless of the system size.

P. W. Brumer: Several of the speakers (namely Profs. B. A. Hess, M. Quack, and M. S. Child), responding to my question, have suggested that something different happens in the classical limit. However, if we accept the idea that quantum mechanics is a generally applicable theory, then it is applicable to macroscopic systems in the classical limit. As such, dynamics and coherences are, as I said, synonymous in both the quantum and the classical limits. I favor the view of Martin Quack that the closeness of the level spacings in large systems simplifies the preparation of a superposition. Hence experiments on larger molecules seem desirable.

G. R. Fleming: When interpreting experimental signals involving coherent motion, it is necessary to distinguish ground- and excitedstate behavior. The experiment is sensitive to lip, and 6p,. Here, lip, looks like an oscillatory wavepacket. Whether 6p, looks like a pure hole, however, depends strongly on the temperature. [See D. M. Jonas et al., J. Phys. Chem. 99,2594 (1995); D. M. Jonas and G. R.Fleming, in Ultrafist Processes in Chemistry and Biology, Blackwell, Oxford, 1995, p. 225.1

J. ”roe: Prof. Zewail, have you analyzed the “coherence” pattern observed in HgI from dissociating IHgI* with respect to the extent of “vibrational adiabaticity” of the motion downhill from the energy barrier? A. H. Zewail: For vibrational adiabaticity we must complete the study of correlation of reaction product distribution to the nature of the initial excitation (see reply to Prof. Marcus below). You may be interested to know that for a given energy within our pulse we see tra-

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jectories to the diatom at low energies (-200-300 fs) and high energies (-1-2 ps).

R. D.Levine: The coherence that is being discussed by Profs. Troe and Zewail is due to a localized vibrational motion in the AB diatomic product of a photodissociation experiment ABC -+ AB + C. Such experiments have been done both for the isolated ABC molecule and for the molecule in an environment. As the fragments recede, effective coupling of the AB vibrational motion to the other degrees of freedom can rapidly destroy the localized nature of the vibrational excitation. This localization can be due to different reasons. In a femtosecond pumping experiment of the type discussed in the lecture of Prof. Zewail, the localization is due to the fast pumping. In our simulations [M. Ben-Nun and R.D. Levine, Chem. Phys. Lett. 203,450 (1993)l the initial state is located at the transition state on the top of an activation barrier and has a thermal distribution in the symmetric stretch motion. This is the motion that correlates with the vibrational coordinate in the products. As the system evolves downhill from the transition state to the products, there are many more states that can be populated because potential energy is being released. In other words, we start the system at the bottleneck in phase space, and as it evolves, the volume in phase space that is available to it grows all the time. One extreme situation is that the system is rapidly spreading over the available phase space. Another extreme is that the system evolves in a strictly vibrationally adiabatic manner, so that at every point along the reaction coordinate the vibrational distribution is a stationary one. What we observe in the simulation is yet another extreme behavior: For a significant duration (on the fast time scales of interest) the system remains quite localized in phase space but in a nonstationary state. In other words, the vibrational phase is very much nonuniformly distributed. In coordinate space this is reflected by a vibrational motion that is quite localized, just as in the femtosecond pumpprobe experiment. We have run the simulations both for a reaction in a liquid and in the isolated gas phase. On the sub-picosecond time scale the localization was essentially the same. There are three quite distinct perturbations that could have caused the vibrational motion to delocalize. The first is that the potential that governs the vibrational motion is not harmonic and it is not harmonic either in the simulations or, of course, in the real molecule. In our experience, under most circumstances this is the effect that comes in at the earliest times. We have seen this to be the case not only for a chemical reaction but also in simulations of femtosecond pumpprobe experi-

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ments (M.Ben-Nun, R. D. Levine, D. M. Jonas, and G. R. Fleming, Chem. Phys. Left. 245,629 (1995)l.The next is the vibrationally nonadiabatic coupling to the motion along the reaction coordinate. Finally there is the perturbation due to the solvent, if any. For a dipolar vibration in a polar solvent or for a vibration that is strongly coupled to the translational motion, the latter two effects will come in earlier in time. Otherwise, it is the delocalization of the vibrational distribution itself that seems to determine the time scale for vibrational coherence. It is interesting that in cases of realistic complexity the coherence can survive long enough to be observable. A. H. Zewail: The “bottleneck” pointed out by Prof. Levine surely is related to the nature of the potential transverse to the reaction coordinate. Do you agree? R. D. Levine: The bottleneck I mentioned is that separating reactants and products of a bimolecular reaction that has an activation barrier, that is, the saddle-point region. As the system descends from the saddle point toward the products’ region, a much larger volume in phase space becomes available to it. It can sample it uniformly or it can fail to do so and remain more or less localized. One manifestation of a final nonuniform distribution that we are long familiar with is that the distribution of product quantum states is nonstatistical. Here we are talking of a complementary manifestation, namely the coherence. The effect requires an initial localization but it does not however require that this is due to the topography of the potential-energy surface. As you have demonstrated, one can create it by a ultrafast optical excitation. R. A. Marcus: The appropriate criterion of vibrational adiabaticity in the IHgI --+ I + HgI reaction studied by Prof. Zewail involves the distribution of vibrational quantum states of the HgI rather than the observation of coherence in the prepared wavepacket. If an IHgI molecule were prepared in a packet of a few vibrational states, vibrational adiabaticity of the motion would imply that the HgI products would be formed in only a few vibrational quantum states. If the motion were vibrationally highly adiabatic, the final HgI packet would display large changes of vibrational quantum number. The extent of vibrational adiabaticity depends on the curvature of the reaction path and how rapidly this vibrational frequency changes along the reaction path (as discussed in some articles I wrote in 1966 [l]). Regardless of whether the reaction IHgI + I + HgI is or is not vibrationally adiabatic, a coherently vibrating wavepacket would still be observed. Only its distribution of vibrational states would differ in the two cases. 1. R. A. Marcus, J. Chem. Phys. 43, 1598 (1965); 45,4493,4500 (1966).

FEMTOCHEMISTRY I

A. H. &wail: With regard to Prof. Marcus's comment, we have observed the coherence-in-products first in the IHgI system where the wavepacket is launched near the saddle point. The persistence of coherence in products is fundamentally due to (I) the initial coherent preparation (no random trajectories) and (2) the nature of the potential transverse to the reaction coordinate (no dispersion). The issue of vibrational adiabaticity in the course of the reaction, as you pointed out, must await complete final-state analysis for well-defined initial energy. However, we do know that for a given energy of the initial wavepacket a broad distribution of vibrational coherence (in the diatom) is observed.

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FEMTOCHEMISTRY: FROM CLUSTERS TO SOLUTIONS

SIZE-DEPENDENT ULTRAFAST RELAXATION PHENOMENA IN METAL CLUSTERS R. S. BERRY Department of Chemistry and the James Franck Institute The University of Chicago Chicago, Illinois

V. BONAeIC-KOUTECK? and J. GAUS Walter Nernst-Znstitut Humboldt- Universitat zu Berlin Berlin, Germany

Th.LEISNER, .I. MANZ, B. REISCHL-LENZ, H. RUPPE, S. RUTZ, E. SCHREIBER, S. VAJDA, R. de VIVIE-RIEDLE, S. WOLF, and L. WOSTE* Institut f i r Experimentaiphysik and Institut Jirr Physikalische und Theoretische Chemie Freie Universitat Berlin Berlin, Germany

CONTENTS I. Introduction 11. Dimers 111. Triatomics A. NeNePo Experiments with Triatomics B. Pump-Probe Experiments of Bound Excited Trimer States C. Time-Resolved Spectroscopy of Bound-Free Trimer Transitions *Report presented by L Woste Advances in Chemical Physics, Volume 101: Chemical Reactions and Their Control on the Femtosecond Zime Scale, XXth Solvay Conference on Chemistry, Edited by Pierre Gaspard, Irene Burghardt, I. Prigopine, and Stuart A. Rice. ISBN 0-471-18048-3 0 1997 John Wiley & Sons, Inc.

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IV. Larger Clusters A. Bound-Free Transitions into Excited States B. NeNePo Experiments References

I. INTRODUCTION Metal clusters epitomize most of the phenomena characteristic of clusters generally, such as a rich variety of locally stable structures for virtually every size, and a wide range of dissociation and ionization energies, including those of the especially stable “magic number” sizes. Additionally, metal clusters exhibit a variety of characteristics of their own that lead us to new insights into the origins of properties of bulk matter. These include the interactions between electronic and vibrational degrees of freedom and the collection of properties whose changes we associate, in bulk matter, with the metal-insulator transition (a problem still unsolved), such as the density of electronic states and electron mobility at the Fermi level. The application of femtosecond spectroscopic probes to the study of metal clusters has opened a new level of understanding of these properties, particularly because it allowed direct connections between experiment and theory. These connections have occurred with Born-Oppenheimer analyses of nuclear structures evolving adiabatically on a potential surface and with dynamical analyses of propagating wavepackets. The most important reason femtosecond studies have been so important in allowing this link is that, as metal clusters add more atoms, their behavior is more and more dominated by the phenomena that occur on that time scale, notably the making, breaking, and shaking of interatomic bonds, which occur whenever a cluster exchanges energy with the outside world. Because so much of the character of metal clusters lies in their slightly delayed response to excitation or deexcitation, the information obtainable from direct, “one-shot” spectroscopic probes, such as photoionization or absorption spectroscopy, is limited. Even resonant two-photon and multiphoton ionization (MPI) have not led very far; MPI spectra of metal clusters larger than trimers have not yet been reported [l]. Depletion spectra, which remove ground-state species to bound states insensitive to ionizing radiation, are sensitive to bound-free transitions and have provided an alternative for measuring absorption spectra of larger aggregates [23. Even within the context of electronic absorption spectra, whether by depletion or otherwise, the transition from one-electron excitations to collective, plasmon excitations has not yet been established definitively, although plasmon modes have been seen in some clusters [3]. Systematic comparisons of depletion

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spectra with ab initio configuration interaction (CI) calculations have shown consistency with the prior calculations of geometric structures and electronic states [4]. However, the observed spectral features are generally very broad and structureless, with no identifiablerovibrational sequences that one would wish for definitive confirmation [ 11. The reasons for the broad, structureless spectra are the intracluster dynamical processes and structural fluctuations characteristic of such floppy species. It is very likely that the metal-insulator transition, the unusual catalytic properties, the unusual degree of chemical reactivity, and perhaps even some of the ultramagnetic properties of metal clusters are all linked intimately with the dynamic, vibronic processes inherent in these systems. Consequently, the combination of pumpprobe spectroscopy on the femtosecond time scale with theoretical calculations of wavepacket propagation on just this scale offers a tantalizing way to address this class of problems [5]. Here we describe the application of these methods to several kinds of metal clusters with applications to some specific, typical systems: first, to the simplest examples of unperturbed dimers; then, to trimers, in which internal vibrational redistribution (IVR) starts to play a central role; and finally, to larger clusters, where dissociative processes become dominant. 11. DIMEW

A. Time-Resolved Spectroscopy of Electronically Excited States Transient two-photon or multiphoton ionization is a powerful tool for gaining insight into the dynamics of excited states [6]. We employed this pumpprobe method to generate the electronically excited A 'qstate of K2 and then to ionize the excited species after a variable delay. The excitation was accomplished by directing pulses of a regeneratively mode-locked titanium sapphire laser onto the potassium dimers in a seeded supersonic beam. The rotational temperature of the K2 particles was about 7 K. The laser operated in a tuning range of 740-910 nm, which covers the vibronic sequence of the K2 A + X transition. The repetition rate of the laser pulses is 82 MHz at a pulse energy of about 20 nJ. The pulse width of 70 fs [full width at half maximum (FWHM)]corresponded to a bandwidth of 170 cm-', which is nearly Fourier transform limited. A second pulse from the same laser, which ionized the excited potassium dimers, arrived after a variable time delay At. This delay was achieved by passing the beam through a Michelson arrangement, as shown in Fig. 1; this allowed us to probe the temporal evolution of the excited particle. The experimental observation was made by mass selectively recording the ion current of K2' as a function of the time

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Figure 1. Experimental set-up for performing transient two-photon ionization spectroscopy on metal clusters. The particles were produced in a seeded beam expansion, their flux detected with a Langmuir-Taylor detector (LTD). The pump and probe laser pulses excited and ionized the beam particles. The photoions were size selectively recorded in a quadruple mass spectrometer (QMS) and detected with a secondary electron multiplier (SEM). The signals were then recorded as a function of delay between pump and probe pulse.

delay A t between the first (excitation) pulse (pump) and second (ionization) pulse (probe). Consequently, the recorded signal intensities reveal the timedependent populations of Franck-Condon windows for ionizing the potassium dimers, which were previously prepared into the electronically excited A state. The results therefore reflect the temporal evolution of the vibrational wavepacket induced in the A state by the pump pulse [7]. The result of such a measurement is shown in Fig. 2, where the temporal evolution of the excited A 'Z: state was recorded for (Fig. ?a) and 39,41K2(Fig. 2b). Both figures show distinct oscillations of similar frequencies, which correspond to the molecular vibration of the excited K2 molecules. Astonishingly, however, the two species show quite different interference patterns, as they fingerprint the excited-state dynamics of apparently similar isotopomers: 39*39K2and 39941K2.The Fourier transforms of the two signals, presented in Fig. 3, provide complementary insight into the dif39739K32

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Figure 2. Transient two-photon spectra recorded at an excitation wavelength of 834 nm and (b) K2.The ionization step (probe) was achieved by a delayed (pump) for (a) 39*39K2 twephoton transition of the same wavelength. Despite their great similarity both molecules exhibit quite individual oscillatory behavior [7]. 39741

ference. Both curves show spectral resonances around 65 cm-', which is close to the vibrational spacing of the A state of K2.The spectral signature, however, should exhibit the anharmonic progression of all those vibronic states coherently excited within the bandwidth of the pump pulse. Such a progression appears in Fig. 3b, which corresponds to the 39,41K2 isotopomer. The corresponding result for the 39,39K2 isotopomer in Fig. 3a, however, shows a quite different result: two widely spaced peaks at the wings of the progression dominate the spectrum. These two dominant peaks are also the cause for the large-amplitude 10-ps modulation, which is visible in Fig. 2a but not in Fig. 2b. Between those peaks in Fig. 3a the residues of a sequence of

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Figure 3. Frequency spectra obtained by the Fourier analysis of the progressions in Fig. 2: Curve (b) shows the characteristic spectrum of an anharmonic progression while curve (a) is obviously perturbed [7].

four suppressed peaks are still observable. The anharmonic progression, as seen in Fig. 3b, is obviously perturbed. The cause of this perturbation can be seen in the energy-level diagram in Fig. 4. The pump pulse coherently excites the K2 molecules from the electronic ground state (x ‘Xi)into the excited state (A ‘Ef).At an excitation wavelength of about 833.7 nm, vibrational levels around u = 12 are populated. In that region the excited “bright” state of 39,39K2,however, is strongly coupled by intersystem crossing with the “dark” K2 ( b 311,) state; this is a spectral coincidence which significantly perturbs resonance positions, lifetimes, and Franck-Condon factors. For 39,4’ K2, however, the spectral overlap for such a coincidence is much less, so no significant perturbations occur [7]. In the experiments just described, the excited particles were ionized by a probe pulse of the same wavelength as the (exciting) pump pulse. As Fig. 4 shows, this is possible energetically only as a two-photon process, which accidentally goes through the higher (2) ‘TIg state of K2. Wavepacket analyses of the process had to take this into consideration. Alternatively, the influence of this state can be eliminated from the experiments by using probe photons of twice the frequency of the pump photons, that is, by doubling the frequency of the probe pulses before they reach their targets. Interposing a doubling crystal of 0-barium borate (BBO) achieved this. The result

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Figure 4. Schematic of the potential energy curves of the relevant electronic states: The .; state pump pulse prepares a coherent superposition of vibrational states in the electronic A E at the inner turning point. Around u = 13 this state is spin-orbit coupled with the dark b 311,, state, causing perturbations. A two-photon probe process transfers the wavepacket motion into the ionization continuum via the (2) 'IIg state [7].

of the consequent two-color experiment is shown in Fig. 5b, which can be compared with the result of the one-color, two-photon ionization in Fig. 5a. Both results show molecular vibration. However, the corresponding oscillation is significantly less pronounced in the two-color experiment, in which the probe pulse reaches the ionization continuum in a single-dipole transition. The corresponding Fourier transforms of the time-dependent ion intensities both show peaks at the fundamental and first-harmonic frequencies, as can be seen in Figs. 5c, d; only the one-color experiments show a peak at the second harmonic. The noise level in the two-color experiments is considerably higher than in the one-color experiments. The reason is that direct ionization out of the A state in the two-color experiments leads to a nearly timeindependent signal because the transition probabilities into the ion state are rather insensitive to internuclear distance. Only slightly increased probabil-

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-. -. ..

Figure 5. Wavepacket motion of the 39939K2A 'qstate: interrogated by (a) a two-photon probe puke via the (2) 'IIg state and (b) a one-photon probe process into the ionization continuum; (c. 4) corresponding Fourier transforms, indicating a stronger second harmonic for the one-photon probe process 171.

ities at the turning points make the wavepacket motion visible in the one-photon ion signal, whereas the two-photon probe of the one-color experiments is effective at only the outer turning point. The two-color experiments have been done only with the 39,39-isotopomer, for which an additional noise factor is the perturbing triplet state, which contributes to all ionization of this isotopomer. The resonant, two-photon ionization of the one-color experiment acts as a filter, allowing only singlet channels to contribute to ionization. The time-resolved spectra of electronically excited dimers recorded at 840 nm for the 39,39-isotopomer mirrors a vibrational wavepacket prepared in an almost unperturbed region of the A state. The experimental data fit well with the simulations by time-dependent wavepackets, as Fig. 6 shows. Although the experimental time-dependent signal is noisier than the simulated signal from theory, the experimental data are dominated by the peaks separated by intervals of -500 fs, the interval between adjacent peaks in the theoretical curve of ion intensity versus time. The theoretical curve was the result of a wavepacket propagation on ab initio potential and dipole transition surfaces [7].The time-dependent Schrodinger equation is solved in the matrix representation, which guarantees the proper description of the possible multiphoton processes. The formalism used to propagate the wavepacket in space is the fast Fourier transform (FFT)method. The free-electron continuum can

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Figure 6. Comparison of theory and experiment of the unperturbed wavepacket dynamics of the transitions shown in Fig. 4 [7].

be simulated by discretizing the corresponding energy range. The pump laser pulse prepares a coherent superposition of vibrational states predominantly molecule. The probe pulse ionizes the molecule in the A state of the 39,39K2 after the variable interval given by the delay shown on the abscissa.

B. Time-Resolved Spectroscopy of Ground Electronic States In the experiments described thus far, the pump laser simply populates the intermediate excited state. The consequence is that the experiment becomes a means to study that excited state. Often we are more concerned with learning about the ground state than about excited states. For this purpose, it is useful to prepare a vibrational wavepacket of that ground state. One useful means to do this is to excite the species of interest to an allowed excited state and then to “down-pump” from that excited state back to the ground state, with a pulse that generates a packet rather than a stationary state. The simplest way to do this currently seems to be to raise the power level of the pulsed pump laser [26]. This process is shown schematically in Fig. 7. Such an experiment for the 39739K2 molecule yielded the results of the upper curve of Fig. 8. A simulation of this experiment provided the lower curve. For about the first 3 ps, the interval between the oscillation periods in both curves is approximately 380 fs, corresponding to the vibrational frequency of the ground state of this molecule. For times later than 3 ps, the

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Figure 8. Comparison of theory and experiment of the ground-state wavepacket for K2 according to the dynamics of the transitions shown in Fig. 7 [7].

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two curves contain two components with intervals in the 380-500-fs range, which get out of phase in about 1.5 ps. This is due to the interference of ground and excited wavepacket signals and an effect of the intense laser field. The more intense laser field of 9 GW/cm2 enhances the resonant, impulsive Raman scattering (RISRS) process between the X and A states. Consequently a wavepacket in the ground electronic state sets off with significant motion. A second ionization pathway at the inner turning point of the ground state reveals this packet. The contribution of the excited-state dynamics to the ion signal complicates the analysis of the ground-state dynamics [8]. How to reduce these effects is one of the open questions about pumpprobe studies of ground states that helped stimulate the next method in this review.

C. NeNePo Studies of Diatomics The need for a way to produce and study neutral clusters of single, known sizes was actually the primary motivation for developing the method we now describe. Since it offers a simple way to study neutral diatomics in their ground states, we begin its description here. The method begins with a beam of negative ions (“Ne”), which passes through a cooling and compressing zone and then a mass-selecting quadrupole spectrometer into an accumulator trap. When an adequate density of negative ions has collected in the trap, a laser pulse photodetaches the electrons of most of the negative ions to make neutrals (“Ne” again). After a chosen delay, a second laser pulse photoionizes the neutrals to make positive ions (“Po”). In the experiments reported thus far, the ionization has been accomplished by two-photon ionization. The positive ions pass immediately out of the trap into a quadrupole mass analyzer and then to a detector. Thus the method has been given the name NeNePo, to indicate the steps of the process, negative to neutral to positive. The sequence of steps appears schematically in Fig. 9, and the core of the apparatus is shown in Fig. 10. Variation of the time interval between detachment and ionization affects the intensity of the positive ion signal through whatever time dependence appears in the ionization cross section as a consequence of the way the neutral is formed by the Franck-Condon detachment process. In the case of a diatomic, if the equilibrium internuclear distances of neutral and negative ion are different and the negative ion is cold, then the neutral appears either in one of several possible excited vibrational states or, if the detachment radiation is broadband, in a nonstationary wavepacket. In the latter case, the populations of the Franck-Condon windows for photoionization of the neutral are, in general, simple oscillatory functions in time, with the frequency of the oscillations equal to those of the center of the correspondingwavepackets. Such an experiment is probably the simplest possible, conceptually, to which the NeNePo process can be applied. It may be considered as a time-dependent extension

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anion

reaction coordinate (arb. units) Figure 9. Principle of NeNePo spectroscopy.The probe pulse detaches the photoelectron from the negative ion, introducing a vibrational wavepacket into the ground state of the neutral particle. Its propagation is interrogated by the probe pulse, which ionizes it to a positive ion.

of the time-independent spectroscopy of neutral systems by photoelectron detachment spectroscopy [27]. This is just what has been done with Ag,. A sputtering source followed by a “phase-space compressor” chamber provided a beam of cooled negative cluster ions of many sizes. From these, the dimers were selected, accumulated in a quadrupole trap, and photodetached with a femtosecond, titanium-sapphire laser. After photodetachment by a 60-fs pulse, the neutral dimers oscillate, causing corresponding oscillations in the ionization cross section, in turn generating the oscillations that dominate the intensity pattern in Fig. 11. This is a simple phenomenon, yielding in a simple way the

113

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Figure 10. Experimental scheme for recording NeNePo spectra. The negative ions are mass analyzed in a quadrupole mass filter and introduced into an ion trap. There they are first neutralized (pump), then reionized to a positive ion (probe), so they can escape the trap and enter into another quadruple mass filter for being detected.

Ion Signal of Ag; ’

*

1

.

,

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1

.

.

.

.

1

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.

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.

.

.

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.

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.

.

.

.

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.

.

.

.

1

.

.

delay time Figure 11. NeNePo spectrum of neutral silver dimers. The progression exhibits the Ag2 vibration in its electronic ground state.

114

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information about the ground state of a diatomic that seemed so difficult to obtain from the intense-field down-pumping experiment of the previous section. It serves a purpose here of demonstrating the consistency and viability of the NeNePo method.

III. TRIATOMICS A. NeNePo Experiments with Triatomics [9] The first NeNePo experiments dealt with silver clusters, Ag,, Ag,, Ag,, and Ag,, particularly with the first of these. The photodetachment and photoionization were done with a single titanium-sapphire laser producing pulses of approximately 60 fs duration. Doubled in frequency, these could be tuned over a wavelength span from above 420 to below 390 nm. As with the dimer, photodetachment was a one-photon process and photoionization a two-photon process. (The clusters of odd numbers of atoms could be studied this way; the even-numbered clusters require at least three photons in the available energy range for photoionization). The interval between pulses could be varied from zero (simultaneous pulses) to 100 ps; the two pulses were made to differ in intensity by about a factor of 2, and either could be the leading pulse. The ion signals from Ag3- are shown in Fig. 12 for wavelengths of 390, 400, 415, and 420 nm, from top down, (a) through (4, in the figure. The zeros are shifted to avoid overlap of the signals. In all cases, the minimum signal appears when the two pulses overlap. The signals reach maxima when the two pulses are about 500400 fs apart, with the higher maximum appearing when the more intense pulse is the later, ionizing pulse. In contrast to the diatomic case, these signals show no sign of simple oscillations (apart from those generated in the zero-time region for 400 and 390 nm, which arise from interference of the two pulses). The signals drop from their maxima as the time interval goes beyond about 600 fs and remain at a plateau out to 100 ps. The interpretation of these signals, at present, is based on the structures of the three species Ag3+, Ag,, and Ag,-, as Fig. 13 shows. The first is linear in its ground state, the second is an isosceles triangle, and the third is an equilateral triangle [lo]. The photodetachment process is presumably a Franck-Condon phenomenon, so that the neutral is generated in or near the linear configuration. We assume that the vertical electron affinity is not an extremely sensitive function of the bending angle, so that the detachment process at all the wavelengths used in these experiments leaves the neutral in its ground electronic state. This point, discussed below, needs experimental substantiation, for example by photoelectron spectroscopy of cooled negative

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115

(d) A = 420 nm (2.95 eV)

I.'..I'...,'...,....I....I....IC

-1.5 -1.0 -0.5

0.0 0.5

delay time At [ps]

1.0

1.5

Figure 12. NeNePo spectra of the silver trimer taken with wave ngths of ( a ) A = 390 nm, (b) h = 400 nm, (c) A = 415 nm, and (6)A = 420 nm. Each curve has its own axis of zero signal. The time-independent background increases steadily with decreasing wavelength. The fine structure around At = 0 is due to the interference of pump and probe pulses [9].

ions. Ab initio calculations predict for the pump transition of Ag,- a vertical detachment energy of 2.45 eV, whereas the electron affinity is only 2.1 eV [lo]. The neutral is thus generated in a highly excited state of its bending mode, a state that has virtually zero overlap with the low-lying vibrational states of the positive ion. Hence the positive ion signal at the detector is very small, when the laser pulses are separated by only a small interval, but grows as the molecule bends. The signal reaches a maximum, when the bending angle reaches a minimum, and then drops as the angle opens again. We presume that the excitation is so high that mode mixing is rapid, so rapid that by the time the bending angle is once more about equal to that of the equilibrium geometry of the neutral the molecule has considerable energy in pseudoro-

116

R. S. BERRY et al.

A‘ t

\

e

\

Figure 13. Geometry changes of Ag3(*) during the NeNePo process.

tation through its equivalent isosceles structures, classically speaking doing a sort of pirouette around the trough of the “Mexican hat” potential of the neutral, with its peak in its center at the equilateral configuration. Recent calculations by Bennemann et al. indicated the possibility of chaotic behavior in the vibrationally excited ground state [ll]. The signal at zero delay between pulses, which is more prominent in the short-wavelength experiments than in those nearer threshold, can only be attributed to at least four-photon absorption. This indicates that the cross section for multiphoton ionization of the silver trimer increases as the photon energy increases. This is an inference that remains to be verified directly. It is highly likely that the neutral is in its ground electronic state when it appears. The vertical detachment energy for the linear negative ion is approximately 2.4 eV, slightly below the energy of the longest wave photons used in these experiments. There are no optically allowed excited states of the neutral within the two-photon range of energy above the bound state of the negative ion, and hence there are no states that might afford photodetachment via a saturated two-photon process. However, the probability of photodetachment is linear in the intensity of the detaching radiation, so the detachment process must be either a one-photon process, and hence to the ground state of the neutral, or a saturated, resonant, two-photon process, which the calculations of the electronic structure seem to rule out. On this basis, we suppose, as

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117

previously stated, that we are observing the neutral in its ground electronic state.

B. Pump-Probe Experiments of Bound Excited W e r States

Transient two-photon ionization experiments on trimer systems were, of course, motivated by a need for time-resolved verification of the pseudorotation motion, which can be considered as a superposition of the asymmetric stretch (ex)and the bending vibration (Q,)[12]. The triatomic molecule with its three modes is quite different from an isolated oscillating dimer, which vibrates in its single mode until eventually it radiates or predissociates. The interplay of vibrational modes in a trimer system can be considered as the prototype of IVR. The experimental approach to the time-resolved observation of an electronically excited trimer state is achieved by means of transient two-photon ionization [13].A pump pulse ( h v , ) coherently excites a wavepacket in the excited electronic state, where it propagates in analogy to the corresponding molecular vibration. The temporal evolution of this wavepacket is probed by a specifically delayed ionization pulse (hv2). A typical result, which was X) transition with transform-limited obtained for the electronic Na3 (B pulses of 70 fs duration (FWHM)is shown in Fig. 14a.The progression indicates a pronounced molecular vibration. The corresponding Fourier transform of this progression is plotted in Fig. 14b.The result indicates only one vibrational mode of 320 fs duration, which corresponds to the breathing mode of Na3 in the B state. There is, however, no indication for the asymmetric stretch, bending mode, namely pseudorotation [ 131. Similar observations were previously also made by Gerber et al. [14].To comprehend the phenomenon, a wavepacket calculation was carried out [13,151 using potential-energy surfaces calculated by means of a full-configuration interaction [ 101.The CI calculations were carried out for the ground and several excited states of Na3 for different values of three vibrations: a bend along the Qx coordinate, antisymmetric stretch along Q,,, and symmetric stretching vibration Qs. The quality of the atomic orbital (AO) basis set has been tested previously on properties of excited states of Na2 and NQ [16,171. The energy surfaces of the ground X(l 2A‘) states and excited B(2A’) states from Fig. 15 clearly show the importance of three-dimensional treatment. The minimum of the ground state (2B2) with the obtuse isosceles triangle corresponds to Qs = 3.626 A. For this value of Qs coordinate the minimum of the excited B state is also associated with an obtuse isosceles geometry, as can be depicted from the upper part of Fig. 15.On the contrary, for large values of Qs (3.889 A) the geometry corresponding to the minimum of the B state has changed into the acute triangle (lower part of Fig. 15).

.-

R. S. BERRY et al.

118

I

-2

1.o

a,

-1

0

1

At 1 ps

2

3

4

T m 320 fs (breathing mode)

0.8

-0

.3s0.6

E"rn

0.4

0.2 0.0

a/crn-' Figure 14. (a)Transient two-photon ionization spectrum of Na3 recorded with transformlimited 60-fs pulses. (b)The corresponding Fourier transform shows only the breathing mode of the relating B state [13].

119

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1?A'

QY

62A' 1 .o

1 .o

A

0.

0.

-1.0

-1.0

-1

0

-1

1

0

Qs = 3.626'4

1.2A' 10

10

0

0

-1 0

-1 0 -1

0

1

n -1

0

1

Qs = 3.88981

Figure 15. Ab initio full CI energy surfaces for the ground X(l ' A ' ) and excited state B(6*A') of Na3 drawn as equidistant contours V ( Q x ,Q y ) = const for Qs values comes onding to the minimum of the X state (3.626 A) and to the minimum of the B state (3.889 ).

1

The wavepacket calculation for the femtosecond pumpprobe experiment presented in Fig. 16 (bottom) is the result of the first consistent ab initio treatment for three coupled potential-energy surfaces in the complete threedimensional vibrational space of the Na3 molecule. In order to simulate the experimental femtosecond ion signal, the experimental pulse parameters were used duration Atfwhm= 120 fs, intensity I = 520 MW/cm2, and central

R. S. BERRY et al.

120

wavenumber ii = O.O73Eh/h = 16021 cm-’ . For comparison both the experimental (top) and theoretical results are included in Fig. 16. Both curves are in good agreement and clearly exhibit the dominant oscillation with a period of approximately 310-320 fs. The theoretical ab initio result thus confirms the assumption that the oscillation period corresponds to the breathing mode. As demonstrated in earlier studies [18], the 120-fs pulse excitation drives the excited wavepacket along the Qscoordinate, and this oscillation period is reflected in the ion signal during the first 4-5 ps. An “approximate” method, described in detail in Ref. (15), was applied to simulate a complementary pumpprobe experiment performed with picosecond laser pulses. In this method the interaction with the probe laser pulse is approximated. A complete three-dimensional ab initio simulation, as carried out for the femtosecond experiment, is hardly possible for the picosecond experiment with the computers available today. The free laser pulse parameters were taken from the picosecond experiment: duration Atfwhm = 1.5 ps, intensity I = 300 MW/cm*, and central wavenumber 7 = 0.073Eh/h = 16021 cm-I. The dynamics induced by such a laser pulse are illustrated by

0.0

0.5

1.0

1.5

At / ps

2.0

2.5

Figure 16. Comparison of experimental (upper) and theoretical (lower) curve for a 120-fs excitation of the pseudorotating Na3 B state. The excitation wavelength was 624.2 nm [13].

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t = 1.5 DS Q, = 7.k &

t = 0.0 DS

Q = 7.i5a,

t = 3.0 DS

Q, = 7.h 4

B

X

-

2

0

Q,

I all

2

-

2

0

Qx

I a0

2

0

-2 Qx

2

1 a0

Figure 17. Snapshots of wavepacket propagation during the picosecond pumpprobe excitation for Na3 for selected delay times between pump and probe pulse [MI.

means of representative snapshots in Fig. 17. The graphics show different time evolutions of the wavepackets propagating in the potential-energy surfaces of the well-localized ground state X of Na3 and on the rather delocalized B state. The initiation of the wavepacket (pump) occurs from a Franck-Condon window close to the minimum potential (first snapshot), corresponding to the configuration of an obtuse triangle. This region also serves as Franck-Condon window for the subsequent excitation from the B state into the ion state. Within a time of 1.5 ps the wavepacket has propagated along the potential-energy surface of the excited B state to the other side of the trough, which corresponds to the configuration of the acute triangle. At 3.0 ps the main part of the wavepacket has returned to its initial position, corresponding again to the configuration of the obtuse triangle. The propagation corresponds to molecular pseudorotation; the resulting theoretical pumpprobe signal is plotted in Fig. 18 (bottom) and compared to the related time-resolved experimental signal (Fig. 18, top). The overall agreement is satisfactory. Interestingly, the temporal progressions are very different from the femtosecond result in Figs. 14u and 16. They show the distinct oscillation of 3 ps periodicity, indicating the temporal behavior of molecular pseudorotation [19]. The period of about 3 ps can be identified with the vibration from the obtuse to the acute geometry [15] (including u = 1, 2 with an approximate energy spacing of 13 cm-' of the angular dependent pseudorotational mode) presented in Fig. 17.

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experimental

I

0.0

0.5

1.0

1.5

2.0

2.5

:

0

At ( P S I

Figure 18. Comparison of experimental (upper) and theoretical (lower) curve for a 1.5-ps excitation of the pseudorotating Na3 B state. The excitation wavelength was 624.2 om.

The only evident difference between experimental and theoretical progression is a pronounced attenuation,which only occurs for the experimentalcurve on a time scale of 6 ps. On the basis of our calculations [15] we conclude that the decay observed in the experimental signal originates from the low pulse intensities applied and basically reflects the overlap of pump-probe laser pulses. The decay does not occur in the calculated ion signal because the probe laser pulse is not taken into account explicitly in the present simulations. In addition to ab initio quantum simulations of the experimental femtosecond and picosecond pump-probe spectra, traditional continuous-wave (CW) spectra could also be simulated using the time-dependent approach to absorption spectroscopy [131. The results show that femtosecond/picosecond versus CW spectroscopy is complementary; in the present case, the radial and angular pseudorotation and the symmetric stretch are observed, and simulated with preferential sensitivity using CW picosecond, and femtosecond spectroscopy, respectively.

C. Time-Resolved Spectroscopy of Bound-Free '&her

Transitions

Stationary spectroscopy on the C and D states of Na3 already indicated the onset of photoinduced fragmentation. Fragmentation becomes more important as the cluster size increases. As a result, nondissociative electronic excitation processes have not yet been observed for free metal clusters larger than trimers [20]. An alternative to conventional spectroscopy of such boundfree transitions was provided by depletion spectroscopy [2]. A deep insight into the dynamics of such photoinduced cluster fragmentation, however, is obtained with ultrafast observation schemes. The principle of such an

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reaction coordinate Figure 19. Excitation scheme for probing bound-free transitions. Initially the photoion of the mother molecule appears. After dissociation, however, the relating fragment ion is observed.

experiment is indicated in Fig. 19: The particles are electronicallyexcited with pulses of a femto- or picosecond laser (pump) into a predissociated state. There they oscillate a few times and then dissociate or they dissociate directly. The temporal behavior of the sequence is monitored with the probe pulse, which interrogates the system by ionizing the excited particles after a variable time delay A f .The result of such an experiment performed on K3 is shown in Fig. 20 for still recording spectra [21]. At At = 0, the signal is a maximum. This represents the cross-correlation between pump and probe pulse. For a delay time A t S 10 ps a pronounced oscillation occurs, which reflects the wavepacket oscillation in the excited state. A magnified segment of this oscillationis shown in the left insert in Fig. 20, whereas in the right insert in Fig. 20 the Fourier transform of the result is shown. It indicates three vibrational modes that correspond to the K3 normal vibrations with vs = 109em-', vx = 82 cm-', and vY = 66 cm-' . Superimposed on these oscillations there is an ultrafast unimolecular decay with a lifetimeof 450 fs, which indicatesthat the observed state predissociates. So far this fact prevented the observation of this excited state by means of resonant multiphoton ionization. The stationary excitation spectrum of the predissociated C state of Na3 is presented in the insert of Fig. 21. It shows a pronounced vibrational

R. S. BERRY et al.

124

I.

I

I

-10

-20

I

0

At / ps

1

I

10

20

I

Figure 20. Transient two-photon ionization of K3.For A t I10 ps a pronounced oscillation appears, indicating the three normal vibrations (see inserts). Superimposed to the vibrations of 450 fs a fast unimoleculax decay time occurs [21].

90

..-o

N Q)

0

60-

401

2.58 2.60 2.62 2.64 2.66

EfeV

30

2.6 ~

2.58

2.60

2.62

E/eV

Figure 21. Fragmentation rates of Na3 C state vibrational bands versus excitation energy 1221. Insert: Highly resolved TPI (lower) and depletion spectrum (upper) of the Na3 C state [81.

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sequence. In order to determine the lifetime of each individual vibrational mode, an experiment with sufficient spectral resolution had to be performed. This did not allow the use of femtosecond pulses. A proper spectral and temporal resolution required a two-color, transform-limited picosecond pumpprobe experiment. For this reason we synchronizedtwo independently operating titanium-sapphire lasers. The experimental result of these lifetime measurements is illustrated in Fig. 22. Each vibronic level labeled u shows a distinct decay time, but there are no wavepacket oscillations. At u = 0 a rather "normal" lifetime of 1.2 ns appears. Apparently this vibrational level does not yet overlap with the repulsive electronic state. At u = 1 the lifetime of the vibronic level is already considerably shortened to 91 ps. At higher vibrational quantum numbers the observed lifetimes become even smaller, reaching 12 ps for u = 4 [22]. The correlation between vibrational mode and dissociation time is plotted in Fig. 21. The result clearly affirms that lifetimes in a predissociated excited state strongly depend on the vibrational excitation

I

1120

0

26

3

12

4

R

J I

0

.

I

100

.

I

.

I

200 300 At 1 pS

.

I

400

Figure 22. Transient TPI spectra of different vibrational bands of the Nas C state cornpared with the fit function f ( r ) = NoP-'/'. The listed lifetimes r of the different vibrational bands were obtained by a least-squares fit procedure [22].

R. S. BERRY et al.

126

of the electronic state. This is particularly important for larger metal clusters, in which vibronic coupling becomes a dominant dynamic process.

IV. LARGER CLUSTERS A. Bound-Free Transitions into Excited States The application of pumpprobe spectroscopy to electronic transitions of larger aggregates reveals the rapidly growing number of different dissociation channels. The result of two-color pumpprobe femtosecond experiments performed on potassium clusters K,,with 3 5 n 5 9 is shown in Fig. 23 [23]. For At > 0 the energy Epump was 1.47 eV, whereas Eprok, the energy of the probe pulse, was 2.94 eV. Time delays with At < 0 inverted this sequence to Epump = 2.94 eV and Eprok= 1.47 eV. In order to describe the features, which appear in Fig. 24, several processes must be taken into account: We call fragmentation type I the direct fragmentation channel of the chosen cluster dissociated by the pump pulse with a decay time 71. We call type I1 those fragmentation processes that occur to particles that have populated the observation channel temporarily with fragments of larger clus-

-10 -5

0

5

10

-5

0

5

$0

Figure 23. Pump and probe spectra of a two-color experiment probing bound-free transitions in Kn(3 5 n I9). For Ar < 0, Epump= 1.47 eV and E p r h = 2.94 eV, for Ar > 0, Epump = 2.94 eV and Eproh= 1.47 eV [23].

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I

K , '

probe

K"

'

127

i probe

KP

Figure 24. Fragmentation model for computing contributions of different fragmentation channels to the observed signal intensities.

ters, before they fragment again. The fragmentation time for the larger clusters is called 7,,,and the refragmentation time for the fragments is 7 2 . These two decay channels are shown graphically in Fig. 24. With the numbers of clusters involved in the type I and type I1 dissociation process being called nl (r) and n2(r) the temporal evolution of the total photodissociation can be described by

(

n(r) = nl(t)+n*(t)= N1 exp --+ T m - 7 2 [exp(-$) Mi72

where M I and N Iare the initial populations of the excited states prepared by the pump pulse, which is assumed to have a &function shape. The measured ion signal nion(r)can be estimated by a convolution of n(t) with the cross correlation of pump and probe pulse, which is the overall system response to the laser pulses s(t):

The function nion(r)was fitted to the experimental data by means of a leastsquares routine. The corresponding curves are shown in Fig. 23 as solid lines. Obviously there is a good agreement between measured and fitted curves. An implemented mathematical filter allows us to extract from the measured transition spectrum the relevant type I contribution, containing the essential

R. S. BERRY et at.

128

information about the excited state. Figure 25 shows a result of this procedure applied to & and the different contributions to the ion signal. From the three decay times, which each measurement provides, the main point of interest how is the type I quantity 71. It characterizes the dynamics of the relevant electronic state of the cluster size under investigation. The data, therefore, allow us to determine directly the photodissociation probabilities 1/71 of the observed clusters excited at the energies of the photon irradiation. The corresponding results reflect the stability of the clusters, as graphically presented in Fig. 26. For all measured cluster sizes the fragmentation probabilities at E = 2.00 eV are smaller than those for the other photon energies (Figs. 26u, b). For E = 2.00 eV and E = 2.94 eV the curves of the dependence of the photodissociation probability on the cluster size have similar shapes. In Fig. 26b both curves show a particular instability for Ks, which

.. -fitted

measured data curve

type I contribution

-10

-5

!i

0

5

10

At 1 PS Figure 25. Deconvolution of various types of fragmentation from the ion signal in the case of Q [23].

SIZE-DEPENDENT ULTRAFAST RELAXATION PHENOMENA I

I

a)

I

I

I

I

129

l

- - C I A 7 eV

I

3

b)

I

3

I

I

4 5 -2.94eV --c 2.00 eV

1

4

I

5

I

6

I

6

I

7

I

7

I

8

I

0

I

9

I

9

cluster size n

Figure 26. Fragmentation rates 1,kl of the observed mother molecules (type I) recorded

at excitation energies of ( a ) 1.47 eV and (b) 2.94 and 2.00 eV [23],

contradicts expectations from jellium model considerations [3]. At E = 1.47 eV, however, the expected higher stability is indeed found for K8.A complete interpretation of the experimental results cannot yet be given, since the influence of ionic fragmentation on the described measurements is still an open question: With the experimental set-up used here dissociation processes occurring after ionization cannot be distinguished. Excellent measurements with respect to this have recently been performed by Haberland et al. [24].

B. NeNePo Experiments The results of NeNePo with clusters of five, seven, and nine silver atoms are graphically presented, together with the relevant geometries that resulted from ab initio calculations [25], in Fig. 27. The temporal progressions have

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NEgativ

*c @!&

NEutral

%./

0

3

‘ta

*a

-#

6

+g@ -1

POsitiv

.+

+

1

delay time {ps)

Figure 27. NeNePo signals and relating geometry changes for Ag3, Ag5, and Ag9.

shown maxima only at early times, that is, when the pulses of radiation essentially overlap. This indicates that the only major ionization process contributing to the signals for these species is direct, simultaneous, multiphoton detachment and ionization. They seem to show no significant evolutionary processes that enhance photoionization. While this is not surprising for Ag,, it is a bit surprising for Ag5, because of structural considerations. The negative ion of this cluster has, according to the calculations 1251, a planar, trapezoidal geometry; its positive ion is trigonal bipyramid. The neutral has two, nearly degenerate stable structures: a planar trapezoid like the negative ion and a trigonal bipyramid like the positive ion. Hence one suspects that the positive ion signal should grow from a small value when the pulses are almost simultaneous to a much larger value as the neutral equilibrates into its two, nearly degenerate geometries. As yet, the barrier between these two forms is probably about 0.2 eV, high enough to make equilibration of the two forms very slow in states with little or no vibrational excitation. Small degrees of heating of the negative ions have failed to generate a “late” maximum in the positive ion signal.

Acknowledgments The financial basis for the work presented here was provided by the Freie UniversitA Berlin and the Deutsche Forschungsgemeinschaft, who supported us in the frame of the Sonderforschungsbereich 337: Energie- und Ladungstransfer in molekularen Aggregaten. One of us (R.S. B) would like to express his thanks to the Alexander von Humboldt-Stiftung for its support. Financial support for the theoretical work from the Verband der Chemischen lndustrie is also gratefully acknowledged.

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References 1. J. Blanc, M. Broyer, J. Chevaleyre, Ph. Dugourd, H. Kuhling, P. Labastie, M. Ulbricht, J. P. Wolf, and L. Woste, 2. Phys. D 19,7 (1991). 2. M. Broyer, G. Delacr6taz. P. Labastie, J. P. Wolf, and L. Woste, Phys. Rev. Lett. 57, 1851 (1986). 3. C. Brechignac, Ph. Cahuzac, F. Carlier, M. de Frutos, and J. Legnier, Chem. Phys. Lett. 189, 28 (1992). 4. V. Bonacic-Koutecky, M. Kappes, P. Fantucci, and J. Koutecky, Chem. Phys. Lett. 170, 26 (1990).

5. J. Manz, and L. Woste, Eds., Femtosecond Chemistry, Vols. 1 and 2, VCH Verlagsgesellschaft, Weinheim, 1995, 6. T. Baumert, M. Grosser, R. Thalweiser, and G. Gerber, Phys. Rev. Lett. 67, 3753 (1991). 7. S. Rutz. E. Schreiber, and L.Woste, in Ultrafast Processes in Spectroscopy 95,O. Svelto, S . de Silvestri, and G. Denard6, Us., Plenum, New York, 127 (1996); R. de Vivie-Riedle, B. Reischl, S. Rutz, and E, Schreiber, J. Phys. Chem. 99, 16829 (1995); S. Rutz, R. de Vivie-Riedle and E. Schreiber, Phys. Rev. A 54, 306 (1996). 8. E. Schreiber, S. Rutz, and R. de Vivie-Riedle, in: W. Waidelich, H. Hagel, H. Opower, H.Tiziani, R. Wallenstein, and W. Zinth (eds.), Laser in Forschung und TechniklLaser in Research and Engineering (Springer Verlag, Berlin, 1996) p. 203. 9. S. Wolf, G. Sommerer, S. Rutz. E. Schreiber, T. Leisner, L. Woste. and R. S . Berry, Phys. Rev. Lett. 74,4177 (1995). 10. V. Bonacic-Koutecky, P. Fantucci, J. Pittner, and J. Koutecky, J. Chem. Phys. 100, 490 (1994). 11. H. Jeschke, M. E. Garcia, and K. H. Bennemann, Phys. Rev., A. December (1996). 12. G. Delacr6taz. E. Grant, R. Whetten, L. Woste, and J. Zwanzinger, Phys. Rev. Lett. 56, 2598 (1986). 13. R. de Vivie-Riedle, J. Gaus, V. Bonacic-Koutecky. H. Kuhling, J. Manz, B. Reischl, S. Rutz, E. Schreiber, and L. Woste, in Femtochemistry, M. Chergui, Ed., World Scientific, Singapore, 225 (1996). 14. T. Baumert, R. Thalweiser, and G. Gerber, Chem. Phys. Lett. 209, 29 (1993). 15. B. Reischl, R. de Vivie-Riedle, S. Rutz, and E. Schreiber, J. Chem. Phys. 104,8857 (1996). 16. V. Bonacic-Koutecky, P. Fantucci, and J. Koutecky, Chem. Phys. Lett. 166, 32 (1990). 17. V. Bonacic-Koutecky, P. Fantucci, and J. Koutecky, Chem. Rev. 91, 1035 (1991). 18. B. Reischl, Chem. Phys. Lett. 239, 173 (1995). 19. K. Kobe, H. Kuhling, S. Rutz, E. Schreiber, J. P. Wolf, L. Woste, M. Broyer, and Ph. Dugourd, Chem. Phys. Lett. 213, 554 (1993). 20. G. Delacr&taz,P. Fayet, J. P. Wolf, and L. Woste, in Elemental and Molecular Clusters, Springer Series in Material Science, G. Benedek, T. P. Martin, and G. Paccione, Eds., Springer-Verlag, Berlin, Heidelberg, New York, 1988. 21. E. Schreiber and S. Rutz, in: M. Chergui (ed.), Femtochemistry-Ultra~ast Chemical and Physical Processes in Molecular Systems (World Scientific, Singapore, 1996) p. 2 17. 22. E. Schreiber, K. Kobe, A. Ruff, S. Rutz, G. Sommerer, and L. Woste, Chem. Phys. Lett. 242, 106 (1995). 23. A. Ruff, S. Rutz, E. Schreiber, and L. Woste, Zeitschr$t f u r Physik D, 37, 175 (1996).

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24. Th. Reirners, W. Orlik, Ch. Ellert, M. Schmidt, and H. Haberland, Chem. Phys. Lett. 215, 357 (1993). 25. V. Bonacic-Koutecky, L.Cespiva, P. Fantucci, and J. Koutecky, J. Chern. Phyys. 98,7981 ( 1993). 26. B. Hartke, R. Kosloff and S. Ruhman, Chern. Ph. Letter 158,238 (1989). 27. D. Neumark, Acc. Chern. Res. 26, 33 (1993).

DISCUSSION ON THE REPORT BY L. WOSTE Chairman: c! S.Letokhov

G. Gerber: In our experiment we have observed that at moderate laser intensities the symmetrical stretch motion in the excited Na3(B) state is always present, in addition to the pseudorotational components, from t = 0 until maximum delay between pump and probe laser. We have no indication that the symmetrical stretch motion of the B state should decay and evolve into the pseudorotational motion. L. Woste: Our femtosecond excitation on the Na3(B) state exhibits the symmetric stretch motion, which phases out after about 2 ps. If, however, we use higher resolved picosecond excitation, this symmetric stretch vibration is not observed any longer, but a 1.5-ps modulation is visible, which we attribute to the pseudorotation. The modulation time agrees well with the calculations presented by Prof. Manz. G. Gerber: In our time-resolved experiments on the Na3(B) state we observe the symmetric stretch even for long delay times. From nanosecond laser and CW laser spectroscopy it is well known that the B state does not decay on femtosecond or picosecond time scales. So I do not see how the decay in the picosecond experiment by Prof. Woste can be understood and how the evolution of the B state symmetric stretch into the pseudorotation and the radial motion can occur. R. de Vivie-Riedle, J. Manz, B. Reischl-Lenz, and L. Woste:' The relation of our femtosecond, picosecond, and CW results for Na3(B) is as follows: The experimental pump and probe spectra of the Na3(B) state, as observed by the Woste group, yield the following results for low or moderate laser intensities: (a) For pulse duration tp = 120 fs, the symmetric stretch vs (7, = 320 fs) is dominant, at least during initial times t d C 3 ps [I]. (b) For longer pulse duration t,, = 1.5 ps, the angular ((p) pseudoro*Comment presented by J. Manz.

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tation vo is dominant. The apparent vibrational period is T~ = 3 ps, with an apparent decay time of the signal &cay = 6 ps [2]. (c) In addition the CW three-photon ionization spectrum, which may be interpreted as a limiting case of pump and probe spectra with 00, exhibits two dominant proinfinitely long pulse durations t p gressions for the angular ((p) and radial (r) pseudorotation yo and vr, respectively [3]. This empirical evidence (axe) of the observation of different vibrational modes, depending on the pulse durations tp of the pump and probe laser pulses, is well reproduced by our ab initio threedimensional (3d) simulations of all three cases (a), (b), (c), except for the missing decay in the theoretical spectra in case (b); see Refs. 4, 5 , 6, respectively (see also Ref. 7). Accordingly, our interpretation of the t,-selective sensitivities of the pump and probe spectra is as follows: (i) Each pump pulse prepares a wavepacket $s(Qs,r,(p,t)representing Na3 in the B state, and this is a characteristic superposition r, (p) with energies E,,,,,,, and of vibrational eigenstates, +,,,,,,,
-

with coefficients c,,,,,~ depending on the laser parameters of the pump pulse, in particular, on the pulse duration t,,. is (ii) The decomposition of $g(Q,, r, (p, t ) in terms of the valid for all times t > t p . In particular, the weight of the partial waves

does not change for t > t,,; for example, an initially (t = t p ) dominant population of the symmetric stretch (e.g., P,,oo >> PO,,,,) will not change into dominant excitations of the pseudorotations, P p s<< ~ PO,^,,, and vice versa. As a consequence, there is no intramolecular vibrational energy redistribution after the pump pulses. (iii) The time-dependent phases of the partial waves in Eq. (1) do, however, imply that the densities pg in vibrational coordinate space r,!Q},

tes,

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do change with time. For example, an initially dominant vibrational motion along the symmetric stretch Qs may be reduced at the expense of increasing vibrations along the pseudorotational coordinates r and cp, and the wavepacket may even spread apparently over the available coordinate space {Qs,r, PIE}, which is defined by

in terms of the potential-energy surface Ve(Q,,r , p ) of Na@) and the total energy E. These effects may be called intramolecular vibrational density redistribution (IVR), and this is observed, indeed, in our movie simulations of the density ps(Qs,r , p , t ) ; see the snapshots in Fig. 1.

1

Qx

~

,

t=0.31 ps

Qx

t=4.78ps

X

1

I

Qx

Qx

Figure 1. Intramolecular vibrational density redistribution IVR of Nag@). The threedimensional (3d) ab initio dynamics of the representative wavepacket $ B ( Q ~ ,r,p,t) is illustrated by equidensity contours ps(Qs, r,p) = I $ B ( Q ~ , r,p, r ) I 2 = const in vibrational coordinate space Qs, Qx = r cos p. Qy = r sin p for the symmetric stretch and radial (r) plus angular (p) pseudorotations, viewed along the Qy axis. The IVR is demonstrated exemplarily by four sequential snapshots for the case where the initial wavepacket ( 1 = 0) results from a Na3(8); similar results are obtained for the 120-fs Franck-Condon (FC) transition Na3(X) laser pulse excitation (A = 621 nm, I = 520 MW/cm*) [ I , 4, 51. The subsequent dynamics in vibrational coordinate space displays apparent vibrations along the symmetric stretch coordinate Qs (7, = 320 fs), followed by intramolecular vibrational density redistribution to the other, i.e., pseudorotational vibrational degrees of freedom. This type of IVR does not imply intramolecular vibrational energy redistribution between different vibrational states of Na3 ( B ) , i.e., the wavepacket shown has the same expansion, Eq. (I), for all times. The snapshots are taken from a movie prepared by T. Klamroth and M. Miertschink.

-

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(iv) The intensities of the pump and probe spectra depend not only , but also on the pump pulse and the resulting wavepacket $ B ( Q ~r,p) on the probe pulse. The latter defines the Franck-Condon window WFC(Q,,r, cp) for ionization, that is, for optimal transitions between r,p) Na3(B) and Na3+ (X)+ e. This Franck-Condon window WFC(Q$, depends on the laser parameters of the probe pulse, in particular on its frequency. For example, the specific wavelength A = 618 nm of the probe pulse yields the Franck-Condon window shown in Fig. 2. The pump and probe spectrum S(t) is then proportional to the overlap of the density ps(QS,r, p, t ) and the Franck-Condon window W ~ c ( Q ~ , r , p which ), depend on the pump and probe laser pulses, respectively:

W )= const x

dQs r dr & P S ( Q ~ ,r , ~~ ,) W F C ( r, Q~,

(5)

(v) Our three-dimensional ab initio simulations [4-6] agree well with the experimental ones [l-31, and this may be considered, first of all, as an indication of the consistency of the approaches involved. Furthermore, we also have to accept the empirical, that is, both experimental and numerical result that different laser pulse durations tp of the pump pulse yield different sensitivities for different vibrational modes of the excited Na3(B). The interpretation of this result in terms of either excitations of specific vibrations in ps(Qs,r,cp,t)by the pump pulse or/and by different sensitivities of the Franck-Condon window WFC(Q~,r,p) determined by the probe pulse is less obvious in view of the overlap integral ( 5 ) of p~ and WFC.For example, different parameters of the probe pulse may yield different Franck-Condon windows and, therefore, different pump and probe spectra, and this may well be the reason for some of the differences in the pump and probe spectra of the Woste group [l, 21 and the Gerber group [8, 91. (vi) For further discussions of this issue, see present chapter (supported by DFG project SF6 337). I. B. Reischl, R. de Vivie-Riedle, S. Rutz, and E. Schreiber, J. Chem. Phys. 104,8857 (1996). 2. E. Schreiber, H. Kuhling, K. Kobe, S. Rutz, and L. Woste, Eer. Eunsenges. Phys. Chem. 96, 1301 (1992); K. Kobe, H. Kuhling, S . Rutz, E. Schreiber, J.-P. Wolf, L. Woste, M. Broyer, and Ph. Dugourd, Chem. Phys. Lett. 213, 554 (1993). 3. G . Delacktaz, E. R. Grant, R. L. Whetten, L. Woste, and J. Zwanziger, Phys. Rev. Lerr. 56, 2598 (1986). 4. B. Reischl, Chem. Phys. Lett. 239, 173 (1995); R. de Vivie-Riedle, J. Gaus, V. BonaCiC-Kouteck9, J. Manz, B. Reischl, S. Rutz, E. Schreiber, and L. WOste, in Femtosecond Chemistry and Physics of Ultrafast Processes, M. Chergui, Ed., World Scientific, Singapore, 1996, p. 319.

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136

0, = 7.223 a. t = 653fs 2 0

0

\ o U”

-2 2 0

0

\ o U”

-2

2 0

0

\ o U”

-2 0

-2 Qx

/

2 a0

Figure 2. FranckCondon windows WFC(Q~,r,(p) for the Na3(X) -c Naj(E) and for the Na3(B) Na3+ (X)+ e transitions, X = 621 nm. The FC windows are evaluated as rather small areas of the lobes of vibrational wavefunctions that are transferred from one electronic state to the other. The vertical arrows indicate these regions in statu nascendi; subsequently, the nascent lobes of the wavepackets move coherently to other domains of the potential-energy surfaces, yielding, e.g., the situation at t = 653 fs, which is illustrated in the figure. The snapshots of three-dimensional (3d) ab initio densities are superimposed on equicontours of the ab initio potential-energy surfaces of Na3(X), Na3(E), and Na3+ (X), adapted from Ref. 5 and projected in the pseudorotational coordinate space Qx = r cos (p, Qy= r sin (p. A complementary projection along the Qs coordinate is presented in Ref. 4. The present FC windows are for X = 621 nm, and the time delay td = 630 fs used in the simulation corresponds to a maximum in the pump-probe spectrum; cf. Refs. 1 and 4. -t

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5. B. Reischl, Ph.D.Thesis, Berlin, 1995. 6. R. de Vivie-Riedle, J. Gaus, V. BonaEiL-Kouteckj, J. Manz, B. Reischl-Lenz, and P.Saalfrank, Chem. Phys., submitted. 7. J. Gaus. K. Kobe, V. BonaEif-Kouteck9, H. Kuhling, J. Manz, B. Reischl, S. Rutz, E. Schreiber, and L. Woste, J. Phys. Chem. 97, 12509 (1993). 8. T. Baumert, R. Thalweiser, V. Weiss, and G . Gerber, 2 Phys. D 26, 131 (1993); T. Baumert, R. Thalweiser, and G. Gerber, Chem. Phys. Lett. 209,29 (1993). 9. Y. J. Yang, L. E. Fried, and S. Mukamel, J. Phys. Chem. 93, 8149 (1989).

S. A. Rice: Prof. Woste, your data indicate that rotational dephasing of the coherent wavepacket is unimportant for the time regime you have studied. Unless your beam has an unusually low rotational temperature, it is to be expected that a heavy molecule such as K2 will have many rotational states excited. Because the different isotopic species you have studied, one homonuclear and the other heteronuclear, would then have different numbers of rotational states in the initial wavepacket, one should expect to observe different rotational dephasing times for the two species. What is the effective rotational temperature of your beam? Is it likely that only a very few rotational states are present in the initial wavepacket? L. Woste: The rotational temperature of dimers in our molecular cluster beam was measured to be about 7 K. So only very few rotational states up to about J = 9 are significantly populated. The great difference between the spectra of 39939K2 and 39*41K2is due to a perturbation induced by a superimposed 311 state, which is of importance in the 39v39K2 molecule, but not so much in the other isotopomer. Of course, we can also visualize one as a homonuclear and the other one as a heteronuclear molecule; but we have not observed effects of this origin in rotational spectra. For this our observation time was too short. A. H. Zewail: I have two questions to Prof. Woste, one related to Prof. Rice's question and the second related to the dynamics in Na3 :

1. Did you use polarized pulses to separate vibrational and rotational dynamics? 2. Did you consider the coupling between the stretch motion and the pseudorotations, hence the relevance of IVR?

L. Woste 1. We have tried the experiment at 90" crossed polarization between pump and probe. There was no more modulation visible. The transients

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only appeared as clearly as shown when “pump” and “probe” laser had the same polarization. 2. Yes, we have considered the possibility of energy transfer from vibration to pseudorotation, and the results give a strong indication for this effect, in agreement with theory.

R. A. Marcus: Prof. Woste, can you go to high enough energies to reach the conical intersection? If so, what effect does that have on your experiments? L. Woste: We did perform excitation experiments into the conical intersection of the Na3(B’) state by means of stationary CW spectroscopy. The results exhibit a very rich structure of highly congested lines, which have not been interpreted so far. For this reason we have not yet tried a femtosecond experiment; but in principle this is possible. D. J. Tannor: Some years ago we looked at the photodissociation of the ozone molecule [J. Am. Chem. Soc. 111, 2772 (1989)l. Ozone ( 0 3 ) can be viewed in much the same way as Na3, in that it has an initial wavepacket with three lobes, one located in each of three symmetrically equivalent wells (see Fig. 1). In the excited state, at the same locations, the potential has saddle points. As a result, each of the component wavepackets bifurcates, and there is a “change of alliances”: The left portion of one wavepacket joins up with the right portion of another wavepacket, and they tunnel into the same product arrangement channel. Ozone may be a bad example because the splitting between initial states of different symmetry cannot be resolved. However, in Na3, if

@b Figure 1.

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one could excite to a dissociative excited state, perhaps this effect could be observed. L. Woste: In relation to Prof. Tannor’s comment, let me remark that the pseudorotating B state of Na3 is bound and there are no indications for fragmentation processes. In Li3, however, there are predissociated pseudorotating states. We have observed and identified them. Li3 has the advantage to be a homonuclear molecule, which however can have different isotopes. This induces an asymmetry, which can be exploited for pump and control. Later we intend to apply this to heteronuclear trimers like NazK, where we hope to split off selectively either an Na or a K atom as a function of the initial conditions. T. Kobayashi: I have a comment and a question to Prof. Woste:

1. I think it is possible to excite either a clockwise or an anticlockwise pseudorotation state by using a circularly polarized beam and also to probe with a circularly polarized beam to detect the circular dichroism induced by the pump beam. 2. Why does the time dependence of odd- and even-numbered clusters Nan show different features?

L. Woste 1. It is a very interesting idea to give a preferred direction to the system’s pseudorotation. We have not tried this yet. We will however examine your suggestion with interest. 2. The phenomenon of odd-even alternations in the fragmentation process of excited Nan..3-10 clusters can be explained as follows: There are mainly two different dissociation channels. The Nan clusters with even n dissociate into an odd-numbered cluster and one Na atom,

* +Nan* - I + Na

Na,

while odd-numbered clusters also dissociate into an odd-numbered cluster but one dimer,

*

*

Na, -. Nan- 2 + Na2 Therefore, in both cases, odd-numbered daughter clusters are generated preferentially. This phenomenon has previously also been observed in nanosecond fragmentation experiments performed by C. Brechignac et al.

FEMTOSECOND CHEMICAL DYNAMICS IN CONDENSED PHASES G. R. FLEMING*and T.JOOt

Department of Chemistry and James Franck Institute University of Chicago Chicago, Illinois

M.CHO'

Department of Chemistry Massachusetts Institute of Technology Cambridge, Massachusetts

CONTENTS I. Introduction

11. Vibrational Dynamics A. Multilevel Redfield Theory

B. Experimental Studies

111. System-Bath Interactions

A. Line Shape Function B. Echo Spectroscopies IV. Discussion References

I. INTRODUCTION Attempts to study or manipulate chemical processes in the condensed phases are inevitably complicated by the spectral broadening induced by the sur'Report presented by G. R. Fleming

*Resent address: Department of Chemistry, Pohang University of Science and Technology, Pohang, Kyungbuk, 790-784, Korea *Present address: Department of Chemistry, Korea University, Seoul 136-701, Korea Advances in Chemical Physics, Volume 101; Chemical Reactions and Their Control on the Femtosecond Erne Scale, XXth Solvay Conference on Chemistry, Edited by Pierre Gaspard, Irene Burghardt, I. Prigogine, and Stuart A. Rice ISBN 0-471-18048-3 0 1997 John Wiley & Sons, Inc.

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G . R. FLEMING, T.JOO AND M. CHO

rounding medium. The intermolecular interactions provide a screen behind which much of the dynamics is masked and in addition produce new processes such as energy and momentum relaxation, dephasing, and coherence transfer that are specific to ensembles. Ultrafast nonlinear spectroscopy [ 11 gives the spectroscopist the capability to both see behind the mask of spectral broadening and, via echo spectroscopy, reveal the mechanisms and time scales involved in the broadening. The comparison of the low-resolution gas phase and room temperature hexane solution spectra of iodine in Fig. 1 makes the obfuscation of linear spectra clear. Manipulation (see S. A. Rice, this volume, Perspectives on the Control of Quantum Many-Body Dynamics: Application to Chemical Reactions) and sensitive probing [ 11 of molecular dynamics both rely on the exploitation of superposition states (coherences). If a dephasing time is defined operationally as the inverse width of the solution spectrum, this corresponds to only a few femtoseconds, and it would seem that any attempt to use superposition states for the control or exploration of the electronic states (we discuss vibrational states later) of such a system would be futile. We will see that this may not necessarily be the case, but to follow the argument, we must first discuss spectral broadening in general. The spectral broadening may have both static and dynamic contributions. Static contributions to the line broadening can generally be removed by some form of nonlinear spectroscopy in either the time domain (e.g., echo spectroscopies) or the frequency domain (e.g., hole-burning spectroscopy). The time scale of the dynamical processes will, in general, limit the time scale on which the molecular dynamics can be followed in a deterministic, as opposed to statistical, manner. Consider, for example, two coupled electronic states for which the coupling may be via the radiation field or perhaps via some exchange term leading to electron transfer. For sake of definiteness we will consider the optically coupled case, but essentially identical considerations apply to optically initiated electron transfer, isomenzation, and so on. In the simplest description both states are coupled to a solvent “bath’ whose own relaxation is very much faster than that of the system. In this case irreversible dephasing occurs on a time scale given by T2 and the “homogeneous” linewidth [full width at half maximum (FWHM)] is given by Av = (aT2)-‘. The loss of information (memory of the Bohr frequency) commences immediately after the system is excited. Note also that in this model the bath influences the system but not vice versa. Such a time scale separation between system and bath may often be appropriate when dealing with intramolecular vibrational motions of molecules but is likely never appropriate for electronic transitions in solution near room temperature. In the past 10 years much effort has been devoted to dynamical aspects of the solvation process in polar liquids utilizing experiments [2-4], theory [5, 61, and computer simulations of molecular dynamics [7-lo]. The

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Wavelength, nm

lo00 800

f; E el

I

I

I

I

I1

1

450

400

-

400200 0

I

I00

650

600 550 500 Wavelength. nm

Figure 1. Top: A portion of the medium-resolution spectrum of the visible B c X 12 absorption spectrum with assignments for the overlapping progressions for Y" = 0, 1,2. The upper state 'Y values are indicated at the estimated band-head positions on the short-wavelength side of each transition; the band maxima are at the fop of the figure (From Ref. 53.) Bottom: The absorption spectrum of I2 in n-hexane.

experiments are based on time resolving the fluorescence Stokes shift, that is, the shift in fluorescence frequency to longer wavelength as polar solvent molecules reorganize to accommodate the newly created excited-state charge distribution. The mere observation of such a shift makes the viewpoint of

G. R. FLEMING, T.JOO AND M. CHO

144

an uninfluenceable solvent untenable, and it is immediately clear that both system and bath influence each other and must be dealt with consistently. Both experiments and theory show that solvation may proceed exceedfor ingly rapidly. Figure 2 shows the fluorescence Stokes shift function s(?) a coumarin dye in water obtained by experiment and simulation [4]. Here, S ( t ) is a normalized function describing the progress toward equilibrium:

where v ( t ) is a characteristic frequency of the fluorescence spectrum, for example, its first moment, at time t. The strongly bimodal character of the

v u

h

0.5

h Y Y

to

0.0

0.0

0.5 Time (ps)

1.o

Figure 2. Experimental and simulated fluorescence Stokes shift function S(r) for coumarin 343 in water. The curve marked A q is a classical molecular dynamics simulation result using a charge distribution difference, calculated by semiempirical quantum chemical methods, between ground and excited states. Also shown is a simulation for a neutral atomic solute with the Lennard-Jones parameters of the water oxygen atom (9). (From Ref. 4.)

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relaxation is typical for many liquids [2]. Aside from the good correspondence of the simulation and experiment, it is noteworthy how rapid the initial phase of the relaxation is. More than 50% of the relaxation is complete in less than 55 fs. As far as the experimental fluorescence data or the extracted response shown in Fig. 2 reveal, we are observing an irreversible relaxation toward equilibrium. For the ensemble of fluorescing dye molecules this is certainly true. However, initially simulation [7-101 followed by theory [5, 61 and finally by experiment [ 2 4 shows that most and probably all of the rapid (-50 fs in water and -100 fs in alcohols) relaxation is Gaussian in nature. it results from free, inertial, small-amplitude motions of the solvent molecules within the potential wells that they initially occupy. This relaxation is then reversible in the sense that each individual molecule is evolving freely, without dissipation, during this period. It is simply the ensemble-averaging effect of the broad distribution in frequencies that leads to the apparently irreversible relaxation. Following the sharp break in the response, the solvent begins to restructure itself (motion now occurs between wells), and energy flows irreversibly into the solvent and the behavior is now dissipative. These remarks can be made much more quantitative and formal, for example, via instantaneous normal-mode models [ll-131; however, for the present we simply summarize the points of this discussion: Broad and featureless absorption spectra do not necessarily imply that all information of the optical transition frequency is dissipated irreversibly on the time scale of 2 r Au, where Av is the spectral width. This lack of dissipation during the initial solvation epoch is of significance for chemical processes in which coherent contributions can be important. Examples are electron or energy transfer processes mediated by “bridge” states. Experiments that monitor the duration of the ultrafast (reversible) epoch of the system-bath interactions will be described in detail in this review, but first we turn from the electronic motions to the nuclear dynamics of solute molecules. it is perhaps not surprising that an ultrashort pulse applied to a discrete spectrum such as the upper I2 spectrum in Fig. 1 will produce a signal with modulation frequencies characteristic of the energy differences in the spectrum. It is a little less obvious that the same effect will occur in cases where the spectrum looks like the lower I2 spectrum of Fig. 1, that is, diffuse and devoid of structure. The key in both cases is that femtosecond laser pulses are shorter than the periods of lower frequency (hundreds of reciprocol centimeter) molecular vibrations. Thus excitation produces wavepackets in the excited state (and “holes” in the ground state). Figure 3 sketches the interactions leading to vibrational superpositions in the excited and ground states and shows that following the two interactions with the light field the excited and ground states are effectively decoupled from each other, and coherent vibrational motion can be observed even in the presence of very large distri-

G . R. FLEMING, T.JOO AND M.CHO

146

l‘r>==lg>

IY>-lg>+le> =<eIe>. < e l f > ,

Electronlc

Coherence

Eiectronlc Population

<$Ih>

Vibrational Coherences

Figure 3. Field-matter interactions for a pair of electronic states. The zero and first excited vibrational levels are shown for each state ( A ) . The fields are resonant with the electronic transitions. A horizontal bar represents an eigenstate. and a solid (dashed) vertical arrow represents a single field-matter interaction on a ket (bra) state. (See Refs. 1 and 54 for more details.) A single field-matter interaction creates an electronic superposition (coherence) state ( B ) that decays by electronic dephasing. Two interactions with positive and negative frequencies create electronic populations ( C ) or vibrational coherences either in the excited ( D )or in the ground ( E ) electronic states. In the latter cases ( D and E ) the evolution of coherence is decoupled from electronic dephasing. and the coherences decay by the vibrational dephasing process.

butions of ground-excited electronic energy gaps. The diagram also shows that both excited- and ground-state vibrational superpositions will inevitably be generated by vibrationally abrupt excitation. The pumpprobe signals generated by wavepackets are surprisingly complicated even in a diatomic molecule. Temperature plays an important role in making the ground-state pumpprobe signal tend toward pure classical behavior. In particular, Jonas et al. [14] showed that bpgg (the pump-induced change in the ground state density operator) tends toward a negative operator (“pure ground-state hole”) with increasing temperature, meaning that many room temperature pumpprobe signals can be broken down into ground and excited contributions, each signal being characterized with a definite sign with respect to zero modulation of the probe light. Wavepacket motion is now routinely observed in systems ranging from the very simple to the very complex. In the latter category, we note that coherent vibrational motion on functionally significant time scales has been observed in the photosynthetic reaction center [15], bacteriorhodopsin [16], rhodopsin [17], and light-harvesting antenna of purple bacteria (LHl) [18-201. Particularly striking are the results of Zadoyan et al. [21] on the

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caging of iodine in solid krypton matrices where very rapid energy relaxation occurs with significant retention of vibrational phase. Indeed, at low temperature, iodine molecules re-formed as a result of rebounding from the krypton cage partly retain the phase of the initially excited molecule for times up to 4 ps. These studies clearly challenge the conventional descriptions of vibrational and electronic relaxation based on the optical Bloch equations. The observation of vibrational quantum beats has generally been taken to imply that vibrational energy relaxation is slow. Is this generally so, or do relaxation pathways neglected at the optical Bloch level, that is, coherence transfer terms, play an important role? A related question raised by the experiments is: Can vibrational coherence be created in a reactive event (e.g., curve crossing) even if the initial state has little or no coherence, or does such an observation imply a coherently vibrating reactant? Further, can a nonadiabatic process lead to coherence in the product or does this require that the process proceed on a single electronic surface? If strong electronic coupling is required for coherence to be created in the product, are electronic recurrences to be expected in addition to the vibrational beats? Finally, if a vibrational wavepacket is observed in the reactant but not the product, does this mean that the coherent motion is not important in the reaction? These questions have been addressed in a series of papers by Jean [22-251 based on applications of multilevel Redfield theory [26]. Jean shows that coherence transfer effects are critical for the interpretation of many femtosecond transient absorption experiments.Thus, again the standard picture used to describe condensed-phase dynamics requires revision. At this point, it is appropriate to attempt to avoid a semantic confusion that often arises from the use of such phrases as “vibrationally coherent reaction.” The observation of coherence [i.e., a temporally modulated signal with a frequency characteristic of the nuclear motion(s)] is a feature of the way the experiment is performed, that is, with short pulse excitation and detection. However, the fact that coherent behavior can be observed reveals information on curve crossing, dephasing, and relaxation dynamics that is relevant no matter how the reaction is initiated. Having described how standard pictures of condensed-phase dynamics are under revision, largely as a result of the fruitful interplay of ultrafast spectroscopy with theory and simulation, we now turn to more detailed descriptions of intramolecular nuclear motion (Section 11). Following a brief discussion of Redfield theory, two experimental systems-iodine in hexane solutions and the bacterial light-harvesting antenna LHl-are used as illustrations. In Section I11 we discuss how nuclear motions modulate electronic energy gaps. After establishing the basic concepts, the results from experimental studies are described. We conclude with a brief survey of the implications of the results and concepts presented in these two sections.

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G. R. FLEMING, T. JOO AND M.CHO

11. VIBRATIONAL DYNAMICS

A. Multilevel Redfield Theory

The observations of vibrational coherence in optically initiated reactions described above clearly show that the standard assumption of condensedphase rate theories-that there is a clear time scale separation between vibrational dephasing and the nonadiabatic transition-is clearly violated in these cases. The observation of vibrational beats has generally been taken to imply that vibrational energy relaxation is slow. This viewpoint is based on the optical Bloch equations applied to two-level systems. In this model, the total dephasing rate is given by

or

depending on whether the lower level is infinitely long-lived or not. Here T I , and Ttb are the population lifetime of levels a and b and T : is the pure dephasing time. Thus T2 can never exceed 2Tl or

A more general approach is required to interpret the current experiments, Jean and co-workers have developed multilevel Redfield theory into a versatile tool for describing ultrafast spectroscopic experiments [22-251. In this approach, terms neglected at the Bloch level play an important role: for example, coherence transfer terms that transform a coherence between levels i and j into a coherence between levels j and k (Jk- il = 2 ) or between levels k and 1 (Il - j l = 2, )k - j l = 2) and couplings between populations and coherences. Coherence transfer processes can often compete effectively with vibrational relaxation and dephasing processes, as shown in Fig. 4 for a single harmonic well, initially prepared in a superposition of levels 6 and 7. The lower panel shows the population of levels 6 and 7 as a function of time, whereas the upper panels display off-diagonal density matrix ele-

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d o ....

I

I

-I

-0.1

I

0

dl

2

4

6 8 Time (pr)

10

12

Figure 4. Time dependence of selected density matrix elements for a harmonic oscillator obtained using the full Redfield tensor. The oscillator is described by o = 100 cm-I, Tl(1 -0) = 2.0 ps, and T;(A.n = 1) = w, where n denotes vibrational levels. The system is initially prepared in a superposition of levels 6 and 7. (a) p67; (b) p34; (c) p o l ; ( d ) dashed line, pw and the solid line, p77. (From Ref. 24.)

ments. Clearly, significant coherence is preserved on time scales much longer than the population relaxation of the initially formed levels. The presence of pure dephasing does not destroy the coherence transfer since the concomitant two-phonon relaxation process enhances coherence transfer. Anharmonicity will reduce the degree of coherence that can be transferred down a well, but as calculations with double-well systems show [24, 251, it is not likely to remove the effect entirely. Thus the generally made assumption that observation of oscillatory contributions to pumpprobe or fluorescence data implies

G. R. FLEMING,T. JOO AND M.CHO

150

slow vibrational energy relaxation may not be generally supportable. Conversely, it may often be possible to perform experiments relying on coherent nuclear motion even in the presence of significant vibrational energy relaxation. The Redfield approach has been applied to various two-state problems that can be considered as simplified models for bond breaking, isomerization [25], electron transfer 122, 231, or energy transfer 127). Figure 5 shows

5

5:

e"

0.8 0.7 0.6 0.5 0.4 03 0.2 0.1 0 15

I

I

a

I

1.25

# A

5:

1

0.75 05 0.25

0

9

0.2

lo 93 -0.4

1

2

W)

3

4

Figure 5. Quantum dynamics for an asymmetric double well under coherent or thermal pre arations. Vertical energy separation between the two wells (Ar) is 300 cm', o = 100 cmp, J = 50 cml , TI (1 0) = 2.8 ps. and T;(An = 1) = 6.0 ps. (-) Coherent preparation; (---) thermal preparation. (a)Nonequilibrium population dynamics of the initially prepared electronic state. (b) Product coordinate trajectory, (Qz(t)). ( c ) Vibrational contribution to the product coordinate, (Q2(t)- A2). The crossing of the diabatic surfaces occurs at Q = -1.10. (From Ref. 24.)

-

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results for an asymmetric double-well system in which the initial state is prepared in one of two ways: (1) coherently vibrating or (2) with a thermal density matrix at t = 0. These two cases correspond to short-pulse excitation when the excited state is or is not significantly displaced from the ground state. For example, in the former case, if the bond length is significantly longer in the excited state than in the ground state, the initial wavepacket will be created near the left turning point of the excited-state potential and will immediately commence to move. By contrast, for very small displacement(s) the initially excited-state wavepacket produced by a short pulse will closely resemble the equilibrium distribution of both states and will not move. The calculations in Fig. 5 use a 20-fs pulse and demonstrate that coherence (i.e., oscillatory behavior in the ensemble-averaged value of the coordinate in the product state, (Qz(r))) is created for both initial conditions, that is, even when the initial state does not contain coherence. The product coherence persists as relaxation proceeds within the product to levels well below the crossing region proceeds, again demonstrating the role of coherence transfer. Such effects are very clearly demonstrated in the calculation of Jean [25] on a model for barrierless isomerization, intended to capture some of the essential features of isomerization in rhodopsins [17]. A 25-fs pulse populates the initial state and Fig. 6 shows the decay of population of that

0.0

0.5

1.0

1.5 Time (ps)

2.0

2.5

Figure 6. Population decay of the initial state in a barrierless double-well system calculated using multilevel Redfield theory [25]. The vibrational frequency is 60 ern'. (-) Coherent preparation of the initial state; (---) thermal preparation of the initial state. (see Ref. 25 for more details). (From Ref. 25.)

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G. R. FLEMING, T. JOO AND M.CHO

state again for initially coherent or thermal (i.e., zero displacement from the ground state) systems. In both cases the population decays essentially irreversibly with an l/e time of -190 fs for coherent preparation and -240 fs in the thermally prepared case. In both cases coherent motion is generated in the product. The coherence persists for -2 ps whereas energy relaxation occurs in -600 fs, which accounts for the lack of electronic recurrences in Fig. 6 and the irreversible nature of the reaction. The curve-crossing dynamics is shown pictorially in Fig. 7, which shows the coordinate probability distributions for reactant and product at various times. The product has lost a substantial fraction of energy to the bath after only a quarter of a vibrational period (0.14 ps) while the distribution of coordinates remains reasonably localized for about two periods. To summarize, Jean shows that coherence can be created in a product as a result of nonadiabatic curve crossing even when none exists in the reactant [24, 251. In addition, vibrational coherence can be preserved in the product state to a significant extent during energy relaxation within that state. In barrierless processes (e.g., an isomerization reaction) irreversible population transfer from one well to another occurs, and coherent motion can be observed in the product regardless of whether the initially excited state was prepared vibrationally coherent or not [24]. It seems likely that these ideas are crucial in interpreting the ultrafast spectroscopy of rhodopsins [17], where coherent motion in the product is directly observed. Of course there may be many systems in which relaxation and dephasing are much faster in the product than the reactant. In these cases lack of observation of product coherence does not rule out formation of the product in an essentially ballistic manner.

B. Experimental Studies Two experimental systems will be briefly described to illustrate some of the ideas presented in the previous section. The examples span the range of system complexity from a diatomic molecule (12) [28] to a supramolecular pigment-protein complex (the core light-harvesting antenna of photosynthetic bacteria, LH1 [18, 191). In solution when iodine is excited to the bound excited state, dissociation and recombination processes occur. The dissociation is the result of solvent-induced curve crossing to the dissociative 2 state, the recombination a result of momentum reversals arising from “collisions” with the surrounding solvent molecules. Eigenstates of the state will decay in a continuous manner, whereas wavepackets-if the curve-crossing probability is less than unity-dezay in a stepwise manner, giving rise to successive pulses of product. The B and ii curves cross near the center of the h state, whereas the B state wavepacket is initially created near the left turning point; thus there

FEMTOSECOND CHEMICAL DYNAMICS IN CONDENSED PHASES

-6-4-2

0

Q

2

4

6-6-4-2

0

Q

2

4

153

6

Figure 7. Reaction coordinate probability distributions at the indicated times (in units of the vibrational frequency, 60 cm-I). (--) Reactant diabatic state; (---) product diabatic state. The distribution at t = 0 is divided by a factor of 3. The positions of the distributions on the vertical axis are determined by the average vibrational energy for that state. (From Ref. 25.)

will be two exits per state vibrational period, one on the outward passage and a second as the contracting molecule passes through the crossing again. In this latter case the separated atoms are initially launched with the wrong sign for their momentum and have to reverse their direction to break the bond [29].

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G . R. FLEMING, T.5 0 0 AND M.CHO

The use of wavepacket spectroscopy to follow the solvent-induced dissociation of iodine in solution has been described in detail by Scherer, Jonas, and co-workers [18, 28, 301. Recently the role of the solvent in inducing the curve crossing has been examined by simulation [29]. Remarkably, the experiments show that the wavepacket survives the solvent-induced curve crossing an$ appears intact (i.e., the atoms are separating ballistically) up to at least 4 A separation [28]. The simulations imply that destruction of the wavepacket by the solvent “cage” (polarizable Ar atoms in this case) occurs between 1-1 separation of 5-6 A [29]. Figure 8 shows the polarization-detected pump-probe signal for iodine in hexane solution for a probe wavelength of 475 nm. The signal contains both ground and excited (2i to a charge transfer state 11291) contributions. The excited-state signal appears after a delay of 220 fs, which corresponds to a bond length of 3.75 A. Singular-value decomposition yields a frequency of 102 cm-’ for the excited-state signal, very similar to the 6 state vibrational frequency. This is the expected value for a single exit per 8 state period rather than the two discussed above. The simulations of Ben-Nun et al. [29] for IZ in polarizable rare-gas solvents shed light on this issue. Figure 9 shows histograms of the population on the 2i state at four times chosen such that the mean intermolecular separation is 4 A 1291. In accord with the experimental results, population takes -200 fs to appear in this region (lower panel). As the dissociative population moves out along the 2i state it starts to spread (second panel of Fig. 9). This is the result of the two effects: the repulsive form of the potential and the almost immediate onset of the interaction with the surrounding solvent. The appearance of the second burst of the dissociative population (third panel of Fig. 9) occurs later than-one might expect for two reasons. First the motion in the outer half of the B state is slowed by interaction with the solvent, compared to that in the inner half of the potential. Second the dissociation generates population moving toward shorter bond lengths. In fact, a close examination of even versus odd dissociative distributions shows that the latter are somewhat less localized than the former. This we attribute to the longer time spent on the 5 state before arriving in the probing window for even curve-crossing events. Finally, a thifd product burst appears, following motion to the inner turning point of the B state and motion back out to the crossing point (upper panel in_Fig. 9). The combination of the more rapid motion on the inner half of the B state and the fact that the 2i state is now launched with momentum in the stretching direction makes the third exit population overlap significantly with the second exit population. We believe that this is why the experimental fits appear consistent with a single exit per period [28, 291. Thus simulation and experiment provide a consistent picture of the dissociation: The wavepacket, whose motion is influenced by both the intramolec-

155

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Transient Dichroisrn of Iodine in Hexane 580 nm pump, 475 nm probe

-.--I

4.6

4.3

0.0

0.3

0.6.

0.9

1.2

Oidrroism Crosscorrelation

1.5

1.8

21

2.4

Delay Time Ips)

T i m @r)

Figure 8. Top: Transient dichroism signal of I2 in n-hexane. Bottom: Linear prediction singular-value decomposition (LP-SVD) for the pump and probe wavelengths of 580 and 475 nm, respectively. Note the overall delay of the negative signal, which is modulated by the B state frequency. (From Ref. 28.)

156

G. R. FLEMING,T.JOO AND M.CHO

t

s4o&cc

second exit

1 0

,

2

4

6

8

I-I Distance / A

1

0

Figure 9. Histograms of the ii state population vs. atom-atom separation at four time points. Different shadings are used to represent the different exits from B at the i state. (In

binning the results each trajectory is weighted by its population.) Shown are the first three exits (black, gray, and blank), which correspond to the first three steps as described in the text. The time periods at which the first, third, and fourth panels are drawn is such that the first, secon!, and third dissociative populations have reached an average intramolecular separation of -4 A. (This distance is about the upper limit of the experimental probing window [29].) The initial localized n_atureof the dissociative population is a direct result of the vibrational localization on the parent B state. Evidence for the spreading of the population of the 7r state is apparent already 60 fs after the first appearance of population at a distance of 4 8, (second panel). This is due both to the repulsive anharmonic shape of the ii potential and to the cage of the surrounding liquid atoms. As time evolves, the spreading in the population becomes more pronounced, and in the third and fourth panels one observes trajectories that reversed the sign of their momentum. The time instants (190, 480, and 540 fs) at which the dissociative populations ypear at a distance of 4 A are not equ_allyspaced. This is due to both the asymmetry of the B state dynamics with respect to the B-ii curve crossing point and the negative momentum with which even-numbered ii states are being spawned. (From Ref. 29.)

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157

ular potential and solvent, splits on each passage through the region of the B/ii state crossing point. The newly formed dissociative packet continues unhindered for several angstroms before caging occurs. The origin of the solvent-induced coupling of the B and ii states (which are of u and g symmetry, respectively) is not fully settled. One possibility [3 11 is that the static electronic field of the solvent couples to an electric dipole transition from the B to ii state at the crossing point of the surfaces. An alternative model uses an analogy with collision-induced predissociation in gas phase systems [3 11. In either case the matrix element coupling the states should be proportional to the solvent polarizability and inversely proportional to the sixth power of the iodine-solvent separation. As simulations show, the combination of packing (i.e., radial distribution functions) and changes in polarizability can produce complex effects [31]. A detailed discussion of these effects is given in Ref. 31. We now turn to a system of much greater complexity, the core light-harvesting protein of purple photosynthetic bacteria, LH1. This complex forms a highly symmetric circular structure that is believed to surround the reaction center, thus providing excitation energy to the special pair [32], with which to initiate primary photosynthetic electron transfer. The structure of LH1 has been determined to only 8.5 A resolution; however, a high-resolution structure of the related LH2 protein is available and allows detailed structural conclusions to be drawn. The crystals of LHl form circular rings containing 32 bacteriochlorophyll (Bchl) molecules with a radius of 48 A to the center of the Bchls. Exceedingly rapid energy migration occurs around the ring and can be followed by fluorescence polarization spectroscopy [18]. The fluorescence anisotropy is defined as

In a simple system the initial value of r, r(O), will be 0.4, and as orientation within the plane of the ring is lost, the anisotropy will fall to 0.1. Figure 10 shows the fluorescence anisotropy for LHl in a mutant system containing only this protein, recorded via fluorescence up conversion [18]. The major decay component has a time constant of 110 fs. A slower component is also present in the anisotropy, and it is also apparent that the final value is a little less than 0.1 expected for randomization in a plane. Both of these effects most likely result from a small distribution (-250 cm-' ) in site energies in the ring. We have modeled the anisotropy decay for an inhomogeneousring system in which incoherent hopping takes places between 16 dimers comprising the ring. Support for the dimer model can be obtained from the LH2 struc-

158

G. R. FLEMING, T. JOO AND M. CHO

0.40 7

0;oo

-1000

0

i000

2000

time (fs) Figure 10. The raw experimental fluorescence anisotropy function r(r) derived from the parallel and the perpendicular (with respect to the pump polarization) fluorescence signals from M2192 membrane, a mutant LHl antenna system devoid of peripheral antenna (LH2) and reaction centers. The membrane is excited at 860 nm and detected at 943 nm. (From Ref. 18.)

ture in which an alternating spacing between Bchl molecules is found [33]. We have investigated an exciton model including inhomogeneous broadening (“diagonal disorder”). This disorder produces some localization even at low temperature, and while we have not yet included phonons in the calculation, we propose 118, 341 that a model of hopping between dimers is likely to be appropriate at physiological temperatures. This dimer-hopping model

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159

..

4-.

1-2.

4-J

20000

1

10000

-500

0

500

lo00

1500

2000

time ( fs ) Figure 11. Magic-angle time-resolved fluorescence from M2 192 membranes excited at 860 nm and detected at 943 nm (dots) and fits with (solid) and without (dashed) damped cosinusoid. Inset shows oscillations in more detail. Top: Residuals to fit: upper panel shows residual to fit without cosine term and a line showing the cosine term that is added into the second fit form; the lower panel shows the residuals to a full fit including the damped cosinusoid. (From Ref. 18.)

yields a hopping time between dimers of 80 fs. A noticeable aspect of the anisotropy decay in Fig. 10 is that no oscillations are evident. Yet both parallel and perpendicular decay curves show pronounced oscillations that presumably cancel in the construction of r(t). Figure 11 shows the magic angle decay in which oscillations with a frequency of -105 cm-' are clearly evident. A similar frequency (1 10 cm-') was observed by Chachisvilis et al. in

160

G.R. FLEMING, T.JOO AND M.CHO

pumpprobe studies of LHl [19]. The absence of such beats in the anisotropy decay indicate that the oscillations arise from nuclear wavepacket motion, just as in iodine, rather than electronic recurrences. The striking feature of the beats, however, is that they persist for at least 500 fs (l/e time), much longer than the hopping time scale extracted above. Just as in the diatomic system, then, the nuclear coherence survives the reactive event, in this case energy transfer. Thus, again, in the language of Redfield theory, coherence transfer terms are important and vibrational relaxation and dephasing are not rapid compared to the energy transfer process. The coherence transfer is simply a result of the energy transfer being rapid compared to the vibrational period. Once the total elapsed time scale (e.g., two to three hops) becomes equal to about half a period, the nuclear coordinate will have sampled the full range of coordinates prior to the next hop and no vibrational coherence will remain. This qualitative argument is borne out quantitatively in the calculations of Jean and Heming 2241 and Jean I251 described above. The origin of the lOS-cm-' mode is not known with certainty, but it is tempting to associate it with intradimer motion as is done for the very similar frequency observed in the special pair of reaction centers [Is].Remarkably, Vos et al. 1151 and Chachisvilis et al. [191 find that damping rate is independent of temperature. The absence of pure dephasing on a picosecond time scale is unexpected and implies very weak coupling of this motion to the protein photons. To summarize, we find that for two very different systems coherent nuclear motion can survive surface-hoppingevents and persist in condensedphase systems for comparatively long times. We now turn to a discussion of how nuclear motion influences electronic energy gaps.

III. SYSTEM-BATH INTERACTIONS A. Line Shape Function The approach here is based on the work of Mukamel [l], to which the reader is referred for a much more complete description. Consider an optical transition between the ground and excited states of a chromophore in solution. The optical transition frequency of the ith chromophore can be written as

where (oeg)is the average transition frequency, Awi(t) a fluctuating term induced by both intramolecular and solvent degrees of freedom, and E i is a static offset from the mean as a result of the local environment of chro-

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161

mophore i. In general, it may not be possible to separate Awi(t) and and we will return to this point later. The average (we& is taken over the whole ensemble so that ei is zero centered. It is generally assumed that the fluctuating term is common to all chromophores, in which case one can speak of “homogeneous” dephasing: Ao(t) = Awi(t). Two points are worth noting here: First, the initial dynamics of A@(?)are Gaussian, and in this sense the system evolves inhomogeneously at short times; this is of major significance in interpreting echo experiments. Second, in comparing echo measurements to fluorescence Stokes shift experiments, which also probe chromophore-solvent dynamics, the fluorescence measurements do not average over the ensemble and are thus independent of the parameter ei. The correlation of Aw(t), M ( t ) , plays a key role in the description of experiments and also enables connections to be established between different types of experiments: 1 M ( t ) = (Au(t)Ao(O))= - (6 AE(t)GAE(O)) A2

(7)

where

is the Heisenberg operator of the fluctuating energy difference between the ground and excited states. [Here h,(h,) is the nuclear Hamiltonian of the excited (ground) state.] The term M ( t ) as written in Eq. (7)is complex. In the response function formalism developed by Mukamel [l], all four wave-mixing spectroscopies are described by four response functions, R I,...,R d , and their complex conjugates. Double-sided Feynman diagrams are shown in Fig. 12 representing these response functions. The response functions in turn are described by a single line shape function g ( t ) given by

The real and imaginary parts of g(t) are related to each other by the quantum fluctuation-dissipation theorem. The imaginary part of g(t) gives the energy shift induced by the Stokes shift, whereas a purely real g(t) gives spectral broadening but no progressive relaxation to, for example, longer fluorescence frequencies as solvation proceeds.

162

G. R. FLEMING. T. JOO AND M. CHO

Figure 12. Two-level system (excited, le), and ground, 18) states) double-sided Feynman diagrams for third-order nonlinear optical spectroscopies in the rotating-wave approximation. The left and right vertical lines represent the ket and bra of the density matrix, respectively. Time increases from bottom to top. The arrows on the left and right vertical lines represent the field-matter interactions with either positive (0) or negative (-a)Fourier components on the ket and bra states, respectively. (For a complete description of the Feynman diagrams, refer to Refs. 1 and 7.) Here, S denotes the resulting third-order signal polarization; pij represents a nonzero density matrix element during the time interval. After the second field-matter interaction, Rl and R2 create an excited-state population while R3 and R4 create a ground-state population.

To establish the connection between M(t) and the fluorescence Stokes shift function S(t) defined in Eq. (l), we first rewrite S(t) as S(t) =

AE(i) - AE(-) AE(0) - AE(<-)

where AE(t) is the time-dependent energy difference between ground and excited states and is directly proportional to the center fluorescence frequency as in Eq. (I). The energy difference operator can be divided into an average value and a fluctuating part:

Linear response theory 1351 then leads to

where the response function G(r)is an antisymmetrized correlation function

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163

It will also be useful to define the symmetrized correlation function C(t):

and to introduce a spectral density p f o ) :

where

6 ( w ) is the Fourier-Laplace c(w) =

f:

transform of the response function,

dtexp(iwt)G(t)

[a.

( 1 3 1 allows to connect the varThe definition of the spectral density ious correlation functions relevant to spectral broadening and spectral diffusion. For example, the fluorescence Stokes shift function S(t) can be written as

where the normalization constant X is identical to the solvent reorganization energy:

Thus we see that the first moment of the spectral density multiplied by A is the reorganization energy (i.e., one half of the Stokes shift magnitude), whereas the time dependence of the first moment of p ( w ) corresponds to the fluorescence Stokes shift. Thus the time dependence of S(t) is determined entirely by the spectral density. At high temperature [i.e., when p(w ) contains frequencies less than 2ke TI, S(r) becomes the classical correlation function [36] used by many previous authors [7-lo]. This follows from

G . R. FLEMING, T. JOO AND M. CHO

164

where @ = l/kBT, so that

In this limit S(t) and the normalized form of M ( t ) [Eq. (7)] are identical, and we will simply use S(t) from now on. Turning to the line shape function g(t), it may be written in general as

g(t) =

1

5

1;

d7

d7’{C(7’) - iG(7’))

(21)

j;

dop(o) sin(wt)

or in terms of the spectral density

g(t)=

-$+I,”

dwp(w)coth

tto

[ 1 - cos(ot)] + i 2ksT

Thus Eqs. (17) and (22) establish the explicit correlation between the fluorescence Stokes shift function and the line-broadening function that gives linear absorption and emission spectra via

where the star denotes complex conjugate. Thus if p(w), or at high temperature S(t), is known, then all linear and third-order resonant nonlinear spectroscopies can be calculated. In practice this can be quite a laborious task when finite laser pulses are included in the calculation [37]. In the following section we explore the reverse procedure, that is, attempts to obtain S(t) and p ( w ) from experiment.

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165

B. Echo Spectroscopies Figure 13 shows a general arrangement for carrying out four- or six-wave mixing spectroscopy. The ultrashort pulses are incident on the sample, and a variety of signals are generated in different phase-matched directions. An exceedingly useful aspect of the double-sided Feynman diagrams (Fig. 12) is that (when time ordering of the pulses is enforced) they can be used by simple inspection to determine which response functions contribute to a signal in a particular direction. Pulses of 16 fs duration and energies up to 40 nJ are generated at repe-

BS1

Pulse 1

BS2

Pulse 2

Pulse 3

P%)

Figure 13. Top: Schematic of the experimental setup showing the signal phase matching directions: BSI and BS2.40 and 50% beamsplitters, respectively; lens, 10 cm focal length, A (2kl - k2) and A' (-kl + 2k2). two-pulse photon echo signals generated from pulses I and 2; B (kl - k2 + k3) and B' (-kl + k2 + k3). three-pulse photon echo and transient grating signals generated from pulses I, 2, and 3; F (2kl - 2k2 + k3) and F' (-2kl + 2k2 + k3). fifth-order three-pulse photon echo signals. Transient grating and three-pulse echo experiments differ only in the delays between the pulses. Bottom: Time ordering of the pulse sequences for three-pulse photon echo measurements (T > 0 case). The symbols T and T denote the center-to-center distances between the pulse pairs ( I , 2) and (2, 3), respectively.

G. R. FLEMING, T. JOO AND M. CHO

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tition rates up to 1 MHz from a cavity-dumped titanium-sapphire laser. To minimize thermal gratings and interference from higher order signals, pulses are attenuated to -1 nJ before the beamsplitters, and a repetition rate of 152 kHz is typically used [37]. The first type of experiment we describe is a stimulated three-pulse echo (3PE) peak shift measurement. In this experiment, the two echo signals symmetrically placed at -kl + k2 + k3 and kl - k2 + k3 are recorded simultaneously. A sequence of such signals recorded at different values of the population period T is shown in Fig. 14 (see Fig. 12; during the period T,

\

\

0.4

0.2

/

\

0.0

Y

\

\

I

-50

0

SO

-50

0

50

1.O 0.8

‘5; 0.6

c

3

bs 0.2 0.4

0.0

1.o 0.8 0.6 0.4 0.2 0.0

7. fs

Figure 14. The 3PE signals for IR144 in ethanol in two different phase-matching directions, -ki + k2 + k3 (solid line) and ki - k2 + k3 (dashed line); ( a ) T = 0 fs, (b) T = 40 fs, and ( c ) T = 1.3 ps.

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the system propagates according to a diagonal density matrix element pee or pgg). First note that the signals are time-reversed images of each other and do not peak at zero delay. The purpose of measuring both signals is to determine the shift of the peak of their signals from zero delay with high precision. In Fig. 14, the shift from zero can be measured to kO.3 fs (SO0 as!). Second, note that the shift decreases for increasing values of the population period T. The shift arises from the fundamental asymmetry of the two sides of the echo signal and reflects the potential for rephasing, that is, the degree of retention of memory of the transition frequency. The shift occurs because the signal involves an integration over the time period t’ (see Fig. 12)

To see why this is so, consider two further figures. In Fig. 15, we show two Feynman diagrams: The right-hand one represents the situation when pulse 2 arrives before pulse 1 (7 c 0), whereas the left hand diagram is for 7 > 0. Only for 7 > 0 are the two coherence periods complex conjugates (Ig)(el, le)(gl) of each other. For 7 c 0 no rephasing is possible. Now consider Fig. 16, which shows the absolute square of a rephasing diagram for different (positive) values of 7 as a function of t’. Clearly the area under these curves initially increases with 7 before beginning to decrease as a result of irreversible dephasing. On the other hand, no such increase in area occurs

2>0

2<0

Figure 15. Feynman diagrams for T > 0 where rephasing is possible and 7 < 0 where rephasing is not possible. Note that for T > 0 the density matrix element during T and r’ are complex conjugate of each other, but they have the same phase for T < 0.

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G. R. FLEMING, T. JOO AND M. CHO

0

20

40 t’, fs

80

Figure 16. Demonstration of the peak shift to nonzero T values in a three-pulse echo signal (3PE vs. 7 ) . The rephasing response function IRI1’ is plotted vs. 1’ for several different i values. The signal is calculated using a near critically damped Brownian oscillator model M ( f ) ;the coupling strength (A), frequency (w), and damping (7)of the Brownian oscillator are 200, 115, and 240 cm-’, respectively. Here, T is set to zero. For small T values, the curve is peaked at t’ = T as a result of rephasing. Note that the area under each curve (i.e., the echo signal integrated over t’) initially increases with increasing T as a result of the rephasing even though the maximum value of the rephasing function decreases (From Ref. 37.)

for the nonrephasing (7 < 0) response function, and as a result the echo is asymmetric as long as rephasing is possible. When rephasing is no longer possible, the two sides (7 > 0, 7 c 0) of the integrated echo signal will be identical and the peak shift will be zero. Thus a plot of peak shift versus T maps out the loss of rephasing ability as a function of time. Of course, it is possible to be much more quantitative. One approach is to model S(t) or p ( w ) and fit the parameters to the experimental data. In the course of many such simulations we observed that the peak shift from zero time, 7*(7‘), was very similar to the S(t) function itself. By making a short-time approximation with respect to the first delay period 7 , it is possible to obtain approximate analytical expressions for 7*(T>, which turn out to be very accurate for times longer than the bath correlation

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time [38]. To begin, the three-pulse photon echo signal for 7 > 0 is written in the impulsive limit as ~ 3 p E ( 7 ,T ) =

JI

dr’ I(t’ -- ~)exp(-2Re[g*(~) - g(T) + g*(t’)

+ g(T + t’) - g*(T + T + t’)]} where I( t’ function:

- 7)

+ g*(7 + T ) (26)

represents the inhomogeneous contribution to the response

where Ain is the width of the inhomogeneous distribution [i.e., the FWHM off(€) = 2 d G T A i n ] . In Ref. 38 expressions for 7 * ( T ) are given without making a high-temperature approximation. In the following, however, we assume that 2ksT > Aw. When the inhomogeneous width is very small, 7 * ( T ) becomes (in the hightemperature and impulsive limits)

where I’ = 2X/A2@andf(T) = (x/A)2[1 - C(T)/C(0)I2.In this case a plot of 7* versus T should be identical to S(r) for times greater than the bath correlation time, aside from a proportionality constant given by the rootmean-square fluctuation amplitude. In this limit both S(t) and 7 * ( T ) vanish at long times. A particularly interesting case is when the inhomogeneous width is comparable to the root-mean-square of the fluctuation amplitude (A;n I2XkeT). In this case the interpretation of the simple two-pulse echo measurement is ambiguous [39, 401. Again, in the classical limit, in this region,

and

G. R. FLEMING, T. JOO AND M.CHO

170

Thus in this case the peak shift 7* does not decay to zero. Figure 17 shows three-pulse echo peak shift data for a dye molecule, IR144, dissolved in ethanol and a plastic matrix (glass), PMMA, at room temperature. At long

: G E

20 0

20

I

60

40

I

T,PS I

T,fs

100

80

I

I

I

Figure 17. 3PEPS vs. T for IR144 in ethanol (solid line) and in polymer (PMMA) glass host (dotted line) at room temperature. Note that the three-pulse echo peak shifts are remarkably similar for short values of T (bottom), but they are significantly different at longer times (bottom). The 2-ps time scale components (3 and 27 ps) present in the ethanol data are absent in PMMA, and the PMMA data have a large offset at large T (6 fs) indicating the distribution is static in the glass (inhomogeneous broadening).

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times the peak shift in the fluid solution is very small (-1 fs) whereas the glassy solvent shows an asymptotic value of 7 fs (constant between 10 and 500 ps), clearly indicating the presence of the inhomogeneous broadening. The decay of the peak shift (and the 7-dependent photon echo signals in general) is the result of the loss of the rephasing ability of the system. Dephasing, in general, is the consequence of nuclear degrees of freedom (be they intramolecular to intermolecular) being coupled to the electronic transition, that is, having different equilibrium positions in the two electronic states. At high temperatures, there is no necessity for coupling between the system nuclear coordinates (Q) and the bath coordinates (4); that is, energy flow from Q to the q coordinates is not required. This picture makes clear the ensemble nature of “dephasing.” At very low temperatures where even bath modes are not thermally populated, coupling between the two systems is necessary for dephasing. An interesting analysis of temperature-dependent holeburning data based on this latter mechanism has recently been provided by Small and co-workers 1411. For the present discussion, we will focus solely on a displaced set of harmonic oscillators giving p ( w ) , which may in general contain both intra- and intermolecular modes, as the origin of the dephasing. This model has much in common with the discussion of wavepacket dynamics in the previous section. However, solvent modes coupled to the electronic transition are likely to have a broad distribution of frequencies. Thus, impulsive excitation of wavepackets in these modes will rapidly lead to destructive interference in the ensemble and apparently irreversible relaxation. However, an individual molecule is (during the period for which this picture applies) evolving freely with retention of its Bohr frequency. During this period, it should be possible to rephase the ensemble with an appropriately chosen pulse sequence. Third-order nonlinear experiments such as the peak shift studies described here are rather insensitive to the nature of initial dephasing, that is, whether it is Gaussian or exponential. We have used higher order (fifth-order) echo measurements [39] to show that the initial dynamics of the dye 1, l’, 3,3,3’, 3’ hexamethylindotricarbrbocyanine iodide (HITCI) in ethylene glycol are Gaussian rather than exponential for -80 fs [40].HITCI has a significantly smaller contribution from intramolecular modes in the dephasing than IR144 and in Ref. 40 we assumed that intermolecular contributions to the dephasing were dominant. Further work along these lines is clearly desirable since it seems quite conceivable that, to borrow terminology from gas phase spectroscopy, “intramolecular” vibrational redistribution (IVR)between solvent modes may be rapid. If it is rapid enough, ultrafast dephasing [Eq.(22)] and solvation [Eq.(17)] can still occur, but rather than being Gaussian, the dynamics will be dissipative. Olender and Nitzan have recently provided an interesting discussion of this point [42]. The validity of a harmonic approximation for solvent modes over a physi-

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G . R. FLEMING, T. JOO AND M. CHO

cally significanttime range has important implications for the use of instantaneous normal modes (INMs) [ 11-13] to capture the liquid dynamics. Instantaneous normal modes are attractive descriptions of ultrafast liquid dynamics since they allow a breakdown of the coupled modes into types of motion, for example, rotational versus translational and even in the former case further decomposition into motions about which principal axis the motion occurs [ll]. Indeed, an INM analysis of the optical Kerr signal of acetonitrile by Ladanyi and Klein [43] coupled with a similar analysis of solvation dynamics [12] shows that both processes are dominated by rotational motions and further that p(w) is essentially identical in both cases, providing theoretical backing for earlier guess of Cho et al. that both optical Kerr and Stokes shift responses could be described by a common p ( w ) in this solvent [a]. Calculations based on computer simulations for water [45] and acetonitrile [12] suggest that the entire ultrafast component in the solvation dynamics of, for example, water (Fig. 2) arises from destructive interference of the harmonic normal modes and thus that this epoch is fundamentally nondissipative. Returning to the 3PE peak shift measurements for IR144 in ethanol, we see that the peak shift (Fig. 17) shows at least four widely spaced time scales. Before discussing these time scales, it is necessary to make two comments. First, numerical studies show that the time constants extracted from peak shift measurements are remarkably insensitive to laser pulse duration [37]. The only significant effect of pulse duration over a wide range is to produce an offset in the r*-versus-T curve [37]. Thus, sub-pulse-width time constants may be extracted with confidence. Second, we have already noted that Eqs. (28) and (29) are not accurate for T less than the bath correlation time. Numerical modeling, based on an assumed form for S(t) [ M ( t ) ] ,shows that the amplitude of the fastest component in S(t) is overestimated, and its time scale underestimated by fitting peak shift data to a sum of exponential terms [37]. However, as noted above, the longer time constants obtained in such a fit should be directly comparable to those obtained independently in fluorescence Stokes shift studies. The peak shift data in Fig. 17 show oscillatory character, as is our first two examples (I2 and LHl). This arises from vibrational wavepacket motion. In addition, the very fast drop in peak shift to about 65% of the initial value in -20 fs results from the interference between the wavepackets created in different intramoleculear modes. This conclusion follows directly from obtaining the frequencies and relative coupling strengths of the intramolecular modes from transient grating studies of IR144, carried out in the same solvents (data not shown). Thus, by visual inspection of Fig. 17, an answer to a long-standing question-What fraction of the spectral width arises from intra- and intermolecular motion?-is immediately apparent.

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The remaining components in the peak shift for IR144 in ethanol have time scales of -100, 3, and 27 ps [37]. The two longer time scales compare with 5 and 29.6 ps obtained by Maroncelli and co-workers in their recent very comprehensive fluorescence Stokes shift study of coumarin 153 in a wide range of solvents [2], although in the latter case the amplitude of the slowest component is significantly greater than is observed here. (In methanol, however, both time constants and amplitudes of the two slowest components are in good agreement [2, 371.) These picosecond time scale relaxations undoubtedly arise from the dielectric relaxation process and can be quantitatively predicted from frequency-dependent dielectric data via the dynamical mean spherical approximation [46] or more sophisticated theories such as those of Raineri et al. [6] or Bagchi and co-workers [ 5 ] . Simulations based on detailed molecular charge distributions [3b, 101 imply comparatively small differences in dynamics between different (large) solutes, so that exact correspondence between the results of Maroncelli and co-workers for coumarin 153 and our own data for IR144 should not be expected. The ultrafast component in solvation dynamics was first observed by Rosenthal et al. in acetonitrile [3a] and methanol [3b] and by limenez et al. in water [4]. Optical Kerr [47], far infrared absorption spectra [48], and a large body of simulation [7-101 and theory [5, 61 also support the idea of the ultrafast component arising from librational (i.e., rotational) motions and dominated by the first solvation shell. Simulations and instantaneous normal-mode analyses imply that this ultrafast component is entirely Gaussian in nature and results from free (small-amplitude) motions of solvent molecules that can be regarded equivalently from single-molecule or collective perspectives 1131. The ultrafast component in the peak shift for IR144 has an exponential time constant of 61 fs. If S(t) initially decays as a Gaussian, exp[-(t/~1,)~],our numerical modeling [37] shows that 61 fs corresponds to a qgof -100 fs. Similarly, we obtain 71g = 105 fs for methanol, in reasonable accord with the simulation of Kumar and Maroncelli [lo], who found 71, - 80 fs in this solvent. The fluorescence data [2] are recorded at lower time resolution; however, the average of the first two time constants of Horng et al. is 98 fs for methanol and 92 fs for ethanol [2]. In both solvents, the ultrafast (Gaussian) component accounts for roughly half of the total solvation energy. Very similar time scales for the ultrafast component are found in polar aprotic solvent, such as acetonitrile, chloroform, and benzonitrile 123. Of course, the slower components that involve restructuring of the liquid are strongly solvent dependent. In striking conformation of the harmonic model (or equivalently the “dephasing” of the broad spectrum of the wavepacket), even PMMA, a glass at room temperature, displays a similar ultrafast time scale [49]. The amplitudes of the ultrafast component with respect to the picosecond components decrease with increasing size in the

G . R. FLEMING, T. JOO AND M.CHO

174

n-alkanol series [Z,371, but the amplitude of this component in peak shift data always appears larger than in corresponding fluorescence data [Z,371. By way of summarizing this discussion, Fig. 18 shows the S(t) [= M ( t ) ] function that was extracted from the peak shift data, and its breakdown into intramolecular and intermolecular contributions. For data fitting we write S ( t ) as

where the oscillatory term contains the contribution of the intramolecular modes [373. The Fourier transform of such a curve [see Q.(1 7)] corresponds directly to the spectral density p f u ) . The S(t) obtained in Fig. 18 can be used

1.o

0.0

0

200

400 t,

fs

600

800

Figure 18. The normalized electronic transition frequency correlation function M(r) I= SO)] obtained from the experimental three-pulse photon echo peak shifts and transient grating data for IR144 in ethanol: (---) total M(t); (...) ultrafast Gaussian component in M(r); (-) oscillatory component that arises from intramolecularvibrational motion.

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1.o 0.8 .c)

‘8 0.6 E

B

5

0.4 0.2 0.0

‘5: 0.0

3 0.4 fi 0.2 E

0.0

T,fs

Figure 19. Comparison of calculated and measured signals using the M ( t ) shown in Fig. 18. (a)Three-pulse echo peak shift, (b)transient grating, and (c) transient absorption. A pulse duration of 16 fs (20 fs for transient grating) and a detuning of 250 cm-’ are used in the calculated signals. The peak near T = 0 in the transient grating and transient absorption signals, usually refened to as the “coherent artifact,” arises from the ultrafast decay (sum of intramolecular vibrational contribution and -100 fs ultrafast solvation dynamics) in M I ) .

to calculate pumpprobe (transient absorption) and transient grating signals for the same solvent-solute system. Figure 19 shows the comparison with the appropriate experimental signals. The agreement is excellent and reveals (as is discussed in detail elsewhere [37]) that the short-time portions of both transient absorption and transient grating signals also reports on the ultrafast component of solvation dynamics.

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G . R. FLEMING, T. JOO AND M.CHO

IV. DISCUSSION

The descriptions given in the previous sections of ultrafast intramolecular and intermolecular dynamics are remarkably similar. Nuclear dynamics on chemically significant time scales and over chemically significant distances is often free ballistic motion, with apparently irreversible relaxation effects arising from averaging over the ensembles. Electronic phase in condensedphase systems remains well defined for 50-100 fs, whereas the vibrational phase is often very robust, surviving well into the picosecond regime. In particular, the vibrational phase survives curve crossing, anharmonicity of potential surfaces, and substantial amounts of energy loss to the surroundings. In this chapter we have exploited the above facts in our studies of molecular dynamics. In particular, the retention of vibrational coherence, that is, well-defined internuclear positions in the ensemble, allows chemical dynamics, such as bond breaking, caging, and so on, to be observed in real time. Not merely does this enable studies of chemical and biological dynamics to be performed at an unprecedented level of detail, but it also opens up the prospect of new classes of experiments. Ultrashort-pulse lasers enable creation of wavepackets on excited-state potential surfaces, and they also allow for the possibility of transferring a wavepacket with a selected position and momentum to a second surface. Jonas has pointed out that stimulated emission pumping can be exploited to position wavepackets on ground-state surfaces, perhaps moving toward transition states. Such an approach may allow real-time study of thermal reactions (i.e., those occumng on the ground state) as opposed to the photochemical processes universally studied at present. The realization that a significant fraction of the solvation energy in small, highly polar solvents such as water-acetonitrile arises from free motions has allowed intuitively satisfying molecular rather than the traditional continuum descriptions of this aspect of liquid state dynamics. In particular, the instantaneous normal-mode approach provides clear insight into the types of nuclear motion involved and the number and location of the solvent molecules involved [11-131. Since no information is destroyed during the Gaussian period of relaxation, the ever-increasing sophistication of pulse-shaping techniques [501 suggests the possibility of much more refined studies of system-bath interactions. For example, optical correlations in static and dynamic properties may be obtainable. The response function formalism described in Section 111 is not able, with current generation experiments, to distinguish between a single homogeneous response function R ( t l , t 2 , t 3 )and one that arises from a distribution of “homogeneous” response functions, that is,

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where I’ might represent a particular liquid configuration and p ( r ) is the distribution function. The impact of coherent nuclear motion on studies of dynamics is, we hope, clear. However, since the coherent motion is observed only because ultrashort pumping and probing is done, the question naturally arises: Is the observation of, say, vibrational coherence in a reactive event of significance in such processes as vision or photosynthesis, which result in the natural world from absorption of incoherent sunlight? Aside from the somewhat obvious points that the time scales of energy relaxation and dephasing are frequently much longer than has previously been assumed and that if the process proceeds ballistically, the typical pictures drawn for, for example, isomerization reactions in solution are literally incorrect, quantum interference does significantly influence the time scale of curve-crossing processes. The standard Golden rule expression for a process is

where a’ and 6’ refer to reactant and product levels and Pa‘ is the population of level a’,J is the electronic coupling, l(~’lb’)[~is the Franck-Condon factor, and the dephasing rate rsu = 1/Tz is as defined in Eq. (3). Equation (33) assumes that raq,t is large compared to 21 (i.e., no electronic and vibrational recurrences). In addition, Eq. (33) deals only with population dynamics: Interferences between different Franck-Condon factors are neglected. These interferences do influence the rate, and the interplay between electronic and vibrational dynamics can be quite complex [25]. Finally, as discussed by Jean et al. [22],Eq.(33) does not separate the influence of pure dephasing ( T a and population relaxation (TI). These two processes (defined as the site representation [ 2 2 ] )can have significantly different effects on the overall rate. For example, when (TI)-’ becomes small compared to ( T a - ’ , Eq. (33) substantially overestimates the rate compared to the value calculated using full Redfield equations [22]. A critical point, in our view, is that performing experiments with short pulses allows one to observe the behavior of individual molecules by forcing the whole ensemble to “travel together” (this is literally what coherence means) until dissipation sets in. But when the process is initiated with incoherent light, individual molecules behave in the same way as they did in the short-pulse experiment, except that now there is a stochastic distribution of

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G . R. FLEMING, T. JOO AND M. CHO

start times. This washes out any sign that, for instance, the molecules travel through a curve-crossing region ballistically. To take a concrete example, the isomerization of 11-cis retinal to all-trans retinal in rhodopsin proceeds with retention of vibrational coherence in the all-trans product when studied by ultrafast spectroscopy [ 171. Basing his argument on a simple Landau-Zener model, Mathies suggests that the ballistic first passage through a crossing region determines the relatively high (0.66) quantum yield of all-trans formation. He suggests that if the isomerizing rhodopsin lost its kinetic energy before reaching the crossing region, the quantum yield and hence the sensitivity of vision would be dramatically reduced. So far the retention of vibrational coherence has been emphasized. In transfer (e.g., electron or energy) processes mediated by bridge states (i.e., where superexchange processes contribute significantly to the rate), the rate can be sensitive to the time scale over which the electronic phase is maintained. This is because the rate will contain contributions from terms only involving coherences between initial and bridge and bridge and final states. The rate of distribution of such superpositions will determine the magnitude of the contribution of these terms to the overall rate. This, as was originally noted by Hu and Mukamel [51], is analogous to influencing the ratio of Raman (coherent) to fluorescence (incoherent) emission by changing the dephasing time scale. The work of Wiersma and co-workers [52] on azulene elegantly demonstrate this point experimentally. To sum up, this chapter has endeavored to show that chemical processes in solution often proceed in a deterministic fashion over chemically significant distances and time scales. Ultrafast spectroscopy allows real-time observation of relative motions even when spectra are devoid of structure and has stimulated moleculear level descriptions of the early time dynamics in liquids. The implication of these findings for theories of solution phase chemical reactions are under active investigation. Acknowledgements We thank John Jean, R. D. Levine, M. Ben-Nun, David Jonas, Steven Bradforth, Ralph Jimenez, Yutaka Nagasawa, and Sean Passino for allowing us to use their results in this review and for much valuable scientific input. M. C. thanks Robert Silbey for support and G . R. F. thanks the Department of Chemistry at Cornell for their hospitality while this chapter was being prepared. This work was supported by the National Science Foundation and in part by the American Chemical Society Petroleum Research Fund.

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(b) S. J. Rosenthal, R. Jimenez, G. R. Fleming, P. V. Kumar, and M. Maroncelli, J. Mol. Liq. 60, 25 (1994). 4. R. Jimenez, G. R. Fleming, P. V. Kumar, and M. Maroncelli, Nature 369, 471 (1994). 5. S. Roy and B. Bagchi, J. Chem. Phys. 99,9938 (1993). 6. F. 0. Raineri, H. Resat, B.-C. Perng, F. Hirata, and H. L. Friedman, J. Chem. Phys. 100, 477 (1 994). 7. M. Maroncelli and G. R. Fleming, J. Chern. Phys. 89, 5044 (1988). 8. M. Maroncelli, J. Chem. Phys. 94,2084 (1991). 9. E. A. Carter and 1. T. Hynes, J. Chem. Phys. 94, 5961 (1991). 10. P. V. Kumar and M. Maroncelli, J. Chem. Phys. 103,3088 (1995). 11. M. Cho, G. R. Fleming, S. Saito, I. Ohmine, and R. Stratt, J. Chem. Phys. 100, 6672

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32. S. Karasch, P. A. Bullough, and R. Ghosh, EMBO J. 14, 631 (1995). 33. D. McDemott, S. M. Prince, A. A. Freer, A. M. Hawthornthwaite-Lawless,M. Z. Papiz, R. J. Cogdell, and N. W. Isaacs, Nature 374, 517 (1995). 34. R. Jimenez, S. N. Dikshit, S. E. Bradforth, and G. R. Fleming, J. Phys. Chem., 100,6825 (1 996). 35. L. D. Landau and E. M. Lifshiz, Statistical Physics Part I , Pergamon, Oxford, 1963. 36. D. Chandler, Introduction to Modem Statistical Mechanics, Oxford, New York, 1987. 37. T. loo, Y. Jia, J.-Y. Yu, M. J. Lang, and G. R. Fleming, J. Phys. Chem. Phys. Chem., 106, 6089 (1996). 38. M. Cho and G. R. Fleming, J. Phys. Chem. 98,3478 (1994). 39. M. Cho, J-Y Yu, T. Joo, Y. Nagasawa, S. A. Passino, and G. R. Fleming, J. Phys. Chem. 100, 11944 (1996). 40. T. loo, Y. ha, and G. R.Fleming, J . Chem. Phys. 102,4063 (1995). 41. T. Reinot, W.-H. Kim,J. M. Hayes, and G. J. Small, J. Chem. Phys., 104, 793 (1996). 42. R. Olender and A. Nitzan, J. Chem. Phys. 102,7180 (1995). 43. B. M. Ladanyi and S. Klein, J. Chem. Phys. 105, 1552 (1996). 44. M. Cho, M. Du,N. F. Scherer, L. D. Ziegler, and G . R. Fleming, J. Chem, Phys. 96,5033 ( 1992). 45. M. Cho. Ph.D. Thesis, University of Chicago, (1993). 46. P.G. Wolynes, J. Chem. Phys. 86,5133 (1987). 47. S. Ruhman, B. Kohler, A. G. Joly, and K. A. Nelson, J. Chem. Phys. 141, 16 (1987); S. Ruhman, A. G. Joly, and K. A. Nelson, IEEE J. Quant. Electron. QE-24, 470 (1988); S. Ruhman and K. A. Nelson, J. Chem. Phys. 94, 859 (1991); J. Etchepare, G. Frillon, G. Hamoniaux, and A. Orszag, Opt. Commun. 63, 329 (1987): Y. J. Chang and E. W. Castner, Jr., J. Chem. Phys. 99, 11 3 (1993). 48. G. 1. Davies and M. Evans, J. Chem. SOC.Faraday II,72, 1194 (1975). 49. Y. Nagasawa, S. Passino, T. Joo, and G. R. Fleming, J. Chem. Phys. in press. 50. M. M. Wefers, H. Kawashiria, and K. A. Nelson, J. Chem. Phys. 102,9133 (1995). 51. Y. Hu and S. Mukamel, J. Chem. Phys. 91,6973 (1989). 52. E. T. J. Nibbering, K. Duppen, and D. A. Wiersma, J. Chem. Phys. 93, 5477 (1990). 53. D. P. Shoemaker, C. W.Garland, and J. W. Nibler, Experiments in Physical Chemistry, McGraw-Hill, New York, 1989. 54. D. Lee and A. C . Albrecht, in Advances in Infrared and Raman Spectroscopy, Vol. 12, R. 1. Clark and R. E. Hester, Eds., Wiley Heyden, New York. 1985.

DISCUSSION ON THE REPORT BY G. R. FLEMING Chairman: V. S. Letokhov

A. H. Zewail: I thought that on the femtosecond time scale the separation of homogeneous and inhomogeneous relaxations is fuzzy. But now you are making a separation in definition. Could you elaborate on their meaning?

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G. R. Fleming: A general way to avoid the semantic difficulties produced by these terms is to use a spectral density p ( w ) that contains all the time scales present in the system-bath interaction. A curious result of the fact that the short-time solvent motion is inertial is that the largest ultrafast contributions to the line broadening arise from a Gaussian component and that the slower diffusive motions give rise to Lorentzian broadening. This is rather the reverse of the conventional picture in which a “homogeneous” Lorentzian line is broadened by a very slow or static Gaussian component. R. A. Marcus: Prof. Fleming has shown, I gather, that apart from the behavior at very short times the harmonic oscillator approximation breaks down. Are there any implications for one current formal treatment of the liquid as a harmonic bath that interacts bilinearly with the solute? Did the discrepancy merely reflect the absence of hypothetical low-frequency modes? G. R. Fleming: Though I think a displaced harmonic bath can be a very general model, there is a problem in attaching meaning to the oscillators on long time scales. The instantaneous normal-mode approach provides an appealing physical picture on short time scales. The distinction is whether the harmonic bath is regarded as a basis set or whether the harmonic modes are viewed as having specific physical significance. This latter viewpoint can only hold at short times, before the liquid has restructured itself. E. Pollak Instantaneous normal modes are reasonable for short times. For long times, one may try to use a generalized Langevin equation representation. Sometimes it will work but sometimes not. When yes and when not is not well understood for liquids. D. J. Tannor: The spectral density has a significant contribution from imaginary frequencies. Yet, if I understand correctly, these imaginary frequencies are ignored in the calculation of the correlation function. G. R. Fleming: The point raised by Prof. Tannor is not a problem in water. There, the solvation dynamics has to do with rotations and there are essentially no imaginary rotational modes. So leaving them out does not produce an error. In acetonitrile, rotational-type motions also dominate the solvation dynamics. There, however, the contribution of imaginary frequencies has been considered in Ref. 1. 1. B. M. Ladanyi and R. M. Stratt, J. Phys. Chem. 99,2502 (1995).

D. J. Tannor: If one were to do the multimode harmonic analy-

sis using Gaussian wavepackets, one would have no trouble including

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the imaginary frequencies on the same footing with the real ones. One might imagine that the contribution from these imaginary frequencies will not be in the shortest time dynamics. The shortest time decay of the correlation function should come from the well dynamics, where the slope is steeper. The contribution from barrier dynamics should only enter later, due to wavepacket spreading, leading to irreversible decay of the correlation function. G. R. Fleming: The suggestion by Prof. Tannor that you can use imaginary modes to find longer time behavior is interesting.

S. Mukamel 1. Prof. Fleming, the expressions you are using for the nonlinear response function may be derived using the second-order cumulant expansion and do not require the use of the instantaneous normalmode model. The relevant information (the spectral density) is related to the two-time correlation function of the electronic gap (for resonant spectroscopy) and of the electronic polarizability (for off-resonant spectroscopy). You may choose to interpret the Fourier components of the spectral density as instantaneous oscillators, but this is not necessary. The instantaneous normal mode provides a physical picture whose validity needs to be verified. Does it give new predictions beyond the second-order cumulant approach? The main difficulty with this model is that the modes only exist for a time scale comparable to their frequencies. In glasses, they live much longer and the picture may be more justified than in liquids. 2. With regard to the previous question of Prof. Zewail: It is possible to decompose the spectra1 density into homogeneous and inhomogeneous parts. This may be done by parameterizing it using, for example, the Brownian oscillator model and then adding an inhomogeneous distribution of these parameters. By doing so, it is possible to show how high-order response functions x ( ~ )x,‘ ~ ).,.. , could provide some information not contained in x(3)measurements. The nonlinear response function to arbitrary order x ( ~ was ) calculated in Refs. 1-3. New types of photon echoes that provide a multidimensional (electronically resonant or off-resonant) spectroscopy were predicted. 1. Y. Tanimura and S. Mukamel, J. Chem. Phys. 99,9496 (1993). 2. V. Khidekel and S. Mukamel, Chem. Phys. Lett. 240, 304 (1995). 3. S. Mukamel, Principles of Nonlinear Optical Spectroscopy, Oxford University Press, New York, 1995.

G. R. Fleming: Prof. Mukamel’s first point is correct. The value of the instantaneous normal-mode approach is to provide a molecu-

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lar picture of the short-time dynamics. It is certainly not a necessary assumption for the analysis. Regarding the second point, as we have both pointed out, resonant third-order experiments are not sensitive to whether the response function is homogeneous or results from the convolution of a distribution function in the parameters on which the response function depends with the response function itself. The nonresonant fifth-order experiment proposed by Tanimura and Mukamel, as well as seventh-order nonresonant experiments do have the capacity to address these issues. It will be interesting to see how these studies develop.

FEMTOSECOND LASER CONTROL OF ELECTRON BEAMS FOR ULTRAFAST DIFFRACTION V. S.LETOKHOV Institute of Spectroscopy Russian Academy of Sciences Troitzk, Moscow Region, 142092 Russia

Femtosecond laser excitation makes it possible to produce in a synchronous manner accurate to within a few femtoseconds an ensemble of molecules in an excited state and observe thereafter the evolution of this ensemble in the subsequent processes of decay, relaxation, and so on, by means of other femtosecond pulses. Another femtosecond pulse is usually used as a probe pulse [l]. However, one can directly observe changes in the geometry of molecules, specifically in molecular vibrations, by the method of electron diffraction using ultrashort electron pulses. This was successfully demonstrated in Ref. 2. Whereas the production of synchronous probe laser pulses is a standard technique, the situation with femtosecond electron pulses is more complicated. I would like to call attention to the possibility of using intense femtosecond laser pulses to control electron beams, specifically to obtain femtosecond electron pulses and to focus and reflect them, and so on [3, 41. This possibility is based on the use of what is known as the dipole (gradient) force that I proposed many years ago to control the motion of neutral atoms [5]. In the case of atoms in a light field in the vicinity of the resonance frequency wo, this force may be expressed in the form

Advances in Chemical Physics, Volume 101: Chemical Reactions and Their Control on the Femtosecond Time Scafe, XXth Solvay Conference on Chemistry, Edited by Pierre Gaspard, Irene Burghardt, I. Prigogine, and Stuart A. Rice. ISBN 0-471-18048-3 0 1997 John Wiley & Sons, Inc.

185

186

V. S . LETOKHOV

where a(@) is the polarizability of an atom in an optical field and (E2)av is the square of the field amplitude averaged over the optical period. The polarizability of a free electron in an optical field of frequency o is described by (y

e2 ma2

= --

According to Eqs. (1) and (2), the electron is repelled from the strongfield region into the low-intensity region. This repulsion might be exploited to change the trajectory of the electron. One can also consider the “light” medium (at the optical wavelength hopt) as a medium with a reduced refractive index of n(w) for nonrelativistic u << c given by the expression

where p 2 is the dimensionless intensity parameter given by the expression

where re = e2/mc2= 2.8 x cm is the classical radius of an electron, m is the electron rest mass, A is the optical wavelength, and I is the intensity of light. This means that for electrons with an energy of &in = 50 eV such a “photon” medium at a laser intensity as high as lOI5 W/cm2 at A , = tpm has a refractive index differing substantially from unity. Electrons will be pushed out of such a medium. And it is exactly femtosecond laser pulses that can help obtain the necessary high laser intensities quite easily. In essence, by using high-intensity ultrashort laser pulses, in a vacuum one can form regions of desired shape with a reduced refractive index for a specified ultrashort period of time. It is precisely this fact that provides the basis for developing ultrafast laser-induced electron optics, exploiting the specific features of the laser light: high intensity, short pulse duration, and high concentration in a small region of space. With an appropriate (quadratic) field profile E(r), where r is the radial displacement, it might be exploited to focus an electron beam. Quite sufficient for this purpose would be, for example, the transverse laser mode TEM;,,

187

FEMTOSECOND LASER CONTROL OF ELECTRON BEAMS

which has a quadratic profile along the coordinate near the beam propagation axis and an intensity minimum at the axis (Fig. la):

wg 2r2 Z(r, z ) = 410 -w2(z) w2(z)

where wo is the radius of the beam waist in the z = 0 plane, w(z) = wo(1 + Z ~ / Z ~ ) ' is / ~the radius of the waist in the z plane, ZR = (n/x)w& wo is the Rayleigh length, I0 = P 0 / 2 m 4 is the intensity of the electromagnetic radiation, and PO is the power of this radiation. This field configuration was discussed in Ref. 6 for sharp focusing of an atomic beam. The diameter of the spot in which the electron beam is focused, 2a,l, is related to the focusing diameter of the laser beam, 2aopt= 2w0, by simple equation,

tr

I h

(a) Figure 1. Various configurations of an optical field for laser-induced electron optics: ( a ) focusing of electrons by the TEMtl mode configurationof the laser field; (b) reflection of electrons by the evanescent laser field; (c) reflective focusing of electrons by the curved evanescent laser field.

188

V. S. LETOKHOV Z’

Electron beam, e-

Electron beam, e’

”?

(e 1

Figure 1. (Continued)

where hdBr = Zxii/mu is the de Broglie wavelength of an electron, Ekin =

m 3 / 2 is the kinetic energy of an electron, and 6 n is the change of the effec-

tive refractive index. Let us consider the focu$ng of electrons with an energy Ekin = 100 eV (u = 5 . 9 lo8 ~ cm/s, hdBr = 1.2 A) by electromagneticradiation at A = 10 pm with an intensity I0 = 0 . 5 10l2 ~ W/cm2 at the focus. In this case the coefficient 6 n

FEMTOSECOND LASER CONTROL OF ELECTRON BEAMS

189

in Eq. (6) is 6n = 0.12; with sin poPl= 0.3,. . . ,0.5, for example, the electron beam can be focused to a spot with a radius uel = 10,. .. ,15 A. The transit time of the electron through the focusing region is 7int = 5 ps, so focusing could be achieved even with picosecond pulses with a length of 100 ps and J. an energy of only El, = X2Z~7pulse= A laser lens for electrons has spherical aberration, which has a negligible effect on estimate (6), and also chromatic aberration, which is a more serious matter. For the particular numerical example we are discussing here, the electrons would have to be monochromatic within something on the order of 0.01%. The same stringent requirement would be imposed on the stability of the pulse intensity. We would thus need a square pulse with minimal rise and decay times (qr
where zo = (X/29r)(ni sin28 - 1)'l2 is the characteristic light field decay length in the z direction normal to the dielectric-vacuum interface, 10 is the field intensity at the interface, X is the optical wavelength, no is the optical refractive index, and 8 is the radiation incidence angle at the interface. The character of reflection of electrons from the evanescent wave strongly depends on the relationship between the duration 7 of the laser pulse and the time of flight of an electron through the laser wave, T ~It~may . be shown that when the laser pulse duration is much longer than the characteristic transit ~ , character of reflection of the electrons is close to the mirror. time T ~ the Where the relationship between these times is reversed, the mirrorlike character of reflection is disturbed. Let us make some simple estimates of the laser field and electron beam parameters with which the reflection of electrons is possible. In the case 7 cc T~~we can write the following expression for the variation of the normal electron velocity component:

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V. S . LETOKHOV

where y = e2Ei/2m2wih and EO is the field amplitude at the surface of the dielectric, It can be seen that the velocity variation of the electron depends on its coordinate at the instant the laser pulse arrives at the dielectric surface. Since the angle of reflection of the electron is equal to the ratio between its normal velocity component ul and the longitudinal velocity component uii, it is clear that this angle is not a constant but a variable ranging between the maximum value of (om,, = - ( y ~ / u l l )and zero with the incidence angle remaining the same. With the laser pulse energy W = J, the laser spot diameter d = 1OX on the dielectric surface, and the light frequency v = 5x 1014s-l, the electron velocity variation is AVL= 2 x lo8exp(-z/zo) centimeters per second. This means that an electron beam with an energy E = 100 eV (u = 5.9 x lo8 cm/s) reflects at a substantial angle of (a = 0.3 rad from the evanescent wave produced by a femtosecond laser pulse. The reflection of the electron should be expected in the case r >> rtr when the maximum potential energy of the electron in the evanescent wave is higher than its kinetic energy associated with its normal motion toward the dielectric surface (i.e., Urn,, 2 E l ) . The maximum normal velocity component of the reflected electrons is given by

s (all the other parameters remaining the At a laser pulse duration T = same) the corresponding velocity urn,,= 6 x lo7 cm/s; that is, as one would expect, it is lower than in the case r << rt,. The possibility of reflection of electrons by an evanescent wave formed upon the total internal reflection of femtosecond light pulses from a dielectric-vacuum interface is quite realistic. The duration of the reflected electron pulses may be as long as 100 fs. in the case of electrons reflecting from a curved evanescent wave, one can simultaneously control the duration of the reflected electron pulse and affect its focusing (Fig. lc). Of course, one can imagine many other schemes for controlling the motion of electrons, as is now the case with resonant laser radiation of moderate intensity [9, lo]. In other words, one can think of the possibility of developing femtosecond laser-induced electron optics. Such ultrashort electron pulses may possibly find application in studies into the molecular dynamics of chemical reactions [I, 21.

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191

References I. J. Manz and L. Woste, Eds., Femtosecond Chemistry, Vols. 1 and 2, Verlag Chemie, Weinheim, 1995. 2. J. C. Williamson, M. Dantus. S. B. Kim, and A. H.Zewail. Chem. Phys. Lett. 196, 529 (1992). 3. V. S . Letokhov. JETP Lett. 61, 805 (1995). 4. V. I. Balykin, M.V.Subbotin, and V. S. Letokhov. Opt. Commun., 119, 727 (1996). 5. V. S. Letokhov, JETP Len., 7 , 272 (1968). 6. W. W. Rigmd. Appl. Phys. Lett 2,51 (1963). 7. R. J. Cook and R. K. Hill. Opt. Commun. 43,258 (1982). 8. V. I. Balykin, V. S. Letokhov. Yu B. Ovchinnikov, and A. I. Sidorov, JETP Lett. 45, 282 (1987); Phys. Rev. Lett. 60, 2137 (1988); Errata, 61, 902 (1988). 9. C. S. Adams, M. Sigel, and J. Mlynek. Phys. Rep. 200, 143 (1994). 10. V. I. Balykin and V. S . Letokhov, Atom Optics with Laser Light, Harwoad, Harwood Acad. Publ. (Chur, 1995). 11. M. Born and E. Wolf. Principles of Optics, Pergamon, New York, 1970.

DISCUSSION ON THE COMMUNICATION BY V. S. LETOKHOV A. H. Zewail: I am pleased to hear your idea about the focusing of electrons by photons. I would like to mention that in the course of our work on ultrafast electron diffraction we have observed focusing in the elastic scattering region when the electron and photon beams overlap in the molecular beam. The pulse duration and photon pulse energy are in the range you mentioned, and we should examine our results in light of your proposal. The phenomenon was reported recently [J. Phys. Chem. 98,2782 (1994)], and we gave a discussion in relation to a lensing effect, but we should consider your proposal since the effect should be most pronounced at low electron density where focusing is expected and electron repulsion is minimum.

T. Kobayashi

1. Concerning the simulation by Prof. Letokhov, I would like to ask about the monochromaticity of the electron beam and the coherence length. 2. To obtain a longer interaction, a longer pulse is better if an intense enough pulse can be utilized to generate the evanescent field.

V. S. Letokhov: Let me answer Prof. Kobayashi’s question as follows:

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V. S.LETOKHOV

The coherence length of the electron beam is determined by the de Broglie wavelength of the electrons, AdBr, and the degree of velocity monochromaticity of the electron beam, Au/v:

For example, for a low-energy electron (100 eV) hd& = 1 A, and for a the coherence length will be monochromatic beam with Au/u = about 104 A = 1 pm.

B. Kohler: Your prediction of a focused spot size of 1-2 nm is very interesting. Are your calculations valid for high fluxes? That is, does your modeling consider space charge effects? V. S. Letokhov: We cannot avoid Coulomb repulsion, which will limit the electron density or electron current in the focal points to a size of a few nanometers.

Advances In Chemical Physics, Volume101 Edited by I. Prigogine, Stuart A. Rice Copyright © 1997 by John Wiley & Sons, Inc.

GENERAL DISCUSSION ON FEMTOCHEMISTRY: FROM CLUSTERS TO SOLUTIONS Chairman: K S.Letokhov

S. A. Rice: I find the use by Prof. Fleming of the normal-mode representation of very short time liquid dynamics interesting. As you also indicated during your presentation, even though the contribution from the complex modes to the solvation dynamics is not important for water in the regime investigated by your experiments, it is desirable to have a better understanding of when those modes do influence the process studied and in what form that influence is signaled. With that goal in mind, I call your attention to the high-frequency viscoelastic behavior of liquids, as revealed by Brillouin light-scattering spectroscopy. In the 1960s and the 1970s the Brillouin spectra of a very large number of complex liquids were measured. The interpretation of these spectra was usually based on a viscoelastic representation of the dynamics, so there should be data available for, for example, the frequency dependence of the viscosity for a frequency range not too different from that probed in your experiments. Is there a correlation between the frequency dependence of the viscosity and the distribution of complex instantaneous normal modes of a liquid? G. R. Fleming: We have not yet looked at the issue raised by Prof. Rice. We understand much more about dielectric friction than about collisional friction at short times. The calculation you suggest would be very interesting.

T.Okada: 1 am very much interested in the results by Prof. Heming showing that very rapid energy relaxation occurs while the memory of phase still remains. This probably indicates that the solvent reorientational relaxation may not proceed so much during the energy relaxation. My question is: Have you measured the relaxation time of the band shape as well as the energy relaxation? 193

194

GENERAL DISCUSSION

We have done a study by time-resolved hole-burning spectroscopy for dye molecules in polar solvents and found that the time correlation function of the hole width decays much slower than that of the peak shift of the hole, which occurs very rapidly, as you observed in the case of the fluorescence Stokes shift [K. Nishiyama, Y. Asano, N. Hashimoto, and T. Okada, J. Mol. Liquids 65/66,41 (1 9931. Could you give me some ideas to explain the discrepancies of relaxation times between energy and distribution relaxations?

G. R. Fleming: The point mentioned by Prof. Okada is not something studied, though I am aware of your work in this area. Hynes has shown how bandwidth changes can reflect deviations from linear response. A second aspect is that, in an inhomogeneous ensemble, different members may have different lifetimes, so that the spectrum will change moreover from this effect.

R. A. Marcus: From the peak related to the inhomogeneous contribution, how large was the range of inhomogeneity in the AG changes, in absolute amount, in the system treated by Prof. Fleming? Did this result refer to a highly viscous (e.g., “frozen”) system? G. R. Fleming: In simple liquids such as methanol and ethanol there is no evidence for relaxation times slower than expected from dielectric measurements. Glasses at room temperature clearly show time scales that are infinite on our measurement time scale. In complex liquids such as glycerol-water mixtures and ethylene glycol, we may observe time scales that are longer than dielectric relaxation, but further studies are required to confirm this. J. Troe 1. In our experiments on the energy transfer of vibrationally highly excited polyatomic molecules in compressed gases and compressed liquids, we find that large-amplitude modes lose their energy much faster than small-amplitudemodes. Prof. Fleming, did you find a similar relation between energy loss and vibrational amplitude? 2. Has one looked in the simulations into the range of the spreading of energy around the central energy-donating molecules? Up to which distances is this ballistic if this is ballistic?

G. R. Fleming: We have not made detailed studies (either experimental or via simulation) on the energy dependence of vibrational

FEMTOCHEMISTRY I1

195

relaxation. This may be an excellent way to get at the time dependence of mechanical friction.

R. D. Levine: In the simulations [M. Ben-Nun, R. D. Levine, D. M. Jonas, and G . R. Fleming, Chem. Phys. Lett. 245,629 (19931, the interaction of the iodine molecule in the B electronic state with the surrounding solvent is found to be quite strong. The reason is the unusually long range of the potential as compared to iodine or other diatomics in the ground electronic state. This means not only that the highly vibrationally excited isolated molecule spends most of its time in an extended configuration near its outermost turning point, where over a wide range of internuclear separations its velocity is low. When the molecule is femtosecond excited from the ground state in the presence of the solvent, the motion out of the initially accessed Franck-Condon region results in the iodine atoms experiencing a strong interaction with the surrounding solvent, which is still in its equilibrium configuration. A. H.Zewail

1. Concerning Prof. Levine’s comment, let me add that in our study of Iz in Ar clusters and fluids we see long-time coherence in the solvent cage, but it depends crucially on the rigidity of the solvent cage. [See J. K. Wang, Q. Lin, and A. H. Zewail, 1. Phys. Chem. 99,11309, 11321 (1995); C. Lienau and A. H. Zewail, Chem. Phys. Lett. 222,224

(1994).] 2. I also have a question for Prof. Fleming: When you treated the

problem of coherence transfer phenomenologically using Redfield’s equation, did you examine details of the potential that leads to this robustness?

G. R. Fleming: An interesting feature of the Redfield theory calculations is that attempts to stop coherence transfer by increasing the dephasing rate also increases the coherence transfer rate. In addition, two-state calculations [M. Jean and G. R. Fleming, J. Chem. Phys. 103, 2092 (1995)] show that the coherence transfer can survive reasonable amounts of anharmonicity. it appears to be quite robust. D. M.Neumark: I have a question for Profs. Woste and Manz. The simulation of the K2 experiment at low intensity shows that a hole is created on the ground K2 surface. Is there any manifestation of this in your experiment? The results you showed (at low intensity) appeared to only reflect vibrational motion in the K2(A) state.

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GENERAL DISCUSSION

R.de Vivie-Riedle and J. Maw:* Prof. Neumark‘s question about detecting the “hole burning” in the nuclear wavepacket of the electronic ground state is very stimulating. In this context, we have developed a scheme for detecting the “hole” in the wavepacket by a femtosecond chemistry laser experiment that involves two laser pulses: Our explanation will be for the specific system K2,but more general applications for other systems are obvious: (i) The first laser pulse 1 is used to “burn the hole” in the nuclear wavepacket of the electronic ground state X. Subsequently, this “hole” is moving coherently on the potential energy surface V&). The intensity of the first laser pulse should be sufficiently high such that the hole is a substantial part that is “burnt o f f from the initial wavepacket. (ii) The second laser pulse 2 should be applied with time delay A? such as to achieve direct three-photon ionization, as in the pump and probe experiments of Ref. 1 (see also Ref. 2). The time delay can then be varied such as to monitor the motion of the hole on V&) as follows: for times when the hole is located close to the Franck-Condon window for laser pulse 2, the overall ion signal will be small; else it will be large. Increasing values of A? will therefore yield an oscillatory overall ion signal reflecting the oscillatory motion of the hole on Vx(q).The analysis of the signal may be complicated by interfering effects of the wavepacket that are initialized on the excited electronic states by laser pulses 1 and 2. The sequence of these laser pulses for hole burning and probing may be called “burn and probe.” It is illustrated by model simulations in Fig. 1 (see pages 197 and 198). Our scheme is an extension of the pioneering work of Kosloff et al. on hole burning in nuclear wavepackets 13-91; see also Ref. 10 (supported by Deutsche Forschungsgemeinschaft). 1. R. de Vivie-Riedle, J. Manz, W. Meyer, B. Reischl, S. Rutz, E. Schreiber, and L. Woste, J. Phys. Chem. 100,7789 (1996). 2. R. de Vivie-Riedle, B. Reischl, S. Rutz, and E. Schreiber, J. Phys. Chem. 99,16829 (1995). 3. B. Hartke, R. Kosloff, and S. Ruhman, Chem. Phys. Lerr. 158,238 (1989). 4. W. T. Pollard. S.-Y. Lee, and R. A. Mathies, J . Chem. Phys. 92,4012 (1990). 5 . R. Kosloff, A. D. Hammerich, and D. J. Tannor, Phys. Rev. Leu. 69, 2172 (1992). 6. T. Baumert, V. Engel, Ch. Meier, and G . Gerber, Chem. Phys. Lett. 200, 488 (1992).

*Comment presented by J. Manz.

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FEMTOCHEMISTRY I1

(a) 0.12

. 5 w‘

g P)

0.03

I

I

-0.07 I

0.15

. B wi

f

0.05

-0.05

5

10 bondlength R (a,)

5

10

bondlength R (aJ

15

Figure la. Burn-and-probe spectroscopy of hole burning in wavepackets: model simulation for K2. with parameters A = 840 (406) nm, I = 2.2 (2.2) GW/cm2, and TWHM = 60 (60)fs for the wavelengths, intensities, and durations of the “burn” (pump) or probe pulses. The top and bottom panels illustrate the wavepacket dynamics induced by burn-and-probe laser pulses with time delays At of 220 and 310 fs, respectively. The snapshots show the absolute values of the wavepacket densities for the X, A, n, and 3 ‘Elelectronic states of K2 and the main contribution from the continuum superimposed on the Kt ion ground state. The potential energy curves are adapted from Ref. I. The “burn” laser pulse burns a hole in the wavepacket of the X state by resonant impulsive stimulated Raman scattering (RISRS) [2-101 via a coherent wavepacket created in the A state [I]; another marginal wavepacket is also born in the II state. Afterwards the wavepacket in the X state moves coherently such that the hole oscillates and provides alternating large and small lobes at shorter and larger bond lengths. Eventually, these lobes enter the Franck-Condon (FC) windows for the resonant X C ion transition. These transitions can be induced by the probe pulse. To illustrate the transition mechanism, the snapshots are taken at the beginning and at the maximum of the probe laser pulse (left and right panels).

-.-

198

GENERAL DISCUSSION

( b) 0.040

0.030

Ec 0.020

0.010

50

150

250

350 delay (fs)

450

550

Figure lb. Shows the resulting ion yield, as measured by the “norm” (i.e., the integrated density) of the wavepacket in the ion ground state, depending on the delay time At of the bum-and-probe laser pulses. The oscillatory pattern of the norm reflects the motion of the hole in the wavepacket of K 2 ( X ) . The beat period of approximately 190 fs corresponds to half of the vibrational period in the X state. Large and small maxima correspond to large and small lobes entering the FC regions, respectively.

7. A. Bartana, R. Kosloff, and D. 1. Tannor, J. Chem. Phys. 99, 196 (1993). 8. U.Banin, A. Bartana, S. Ruhman, and R. Kosloff, J. Chem. Phys. 101,8461 (1994). 9. A. Bartana, U. Banin, S. Ruhman, and R. Kosloff, Chem. Phys. Len 229, 211 ( 1994).

10. D.J. Tannor, R. Kosloff, and S. A. Rice, J. Chem. Phys. 85, 5805 (1986).

T. Kobayashi: I have a comment and a question for Prof. Reming: 1. There are several pathways to reach the fifth-order nonlinearity with kl - 2kz + 2k3.Are there any interference effects observed among these processes? Even with the phase matching to observe the k, 2k2 + 2k3 component there are several paths contributing. 2. In view of the arguments relating to the coherent state picture, is there actually an experiment that shows that the state prepared by a short pulse is really a coherent state with minimum uncertainty of two conjugate observables? In the general case, using Fourier-transform-

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limited pulses, a deformed state with a product of variances of conjugate variables being larger than that of a vacuum state will result. To verify the state to be a coherent state, the uncertainties AX+,AX- must be those of the vacuum state, as shown in Fig. 1. bisector

Figure 1. The vacuum or coherent states are denoted by V, C, the n-photon or thermalized states by N. T ; the asymmetric states by A; the squeezed states by S, PS.

G. R. Fleming: Regarding Prof. Kobayashi’s first question, I believe the interferences are correctly accounted for in Prof. Mukamel’s formalism.

V. S. Letokhov: Regarding the issue of coherence, in quantum optics the term coherent state of laser light means the particular combination of “Fock” states (quantum state with given occupation number of photons (n)) with minimum uncertainty. In coherently driven photochemistry we are assuming that any combination of quantum eigenstates with synchronized phases of the wave functions that form the wavepacket means coherence. Perhaps we should determine the degree of coherence, following quantum optics terminology. B. A. Hess: The application of the Redfield equation requires that the time scales involved are well separated. Is this the case in the example shown by Prof. Fleming? G. R. Fleming: Using a Markovian approximation for the bath is always likely to be a very poor approximation for electronic dephasing. For vibrational dynamics, while the issue needs careful examination,

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it may often be possible to use the Markov approximation for the bath as we did in the Redfield calculations.

S. Mukamel: In order to represent situations in which nuclear and electronic dynamics take place on the same time scale, one needs to incorporate nuclear degrees of freedom into the description. A frequency-dependent Redfield superoperator can capture some effects, but in general is very limited and may even yield negative probabilities. A method for decomposing a given spectral density into a few collective coordinates and identifying these coordinates was presented in Ref. 1. 1. V. Chernyak and S. Mukamel, J. Chem. Phys. 104,444 (1996). A. H. Zewail: Prof. Manz, the experiments are now into the realm of complex systems. How many degrees of freedom can theory now treat to compare with experiment?

J. Manz: The state of the art in the quantum dynamical simulations of molecular dynamics is, to the best of our knowledge, as follows: (i) One-, two-, and even three-dimensional (Id, 2d, 3d) model calculations using brute-force fast Fourier transform (FFT) propagation techniques for time-dependent wavepackets are by now standard. The earliest applications have been carried out by R. Kosloff and D. Kosloff for bimolecular model systems [I]; by M. D. Feit, J. A. Fleck et al. [2], as well as by R. H. Bisseling, R. Kosloff et al. [3] for unimolecular model systems; by T. Joseph and J. Manz [4] for molecular model systems in laser fields, including laser control of decay rates; by D. J. Tannor, R. Kosloff, and S. A. Rice I S ] for the laser control of the branching ratio of competing chemical products; and by B. Reischl, R. de Vivie-Ride et al. [6] for the first 3d ab initio simulation of a femtosecond pump and probe experiment. (ii) The first 4d simulations have been published by U. Nielson, S. Holloway, J. K. Ngrskov et al. [7] and by A. E. Janza, W. Karrlein, and J. Manz [8] for simulations of molecule-surface reactions and for unimolecular reactions, respectively. (iii) The present “world record,” that is, numerically converged 5d and even 6d simulations, is held by L. S. Cederbaum, H.-D. Meyer, U. Manthe et al. [9] using the multiconfiguration time-dependent Hartree (MCTDH) approach. (iv) There are fundamental numerical reasons for assuming that these types of brute-force FFT propagations [l-81 cannot be extended

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20 1

to model systems with more than eight or perhaps nine dimensions, corresponding to the vibrational degrees of freedom of a 5-atomic molecule [lo]. The same limit applies to other methods, for example, expansions in terms of zero-order eigenstates [ 111. (v) In order to go beyond the “6d-8d brute-force FFT-limit,” one has to employ numerical approximations and/or restrict applications to specific simple systems, for example, harmonic oscillators with bilinear couplings. Promising developments with applications to dozens or even several hundreds of degrees of freedom (d.0.f.) include modem path-integral techniques developed by W. Domcke et al. [12] and time-dependent self-consistent-field (TDSCF) methods pioneered by H.Metiu et al. (131 and by R. B. Gerber et al. [14]. Recently, these techniques have even been extended to TDSCF-CI methods, including time-dependent configuration interactions (CI); see also the new coupled classically based separable-potential (CSP) method of R. B. Gerber and P. Jungwirth [15], the semiclassical approach of G. Stock [16], and the quantum Monte Car10 method as developed by M. Suhm [17]. These new developments [12-171 are fundamental and promising, that is, they open the way to model simulations of large quantum systems. The numerical approximations involved may, however, restrict these applications to specific situations, for example, systems with few reactive d.0.f. which are weakly coupled to many harmonic or slightly anharmonic vibrations [121 or weak couplings between many d.0.f. [13-151 during short times (<1 ps or so), excluding interference and other effects at longer times. For reviews, see Ref. 18. 1. R. Kosloff and D. Kosloff, J. Chem. fhys. 79, 1823 (1983). 2. M. D. Feit, J. A. Fleck, Jr., and A. Steiger. J. Comput. Phys. 47,412 (1982). 3. R. H. Bisseling, R. Kosloff, J. Manz, and H. H. R. Schor, Be,: Bunsenges. fhys. Chem. 89,270 (1985). 4. T. Joseph and J. Manz, Mofec. Phys. 58, 1149 (1986). 5. D. J. Tannor, R. Kosloff, and S. A. Rice, J. Chem. Phys. 85,5805 (1986). 6. B. Reischl, Chem. Phys. Lett. 239, 173 (1995); R. de Vivie-Riedle, J. Gaus, V. BonJif-Kouteck9, J. Manz, B. Reischl, S. Rutz, E. Schreiber, and L. Woste, in Femtosecond Chemistry and Physics of Ultrafast Processes, M. Chergui, Ed., World Scientific, Singapore, 1996, B. Reischl, R. de Vivie-Riedle, S. Rutz, and E. Schreiber, J. Chem. Phys. 104, 8857 (1996). 7. U. Nielsen, D. Halstead, S. Holloway, and J. K. Nerskov, J. Chem. Phys. 93,2879 ( 1990). 8. A. E. Janza, W. Karrlein, and J. Manz, Faraday Discuss. Chem. Soc. 91, 134 (1991). 9. H.-D. Meyer. U. Manthe, and L. S. Cederbaum, Chem. Phys. Lett. 165,73 (1990); U. Manthe, H.-D. Meyer, and L. S. Cederbaum, J. Chem. fhys. 97, 3199, 9062 (1992). For recent applications to even larger albeit special systems, e.g., four

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modes coupled weakly to a 20-mode bath, see G. A. Worth, H.-D. Meyer, and L. S. Cederbaum, J. Chem. fhys., in press. 10. R. H. Bisseling, Ph.D. Thesis, The Hebrew University, Jerusalem, 1986. 11. M. J. Bramley and T. Carrington, J . Chem. Phys. 99,8519 (1993); M. J. Bramley, J. W. Tromp, T. Carrington, and G. C. Corey, J. Chem. fhys. 100,6175 (1994). 12. S. Krempl, M. Winterstetter, H.Plohn, and W. Domcke, J. Chem. fhys. 100,926 (1994). 13. N. P. Blake and H. Metiu, in Femtosecond Chemistry, J. Manz and L. Woste, Eds., Verlag Chemie, Weinheim, 1995, p. 533. 14. R. B. Gerber, A. B. McKoy, and A. GarciCVela, in Femtosecond Chemisrry, J. MMZ and L. Woste, a s . , Verlag Chemie, Weinheim, 1995, p. 499. 15. P. Jungwirth and R. B. Gerber, J. Chem. Phys. 102, 6046 (1995). 16. G. Stock, J. Chem. Phys. 103, 1561, 2888 (1995). 17. M. A. Suhm, Chem. fhys. Lett. 214,273 (1993); Ber: Bunsenges. Phys. Chem. 99, 1159 (1995). 18. R. Kosloff, J . fhys. Chem. 92,2087 (1988); V.Mohan and N. Sathyamurthy, Comput. Phys. Rep. 7,213 (1988); R. Kosloff, Ann. Rev. Phys. Chem. 45, 145 (1994); N. Balakrishnan, C. Kalyanaraman, and N. Sathyamurthy, fhys. Rep. (1996). in press.

M. Quack: Prof. Manz, when I saw the wonderful 3-d quantum mechanical simulation of vibrational motion, I wondered to what extent you have included a realistic simulation of the defection process used in the experiments. Could you comment on this further (how the ionization is treated and how many electronic states were included)? This question is actually the same as the one addressed this morning to Zewail and Gerber. R. de Vivie-Riedle and J. Manz:* The Id ab initio simulations by R. de Vivie-Riedle, J. Manz, and B. Reischl of the pump and probe investigations of the rnultiphoton ionization of K2 [ l ] employ five quantum ab initio potential energy surfaces for the X 'E;, A 'Ei,4 'Ce, 2 I l l g states of K2 plus the electronic ground state (X) of KZ, together with the ab initio dipole coupling between the neutral states. These results have been evaluated by W. Meyer [l]. In addition, we assume the Condon approximation for the dipole coupling between states K2(2 'IIg) and Ki(X). On this basis, we solve the time-dependent Schriidinger equation for the nuclear wavepackets on all (five) electronic states with semiclassical dipole coupling to the laser fields of the pump and probe laser pulses using the experimental laser pararneters 11, 21. The electronic continuum of the photoionized products *Comment presented by J. Manz.

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K;(X) + e is discretized, as suggested in Ref. 3; careful numerical tests show that the experimental results [l, 21 are well reproduced by taking into account just one dominant continuum state, in accord with the Franckxondon principle. The 3d ab initio simulations [4] for Na3 are based, in a similar way, on three ab initio potential-energy surfaces for Na3(X), Na3(B), and Na;(X), with 3d ab initio dipole coupling between Nas(X) and Na3(B) evaluated by V. Bona%-Kouteck$ et al. [5] plus Condon-type coupling between Na@) and Na;(X). Additional potential-energy surfaces interfere at the conical intersections of the pseudo-Jahn-Teller distorted Na3(B) state (see Ref. 6), but we have tested carefully [4] that these interferences are negligible in the frequency domains of the experimental femtosecond/picosecond laser pulse experiments [7]as well as in the continuous-wave experiments [8]. These 3d ab initio simulations turn out to be extremely demanding; for example, it took us about 300 h of CRAY-YMP computing time to produce the theoretical pump and probe spectra for a laser pulse with duration t p = 120 fs and delay times td < 3 ps [4]. For longer pulse durations tp = 1.5 ps, these simulations would have been prohibitively expensive. Therefore we have simulated the experimental pump and probe spectra by a less demanding approximation;that is, the signal is proportional to the overlap of the time-dependent 3d nuclear wavepacket generated by the pump pulse and propagated on the 3d ab initio potential-energy surface of Na3(B). As for the Franck-Condon window for the Na3(B) ---c Na3(X) transition, which is defined by the probe pulse [4], compare with Ref. 9 and with the preceding comment by R. de Vivie-Riedle, J. Manz, B. Reischl, and L. Woste. I. R. de Vivie-Riedle, J. Manz, W. Meyer, B. Reischl, S. Rutz, E. Schreiber, and L. Woste, J. Chem. Phys. 100, 7789 (1996). 2. S. Rutz, E. Schreiber, and L. Woste, Surf: Rev. Lett., in press; E. Schreiber, S . Rutz, and L. Woste, in Fast Elementary Processes in Chemical and Biological Sysferns, Proceedings, C. Troyanowsky, Ed., American Institute of Physics, New York, 1995. 3. R. de. Vivie-Riedle, B. Reischl, S . Rutz, and E. Schreiber, J. Phys. Chem. 99, 16829 (1995). 4. B. Reischl, Chem. Phys. Leu. 239, 173 (1995); R. de Vivie-Riedle, J. Gaus, V. BonaEif-Kouteckf, J. Manz, B. Reischl, S. Rutz, E. Schreiber, and L. Woste, in Femtosecond Chemistry and Physics of Ultrafast Processes, M. Chergui, Ed.,World Scientific, Singapore, 1996; B. Reischl, R. de Vivie-Riedle, S. Rutz, E. Schreiber, J. Chem. Phys. 104, 8857 (1996). 5. R. de Vivie-Riedle, J. Gaus, V. BonaWKouteckf, J. Manz, B. Reischl-Lenz, and P. Saalfrank, Chem. Phys., submitted; J. Gaus, K. Kobe, V. Bonazif-Kouteckf, H. Kiihling, J. Manz, B. Reischl, S . Rutz, E. Schreiber, and L. Woste, J . Phys. Chem. 97, 12509 (1993).

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6. J. Schon and H. Koppel, Chem. Phys. Leu. 231,55 (1994); J. Schon and H. Koppel, J. Chem. Phys., in press. 7. E. Schreiber, H. Kuhhng, K. Kobe,S. Rum, and L. Woste. Ber. Bunsenges. Phys. Chem. 96, 1301 (1992); K. Kobe, H. Kuhling, S. Rutz, E. Schreiber, J.-P. Wolf, L. Woste, M. Broyer, and Ph. Dugourd, Chem. Phys. Lett. 213, 554 (1993). 8. G. Delacrhz, E. R. Grant, R. L. Whetten, L. Woste, and J. Zwanziger, Phys. Rev. Lerr. 56, 2598 (1986). 9. R. Bersohn and A. H. Zewail, Ber: Bunsenges. Phys. Chem. 92,373 (1988).

G. R. Fleming: With regard to the question of Prof. Quack, let me add that Jeff Cina has incorporated a quantum detector into his model of fluorescence up-conversion experiments [11. 1 . A. Matro and J. Cina, Adv. Chem. Phys., in press.

D. J. Tannor: To understand the role of dissipation in quantum mechanics, it is useful to consider the density operator in the Wigner phase-space representation. Energy relaxation in a harmonic oscillator looks as shown in Fig. 1, whereas phase relaxation looks as shown in Fig. 2; that is, in pure dephasing the density spreads out over the energy shell (i.e., spreads in angle) while not changing its radial distribution

PI.

Figure 1.

Figure 2.

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205

Recently, in trying to understand certain features of Redfield theory, we have transformed the solution of the Redfield equations from the (E, E’) representation to the Wigner (p, g) representation. For an initial Gaussian in (p, g) with no pure dephasing processes, the distribution remains Gaussian for all time. The first moments obey the same equations as those of the distribution in the classical Fokker-Planck equation, whereas the second moments have one small difference from the classical equations that ensures that the Heisenberg uncertainty principle is satisfied. These equations for the moments correspond to a special case of the equations in the Gaussian-Wigner ansatz of Yan and Mukamel. Since the first moments are the same as the classical ones, p and q are not symmetrically related: The frictional force opposes p but is independent of g (see Fig. 3). P4

energy dissipation slower

‘c\ energy dissipation faster

Figure 3.

Figure 4.

Defining 8 = tan-’p/q and w ( t ) = de/dr [w(t) is periodic but time dependent], the frequency of spiraling in phase space is modulated by the frictional force. Moreover, the rate of energy dissipation,

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GENERAL DISCUSSION

l/Tl = ( l / E ) d E / d t , also depends on time and is maximal when Ip1 is maximal and minimal when IpI is minimal. In many treatments of the quantum damped oscillator (e.g., with the semigroup formalism or within the ubiquitous rotating-wave approximation) p and q are treated symmetrically and there is no modulation of w or l/T1. Two other effects can be seen immediately from the phase-space representation. The Bloch approximation Prof. Fleming mentioned, which assumes no coherence transfer during the relaxation process, corresponds to a picture similar to that of the pure dephasing situation (see Fig. 4); that is, as soon as energy relaxation begins, the coherence between vibrational levels (which is the source of the angular localization in phase space) cannot follow and complete angular delocalization ensues. A second effect that can be seen easily in the phase-space representation is that, in the exact solution, when energy relaxation is complete, there must also be complete angular delocalization. Disappearance of angular localization must accompany energy loss. This is a pictorial representation of the statement that T2 5 2T, (see Fig. 5). 1. D. J. Tannor, unpublished (1984); D. Kohen, Phase-Space Distribution Function Approach to Molecular Dynamics in Solutions, Ph.D. Thesis, University of Notre Dame, 1995.

Figure 5.

T. Kobayashi: Let me add the following to the comment by Prof. Tannor: 1. In order to discuss vibrational coherence transfer, it is necessary to use two coordinates and two momenta, as shown in Fig. 1. 2. A two-pendulum model with obscured (uncertain in t e r m s of quantum mechanical uncertainty) position can explain the maintenance of the phase relationship between pendulum 1 and pendulum 2 (Fig. 2) (see Figures 1 and 2 on page 207).

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"A" Figure 1.

Figure 2.

Bi

is the phase in the phase space ( p i . 4;)for i = 1,2.

potential curve 2

9 Figure 3.

3. In general, coherence destruction is due to elastic collisions, depopulation, or scattering by phonons. However, the dephasing in the context of wavepackets is not due to such collision, scattering, or depopulation processes but can be explained as follows. Transfer of a part of the wavepacket takes place at different positions near the crossing point of two potential curves 1 and 2 (Fig. 3). Since the speed of the

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GENERAL DISCUSSION

wavepacket along the surfaces of potential curves 1 and 2 is different, the wavepacket spreads. However, if the period of time of crossing is short enough, the spread will be negligibly small and the wavepacket starts to oscillate with the frequency w2 of the potential curve 2. M. S. Child: Following the comment by Prof. Kobayashi, let me further remark that the proper phase portrait for the perturbation model illustrated in Fig. 1 is shown in Fig. 2. Thus a wavepacket initiated in well A passes to well B by a curve crossing. Prof. Fleming showed an interesting case of persistent coherence in such a situation, despite the erratic pattern of the eigenvalue separations. An alternative, possibly more revealing approach, is to employ Stuckelberg-Landau-Zener theory, which relates the interference (i.e.. coherence) in the two different wells via the area shown in Fig. 2.A variety of applications to time-independent problems may be found in the literature [l]. 1. M. S . Child, Semiclassical Mechanics wirh Molecular Application, Oxford University Press, New York, 1991, Chapter 3.

I

Figure 1.

r

0 A

I

r

Figure 2.

B.Kobler: Prof. Fleming, your experimentalresults clearly indicate that in the case of I2 in hexane the vibrational coherence of an initially prepared wavepacket persists for unexpectedly long times. However, quantum dynamical calculations show that wavepacket spreading due to anharmonicity can be very substantial even for isolated molecules

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on the time scale of a single vibrational cycle. How sensitive are your experiments to dispersive spreading in the wavepacket?

G. R. Fleming: Sure the wavepacket spreads, but not as much as you would think on this time scale. What can be done experimentally to get precise data is to do wavelet analysis to see what shape it had. That is a realistic goal for a simple system using solid-state lasers. A. H. Zewail: If the curve-crossing problem of 12 in hexane is similar to NaI, then it is not surprising that the wavepacket survives the crossing for up to -1 ps; in NaI it is up to -10 ps. If the interaction with the solvent is weak, coherence persists, as is certainly the case for rhodopsin and others.

J.-L. Martin: Prof. Fleming, you pointed out the importance of having ballistic motions along the reaction coordinate: “If the motion was not ballistic through the transition region, the quantum yield could be much less ..,.” Along this line, what would be the influence of “ballistic” motions (versus stochastic motions) along coordinates orthogonal to the reaction coordinate? Do you agree with the idea that, in principle, the dynamical behavior of the system along the orthogonal coordinates should also have an influence on the reaction rate and efficiency? G. R. Fleming: Yes, the role of orthogonal coordinates could be significant. I believe that Prof. J. Jean (Washington University, St. Louis) is beginning to address this issue via Redfield theory.

P.Gaspard: Concerning multiple-pulse echo experiments, I would like to know if there are results on the decay of the amplitude of the echo as the number of pulses increases with equal-time spacing between pulses. If the decay is exponential, the rate of decay may characterize dynamical randomness since it is closely related to the socalled Kolmogorov-Sinai entropy per unit time [see P. Gaspard, Prog. Theol: Phys. Suppl. 116, 369 (1994)l. S. Mukamel: In general, multipulse experiments depend on a multitime correlation function of the dipole operator [l]. The term x(”) depends on a combination of n + 1 time correlation functions. Their behavior for large n will depend on the model. In some cases (e.g., the accumulated photon echo used by Wiersma) the multiple-pulse sequence is simply used to accumulate a large signal and the higher

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GENERAL DISCUSSION

order correlation functions factorize into a product of lower ones. In other situations, we obtain a multidimensional spectroscopy and the amount of microscopic information grows with the number of pulses

t11.

1. S.Mukarnel, Principles ofNonlinear Optical Spectroscopy, Oxford University Press, New York, 1995.

LASER CONTROL OF CHEMICAL REACTIONS

PERSPECTIVES ON THE CONTROL OF QUANTUM MANY-BODY DYNAMICS: APPLICATION TO CHEMICAL REACTIONS S. A. RICE Department of Chemistry and The James Franck Institute The University of Chicago Chicago, Illinois

CONTENTS I. Introduction 11. General Considerations 111. The Brurner-Shapiro Method IV. The Tannor-Rice-Kosloff-Rabitz Method V. Generic Conditions for Control of Quantum Dynamics VI. How Much Control of Quantum Many-Body Dynamics Is Attainable? VII. Reduced Space Analyses of the Control of Quantum Dynamics A. Reduced Representation in State Space B. Reduced Representation in Coordinate Space C. Reduction by Factorization: Time-Dependent Hartree Approximation VIII. The Control of Dynamics-Inverse Scattering Duality IX. Conclusions References

I. INTRODUCTION One measure of scientific and technological progress is the succession of advances in our ability to control the outcome, and in some cases the paths, of processes of interest. In this record one can find, at the macroscopic level, both the application of mechanics to engineering design and the second law Advances in Chemical Physics, Volume 101: Chemical Reactions and Their Control on the Femtosecond Time Scale, XXrh Solvay Conference on Chemistry, FAited by Pierre Gaspard, Irene Burghardt, I. Prigogine, and Stuart A. Rice. ISBN 0-471-18048-3 0 1997 John Wiley & Sons, Inc.

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of thermodynamics. Taken together, the former provides the understanding needed to design the apparatus that can transform energy changes into various kinds of work, and the latter defines the limits to the transformation of energy changes into work that an apparatus can achieve. The record also contains examples of control of microscopic dynamics, for which quantum mechanics is the relevant theory. For example, methods have been developed for the precise control of the orbits of elementary particles in accelerators and in electron microscopes and for the precise control of excitation of nuclear spins in magnetic resonance imaging. However, it is only in the last few years that it has become apparent that it is possible to think about developing generic tools to control the quantum dynamics of microscopic many-body systems. This chapter provides an overview of contemporary understanding of the opportunities for control of the dynamical evolution of a quantum many-body system, with emphasis on using that control to influence the selectivity of product formation in a chemical reaction. Finding ways to control the selection of products of a chemical reaction is, arguably, the essence of chemistry. The intensive studies of synthetic methodology carried out over the past two centuries have led to the development of numerous methods for generating desired chemical species. Typically, chemical synthesis is carried out in liquid media, and the overwhelming majority of the methods used rely on amplifying the yield of the desired product by adjusting the equilibrium between reactants that do and do not form the desired species so as to favor the former, or by adjusting the rates of competing reactions that form different species from the same reactant so as to enhance the formation of the desired species, or by combinations of these methods. All of the extant methods are fundamentally macroscopic in the sense that they depend on the statistical, incoherent properties of a many-molecule system, for example, collisions between reactant molecules and between reactant and solvent molecules. In contrast, this chapter discusses the influence on product selectivity generated by active control of the molecular dynamics and, more generally, the active control of the temporal evolution of complex molecular systems. Much of the study of active control of quantum molecular dynamics has been stimulated by advances in laser technology, in molecular spectroscopy and in our understanding of molecular dynamics. The developments in laser technology we refer to include methods for the generation of very short pulses of light, of shaped pulses, of pulses with a well-defined phase relationship, of very pure monochromatic light fields, and of very high intensity light fields. The application of these and other laser technologies to molecular spectroscopy has yielded both a wealth of information about the properties of molecular potential energy surfaces and an increased awareness that interference effects can be used to guide system evolution. Simply put, it

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215

is now recognized that the dynamics of a strongly coupled light-matter system can be influenced by alteration of the temporal and spectral distributions of the radiation coupled to the system. The underlying principle of the new approach to controlling product selectivity in a reaction is different from that used in earlier attempts to achieve “bond selective chemistry.” The initial attempts to control product selectivity in a photoinduced chemical reaction were based on one or the other of two characteristics of laser light sources, namely, (i) the capability to produce light fields with very small spectral bandwidth and (ii) the capability to produce light fields with very short duration and very high intensity. In the former case the underlying concept is that excitation of different rovibrational modes of a reactant molecule will, if the energy remains localized for a long time relative to the time required for reaction, generate different products. In the latter case the underlying concept is that excitation of a particular bond, which is achieved by excitation of a high overtone of a localized molecular vibration, will increase the rate of bond breakage sufficiently to exceed the rate of energy transfer from the excited bond to the rest of the molecule. With the exception of a few very special cases, both methods fail. The source of this failure is simply stated: The rate of intramolecular energy transfer is large enough to invalidate the assumption on which the method is based. At a deeper level, the source of this failure can be traced to the choice of an inappropriate representation of the relationship between reactants and products. That is, the specificationthat excitation of a particular bond will lead to breaking of that bond, without reference to the influence of and the effects on the remainder of the molecule, does not properly account for the properties of the molecular Hamiltonian and the potential-energy surface, and hence does not adequately define the pathway that connects reactants and products. The new approach to controlling the selectivity of product formation in a chemical reaction makes full utilization of the relevant properties of the molecular Hamiltonian and the potential energy surface, and it does not presuppose a particular reaction pathway specified by a particular degree of freedom of the molecule. Rather, the new approach is based on exploitation of quantum interference effects [ 1-1 13. Although the several original proposals for using quantum mechanical interference to control product selectivity in a chemical reaction at first appeared to be quite different, it is now apparent that they merely emphasized different aspects of a general methodology. This chapter will describe the basic principles of each of the methods proposed, comment on the current state of theory and experiment, and sketch the problems that must be resolved to further develop both our understanding of the control of many-body quantum dynamics and its application to product enhancement in chemical reactions.

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11. GENERAL CONSIDERATIONS

We begin with a broad-stroke description of the different strategies for using quantum mechanical interference to control product selectivity in a chemical reaction. Consider a branching chemical reaction, that is, one in which the excited reactant molecule can form at least two distinct product species. Brumer and Shapiro [l-31 observed that if there are two independent excitation routes between a specified initial state of the reactant and a specified initial state of the products, then two monochromatic coherent excitation sources can be used to influence the relative concentrations of the products formed. Control of the ratio of product concentration is possible because quantum theory requires that the probability of forming a specified product is proportional to the square of the sum of the transition amplitudes for the two pathways from the initial state to that product; because the amplitudes can have different signs, the magnitude of that probability is determined by the extent of their interference. For example, when one- and three-photon transitions generate the independent pathways between the initial and final states, the extent of interference can be controlled by altering the relative phase of the two excitation sources. The situation is analogous to the formation of a diffraction pattern in a two-slit experiment in that the excited-state amplitude in each molecule is the sum of the excitation amplitudes generated by two routes that are not distinguished from each other by measurement. Using this method, Gordon and co-workers [ 12-16] have reported control of the population of an excited-state level in HCl, control of the photofragmentation yield of CHJI CH3 + I*, and control of the ratio of concentrations of the products in the branching ionization and photodissociation reactions HI -+ HI+ + e versus HI + H + I. These results provide an experimental confirmation of the Brumer-Shapiro control scheme. The selectivity of product formation in a chemical reaction can also be influenced via interference in the time domain. Consider the case when only two electronic potential-energy surfaces are involved. In the simplest realization of the Tannor-Rice control scheme [4, 51, attention is focused on amplitude control by altering the separation in time of “pump” and “dump” pulses. In this simplest scheme an incident (first) pulse of light transfers probability amplitude from the electronic ground state to the excited state, creating a wavepacket on the excited-state potential-energy surface. In general, the excited state of the molecule has slightly different bond lengths and bond angles, so the wavepacket created on the excited-state potential-energy surface cannot be stationary with respect to the excited-state Hamiltonian. Necessarily, then, the wavepacket on the excited-state potential-energy surface evolves by translation on that surface and by dephasing of the compo-

-

CONTROL OF QUANTUM MANY-BODY DYNAMICS

217

nents of the wavepacket. A second pulse of light, incident after an interval t , will, depending on the position and momentum of the wavepacket, dump some of the amplitude from the excited-state potential-energy surface into a selected reaction channel on the ground-state potential-energy surface. The idea is to dump the amplitude beyond any barrier obstructing the exit channel on the ground-state potential-energy surface. To second order in perturbation theory the transfer of amplitude from the excited state to the ground state is not sensitive to whether that amplitude is in phase or out of phase with the preexisting amplitude. This control scheme has been demonstrated by Gerber and co-workers [17] with respect to the competition between ionization NaZ+ + e versus Na2 and dissociative ionization of Naz, namely Na2 4 Na' + Na + e, by varying the time delay between the first and second pulses. This result is an experimental confirmation of the simplest version of the Tannor-Rice control scheme. A more sophisticated version of the Tannor-Rice scheme exploits both amplitude and phase control by pumpdump pulse separation. In this case the second pulse of the sequence, whose phase is locked to that of the first one, creates amplitude in the excited electronic state that is in superposition with the initial, propagated amplitude. The intramolecular superposition of amplitudes is subject to interference; whether the interference is constructive or destructive, giving rise to larger or smaller excited-state population for a given delay between pulses, depends on the optical phase difference between the two pulses and on the detailed nature of the evolution of the initial amplitude. Just as for the Brumer-Shapiro scheme, the situation described is analogous to a two-slit experiment. This more sophisticated Tannor-Rice method has been used by Scherer et al. [181 to control the population of a level of 12. The success of this experiment confirms that it is possible to control population flow with interference that is local in time. In principle, the methods available for guiding the evolution of a quantum system by coupling it to an external field are not restricted to the use of a time-independent field or a simple pulse sequence. If the goal to be achieved is, say, maximization of the amount of a product in a reaction, the design of the external field that accomplishes the goal is an inverse problem: Given the target product and the quantum mechanical equations of motion, calculate the guiding field required. The solution to this inverse problem is very likely not unique, which for the case under consideration is a strength since it is then plausible that one of the possible guide fields is more easily generated than others. The methodology used in calculations of the field required to maximize a particular product yield is optimal control theory. The first application of a full version of optimal control theory to the quantum dynamics of molecular

-

218

S. A. RICE

systems was published by Pierce et al. [8]; Tannor and Rice [4] had earlier formulated the search for an optimal dump pulse, for fixed pump pulse shape, within the framework of perturbation theory, as a problem in the calculus of variations. Later, Kosloff et al. [6] also introduced optimal control theory to design the field which guides a branching reaction to generate maximum product of a particular species. In the model problems studied to date it is predicted that the use of an optimal guide field can increase the desired product yield by many orders of magnitude relative to the yield from a two-pulse control field [19]. It is usually found that the optimal guide field has a complicated spectral and temporal structure whose efficiency is determined by the extent of interference between the amplitudes associated with its different spectral and temporal components. The initial formulation of the theory of control of quantum many-body dynamics has been extended to include the following: A formulation of the conditions that define control of dynamical processes, for example, population transfer between two potential-energy surfaces, in generic terms, without specific reference to the particular properties of the molecule [20] A procedure that inverts the control field for a given potential-energy surface to learn about the potential-energy surface [21-231 An approximate but accurate reduced representation of the n-body dynamics, so that the control process can be defined in a subspace of rn c n degrees of freedom [20, 24, 251 Despite these extensions, the formal theory of control of quantum manybody dynamics does not yet directly address a number of important issues. Amongst these are the following: What are the limitations to the control of quantum many-body dynamical processes? How does the efficiency of a control process depend on the number of degrees of freedom of the controlled molecule? What is the maximum efficiency of field-induced control of molecular dynamics; that is, is there an analogue of the second law of thermodynamics that specifies the maximum efficiency for a process in terms of properties of the system? How sensitive is the computed control field to fluctuations in the field source and to uncertainties in the molecular potential-energy surface? Does the presence of a thermal distribution of initial states of the reactant molecule seriously compromise the extent of possible control of the molecular dynamics?

CONTROL OF QUANTUM MANY-BODY DYNAMICS

219

Can one systematically incorporate the constraints on experimental realization of a control scheme and a feedback mechanism into the theory? We now turn to a more detailed analysis of the theory of control of quantum many-body dynamics, focusing attention on the particular case of control of product formation in a photoinduced unimolecular reaction.

III. THE BRUMERSHAPIRO METHOD Brumer and Shapiro suggested, in 1986, that control of the branching into different products in a chemical reaction could be influenced by taking advantage of three fundamental characteristics of the quantum mechanical description of the reactant-product system [ 11. Although these characteristics will appear to be “trivially obvious,” taking account of the consequences they imply is a profound observation. To be definite, we consider the case that the products of the reaction are fragments of the reactant molecule. The first characteristic feature we cite is that the eigenstate solutions to the Schradinger equation, as a function of energy, provide a complete description of a system. Then any solution to the time-dependent Schroedingerequation can be represented as a superposition of the time-independent eigenstate solutions, each associated with the time dependence exp(- iEr/h). Second, despite the degeneracy of the eigenstates in the continuum representing the reaction products, it is possible to uniquely comelate specific product state wave functions with total system eigenfunctions; this correlation can be specified by a boundary condition on the total system wave functions. Third, if one can find two independent pathways that connect the same initial and final states of the system, then one can modulate the probability of formation of a specific final state because that probability is proportional to the square of the sum of the amplitudes associated with the individual transitions from the initial to the final states. Note that this representation of the probability of forming a specific product molecule from a particular initial state of the reactant molecule bypasses any necessity to discuss the intermediate dynamics. Put another way, when looked as indicated above, there is no need to consider competition between processes such as intramolecular energy transfer and bond breaking. Brumer and Shapiro have examined several scenarios for exploiting the consequences of the characteristic features of a reacting system described above [26]. We shall consider only one of these, namely, exploitation of interference between one- and three-photon pathways between a specified initial bound state of a molecule and specified product states in the continuum, siFce thaLis the case that has been subject to experimental verification. Let H, and H e be the Born-Oppenheimer Hamiltonians for the ground and

220

S. A. RICE

excited states of a molecule, rppectively. The initial state of the molecule is taken to be an eigenstate of Hg with energy Ei, and the final state is in the continuum with energy E. The continuum eigenstate with energy E is defined with incoming boundary conditions and is represented by the ketlE, n,q-), where n denotes all of the quantum numbers of the system other than the energy and q- denotes the particular product channel with the superscript identifying the incoming boundary conditions. Suppose the molecule, in its initial state, is irradiated with two electromagnetic fields: E(t)=El C O S ( U I I + ~*~R + O l ) + E 3 ~ 0 ~ ( w 3 t +*R+O3) k3

(3.1)

We consider the case when w3 = 3wl. In (3.1) E, = lEa1Za,a = 1, 3 with lEal the magnitude and z, the polarization of the incident field. Let k3 = 3k1, corresponding to parallel incident fields. Given the initial state of the molecule, the probability of forming a product molecule in exit channel q with energy E is

with W,(E, q; Ei), cx = 1, 3, the probabilities of fragmentation arising from one- and three-photon absorption, respectively, and W13(E, q; Ei) describes the interference between the one- and three-photon fragmentation pathways. In the weak-field limit it can be shown that

Provided that the fragmentation reaction requires more than two photons of energy Aw 1, Wl (E, q;Ei) can be shown to have the form

with

and the interference term

W13

can be shown to be

CONTROL OF QUANTUM MANY-BODY DYNAMICS

where the phase

S$) and the amplitude

22 I

are defined by

Using (3.3)-(3.7) the branching ratio for formation of products in exit channels q and q’ takes the form

R,,) =

W(E,4; Ei) W(E,q’; Ei)

(3.8) where

and similarly for q and 9’. Equation (3.8) is the key relation of the Brumer-Shapiro scheme for controlling the selectivity or product formation in a branching chemical reaction. Note that both the numerator and denominator of (3.8) contain a sum of contributions from the independent one- and three-photon pathways and a contribution from the interference term. Since the latter can be altered by changing the relative phase of the one- and three-photon pathways (03 - 381) and their relative amplitudes, the product ratio Rqqt can be controlled over some range of values. To determine how effective the method described is in controlling the concentrationsof the products of a branching chemical reaction, Brumer and Shapiro have carried out [27] an extensive study of the branching reaction

222

S. A. RICE

The potential-energy curves for IBr are displayed in Fig. 1. The calculation reported by Brumer and Shapiro includes the effects arising from rotational motion of the molecule. Figure 2 shows the results obtained when the projec-

25 oo(

20oo(

-- 1 5 W I

E

0

Y

> 10 oo(

-A

5 OO(

0

0

0.2

1 1 1 1 , 1 0.4

0.6

0.8

R (nm)

Figure 1. The IBr potential energy curve. The arrows depict the one- and three-photon pathways that interfere with and whose phase difference is used to control the product selection. (From Ref. 27.)

223

CONTROL OF QUANTUM MANY-BODY DYNAMICS

)

Figure 2. Contour plot of the percentage of Br*(2Plp) in the photodissociation of IBr from an initial bound state in the vibrationless ground state with j i = 1 and mi = 0. (From Ref. 27.)

tion of the molecular angular momentum along the z axis is fixed, and Fig. 3 shows the results obtained when the projection of the angular momentum along the z axis is randomized for fixed angular momentum. These results clearly demonstrate that, in principle, the ratio of the amounts of products in this branching reaction can be controlled over a considerable range by varying the intensities and relative polarization of the one- and three-photon fragmentation pathways . An example of one photon-three photon continuous-wave (CW) interference control of the product distribution in the branching photofragmentation and photoionization reactions

HI HI

--c --c

HI'+e H+I

has been reported by Zhu et al. [15]. Figure 4 displays the HI and I yields

S. A. RICE

224 36

28

6 rc,

21

I

8

I4

7

Figure 3. Contour plot of the percentage of Br*(2Pl,z) in the photodissociation of IBr from an initial bound state in the vibrationless ground state with j i = 42 and m-averaged. (From Ref. 27.1

(measured by the ion concentrations HI+ and I+) as a function of the phase difference between the three-photon and one-photon fields (proportional to the pressure of hydrogen through which the two beams pass, taking advantage of the difference in refractive indices at the frequencies w3 and w I ). Note that the signals from HI+ and I+ are out of phase (about 150"), which is the signature of the phase-modulated control of the yields of the products in this branching reaction. There are many different variants of the Brumer-Shapiro control scheme; the reader is referred to the original publications for discussions of these variants. Bmmer and Shapiro have also discussed the selection rules relevant to interference control of product selectivity and the influence of sources of incoherence on the quality of interference control achievable [26].

CONTROL OF QUANTUM MANY-BODY DYNAMICS

I'

14

i

O

L

1

2

3

Internuclear Distance (A) (a1 Figure 4. ( a ) Potential-energy diagram for HI, with arrows showing the one- and threephoton paths whose interference is used to control the ratio of products formed in the branching reactions HI I+ + e and HI H + I. (b) Modulation of the HI' and I+ signals as a function of phase difference between the one- and three-photon pathways (proportional to the HZ pressure in the cell used to phase shift the beams). (From Ref. 15.)

-.

-.

226

S. A. RICE

'p .-V

> c

0

I I

0.I25

1

2

3

4

5

6

7

8

H, Pressure (Torr) (b) Figure 4.

(Continued)

IV. THE TANNOR-RICE-KOSLOFT-RABITZ METHOD In 1985 Tannor and Rice [4] showed that it is possible to control the relative yields of products in a branching chemical reaction by varying the interval between an initial pump pulse of radiation that transfers amplitude from the ground-state potential-energy surface to an excited-state potential-energy surface and a later dump pulse of radiation that transfers amplitude in the opposite direction. They represented the search for the optimal dump pulse shape for a given pump pulse shape and total number of photons as a problem in the calculus of variations. The pump-dump scheme was first formulated using second-order perturbation theory to describe the time evolution of the wavepacket amplitude on the ground and excited potential-energy surfaces. This formulation leads to a nonlinear integral equation for the optimal pulse shape. The utility and power of optimal control theory for the calculation of the temporal shape and spectral content of the pulse that maximizes the yield of the specified product were first recognized by Rabitz and coworkers [7, 81 and, independently and a little later, by Kosloff et al. 161.

CONTROL OF QUANTUM MANY-BODY DYNAMICS

227

Reformulations and/or extensions of this work have been reported by several groups. Because of the transparency of interpretation, it is convenient to start with the original Tannor-Rice perturbation theory treatment of control of molecular dynamics. We consider a molecule that can undergo fragmentation to produce two products: ABC ABC

-+

-+

BC+A AB+C

For simplicity, the fragmentation is assumed to occur on the ground-state potential-energy surface. If the bond lengths and bond angles of the molecule are different in an excited state than in the ground state, the potential-energy surface for that excited state will be translated and rotated relative to the ground-state potential-energy surface. Consequently, amplitude placed on the excited-state potential-energy surface by a photoinduced Franck-Condon transition will not be stationary; it is the evolution of the amplitude with increasing time that is used to alter the probability of formation of the two products of the reaction. The Hamiltonian for the system we are considering can be represented as a 2 x 2 matrix of operators:

and the time-dependent Schriidinger equation reads

We assume that at t = 0 the system is in the ground state, so

Equation (4.2) can be transformed into two coupled integral equations for the amplitudes on the ground- and excited-state potential-energy surfaces, namely,

228

S . A. RICE

(4.3)

To second order in perturbation theory (the weak-field regime) we have

which contains the field only to second order. Equation (4.4) has the following simple interpretation: The amplitude $g(0)evolves on the ground-state potential-energy surface from t = 0 to t = t 1 . At f 1 the ground-state amplitude is transferred, by a vertical transition, to the excited-state potential-energy surface, on which it propagates until t2. At t 2 the amplitude on the excitedstate surface is transferred, by a vertical transition, back to the ground-state surface, on which it evolves until t. Because the fields E(t1) and E(t2) are distributed in time, we must integrate over 41 the instants at which the up and down transitions can occur. Let FA and Pc be the projection operators onto the exit channels for products BC + A and AB + C, respe5tively. Given $&), the probability for forming product A is limt- & g ( t ) l P A I$&)). We can gain insight into how pulse shape and amplitude propagation are related by consideration of a simple example. Suppose, starting from an initial state with wave function $', we wish to populate a particular discrete state of the molecule, with wave function $f,by a two-photon process. Given the shape of the pump pulse El(t,), what shape must the dump pulse E2(t2) have to maximize the population of the target state? Tannor and Rice showed that for given El(tl)

that is, the optimal dump pulse shape is matched to the convolution of the

CONTROL OF QUANTUM MANY-BODY DYNAMICS

229

excitation pulse shape with the wavepacket propagated on the excited-state potential-energy surface. A similar result is found when the final state is in the continuum, although in that case the formal representation of the dump pulse shape involves a Fredholm integral equation of the second kind. To determine how effective pump-dump pulse separation is as a tool for altering the ratio of concentrations of products in a branching reaction, Tannor et al. [5] studied a model system with a ground-state potential-energy HH + D and surface resembling that for the branching reactions HHD HHD DH + H. The model calculations did not include the effects of molecular rotation. We show in Fig. 5 the results of that calculation. Clearly, the ratio of product concentrations can be varied over a considerable range by altering the time between the pump and dump pulses. The pulse delay control scheme has been demonstrated by Baumer and co-workers [171 with respect to the competition between ionization and dis-

-

-

I

Quantum branching ratios

I= Channel 1 0 = Channel 2

210

410

610

810

1010

Stimulation time (au)

-

Figure 5. Product yield as a function of pulse separation for the model branching reaction patterned after HHD + HH + D and HHD DH + H. Channel 1 corresponds to formation of D.

230

S. A. RICE

4.5

+

(D

+N

z"

3.0

1.5

I 0

I

*

I

2000

loo0 Pump-probe delay (fs)

Figure 6. Ratio of the concentrations Na2+/Na+ as a function of pulse delay for the competing reactions Naz --c Na2' + e and Na2 + Na+ + Na + e. (From Ref. 17.)

-

sociative ionization of Na2, namely Na2 Na2+ + e versus Na2 Na' + Na + e, by varying the time delay between the first and second pulses. Figure 6 displays the ratio of concentrations Na2+/Na+ found in their experiments. It is easy to see that if one wishes to use the Tannor-Rice scheme to generate a large concentration of a particular reaction product, it is necessary to have most of the wavepacket amplitude on the excited-state potential-energy surface simply and compactly distributed over that product exit channel on the ground-state potential-energy surface. However, in the typical case, the evolution of the wavepacket on the excited-state surface generates a very complicated distribution of amplitude; hence a simple dump pulse cannot efficiently transfer amplitude to the exit channel on the ground-state surface. We can determine what initial amplitude distribution on the excited-state surface will evolve to the amplitude distribution over the exit channel that we seek by integrating the Schriidinger equation backward with the desired final amplitude distribution as an initial condition. The result of this calculation is an initial distribution of amplitude so complicated that it cannot be created by a Franck-Condon transition from the ground state. Nevertheless, this calculation conveys an important message. If, instead of using separated pump and dump pulses, we use a temporally and spectrally shaped field, it should be possible to continuously transfer amplitude back and forth between the --+

CONTROL OF QUANTUM MANY-BODY DYNAMICS

23 1

ground-state and excited-state surfaces as the wavepackets move about on these surfaces. It is this observation that led Kosloff et al. [6] to develop the formalism for optimal pulse shaping. Consider again the system studied by Tannor and Rice. We now seek to maximize

at some specified time t = T. Here, J is a functional of the radiation field E(t); hence the maximization is to be carried out with respect to variation of the functional form of E(t), that is, its temporal shape and spectral content. The time T is chosen such that the outgoing wavepacket amplitude on the excitedstate surface is beyond any barriers in the ground-state surface exit channel for the particular product selected. When properly chosen, J is independent of T. The formalism used to calculate the pulse shape that maximizes J is optimal control theory. This formalism can be considered to be an extension of the calculus of variations to the case where the constraints include differential equations. In general, the constraints expressed in the form of differential equations express the restriction that the amplitude must always satisfy the Schrodinger equation. In addition, there can be a variety of other constraints, such as a restriction on the total energy in the pulse or on the shape of the pulse. These constraints are accounted for by the method of Lagrange multipliers, which modify the objective functional (4.6) and thereby permit the calculation of the unconstrained maximum of the modified objective functional. When the only constraints are satisfaction of the Schrdinger equation and limitation of the pulse energy, the modified objective functional can be written in the form

where

232

S. A. RICE

The variation of 5, is taken with respect to E and J,.We require that 67 = 0 for all 6 Re E, 6 Im E, and 6$. Then one finds that we must simultaneously satisfy

subject to

and

a*

i A -= H $

(4.1 1)

J,(O) = *o

(4.12)

%,

at

subject to

Equation (4.9) is the equation of motion of the Lagrange multiplier that restricts the solution to satisfy the Schriidinger equation; it is to be solved subject to the final-state condition (4.10). Equation (4.11) is the Schrodinger equation for our system; it is to be solved subject to the initial condition (4.12). The field that results from these calculations is given by

with

Note that the calculation of the optimal pulse shape is a double-ended boundary-value problem: J , is known at t = 0 while x is known at t = T. This aspect of the calculation of the optimal pulse shape mirrors the considerations advanced concerning the competition between spreading of the wavepacket as it moves on the potential-energy surface and the use of interference between pump and dump fields to counteract that spreading.

CONTROL OF QUANTUM MANY-BODY DYNAMICS

233

The solution to Eqs. (4.9)-(4.14) must, in general, be obtained by numerical analysis. The reader is referred to the literature for the techniques used. We consider, as an example, calculation of the optimal pulse shapes for generation of products for (essentially) the same model system studied by Tannor, Kosloff, and Rice (see above). Some of the results [6] of the calculations are shown in Figs. (7) and (8).

.075 h

c

0.5 1 e

s Q)

0.3

0

;F

f

-.025

- .05C

300 600 900 1200 1500

1 .a

E

B z

b

a

J

- .5

0 300 600 900 1200 1500

Time (a.u.)

0.6;

0.3: -0.01

J

10

C

375

750

x 10' Time (a.u.)

113

J

150

Figum 7. (a) Pulse sequence resulting from the optimization of the control field for the HH + D and HHD reaction surface patterned after that for the branching reactions HHD DH + H.This pulse sequence is intended to maximize the formation of D. (b) The Husimi transform of the pulse sequence shown in (a). (c) Time dependence of the norms of the groundstate and excited-state populations as a result of application of the pulse sequence shown in (a). Absolute value of the ground-state wave function at 1500 au (37.5 fs) propagated under the pulse sequence shown in (a). shown superposed on a contour diagram of the groundstate potential-energy surface. (From D. J. Tannor and Y. Jin, in Mode Selective Chemistry, B. Pullman, J. Jortner, and R. D. Levine, Eds. Kluwer, Dordrecht, 1991.)

-

+

234

S. A. RICE

0.5 ?

3

0.3

0.1

c 0 a -.l U

E

LI

. 0 7 5 -

a

300 600 900 1200 1500

0

Time (a.u.)

-

b 5 1 0. 300. 600. 900.1200. 1500 Time (a.u.)

I

1.00 1

-0.00

- .3

-

OOo

375

750

x 10' Time ( a u )

113

,501

Figure 8. ( a ) Pulse sequence resulting from optimization of the control field to generate H in the same reaction as studied in Fig. 6. (b) The Husimi transform of the pulse sequence shown in (a).(c) Time dependence of the norms of the ground-state and excited-state populations as a result of application of the pulse sequence shown in (a). Absolute value of the ground-state wave function at 1500 au (37.5 fs) propagated under the pulse sequence shown in (a), shown superposed on a contour diagram of the ground-state potential energy surface. (From D. I. Tannor and Y. Jin, in Mode Selecrive Chemistry, B. Pullman, J. Jortner, and R. D. Levine, Eds. Kluwer, Dordrecht, 1991.)

In general, the results of the calculations establish that it is possible to guide the reaction to preferentially form one or the other product with high yield. Note that, unlike the original Tannor-Rice pump-dump scheme, in which the pulse sequences that favor the different products have different temporal separations, the complex optimal pulses occupy about the same time window. Indeed, the optimal pulse shape that generates one product is very crudely like a two-pulse sequence, which suggests that the mechanism of the enhancement of product formation in this case is that the time delay between the pulses is such that the wavepacket on the excited-state

CONTROL OF QUANTUM MANY-BODY DYNAMICS

235

potential-energy surface is located over the ground-state exit channel and has outgoing momentum when the second pulse stimulates transfer to the ground-state surface. On the other hand, the optimal pulse shape that generates the other product has a more complex structure and, in fact, indicates there is continuous excitation and deexcitation, albeit with amplitudes that vary with time. The mechanism of enhancement of product formation in this case appears to be cyclical repetition of the transfer of amplitude between the two potential-energy surfaces in a fashion that minimizes the destructive interference in the transfer of the wavepacket placed over the exit channel on the ground state. The plots displayed in Figs. 7 and 8 do not show the phases of the optimally shaped pulses; these are likely significantly different. Finally, we note that the optimally shaped pulses convert a large fraction of the reactant to the desired products, unlike the simple Tannor-Rice scheme, which generates high selectivity of product formation but only small yields of products. We will return to the issue of complete transformation of a reactant to a selected product later in this review. Wilson and co-workers [28-321 have proposed a slightly different formulation of the calculation of the field required for optimal control of a quantum many-body system. This formulation is more general than the analysis described above in the sense that it is based on the equation of motion for the density matrix. The advantage that the density matrix representation of the system affords is the ability to study mixed states of the system, such as are characteristic of a thermal ensemble of molecules. Most of the applications of control theory to molecular dynamics reported by Wilson and co-workers are based on the assumption that the yield is determined by linear response theory, which restricts the cases that can be considered to those with weak fields. This version of the control theory is less general than the analysis described above. Indeed, this version of the control theory is very similar to the second-order perturbation theory analysis of Tannor and Rice. For example, with (4.6) as target functional and (4.9) as the constraint on the pulse energy, the yield is determined by the solution of an eigenvalue problem in which the optimal field is the eigenfunction belonging to the maximum eigenvalue of a Fredholm linear integral equation whose kernel depends on the evolution of the amplitude is an undriven system. Specifically, to second order in the field strength the density matrix is given as

where, for a system with two electronic states,

236

S . A. RICE

(4.16)

and 60is the (Liouville space) propagator associated with the field-free system Hamiltonian. The interpretation of (4.15) is very much like the interpretation of (4.4). The initial state of the system is represented by &), and it propagates freely until t l , at which time an interaction with the field occurs (e.g., a photon is absorbed). The excited system then propagates freely from r 1 to 12, at which time a second interaction with the field occurs (e.g., there is stimulated emission). The integrations account for all possible times at which the two interactions of the molecule with the field can occur, With (4.6) as the target functional the optimal control field is determined by

In this weak-field limit the relationship between the yield of product and the optimal field is

(4.18)

We call the reader’s attention to the similarity between Eqs. (4.17) and (4.5). The result obtained by Wilson and co-workers is more general that the Tannor-Rice result in that the latter calculates the field that maximizes the product yield for a pump pulse with given shape whereas the former makes no restriction concerning the shape of the pump pulse and does not assume that the pump and dump pulses can be distinguished from one another. Wilson and co-workers have also considered optimal control of molecular dynamics in the strong-field regime using the density matrix representation of the state of the system 1321. This formulation is also substantially the same as that of Kosloff et al. [6] and that of Pierce et al. [8, 91. Kim and Girardeau [33] have treated the optimization of the target functional, subject to the constraint specified by (4.8), using the Balian-Veneroni [34]variational method. The overall structure of the formal results is similar to that we have already described.

CONTROL OF QUANTUM MANY-BODY DYNAMICS

237

V. GENERIC CONDITIONS FOR CONTROL OF QUANTUM DYNAMICS We now describe a generic formalism for active control of a molecule coupled to the radiation field. That is, we examine how the control conditions for a variety of circumstances can be expressed in terms of the phase of the external field and the phase of the relevant dynamical variables. For simplicity, we consider a simple case, namely, when only two electronic states of the molecule play roles in the reaction dynamics; we take these to be the ground electronic state and the first excited electronic state. The radiation that couples the two surfaces is the means of control. The internal state of the molecule is defined by the density operators i,,j E g, e, where g and e denote the ground and excited states, respectively. The combined density operator describing the state of the system can be represented as

where the symbol 8 denotes the-tensor product, f i j is a projection operator on surfacej E g, e, and the & are raising and lowering operators that transfer amplitude from one surface to another. The first two terms in Eq. (5.1) represent the state of the molecules with population on the ground and excited surfaces, whereas the last two terms represent the electronic coherence induced by the radiation field. The Hamiltonian of the system consists of the sum of internal Hamiltonians and an interaction term

where the zero-order Hamiltonian is given by Ho=

fi* fig B i g+ H e63 Pe = 8 1 + vgig + veie 2m II

(5.3)

Of course, each of the terms in (5.3) is a function of the internal coordinates of the molecule. For the case we consider, the interaction terms in (5.2), which control the transfer of amplitude between the two electronic manifolds, only contain the radiative coupling term

V , = -F @ {i+E(t)+ kE*(t)}

(5.4)

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S. A. RICE

where is the transition dipole moment operator and E(t) represents a semiclassical time-dependent radiation field. It is via control of the spectral composition, the time profile of the field amplitude, and the phase of the field that we can control the evolution of the molecule. Although not accounted for in the Hamiltonian considered thus far, when intramolecular coupling of electronic manifolds is included in the Hamiltonian, radiationless transitions within the molecule can be included in the group of dynamical processes to be controlled. The evolution of the molecule is described by the generalized Liouville-von Neumann equation 135, 361

where is an operator representing the dissipative coupling of the system to background states. Equation (5.5) describes the dynamics of an open quantum mechanical system under the assumption that the evolution operator d:fines a dynamical semigroup [36-391. The source of the dissipative term LD; is the reduction of the combined system and bath dynamics to the dynamics of the system only. Th: semigroup formalism provides an explicit form for the dissipative operator Lo, but we shall not need that detailed form. Note that the first term in (5.5) describes the unitary dynamics supported by the Hamiltonian. The equation of motion of an explicitly time-dependent operator is given as

whe;e the first term represents the explicit time depeFdence of the operator A, the second term the Hamiltonian evolution of A, and the third term the Heisenberg representation version of the dissipative superoperator in Eq. (5.5). The mechanism by which control of the dynamical evolution of our model molecule is achieved is the alteration, by variation of the external field, of population and energy transfers between its two electronic states. This mechanism is, in a sense, analogous to the control of transformation of the equilibrium states of a macroscopic system by altering population and energy transfers between macroscopic states via variation of external parameters. Accordingly, it is interesting to examine the exchange of energy between the molecule and the external field and to relate that energy exchange to

CONTROL OF QUANTUM MANY-BODY DYNAMICS

239

alteration in the populations of the molecular states. The rate of change of energy is, using the equation of motion described above,

(5.7) since [h,k]= 0. Equation (3.1) is a version of the first law of thermodynamics [40-43], written in terms of the time rate of change of the energy and the power,

which is the time derivative of the work, and the heat flow,

dQ = (iEh) dt

(5.9)

With these definitions, the power absorbed from the field into the system becomes (5.10) In (5.10), @ Q i+) is the expectation value of the instantaneous transition dipole moment; variation of its value provides the means for controlling the molecular evolution. For the system under consideration, with only two electronic potentialenergy surfaces, the conservation of population implies that d N g + dN, = 0

(5.11)

The flow of population from one electronic surface to the other can then be calculated using Eq.(5.6): (5.12) Ignoring nonradiative couplings between the ground- and excited-state sur-

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S. A. RICE

faces implies setting LDP, = 0, whereupon the ground-state surface population change becomes * * A

-d N - ,- - 2 Im(G 63 5+) E(t)) dt h

(5.13)

The flow of energy from the ground state can also be calculated when it is assumed that the rate of electronic dephasing is smallA(i.5.,LEk, = 0) and/or the rate of pure vibrational dephasing is small (i.e., LZH, = 0). These conditions apply when the rate of relaxation to equilibrium is small relative to the rate of loss of phase coherence. Under these conditions (5.14)

Consider the case that the external field is pulsed so E(t) = 0 when t = f=. Energy balance requires that

which is a formal statement of the fact that the power uptake can be distributed into the ground and excited states as yell as into energy stored by the interaction. The boundary term -2Re{G@S+) *E(t)},which goes to zero when deriving Eq. (5.15), represents the transient loading of the system. The energy balance equation is one of the main tools of the thermodynamic representation of the control process. The spatial derivative of the loading term represents the internal force that the electromagnetic field exerts on the molecule. This force is (5.16)

where x represents the internal coordinates of the molecule. This interpretation of the interaction between an external field and a molecule leads to the possibility of applying a directional force on the molecule. Molecular transfer processes can be promoted either by controlling the field E ( t ) or its time derivative. We note that the transfer equations (5.13) and (5.14) have similar structure, namely, each contains the imaginary part

CONTROL OF QUANTUM MANY-BODY DYNAMICS

24 1

of a product of a molecular expectation value (X) and the field E(t). Equation (5.10) has an analogous structure; transfer is controlled by the real part of the product of a molecular expectation value and the time derivative of the field. For convenience we rewrite (5.13) in the form (5.17)

where $p is the phase angle of the instantaneous dipole moment and +E is the phase angle of the radiation field. The overall phase angle in (5.17) is the sum of the phase angle of the induced polarization of the molecule and the phase angle of the polarization of the light. In a similar way (5.18)

and (5.19) where +,,H is the phase angle of

GH, @ $+). We also have (5.20)

+,,.

where is the phase angle of (@;/ax) @ g+). Equations (5.17)-(5.20) clearly show that the most important control parameter for transfer of energy or population between the energy surfaces of a molecule is the phase angle. Consider the case when the external field is a monochromatic circularly polarized pulse, IE(r)l = A exp(-iwt), where A is a slowly varying envelope function. For this pulse the phase angle of dE/& is rotated by 1~/2from the direction of E. From Eqs. (5.17) and (5.18) we then find d% P=-Aw dt

(5.21)

so the power absorbed is proportional to the population transfer. When more general pulse shapes are used, the character of the control can be active or passive. For active control conditions the phase angle is given as

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S. A. RICE

4~ 4' = +

4p

+4E

{0

for maximum energy transfer for maximum energy emission

r

(5.22)

U for maximum positive population transfer

=

{-:

(5.23) for maximum negative population transfer

U -

maximum energy transfer to the ground-state surface for maximum energy removal from the ground-state surface

(5.24) $1' -k

6E=

0 ?r

for maximum positive force for maximum negative force

(5.25)

Active control of population transfer using the control relation displayed in Eq. (5.23) has been demonstrated experimentally by Sherer et al. [18]. In this experiment gaseous 12 was irradiated with two short (femtosecond) laser pulses; the first pulse transfers population from the ground-state potential-energy surface to the excited-state potential-energy surface, thereby creating an instantaneous transition dipole moment. The instantaneous transition dipole moment is modulated by the molecular vibration on the excitedstate surface. At the proper instant, when the instantaneous transition dipole moment expectation value is maximized, a second pulse is applied. The direction of population transfer is then controlled by changing the phase of the second pulse relative to that of the first pulse. Another interesting possibility that emerges from examination of Eqs. (5.17H5.25) is the induction of stimulated emission; this is predicted to occur when the combined phase angle in Eq. (5.22) is u.In principle, then, provided the phase relation is right, a pulsed laser can be built from an ensemble of molecules without the usual condition of population inversion. For some purposes, passive control of molecular evolution can be more important than active control; one such case occurs when we wish to prevent transfer of population or energy. The phase angle relations are

4,, + 4 k = +r

for zero total energy transfer

4,, + 4E= 0, ?r for zero population transfer

(5.26)

(5.27)

CONTROL OF QUANTUM MANY-BODY DYNAMICS

&, +

= 0, T

+ t$E = +r

243

for zero change in the ground-state energy (5.28) for zero force

(5.29)

Examination of the passive control conditions in Eqs. (5.26H5.29) shows that there are two values of the sum of phase angles for which zero transfer occurs. In principle, then, one can simultaneously block the transfer of, say, the energy and select the direction of the transfer of the population. One particularly interesting case is the definition of the phase angles for zero total power absorption. Since no energy is absorbed or emitted from the field these conditions define laser catalysis [a]. A note of caution must be inserted at this point. It appears, at first sight, that there is a meaning that can be attached to the absolute phase of the field and to the phases of the molecular expectation values. However, it must be remembered that the phase of the molecular quantity is induced by the radiation field prior to the present time. Therefore all phases must be related to the phase of a previous pulse that synchronizes the molecular clock with the field clock. With this synchronization it is possible to understand how quantum mechanical interference between events induced in the past propagates and can be used to control energy and/or population transfer at a later time. The generalization to the control of the dynamics of a molecule with n electronic states is straightforward. For the purpose of deducing the control conditions we will examine the extreme case in which every possible pair of these electronic states is connected via the radiation field and a nonzero transition dipole moment. If the molecule is coupled to a radiation field that is a superposition of individual fields, each of which is resonant with a dipole allowed transition between two surfaces, the density operator of the system can be represented in the form

yhere kii is the projection operator onto surface i, i E 1, .. . , n, and $, and

qjare lowering and raising operators.

The various transfer equations for the control of molecular dynamics can

be worked out as before, leading to the following phase angle conditions for

active and passive control:

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S . A. RICE

$ij

+ &..

0 for maximum energy absorption T for maximum energy emission

=

(5.31)

for I c i I n, where Ei, is the resonant electric field component between states i and j;

‘pli

+ ’Eli =

{

for maximum positive population transfer for maximum negative population transfer

$r

(5.32)

where Eli is the resonant field component between the ground state and excited state i; ’ $T

-

‘ p ~ ‘ ~ i j= +

{

+

for maximum energy transfer to ground-state surface (5.33) for maximum energy removal from ground-state surface

0 for maximum positive force r for maximum negative force

(5.34)

The above relations define the conditions for concurrent control of population and energy transfers between all of the states of the system that are connected by dipole allowed transitions. It is unlikely that a situation that complicated will ever be encountered. In the n-state molecule language, typically, not all pairs of states of the molecule are connected with nonzero transition dipole moments. In the skeleton spectrum language, there is usually only a small subset of dipole coupled “doorway states.” In both cases, of course, when only some pairs of states are coupled with nonzero transition dipole moments, the appropriate control conditions are simplified. We now consider, as an example, the construction of a globally optimym control field that affects the transfer of a ground-state quantity 8 = (A,) with minimum power consumption under the restriction of zero population transfer between the two electronic states of the system. As before, this field is obtained by varying 9 at a specific final time T with the following constraints: (a) The evolution of the system is governed by the Liouville-von Neumann equation (5.5). (b) There is zero population transfer so dN, = 0.

CONTROL OF QUANTUM MANY-BODY DYNAMICS

(c) The power consumption is bounded by (4.9).

245

Taking account of the constraints by the method of Lagrange multipliers, the functional to be minimized takes the form

8*= Tr[i, 8 f',$(t)]

+

1: [($ dt Tr

- I$)

h + h(E(t)(']

(5.35)

where h is an operator Lagrange multiplier and X is a scalar Lagrange multiplier. The variation of 8*is with respect to ;and IEI. The condition dN,/dt = 0 determines the phase of the optimal field through Eq. (5.27). It therefore is omitted from the variation. Taking the variation of (5.35) and integrating by parts leads to the following equations: (a) A forward equation for the density operator,

a; = L'; A

at

(5.36)

subject to the initial condition $ = b(0). (b) A backward equation for the Lagrange operator h, dt

(5.37)

subject to the final condition h ( T ) = A, 03 b,. The dissipative part of (5.37) is symmetric in time, meaning that dissipation takes place in the forward as well as in the backward evolution. (c) A condition on the field:

Equation (5.38)can be interpreted as the scalar product of a forwardmoving density and a backward-moving time-dependent operator. The optimal field at time t is determined by a time-dependent objective function propagated from the target time T backward to time t . A first-order perturbation approach to obtain a similar equation for optimal chemical control in Liouville space has been derived in a different method by Yan et al. [28].

246

S . A. RICE

In dissipative dynamics, the backward-propagating target operator decays into a stationary operator, and therefore, i*b(--) = 0. This leads to loss of control, as can be seen from Eq. (5.38). If the goal of the control process is the reduction of the epergy on the ground-state potential-enepy surface, the target operator is H,(T). When the control field is weak, B(t) can be expanded in powers of the field. For the energy reduction scheme under consideration the target fynction then becomes time dependent and is just the Schrodinger operator H,. Thus far we have not made explicit use of the phase constraint that defines the control of a particular dynamical process. Although we have use! a c9nstraint on the energy while minimizing, via the variational calculus, (Ag@Pg) on the ground-state surface, this procedure yields only the variational solution for the amplitude of the electric field; the variational solution for the field phase is lost. However, the phase of the field is constrained by the condition on dNg/dt through Q. (5.27). In general, if the goal is to minimize some dynamical function without consideration of the changes of any other observables, then we do not need to explicitly specify any of the phase relationships exhibited by the field. We now examine the formalism needed to explicitly include a constraint on the phase of the field in the optimization procedure [20]. Consider the case where we try to minimize the ground surface energy under the condition of zero population transfer, for which we have the phase relation

To incorporate (5.39) as a constraint in the variational calculation of the optimal field, we represent the electric field in the form E(t) = A&) exp[i+~(t)] and the objective functional in the form

where XI and A2 are two scalar Lagrange multipliers. We note that the time average of 4~ + 4p vanishes but the time average of ( 4 +~4J2 is positive definite, hence the form that a ears in (5.40). Taking the variation of 8 with respect to i , AE and ~#JE leads to the

P

CONTROL OF QUANTUM MANY-BODY DYNAMICS

247

folloying equation for the phase, in addition to (5.36), (5.37), and (5.38)for

PO), B W ,and A&):

(5.41)

Equation (5.41) explicitly describes how the time evolution of the phase angles must vary so as to minimize the value of (&c + c#J,,)~to satisfy the constraint of zero population transfer between potential-energy surfaces. Note that control of the field phase is critically influenced by the backward propagation of the target function B(t).

VI. HOW MUCH CONTROL OF QUANTUM MANY-BODY DYNAMICS IS ATTAINABLE? Thus far our examination of the quantum mechanical basis for control of many-body dynamics has proceeded under the assumption that a control field that will generate the goal we wish to achieve (e.g., maximizing the yield of a particular product of a reaction) exists. The task of the analysis is, then, to find that control field. We have not asked if there is a fundamental limit to the extent of control of quantum dynamics that is attainable; that is, whether there is an analogue of the limit imposed by the second law of thermodynamics on the extent of transformation of heat into work. Nor have we examined the limitation to achievable control arising from the sensitivity of the structure of the control field to uncertainties in our knowledge of molecular properties or to fluctuations in the control field arising from the source lasers. It is these subjects that we briefly discuss in this section. We consider, first, whether it is in principle possible to control the quantum dynamics of a many-body system. The goal of such a study is the establishment of an existence theorem, for which purpose it is necessary to distinguish between complete controllability and optimal control of a system. A system is completely controllable if an arbitrary initial state can be transformed, without loss to other states, into an arbitrary final state at some time T. A system is strongly completely controllable if an arbitrary initial state can be transformed, without loss to other states, into an arbitrary final state at a specified time T. Optimal control theory designs a field, subject to specified constraints, that guides the evolution of an initial state of the system to be as close as possible to the desired final state at time T. The work of Huang, Tarn, Clark et al. [46-SO] deals with a general for-

248

S. A. RICE

mulation of the complete controllability problem for quantum many-body dynamics and includes an existence proof that establishes sufficient (but not necessary) conditions for complete co~trol.They consider a system [45] that can be described by the Hamiltonian Ho in the absence of control fields; the system is assumed to have a discrete, but not necessarily bounded, spectrum (e.g., a harmonic oscillator). To control the dynamical evolution of the system, external fields are applied and the Schrodinger equation has the form

where the uk are real functions of the time and the f i k are linear Hermitian operators. Note that the U k needed to control the evolution of the state of the system depend on the state of the system, so Eq. (6.1) is, in fact, strongly nonlinear. The Huang-Tam-Clark theorem [47] is developed for the case that the fik are independent of the time and the control amplitudes U k are piecewise constant functions of the time. The proof involves many technical details; we will be concerned only with a loose paraphrasing of the results. Huang et al. show that, for a system with a discrete spectrum, under the conditions stated for the control field amplitudes and the corresponding operators, one can always find a set of field amplitudes that will guide the evolution of an initial state0' to come arl$rarily close to a chosen final state at some time T. If the set of operators H k generates an infinite dimensional Lie algebra, an infinite set of switchings of the piecewise constant fields is required to achieve the final state. However, even though an infinite set of switchings is needed to reach the limit of complete transformation of the initial state to the final state, a very large fraction of that transformation may be achieved in a small n%mber of switchings. If the Lie algebra generated by the set of operators H k has finite dimension, it can be shown that a system with a discrete nondegenerate spectrum is completely controllable in the sense that an arbitrary initial state can be transformed into an arbitrary final state at some later time. The scope of the Huang-Tam-Clark theorem is not strictly restricted to systems with a discrete spectrum, although only one very simple example [47]of control of a system with a continuous spectrum has been discussed. Rarnakrishna et al. [5 I] have studied the controllability of quantum manybody dynamics of systems with a finite number of levels from a point of view that is somewhat different from that used by Huang et al. [47].For

CONTROL OF QUANTUM MANY-BODY DYNAMICS

249

the purpose of investigating controllability, Huang et al. interpret (6.1) as an infinite dimensional bilinear system. Ramakrishna and co-workers [5 I] instead express the Schrodinger equation with included control field in terms of the eigenstates of an operator of interest. This approach yields, for a finite set of states, a finite dimensional bilinear control representation. We refer the reader to the original publication for the technical details of the analysis. A loose paraphrasing of the results of Ramakrishna et al. is that in a system with a finite number of nondegenerate discrete levels it is always possible to completely control the evolution of an arbitrary initial state to a selected final state. This result confirms the inference drawn by Tersigni et al. [52] from a study of the optimal control fields that transform various initial states to selected final states in a model five-level system. Shapiro and Brumer [53]have examined a system in which the eigenstates of the Hamiltonian are subdivided into three sets, with dimensions Mo,M I and M2,and ask if it is possible to transform a specified initial state of a system that lies in the subset of states with dimension M O into a specified final state of the system that lies in the subspace of states with dimension M 1 without passage through the states of the system that lie in the subspace with dimension M2.It is shown that if Mz 2 Mo stringent restrictions are required to prevent involvement of the states in M2 in the specified transformation. This inability to direct the evolution of the state of the system away from a specified set of substates does not contradict the Huang-Tam-Clark theorem, since that theorem does not admit constraints on the evolution pathway of the state of the system. Establishing the complete controllability of the quantum dynamics of a many-body system is an important backdrop to the development of practical algorithms for generating that control. However, the extant existence theorems give no hint as to how such algorithms can be formulated or how the controllability is influenced by constraints on the applied fields and/or on the evolution pathways that can be used. It is just the latter issues that play a central role in the optimal control theory analysis of the guided evolution of quantum many-body dynamics. It is important to note that setting up a calculation of the optimal control field for the transformation of an initial state of a system into a particular final state of that system does not guarantee that such a field exists. And, it must be remembered, that even when the optimal control field can be found, it does not, in general, provide complete control, since the latter implies that the norm of the difference between the final state reached with the optimal field and the target final state can be made arbitrarily small with respect to variation across admissible controls, not merely a minimum. Peirce et al. [54], Zhao and Rice [55], and Demiralp and Rabitz [56) have studied the existence of optimal control fields with respect to quan-

250

S. A. RICE

tum dynamics. For the case that the control field is bounded and can be expressed as an integral operator of the Hilbert-Schmidt type, the following results have been obtained: (i) Peirce et al. [54] proved that for a spatially bounded quantum system, which necessarily has spatially localized states and a discrete spectrum, optimal control of the evolution of a state is possible. (ii) Zhao and Rice [55] adapted the analysis of Pierce et al. [54] to show that in a system with both discrete and continuous states optimal control of evolution in the subspace of discrete states is possible. (iii) Zhao and Rice [55] also showed that evolution can be optimally controlled in the subset of continuum states that can be transformed to be L2 integrable by a complex rotation (wavepacket states that can be transformed to surrogate localized states). (iv) Demiralp and Rabitz [56] have shown that, in general, there is a denumerable infinity of solutions to a well-posed problem of control of quantum dynamics; the solutions can be ordered in quality according to the magnitude of the minimum of the objective functional J (see Section IV) that is achieved. In all of these cases what is established is the existence of a control field that will minimize the objective functional J . If the value of the minimum attained is zero, the system is completely controllable; if not, the optimal solution defines the maximum attainable control of the evolution of the system for the given set of control functions. The preceding analyses assume that we possess complete knowledge of the molecular Hamiltonian of the system we wish to control and that no disturbances will generate fluctuations in the control field used to guide a particular state-to-state evolution. In practice, neither of these assumptions is completely correct. Except for the smallest molecules we have imperfect knowledge of many-body potential-energy surfaces, of state-to-state coupling energies, and so on, and real experiments are always subject to a variety of mechanical, electronic, and optical disturbances. It is therefore desirable to design a control field that will generate the desired transformation of an initial state to a final state while being robust with respect to uncertainties in the molecular Hamiltonian and to disturbances that generate control field fluctuations. Rabitz and co-workers [57-591 have devoted considerable effort to devising a method for reducing the sensitivity of the field that controls a stateto-state transformation to various uncertainties and disturbances. The general scheme they develop introduces the constraints implied by the existence of the disturbances into the functional J and then uses minimax

CONTROL OF QUANTUM MANY-BODY DYNAMICS

25 1

analysis to determine the optimal control field. T h ~ s , ~the i f effects of the disturbances are collected in a pseudo-Hamiltonian Hdist(x,t , u(t), w(x,t)), which can be a function of any or all of the molecular coordinates x, the field u(r), the disturbance w(t), and the time, the optimal field is determined from min,max,J(u(t), w(x,t) subject to the dynamical constraint that the Schradinger equation be satisfied and a constraint on the disturbances of the form F(w(x,t ) ) = 0. The minmax point Urn(?) and the function W m ( X , t) satisfy

subject to the constraints mentioned, with urn(?)the best optimal field and wm(x,t ) the worst disturbance. Clearly, J(u,(t), w,(x, t)) determines the best control field for the worst disturbance, and hence is a very conservative representation of the limitation of the extent of control generated by disturbances. Zhang and Rabitz I57,SSl have applied this analysis to several simple cases. The results obtained illustrate that robust control fields can be designed under a variety of conditions and for a range of types and magnitudes of disturbances; they also suggest that there can be circumstances in which the magnitude of a disturbance is sufficient to destroy the possibility of control of the quantum dynamics. All of the analyses described above are used in a predictive mode. That is, given the molecular Hamiltonian, the sources of the external fields, the constraints, and the disturbances, the focus has been on designing an optimal control field for a particular quantum dynamical transformation. Given the imperfections in our knowledge and the unavoidable external disturbances, it is desirable to devise a control scheme that has feedback that can be used to correct the evolution of the system in real time. A schematic outline of the feedback scheme starts with a proposed control field, applies that field to the molecular system that is to be controlled, measures the success of the application, and then uses the difference between the achieved and desired final state to design a change that improves the control field. Two issues must be addressed. First, does a feedback mechanism of the type suggested exist? Second, which features of the overall control process are most efficiently subject to feedback control? Judson and Rabitz [60] have provided a numerical demonstration of an existence theorem for feedback control in the guiding of the evolution of the state of a system. The example they consider is the transfer of 100% of the population from the vibrationless ground rotational state of KCl to the vibrationless state withj = 3, m = 0, by a suitable field. The novel idea they exploit is to use the population transfer generated by a trial field as input to an adaptive learning algorithm for comparison with the desired popula-

252

S . A. RICE

tion transfer. The learning algorithm then defines changes in the trial field that are intended to decrease the difference between the population transfer generated by the new trial field and the desired population transfer. The specific learning algorithm they adopt is the so-called genetic algorithm, by means of which an analogy is exploited between the search for fields that enhance population transfer and survival of the fittest in a Darwinian evolution. This analogy is exploited by starting the search for the control field with a large family of fields constructed from pulse sequences; the initial pulse sequences can be defined in any fashion, even by use of a random-number generator. After application of the initial family of fields to the molecular system, all fields except the two that generate the largest population transfer are rejected. The two selected fields form the basis for a new set of fields generated by rearranging the pulses of the selected fields in all possible ways. Then the new family of trial fields is applied to the molecular system and all except the two fields that generate the largest population transfer are discarded. This process is continued until the desired population transfer is achieved. We consider the Judson-Rabitz result to be extremely important because it establishes that feedback can be used to correct the evolution of an applied field to become an optimal control field. However, the manner in which the feedback is implemented, namely by what is equivalent to a very rapid random search of the appropriate parameter space, gives no clues concerning the relationships between the physical parameters of the system and the optimal field needed for a particular transformation of the state of the system. A different demonstration of the value of feedback in the generation of an optimal control field has been reported by Amstrup et al. [59]. They studied the optimal control of a pump-dump experiment involving CsI (see Section IV). In this example the population is first transferred to the excited (dissociative) state and then, after a delay, transferred back to the ground state. The goal is to prepare a large ground-state population of molecules with some large internuclear separation. The pump and dump fields were separately optimized, assuming that either the time between pulses or the amplitudes of the pulses were uncertain. A chirp representation of the electromagnetic field was used, since this is experimentally realizable and has only a few parameters. A “generalized genetic algorithm” was used to generate the feedback for driving the evolution of the field, with special attention paid to the role of uncertainties in pulse delay and pulse amplitudes. The results obtained show that uncertainties in pulse delay and pulse amplitudes can be effectively mitigated by the use of feedback. Finally, we remark that the use of an optimal control field to enhance achievement of a particular quantum dynamical transformation usually increases the efficiency of that transformation by several orders of magni-

CONTROL OF QUANTUM MANY-BODY DYNAMICS

253

tude. And, it is usually the case that the optimal control field has a complicated structure. However, a good approximation to the major features of the optimal control field can usually be generated from a few Fourier components of that field, and in test cases it is found that this approximate field only degrades the efficiency by of order 50%.That degradation is acceptable when the gain in efficiency obtained from the optimal control field is 104, as in the pumpdump enhancement of the photodissociation of 12 studied by Amstrup et al. [61].

VII. REDUCED SPACE ANALYSES OF THE CONTROL OF QUANTUM DYNAMICS The analyses of control of the quantum dynamics of a many-body system given above are not, in principle, restricted by the complexity of the system. Nevertheless, the calculations required to design an optimal control field or to identify competing pathways between the same initial and final states become more difficult very rapidly as the number of degrees of freedom increases. For that reason done it is desirable to have a reduced space formulation of the control process. In addition, our intuition about chemical reactions is rooted in the similarities between functional group properties across a variety of molecules. In this sense much of chemistry is “local,” and we expect a suitable reduced-space representation to be applicable to the design of control process. In this section we sketch three reduced representations of quantum dynamics that have been suggested for use in the design of optimal control fields.

A. Reduced Representation in State Space We first examine the quantum dynamics of a system whose properties are represented by an n-state spectrum. In general, these states cannot be uniquely associated with individual degrees of freedom of the system. Indeed, we must expect that the amplitude associated with a state will, typically, depend on many (possibly all) of the degrees of freedom of the system. We seek a representation of the population dynamics of this n-state system in terms of the properties of a surrogate system with fewer states. The states of the surrogate system must, of course, be defined in terms of the n states of the full system, so we are really developing an alternative representation of the n-state system. However, we expect the formal relations that define the reduction of n to, say, rn states to be of value in guiding the generation of accurate approximations to the dynamics of the full n-state system. Tang et al. [20] have analyzed a reduction procedure using the Schrijdinger representation of the dynamics of the n-state system. The molecular wave function of the n-state system can be written as a superposition

254

S . A. RICE

of the eigenfunctions $ j , with coefficients C j ( t ) e - i w j ' . Substitution of this expansion into the Schrijdinger equation yields the equation of motion for the coefficients: (7.1)

where V j k = ( U j l V l U k ) and W k j = W k - W j . We assume that the molecule-field coupling is dominated by the dipole transition interaction and represents the resonant continuous electric field e(t) in the form

Adopting the rotating-wave approximation (RWA) and introducing the detuning frequency A o j k = W j k - y j k and the Rabi frequency M j k = - p j k l 4 0 l / k we find cj(t) +i

E

ke (I,

Mjk(t)Ck(t)

(7.3)

...,n )

where the M j k ( t ) are the elements of the n x n time-dependent matrix

If all states of the system are strongly coupled to each other, the system dynamics can only be described by completely solving the above equations. However, it is extremely unlikely that this is the case. Rather, it is commonly found that some pairs of states are strongly coupled and other pairs of states are weakly coupled. Then we expect that the population transfers among strongly coupled states dominate the system dynamics and that it should be possible to study the n-state system dynamics in the subspace of strongly coupled states with a correction from the influence of the weakly coupled states.

CONTROL OF QUANTUM MANY-BODY DYNAMICS

255

The reduction schemes used by Tang et al. [20] to define the surrogate fewer state system follows the method proposed by Shore [62]. The scheme has a cFmpact form when we introduce two orthogonal projection operators P and Q and work in the frequency domain instead of the time domain. The time evolution matrix for the n-state system dynamics, U(t), and its Fourier transform, G(w), satisfy the following equations:

U(t) = 0;

(wI - M ) G ( w ) = 1

(7.5)

where w is the frequency-domain variable and 1 is the n x n unit matrix. Let F be the projection operator onto the subspace composed of the states having stronger cyuplings within which we try to approximate the system dynamics, and let Q be the projection operator onto the remaining states. We are :Interested in evaluating the matrix elements of G(w ) within the subspace of P states in the frequemy dcmain. Multiplying both sides of the equation for G ( w ) in Eq. (7.5) by P + Q = 1, we find

Further multiplication by k from the right and Q from the left, followed by some rearrangement, yields &(w)F

= Q [ Q ( w l - M)Q]-'Q&G(u)F

(7.7)

Note that we require the inverse of the matrix (01 ; M ) within theAQsub; space. Now multiplying both sides of Eq.17.6) by P, substituting Q G ( u ) P from Eq. (7.7), and again multiplying by P on both sides yields P{wl

-

PMF - PM&Q(wl- M)Q]-'QMF}PG(w)b = P

(7.8)

We now write M ( w ) = PMi, + PMQ[Q(wl- M)Q]-'QMF

(7.9)

which i,s a representation of the frequency-domain time evolution operator within P space: FG(w)F = F[ol- M(w)]-'P

(7.10)

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S. A. RICE

From (7.10) we can get the time evolution operator bU(t)k by use of a Fourier transform. These localized operators permit the construction of thos? portions of the time-evolved state v e c t p that lie within the subspace of P states. The influence of the remaining Q states occurs through the action of the operator M(w). The preceding analysis is just a transformation of one representation of the n-state problem to another representation. To be useful, the new representation must admit simplifying approximations not suggested by the original representation. One such approx(mation is to replace the frequency variable w in M(o) in (7.10) by a typical P space eigenfrequency, say MY. We thereby obtain the frequency-independent effective operator

M = bMF + kMQ[Q(oY - M)Q]-'Q&

(7.11)

Viewed in the time domain, the replacement of M ( w ) by M washes out the details of the time variation within Q space. For this appryximation to be useful, all strongly coupled states should be included in the P space !nd the Q space should not include any states that couple strongly to the P space (weak coupling assumption). We now find that the population dynamics of the m levels within the P space is governed by the equations of motion kim

(7.12) k e ( I , ...,m )

We now connect the analysis given above with the equation of motion displayed in Eq.(5.5). That equation of motion follows from subdivision of a system into an open subsystem S and a complementary reservoir R. When the coupling between S and R is weak, the evolution of the open system S, due to the internal dynamics of S and the interaction with the reservoir R, can be described in density matrix form by Eq. (5.5). Now writing

we find (7.14)

CONTROL OF QUANTUM MANY-BODY DYNAMICS

257

which we require to satisfy the semigroup condition [38, 391 =

Al

(7.15)

+

Hence i is a semigroup generator. Returning to the formal reduction procedure described at the beginning of this section, we note again that the operator M ( w ) incorporates all of the dynamics associated with evolution of the n-state system, and the formalism merely reorganizes the exact representation of the n-state population transfer ciynamics. The formalism, as such, does not demand that the m states in the P subspace are strongly coupled to each other and htt: the n - M states in the Q subspace are weakly coupled to those in-the P subspace. The use of a typical unperturbed eigenfrequency of the P subspace, to replace the variable frequency w in Mid),by virtue of washing out the details of the time variation within the Q subspace, generates the separation of the total system into a strongly coupled subsystem that is weakly coupled to a reservoir. In general, we expect this approximation will lead to a loss of time revepibility, and hence can be used to obtain an explicit form for the operator LD. Given (7.12) it is straightforward to obtain the corresponding density matrix form of the equation of motion. The time depeFdence of the density matrix element for the mth-state population in the P subspace is found to be

~7,

A comparison of (7.16) and (5.5) yields

where, in the RWA,io has the representation

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S . A. RICE

Tang et al. [20] have examined the population dynamics in a three-level system, and its representation in a surrogate two-level system, to test the scheme outlined above. In the model system considered state 3 is weakly coupled with states 1 and 2, so that population transfer between states 1 and 2 should dominate the dynamics, with only a small contribution from population transfer to and from state 3. The coupling of state 3 with states 1 and 2 was taken to be one-tenth of the coupling between states 1 and 2, that is, Mi3 = M23 = M12/10 = - 1/10. Using the formalism sketched above, the exact system dynamics is governed by the coupled equations of motion for the three states, t l ( t ) = 0 . 5 i ! ~ ~ (+t o) . ~ c ~ ( ~ ) I &(r) = 0 . 5 i [ ~ , ( +t )0.1~3(t)l

C3(r) = o . s ~ [ o . ~ c ~+( ~ o .)I c ~ ( ~ ) I

(7.19)

and the approximate system dynamics is governed by the two coupled equations of motion for the two surrogate states,

C1 ( t ) = i[0.o05c1( t )+ 0.505~2(t)l c * ( r ) = i [ 0 . 5 0 5 ~ ~+( to.oo5c2(t)l )

(7.20)

The values of (Cl(t)I2and IC2(t)I2 obtained from (7.19) and (7.20) are compared in Figs. 9 and 10. The amplitudes and periods of the temporal evolution predicted by the two approaches to the system dynamics are seen to agree quite well. The differences seen in the amplitudes shown in Fig. 9 are a consequence of the replacement of the exact eigenfrequ%nciesof the Rabi frequency matrix with a typical eigenfrequency from the P subspace. Kaluza and Muckerman [63] have suggested a reduced representation in state space that is similar in spirit to that proposed by Tang et al. [20] but that also has some distinctive differences from their analysis. Kaluza and Muckerman [63] examine the states of a molecule populated by pulsed laser multiphoton excitation; they call these states “primitive bright states.” The primitive bright states are not, in general, eigenstates of the Hamiltonian. A set of active bright states is constructed by orthogonalization of the states generated by successive application of a pulsed field, at discrete time intervals, to the initial state of the molecule. Then the overlap matrix of the states so generated is calculated and diagonalized, and only that subset of the active bright states with largest overlap is selected to describe the time evolution of the state of the molecule. Kaluza and Muckerman have illustrated their method of analysis by examining the selective excitation of acetylene. Some of their results are shown in Fig. 11. Clearly, the Kaluza-Muckerman

259

CONTROL OF QUANTUM MANY-BODY DYNAMICS

0.8

-

0.6

-

0.4

-

0.2

0

3

9

6

12

time

Figure 9. Population of the ground-state surface obtained from the exact three-state dynamics (-) and surrogate two-state dynamics (-). The initial conditions are Cl(0) = 1 and C2(0) = C3(O) = 0.

reduction scheme is quite accurate and is useful for calculation of control fields.

B. Reduced Representation in Coordinate Space A very perceptive treatment of chemical reaction dynamics, called the reaction path Hamiltonian analysis, states that the reactive trajectory is determined as the minimum energy path, and small displacements from that path, on the potential-energy surface [64-711. The usual analysis keeps the full dimensionality of the reacting system, albeit with a focus on motion along and orthogonal to the minimum energy path. It is also possible to define a reaction path in a reduced dimensionality representation. The definition of a reduced dimensionality reaction path starts with the full Cartesian coordinate representation of the classical N-particle molecular Hamiltonian,

c p' 3N

H(p,x) =

i= 1

2mi

+ " ( X I , . .. , X 3 N )

(7.21)

where x and p are the 3N-dimensional coordinate and conjugate momentum

260

S. A. RICE

0

3

6

9

12

time Figure 10. Population of the first excited-state surface obtained from the exact three-state and surrogate two-state dynamics (-). The initial conditions are Ci(0) = 1 dynamics (-) and Cz(0) = C3(O) = 0.

vectors. Let a = ( a , , .. ., ~ 3 be~ a )vector on the reaction path. Then the potential-energy function V(x) can be expanded near the reaction path in powers of x - a. When only terms to second order are retained,

V(x) = V(a) + VV(a) . (x - a) + ;(x

- a)

- F - (x - a) + . ..

(7.22)

where F is the force constant matrix. Because the displacement vector x - a is orthogonal to VV(a) in the 3N-dimensional vector space, the linear term in Eq. (7.22) vanishes. For simplicity, and because they are not of interest for our purposes, the motions corresponding to molecular rotations and translation of the center of mass are removed by use of the projector

FP = (1 - P*) * F (1 - PT)

(7.23)

-8

-2:

v)

Figure 11. Comparison between the expectation values of several observables pertaining to the state of the linear HCCH molecule at the end of the pulse for various transform-limited pulses. A 91-state active bright-state basis approximation is compared with the result of a full calculation (labeled DVR). ( a ) Total energy at the end of the pulse. (6) Norm of the wavepacket. ( c ) Kinetic energy in the CH, stretching motion. ( d ) The CH, bond length. (e) Kinetic energy in the CC stretching motion. (f)Expectation value of the CC distance. (From Ref. 63.)

- 5\o:

26 1

262

S. A. RICE

where iRT is the projection operator for the translational and rotational a normal-mode analysis is carried motions. Following the application of iRT, out. For a polyatomic reactant with many degrees of freedom the numerical calculations required to execute the program outlined above can easily achieve a scale that is impossible to handle even with a vectorized parallel processor supercomputer. The simplest approximation that reduces the scale of the numerical calculations is the neglect of some subset of the internal molecular motions, but this approximation usually leads to considerable error. A more sophisticated and intuitively reasonable approximation [72, 731 is to reduce the system dimensionality by placing constraints on the values of the internal molecular coordinates (instead of omitting them from the analysis). Assuming that most of the atomic displacements in the reactant molecule are small, the obvious choices for constraints on the internal coordinates not directly participating in the reaction are fixed values of the bond lengths and bond angles (say, f in number). The technical details concerning the construction of mathematical representations of these constraints and their incorporation into the appropriate projection operators can be found elsewhere [24]. The result is, formally,

A normal-mode representation of the Hamiltonian for the reduced system involves the diagonalization of the projected force constant matrix, which in turn generates a reduced-dimension potential-energy surface in terms of the mass-weighted coordinates of the reaction path [64]:

As usual, the Qk and wk (k = 1, ... , 3N-f - 7) are the normal coordinates and the corresponding normal-mode frequencies. The kinetic energy is then

where the Bkf are coupling constants.

CONTROL OF QUANTUM MANY-BODY DYNAMICS

263

Suppose the reactive polyatomic molecule of interest can undergo unimolecular reaction to form several products, and we imagine carrying out a constrained reaction path analysis for each of the product channels. To carry out the analysis of a particular constrained reaction path, Zhao and Rice adopted a system-bath model [74] in which the reaction path coordinate defines the system and all other coordinates constitute the bath. The use of this representation permits the elimination of the bath coordinates, which then increases the efficiency of calculation of the optimal control field for motion along the reaction coordinate. Miller and co-workers [64,671 have shown that a canonical transformation of the reaction path Hamiltonian yields the form

(7.27)

with the Gkl coupling constant between the normal modes k and 1. After expansion of the first term in (7.27)to terms of quadratic order, and assuming vibrational adiabaticity, it is found that 2

H eff -- ps + 2

3N-f-7

k= I

(7.28)

264

S . A. RICE

with E the total system energy. We note that this effective Hamiltonian treats the bath as a set of linearly shifted harmonic oscillators. The harmonic bath coordinates can be eliminated by use of the vibrational adiabatic approximation [75] or, alternatively, the use of a shifted linear oscillator basis set to construct the Hamiltonian in the matrix representation, which renders the Hamiltonian matrix function a function of only the system reaction coordinates s. To use the reaction coordinate representation in the optimal control theory analysis, Zhao and Rice suppose that the ground- and excited-state potentialenergy surfaces are similar enough to each have product channels that yield, respectively, ground and excited states of the same product. The key approximation they introduce is the assumption that, using the same coordinates, the reaction paths on the two surfaces are roughly parallel. If that assumption is valid, the projection of the ground-state reaction path on the upper surface will usually lie in the valley that defines the upper surface reaction path, and vice versa. By “in the valley” we mean that the projected amplitude is in the neighborhood of the reaction path, although likely displaced to higher energy. We then expect that it will be possible to design a control field that transfers amplitude back and forth between the two potentialenergy surfaces and forces the amplitude to follow the general direction of the reaction path, although not the detailed reaction path itself. ClearIy, this approximation leads to decrease in the efficiency of product formation from that achievable with a field optimized with respect to the full space of the potential-energy surface. The calculations required to determine the (optimal) field that maximizes the yield of a particular product are similar to those we have described earlier, except for the added complexity of working in the reaction path representation. The major change is that the reaction path Hamiltonian replaces the full Hamiltonian in defining the constraint on the system dynamics. Looked at in general terms, the scheme Zhao and Rice have proposed [74] should be applicable, by construction, to the class of reactive systems in which the reaction paths on the upper and lower potential energy surfaces are similar. Actually, we expect this scheme to be applicable to a larger class of systems, since all that is required of the upper potential energy surface is that it not have a structure with a valley that is everywhere orthogonal to the projection of the lower state reaction path. However, the more similar are the contours of the two potential energy surfaces, the more efficient will be the optimization of the selected product yield. We then expect, if the condition described in the following paragraph is satisfied, that the scheme can be used to design a field that optimizes the product selectivity from the reaction of a polyatomic molecule. At present there are no reported tests of the accuracy of the Zhao-Rice

CONTROL OF QUANTUM MANY-BODY DYNAMICS

265

reduction scheme. Clearly, the reaction path formalism replaces the true potential energy surface with a simpler surface. The most important question raised by the use of this representation, which involves approximations, is whether it is sufficiently accurate to preserve the phase relations that characterize the transfer of amplitude between the true upper and lower potential energy surfaces.

C. Reduction by Factorization: Time-Dependent Hartme Approximation

The formulation of the calculation of the optimal control field that guides the evolution of a quantum many-body system relies, basically, on the solution of the time-dependent Schrodinger equation. Messina et al. [25] have proposed an implementation of the calculation of the optimal control field for an n-degree-of-freedom system in which the Hartree approximation is used to solve the time-dependent Schriidinger equation. In this approximation, the n-degree-of-freedom wave function is written as a product of n single-degree-of-freedom wave functions, and the factorization is assumed to be valid for all time. Messina et al. consider a system with two electronic states Jg) and )e). The system is partitioned into a subset of degrees of freedom that are to be controlled, labeled Z, and a background subset of degrees of freedom, labeled x; the dynamics of the 2 subset, which is to be controlled, is treated exactly, whereas the dynamics of the n subset is described with the time-dependent Hartree approximation. The formulation of the calculation is similar to the weak-response optimal control theory analysis of Wilson et al. described in Section IV [28-321. The solution of the time-dependent Schrodingerequation for this system can be represented in the form

The reduced density matrix for the system is defined by

with (7.3 1)

and similarly for j e ( Z ,Z’, t). Following the procedure sketched in Section IV, it is found that the weak-response optimal control field is determined by

266

S. A. RICE

(7.32) J to

where

A ( t f ) Tr[&;,(tf)]

dt dt’ Ec(t)MFd(t,t’)E:(t’)

=

(7.33)

is the target yield,

dZ dZ‘&(Z,Z’)

d x $bo’*(Z,x,tf - t)$bo)(Z’,x,tf

- t’)

(7.34) is the reduced material response function, and E&) is a slowly varying complex field from which the high-frequency component oeg(the resonance frequency) has been removed. The Hartree approximation is now introduced for the purpose of calculating the reduced material response function (7.34). The fashion in which this is carried out is by writing the overall bath Hamiltonian as a sum of the Hamiltonians for each bath degree of freedom, which implies that the overall bath wave function is a product of the wave functions for each bath degree of freedom. It can then be shown that

(M7d(t,~’))TDH = Q ( t , t‘)

J

dZ dZ‘ A,(Z, Z’)&(Z, tf

- t)&.(Z’, tf - t’)

with 8 the phase angle

(7.36)

CONTROL OF QUANTUM MANY-BODY DYNAMICS

267

Messina et al. [25] test the time-dependent Hartree reduced representation with a simple two-degree-of-freedom model consisting of the I2 vibration coupled to a one-harmonic-oscillator bath. The objective function is a minimum-uncertainty wavepacket on the B state potential curve of 12. Figure 12, which displays a typical result, shows that this approximate representation gives a rather good account of the short-time dynamics of the system.

VIII. THE CONTROL OF DYNAMICS-INVERSE SCATTERING DUALITY Thus far we have examined the determination of a field that will control the quantum many-body dynamics of a system when all that is specified is the initial and final states of the system and the constraints imposed by the equations of motion and physical limitations on the field. When posed in this fashion, the calculation of the control field is an inverse problem that has similarities to the determination of the interaction potential from scattering data. Despite the similarities, the mathematical methods used are very different. Because only the end points of the initial-to-final state transforma-

C

H

0)

=I

I&

18.0:

Figure 12. Magnitude of the response function for the stretched I2 target state. The solid line is the result of an exact calculation; the dotted line is the result of the use of the Hartree approximation. The parameter c is the coupling constant between the I2 molecule and the bath oscillator. (a) Bath oscillator frequency of 50 cm' . (b) Bath oscillator frequency of 100 cm' . (From Ref. 25.)

268

S. A. RICE

tion are specified, it is necessary to use a variational approach to calculate the optimal field that generates that transformation. Note, however, that a given applied field will uniquely determine the time evolution of the expectation value of an observable. This observation suggests that if, in addition to the initial and final states, extra information in the form of the trajectory of an observable is specified, the calculation of the field should become very much simpler. This approach is called inverse control and is obviously complementary to the optimal control approach with which we have been concerned to this point. The theoretical basis for inverse control of the quantum dynamics of a many-body system was established by Ong et al. [481. These investigators established the sufficient and necessary conditions for the existence and uniqueness of the solution to the inverse-control equations. The application of the inverse-control analysis to molecular problems has been advanced by the work of Rabitz and co-workers [76-781. The general concepts of inverse control are simple to grasp, Suppose we specify, a,priori, the time dependence of the expectation value of some observable, (0) = y ( t ) . The Schrodinger equation for the system with applied control field is just

and the equation of motion for an observable is

CONTROL OF QUANTUM MANY-BODY DYNAMICS

269

Since y ( t ) is known by definition, (8.2) may be directly inverted to calculate

u(t) except at points at which ( [ O , H l ] )= 0:

Indeed, even when ( [ b , H l ] )= 0 there is a prescription focf calculating u(t) from an equation for a higher order time derivative of (0) [76]. We note that the structure of (8.3) defines a form of feedback of information from the system to the field since the evaluation of the expectation values of the commutators requires knowledge of the system wave function. The methodology has been extended to the case where the field to be calculated is constrained to best reproduce the trajectories of several expectation values [77]. Rabitz and co-workers have provided interesting examples of the use of the procedure described to determine the field that drives a prescribed time evolution of the internuclear separation of a nonrotating diatomic molecule, a field designed by use of energy tracking that dissociates the HF molecule, and a field designed by use of competitive atomic acceleration tracking to selectively break the stronger bond in a nonrotating linear triatomic molecule. What is arguably the most interesting potential application of the analysis inherent in (8.3) is the suggestion, by Lu and Rabitz [78],that tracking be used to invert laboratory data to obtain, for example, the potentialenergy curve of a nonrotating diatomic molecule. In this case the laboratory data are measured energy-level positions and Franck-Condon factors for transitions between levels, which can be used to synthesize the trajecl($(0)(9i)(2 exp(-iEit/Fi). Application of this methodology tory y(r) = to multidimensional potential-energy surfaces is also, in principle, possible. The results just described establish the existence of a correspondence between the control of molecular dynamics when only the initial and final states are specified and the determination of a field that generates a defined trajectory of an observable. The distinction between the approaches arises from the point of view taken. This correspondence also provides, in a sense, a connection between the modem view of the control of molecular dynamics and the intuitive “driven bond-breaking” view.

xi

270

S. A. RICE

IX. coNcLusIoNs This overview was designed to illustrate that control of quantum manybody dynamics is in principle both possible and experimentally feasible. At present, the analyses described are likely to be most important as tools for learning more about molecular dynamics and for testing concepts advanced to describe aspects of molecular dynamics. The cost of photons is sufficiently great that it seems unlikely that commercial chemical syntheses can be based on the enhanced selectivity of product yield generated by control fields. However, there are likely other practical applications. For example, Kuriziki et al. [79] have shown how interference effects can be used to generate a fast semiconductor optical switch, and this prediction has been verified [80]. Since the underlying principles of control theory as applied to quantum dynamics are very broadly applicable, it is likely that many other applications will be developed in the near future. Indeed, as laser technology improves and as our understanding of complex molecules advances, it is likely that control of quantum many-body dynamics will become a mainstream component of chemical physics.

Acknowledgments The work described in this chapter could not have been carried out without the intellectually stimulating interactions I have had with my co-workers David Tannor, Ronnie Kosloff, Pierre Gaspard, Sam Tersigni, Andras Lorincz, Bjame Amstrup, Roger Carlson. and Meishan Zhao. I have also had many fruitful discussions with Paul Brumer, Moshe Shapiro, Kent Wilson, Herschel Rabitz, Jeff Cina, and Graham Fleming. This research was supported by grants from the National Science Foundation.

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(1 989). 7. S. Shi, A. Woody, and H. Rabitz, J. Chem. Phys. 88, 6870 (1988). 8. A. P. Peirce, M. A. Dahleth, and H. Rabitz, Phys. Rev. A 37, 4950 (1988). 9. A. P. Peirce, M. A. Dahleh, and H. Rabitz, Phys. Rev. A 42, 1065 (1990). 10. S. Shi and H. Rabitz, Comp. Phys. Commun. 63,71 (1991). 11. P.Gross, D. Neuhauser, and H. Rabitz, J. Chem. Phys. 96, 2834 (1992). 12. S. M. Park, S-P. Lu, and J. Gordon, J. Chem. Phys. 94, 8622 (1991). 13. S-P.Lu, S. M. Park, Y. Xie, and R. J. Gordon, J. Chem. Phys. 96,6613 (1992). 14. V. Kleiman, L. Zhu, X. Li, and R. J. Gordon, J. Chem. Phys. 102,5863 (1995).

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45. J. W. Clark, in Condensed Matter Theories, Vol. 11, E. V. Ludena, Ed., Nova Scientific Publishers, Cammack, New York, 1996. 46. T. J. Tam,G. Huang, and J. W. Clark, Math. Modelling 1, 109 (1980). 47. G. M. Huang, T.J. Tam,and J. W. Clark, J. Math. Phys. 24, 2608 (1983). 48. C. K. Ong, G. M. Huang, T. J. Tam, and J. W. Clark, Systems Theory 17, 335 (1984). 49. J. W. Clark, C. K. Ong, T. J. Tam, and G. M. Huang, Math. Systems Theory 18,33 (1985). 50. T. J. Tam, J. W. Clark, and G. M. Huang, in Modeling and Control of Systems, A. Blaquiere, Ed., Springer, Berlin, 1995. 51. V. Ramakrishna, M. V. Salapaka, M. Dahleh, H. Rabitz, and A. Peirce, Phys. Rev. A 51, 960 (1995). 52. S. Tersigni, P. Gaspard, and S. A. Rice, J. Chem. Phys. 93, 1670 (1990). 53. M. Shapiro and P. Brumer, J. Chem. Phys. 103,487 (1995). 54. A. P. Peirce, M.A. Dahleh, and H. Rabitz, Phys. Rev. A 37, 4950 (1988). 55. M. Zhao and S. A. Rice, J. Chem. Phys. 95, 2465 (1991). 56. M. Demiralp and H.Rabitz, Phys. Rev. A 47, 809 (1993). 57. H. Zhang and H. Rabitz, Phys. Rev. A 49,2241 (1994). 58. H. Zhang and H. Rabitz, J. Chem. Phys. 101,8580 (1994). 59. B. Amstrup, G. 1. Toth, G. Szabo, H. Rabitz, and A. Lorincz, J. Phys. Chem. 99, 5206 (1995). 60. R. S . Judson and H. Rabitz, Phys. Rev. Lett. 68, 1500 (1992). 61. B. Amstrup, R. J. Carlson, A. Matro, and S. A. Rice, J. Phys. Chem. 95, 8019 (1991). 62. B. W. Shore, The Theory of Coherent Atomic Excitation, Vol. 2, Wiley, New York, 1990. 63. M. Kaluza and J. T. Muckeman, Chem. Phys. Lett. 239, 161 (1995). 64. W. H. Miller, N. C. Handy, and J. E. Adams, J. Chem. Phys. 72,99 (1980). 65. W. H. Miller, J. Phys. Chem. 87, 3811 (1983). 66. W. Miller, in The Theory of Chemical Reaction Dynamics, D. C. Clary, Ed., Reidel, Boston, 1986, p. 27. 67. W. Miller, in Potential Energy Sulface and Dynamics Calculations, D. G. Truhlar, Ed., Plenum, New York, 1981, p. 243. 68. D. G. Truhlar, A. D. Isaacson, R. T. Skodje, and B. C. Garrett, J. Phys. Chem. 86,2252 (1982). 69. D. G. Truhlar, A. D. Isaacson, and B. C. Garrett, in Theory of Chemical Reaction Dynamics, Vol. 4, M. Baer, Ed., CRC, Boca Raton, FL, 1985, p. 65. 70. B. C. Garrett and D. G. Truhlar, in Potential Energy Surface and Dynamics Calculations, D. G. Truhlar, Ed., Plenum, New York, 1981, p. 897. 71. D. G. Truhlar, F. B. Brown, R. Steckler, and X . Isaacson, NATO AS1 Sex C 170, 285 (1986). 72. D. Lu, M.Zhao, and D. G. Truhlar, J. Compr. Chem. 12,376 (1991). 73. D. Lu and D. G. Truhlar, J. Chem. Phys. 99,2723 (1993). 74. N. Makri and W. H. Miller, J. Chern. Phys. 86, 1451 (1987). 75. B. C. Garrett and D. G. Truhlar, J. Phys. Chem. 86, 1136 (1982). 76. P. Gross, H.Singh, H. Rabitz, K. Mease, and G. M.Huang, Phys. Rev. A 47,4593 (1993). 77. Y.Chen, P. Gross, V. Ramakrishna, and H. Rabitz, J. Chem. Phys. 102, 8001 (1995).

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78. Z-M. Lu and H. Rabitz, J. Phys. Chem. 99, 13731 (1995). 79. G. Kurizki, M. Shapiro, and P.Brumer, Phys. Rev. B 39, 3435 (1989).

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E.Dupont, P.B. Corkum, H. C. Liu, M. Buchanan, and 2. R. Wasitewski, Phys. Rev. k t t . 74,3596 (1995).

DISCUSSION ON THE REPORT BY S. A. RICE Chairman: M. Quack

B. Kohler: I would like to make a comment about the experimental status of shaped pulse control. A series of experiments have been performed in Prof. Kent Wilson’s laboratory that directly validate the use of tailored light fields for controlling molecular quantum dynamics. In one experiment we sought to control the evolution of a vibrational wavepacket in the bound region of an electronically excited state of the 12 molecule [I]. Specifically, we chose to produce a focused, minimumuncertainty wavepacket centered about a particular internuclear separation at a particular time after the application of a control pulse. To achieve this target, it is necessary to overcome the wavepacket spreading that is usually observed when an ultrashort but otherwise untailored pulse prepares a nuclear wavepacket on an anharmonic potentialenergy surface. Optimal control theory showed that the best light field in the linear response limit for achieving our target consists of a single, approximately Gaussian-shaped pulse with substantial phase modulation in the form of a pronounced negative-frequency chirp. This pulse was subsequently synthesized as well as possible in the laboratory, and the resultant wavepacket focusing at the target time was confirmed by monitoring the total fluorescence induced by a second, variabiy delayed probe pulse. These results demonstrate the feasibility of using tailored light fields designed by optimal control theory to control the quantum evolution of molecules. Finally, I would like to point out that the ability to produce focused wavepackets and thereby localize a molecule in a chosen region of phase space at a chosen target time can play a significant role in controlling the outcome of chemical reactions using two or more light pulses as originally proposed by Tannor and Rice 121. Our results on iodine demonstrate for the first time that “active” localization with shaped pulses is experimentally achievable. In fact, in the most recent experiments carried out in San Diego a two-pulse sequence has been used to control the predissociation of gaseous sodium iodide. In this experiment the second pulse was timed to transfer a wavepacket that had been focused by an initial, tailored control pulse to a higher

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potential-energy surface and thereby reduce the amount of neutral product formation due to predissociation.

1. B. Kohler, V. V. Yakovlev, J. Che, 1. L. Krause, M. Messina, K. R. Wilson, N. Schwentner, R. M. Whitnell, and Y. Yan, Phys. Rev. Letr. 74, 3360 (1995).

2. D. J. Tannor and S. A. Rice, J. Chem. Phys. 83,5013 (1985).

J. Manz: Prof. S. A. Rice has given a comprehensive survey and classification of various theoretical strategies for laser control, together with important experimental examples. The most fundamental distinction is between (i) the strategies of Brumer and Shapiro [l], who employ phase control of interfering excitation pathways from reactants to products using continuous-wave lasers, and (ii) the strategies of Tannor et al. [Z] and Rabitz [3], who employ ultrashort laser pulses to drive the system coherently from the reactant to the product channels (see current chapter). I would like to point to some additional work pioneered by Paramonov and Savva already in 1983 [4] and developed further in our group [5] for laser control of vibrational transitions and isomerizations (see the survey by Korolkov et al., this volume). Furthermore, we have demonstrated the possibility of laser control of reaction rates in Ref. 6. These theoretical studies [MI employ (series of) infrared (IR) femtosecond/picosecond laser pulses in order to drive the nuclear wavepacket from the reactant to the product on a single electronic state, typically the ground state. In contrast, Tannor et al. [2] employ ultraviolet/visible (UV/VIS) femtosecond laser pulses inducing electronic transitions. We have also developed an IR laser pulse variant of optimal control theory [7],similar to the theory pioneered by Rabitz et al. [3]. All strategies [4-71 employ ultrashort infrared laser pulses; they may be classified, therefore, as variants of strategy (ii), according to the classification scheme of S. A. Rice. My question to Prof. B. Kohler (as representative of the group of K. R. Wilson) is whether he would agree with S. A. Rice’s classification that puts the technique of K. R.Wilson et al. [8] into strategy (ii)? What are the fundamental analogies and what are the differences between their approach [8] and the Tannor-Rice-Kosloff-Rabitz approach (see Refs. 2 and 3 and current chapter)? Finally, I should like to point to another strategy (iii) of laser control by vibrationally mediated chemistry that is achieved by IR + UV continuous-wave (CW) multiphoton transitions (see the pioneering papers by Letokhov [9] and sequel theoretical developments [101 and experimental applications [1 11). 1 . M. Shapiro and P. Brumer, J. Chem. Phys. 84, 4103 (1986); P. Brumer and M. Shapiro, Acc. Chem. Res. 22,407 (1994).

2. D. J. Tannor and S. A. Rice, 1. Chem. Phys. 83, 5013 (1985); D. J. Tannor, R.

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Kosloff, and S. A. Rice, J. Chem. Phys. 85, 5805 (1986); D. J. Tannor and S. A. Rice, Adv. Chem. Phys. 70,441 (1988). 3. S. Shi, A. Woody, and H.Rabitz, J. Chem. Phys. 88, 6870 (1988); W. S. Warren, H.Rabitz. and M. Dahleh, Science 259, 1581 (1993). 4. G. K. Paramonov and V. A. Savva, Phys. Lett. A 97, 340 (1983); G. K. Paramonov, V. A. Savva, and A. M. Samson, Infrared Phys. 25, 201 (1985); G. K. Paramonov. Phys. Lett. A 152, 191 (1991); G. K. Paramonov, in Femtosecond Chemistry, Vol. 2, J. Manz and L. Woste, Eds., Verlag Chemie, Weinheim, 1995, p. 671. 5. J. E. Combariza, B. Just, J. Manz, and G. K. Paramonov, J. Chem. Phys. 95, 10355 (1991); M. V. Korolkov, Yu. A. Logvin, and G. K. Paramonov, J. Chem. Phys., in press; M. V. Korolkov and G. K. Paramonov, Phys. Rev. A, submitted; M. V. Korolkov, G. K. Paramonov, and B. Schmidt, J. Chem. Phys., in press; M. V. Korolkov, J. Manz, and G. K. Paramonov, J. Chem. Phys., in press; M. V. Korolkov, J. Manz, and G. K. Paramonov, J. Phys. Chem., in press. 6. T. Joseph and J. Manz, Molec. Phys. 58, 1149 (1986). 7. W. Jakubetz, J. Manz, and H.-J. Schreiber, Chem. Phys. Lett. 165, 100 (1990); W. Jakubetz, E. Kades, and J. Manz, J. Phys. Chem. 97, 12609 (1993). 8. B. Kohler, J. L. Krause, F. Raksi, C. Rose-Petruck, R. M. Whitnell, K. R. Wilson, V. V. Yakolev, Y. J. Yan, and S. Mukamel. J. Phys. Chem. W, 12602 (1993); J. L. Krause, R. M. Whitnell, K. R. Wilson, and Y. J. Yan, in Femtosecond Chemistry, Vol. 2, J. Manz and L. Woste, Eds., Verlag Chemie, Weinheim, 1995, p. 743; B. Kohler, V. V. Yakovlev, J. Che, J. L. Krause, M. Messina, K. R. Wilson, N. Schwentner. R. M. Whitnell, and Y. J. Yan, Phys. Rev. Lett. 74, 3360 (1995). 9. V. Letokhov, Science 180,451 (1973). 10. E.Segev and M. Shapiro, J. Chem. Phys. 77,5604 (1982); V. Engel and R. Schinke, J. Chem. Phys. 88, 6831 (1988); D. G. Imre and J. Zhang, J. Chem. Phys. 89, 139 (1989).

11.

F. F. Cnm, Science 249, 1387 (1990).

B. Kohler: The approach of Wilson and co-workers to the control of quantum dynamics within the linear response approximation has, I believe, much in common with the perturbation theory results of Rice and others. But as Prof. Rice points out in his report, by recasting the control problem in terms of the dynamics of a density matrix in phase space, the Wilson group’s formulation permits the treatment of mixed states as well as the treatment of external interactions with the environment (i.e., solvent interactions) in a particularly straightforward manner. In addition, no restrictions are imposed on the structure of the electric field of the light (other than the requirement that the field be zero outside of a temporal interaction window), and the solutions are globally optimal in the weak-field sense. Three interesting aspects of the optimal fields that were computed in San Diego for several timedomain control scenarios are (1) the tailored fields are relatively simple; (2) the control is quite robust to changes to the tailored fields [ 11; and

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(3) these weak-field solutions may be used to control a large molecular population either by scaling their intensity or by using them as a starting point in an iterative search for strong-field solutions [2]. 1. J. L. Krause, R. M. Whitnell, K. R. Wilson, Y. Yan, and S. Mukamel, J. Phys. 99, 6562 (1993); B. Kohler, J. L. Krause, F. Raksi, C. Rose-Petruck, Whitnell, K. R. Wilson, V. V. Yakovlev, Y. Yan, and S . Mukamel, J. Phys. 97, 12602 (1993). 2. J. L. Krause, M. Messina, K. R. Wilson, and Y. Yan, J. Phys. Chem. 99, (1995).

Chem.

R. M. Chem. 13736

S. A. Rice: I agree with Prof. Kohler that the use of a density matrix formalism by Wilson and co-workers generalizes the optimal control treatment based on wave functions so that it can be applied to, for example, a thermal ensemble of initial states. All of the applications of that formalism I have seen are based on perturbation theory, which is less general than the optimal control scheme that has been developed by Kosloff, Rice, et al. and by Rabitz et al. Incidentally, the use of perturbation theory is not to be despised. Brumer and Shapiro have shown that the perturbation theory results can be used up to 20% product yield. Moreover, from the point of view of generating an optimal control field, the perturbation theory result can be used as a first guess, for which purpose it is very good. S. Mukamel: The phase-space formulation of coherent control made using the density matrix and its Wigner representation generalizes the wave-function-based formulation in several important aspects: It allows performing calculationsat finite temperatures,the development of reduced descriptions, and the incorporation of more general constraints and “objectives.” In the perturbative limit (which is not essential to this formulation) the results were explained using molecular nonlinear (multitime) response functions that do not depend on the driving field. In addition, this approach provides an important semiclassical intuition and analytical expressions for the globally optimized solution. S. A. Rice: I agree with Prof. Mukamel’s remarks. P. W.Brumer: I would tend to agree with Prof. Rice that producing tankerloads of molecules by coherent control methods is probably far-fetched due to the cost of photons. However, we all realize that the possibility exists of producing expensive molecules in this way, or possibly catalysts, where small quantities suffice. However, I do not agree with Prof. Rice’s suggestion that the Brumer-Shapiro schemes, as the optimal control schemes, become more difficult to apply as the molecules increase in size. At the moment, barring the question of the role of overlapping levels, where

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the overlap is due to the natural linewidths, larger molecules should behave the same way, with respect to coherent control, as small ones. In particular, note that many of our proposed control scenarios provide the experimentalist with a clear-cut statement of which parameters need to be varied to achieve control. Further, they tend to utilize relatively simple laser pulse (or CW) characteristics. Thus, our approach would apply to larger molecules in the same way as to smaller molecules; that is, the experimentalist needs only to vary the indicated parameters (e.g., laser intensities, phases, etc.) and search for control in this parameter space. I note, however, the caveat above: Our work to date has ignored the possibility of overlapping levels due to the high density of molecular states coupled with the background radiation field. We expect to examine these cases shortly.

M.Quack I should like to point out that, in spite of the pessimism expressed by Stuart Rice (and in the discussion by Paul Brumer) concerning possible applications of reaction control on a large-scale chemical production basis, one may have a much more optimistic outlook in the following sense: One can think of a future use of the reaction control on biomolecules (enzymes, DNA, RNA, etc.) in order to design specific biomolecules by selective reactions. Currently such design is carried out by “wet biochemistry” techniques. Future design would perhaps be by physical (laser) techniques. The large-scale chemical production occurs then subsequently by ordinary biomolecular multiplication and use in biotechnological processes. The cost of the photons (or the whole initial process) in generating the modified biocatalysts would play no role in such a scheme. The potential advantage of physical methods in molecular design would be their anticipated better general applicability. Of course, the exact form of the methods of physical (laser chemical) molecule design may differ from what we discuss now. I often like to make the analogy with the first magnetic resonance experiments in atomic beams following Stern and Gerlach and I. Rabi. At that time one would not have predicted today’s general use of magnetic resonance in organic, inorganic, and biochemical analytical laboratories (and in medicine). Yet this is exactly what happened, but of course not with molecular beams. Thus, future methods of physical (laser) reaction control may look a bit different from what is discussed now. One must distinguish principles from practical realization.

S. A. Rice: I do not wish to play the role of cynic with respect to the potential commercial use of active control of product selection in a reaction, but I suspect the path to such commercial exploitation

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of the basic ideas and to the creation of viable applications has many potholes and detours, some likely minor and some likely serious. I find Prof. Quack’s suggestions (and enthusiasm) for possible amplification schemes that decrease the cost of active control to be stimulating, and I hope they will prove to be workable. M. Shapiro: I am more optimistic about the possible practical applications of coherent control. In particular, if one stays within the confines of perturbation theory, using two-path control, there exists a generic expression, in the form of a quadratic polynomial, for the probability of observing a given product. Provided one can identify spectroscopically the two intermediate states needed for this type of control, one need not do any calculation. One can hone in on the desired field parameters by simply measuring the product yield at three independent combinations of field parameters. In this way one completely determines the quadratic polynomial coefficients in the known generic form, thereby knowing the behavior of the system for all sets of field parameters, including the ones that maximize the probability of observing a given product or quantum state. S. A. Rice: I agree with the optimism of Prof. Shapiro concerning the broad range of opportunities for use of optimal control methods. The point I wished to make concerning large molecules is that their spectra are complex because there are many degrees of freedom. It is likely that states that yield different products are intermingled. With a two-path interference control scheme these must be identified so that a small change in excitation wavelength does not access unwanted species, as was seen by Gordon even for the simple molecule HI. There is a corresponding problem with the optimal control scheme, since it is difficult to visualize the dynamics in the full dimensionality of the potential-energy surface. Accordingly, I believe we need a reduced description to find our way through the complicated dynamics or to locate the states that produce the desired products. R.A. Marcus: I have a question for Prof. Rice: Do most industrial photochemical processes involve chain reactions? If so, one suspects that they will be less susceptible to “coherent control.” Are there some industrial photochemical processes not involving free radical chains? I gather that you hold the view, which seems reasonable, that the opportunities for coherent control are greater in devices such as electronic switches and generally in telecommunications? S. A. Rice: I think the first industrial applications of control techniques such as I have discussed will be for optoelectronic switching and, possibly, other optoelectronic devices. The use of control methods

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to synthesize high-value complex molecules, assuming we learn how to predict the general features of the control field and develop suitable feedback methods to generate the actual field needed, is in my opinion not yet in sight. I understand that with modem biotechnology amplification methods it is in principle possible to generate a very special complex intermediate by use of optimal control methods and then use those amplification tools for production. I know nothing about the necessary techniques, costs, and so on, so I leave speculation concerning that matter to others.

K. Schafber: Prof. Marcus’s inquiry whether there are any examples of industrial photochemical nonradical chain production processes can be answered positively, although one has to admit that the usage of this sector of synthetic photochemistry has been quite limited, much to the disappointment of organic photochemists. The actual progress in applied photochemical technology can be learned from publications by Roberts et al. [I] and Braun et al. [2]. The most frequent arguments against photochemical production, whenever alternatives to photochemical processes are at hand, concern the high cost of electric energy, the need to invest in new plant facilities (arguments that occasionally are debatable), and the lack of experience in the design of lamp sources and reaction vessels. These latter arguments are unfortunately true to an often embarrassing extent. The fact is that even in large chemical companies the know-how in design of photochemical plants is frequently still in its infancy. Savings in ever-increasing costs of working up procedures owing to substantial decreases in pollutants very often do not seem to counterbalance the arguments against photochemistry. As a general rule, production quantities for photoreactions with quantum yields of < 1 are limited to around 100 tons per year, and photoreactor design tends to limit the size to units of about 20 kW. As a consequence, productions concentrate mostly on small batches of highly prized intermediates and end products, predominantly in pharmaceutical and perfume industries. Photophysical and photochemical selectivities are often provided for by the structural complexity of relatively large organic molecules, the phototransformationsof which most often are independent of excitation wavelength (practically all preparative reactions are carried out in solution and occur from the lowestlying excited state of a given spin multiplicity; such cases have not been addressed by Prof. Rice). For example, internal conversion to ground state and fluorescence may be the only depletion channels competing T I intersystem with a reaction from the SI excited state, or SI crossing may be predominant and one uni- or bimolecular triplet pro-

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12. Prigogine, I., Bellemans, A., and Naar-Colin, C., J . Chem. Phys. 26, 751 (1957). 13. Prigogine, I., The Molecular Theory of Solutions, North-Holland Publ. Comp., 1957, Chapters XVI and XVII. 14. Mathot, V., Comfit. rend. Riunion sur les Changements de Phases, Paris, 1952, p. 115. 15. Prigogine, I., Trappeniers, N., and Mathot, V., Discussions Faraday SOC. 15, 93 (1953);J. Chem. Phys. 21, 559-60 (1953). 16. Lennard-Jones, J. E., and Devonshire, A. F., Proc. Roy. SOC., A163,63 (1937) and A164,l (1938). 17. See for instance Guggenheim, E. A., Mixtures, Oxford University Press, 1952, Chapter X. 18. Hijmans, J., Physica, 27, 433 (1961). 19. Holleman, Th., and Hijmans, J., Physica, 28, 604 (1962). 20. American Petroleum Institute, Research Project 44, Pittsburg, 1953, Selected values of physical and thermodynamic properties of hydrocarbons and related compounds, Table 20d. 21. Reference 20, Table 2Om. 22. Waddington, G., and Douslin, D. R.J. Am. Chem. SOC. 69,2274 (1947); Osbome, N. S., and Ginnings, D. C.,J. Res. Nut. Bur. Stand., 39,453 (1947); Waddington, G., Todd, S. S., and Hufhan, H. M., J . Am. Chem. SOC. 69, 22 (1947). 23. Reference 20, Table 20k. 24. McGlashan, M. L., and Potter, D. J. B.,Joint Conference on Thermal and Transport Properties of Fluids, London, 1957, session 1, paper 10. 25. Boelhouwer, J. W. M., Physica 26, 1021 (1960). 26. Boelhouwer, J. W. M., Physica 34, 484 (1967). 27. Flory, P. J., Orwoll, R. A,, and Vrij, A., J. Am. Chem. SOC.86, 3507 (1964). 28. Brensted, J. N., and Koefoed, J., Kgl. Danske Videnskab. Selskab. Mat. Fys. Medd. 22, No. 17, 1 (1946). 29. Hijmans, J., Mol. Phys. 1, 307 (1958). 30. Longuet-Higgins, H. C., Discussions Faraday SOC.15, 73 (1953). 31. Dixon, J. A . , J . Chem. Eng. Data 4, 289 (1959). 32. See for instance : CarathCodory, C., Conformal Representation, Cambridge University Press, 1952, Chapters 10 and 11. 33, Desmyter, A., and van der Waals, J. H., Rec. Trav. Chim. 77, 53 (1958); van der Wads, J. H., private communication. 34. Holleman, Th., Physica 29, 585 (1963). 35. Hijmans, J., and Holleman, Th., Mol. Phys. 4, 91 (1961). 36. Van der Waals, J. H., and Hermans, J. J., Rec. Truer. Chim. 69, 949 (1950); van der Wads, J . H., Thesis, Groningen, 1950. 37. McGlashan, M. L., and Morcom, K. W., Trans. Faraday SOC. 57, 907 (1961). 38. Bhattacharyya, S. N., Patterson, D., and Somcynsky, T., Physica 30, 1276 (1964): 39. Holleman, Th., Physica 31, 49 (1965). ,

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J. Manz 1. In this chapter Prof. Rice has advocated two techniques that should be useful for evaluations of optimal fields for laser control of chemical reactions: (i) reduced space of eigenstates for representations of nuclear wavepackets and (ii) the use of effective reaction coordinates. Both techniques have already been used for efficient evaluations of reaction probabilities in model reactions. See, for example, Ref. 1 for the prediction of population inversion and Ref. 2 for the demonstration of rather strong deviations of chemical reactions from the reaction path, specifically in the case of hydrogen transfer reactions. 2. Very recently, Jakubetz et al. have extended the applications of our variant of Rabitz’s theory of optimal control by IR femtosecond/picosecond laser pulses [3] from vibrational transitions to isomerizations, specifically for the HCN + CNH reaction [4]. 3. Prof. S. A. Rice has pointed to another experimental verification of the Tannor-Rice-Kosloff scheme, carried out by Prof. G. R. Fleming. I would like to ask Prof. Fleming whether he could explain to us his experiment, that is, how are the two pump and control laser pulses used to control the branching ratio of competing chemical products? 1. J. Manz and H. H. R. Schor, Chem. Phys. LRtt. 107, 549 (1984). 2. B. Hartke and J. Manz, J. Am. Chem. Soc. 110,3063 (1988).

3. S. Shi, A. Woody, and H. Rabitz, J. Chem. Phys. 88, 6870 (1988); W.Jakubetz, J. Manz, and H.-J. Schreiber, Chem. Phys. Lett. 165, 100 (1990); W. Jakubetz, E. Kades, and J. Manz, J. Phys. Chem. 97, 12609 (1993). 4. W. Jakubetz, B. L. Lan, and V. Parasuk, in Femtosecond Chemistry and Physics of Ultrafast Processes, M. Chergui, Ed., World Scientific, Singapore, 1996; W. Jaku-

betz and B. L. Lan, in preparation.

S. A. Rice: I agree with Prof. Manz that reduced representations

of many-body dynamics have been introduced many times in other

problems. Such representations are common in statistical mechanics; for example, the Boltzmann equation describes the time evolution of a gas at the level of the single-particle distribution function, due to collisions that are representative of some of the properties of the twoparticle dynamics, and with neglect of all higher order dynamics. In general, a reduced representation is an exact transfonnation of the Nbody dynamics from one formalism to another formalism. The value of the reduced representation is that it suggests new approximations that could not easily be envisaged in the original N-body formulation; I expect the same advantage to apply to reduced representations used in control theory.

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G. R. Fleming: In reply to Prof. Manz let me say that our experiments [J. Chem. Phys. 93, 856 (1990); 95, 1487 (1991); 96, 4180 (1992)l were the first femtosecond experiments in which the optical phase was explicitly controlled. The trick amounts to figuring out how to vary the delay in integer (integer +; or integer +$) multiples of the wavelength. D. J. Tannor: I would like to point out that the Scherer-Fleming wavepacket interferometry experiment is very different from the Tannor-Rice pump-dump scheme, in that it exploits optical phase coherence of the laser light (optical phase coherence translates into electronic phase coherence between the wavepackets on different potential surfaces). However, there was a paragraph in the first paper of Tannor and Rice [J. Chem. Phys. 83,5013 (1985), paragraph above 4. (ll)] that did in fact discuss the role of optical phase and suggested the possibility of experiments of the type performed by Scherer and Fleming. M. Quack: Prof. Rice has given a superb review and insight into the field. In a sense, it is a wonderful complement to his lecture on dynamics of molecules in Schliersee, published just 20 years ago [ 13. I would like to ask Prof. Rice how he would fit in the scheme proposed many years ago by Tom George, who suggested manipulating effective Bom-Oppenheimer potential surfaces for bimolecular reactions with strong laser fields in order to achieve reaction control. I realize that this was very hypothetical, requiring very intense fields that would presumably lead to ionization, but it still seems to me interesting just as a concept. 1 . S. A. Rice, in Excited States, Vol. 2, E. C. Lin, Ed., Academic, New York, 1975, p. 111.

S. A. Rice: The coupled matter-radiation system considered in the control schemes can, indeed, be studied from the point of view of dressed potential-energy surfaces, as suggested by the remark by Prof. Quack. We find it more convenient to use the equivalent point of view of continuous transfer of amplitude back and forth between the undressed potential-energy surfaces, because the formalism we have developed calculates the temporally and spectrally shaped field for that dynamical representation. P. W. Brumer: Tom George did indeed do significant work in control of reactions. However, if I recall correctly, his approach required extremely high field strengths and, as such, induced undesirable processes that competed with control (e.g., molecule ionization).

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G. Gerber: The w3 = 3 0 and w scheme has been experimentally explored by Dan Elliott using CW laser radiation and by Bob Gordon using nanosecond radiation. I would like to know the viewpoint of Prof. Rice about what we would learn additionally by using ultrafast laser pulses. In ultrashort laser excitation, the individual levels under consideration are coherently coupled. S. A. Rice: To answer the question by Prof. Gerber, I should emphasize that the key element of the Brumer-Shapiro scheme is the exploitation of interference between two pathways connecting the same initial and final states. Unless the excitation sources connect the same initial and final states, and not different ones for the two pathways, the interference effect is compromised. The use of short laser pulses for the two excitation pathways has the potential disadvantage of providing bandwidth for the excitation of other pathways in addition to the desired pathways; hence the pulse must be kept long enough to avoid this possibility.

EXPERIMENTAL OBSERVATION OF LASER CONTROL: ELECTRONIC BRANCHING IN THE PHOTODISSOCIATION OF Na2 A. SHNITMAN, I. SOFER*, I. GOLUB, A. YOGEV* and M. SHAPIRO** Departments of Chemical Physics and *Energy and Environment The W e i m n n Institute of Science Rehovot, Israel 76100, 2. CHEN and P. BRUMER

Chemical Physics Theory Group and The Ontario Laser and Lightwave Research Centre University of Toronto, Toronto MSS IAI, Canada.

Control over the product branching ratio in the photodissociation of Naz into Na(3s) + Na(3p), and Na(3s) + Na(3d) is demonstrated using a two-photon incoherent interference control scenario. Ordinary pulsed nanosecond lasers are used and the Na;! is at thermal equilibrium in a heat pipe. Results show a depletion in the Na(3d) product of at least 25% and a concomitant increase in the Na(3p) yield as the relative frequency of the two lasers is scanned. We report the experimental observation of laser control over a branching photochemical reaction. The reaction studied is the 2-photon dissociation of **Communication presented by M. Shapiro Advances in Chemical Physics. Volume 101: Chemical Reactions and Their Control on the Femtosecond Time Scale, XXth Solvay Conference on Chemistry, Edited by Pierre Gaspard, Irene Burghardt, I. Prigogine, and Stuart A. Rice. ISBN 0-47 1 - 18048-3 0 1997 John Wiley & Sons, Inc.

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the Na2 molecule at energies where one Na atom is in its ground state and one Na atom is in the 3p, 4s or 3d states, i.e.

Na2

2hv ---t

Na(3s)+ Na(3p), Na(3s)+ Na(4s), Na(3s)+ Na(3d).

Control is demonstrated over the Na(3d), Na(3p) branching ratio. Achieving laser control over dynamical processes has been a longstanding goal of both physicists and chemists. Recent theoretical work [1]-[3] has shown that this goal may be achieved by manipulating quantum interferences, an area of research known as coherent control. Experimental verification of the basic principles of coherent control have followed [4]-[ 121 showing, for example, that total ionization rates can be coherently modulated [6]-[9] and that current directionality can be phase-controlled [7]-[ 121. However, there has only been one very recent report [131 of the primary aim of coherent control: to successfully manipulate integral yields into diferent competing product channels. Here we present an experimental demonstration of such control. Our approach is based upon our recent theoretical prediction [I41 that laser induced continuum structure (LICS) [15] can give rise to final channel selectivity. In this arrangement one gives structure to the continuum by optically dressing it with a bound state. We showed theoretically [ 141 that if we dress the continuum with an initially unpopulated bound state using a laser field of frequency w2, while exciting a populated bound state to this dressed continuum using a laser field of frequency w 1 , then a quantum interference arises whose destructive or constructive character depends upon the final channel. (An illustration of this scenario as it applies to Na2 is shown in Fig. 1). Theoretical studies [14] further showed that the character of this interference depends on the relative frequency between the two light fields, and that selectivity between the Na(3p) and Na(3d) channels [Eq.(I)] can be achieved by varying w1 or w2. This effect is virtually independent of the relative phase between the two light fields, i.e. the light fields need not be coherent. Thus, although the control depends on quantum interference, these interferences are not destroyed by incoherence of the incident laser radiation. The fact that this control scenario does not require laser coherence makes it especially attractive for laboratory use since generally available, non-transform limited, nsec dye lasers can be used. In our experiment we use two dye lasers pumped by a frequency-doubled Nd-Yag laser. One dye laser, whose frequency w2 was tuned between 13,312 crn-land 13,328 cm-I, was used

287

F'HOTODISSOCIATION OF Na2

.15 ?

2

W

v1

.lo-

4

a)

3 .05-

8

4

0

a

0-

XIC,

Figure 1. Incoherent Interference Control (IIC) scheme and potential energy curves for Na2. This scheme is composed of a 2 w i photon process proceeding from an initial state, assigned here as ( u = 5, J = 37). via the u = 35, J = 36, 38 levels, belonging to the interacting A' /311u electronic states, and a one 0 2 photon dresses the continuum with the (initially unpopulated) u = 93, J = 36 and u = 93, J = 38 levels of the A ' /311, electronic states.

xu

xu

to dress the continuum with a vib-rotational state of the A' C,/311umixed electronic state [16] of N g . The other dye laser, whose frequency W I was fixed at 17,474.12 cm-', was used to induce a 2-photon dissociation of the u = 5 , J = 37 ground state of Na2, through intermediate resonances (assigned as u = 35, J = 38 and u = 35, J = 36) of the A' C,/311, mixed state. Our W I and w2 pulses, both of -5 nsec duration with the stronger amongst them ( w 2 ) having an energy of -3.5 d, were made to overlap in a heat pipe containing Na vapor at 370 - 410°C. Spontaneous emission from the excited Na atoms [Na(3d) -+ Na(3p) and Na(3p) -+ Na(3s)l resulting from the Na2 photodissociation, was detected and dispersed in a spectrometer and a detector with a narrow bandpass filter. Figure 2 shows experimental Na(3d) and Na(3p) emission as a function of w2 at a fixed w 1 . Each point represents an average over a few hundred laser shots, each chosen to have an w 2 pulse energy which deviates by less than 5% from 3.5 mJ. At these energies we estimate our pulse intensity to be -10' Watts/cm2. We see that when the Na(3d) yield dips, the Na(3p) yield peaks, in accordance with theoretical expectation [171. The controlled modulation of the Na(3p)/Na(3d) branching ratio is seen to exceed 30%.

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-

Figure 2. Experimental Na(3d) fluorescence (solid), and Na(3p) fluorescence (dashed), (both uncalibrated), for the Na2 Na(3s) + Na(3d). Na(3p) IIC scenario whose details are given in Fig. 1, as a function of the w:! frequency. The w l frequency is fixed at 17,474.12 cm-'.

The theoretical calculations [ 171 of the Na(3d) yield resulting from photodissociation of a single initial Na2 bound state are presented in Fig. 3 and contrasted with the experimental results of Fig. 2. The same is done in Fig. 4 for the Na(3p) yield. We see two major Na(3d) dips, accompanied by Na(3p) peaks, in good agreement with the experiment. The calculations were done for the initially unpopulated u = 93, J = 31 and u = 93, J = 33 levels of the mixed A' /311, electronic state, accessed via the u = 33, J = 31 and u = 33, J = 33 intermediate resonances by the 2w 1 photon process. These u, J values differ slightly from those which we experimentally assigned (u = 35 and J = 36 and 38), but the lineshapes were found to change very little with small changes in u,J. Considerng the uncertainties in the theoretical potentials used [ 17, 181, the agreement between theory and experiment (especially in the Na(3d) signal) is impressive. Additional computations [ 141 suggest that the observed experimental sub-structures may be due to the excitation of numerous additional, as yet unassigned, thermally populated vib-rotational Na2 energy levels. The Na(3p) experimental signal is superimposed on a high background due to population of the Na(3p) state by emission from the Na(3d) and Na(4s) states, and due to direct population of Na(3p) from of the Na2 molecule by an 0 1 + 0 2 absorption (not possible energetically for the Na(3d) channel).

xu

289

PHOTODISSOCIATION OF Na2 NaZ-Na+Na(3d);

13310

13315

Theory a n d experiment

13320 13325 oz (cm-')

13330

-

13335

Figure 3. Comparison of the experimental and theoretical Na2 Na(3s) + Na(3d) yields as a function of 0 2 . In the calculation. an intermediate u = 33, J = 31, 33 resonance is used and 0 1 is fixed at 17,720.7 cm-.'. The intensities of the two laser fields are I(w1) = 1 . 7 2 ~lo8 Watts/cm2 and I(w2) = 2 . 8 4 ~10' Watts/cm2. The w2 frequency axis of the calculated results was shifted by -1.5 cm-' in order to better compare the predicted and measured lineshapes.

We also had to overcome radiation trapping effects by monitoring the Na(3p) -+ Na(4s) emission off line-center. The results shown in Figs. 2 and 4 are obtained by subtracting the contribution of these processes from the observed Na(3p) signal. To do so we calibrated the contribution from the Na(3d)

-

2

P

m

68

?

"* 13315

13i20

I

13325 o2 (cm-')

I

I

13330

Figure 4. Comparison of the experimental and theoretical Na2 as a function of 0 2 . with parameters, as in Fig. 3.

13335 -c

Na(3s) + Na(3p) yields

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SHNITMAN et al.

Na(3p) emission via a separate experiment, where we monitored the Na(3p) signal resulting from the direct 2-photon excitation of the Na(3d) state. The direct w I + w2 contribution was accounted for by measuring the Na(3p) signal at different w I + w2 intensities. Because we saturate the one photon w 1 resonance, the dependence of the w I + w2 process was found to be linear in the w2 intensity. Hence, determining the slope and intercept of this linear dependence allowed us to subtract out its contribution at the experimental W I and w 2 intensities. To confirm that the observed Na(3d) dip and Na(3p) peak structures are indeed due to incoherent interference control, i.e., the interference between the w1 and the w2 induced optical processes, we ran the following checks: 1. We verified that what we are seeing is a strong field effect by changing the w2 power. Reducing the power by a factor of -50 resulted in the complete vanishing of the dip/peak structure. 2. We verified that the observed structures are due to the combined action of the two lasers by delaying the W I pulse relative to the w2 pulse. A delay of 339 nsec, guaranteeing no overlap between the pulses, completely eliminated the Na(3d) dips. 3. Real time measurements of the rise and decay of the Na(3d) and Na(3p) signals were performed. If the observed structures are due to an accidental secondary transfer of population from the Na(3d) to the Na(3p) state then such a mechanism would be reflected in the time dependence of the Na(3d) signal. Specifically, if a (collisional or other relaxational) mechanism was in effect at one 0.9 frequency and not at another, thus giving rise to the observed Na(3d) dip and Na(3p) peak at that particular w2 frequency, we would see a faster decay of the Na(3d) signal at that frequency relative to the other. Our findings show an identical decay curve for the Na(3d) signal at all w2 frequencies probed. The only thing which changes is the area under the decay curve. This indicates that it is the actual production of the Na(3d) [Na(3p)] state which is affected by changing w2, and not its subsequent decay [buildup]. 4. We shifted the w I frequency (by -0.19 cm-' and by -0.40 cm-') and examined the effect of such shifts on the dependence of the Na(3d) and Na(3p) yields on w2. Since the 2-photon w 1 absorption is mediated by a (saturated) intermediate I-photon resonance, the w 2 dependence of the Na(3d) and Na(3p) structures should move by an amount equal to the W I shift. This is indeed the case, as demonstrated in Fig. 5, where the updependent structures are seen to red shift, respectively by 0.23 cm-' and 0.37 cm-' . These values are, within our frequency resolution of fo.04cm-' ,in perfect accord with the above expectations. A similar

29 1

PHOTODISSOCIATION OF Na2 I

I

I

I

, The Na(3d) fluorescence as a function of w 2 for three different w1 frequencies. The lowest trace corresponds to an w l value of 17,474.12 Em-'. The upper traces result from red shifting the w l frequency by .I9 cm-I and .40 cm-l. We observe a red shift in the w 2 dependence of the Na(3d) yield of, respectively, .23 cm-l and .37 cm-', each being, within our experimental uncertainty of identical to the respective o 1 shift.

shift of the Na(3p) peaks was also observed, thus verifying the optical origin of the effect. In summary, we have experimentally demonstrated laser control of a branching photochemical reactions using quantum interference phenomena. In addition we have overcome two major experimental obstacles to the general implementation of optical control of reactions: (a) we have achieved control using incoherently related light sources, and (b) we have affected control in a bulk, thermally equilibrated, system. Acknowledgments This work was supported by the Minerva Foundation, Germany, by the Israel Academy of Sciences and by the U.S. Office of Naval Research under contract number N00014-90-J-1014.

References 1. For recent reviews, see M. Shapiro and P. Brumer, Int. Reviews Phys. Chem. 13, 187 (1994); P. Brumer and M.Shapiro, Ann. Rev. Phys. Chem. 43, 257 (1992). See also, P. Brumer and M.Shapiro, Chem. Phys. Lett. 126, 541 (1986). 2. See, D. J. Tannor, and S. A. Rice, Adv. Chem. Phys. 70, 441 (1988); R. Kosloff, S. A. Rice, P.Gaspard. S. Tersigni, and D. J. Tannor, Chem. Phys. 139,201 (1989); S. Shi and

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H. Rabitz, Chem. Phys. 139, 185 (1989); Y. Yan, R. E. Gillilan, R. M. Whitnell and K. R. Wilson, J. Phys. Chem. 97, 2320 (1993). 3. A. D. Bandrauk, J-M. Gauthier and J. F. McCann, Chem. Phys. Lett. 200,399 (1992); M. Yu. Ivanov, P. B. Corkum and I? Dietrich, Laser Physics 3, 375 (1993). 4. C. Chen, Y-Y. Yin, and D. S. Elliott, Phys. Rev. Lett. 64, 507 (1990); ibid, 65, 1737 ( 1990). 5. S. M. Park, S-P. Lu, and R. 3. Gordon, J. Chem. Phys. 94, 8622 (1991); S-P. Lu, S. M. Park, Y. Xie, and R.J. Gordon, J. Chem. Phys. 96,6613 (1992). 6. V. D. Kleiman, L. Zhu, X. Li and R. G. Gordon, J. Chem. Phys. 102,5863 (1995). 7. G. Kurizki, M. Shapiro, and P. Brumer, Phys. Rev. B 39, 3435 (1989). 8. 8. A. Baranova, A. N. Chudinov, and B. Ya Zel’dovitch, Opt. Comm., 79, 116 (1990). 9. Y-Y. Yin, C. Chen, D. S. Elliott, and A. V. Smith, Phys. Rev. Lett. 69, 2353 (1992). 10. E. Dupont, P. B. Corkum, H.C . Liu, M. Buchanan and 2. R. Wasilewski, Phys. Rev. Letters 74, 3596 (1995). 11. B. Sheeny, B. Walker and L. F. Dimauro, Phys. Rev. Lett. 74,4799 (1995). 12. Y.-Y. Yin, R. Shehadeh, D. Elliott, and E. Grant, Proceedings, OSA Annual Meeting, Dallas (1994). 13. L. Zhu, V. Kleiman, X. Li, S. P. Lu, K. Trentelman and R. J. Gordon, Science 270, 77 (1995). 14. Z. Chen, M. Shapiro and P. Brumer, Chem. Phys. Letters 228,289 (1994); J. Chem. Phys. 102, 5683 (1995). 15. See, for example, P. L. Knight, M.A. Lauder and B. J. Dalton. Phys. Rep. 190, 1 (1990). and references therein; 0. Faucher, D. Charalambidis, C. Fotakis, J. Zhang and P.Lambropoulos, Phys. Rev. Lett. 70, 3004 (1993). 16. Z. Chen, M. Shapiro and P. Brumer, J. Chem. Phys. 98, 8647 (1993); J. Chem. Phys. 98, 6843 (1993). 17. Theoretical calculations reported here are for CW excitation. Pulsed laser based computational studies are ongoing. 18. The potential curves and the relevant electronic dipole moments are from I. Schmidt, Ph.D. Thesis, Kaiserslautern University, 1987.

DISCUSSION ON THE COMMUNICATION BY M. SHAPIRO Chairman: M. Quack B. Kohler: Prof. Shapiro, could you add some “nonarbitrary” units

to the y-axis of the data from your interesting Na2 control experiment?

What percentage change in the yield are you able to obtain? M. Shapiro: The change in the Na(3d)/Na(3p) branching ratio is typically 30%-40%.We cannot give absolute units for the yield of each channel because we have not calibrated the fluorescence emission signal. M. S. Child: It seems to me that control by changing the frequency

PHOTODISSOCIATIONOF Na2

293

of the laser is quite different from other control schemes. How does this differ in principle from one photon control by turning from one electronic state to another? M.Shapiro: The method of incoherent interference control used in our experiment is completely general and allows us to use Fano type interferences to control final states even if such lines do not naturally exist. Of course if the molecule accommodates you (as FNO)you do not need this but this is a rare situation. For other situations, especially when the material continuum is slowly varying and tiny changes in the laser frequencies in the one photon transition have absolutely no effect on the product ratios, our method allows for control, via the optical induction of resonances, in complete generality.

COHERENTCONTROLOF BIMOLECULAR SCATTERING P. BRUMER* Chemical Physics Theory Group, Department of Chemistry University of Toronto Toronto, Canada

M.SHAPIRO Department of Chemical Physics The Weizmann Institute of Science Rehovot, Israel

CONTENTS I. Introduction II. Control of Collisions References

I. INTRODUCTION Coherent control of molecular processes has seen enormous progress in the last half decade. Advances have been summarized in a number of recent reviews (see S. A. Rice, this volume; see also Refs. 1 and 2). Consideration *Communication presented by I? Brumer Advances in Chemical Physics, Volume 101: Chemical Reacrions and Their Control on the Femtosecond Time Scale, XXrh Solvay Conference on Chemistry, Edited by Pierre Gaspard, Irene Burghardt, I. Prigogine, and Stuart A. Rice. ISBN 0-471-18048-3 0 1997 John Wiley & Sons, Inc.

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of these papers shows that the vast majority of control papers address the problem of controlling bound-molecule dynamics or photodissociation. By contrast, bimolecular processes, which form the bulk of interesting chemistry, has only received theoretical attention in one paper [3]. In that paper we showed that a variant of the two-level pump-dump control scenario [4] could be used to control bimolecular reactions, Unfortunately, the controlled yield above the reaction threshold was small compared to the natural reactive probability. As such, the proposed scheme was useful primarily below the threshold for reaction. In this chapter we briefly summarize the essence of our recent work [ 5 ] , which introduces a coherent control strategy to control bimolecular collisions, Computations designed to examine the range of control possible with this scenario are currently being carried out [6]. The essential principle of coherent control in the continuum is to create a linear superposition of degenerate continuum eigenstates out of which the desired process (e.g., dissociation) occurs. If one can alter the coefficients a; of the superposition at will, then the probabilities of processes, which derive from squares of amplitudes, will display an interference term whose magnitude depends upon the a;. Thus, varying the coefficients a; allows control over the product properties via quantum interference. This strategy forms the basis for coherent control scenarios in which multiple optical excitation routes are used to dissociate a molecule. It is important to emphasize that interference effects relevant for control over product distributions arise only from energetically degenerate states [7], a feature that is central to the discussion below. Consider now the bimolecular reaction B + C F + G, where B, C, F, and G are atoms or molecules. Here the goal is to control the reactive versus nonreactive probabilities. Creating the required superposition of degenerate eigenstates of B + C means creating continuum states with a welldefined relationship between the internal energies of B, C and the B, C relative translational energy. Optical control of the kinetic energy of a colliding pair can in principle be achieved by excitations of continuum states while a collision takes place [3]. However, such control is difficult because the time spent by the colliding partners in the region where they strongly interact is very short (typically 0.1-1 ps). Indeed, there are very few examples in which direct optical excitation of continuum states in the strong-interaction region (the so-called transition-state spectroscopy) has been achieved [81. A practical way of forming superposition states of correlated scattering states, thereby achieving control, is the topic of this chapter. Space limitations prevent more than a sketch; details can be found elsewhere [5].

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COHERENT CONTROL OF BIMOLECULAR SCATTERING

297

II. CONTROL OF COLLISIONS The approach we advocate is straightforward. Consider the bimolecular process

B(i) + C ---t F + G

(1)

Here we assume that C is an atom, and we have explicitly indicated the internal state of molecule B using the label i. The energy of this state is given by E = EC(O) + €B(i) + EF"(i), where EC(O) is the internal energy of C, E B ( ~ is ) the internal energy of B(i), and E;"(i) is the center-of-mass kinetic energy in the reactant channel, which we have labeled the 0 channel. For two different eigenstates, comprised of B(I) + C and B(2) + C, to be of the same energy requires

Thus, to attain control, we wish to create a superposition of states of B(1) + C and B(2) + C:

with kinetic energies satisfying Eiq. (2). Here l i , X ) with X = B, C are eigenstates, of energy e x ( i ) , of the internal Hamiltonians he and hc of B and C. The IEF(i)) denote plane waves describing the free motion of B relative to C, that is, (RIEF"(i)}I exp(iki . R), where lkil = { 2 p ~ ~ E y ( i ) ) ' / * / h and p ~ =c mBW/(mB + W) is the reduced mass of the BC pair. Then, in a manner analogous to unimolecular control, varying ai will alter the product distributions [ 5 ] . Control over the ai and production of the desired superposition states can be achieved by several routes. One nice way is to utilize the reactants from an earlier photodissociation step, altering the ui by any of a number of coherent control scenarios [2] for this prereactive step. Consider then preparing In, /3) via a "prereactive" stage in which an adduct AB, made up of a structureless atom A and the molecular fragment B, is photodissociated. The AB is assumed to be initially in a pure state of energy Eg and the photodissociation is carried out with a coherent source. Under these circumstances photodissociation produces B in a linear combination of internal states. For

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convenience we assume two such internal states:

AB

--. ho

A+B(i)

i = 1,2

(4)

The total energy in the photolysis stage, denoted EAB, is EAB = E, + ho, where w is the frequency of the photolysis laser. Hence, following the photodissociation process, fragment B is described by In,B) =

C ajli,B)IEg"(i))

i = 1.2

where

where ~ A =B mAmg/(mA + m g ) is the reduced mass of the AB pair and m A and me are the masses of the A and B fragments. Preparing &. ( 5 ) is not, however, sufficient to produce the desired In,@ state [Eq.(3)] since one needs contributions to In, 0) to be of the same energy. This can be achieved by colliding B and C at a specific angle. To see this, consider the following kinematic argument in order to choose 6 , the incident angle of C, and x , the direction of the photolysis fragment B (both defined with respect to V A Bthe , AB center-of-mass velocity vector). We denote the velocity of the jth particle in the AB center of mass system by u, and its laboratory velocity by v,. For particle B we have laboratory velocities v g ( I), vB(2) and corresponding center-of-mass velocities uB( I), ug(2), where, following the photolysis prereactive step, the velocity of the B fragment in the AB center-of-mass system is given, according to EQ.(6), as

We choose is

uc collinear with ue(i) so that the relative BC velocity, UB&,

We focus only on the plus solution, the minus sign giving nonphysical results. Using Eq. (8), the kinetic energy of B relative to C is given as

COHERENT CONTROL OF BIMOLECULAR SCATTERING Ekin

i

6 ()=

ic~Bcv2,c(i) = ; P B c { ~ ~ + u 3 i ) + 2ucue(i))

299

(9)

Imposing the condition in Eq. (2) gives, using Eqs. (2), (7),and (9), that

where ~ C - A B = mC(mA + mB)/(mA

+ mB + mc). Knowing UC, we have that

and

X

sin 8 = uc sin VC

(12)

This then provides the desired 8 and x to ensure that the elements of the superposition state [Q. (3)] are degenerate. The above formalism can be readily extended to general superposition states of the form

i,1

with E F ( i , I ) = E - ~ c ( 1 )- € ~ ( i )Here, . the I E F ( i , 1)) states are plane waves describing the free motion of B relative to C [(RJEp(i,l)) = exp(iki,.R)], where fkirl = {2jtBCEp(i, Z))’/2/h. That is, we can show that such a superposition leads to interference and hence to the possibility of control over the reaction cross sections. However, the experimental realization of such states presents a serious challenge. First, it is entirely unclear how to produce general controllable ail. Second, a typical experiment produces wave functions where the eigenstates of HB, hc, and Kp are uncorrelated and spread out over a considerable energy range. Under these circumstances any quantum interference terms are negligible compared to the uncontrolled contributions. Indeed, the inability to create controllable states such as Eq. (1 3) should serve to emphasize the insight required to design adequate scenarios. Note added in proof: Careful consideration has shown that the above approach pays insufficient attention to the motion of the center of mass

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associated with the B-C collision when B is prepared as described above. Appropriate consideration of this aspect of the collision shows that an alternate scenario is necessary to ensure that quantum interference, and hence control, survives. See references [5] and [6] for details.

References 1. 2. 3. 4. 5. 6. 7. 8.

P. Brumer and M. Shapiro, Sci. Am. 272, 56 (1995). M. Shapiro and P. Brumer, Int. Rev. Phys. Chem. 13, 187 (1994). J. L. Krause, M. Shapiro, and P. Brumer, J. Chem. fhys. 92, 1126 (1990). D. J. Tannor and S. A. Rice, J. Chem. Phys. 83, 5013 (1985); T. Seideman, M. Shapiro, and P.Brumer, J. Chem. Phys. 90, 7132 (1989). M. Shapiro and P. Brumer, Phys. Rev. Lett., 77, 2574 (19%). D. Holmes, P. Brumer, and M. Shapiro. J. Chem. Phys. 105,9162 (1996). P. Brumer and M. Shapiro, Chem. Phys. 139, 221 (1989). B. A. Collings, J. C. Polanyi, M. A. Smith, A. Stolow, and A. W. Tarr, Phys. Rev. Lett. 59, 2551 (1987).

LASER HEATING, COOLING, AND TRANSPARENCY OF INTERNAL DEGREES OF FREEDOM OF MOLECULES D.J. TANNOR* Department of Chemical Physics Weizmann Institute of Science Rehovot Israel R. KOSLOFF AND A. BARTANA Department of Physical Chemistry and the Fritz Haber Research Center The Hebrew University Jerusalem Israel

CONTENTS I. Introduction

11. Instantaneous Dipole Moment: Generalized Einstein B Coefficient 111. Vibrational Heating Using Nondestructive Optical Cycling

IV. Nonevaporative Cooling References

I. INTRODUCTION We are in the infancy of a new era in the physical sciences. It is probable that in the coming years we will learn general strategies for manipulating the *Communication presented by D. J. Tannor Advances in Chemical Physics, Volume 101: Chemical Reactions and Their Control on the Femtosecond Time Scale, XXth Solvay Conference on Chemistry, Edited by Pierre Gaspard, Irene Burghardt, I. Prigogine, and Stuart A. Rice. ISBN 0-471-18048-3 0 1997 John Wiley & Sons, Inc.

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precise quantum states of atoms and molecules using laser light [l-5, 181 (see also S. A. Rice, Perspectives on the Control of Quantum Many-Body Dynamics: Application to Chemical Reactions). The possibilities inherent in manipulation go far beyond simply the excitation of particular quantum states that might otherwise be hard to access but, in addition, to controlling the phase coherence of superposition states in order to achieve desired chemical and physical purposes. This phase coherence can at times be responsible for the most peculiarly quantum effects and at other times for the building of quantum mechanical wavepackets and the recovery of classical-like behavior. One of the new areas to exploit the assumption of optical phase coherence is quantum computing 161. The key concept is that in a set of N two-level systems there are 2'"' basis functions that can be manipulated independently using the laser and that are available for computation, as compared with the N elements that are up or down in conventional computing. A second new area is laser-induced transparency: An ordinarily opaque material can become transparent in the presence of laser light [7]. Again, the idea is simple: If the laser prepares a coherent superposition of states that connect via opposite signs to the excited state, the net absorption will be zero. This same concept is behind lasing without population inversion and population inversion without lasing [8]. Moreover, the same basic concept of preparing a coherently trapped superposition state underlies one of the most successful methods of atom cooling [9, 101. Here, a superposition of translational states is created that is dark for u = 0 and not dark for nonzero u's. As atoms get cooled u 0 and then they no longer interact with the light! This cooling mechanism has allowed cooling to 2 pK, below the limit dictated by random recoil from spontaneous emission. In this brief chapter we discuss two applications of phase tailored pulses to the manipulation of molecules: vibrational heating without demolition and laser cooling of internal molecular degrees of freedom. In both these applications the laser field is used to maintain orthogonality of a lower and upper electronic wavepacket, and as a result there is no absorption. In some sense this work can be viewed as a generalization of the principle of laserinduced transparency and its relatives, discussed in the previous paragraph, from three-level systems to multilevel wavepackets. However, there is a richness in the molecular multilevel case that is absent in the atomic case. The coordinate dependence of the wavepacket in general has a nontrivial spatial dependence that can be manipulated while maintaining the conditions of zero absorption. 3

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II. INSTANTANEOUS DIPOLE MOMENT: GENERALIZED EINSTEIN B COEFFICIENT The time-dependent Schriidinger equation for a two-electronic-state system with transition dipole moment p can be written as

The subscripts g and e refer to the ground and excited electronic state indices, respectively. The term Hg/e refers to the Born-Oppenheimer Hamiltonian for the ground/excited electronic state, respectively; $g,e is the wave function (wavepacket) associated with the ground/excited Bom-oppenheimer potential-energy surface. The two electronic states are coupled by the transition dipole moment p , which interacts with the electric field e ( t ) associated with the laser pulse. Complex values of the field are considered admissible by associating the real and the imaginary parts of the field with the two independent polarizations of the laser light perpendicular to its direction of propagation. It is straightforward to show that Eq. (1) leads to the following equation for the rate of change in excited-electronic-statepopulation:

The quantity peg = (+e\p\+g) is the instantaneous (complex) transition dipole moment, also called the polarization. It is a generalizationof the constant peg, which is essentially the Einstein B coefficient. In the standard Einstein treatment the B coefficients for absorption and stimulated emission are identical. In fact, however, peg is complex and not equal to pge; moreover, it is time dependent. An example of the use of Eq. (2) is provided by the wavepacket interferometry experiments of Scherer, Fleming et al. [ 111. These workers have demonstrated that the phase of the light can be used to control constructive versus destructive interference of wavepackets in the excited electronic state. An alternative way of interpreting their experiment is that the phase of the second pulse relative to the first determines the direction of population transfer between the two electronic states. In the spirit of the present discussion, absorption versus stimulated emission is being controlled by the choice of phase of the light relative to the instantaneous p g e $ peg! Since the direction of population transfer is not determined in this case by population inversion

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alone, it is a short step to consider the possibility of lasing without population inversion and population inversion without lasing using phase-locked laser pulse sequences. It is possible to view Eq. (2) as a generalization of the third of the optical Bloch equations, that is, the equation for population transfer. This leads us to consider analogies between coherence phenomena in two-level systems and the case where there is a wavepacket on each of two electronic state potential-energy surfaces. Moreover, the Feynman-Vernon-Hellwarth (WH) geometric picture, which is so useful for two-level systems [121, may be expected to be useful in the wavepacket context as well. In particular, the applications below to heating and cooling can be viewed as wavepacket generalizations of photon locking in two-level systems [13].

III. VlBRATIONAL HEATING USING NONDESTRUCTIVE OPTICAL CYCLING Several years ago, Nelson posed the following problem [14]: Is it possible to design a pulse sequence, using an impulsive stimulated Raman mechanism, to give large-amplitude vibrational motion on the ground electronic state without creating significant amounts of excited electronic state population? The problem with excited-state population is twofold: It complicates the interpretation of the experimental results, and it is often a precursor to dissociation or ionization. It is simple to see that the condition

where C(t) is some real envelope function (positive or negative) guarantees the condition of zero electronic population transfer [19]. The equation for the energy of the wavepacket on the ground state is given by

The strategy is then to adjust the sign of C(t) in such a way that the vibrational energy on the ground state will monotonically increase. Numerical experiments show that this works beautifully, both for weak fields and strong fields; for strong fields the heating is simply faster (see Figs. l a d ) [19]. It is interesting to contemplate the possibility of extending these ideas

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T

Figure 1. (a) The FVH diagram illustrating photon locking in two-level system. Pseudopolarization (r) vector begins at z = - I , corresponding to all population in ground state, and precesses about field (a)vector (which initially points along x axis) according to equation dr/dt = x r. When r vector reaches x-y plane ( r / 2 pulse), phase of field is changed by b / 2 . This rotates field vector by 90".bringing it into alignment with r vector and making further precession impossible. Physical manifestation of this geometric picture is that although resonant field continues to be applied, there is no further population transferred between two levels. (b)Potential-energystructure of model. Frequencies of ground and excited surfaces are 1.0 and 0.8, respectively. Excited surface shifted by 7.0 units of energy and 3.0 units of distance, which leads to vertical distance of 10.6 units. Dipole operator has slope of 1. Coherent wave function with energy of 1.0 used as initial state on ground surface. Ground and excited absolute values of wave function shown after first exciting pulse (not to scale). (c) Change in population on ground state ( A N g )(dotted line), change in ground-state energy (AE,) (dashed line). and real part of field e (solid line) as function of time. Note monotonic increase in groundstate energy, coming in bursts at times that pulse is on. This is weak-field case; strong-field case has same qualitative behavior but rise in energy is faster and pulse shapes more sharply modulated. ( d ) Power spectrum of pulse in (c). Frequency of vertical transition 9; spectrum shows dips at excited-state vibrational resonances, consistent with condition of zero absorption.

to more than two electronic states. For example, if there is a third higher electronic state that is strongly coupled to the second electronic state, is it still possible to apply the excitation without demolition procedure? The answer when there are three states is manifestly yes; if there are many more excited states, the situation is not clear.

IV. NONEVAPORATIVE COOLING One of the most exciting areas in atomic physics in the last 10 years has been laser cooling of translational motion in atoms to 10-3-10-9 K [9, 10, 151. Several variations on cooling of atomic translational motion have now been proposed, but the generic scheme is as follows: Two monochromatic laser beams are propagated, one along and one opposite an atomic beam; by tuning the frequency of the laser to the red of resonance with a sharp

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30.

20.

10.

0.0

I

-7.0

1

1

-3.0

1.o

time (C)

Figure 1. (Continued)

I

5.0

9.0

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0.0

0

(4

Figure 1. (Conrinued)

electron.,: transition in the atom, those atoms propagating faster an^ therefore having a larger Doppler shift absorb light and are slowed down by the momentum imparted by the photon. Slowly shifting the frequency of the light into resonance progressively slows down more and more of the atoms. An added effect comes from a combination of level splitting in the presence of the light, together with a light-induced polarization gradient, leading to a so-called Sisyphus effect [9]. The cooling limit of this process is dictated by the random recoil imparted by the spontaneous emission of the photon; however, even this limit can be overcome by a method of velocity-dependent coherent population trapping [lo]. It is generally believed that the same schemes for translational cooling cannot be applied to molecules. The reasons are quickly perceived: molecules, even the smallest of them, have a high density of internal states coming primarily from rotations and vibrations, although also from electronic and spin degrees of freedom. The atomic cooling scheme relies on the validity of a two-level system picture, so that a single frequency red shifted from resonance will cool the entire population. In a molecule with a

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congested energy-level spectrum, red shifting from one particular resonance line will cool only those molecules in that one level; then molecules in each level will have to be cooled separately. If there are thousands or millions of internal states occupied, this is quite a challenge. In addition, red shifting away from one level may entail shifting into resonance with another level. Recently, Bahns et al. [161 suggested a strategy for cooling molecules that involves the progressive cooling of rotations, then translations, and finally vibrations. Probably the most challenging of these stages is the first, the cooling of rotations. The Connecticut group proposes to use the molecule itself to generate the laser light at the characteristic absorption frequencies of the rotational manifolds. When this multitude of frequencies impinges on the rotationally hot molecule, it will lead to optical absorption followed by spontaneous emission. If the frequency corresponding to resonant excitation out of the ground rovibrational state is blocked, the population in this state can only grow, through spontaneous emission, and never diminish. One can imagine a variation on this scheme in which all frequencies are at first blocked and then unblocked from lowest to highest according to some schedule. An alternative approach to vibrational cooling uses shaped pulses, again based on photon locking 1171. The goal is to cool an initial thermal rotational or vibrational distribution down to 0 K. The strategy, as in the heating scenario, is to use the excited electronic state as an optical lever for cooling without transferring a significant amount of population upstairs. Formally, the biggest difference from the heating is that the equations must be cast in terms of the density operator to describe the absence of coherence in the initial population and the subsequent incomplete coherence. We begin with some defining equations for the operators when there are two coupled potential surfaces. The density operator is given by p = pg (8 P ,

+ p, @ P , ipi (8 s,

+pt

0s-

where Pele are projection operators for the ground- and excited-state surfaces, respectively, and S, are the raising and lowering operators from one surface to another. The Hamiltonian is given by

H = Ho + V(t) where

(7)

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and the surface Hamiltonians H, and H, are functions of the internal coordinates. Moreover,

The evolution of the density operator is described by the Liouville-von Neumann equation

The equation for population change is now given by 1

d N R - _- . dt A

(p 63 (S+E - SA*))

where ,&t is the phase of the instantaneous dipole and is the phase angle of the radiation field. Moreover, the equation for the change in ground-state energy is now

where t $ , ~ is the phase angle of ( p H R C3 S+).For zero population transfer, choose E = C(r)+((p BS-)), where C(t) is a real function of time and +(X) = X / l X ) = eid' is the phase factor. Under the condition of zero population transfer the change of energy on the ground state becomes

The condition on the function C(t) that will lead to a monotonic decrease in energy is

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310

T +

Imag

--

Q.

L

0

Real

0.5

h

3

cEgz witout dissipation <Eg> with dissipation

0.49

-

I

--------I

9)

Y

6k

0.48

9)

w

C

0.47

0.46

0

10

20

30

40

T h e (fs) (b)

Figure 2. (a) Phase-angle diagram for zero mass transport with monotonic decrease in ground-state energy. Key observation is that zero mass transport condition requires that g u + @ , = 0, n; this determines %e within a sign. Except for pathological case, there will always be one choice of sign that gives %,H + gt < 0, leading to monotonic decrease in ground-state energy; other choice of sign leads to monotonic increase in ground-state energy. (b) Ground surface energy in presence of the cooling pulse as function of time, with and without dissipation. (c) Phase-space density of initial density operator p g ( 0 ) ( Wigner distribution function in position-momentum phase space). Upper panel: stereoscopic projection. Lower panel: contour map. (d) Phase-space display of final density operator without dissipation pg(tf ). Upper panel: stereoscopic projection. Lower panel: contour map.

LASER HEATING, COOLING, AND TRANSPARENCY 6.

4. 2.

0

1

1.5

2

R

(A) (d)

Figure 2. (Continued)

2.5

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These equations are the density matrix analogues of the heating equations in the previous section. Figures 2u-d (see pages 3 10-3 11) show that this strategy is effective in producing monotonic cooling. However, the numerical results indicate that in absolute terms the temperature drop is not dramatic. It turns out that entropy limits the degree of cooling attainable. These entropy considerations, together with alternative laser cooling strategies, will be presented in a separate publication.

Acknowledgment This work was supported by a grant from the U.S. Office of Naval Research.

References 1. A. H. Zewail, Phys. Today 33, 27 (1980). 2. P. Brumer and M. Shapiro, Sci. Am. (March 1995). 3. W. S. Warren, H. Rabitz, and M. Dahleh, Science 259, 1581 (1993). 4. D. J. Tannor, in Molecules in Laser Fields, A. Bandrauk, Ed., Dekker, New York, 1994. 5 . B. Kohler, J. Krause, F. Raksi, K. R. Wilson, R. M. Whitnell, V. V. Yakovlev, and Y. Yan, Accr. Chem. Res. 28, 133 (1995). 6. D. P. DiVincenzo, Science 270,255 (1995). 7. K.J. Boller, A. Imamoglu, and S. E. Harris, Phys. Rev. Lett. 66,2593 (1991); J. E. Field, K. H. Hahn, and S. E. Hams, Phys. Rev. Lett. 67,3062 (1991). 8. M. 0. Scully and M. Fleischhauer, Science 263, 337 (1994); Phys. Today, p. 17 (May 1992). 9. C. N. Cohen-Tannoudji and W. D. Phillips, Phys. Today, p. 33 ( a t . 1990). 10. A. Aspect, E. Arimondo, R. Kaiser, N. Vansteenkiste, and C. Cohen-Tannoudji, Phys. Rev. Lett. 61, 826 (1988). 11. N. F. Scherer, R. J. Carlson, A. Matro, M. Du, A. J. Ruggiero, V. Romero-Rochin, J. A. Cina, G. R. Fleming, and S. A. Rice, J. Chem. Phys. 95, 1487 (1991). 12. L. Allen and J. H.Eberly, Optical Resonance and Two-Level Arums, Dover, New York, 1987. 13. E. T. Sleva, I. M. Xavier, Jr., and A. H. Zewail, J. Opt. Soc. Am. B 3,483 (1986). 14. K. Nelson, in Mode Selective Chemistry, Vol. 24, Jerusalem Symposium on Quantum Chemistry and Biochemistry, by J. Jortner, Ed., Kluwer Academic, Hingham, MA, 1991, p. 527. 15. A. Kastberg, W. D. Phillips, S. L. Rolston, R. J. C. Spreeuw, and P. S . Jessen, Phys. Rev. Lett. 74, 1542 (1995). 16. J. Bahns, P. Gould, and W. Stwalley, ACS Absa (April 1995). 17. A. Bartana, R. Kosloff, and D. J. Tannor, J . Chem. Phys. 99, 196 (1993).

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18. R. Kosloff, S . A. Rice, P. Gaspard, S. Tersigni, and D. J. Tannor, Chem. Phys. 139, 201 (1989). 19. R. Kosloff, A. D. Hammerich, and D. Tannor. Phys. Rev. Leu. 69, 2172 (1992).

DISCUSSION ON THE COMMUNICATION BY D. J. TANNOR Chairman: M. Quack D. M. Neumark We are interested in generating coherent vibrational motion in negative ions, which typically do not have bound excited electronic states. Does your Impulsive Stimulated Raman Scattering (ISRS) scheme work if the excited state is not bound? D. J. Tannor: I think there is a good chance that the scheme, or some variation, may work even with a dissociative intermediate state. The reason is that the excited-state population is small; in some sense it is “virtual” in the old sense of Raman scattering. Moreover, when strong fields are used, the validity of the original Born-Oppenheimer surfaces as a good zero-order picture generally breaks down. In other words, with strong fields, there is no a priori reason to believe the excited-state dynamics will remain dissociative. It would be very interesting for us to do the calculation. J. Manz: Prof. D. Tannor has demonstrated the possibility to achieve laser cooling (or heating) of molecules in the electronic ground state. These effects are achieved, however, by laser-induced transitions to and from the electronic excited state. As a consequence, one may end up with an ensemble of molecules where part of them are cooled (or heated) in the electronic ground state, whereas the rest may be found in the electronic excited state with a rather uncontrolled “mixed” distribution of nuclear states. I would like to ask Prof. Tannor to give us some details about the relative fraction of molecules in the excited state, about their distributions of vibrational states, and about their role in view of the original purpose of laser cooling or heating. D. J. Tannor: In reply to Prof. Manz, I have to emphasize that there are three quantities to examine: 1. Norm (population) in the excited electronic state 2. Energy in the excited electronic state 3. Entropy in the excited electronic state

The scheme guarantees that the norm in the excited state is maintained constant (e.g., at 2%. to avoid sample destruction). The excitedstate energy does not necessarily rise; there is no conservation rule that

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states the sum of the energy in the ground and excited electronic states must be conserved, since the light can carry away energy. However, the entropy in the ground and excited states is in fact conserved, and hence the entropy in the excited state must increase as the entropy in the ground state decreases (in the cooling scenario). This in fact puts limitations on the ability to cool, depending on the excited-electronicstate density of internal vibrational/rotational states.

RAMIFICATIONS OF FEEDBACK FOR CONTROL OF QUANTUM DYNAMICS H. RABITZ Department of Chemistry Princeton University Princeton, New Jersey CONTENTS I. Introduction

11. The Ubiquitous Role of Feedback A. Feedback in the Design of Molecular Controls

B. Feedback in the Laboratory Control of Molecular Dynamics C. Feedback in the Inversion of Molecular Dynamics 111. Conclusion References

I. INTRODUCTION Recent years have seen a flurry of activity in both the theoretical and experimental aspects of control over molecular processes [l] (see also S. A. Rice, “Perspectives on the Control of Quantum Many-Body Dynamics: Application to Chemical Reactions,” this volume). Most of the emphasis has been on the use of optical fields as a means for control, although other approaches can be envisioned in special circumstances 121. The key underlying principle of the overall subject is the achievement of control through the manipulation of quantum wave interferences [ l , 31, although full control will surely not be lost in the incoherent regime. The topic of molecular control is rich in detail as well as in potential appliAdvances in Chemical Physics, Volume 101: Chemical Reactions and Their Control on the Femtosecond Time Scale, Xyth Solvay Conference on Chemistry, Edited by Pierre Gaspard, Irene Burghardt, I. Prigogine, and Stuart A. Rice. ISBN 0-471-18048-3 0 1997 John Wiley & Sons, Inc.

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cations. The purpose of this chapter is to emphasize a few common points of generic interest to molecular control theory and its laboratory implementation. In particular, it will be argued that the principle of feedback underlies both control field design and the implementation of control in the laboratory, especially for complex systems. The notion of feedback leads to a compact nonlinear form for the control Schrodinger equation. In turn, this mode of thinking naturally suggests an important spin-off application of control for inversion (i.e., learning about molecular Hamiltonians from laboratory data). This chapter will present a common framework for all of these topics.

II. THE UBIQUITOUS ROLE OF FEEDBACK The notion of quantum feedback control naturally suggests a closed-loop process in the laboratory to stabilize or guide a system to a desired state. In addition, feedback is important in the design of molecular controls. These points will be made clear below, starting with considerations of design followed by a discussion of its role in the laboratory and finally leading to feedback concepts for the inversion of laboratory data. A. Feedback in the Design of Molecular Controls

Here, we consider the design of an optical electric field ~ ( t interacting ) through a molecular dipole moment p to achieve control over quantum molecular evolution. Various approaches have been taken to the design of e ( t ) to achieve molecular objectives. The most general technique is to pose the task as an optimal control problem [l, 41 (see also S. A. Rice, this volume), where the goal is to design the field e ( t ) to meet the objective in balance with a variety of possible competing deleterious processes. Each physical problem will have its own transcription of this competition. As we a priori do not know the control e ( t ) , it is not sufficient just to consider solving the Schrodinger equation alone,

where Ho is the molecular Hamiltonian and *O is the assumed known initial condition. The optimal design problem, in its simplest form, reduces to solving Eiq. (l), along with the equation d at

ih - += [Ho - pc(t)]+

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where t#J can be thought of as a Lagrange multiplier, having a final condition * T ) = XOQ(T). Here, T is the final target time when we desire the observable (Q(T)lOl\k(T))to take on the value 0,where the objective operator is 0. The coefficient X is the target deviation h = ((*(T)lOlq(T))- 0).Equations (1) and (2) are accompanied by an additional relation for the desired control field,

W))

c (1) = R (W)II.L I

(3)

Equations (1) and (2) are deceptively simple. However, their real nature is revealed by substitution of Q. (3) directly into Eqs. (1) and (2), which yields two coupled cubically nonlinear Schrodinger-type equations. In addition, Eq. (2) has a final condition that depends on the as yet unknown solution * ( T ) to Eq. (1) at the target time T. It can be shown that there are generally multiple solutions to these equations [ 5 ] , with each one corresponding to a local optimal field c ( t ) , producing a particular value (\k(T)lOl\k(T))for the target goal. Thus, we may also view these equations as corresponding to a nonlinear eigenvalue problem, where X is the eigenvalue indicating the quality of the achieved control. Various numerical illustrations of these equations have been performed for a variety of applications [6],with encouraging results for eventual laboratory implementation. Our purpose here is to examine the general structure of these equations. The role o f + deserves special attention. Given the form of Eqs. (1) and (2), appears to be on equal footing with the physical state 9'. However, t#J has the special role of a feedback function, as it is driven by the final condition, which is proportional to the target deviation A. Thus, the solution to Eq. (2) feeds back information on the quality of the achieved final state to in turn alter the control field in Eq. (3), which then manipulates the evolving state in Eq. (1). This convoluted feedback process generally necessitates solving Eqs. (l), (2), and (3), iteratively, but most interesting is the simple observation that a formal solution of Eqs. (2) and (3) followed by substitution into Eq. (1) yields the final effective Schrodinger equation

The dependence of the control field c(\k, A) on the state \k could be quite complex, as a solution of Eqs. (2) and (3) would suggest. In general, this relation may only be revealed numerically. Furthermore, the dependence of the control field on the target deviation eigenvalue h emphasizes the inherent role of feedback in the control design process. This relation also indicates the nonlinear eigenvalue role of X.

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Although little is known about the general nature of the solutions to Eq. (4),some revealing special cases have been examined. In particular, for socalled tracking control 171, where the objective is to follow a given evolution (O(t))= (*(t)lO(*(t)) to the target value 0 , then one may explicitly show that

Here, substitution of Eq. (5) into Eq. (4) produces an integrable nonlinear Schrodinger equation for the state \k. The field exactly meeting the objective is given by substituting \k(t),from solving Eq.(4), back into Eq. (5). Another intriguing case is for the control of a free atom, where Ho is just the kineticenergy operator. In this case, one has the classic nonlinear SchrGdinger equation

where M is the mass of the atom, y is a coupling parameter, and motion in one dimension is considered. Here, the control field is ~ ( xt,) = (!P(x, t ) I 2 . At first sight, it might appear to be impossible to create a field that explicitly depends on the spatial and temporal dependence of the evolving probability density, but a special case with potentially important significance may be practical (81, as mentioned in Section 1I.B. The full ramifications and exploitation of the feedback nature of Eq. (4) has not been explored for design purposes. Although tracking control gives a specific realization of the feedback nature of the field, even there, many variations on the theme arise [7].It may also be possible to use intuition to guide the form of €(*,A) to achieve good quality control, once again keeping in mind the special role of feedback for steering the currently evolving wavepacket to the final desired state. The physical clarity of the feedback process and the potential for computational savings suggest that analysis of this matter may be especially interesting for molecular control.

B. Feedback in the Laboratory Control of Molecular Dynamics

If the various notions and concepts in Section 1I.A can be exercised to yield , serious problems remain. First, precise knowledge a control field ~ ( t )two of the molecular Hamiltonian H is often seriously lacking, although there is typically a qualitative understanding of virtually any Hamiltonian. Sec-

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ond, even in those cases where the Hamiltonian is known and the design equations may be numerically solved to high accuracy, the resultant control ) be subject to some error upon its generation in the laborafield ~ ( twill tory. Given that the inherent molecular control mechanism is manipulation of wave interference [ 1, 31, then the degree of anticipated tolerance to these errors and uncertainties may be low in many cases. This type of circumstance is exactly the regime where laboratory feedback control is nautrally applied in analogous traditional engineering applications [9]. Although there are many nuances to laboratory feedback control, the essential features can be easily understood [lo]. One would typically start with a zero-order design ~ ( tfrom ) the efforts outlined in Section 1I.A and implement the field design in the laboratory for application to the molecular sample. At this stage, the laboratory observation of the target deviation X would then be fed into a learning algorithm, which would suggest a new control ~ ( t )and , the process would be repeated as many times as necessary until convergence occurs. The notion of feedback suggested here is distinct from that often used in traditional engineering control, where real time feedback is employed. Here, we refer to feedback in the sense of a sequential learning process. Naturally, a key issue is the rate of learning or convergence, and the rate may depend on many factors, including the sophistication of the learning algorithm, the flexibility inherent in the laboratory control field adjustments, and the nature and quality of the laboratory observations [lo]. In the most difficult cases, it may be necessary to perform intermediate-state measurements in order to garner information about where the system has evolved to, short of reaching the desired target state. Computer simulations of this learning control process have been very encouraging for its eventual success in the laboratory, and perhaps most intriguing is the suggestion that the feedback process may be self-starting under the best of conditions [lOa] and thus not even requiring a trial design. Precise knowledge of the created control field is not necessary to implement feedback control. It is only necessary that the control laser be stable and reliable, such that its performance does not wander while going from one experiment to the next. Another important point is that the control field frequency bandwidth will not need to become ever broader, as larger molecules are considered for control. The bandwidth should saturate rather rapidly with molecular size, and this suggests that the same apparatus capable of achieving control over simple molecules may be readily adapted to complex ones. We also suggest that laboratory feedback will be the only way to reliably investigate the feasibility of control over polyatomic molecules, as theoretical limitations may prevent a proper analysis. Most encouraging is the fact that all of these issues have analogues in traditional engineering control, where they are successfully exploited almost in a routine fashion [9]. Hopefully, it

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will not be too long before the same capability is present in the molecular control domain. A special case of feedback control arises in consideration of the nonlinear Schrodinger equation (6). A capability now exists for creating travelingwave-like potentials U(x - uf), where u is a characteristic velocity [ll]. Under these conditions, it may be shown that Eq. (6) can be transformed into a stationary Schrodinger equation dependent on y = x - uf admitting a solitonic solution [8]. This situation suggests that it might be possible to create a quantum soliton in the laboratory. However, the realization of such a soliton calls for a special understanding in the realm of feedback control. The potential is the probability density l\k(y)1*, which could, in principle, be measured through ultrafast X-ray or electron beam diffraction [ 121. Thus, one could envision the generation and observation of a feedback-controlled quantum soliton [8]. The nondispersive nature of the wavepacket does not violate any principles of quantum mechanics, as the inherent time dependence of the potential exactly compensates for the naturat tendency of the wavepacket to spread. Such an application of quantum control would be intriguing and, possibly, of some practical consequences.

C. Feedback in the Inversion of Molecular Dynamics As commented in Section IIB, we typically do not know the Hamiltonian H to high precision. A major interest in chemical physics continues to be performance of experiments and computations to better refine molecular Hamiltonians. Some studies have suggested that the control concepts presented above can be turned around and applied to address this important goal of learning about Hamiltonian structure [13]. This viewpoint is especially related to molecular tracking embodied in Eq. (5). For inversion, the goal is to determine some portion of HO(particularly the potential) through laboratory-observed knowledge of the data track (O(t)),which in turn will satisfy Heisenberg's equation of motion

This equation may be viewed as an integral equation, where the unknown is the operator Ho, with the data on the left-hand side being known. Equations (5) and (7)are analogous to each other, except that in Eq. (5) the unknown portion of the Hamiltonian corresponding to the control field was explicitly solved for. In Eq. (7), we obtain an integral equation for Ho serving this purpose. The analogy goes further, in that the right-hand side of Eq. ( 5 ) depends on the unknown evolving state 'k, as does the kernel of the integral

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term on the right-hand side of Eq. (7). Thus, we can create an inversion algorithm based on simultaneously solving Eqs. (7)and (8): iA

a

- \k = H o ~ at

This algorithm may be viewed as “direct,” since the solution will yield the sought-after Hamiltonian Ho. Feedback is present here in that the information learned about the operator Ho in Eq. (7)will depend on the evolving state analogous to the same way that the field depends on the state in Eqs. (3), (4)- or (5). A host of issues need to be explored to establish the full viability of this tracking approach to laboratory data inversion, but preliminary examination of the algorithm has been most encouraging [ 131. It has been referred to as a tracking inversion algorithm, since the data (O(t))are tracked through Q. (7)to yield the sought-after Hamiltonian Ho.This concept naturally admits ultrafast pumpprobe data but is not restricted to it. In particular, it would in principle be possible to generate a “synthetic” track by Fourier transforming traditional continuous-wave spectral data, and a relation analogous to Eq. (7) will result. This prospect is especially interesting, as it would allow for the utilization of abundant high-quality continuous-wave data in a natural time-dependent framework for inversion. It may turn out that one of the most important legacies of molecular control is its linkage to inversion of laboratory data to learn about molecules. The full embodiment of this concept again embraces feedback, as a relay of control and inversion can be envisioned to learn about molecules in the most efficient manner. 111. CONCLUSION

This chapter has emphasized the special and central role that feedback plays in virtually all aspects of control over molecular quantum phenomena. In terms of applications, the manipulation of chemical reactions still stands as a prime historical objective. However, other rich applications abound. For example, the growing interest in the field of quantum computing is a potentially exciting area [ 141, and any practical realization of quantum computers will surely entail control over quantum phenomena. Other unforeseen applications may also lie ahead.

Acknowledgments The author acknowledges support from the Army Research Office and the National Science Foundation.

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References 1. W. S. Warren, H. Rabitz, and M. Dahleh, Science 259, 1581 (1993). 2. P. Gross, V. Ramakrishna, E. Vilallonga, H. Rabitz, M. Littman, S. A. Lyon, and M. Shayegan, Phys. Rev. E 49, 11100 (1994). 3. P. Brumer and M. Shapiro, Ann. Rev. Phys. Chem. 43, 257 (1992). 4. A. Peirce, M. Dahleh, and H. Rabitz, Phys. Rev. A 37,4950 (1988). 5 . M. Demiralp and H. Rabitz, J. Marh. Chem. 16, 185 (1994). 6. R. Kosloff, S. Rice, P. Gaspard, S. Tersigni, and D. Tannor, Chem.Phys. 139,201 (1989); Y.Yan, R. Gillilan, R. Whitnell, K. Wilson, and S. Mukamel, J. Chem. Phys. 97, 2320 (1993); T. Szak6cs. B. Amstrup, P. Gross, R. Kosloff, H. Rabitz, and A. Liirincz, Phys. Rev. A 50, 2540 (1994). 7. P. Gross, H. Singh, H. Rabitz, K. Mease, and G. M. Huang, Phys. Rev. A 47,4593 (1993). 8. M.Demiralp and H. Rabitz, “Feedback Controlled Quantum Solitons,” to be published. 9. K. J. Astrom and B. Wittenmark, Adaptive Conrrof (Addison-Wesley, 1995). 10. (a) R. S. Judson and H . Rabitz, Phys. Rev. Lett. 68,1500 (1992); (b) G. J. T&h, A. Lorincz, and H. Rabitz, J. Cbern. Phys. 101,3715 (1994). 11. R. Graham, M. Schlautmann, and P. Zoller, Phys. Rev. A 45, R19 (1992). 12. M. Dantos, S . Kim, J. Williamson, and A. Zewail, J. Phys. Chem. 98, 2782 (1994). 13. 2.-M. Lu and H. Rabitz, J. Phys. Chem. 99, 13731 (1995). 14. T. Slator and H. Weinfurter, Phys. Rev. Lett. 74, 4087 (1995).

DISCUSSION ON THE COMMUNICATION BY H. RABITZ Chairman: M. Quack T. Kobayashi 1. What is the origin of the nonlinearity introduced in the Schrodinger equation represented by a potential proportional to I$ 12? 2. How can such a potential be introduced experimentally? 3, Where does the soliton propagate? 4. Is the feedback expected to be very fast?

H. Rabitz: In general, nonlinearity arises in quantum control through the design or feedback process. In either case, nonlinearity enters as quantum control is inherently an inverse problem. The physical objective is prescribed, and we seek to find a piece of the Hamilwould tonian (i.e., the electric field). A nonlinearity of the form be introduced in the laboratory by measuring the probability density through, perhaps, electron or X-ray diffraction and feeding that observation back in order to create a potential of that form. Special forms for such potentials are becoming feasible to generate in the laboratory.

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The soliton considered here might correspond to an atom translating in an optical field or perhaps an exciton in a solid. Under these conditions, the feedback control does not necessarily have to be ultrafast, as the propagation and velocity are at our disposal.

R. W. Field: Prof. Rabitz, I like the idea of “sending out a scout” to map a local region of the potential-energy surface. But I get the

impression that the inversion scheme you are proposing would make no use of what is known from frequency-domain spectroscopy or even from nonstandard dynamical models based on multiresonance effective Hamiltonian models. Your inversion scheme may be mathematically rigorous, unbiased, and carefully filtered against a too detailed model of the local potential, but I think it is naive to think that a play-andlearn scheme could assemble a sufficient quantity of information to usefully control the dynamics of even a small polyatomic molecule. In our conversation after the discussion, I became at least willing to believe that we both believe in simplified models. However, you present your control schemes in a way that feels dismissive of all previous efforts by spectroscopists to extract information from spectra. Having in advance a simple picture of what a molecule wants to do and also knowing why the molecule wants to do it seems superior to a picture designed on-the-fly on a purely numerical basis.

H. Rabitz A key motivation in developing the inverse algorithm was to draw a firm linkage between control field design and potential determination. Both are inverse problems aiming to determine a portion of the Hamiltonian, and it is most interesting that a common mathematical formulation now exists for both applications. In particular, the Schrodingerequation for the wavepacket is coupled to an integral equation for the potential (or the control field). Focusing on potential inversion, the temporal data is a source term in the integral equation. The form of the potential extracted from the equation depends on the representation chosen as most appropriate in a given case. In particular, the potential could be sought in coordinate space or as matrix elements in a state space. The wavepacket “scout” wandering about on the surface or in a state space will naturally identify what may be learned regarding the potential. A temporal picture is an inherent component of this analysis. The inverse algorithm is a direct marching process to reveal more about the potential as time evolves and the wavepacket samples more of the potential. Virtually all inverse problems are ill-posed, in that the finite available data will not permit a full determination of the potential. There-

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fore, one needs to regularize (stabilize) the inversion by (1) assuring that the potential is not sought in regions poorly sampled by the data and (2) the finite amount of data is supplemented as necessary by a priori physical knowledge about the potential. The algorithm naturally assures point 1 by only extracting the potential in regions where the wavepacket is significant. A variety of input information can be used to handle point 2. It is essential not to overly constrain the inversion to just give a potential rather than the true potential; at the same time a constraint that is too weak will leave the inversion algorithm unstable, which may render a nonunique potential. In some cases it has been found that merely asking for the potential surface to be smooth, in a differential sense, is sufficient to stabilize the inversion. The goal is to seek stabilization with the mildest criteria possible and then let the algorithm pin down the potential in the regions permitted by the data. Naturally many practical implementation issues arise, including the need to solve the dynamical equations, at least in the regions of importance sampled by the data. In this regard there is a classical mechanical analogue of the coupled dynamical and integral equations. Exploitation of classical inversion may be important, at least as a first step to define the potential in polyatomic cases. The key point at this time is that the new formulation provides a rigorous foundation to build upon for achieving direct practical inversions of temporal and spectroscopic data.

R. W.Field: The goal is not merely to represent the spectrum or the dynamics but to be able to create reduced-dimension pictures that are intelligible to mortals. Pictures lead to insight. Insight leads to more effective control strategies. H. Rabitz: The goals of Hamiltonian information extraction are clear, but prior means for this purpose are generally unsatisfactory in many respects. Various sources of data are available for exploitation. In some applications, reduced models will suffice, while for others, only high accuracy detailed potentials can meet the needs. What is necessary is a rigorous inversion algorithm that can incorporate appropriate physical constraints to reliably extract the Hamiltonian information. D. J. Tannor: Prof. Rabitz, from the time-domain formulation of conventional electronic absorption spectroscopy, we know that the information content in the wavepacket autocorrelation function is identical to that in the high-resolution spectrum. Yet it is clear that the wavepacket autoconelation function only directly probes the times at

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which the wavepacket returns to its initial position and hence is a direct probe only of the Franck-Condon region. We know the amount of time the wavepacket spends away from the Franck-Condon region, but not where it goes. In what way does your tracking algorithm interrogate the wavepacket a b u t where it went when it left the Franck-Condon region?

H.Rabitz: The information in the recurrence time alone is minimal. However, the temporal structure of the recurrence signal contains detailed information on the surface explored by the wandering scout wavepacket during its excursion. Further experiments may be necessary to follow (i.e., track) the wavepacket through its excursion over the potential surface. Such pump-probe experiments go beyond conventional spectroscopy.

THEORY OF LASER CONTROL OF VIBRATIONAL TRANSITIONS AND CHEMICAL REACTIONS BY ULTRASHORT INFRARED LASER PULSES M.V. KOROLKOV, J. MANZ,* and G. K. PARAMONOV Freie Universitat Berlin Institut fur Physikalische und Theoretische Chemie Berlin, Germany

CONTENTS I. Introduction 11. Models and Techniques 111. Applications A. Individual Vibrational-State-to-Vibrational-State Transitions B. Series of Vibrational Transitions C. Vibrational Transitions in Competition with Dissipative Processes D. Above-Threshold Dissociation E. Isomerization IV. Conclusions References

I. INTRODUCTION The theory of laser control of chemical reactions may be classified into two

different domains: Laser control by continuous wave (CW) lasers and by laser pulses. The former includes, for example, the strategies of (i) vibrationally mediated chemistry [ 11 and (ii) coherent superpositions of independent excitation routes [2]; for experimental demonstrations see (i) Ref. 3 and *Communication presented by J. MQnZ Advances in Chemical Physics, Volume 101: Chemical Reactions and Their Control on the Femtosecond Time Scaie, XXth Solvay Conference on Chemistry, Edited by Pierre Gaspard, Irene Burghardt, 1. Prigogine, and Stuart A. Rice. ISBN 0-471-18048-3 0 1997 John Wiley & Sons, Inc.

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(ii) Ref. 4, respectively. The latter involves the theories for selective transitions by series of (iii) ns [S] or (iv) ps/fs [6] ultraviolet (UV) or visible (VIS) laser pulses, by (v) chirped laser pulses [71, and by (vi) optimal Control [S]; for experimental verifications see (iii) Ref. [9], (iv) Ref. [lo], and (v) Ref. [7]. For surveys, see Ref. 11 as well as several other fundamental contributions in this volume (S. A. Rice, Chapter 6, this volume) or elsewhere [121. The purpose of this chapter is to present a brief survey of the theoretical work of our group on another strategy, that is, laser control of vibrational transitions by single IR laser pulses in the femtosecond/picosecond time domain [13] or by selective series of such pulses [14] with applications to isolated molecules or radicals in the gas phase [13, 141 or in an environment (151 and with extensions to selective photodissociation or predissociation [16] and isomerizations [14, 171; see also the review [18] and complementary work on laser control by IR femtosecond/picosecond laser pulse [19, 201. The model systems include simple pseudo-one-dimensional (Id) ones [13-191 as well as more complex (2d, 3d) ones [18,20,21]. In our strategy, all of these applications and extensions, that is, series of vibrational transitions, photodissociation, and unimolecular reactions, are essentially decomposed into sequences of selective vibrational-state-to-vibrational-state or vibrational-state-to-continuum-state transitions, with possible extensions to continuum-state-to-continuum-state transitions (223, which are achieved by corresponding sequences of IR femtosecond/picosecond laser pulses. It will be helpful, therefore, to consider first the fundamental quantum effect of state-to-state transitions induced by individual IR femtosecond/picosecond laser pulses, before presenting extensions to multiple transitions or chemical reactions induced by series of IR femtosecond/picosecond laser pulses. Accordingly, the chapter is organized as follows: In Section 11, we give a brief survey of the fundamental models and theories, followed by theoretical applications to vibrational transitions in closed [13, 141 (Section IILA, B) and open [ 151 (Section 1II.C) systems with extensions to photodissociation [161 (Section 1II.D) and isomerization [ 171 (Section 1II.E). The conclusions are in Section IV. 11. MODELS AND TECHNIQUES The original theory of individual state-selective vibrational transitions induced by IR femtosecond/picosecond laser pulses has been developed by Paramonov and Savva et al. for single laser pulses [13] (see also Ref. 23) followed by more general extensions to series of IR femtosecond/picosecond laser pulses in Refs. 14 and 24. For illustration, let us consider two simple, one-dimensional model systems that are assumed to be decoupled from any

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329

other inter- or intramolecular (vibrational, rotational, etc.) degrees of freedom. Our first example is a Morse oscillator tailored to the OH bond in HOD, (adapted from Ref. 25; see also Refs. 14-16, 21, 23, and 24). This model will serve to explain our strategies for IR femtosecond/picosecond laser-pulse-induced vibrational transitions from initial to final states [e.g., OH(Ui) OH(uf)] and for photodissociation [OH(ui) 0 + HI. It will also serve as reference for more complex systems (e.g., state-selective vibrational transitions in HOD) [18, 211 or OH coupled to other intra- and intermolecular vibrations, which may be represented by a thermal bath [151. The second one-dimensional example is a model system with a shallow, slightly asymmetric double-well potential tailored to substituted semibullvdenes (SBVs) (adapted from Ref. 26). This model will be used to illustrate IR femtosecond/picosecond laser control of unimolecular isomerizations, specifically the Cope rearrangement of the substituted SBV [26]. In brief, the two model systems will be called OH and SBV; they should be considered as representatives for similar systems [ 14, 18-21, 271. The OH and SBV potential-energy surfaces V ( q ) versus bond or reaction coordinates q, together with the vibrational levels E , and eigenfunctions +,,(q), are shown in Figs. 1 and 2, respectively. Our first task will be to design optimal IR femtosecond/picosecond laser pulses for vibrational transitions,

-

-

is an isolated system from the initial (t = 0) state hi to the final (t = t p ) one &, [tp is the duration of the laser pulses and E = exp(-iq) denotes an irrelevant phase factor]. For this purpose, we solve the corresponding timedependent Schodinger equation using the semiclassical dipole approximation and neglecting all other couplings,

subject to the initial condition (1). The standard notations in Eq. (2) denote the molecular Hamiltonian H = T + V and dipole operator p as well as the electric field of the IR femtosecond/picosecond laser pulse, € ( t )= €0 . cos w t * s ( t )

(3)

where €0, w , and s ( t ) are the amplitude, frequency, and shape function. As a convenient example, we employ sin2 pulses,

M. V. KOROLKOV, J. MANZ, AND G. K. PARAMONOV

3 30 20000

m-----OH

10000

0

-5

/-

- 0 +

I

II II

1

H

1

.

-10000

> P a

c

W

-20000 -

\

n n

1 ” T - f

-30000

-40000 0

1

.2

3

qlao

4

5

6

Figure 1. Morse potential V(q), vibrational levels Eu, and wave functions &(q) for the model OH (adapted from Ref. 14). The arrows indicate various selective vibrational transitions as well as above-threshold dissociations (ATDs) induced by IR femtosecond/picosecond laser pulses, as discussed in Sections 1II.A-1II.D; see Figs. 3-5. Horizontal bars on the mows mark multiple photon energies hw of the laser pulses; cf. Table I. The resulting ATD spectrum is illustrated by the insert above threshold.

CONTROL BY ULTRASHORT INFRARED PULSES

33 1

2500 2000 l-l

'E 0

1500

\

h

g G

1000

500 0 -100

-50

50

0

9/Pm

100

Figure 2. Double-well potential V ( q )with corresponding vibrational levels Eu and wave functions &(q) for the model 2,6-dicyanoethylmethylsemibullvalene(SBV) (adapted from Ref. 26). The reaction coordinate q indicates the Cope rearrangement of the model SBV from the reactant (R) isomer versus the transition state f to the product (P) isomer. Vertical arrows f P by two IR femtosecond/picosecond indicate the laser control of the isomerization R laser pulses: cf. Fig. 6 and Table I. +

s ( t ) = sin*

+

( 7) for I I 0

t

tp

(4)

The dipole function p for OH is modeled as a Mecke function (adapted from Ref. 28), for SBV we assume the ubiquitous linear relation p = f . e q with effective chargef e and scaling factorf. Equation (2) can be solved either

-

-

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M. V. KOROLKOV, J. MANZ, AND G. K. PARAMONOV

directly using fast Fourier transform (m) propagation methods [29] or by expanding $4)in terms of vibrational eigenstates &,

and transforming Fq. (2) into the matrix version ihC(t) = [ H - p ( t ) ] * c(t)

(6)

which is then solved, for example, by the Runge-Kutta method [30]. In this representation, the transition (1) is rewritten as C,(t

= 0 ) = 6",i

--t

C U ( t = t p )= 6 , ,

*

f

(7)

or in terms of populations

we have that PU(t= 0) = 6,,

Pu(t = t p )= 6,,

--+

(9)

These selective transitions (l), (7),and (9) may be achieved by proper optimization of the parameters € 0 and w, as described elsewhere [13, 18, 211. Extensions to IR femtosecond/picosecond laser-pulse-induced dissociation or predissociation have been derived in Ref. 16, using either the direct or the indirect solutions of the Schrdinger equation (2); the latter requires extensions of the expansion (5) from bound to continuum states [16,31]. (The consistent derivation in Ref. 16 is based on s. Flugge in Ref. 31). The same techniques can also be used for IR femtosecond/picosecond laser-pulse-induced isomerization as well as for more complex systems that are two dimensional, three dimensional, and so on, at the expense of increasing numerical efforts due to the higher dimensionality grid representations of the wavepackets $ ( t ) or the corresponding expansions (5) (see, e.g., Refs. 18, 20, and 21). Extensions from the preceding ideal, isolated systems to others that are coupled to an environment are quite demanding and nontrivial 1321 because the IR femtosecond/picosecond laser pulse has to achieve the selective vibrational transition (9) in competition against nonselective processes such as dissipation. For simulations, we employ the equation of motion for the reduceddensity operator

CONTROL BY ULTRASHORT INFRARED PULSES

333

The first term on the right-hand side corresponds to Eq. (2), whereas the second term describes dissipative effects that are induced in the system due to its coupling to the environment. The latter is modeled, as usual [32, 331, as the thermal (temperature T) ensemble of harmonic oscillators, with nonlinear coupling A ' Q(q) F ( { q k } ) between the system and the thermal bath, expressed in terms of nonlinear molecular and linear environment coupling operators Q(q) and F ( { q k } ) . As shown in Ref. 15, it is important to describe the dissipative term in Eq. (10) by making use of the non-Markovian expression e

with time correlation function (F(t')F(O)); compare with the Markovian approximation in Ref. 32. The latter may be appropriate for other purposes, for example, IR absorption spectroscopy with traditional CW lasers [33]. In this chapter, we employ the algebraic, vibrational state representation of J2q. (10) [compare with 3.(6)]. Proper optimizations of the IR picosecond laser pulse parameters should then achieve the vibrational transition (9) as well as possible, where PU(t)= p u U ( t ) .

III. APPLICATIONS

A. Individual Vibrational-State-to-Vibrational-Statelkansitions

As our first example, we consider individual vibrational-state-to-vibrationalstate transitions in the model OH (see Fig. 1); specifically,

and OH(U~= 5 ) +OH(U~= 10)

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M. V. KOROLKOV, 1. MANZ, AND G. K. PARAMONOV

The results are shown in Fig. 3 and Table I. Apparently, optimal IR femtosecond/picosecond laser pulses with durations tp = 500 fs may induce nearly perfect transitions (12), (13) in the model OH. Similar examples are documented in Refs. 13, 18, and 23. A detailed discussion of the derivation of the optimal laser parameters, depending on the vibrational level Eu and the transition dipole matrix elements puw, is also given in Refs. 13, 18, and 23. Suffice it here to say that in many (but not all) cases the optimal frequency w is close (but not identical) to the resonance frequency,

250 0

-250

-500

1

1

0

0.5 1 T I M E (ps)

1.5

0

0.5

1 T I M E (ps)

1.5

1

0.8

0.6 0.4

0.2 0 -

5)

-.

Figure 3. Selective vibrational transitions OH@, = O)-OH(uf = 5 ) and OH(vi = OH(vf = 10) induced by two individual IR femtosecond/picosecond laser pulses. The

electric fields E ( t ) and the population dynamics &(t) are shown in panels ( a ) and (b),respectively. Sequential combination of the two individual laser pulses yields the overall transition OH(u = 0) --c OH(u = 5 ) OH(uf = 10);cf. Fig. 1 and Table I. For the isolated system, the population of the target state Pv= &) is constant after the series of iR femtosecond/picosecond laser pulses, t > 1 ps.

-.

335

CONTROL BY ULTRASHORT INFRARED PULSES TABLE I Laser Parameters for Vibrational Transitions in Model Systems OH and SBV Transition

-

OH(u; = 0) + OH(U~ = 5)a OH(u; = 5) OH(U~ = 10)' OH(Q = 10) + OH(U~= 15)' OH(Q = 15) + 0 + H' SBV(U~ = 0) + SBV(U~ = 6)' SBV(U~ = 6) + SBV(uf = I)'

I , (ps)

u12m (cm-')

0.5 0.5

3426.2 2524.6 1625.5 822.5 1252.8 1185.0

1.o

0.5 1.o

1.o

EO

(MV/cm-') 400.8 214.6 148.1 177.3 32.8/f 30.51'

10 (TW/cm2)

2 12Ae 61 .O 29.0 41.7 1.451f 1.251f

=See Figs. 1, 3, and 4. bSee Figs. 1 and 5. 'See Figs. 2 and 6. df is the scaling factor for the dipole function p = f e q of SBV. eIntensities are calculated as follows: 10 = icoce$ they can be reduced by making use of series of vibrational transitions.

with rather high demands for the accuracy, typically 6w/w < lW3. In contrast, the system offers several choices of optimal electric field amplitudes €0, with rather moderate requirements for the accuracies, say, BEO/EO< 0.1. In practice, one should employ the smallest possible choice of €0 in order to avoid competing nonselective processes (e.g., power broadening at larger electric field strengths). The resulting optimal IR femtosecond/picosecond laser pulses may be interpreted as generalized if-pulses [19].

B. Series of Vibrational Wansitions Combination of several individual vibrational transition (Section 1II.A) yields a selective sequence of vibrational transitions induced by series of IR femtosecond/picosecond laser pulses. For example, the two individual transitions (12). (13) may be combined to the sequence

OH(U= 0 )

-

OH(U= 5 ) -+OH(U= 10)

(15)

and the selective overall transition (15 ) is achieved simply by the corresponding series of individual IR femtosecond/picosecond laser pulses; see Fig. 3. Similar series of state-selective vibrational transitions induced by series of IR femtosecond/picosecond laser pulses are documented in Refs. 14, 17-19, and 24; see also Refs. 18, 20, and 21 for applications to more complex sys-

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M. V. KOROLKOV, J. MANZ, AND G . K. PARAMONOV

tems such as HOD. In practice, it may be quite advantageous to employ such series of state-selective IR femtosecond/picosecond laser pulses, because their individual components may have smaller field amplitudes, in comparison with single pulses. On the other hand, series of nonoveralpping IR femtosecond/picosecond laser pulses (such as those shown in Fig. 3) may imply rather long durations of the overall excitation process (in comparison with the duration of their individual components), and this may enhance the probability of competing processes such as IVR. Gratifyingly, however, exceedingly long overall durations may be reduced by partial overlaps of the pulses; see Refs. 14, 18, 19, and 24 and compare, for example, Figs. 3 and 4.

C. Vibrational Transitions in Competition with Dissipative Processes

Next let us extend the strategy of selective sequences of vibrational transitions by series of IR femtosecond/picosecond laser pulses from closed systems (Section IILB) to open ones. For direct comparison with the results shown in Fig. 3, we consider the case of the transition (15) in competition with dissipation to a thermal bath. The results are shown in Fig. 4. Here we employ essentially the same sequence of IR femtosecond/picosecond laser pulses as in Section 1II.B without retuning of the laser parameters but with partial overlaps in order to reduce the overall duration and, therefore, the probability of competing dissipative processes. As a consequence,these laser pulses yield rather successful population transfer even in the open system, in close analogy to the isolated system, that is, the populations of the intermediate and final vibrational levels OH(u = 5 ) and OH(u = 10) approach maximal values of 0.97 and 0.88, respectively, corresponding to the target transition (15). In contrast with the isolated system, however, these maximum populations have rather short lifetimes (400fs) because the coupling to the environment induces competing relaxations to lower levels, for example, OH(u= 1 0 ) + 0 H ( ~ =9 ) - + 0 H ( ~ =8)-

..-

(16)

The decay process (16) is clearly visible in Fig. 4, in particular after the end of the laser pulses (t > 750 fs); ultimately, the population P, approach the thermal Boltzmann distribution (T = 300 K in Fig. 4).

D. Above-Threshold Dissociation

The strategy of multiple vibrational transitions induced by series of IR femtosecond/picosecond laser pulses may also be extended to photodissociation or predissociation. This is documented in Fig. 5 for the case

OH(U= 10)-+OH(u= 15)-O+H

(17)

CONTROL BY ULTRASHORT INFRARED PULSES

-500 I

0

337

1 T I M E (ps)

1.5

1

1.5

0.5

1

-

z 0 +

0.8

3

0.4

-

0 a

0.2

-

v)

e (

4 4

0.6 -

0, Y

0

0

v=o

4

0.5

T I M E (PSI

Figure 4. Series of IR femtosecond/picosecond laser pulses for the sequence of vibrational transitions OH(u = O)+OH(u = S)-OH(u = 10) for the model OH coupled to a thermal heat bath; cf. Fig. 1 and Table I. The notations are as in Fig. 3. The laser pulses compete against dissipation. After the pulses, the coupled system relaxes toward a Boltzrnann distribution (T= 300 K).

which may be considered as an extension of the vibrational transitions (15) to photodissociation; see also Fig. 1. For the present purpose, the most important effect is achieved by the final IR femtosecond laser pulse, which converts the bound state OH(u = 15) into a distribution of continuum states representing 0 + H. In Ref. 16 it has been shown that one can control the resulting populations of continuum states; for example, by proper optimization of the final IR femtosecond/picosecond laser pulse, it is possible to produce nearly monoenergetic products 0 + H, or several wavepackets with specific energies separated by the IR femtosecond laser photon energy h a , and so on.

M. V. KOROLKOV, J. MANZ, AND G . K. PARAMONOV

-300

0

0.5

1 T I M E (ps)

I .5

0

0.5 1 T I M E (ps)

1.5

1

0.8

0.6 0.4

0.2 0

Figure 5. Series of IR femtosecond/picosecond laser pulses for the sequence of transitions OH(u = 10)-OH(u = 15) -0 + H for the isolated model OH; cf. Fig. 1 and Table I. The notations are as in Fig. 3; populations Pwell(r) = P J r ) and Pcont(t)= IP,ll(t) indicate the total populations of bound and continuum states embedded in the potential well and above the dissociation threshold, respectively. The resulting spectrum of ATD is shown in Fig. I .

c$Lo

The resulting phenomena of above-threshold dissociation (ATD) resemble those of above-threshold ionization (ATI); see Ref. 34.

E. Isomerization

Our final example provides the extension from multiple vibrational transitions (Section II1.B) to isomerization. Specifically, we consider the model Cope rearrangement of 2,6-dicyanoethylmethylsemibullvalene(SBV) from the reactant (R) via the transition state d to the product (P) isomer; see Fig. 2. The system has been designed by Quast with specific substitutions

CONTROL BY ULTRASHORT INFRARED PULSES

339

that cause a slightly asymmetric potential-energy surface with rather shallow potential-energy barrier along the reaction coordinate q; for details see Ref. 35. Infrared femtosecond/picosecond laser pulse control of the isomerization is achieved by two subsequent vibrational transitions, similar to expression (15) but with the second transition causing stimulated emission rather than absorption:

SBV(U= 0)

-

SBV(U= 6) +SBV(U= 1)

cf. Fig. 2. The first transition excites the reactant ground state SBV(u = 0) to the delocalized state SBV(u = 6) with energy just above the potential barrier. The second pulse deexcites the intermediate state SBV(u = 6) into the product ground state SBV(u = 1). The corresponding selective IR femtosecond/picosecond laser pulses and population dynamics are shown in Fig. 6. Apparently, the series of two IR femtosecond/picosecond laser pulses induces nearly perfect control of the model isomerization. Similar controls of model reactions are documented in Refs. 14 and 17.

IV. CONCLUSIONS Control of vibrational transitions, ATD, and AT1 by series of IR femtosecondfpicosecond laser pulses is a promising strategy. It should be a challenge for our experimental partners to verify this strategy by applications to real systems. The recent development of powerful IR femtosecond laser pulses [36] is an important step toward this goal. Simultaneously, there are also challenging tasks for theoreticians, for example, further developments of the present strategy to chemical reactions in open systems. Most important, perhaps, will be extensions and tests of our strategy for even more complex multidimensional model systems, including rotations. Specifically, it will be fascinating to see whether the strategy can achieve selective overall transitions in systems with rather large level densities, for example, with vibrational level spacings smaller than the spectral widths of the IR femtosecond/picosecond laser pulses. For practical purposes, it will also be important to compare the present strategy [ 13-24] with others [1-12] (see also S. A. Rice, Chapter 6, this volume). For example, series of IR femtosecond/picosecond laser pulses may be considered as special cases of optimal control [8] subject to the restriction of simple pulse shapes, and this restriction may turn out to be advantageous for experimental implementations but disadvantageous for the aim of designing optimal laser pulses with rather low intensities (compare, e.g., Refs. 21 and 8). For the specific purpose of laser control of isomerization, it is also interesting to compare the efficiency of the present strategy, which employs two IR

340

3w

M. V. KOROLKOV, J. MANZ, AND G. K. PARAMONOV

-50

0

0.5

1

1.5

0

0.5

1

1.5

T I M E (ps)

1 cn

x

0.8

e

0.6

Ll

0.4

0 Y

4

5

a 0 a

0.2 0

T I M E (ps)

-.

Figure 6. Series of IR femtosecond/picosecond Laser pulses for the sequence of vibrational transitions SBV(u = 0) +SBV(u = 6) SBV(u = 1) for laser control of the Cope rearrangement of the model substituted semibullvalene (SBV) shown in Fig. 2 (adapted from Ref. 26). The notations are as in Fig. 3. The electric field is scaled by the scaling factorf of the effective charge associated with the dipole function = f . e . q.

femtosecond/picosecond pump-and-dump laser pulses (see Section IILE), with the strategy of V I S j W femtosecond pump-and-dump laser pulses [6]. We consider the recent experimental verifications [101 of the latter as encouragement for equivalent implementations of our own strategy. Work along these lines is in progress.

CONTROL BY ULTRASHORT INFRARED PULSES

34 1

Acknowledgments We thank our previous coauthors [13-17. 22-27] for their substantial contributions to the development of the strategy of laser control of vibrational transitions and unimolecular reactions by IR femtosecond/picosecond laser pulses. Specifically, we are grateful to M. Dohle for preparing Figures 2 and 6 for this survey (adapted from Ref. 26). Generous financial support by Volkswagen-Stiftung through project I/69348 and Fonds der Chemischen Industrie is also gratefully acknowledged. The computer simulations have been carried out on HP 750 workstations at Freie Universitit Berlin.

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31. S. Flugge, Practical Quantum Mechanics, Springer-Verlag, New York 1974; A. Erdilyi, Ed., Higher Transcendental Functions, Vols. 1 and 2, Bateman Manuscript Project, McGraw-Hill, New York, 1953; A. Matsumoto, J. Phys. B: At. Mol. Opt. Phys. 21, 2863 (1988). 32. 0. Kuhn, D. Malzahn, and V. May, fnl. J. Quant. Chem. 57, 343 (1996). 33. F. Neugebauer, D. Malzahn, and V. May, Chem. Phys. 201,151 (1995);C. Scheurer and P. Saalfrank, J. Chem. Phys. 104,2869 (1996); P. A. Apanasevich, Principles of Interaction of Light with Matter, Minsk, Nauka i Thechnika, 1977 (in Russian); M. Schreiber, 0. Kuhn, and V. May, J. Lumin. 58, 85 (1994); C. Scheurer and P. Saalfrank, Chem. Phys. Lett. 245,201 (1995). 34. Y.R. Shen, in Fundamental Systems in Quantum Optics, Les Houches, Session LIII, J. Dalibard, J.-M. Raimond, and J. Zinn-Justin, a s . , North-Holland, Amsterdam 1992, p. 1049; M.Gavrila, Atoms in Intense Laser Fields, Acadmic, New York, 1992. 35. H. Quast, A. Witzel, E.-M. Peters, K. Peters, and H.G. von Schnenng, Chem. Ber. 125, 2613 (1992); A. Witzel, Ph.D. Thesis, Universitat Wurzburg, 1994; T. Dietz, Ph.D. Thesis, Universitat Wurzburg, 1995. 36. F. Seifert, V. Petrov, and M.Woerner, Opr. Lett. 19, 2009 (1994).

DISCUSSION ON THE COMMUNICATION BY J. MANZ Chairman: M. Quack S. A. Rice: What is the source of dissipation?

J. Manz: The surrounding molecules, for example.

TIME-FREQUENCY AND COORDINATE-MOMENTUM WIGNER WAVEPACKETS IN NONLINEAR SPECTROSCOPY S. MUKAMEL, C. CIORDAS-CIURDARIU, AND V. KHIDEKEL

Department of Chemistry University of Rochester Rochestel; New York

CONTENTS I. Introduction 11. Correlation Function Expression for Spontaneous Light Emission 111. Wigner Wavepackets in Phase Space: The Doorway-Window Picture IV. Nuclear Wavepackets in Pump-Probe Spectroscopy V. Extension to Heterodyne-Detected Four-Wave Mixing Appendix A: Time- and Frequency-Gated Autocorrelation Signals Appendix B: The Signal and the Optical Polarization Appendix C: Four-Point Correlation Function Expression for Fluorescence Spectra Appendix D. Phase-Space Doorway-Window Wavepackets for Fluorescence Appendix E: Doorway-Window Phase-Space Wavepackets for Pump-Robe Signals References

I. INTRODUCTION In the analysis of linear and nonlinear optical spectroscopies, the electric fields and optical gates are commonly represented by their amplitudes. Similarly, the material system is represented by an amplitude as well, the wave function. However, optical signals are given by products of such amplitudes.

Advances in Chemical Physics, Volume 101: Chemical Reactions and Their Contml on the Femtosecond Erne Scale. XXth Solvay Conference on Chemistry, Edited by Pierre Gaspard, Irene Burghardt, I. Prigogine, and Stuart A. Rice. ISBN 0-471-18048-3 0 1997 John Wiley & Sons, Inc.

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S. MUKAMEL, C. CIORDAS-CIURDARIU, AND V. KHIDEKEL

For the material system we need two amplitudes, representing the bra and ket used in the calculation of the polarization matrix element, whereas the number of field amplitudes depends on the order of the particular nonlinear process. An improved intuitive picture can be obtained by grouping the field amplitudes in pairs using a mixed temporal-spectral (Wigner) representation. This often highlights much more clearly the roles of spectral and temporal features of the field and provides valuable information about the signal (e.g., its time-dependent spectrum). Similarly, the pair of wave function amplitudes can be used to form the density matrix, and a Wigner transform with respect to space brings it to a mixed coordinate-momentum (phase-space) representation that closely resembles the classical phase-space distribution.* In this chapter we show how such mixed Wigner representations can be used effectively to compute and interpret optical signals. The development of a mixed time-frequency representation in which both characteristics of the field and the response function are highlighted is currently receiving considerable attention. This activity is triggered by the rapid progress in pulse-shaping techniques, which made it possible to control the temporal profiles as well as the phases of optical fields with a remarkable accuracy [1-4]. These developments have further opened up the possibility of coherent control of dynamics in condensed phases [5-71. Mixed time-frequency measurements were first introduced in acoustics in the analysis of sound and speech [8]. The frequency-resolved optical gating (FROG)technique [9, 101 enables one to measure the FROG spectrograms that contain both temporal and spectral information about the signal. This spectrogram can then be inverted, and the intensity and phase of a pulse is obtained. However, the correspondence between FROG signals and the temporal and spectral profiles of the field is not always unique [7, 91. In this chapter we use the Wigner distribution, known also as the chronocyclic representation [11-15]. The interest in using this distribution, whose properties have been known for a long time, for the description of optical fields was resuscitated in the last few years, when it became apparent that it not only is a theoretical construct but also can be actually retrieved from nonlinear experiments [ 121, providing, therefore, a method for characterizing a signal completely, both in the time and frequency domain. The easiest way to obtain a mixed time-frequency picture (spectrogram) is to pass the signal through a spectral and temporal filter centered at some controlled frequency and time prior to its detection [16]. Phase-space wavepackets for nuclear motions have been applied to the interpretation of nonlinear optical measurements using the Liouville space *The electric field can also be transformed in coordinate space, but this will not be considered here.

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representation of the response function 1171. We shall demonstrate how such wavepackets can be used in the analysis of fluorescence and pumpprobe spectroscopies. In Section I1 we study spontaneous light emission. We consider a setup in which the signal emitted by the sample passes through both a spectral and a temporal gate before it reaches the detector. We show that the detected signal can be written as a convolution of the bare autocorrelation function and a joint characteristic function of both gates. We then develop a correlation function expression for the bare signal and show that this expression is identical to but more compact than the one obtained by calculating the photon emission rate in Liouville space [17]. In Section I11 we rewrite the material response function in phase space using the Wigner representation for the external fields, the gate, and the doorway and window wavepackets. In Section IV we apply the same formalism to develop the doorway-window picture for pumpprobe spectroscopy. Finally, in Section V we show how our pumpprobe expressions may be generalized and applied to the calculation of heterodyne-detected four-wave mixing spectroscopies.

11. CORRELATION FUNCTION EXPRESSION FOR SPONTANEOUS LIGHT EMISSION We consider a single molecule whose interaction with an external radiation field E(r, t) is given by

where V is the molecular dipole operator and r is its position. (The following expressions hold for a real dilute sample made of noninteracting molecules. For simplifying the notation we consider a single molecule.) While discussing the properties of light using a classical description, the formalism is recast as close as possible to the quantum form by introducing the complex analytical signal E(r,t), which may be obtained from the real field E(r, t ) by [ 181

By construction, the Fourier transform of the analytical signal contains only positive frequencies. We further assume that the field comprises a finite number of pulses with wave vectors k,. We then have

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S. MUKAMEL, C.CIORDAS-CIURDARIU, AND V. KHIDEKEL

E(r, t ) =

i

E,(t)eikjr

+ C.C.

where !€,(t) and I , ( t ) are the complex-valued analytic field and polarization, respectively. The field envelopes Ej(t) can be further represented in the form Ej(t) = gj(t)e-iwj',where g,(t) is a slowly-varying envelope and o, is the carrier frequency. In the simplest (homodyne) detection scheme one measures the time-integrated intensity of the field generated by the sample in a given direction k,: (2.3)

Time-integrated detection is commonly u.: xi in wave-mixing experiments, such as photon echoes [19]. However, time-integrated photon echo signals do not contain sufficient information to establish the complete form of the spectral density responsible for optical dephasing [20]. Additionally, most valuable microscopic information may be obtained by time-gated (or timeresolved) detection [21, 221, achieved by overlapping the total response with a narrow gate pulse, which provides the temporal profile of the signal. A number of more elaborate techniques, which provide a richer information, including spectral characteristics of the signal, have been developed. The second-order autocorrelation function (Ej*(tl)T.j(t*)) allows one to obtain more information about the field El81: its amplitude as well as the phase. This can be viewed as the time-resolved spectrum of the light pulse, namely the time-dependent strength of its different frequency components. The autocorrelation signal can be observed by passing the field through two gating devices, a spectral gate whose transmission function is centered at wo and a time gate centered at to. The measured signal (the total energy received by the detector) is given by Eq. (2.3), except that Ej(t) now denotes the gated field Ej. The resulting gated autocorrelation signal becomes a function of both wo and to, thus retaining its temporal and spectral information:

WIGNER WAVEPACKETS IN NONLINEAR SPECTROSCOPY

349

In Appendix A we show that the gated signal I ~ ~ ~ , , 00) f t " can , be recast in the form

where

[-- @*(t - $)$(t 00

? ~ " t ~a) ( t ,=

J

+ i7))eiwrd7

is the ideal (bare) autocorrelation signal, corresponding to a gate with an infinite temporal and spectral resolution, and +(r', w'; to, wo) is the joint gate function that depends on the transmission functions of both gates as well as the order in which they are applied. This function is calculated in Appendix A. The bare signal is not positive definite and it may even assume negative values. This is not the case for the gated signal, which is written as the squared amplitude (of the gated field) and is therefore guaranteed to be nonnegative. The gate function +(t', w'; to, w g ) is usually localized in both time and frequency. However, it cannot be made infinitely narrow in both variables due to fundamental uncertainty of the Fourier transform, &At 2 1. It then follows that mathematically the ideal gate + ( t , w ; t o , w o ) = &(to - t ) x 6(wo - (J) cannot be realized. Physically, however, an ideal gate is possible when both the temporal and spectral profiles of the gate are narrower than certain relevant material scales. The detected signal is then virtually identical to the bare signal. An example of such a limiting case is the snapshot limit of pump-probe spectroscopy [171. By expanding @*(t - ;7@(t + to second order in the external field, the autoc_orrelationsignal IAuro(t,w ) gives the spontaneous light emission, denoted I s ~ ~ a). ( t , In Appendix C we express the polarization in terms of a response function convoluted with the external field. By invoking the rotating-wave approximation, we get

i))

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S. MUKAMEL, C. CIORDAS-CIURDARIU, AND V. KHIDEKEL

where

is the four-point equilibrium correlation function of the dipole operator. V(t) are operators in the interaction picture, evolving in time with respect to the material Hamiltonian H o (with no external field)

V(t) = exp(iHot)Vexp(-iHot)

(2.9)

Hereafter we set h = 1 (unless its inclusion is needed for clarity). The Liouville space path diagram corresponding to this correlation function is shown in Fig. 1. An alternative way to calculate the SLE spectrum is to expand the molecular density matrix to second order in the field and compute the time-dependent photon emission rate. The resulting expression is [23]

This formula, unlike Q.(2.7),maintains a complete bookkeeping of the time ordering of the various interactions with the radiation field [17] and has six terms; the Liouville space paths corresponding to three of them are shown in Fig. 2, and the paths for the complex-conjugate terms are obtained by interchanging the left and the right portions of each path. It can be easily shown that Eqs. (2.7) and (2.10) are identical. Indeed, the three-dimensional integration domain in (2.7)can be divided into six subdomains, each corresponding to a term of (2.10) upon a proper change of time variables:

WIGNER WAVEPACKETS IN NONLINEAR SPECTROSCOPY

35 1

-t

Figure 1. Liouville space diagram corresponding to the only term that contributes to the spontaneous light emission from a two-level system within the rotating-wave approximation [Eq.(2.7)]. Here (8) and le) denote the ground and the excited states, respectively.

(2.1 1)

(a) (b) (c) Figure 2. Liouville space paths for the three terms of Eq. (2.10).

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S. MUKAMEL, C. CIORDAS-CIURDARIU, AND V. KHIDEKEL

r
r>O

Figure 3. Diagrams showing how to divide the triple integral in (2.7) to get the six terms of (2.10).Domains (a),(b),and (c)correspond to the three Liouville space paths given in Fig. 2 and domains ( d ) , (e), and (f)to the complex-conjugate paths.

These integration domains are depicted in Fig. 3. The first three terms in (2.11) represent the third, the second, and the first terms in (2.10), respectively. A simple substitution of time variables, as specified in Table I, finally recovers the six terms of (2.10). We now rewrite Eq. (2.7) by introducing the Wigner distribution for the external field as well,

Wdt,a)=

I

m

E*(? - $7)E(t + 4 7 ) e i W ' d7

(2.12)

-m

Substituting (2.12) into (2.7), we obtain the autocorrelation signal (2.5), where the bare signal is given by TABLE I Correspondence between Time Arguments of Eq. (2.7) and Six Terms of Eq. (2.10)

WIGNER WAVEPACKETS IN NONLINEAR SPECTROSCOPY

353

We have thus expressed the autocorrelation signals using the Wigner representation for both the external fields and the gate. The molecular properties are contained in the response function F(4).In the next section, we show how when the incoming external pulses and the detection gate are temporally well separated, we can use the Wigner representation for the material system as well.

III. WIGNER WAVEPACKETS IN PHASE SPACE: THE DOORWAY-WINDOW PICTURE Assuming that the incoming pulses and the gated emission are temporally well separated, we can recast the general expression for the SLE presented in the previous section in a different form that lends itself more easily to physical interpretation. This will allow us to separate the process into the preparation of a doorway wavepacket by the pump field, a subsequent propagation of this wavepacket for a specified period, and finally observing it through a window wavepacket created by the detection device. By using the Wigner phase-space function for the doorway and the window wavepackets, we obtain an “all-Wigner” representation of the signal. We first rewrite FQ.(2.13) by making the following change of variables 71 + r2 -+ 27’, 71 - 7 2 + 7 + r”. The assumption that the pulses are well separated temporally allows us to extend all time integrals to infinity, resulting in

~ ( ~ -) 7’( t+ W ,t + 47, t 2

i7, t -

7’

- 47’’)

WE(t- r’, w 1)

(3.1)

By introducing a complete basis set in the coordinate representation, we show in Appendix D that the SLE signal can be recast in the form

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S. MUKAMEL, C. CIORDAS-CIURDAFW, AND V. KHIDEKEL

with D,(xx’;to) =

(3.3) we(XX’; 00)=

1

27r

j1

-no

dt dr dw @ ( t ,o;0, oo)eiu’

Here De(xx’;0) is the doonvay wavepacket representing the molecular density matrix created by the external field at t” = 0. It then propagates for the delay period to resulting in D,(XX‘;to). Here, W,(xx’; 0 0 ) is the window wavepacket created by the gate and represents the density matrix responsible for emission at frequency WO. (Note that the time delay evolution is included now in the doorway wavepacket so that the window does not depend on to.) The signal is obtained by calculating the overlap of these two wavepackets in Liouville space. A more intuitive picture of these wavepackets can be obtained by switching to the Wigner (phase-space) representation

and the signal [Eq. (3.2)1 assumes the form

Now the overlap of these wavepackets is calculated in phase space. Note that this is a fully quantum mechanical picture, and no classical approximations were made. Our only assumption is that the excitation and gating processes are well separated. This form, however, allows the development

WIGNER WAVEPACKETS IN NONLINEAR SPECTROSCOPY

355

of semiclassical approximations for computing the doorway and the window wavepackets. When the Wigner distributions of both the external field and the gate function are fast compared with the time scale of nuclear dynamics and have a narrow spectrum compared with the dephasing rate, we can set W&”,wl) = 6(t”) - 6(wl - w,) and @(t,w;O,oo) = 6(t) - 6(w - 0 0 ) [here we is the frequency of the external field E(t)]. Although, as stated earlier, such a form is mathematically not possible, it can still be used approximately since the “narrowness” in time and in frequency are with respect to different molecular quantities. We can then calculate the integrals with respect to t”, 0 1 for the doorway wavepacket and t, w for the window wavepacket and obtain the snapshot limit of both wavepackets: co

d7” eiofT”

D:o)(xx‘; to) =

J

--

The ideal snapshot spectrum is obtained by substituting Eqs. (3.6) and (3.7) in Eq. (3.2).

IV. NUCLEAR WAVEPACKETS IN PUMP-PROBE SPECTROSCOPY

In this section we apply the same formalism to pumpprobe spectroscopy, where one measures the absorption of a probe pulse E 2 ( t ) by a molecule excited by a pump pulse El(t). The pumpprobe signal can be written as

J

--

where 4 ( t ) is the polarization induced in the sample by the external electric field. We start by expanding the polarization to first order in the probe amplitude E?(t):

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S. MUKAMEL, C. CIORDAS-CIURDARIU, AND V. KHIDEKEL

J

-oo

where S(')(t,7) is the response function linear in the probe [171, S")(t, 7 ) = i8(t - 7)([Pl(f),P~(T)])

- 7 ) is the Heaviside step function and !P'(t) is the polarization operator in the Heisenberg picture, calculated with respect to the Hamiltonian H', which includes the pump field but excludes the probe:

O(t

The pumpprobe signal can then be written as

This formula resembles Eq.(2.6) for the autocorrelation signal. We can further expand S ( ' ) ( t ,7 ) to second order in the pump field and express the result in terms of the four-point correlation function (2.8) (see Appendix E). Similar to the correlation measurements discussed in the previous sections, we can define the doorway and window wavepackets and write the signal as their overlap in phase space. The derivation is presented in Appendix E, and we have

ZPP(t0,

wo) =

Jf

dx dr'[W,(xx'; w0)De(xx'; to) + W,(xx'; wo)Dg(xx'; to)] (4.3)

where to is the delay between the pump and probe pulses and wo is the carrier frequency of the probe. Upon transforming from the coordinate to the Wigner phase-space representation, we can write (4.3)as

WIGNER WAVEPACKETS IN NONLINEAR SPECTROSCOPY

357

The wavepackets D,,Wj,j = g, e, can be written in the coordinate representation as

(4.6)

* jjjy- d t d7 dw W&, w)eiwr

W,(xx‘; wo) = 27r

The first term in Eq. (4.3) is reminiscent of Eq.(3.2) for the spontaneous emission spectrum. It represents a doorway wavepacket created by the pump in the excited state, which is then detected by its overlap with a window. The only difference is that the role of the gate in determining the window in SLE is now played by the probe Wigner function 0 2 . In addition, the pumpprobe signal contains a second term that does not show up in fluorescence. This term represents a wavepacket created in the ground state (a “hole”) that evolves in time as well and is finally determined by a different window W, [24]. In the snapshot limit, defined in the preceding section, we have

358

S.MUKAMEL, C. CIORDAS-CIURDARIU, AND V. KHIDEKEL

J

--

V. EXTENSION TO HETERODYNE-DETECTED FOUR-WAVE MIXING The description of pumpprobe signals presented in the preceding section can be immediately generalized to heterodyne-detected transient grating spectroscopy as well as to other four-wave mixing techniques. Heterodyne detection involves mixing the scattered field with an additional heterodyne field E4(t). The signal in the k, direction can then be written in terms of the polarization P&) as

Consider the signal emitted in the direction ks = k3 - k2 + kl. The polarization !P&) is given by &. (El) except that now the two fields that excite the sample are different. We therefore get

* - t 3 - t2 - ti) x E3(t - t3)E,(t - t 3 - t2)E2(t

x E3(t - t3)Z?(t - t 3 - t2)El(t - r3 - t 2 - t i )

(5.2)

WIGNER WAVEPACKETS IN NONLINEAR SPECTROSCOPY

359

Here the time ordering of the El and E2 fields can be arbitrary; we only assume that the field 2 3 comes after El and E 2 and does not overlap with them. We can then follow the calculations of pumpprobe signals in Appendix E and introduce the joint Wigner distribution for the fields El and E2 and for the field fE3 and 2 4 : am

am

Substituting this into (5.1), we get

The doorway-window picture applies here as well. We can use Eqs. (4.5X4.8) provided we replace the pump wavepacket W I by W ~and I the probe wavepacket W 2 by W 4 3 .

APPENDIX A: TIME- AND FREQUENCY-GATED AUTOCORRELATION SIGNALS Autoconelation signals may be observed by passing the field through two gating devices, a spectral and a time gate. Each gate has its transmission function (in the time and frequency domain) centered at 0 0 and to, respectively. The field passing through each of these gates becomes

Here F,(w; W O ) and !Fr(f; to) are the spectral and temporal transmission functions and and E are the fields before and after passing through the gate, respectively. We shall denote them the bare and the gated signal, and throughout this appendix bare quantities will be denoted by a tilde. Note that the order in which the two gates are applied is important, even though both

360

S. MUKAMEL. C . CIORDAS-CIURDARIU,AND V. KHIDEKEL

devices are linear and independently controlled. From a practical point of view, to attain optimal time and frequency resolution, it is advantageous to apply first the gate with the less ideal transfer function [25]. Below we will consider both configurations. The measured signal (the total energy received by the detector) is given by Eq. (2.3), except that the field now passes through gating devices prior to its detection. Therefore, the signal becomes a function of both 00 and to, thus retaining its temporal and spectral information. The autocorrelation signal is defined by Eq. (2.4). We will show in Appendix B that within the slowly varying amplitude approximation the measured field is proportional to the polarization induced in the sample by the external electric fields [Eq. (B4)]. We therefore have

It was shown elsewhere [26] that the gated signal Z ~ ~ ~ ~00) ( t is 0 given , by Eq. (2.5) together with (2.6). The transmission function is expressed in terms of the Wigner functions for the gates,

W$(t,0)=

1

0

Y;(W - $,J)Ts(w

+ $o’)e-io’r dw’

If the spectral gate is applied first, we have

where the subscript st indicates that the signal first passes through the spectral gate and then through the time gate. For the reverse order of the gates,

we have

These two equations can also be written in another form using the definition

WIGNER WAVEPACKETS IN NONLINEAR SPECTROSCOPY

36 1

of the Wiper function, we then have

as&’,w’;

I_., m

to, wo) =

dtl!F,(t; to)12Ws(t - t’, w’; wo)

(A31

Assuming specific forms of the time and spectral gates, we get the signal measured in different experimental setups. If the time gate is infinitely short, F1(f;fo)= &t - to), we obtain the spectrograms discussed in Ref. [16]. If the signal passes only through the time gate, then the gate acts like the reference pulse in the FROG configuration. The joint gate function is centered around the frequency wo and the time to and acts as a filter on the bare signal ?. As an example we shall consider the case when the spectral gate is given by the Fabri-Perot 6talon [16] and the time gate is exponential,

Note that in this case the transmission functions (A5) and (A6) depend on t, to and w, wo only through the differences t - to and w - w g and the joint gate function #sr(t’, w’; to, 0 0 ) only depends on T = f’ - to and Q = w’ - 00. The joint gate function for the gates ( A 3 and (A6), calculated by (A3), assumes the form T>O

In the case y>> I’ the above expression simplifies and can be written in the compact form

362

S. MUKAMEL, C.CIORDAS-CIURDARIU, AND V. KHIDEKEL

For the reverse order of gates, when the signal passes first through the time gate and then through the spectral one, we have

x { [Q - y(y +

r)]cos 2QT - (2y + r)sin 2QITI }

More elaborate gating profiles may be obtained using pulse-shaping techniques [2, 31.

APPENDIX B: THE SIGNAL AND THE OPTICAL POLARIZATION We relate here the scattered field to the polarization induced in the sample by external fields. The radiated field is the solution of the Maxwell wave equation

V x V x z(r, t ) + - I

a2

.2

at2

t)=

4?r

--

c2

a2

g(r, t ) at2

(Bl)

Here g(r,t) is the analytic polarization, which is defined in terms of the real polarization Ztr)(r,t ) in the same way as the analytic signal was defined in terms of the real signal [Eq.(Bl) is usually written for the real field and polarization; however, it follows from their definition that the analytic quantifies satisfy the same equation]. When the analytic polarization P(r, t) is given, the wave equation is linear and inhomogeneous and can be soived exactly in a closed form. The general solution of E!q. B1 is [17]

E(r, t ) =

where

I

dr’ dt‘ G(r - r‘; t - t’)g(r’, t’)

WIGNER WAVEPACKETS IN NONLINEAR SPECTROSCOPY

363

and

We are interested in far-field limit measurements, that is, when the size of the material system is much smaller than its distance from the detector. In this limit, the usual approximation one makes is [27]

lr-r’l=r-r’cos 8 where 8 is the angle between r (the vector giving the position of the observation point) and r‘ (the variable of integration over the volume of the material). In this approximation the Green function G(r - r; w) becomes w2

Gje(r- r‘; w ) = 2 (a,/ c

F)

e i ( a / c ) ( r - r’cos

-

e)

r

We now use this form for the Green function in Eq. (B2), integrating with respect to w and using the relation

J

2e-iCd(f - fo)

d2 dw = -27r 6(t - to) dt2

We then integrate with respect to t’ and obtain

2‘R Ej(r,t)= -C2Y

(6,l-

F)I

dr’

c

Since we are interested in the far-field case, we can approximate in the integrand r - rf cos 0 = r, and therefore, integrate over r’ and obtain the following relation between the scattered field and the polarization:

S . MUKAMEL, C. CIORDAS-CIURDARIU, AND V. KHIDEKEL

364

where ztot(t) is the total polarization of the material. Although we have obtained a formula that takes into account retardation effects, we will neglect them in what follows. We also invoke the slowly-varying amplitude approximation setting B ( t ) = - w 2 p ( t ) . Finally, we can write E,(r,t) =

2uw -

c2r

Assuming that the polarization Pl(r‘, t) has the form of a plane wave along the z direction,

we can integrate over z’ in (B3) and get Ej(r, t ) =

2uw -

rkc2r

where =

j J,

dx’ dy’ p, (x’, y’, t )

APPENDIX C: FOUR-POINT CORRELATION FUNCTION EXPRESSION FOR J?LUORESCENCE SPECTRA We consider a molecular system interacting with an external electric field and described by the Hamiltonian

H = Ho - VE(r, t )

(C1)

where HO is the electronic and nuclear molecular Hamiltonian and the dipole operator V represents its interaction with the field. Throughout this section E(r, r ) denotes the real field. The autocorrelation function of the polarization operators (&r1)&f2)) is expressed in terms of the dipole operator as

WIGNER WAVEPACKETS IN NONLINEAR SPECTROSCOPY

365

where V&) denotes the dipole operator whose evolution is governed by the Hamiltonian H:

V H ( t )= U(t,--)V(--)Ut

--

( t , --)

U ( t ,t’)

= exp

{ j: i

H(t)dt}

--

The autocorrelation function (C2) is characterized by three intervals of evolution: from to t l , from f l to 1 2 , and from t2 to (Fig. 4a). It is inconvenient to use the finite interval ( t l , t2), since the corresponding time integrals will have (variable) finite limits. A more practical way to analyze (C2) is to extend the interval (tl ,t2) to and factorize the evolution operator U(t1,t2) as U(--,t2)Ut (--, t i ) (Fig. 4(b). We then have

--

Perturbative expansion with respect to the external electric field yields,

with

366

S. MUKAMEL, C. CIORDAS-CIURDARIU, AND V. KHIDEKEL

(a)

(b)

Figure 4. ' h o ways to describe the time-evolution of the dipole operators: single ( a ) and double (b) Liouville space diagrams.

Here V ( t ) is the dipole moment given by Eiq. (2.9). For n = 2 the subscriptsj, I , m,i take the values 0, 1, 2 (so that their sum is at most 2), and the result is expressed in terms of the four-point correlation functions:

WIGNER WAVEPACKETS IN NONLINEAR SPECTROSCOPY

367

where

We now consider a two-level system and invoke the rotating-wave approximation, in which we neglect highly oscillatory terms. We can now calculate the evolution operator Uo(t,r’) explicitly, using the Hamiltonian of the ground and excited states of the two-level system. If w is close to the transition frequency of the two-level system, then the only term that will not is the term with C?&. Keeping contain highly-oscillating factors e’(O this term only, we get

This formula can be easily rewritten for the analytic field and polarization. Substitution into Eq. (2.6) finally results in Eq. (2.7).

APPENDIX D: PHASE-SPACE DOORWAY-WINDOW WAVEPACKETS FOR FLUORESCENCE We now derive the expression for the fluorescence signal in terms of the doorway and window wavepackets instead of the four-point correlation function. We start with Eq.(3.1) and write the four-point correlation function F(4) explicitly as the trace with respect to the equilibrium density matrix. We then use the cyclic invariance of the trace and obtain 1 + p, t - 7’ + i7”) = Tr[V(t - $7)V(t + ;7)V(t - 7’ + $‘‘)V(t

~ ( ~’ 7’ ( t- :r”, t -

$7,t

- 7’- $7”))

(Dl)

We now write the equilibrium density matrix in the coordinate represen-

368

S. MUKAMEL, C. CIORDAS-CIURDARIU, AND V. KHIDEKEL

tation and insert in (Dl) the unity operator dx’Ix‘)(x’l. We then have F‘4’(t - 7’

1 1 I ’ I - p”, r - 97, t + 27, t - 7 + 97

’/

)

We assume that the external field is peaked at the time -to and the temporal gate at the time t = 0. Changing in the doorway wavepacket the variable t” = t + to - 7’, we can rewrite equation (D2) as F‘4’(t - 7’ -

iT”,

t - 47,t + 47, t - 7’

+ i7”)

where

is the doorway wavepacket created by the external field, and

is the window wavepacket. The response function is then calculated by the overlap of the window wavepacket with the doorway wavepacket propagated for the time to. These formulas can be rewritten in the Heisenberg picture, pD(Xt x’; f ” , 7”)

vege-iHg(r”/2)poe-iHs(~”/2) v

= (xle iHe(r”/2 - I” + 10)

ge

e i H e ( ~ ” / 2 + r” - 10)

lx’)

P w(x, x’; t , 7 )

eiffg(r/2) iH - (xle-i/fe(r/2 - ?)v eg e

(7/2)v e - i H e ( 7 / 2 + r ) ge

1 4 .

We now return to the expression for the gate signal (2.5) and substitute the bare signal (3.1) with the four-point correlation function F(4)replaced with the doorway and window wavepackets as in (D3). We then separate the four time integrals into two groups: the integrals with respect to t” and 7” for the doorway wavepacket and with respect to t and 7 for the window wavepacket. We can also change all limits of integration to infinity; this is

WIGNER WAVEPACKETS IN NONLINEAR SPECTROSCOPY

369

possible since we assumed that the Wigner function of the incoming field WE and the gate function CP are well separated in time. We then obtain Eqs. (3.2H3.4).

APPENDIX E: DOORWAY-WINDOW PHASE-SPACE WAVEPACKETS FOR PUMP-PROBE SIGNALS We shall calculate here the pumpprobe signal (4.1) using the doorway window wavepackets representation. The polarization P ~ ( tto) third order in the external field is given in Ref. 17 and is shown to be expressed in tenns of the four-point correlation function (2.8):

where R:3’(t3,t2,tl) = F‘4’(t - t3 - t 2 - tl,t,t - t3.t R‘3’ 2 (t3,

- t3 - t 2 )

t2,tl) = F‘4’(t- t3 - t2, t , t - t3, t - t 3 - t 2 - t i )

R(3) 3 (t3, t2, ti) =

F‘4’(t - t 3 , t , t - t 3

- t2,t

- t3 - t2 - ti)

R,( 3 )( t 3 , t 2 , t l ) = F(4’(t- 13 - t 2 - t i , t - t 3 - t 2 , t - t 3 , t )

Substituting this into (4.1) and introducing the Wigner representation for the pump and the probe fields as was done for the fluorescence, we get

370

S. MUKAMEL. C. CIORDAS-CIURDARIU, AND V. KHIDEKEL

Similar to the Auoresence discussed in Appendix D, we can define doorway and window wavepackets and write the signal as their phase-space overlap. Assuming that the delay time between the probe and the pump pulses is to, we obtain Eqs. (4.5)-(4.8).

Acknowledgments

The support of the National Science Foundation and the Air Force Office of Scientific Research is gratefully acknowledged.

References 1. D. J. Tannor and S . A. Rice, J. Phys. Chem. 83, 5013 (1985). 2. A. M. Weiner, Prog. Quantum Electron. 19, 161 (1995). 3. J. T. Fourkas, L. Dhar, K. A. Nelson, and R. Trebino, J. Opt. SOC. Am. B 12, 155 (1995). 4. N. F. Scherer, L. D. Zeigler, and G. R. Fleming, J. Chem. Phys. 96, 5544 (1992). 5. W. S. Warren, H.Rabitz, and M. Dahleh, Science 259, 1581 (1993). 6. B. Kohler, J. L. Krause, F. Raksi, C. Rose-Petruck, R. M. Whitnell, K. R. Wilson, V. V. Yakovlev, Y. Yan, and S. Mukamel, J. Phys. Chem. 97, 12602 (1993). 7. B. Kohler, V. V. Yakovlev, J. Che, J. L. Krause, M. Messina, and K . R. Wilson, Phys. Rev. Lett 74,3360 (1995). 8. W. Koenig, H. K. Dunn, and L. Y. Lacy, J. Acoust. SOC. Am. 18, 19 (1946). 9. D. J. Kane and R. Trebino, IEEE J. Quantum Electron. 29, 2 (1993). 10. R. Trebino and D. J. Kane, J. Opt. SOC.Am. A 10, 1101 (1993). 1 I . J. Paye, IEEE J. Quantum Elecrron. 28, 2262 (1992). 12. J . Paye, in Ultrafast Phenomena, Vol. 9, P. F. Barbara, W. H.Knox, G. A. Mourou, and A. H. Zewail, Eds., Springer-Verlag, New York 1994. 13. L. Cohen, Proc. IEEE 77,941 (1989). 14. L. Mandel and E. Wolf, Eds., Selected Papers on Coherence and Fluctuations of Light, with Bibliography, Dover, 1970. 15. M. G. Raymer, M. Beck, and D. T. Smithey, Phys. Rev. Lett. 70, 1244 (1993). 16. H. Stolz, lime-Resolved Light Scattering from Excitons, Springer, Verlag, Berlin; New York 1994. 17. S. Mukamel, Principles ofNonIinear Optical Spectroscopy, Oxford University Press, New York 1995. 18. H. M. Nussenzweig, Introduction to Quantum Optics, Gordon & Breach, London; New York 1973. 19. E. T. J. Nibbering, D. A. Wiersma, and K. Duppen, Phys. Rev. Lett. 66,2464 (1991). 20. D.C. Amett, P. Vohringer, R. A. Westervelt, M. 3. Feldstein, and N. F. Scherer, in Ultrafast Phenomena, Vol. 9, P. F. Barbara, W. H. Knox, G. A. Mourou, and A. H. Zewail, Eds., Springer-Verlag, Berlin 1994. 21. W. P. de Boeij, M. S. Pshenichnikov, and D. A. Wiersma, Chem.Phys. Letf. 238,l (1995). 22. P. Vohringer, D. C. Amett, T.-S. Yang, and N. F. Scherer, Chem. Phys. Lett. 237, 387 ( 1995). 23. S. Mukamel, Adv. Chem. Phys. 70 (Part I), 165 (1988).

WIGNER WAVEPACKETS IN NONLINEAR SPECTROSCOPY

37 1

24. Y. J. Yan and S. Mukamel, Phys. Rev. A 41, 6485 (1990). 25. V. Wong and 1. A. Walmsley, J. Opt. Soc. Am. B 12, 1491 (1995). 26. S. Mukamel, C. Ciordas-Ciurdariu, and V. Khidekel, IEEE J. Quanrum Electron., 32, I278 ( 1996). 27. J. D. Jackson, Classical Elecrrodynarnics, Wiley, New York, 1975.

Advances In Chemical Physics, Volume101 Edited by I. Prigogine, Stuart A. Rice Copyright © 1997 by John Wiley & Sons, Inc.

GENERAL DISCUSSION ON LASER CONTROL OF CHEMICAL REACTIONS Chairmun: M. Quack

V. Engel: Prof. Manz, 1want to come back to the question of dissipation. We learned from the talk of Prof. Fleming that coherences persist for a while in a liquid surrounding, but not too long. The control schemes we heard about rely on quantum interference or coherence. I would like to know: What are the perspectives of applying the control schemes to chemical reactions in a liquid? J. Ma-: Prof. V. Engel’s question points to the virtues of using (i) ultrashort laser pulses for controlling chemical reactions, in comparison with (ii) continuous-wave lasers. Compare, for example, the strategies of (i) Tannor et al. [l], Rabitz [2], Combariza et al. [3] (see also Korolkov et al., “Theory of Laser Control of Vibrational Transitions and Chemical Reactions by Ultrashort Infrared Laser Pulses,’’ this volume) versus (ii) the strategies of Brumer and Shapiro [4] or Letokhov [5] and others [6]. See the review by S. A. Rice (this volume). Intuitively, one may assume that the advantage of using ultrashort laser pulses should be that they can be faster and, therefore, “beat” competing dissipativeprocesses such as intramolecular vibrational redistribution (IVR). This argument is supported, certainly, by the observation of relatively long coherence lifetimes of reactive wavepackets even in the condensed phase, as has been demonstrated beautifully by G. R.Fleming et al. (“Femtosecond Chemical Dynamics in Condensed Phases,” this volume) and A. H. Zewail[7]. However, this advantage may be valid only for rather strong laser fields, because, otherwise, for weak laser fields, there is a theorem by Brumer and Shapiro saying that time-dependent and time-independent fields will achieve equivalent molecular transitions [81. 1. D. J. Tannor and S. A. Rice, J. Chem. Phys. 83, 5013 (1985); D. J. Tannor, R. Kosloff, and S. A. Rice, J. Chem. Phys. 85, 5805 (1986); D. J. Tannor and S. A. Rice, Adv. Chem. Phys. 70, 441 (1988). 2. S. Shi, A. Woody, and H. Rabitz. J. Chem. Phys. 88,6870 (1988); W. S. Warren. H. Rabitz, and M. Dahleh, Science 259, 1581 (1993); W. Jakubetz, J. Manz, and H.-J. Schreiber, Chem. Phys. Lett 165, 100 (1990): W. Jakubetz, E. Kades, and J. Manz, J. Phys. Chem. 97, 12609 (1993).

373

374

GENERAL DISCUSSION

3. J. E. Combariza, B. Just, J. Manz, and G. K. Paramonov, J. Chem. Phys. 95, 10355 ( 1991). 4.

M. Shapiro and P. Brumer, J. Chem. Phys. 84, 4103 (1986); P. Brumer and M. Shapiro, Acc. Chem. Res. 22,407 (1994).

5 . V. Letokhov, Science 180,451 (1973). 6. E. Segev and M. Shapiro, J. Chem. Phys. 77,5604 (1982); V. Engel and R. Schinke, J. Chem. Phys. 88,6831 (1988); D. G. Imre and J. Zhang, J. Chem. Phys. 89, 139 (1989); M. D. Likar, J. E. Baggott, A. Sinha, T. M.Ticich, R. L. Vander Wal, and F. F. Crim, J. Chem. Suc. Faraday Trans. I1 84, 1483 (1988); F. F. Crim, Science 249, 1387 (1990). 7. A. H. Zewail, in Femtosecond Chemistry, J. Manz and L. Woste, Eds., Verlag Chemie, Weinheim, 1995, p. 15; R.Zadoyan, Z. Li, P. Ashjian, C. C. Martens, and V. A. Apkarian, Chem. Phys. LPtf. 218, 504 (1994). 8. M. Shapiro and P. Brumer, J. Chem. Phys. 84, 540 (1985).

M. Shapiro: In relation to the problem discussed by Profs. Engel and Manz, I should point out that, if there are fast dephasings, one definitely would have to use fast pulses. Graham Fleming’s data suggest, however, that coherences persist for much longer times than anticipated in the past. In order to overcome decoherence, we can consider adiabatic passage techniques (cf., e.g., Bergmann, Reuss, van Linden van der Heuvel, and Neusser) whose advantage is that you can completely empty the ground state, irrespective of the exact pulse shape, provided that the area under the pulse d t E ( t ) p l exceeds the limit imposed by the adiabatic condition. L. Woste: Prof. Manz, when you shape potential-energy curves or surfaces with high-intensity IR laser fields, don’t you believe that these field strengths can also induce electronic transitions leading to entirely different states and potential-energy surfaces, so that the proposed shaping process loses relevance? M. V. Korolkov, J. Maw, and G. K. Paramonov:* We are fully aware of the danger of using exceedingly intense IR picosecond/femtosecond laser pulses for control of molecular vibrations or reactions. As a rule derived from laser interactions with atoms, the intensities should be below the Keldish limit [ 11,

[I

2 *Comment presented by J. Manz.

16Z2

LASER CONTROL OF CHEMICAL REACTIONS

ZKeldish =

2 2

375

W/cm2

where Es is the ionization potential in electron-volts. Otherwise the laser may ionize the atoms, yielding a large displacement ro in the presence of an electric field with amplitude E Oand carrier frequency w [2]:

where I is in watts per square centimeters, h is in nanometers, and is in reciprocal centimeters. We wish to point out, however, that, to the best of our knowledge, there are so far no systematic investigations of similar Keldish-type limits for molecules. We have carried out exploratory test calculations for a model system that indicate that even stronger intensities may not be sufficient for ionization by ultrashort infrared laser pulses (see Fig. 1). In any case, one should always try to keep the laser intensities, specifically the amplitudes

or the corresponding amplitudes of the laser field strength as small as possible. There are several ways (i), (ii) or optimal conditions (iii), (iv), (v) for this purpose: (i) Prolongation of the pulse duration. (ii) Separation of a single IR femtosecond/picosecond laser pulse into a series of IR femtosecond/picosecond pulses [3]. (iii) Applications to systems with rather strong variations of the dipole function ~ ( q along ) the vibrational or reaction coordinate (4).Note that the semiclassical molecule-laser interaction operator is

376

GENERAL DISCUSSION (

INITIAL ELECTRON STATE

a)

...._.___.___.._.._____ ---.... E, = - 1 2 . 9 e V

-10

- 20

.'!

-30

-5

-2.5

(

0

0

COORDINATE

b)

100

2.5

( A )

LASER PULSE

200 T I M E

300 (fs)

400

( c ) ELECTRON MOTION

I .5

5

500

I

0.5

0 -0.5 -1 -1.5

'I

0

I loo

200 T I M E

300

(fs)

( d ) ELECTRON

400

I

SO0

MOTION

-5

-10

-IS

-

I

0

Es=-12.9eV

100

200 T I M E

300
400

I

SO0

Figure 1. Testing the Keldish limit 11, 21 to ionization by intense infrared femtosecond/picosecond laser pulses used for control of chemical reactions [3,4]. ( a ) Electronic ground state embedded in a typical made1 potential curve with the ionization potential Es = 12.9 eV. (b) Intense ( 2 0 = 35.5 CV/m-', 10 = 3.3 x lOI4 W/cm*), ultrashort ($, = 0.5 ps), infrared ( l f i = 3784 cm-') laser pulse. (c) Expectation value for the position of the !lectron, which is driven by the laser field shown in panel (b) [compare with ro = 122 A, Eq. (3)J. ( d ) Electron energy. These model calculations demonstrate that even very intense (I > IK&jish) ultrashort 1R laser pulses may not cause ionization; that is, the simple estimates (1)-(4) [ l , 21 are not applicable.

LASER CONTROL OF CHEMICAL REACTIONS

377

that is, large variations of p(q) may allow use of rather small values of field amplitudes ZO and, therefore, intensities [cf. Eq. (511. (iv) Applications to vibrational transitions between close (say Av = 1, 2, 3, ... , not 10, 20, 30, ...) vibrational levels. (v) For applications to isomerizations, items (iii) and (iv) imply that the reaction should proceed across rather shallow barriers of the potential energy surface, and the reactants and products should have rather different (e.g., with opposite signs) dipole moments. (Seealso Korolkov et al., “Theory of Laser Control of Vibrational Transitions and Chemical Reactions by Ultrashort Infrared Laser Pulses,” this volume; supported by DFG.) 1. L. V. Keldish, Sov. Phys. JETP 20, 1307 (1965); S. Augst, D. D. Meyerhofer, D. Strickland, and S . L. Chin, J. Opt. Soc. Am. B 8, 858 (1991); P. Dietrich and P. B.

Corkum, J. Chem. Phys. 97,3187 (1992). 2. A. D. Bandrauk. in Femtosecond Chemistry, J. Manz and L. Woste, Eds., Verlag Chemie, Weinheim, 1995, p. 261. 3. J. E. Combariza, B. Just, 1. Manz, and G. K. Paramonov, J. Chern. Phys. 95, 10351 (1991); B. Just, J. Manz, and G. K. Paramonov, Chem. Phys. Lett. 193,429 (1992).

M. Quack Somewhat related to the very nice isomerization scheme used by Dohle and co-workers [l], I would like to make a more general comment. In connection to “control” in dynamics I would like to take here a general point of view in terms of symmetries (see Scheme 1): We would start with control of some symmetries in an initial state and follow their time dependence. This can be used as a test of fundamental symmetries, such as parity, P, time reversal symmetry, T, CP, and CPT,or else we can use the procedure to discover and analyze certain approximate symmetries of the molecular dynamics such as nuclear spin symmetry species [2], or certain structural vibrational, rotational symmetries [3]. I would like to draw attention here to some work on chiral molecules, which allows very fundamental tests of symmetries in physics and chemistry. The experiment outlined in Scheme 2 [4] allows us to generate, by laser control, states of well-defined parity in molecules, which are ordinarily left handed (L) or right handed (R) chiral in their ground states. By watching the time evolution of parity, one can test for parity violation and I have discussed in detail [MI how parity violating potentials AEpv might be measured, even if as

378

GENERAL DISCUSSION Control of symmetry of initial state

Time dependence of symmetry properties?

Test of fundamental symmetries in nature (such as P, T, CP, CPT) or Test of approximate symmetries of dynamics

Scheme 1. Control of symmetries in dynamics.

T

w

4

h

rn

>

W

4 ->

R Scheme 2. Control of parity and chirality in the scheme of Ref. 4. L

LASER CONTROL OF CHEMICAL REACTIONS

379

small as J/mol. Time scales corresponding to such symmetry violation are hours to days. On the other hand, one can also generate a very short-lived time-dependent chiral excired state, even if the potential has a minimum at an achiral geometry. If the excited-state potential is harmonic, for example, one may derive interesting results on chirality in relation to harmonic oscillator dynamics [7]. This includes femtosecond evolution of chirality and control of stereomutation in chiral molecules [7]. On the most fundamental level, we have shown how this experimental scheme might be used for a fundamental test of CPT symmetry violation [8]. While still somewhat hypothetical, at present, this would constitute the most sensitive currently proposed test on CPT symmetry. The sensitivity expressed as a baryon mass difference Am between particles and antiparticles (with mass m) would be of the order 181. The best currently proposed other experiment of Am/m = is on antihydrogen spectroscopy at CERN (not yet carried out) with Am/m = and the best existing result for the proton-antiproton pair is h / m I [9]. Relating to the introductory comments by Prof. Prigogine on irreversibility on Tuesday, one might consider time reversal symmetry violation and indeed CPT symmetry violation as a more fundamental approach to this problem. One could then envisage that the second law is not just a de fact0 violation of time reversal symmetry but a de lege violation in the terminology of Refs. 5 and 6. Figure 1 shows the entropy evolution of CF2HCl on the femtosecond time scale, showing relaxation toward maximum entropy. With time reversal symmetry, the time reversed dynamic state would return to zero entropy after the appropriate time. This is the current status of experimental knowledge. However, one might envisage that on long time scales for complex systems time reversal symmetry may not apply and entropy stays near the maximum, as drawn roughly as a second option into the figure. 1. M. Dohle, J. Manz, and G . K. Paramonov, Be,: Bunsenges. Physik. Chem. 99,478 (1995). 2. M. Quack, Mol. Phys. 34, 471 (1977).

3. A. Beil, D. Luckhaus, R. Marquardt, and M. Quack, J. Chem. Soc. Furuduy Discuss. 99, 49 (1994); M. Quack, J. Mol. Srrucr. 347, 245 (1995). 4. M. Quack, Chem. Phys. Lett. 132, 147 (1986).

5. M. Quack, Angew. Chemie 101,588 (1989); Angew. Chemie Inr. Ed. Engl. 28,571 (1 989). 6. M. Quack, in Femtosecond Chemistry, J. Manz and L. Waste, Eds., Verlag Chemie, Weinheim, 1995, Chapter 27. p. 781.

380 t/ps

Figure 1. Time-dependent entropy for the three strongly coupled CH stretching and bending vibrations in CF2HCI[6].

t/ps

LASER CONTROL OF CHEMICAL REACTIONS

381

7. R. Marquardt and M . Quack, J. Chem. Phys. 90, 6320 (1989); J. Phys. Chem. 98, 3486 (1994); Z Phys. D 36,229 (1996). 8. M. Quack, Verhund. DPG VI 28,244 (1993); Chem. Phys. Lett. 231,421 (1994). 9. G. Gabrietse, D. Phillips, W. Quint, H. Kalinowsky, G. Rouleau, and W. Jhe, Phys. Rev. Letf. 74,3544 (1995); J. Groebner, H. Kalinowsky, D. Phillips, W. Quint, and G. Gabrielse, Verhund. DPG VI 28, 315 (1993).

M.E. Kellman: The idea by Prof. Quack of using molecular experiments to test fundamental symmetries is very interesting. In connection with the Na3 pseudorotation experiments we heard about yesterday, have you considered testing permutation symmetry of identical particles, that is, the Pauli principle and nuclear spin statistics? M. Quack: The violation of the principle of nuclear spin symmetry conservation [l] could be seen in a similar scheme as I discussed for parity, but, in contrast to parity violation, it can also be seen by more standard spectroscopic techniques (and has been seen repeatedly). On the other hand, one might also look for violations of the Pauli principle, which in fact we have done [2]. However, it seems unlikely to find such a violation (and nothing of that kind has ever been found), although in principle one must allow even for such a phenomenon. 1. M. Quack, Mol. Phys. 34, 477 (1977). 2. M. Quack, J. Mol. Strucr. 292, 171 (1993); and unpublished results.

M. S. Child: I would like to ask Prof. Quack to what extent tunnelling between the two enantiomers might affect these conclusions? M. Quack Tunneling will be completely suppressed if AEpv >> A&,,. Even if that is not the case, if M,, is negligible, tunneling may be exceedingly slow, as was already discussed by Hund in 1927 (see Refs. 1 and 2 and references cited therein). 1. M.Quack, Angew. Chemie 101,588 (1989); Angew. Chemie Inr. Ed. Engl. 28,571

( 1998). 2. M . Quack, in Ferntosecond Chemistry, J. Manz and L. Woste, Eds., Verlag Chemie, Weinheim, 1995, Chapter 27, p. 781; R. Marquardt and M. Quack, 2 Phys. D 36, 229 (1996).

J. Maw: Prof. M. Quack has just presented to us a fascinating strategy for laser control of chemical enantiomers with different parities [ 13. A complementary approach has been suggested earlier by Brumer and Shapiro, based on their general strategy of laser control [2]. I would like to ask Prof. M.Shapiro whether he could comment on his approach in comparison with that of M.Quack. 1.

M. Quack, in Femtosecond Chemistry, J. Manz and L. Woste. Eds., Verlag Chemie,

382

GENERAL DISCUSSION

Weinheim, 1995, Chapter 27, p. 781, R. Marquardt and M. Quack, Z. Phys. D 36, 229 (1996). 2. M. Shapiro and P. Brumer, J. Chem. Phys. 95, 8658 (1991).

M. Shapiro: Our approach [M.Shapiro and P. Brumer, “Controlled Photon Induced Symmetry Breaking: Chiral Molecular Products from Achiral Precursors” J. Chem. Phys. 95, 8658 (1991)], differs from that of Quack in that we show how to generate chirality (from achiral precursors). T. Kobayashk I would like to make the comment that an interesting application of wavepacket control [I] is phonon squeezing in molecular systems and the creation of the Schrodinger cat state. It was theoretically predicted that there are several mechanisms that lead to squeezing of phonon states. It was found earlier that a sudden frequency change during an electronic Franck-Condon transition leads to special quantum mechanical statistics, called squeezing [2-9], of the molecular vibrations [10-121. A state is termed “squeezed” if some of its characteristics have less noise than the corresponding quantum noise of the vacuum state. The concept of squeezing turned out to be very fruitful in basic research and implies a lot of promising practical possibilities. The above-mentioned mechanism of squeezing the vibrational state prompted some controversial discussion in the literature [13-16]. The phenomenon is caused by the change of the frequency of the molecular vibration provided that the transition takes place in a fraction of time negligibly small as compared with the vibrational period. Recently we have shown that phonon squeezing, connected to the finite duration of the excitation pulse, occurs even in the absence of frequency change. This effect would be rather common in ultrashort laser pulse experiments [17-19]. Non-transform-limited pulses, either chirped or incoherent, are very useful in high time resolution spectroscopy 1201. For phonon squeezing, chirped pulses can also be used as shown in Figs. 1 and 2. With a chirped pulse even a Schrodinger cat state can be obtained, as shown in Fig. 2b. Advantages of using a chirped pulse are also shown in Fig. 2b. As can be seen from the figure, the chirped pulse applied to a periodically oscillating system is equivalent to a double pulse. In such a way, a well controlled chirped pulse may serve a purpose equivalent to a pulse train. The Wiper function corresponding to Fig. 2b is shown in Fig. 3. This means that the coherent states are interfering with each other. In such a way we can control a wavepacket by using a chirped pulse instead of an ultrashort pulse.

0.15

0.1 1

10

100

0.3

0.35 W

0.4

1

1.

chirp rate

0.25

I

I !

I I 1

P

Ph

0.5

t =25T

0.45

0.6

(~=0.0437)

0.55

Figure 1. Uncertainty of the quadrature AX: of the phonons (squeezing occurs if AX becomes less than unity) after a resonant Franck-Condon transition induced by a chirped pulse of moderate duration (u = 0.0437~) as a function of the chirp parameter w, which is in the units of the phonon frequency w. The electron-lattice constant is supposed to be g = 5 . The markers a-d refer to Fig. 2.

0.2

.I

..

I I

no-frequency-change case using chirpea long pulse

pulse width = 40 phonon periods (long pulse)

Figure 2. The Q-functions of phonon states after the electronic transition induced by differently chirped pulses. ( a ) The Q-function at the minimal AX+,that is, maximal squeezing (the chirp parameter w = 0.399; see marker a in Fig. 1). (b) The AX+ is in its next maximum. (c) An intermediate state between a maximum and minimum. As we consider lower and lower chirp parameters, the maxima and minima become less prominent ( d ) ,approaching the number state with equal distribution along a circle.

u = 0.0437 o

LASER CONTROL OF CHEMICAL REACTIONS

385

Figure 3. Wigner function corresponding to Fig. 26.

As a method to control wavepackets, alternative to the use of ultrashort pulses, I would like to propose use of frequency-modulated light. Since it is very difficult to obtain a well-controlled pulse shape without any chirp, it is even easier to control the frequency by the electro-optic effect and also by appropriate superposition of several continuous-wave tunable laser light beams. I . Physics Today 430) (1990), Special Issue on Dynamics of Molecular Systems.

2. D. F. Walls, Nature 306, 141 (1983).

3. R. Loudon and P. L. Knight, J. Mod. Opt. 34, 709 (1987).

4. 5. 6. 7.

M. C. Teich and B. E. Saleh, Quantum Opt. I, 153 (1989). L.-A. Wu, H. J. Kimble, J. L. Hall, and H. Wu, Phys. Rev. Lett 57, 2520 (1986). W. Schleich and I. A. Wheeler, Nafure 326, 574 (1987). J. Gea-Banacloche, R. R. Schilcher, and M. S. Zubairy, Phys. Rev. A 38, 3514 (1988).

8. J. Janzky and Y. Yushin, Phys. Rev. A 36, 1288 (1987). 9. C. K. Hong and L. Mandel, Phys. Rev. A 32,974 (1985). 10. S. Reynaud, C. Fabre, and E. Giacobino, J. Opf. Soc. Am. B 4, 1520 (1987). 11. J. Janzky and Y. Yushin, Optics Commun. 59, 151 (1986). 12. R. Graham, J. Mod. Opt. 43, 873 (1987). 13. H.-Y. Fan and H. R. Zaidi, Phys. Rev. A 37,2985 (1988). 14. 1. Janzky and Y. Yushin, Phys. Rev. A 39,5445 (1989). 15. H.-Y. Fan and H. R. Zaidi, Phys. Rev. A 39, 5447 (1989). 16. Xi Ma and W . Rhodes, Phys. Rev. A 39, 1941 (1989). 17. J. Janzky. T. Kobayashi, and An. V. Vinogradov, Optics Commun. 76, 30 (1990). 18. W. T. Pollard, S.-Y. Lee, and R. A. Mathies, J. Chem. Phys. 92,4012 (1990). 19. J. Janzky and An. V. Vinogradov, Phys. Rev. Left. 64,2771 (1990). 20. T. Kobayashi, A. Terasaki, T. Hattori, and K. Kurokawa, Appl. Phys. B 47, 107 (1988).

386

GENERAL DISCUSSION

S. R. Jain: When Prof. Rice talks about optimal control schemes, his Lagrange function follows a time-reversed Schrxinger equation. Is it assumed in the variational deduction that the Hamiltonian is time reversal invariant; that is, is it always diagonalizable by orthogonal transformations? S. A. Rice: Yes, there is time reversal invariance. S. Mukamel: I would like to make a comment regarding interference effects in quantum and classical nonlinear response functions (1, 21. Nonlinear optical measurements may be interpreted by expanding the polarization P in powers of the incoming electric field E. To nth order we have

Quantum mechanically, the nonlinear response function S(")(t,r,, is given by a combination of 2" terms representing all possible "left" and "right" actions of the various commutators divided by h". The various terms are given by an (n+ 1)-order correlation function of the dipole operator V (with different time arguments). These terms interfere, and this gives rise to many interesting effects, such as new resonances. The (l/tz)"factor indicates that individual correlation functions do not have an obvious classical limit; however the observables which are given by proper combinations of correlation functions are analytic functions of ti. The quantum linear response function is given by

. . . , 71)

The classical linear response function can be written using the fluctuation-dissipation theorem as a single term,

Unlike the quantum response (2), which contains an interference of two Liouville space paths, the classical expression (3) contains no inter-

LASER CONTROL OF CHEMICAL REACTIONS

387

ference and may be directly computed using classical trajectories that sample the initial density matrix. However, classical nonlinear response functions do involve interference between two or more terms. The second-order quantum and classical response functions are given by

and x, are the coordinates and momenta. The matrix where VJ = W/&,

M , which relates small deviations 6xj to 6Xk at different times, is known as the stability matrix. Its elements are defined as

The stability matrix carries the necessary information related to the vicinity of the trajectory and provides an efficient numerical procedure for computing the response function. It plays an important role in the field of classical chaos; the sign of its eigenvalues (related to the Lyapunov exponents) controls the chaotic nature of the system. Interference effects in classical response functions have a different origin than their quantum counterparts. For each initial phase-space point we need to launch two trajectories with very close initial conditions. [For S@) we need n trajectories.] The nonlinear response is obtained by adding the contributions of these trajectories and letting them interfere. It is interesting that the linear response does not depend on M.Nonlinear spectroscopy should therefore be a much more sensitive probe for classical chaos than linear spectroscopy. 1.

S. Mukamel, Principles of Nonlinear Optical Spectroscopy,Oxford University Press,

New York. 1995. 2. S. Mukamel, V. Khidekel, and V. Chemyak, Phys. Rev. E 53, 1 (1996).

388

GENERAL DISCUSSION

S. A. Rice: I would appreciate hearing the comments of Prof. Mukamel concerning the approach to the classical limit from the point of view of his formalism. Is that approach analytic? S. Mukamel: The nth-order response function is given by a combination of 2" correlation functions divided by A". In the classical limit, the combination of correlation functions (but not each one individually) behaves as tiR and ti cancels. It is then possible to expand the response function analytically in h. As long as we expand a physical quantity (i.e., a response function) rather than a correlation function, the result will be analytic. E. Pollak: I am not sure I understand the remark by Prof. Mukamel that chaos does not express itself in the linear response regime. The fluctuations of the exact quantum response about the classical will be described in terms of the Gutzwiller summation and will thus reflect the chaos. S. Mukamel: While there are some signatures of chaos in the linear response, my point is that the nonlinear response carries much more direct and sensitive information. The reason is that the stability matrix enters the nonlinear response directly, reflecting interference of initially close trajectories. Such interference is absent in the linear response. P. W. Brumer: Prof. Mukamel has emphasized that in examining objects as they approach the classical limit one sees essential singularity behavior or not depending upon the object. I would like to add to this remark by indicating that we have recently successfully completed a program designed to demonstrate the emergence of classical mechanics from quantum mechanics for dynamics, be it integrable or chaotic. This approach is an extensive generalization of our earlier papers [C. Jaffe and P. Brumer, J. Chem. Phys. 82, 2330 (1985); C. Jaffe, S. Kanfer, and P. Brumer, Phys. Rev. Len. 54, 8 (19931, where correspondence requires establishing a relationship, as h 0, of the eigenvalues and eigenfunctions of the quantum Liouville operator with the eigenvalues and eigenfunctions of the classical Liouville operator. In addition to producing a completely general approach to classical-quantumcorrespondence,our approach [J. Wilkie and P. Brumer, J. Chem. Phys., submitted] shows that the classical limit emerges by the elimination of essential singularities.

-

INTRAMOLECULAR DYNAMICS

SOLVENT DYNAMICS AND RRKM THEORY OF CLUSTERS R. A. MARCUS Noyes Laboratory of Chemical Physics California Institute of Technology Pasadena, Calgomia

CONTENTS 1. Introduction 11. Microcanonical Solvent Dynamics Modified RRKM Theory A. One-Coordinate Type Treatment B. Vibrational Assistance Treatment 111. Discussion References

I. INTRODUCTION In this chapter we consider the problem of reaction rates in clusters (microcanonical) modified by solvent dynamics. The field is a relatively new one, both experimentally and theoretically, and stems from recent work on welldefined clusters [l, 21. We first review some theories and results for the solvent dynamics of reactions in constant-temperaturecondensed-phase systems and then describe two papers from our recent work on the adaptation to microcanonical systems. In the process we comment on a number of questions in the constant-temperature studies and consider the relation of those studies to corresponding future studies of clusters. A brief review for constant-temperaturecondensed-phase systems is given in Fig. 1. The field of solvent dynamics has grown so extensively that it is

Advances in Chemical Physics, Volume 101: Chemical Reactions and Their Control on the Femtosecond Time Scale, XXrh Solvay Conference on Chemistry, Edited by Pierre Gaspard, Irene Burghardt, I. Prigogine, and Stuart A. Rice. ISBN 0-471-18048-3 0 1997 John Wiley & Sons, Inc.

39 1

392

R. A. MARCUS Solvent Dynamics & Chendcal Rcaciians

Memory

Gmte-Hynes 1980

TrOe

Hochstrasser

I

BT.

Langer

Zusman

1980

Wolynes 1983 Hyncs 1986 Ripsdormer 1987

Refs.

16

McCanunon 1983

Ligand AgmonHopfield 1983

BarrierlessBT. Flemingcral

ET. BerezhLowsLi Sumi1990 1990 Marcus E k f l U N C . 1983 1986

Pollak 1986

Sumi jwm+ 1991

&O-

Quantum

k(x)

Barbara, Weaver 1991-2

51)

Y~~hlhara, Rasaiah 1994

Yoshihara

1991

kwlxm Barbara, Weaver. Yoshihara, Simon

Figure 1. Brief survey of some developments in the solvent dynamics field.

difficult for recent reviews [3-51 to keep pace. Only some representative articles are cited below. The classic paper is solvent dynamics, due to Kramers, appeared in 1940 [6]. Subsequently, apart from some isolated works in the physics literature, such as Langer’s generalization to many coordinates in 1969 [7], there was relatively little follow-up, and particularly little in the chemical literature, until around 1980. The subsequent developments can be classified as being largely of three types: (1) those, like Kramers’s, that are one-coordinate treatments; (2) their many-coordinate extensions; and (3) treatments having one slow coordinate, the remainder being fast coordinates, appropriately averaged.

Kramers’s equation, it may be recalled, is [6]

where P( p, q, t ) is the probability density in phase space, q the reaction coor-

SOLVENT DYNAMICS AND RRKM THEORY OF CLUSTERS

393

dinate, p its conjugate momentum, U(q) the potential energy function, and ( the frictional coefficient. Kramers’s theory for reactions in liquids, which includes both the inertial and the overdamped limits, was extended by Grote and Hynes [8]to include a memory effect, namely, a frequency-dependentfriction f(w). A number of other one-dimensionalextensionshave also been made [9]. Some of the ideas were tested experimentally by many investigators 4, 5 , 10, 11. Another pioneering one-coordinate extension of Kramers’s analysis was made for electron transfer reactions by Zusman [ 121 and Alexandrov [121(1980), and further illuminating developments were made by a number of researchers [13-15]. When there are no relevant “vibrationally assisting” coordinates (examples are mentioned later), a one-coordinate approach suffices. The second approach, a multidimensional one, was given by Langer [7]. Other multidimensional developments were many [16-18]. McCammon [ 171 discussed a variational approach (1983) to seek the best path for crossing the transition-statehypersurface in multidimensionalspace and discussed the topic of saddle-point avoidance. Further developments have been made using variational transition state theory, for example, by Pollak [ 181. The third, and perhaps now the currently major, approach for treating the experimental data on electron transfer reactions assumes that there is one slow coordinate, with the remaining coordinates being fast. The equation used, or coupled with one for the back reaction or further extended by making D a D(t), is

ap

---

at

(

-D-ax a -+-ap ax k a ~””) ax -k(X)P(X)

(1.2)

and is obtained by an averaging over or adiabatic eEimination of the fast (vibrational) variables. Here, P(X)is the probability density along the slow coordinate X, D is a diffusion constant in this X space, G(X)is the free energy to reach any X from the equilibrium value of X,X = 0 for the reactant, and k(X)is a rate constant at any given X for crossing the barrier. The motion along X is, as seen in (1.2), treated as overdamped. Using Eq. (1.2) Agmon and Hopfield (Ref. 19; cf. Ref. 23) treated the dissociation of a ligand from a heme in a protein (1983), and Sumi and Marcus [20] treated electron transfer reactions (1986). For electron transfers the previous (one-coordinate) treatments neglected the very common case that solute vibrations play a major role (vibrational assistance) in the transfer when there are significant changes in vibrational geometry, for example, in bond lengths. The use of Eq. (1.2) removes that defect. Beginning around 1990 Berezhovskii, Zitserman, and co-workers introduced a number of treatments of the type based on Eq. (1.2) [21].

394

R. A. MARCUS

The various treatments in the literature based on Q. (1.2) have differed primarily in two respects: (1) the expression for the rate constant k ( X ) in Eq. (1.2) is specific for the process and so may differ from process to process and (2) the technique for solving Eq. (1.2) differs. For example, Agmon and Hopfield [19] solved Eq. (1.2) numerically, as did Nadler and Marcus [22], Agmon and co-workers [23], and others. Sumi and Marcus [201 introduced, instead, a decoupling approximation, which depended on there being a difference in time scales for the reaction and for the solvent fluctuations. (An excellent summary and an extension of their work is given in Rasaiah and Zhu [24].) Berezkhovskii et al. introduced an approximation that divided the X space into two parts, separated by a value of X at which the escape time 7esc(X) equals the solvent relaxation time T ~ ~ ~[21]. ( X Fleming ) and co-workers treated barrierless electronic energy transfer (1983) [25] and the corresponding barrierless electron transfer (1990) [26]. For electron transfers various extensions and experimental tests of the Sumi-Marcus treatment have been introduced. They include a quantum version for k ( X ) (of particular importance in the “inverted region”) [24, 27, 281, inclusion of forward and reverse reactions [24], and the use of a time-dependent D(t) to allow for several relaxation times [24,27,29]. Other experimental tests or extensions have also been introduced [30]. In experimental tests, effects such as the dynamic Stokes shift of fluorescence in these polar systems have been especially invaluable in providing necessary data for relaxation in these electron transfer systems [3 11. Numerical solutionsby Yoshihara and coworkers [29] and by Barbara and co-workers [27], permitting the inclusion of a D(t), have been important. Use of a D(t) in general had been made in the work of Hynes (141 (cf. Ref. 32). Analyses of solvent dynamics, accompanied by computer simulations, of Chandler and co-workers [33] and of Maroncelli, Fleming, and their co-workers [34] have provided further insight. Inertial effects on solvent dynamics, using a formalism of Mukamel and co-workers [353, have been incorporated by Barbara and co-workers [36]. Further relevant solvent dynamics theoretical analyses [37] and measurements using ultrafast laser spectroscopy have also been described [38]. Earlier theoretical studies of dynamical spectral shifts of solutes had been made by Bakhshiev, Mazurenko, and their co-workers [39] and references cited therein. In the case of electron transfer reactions, besides data on the dynamic Stokes shift and ultrafast laser spectroscopy, data on the dielectric dispersion E ( W ) of the solvent can provide invaluable supplementary information. In the case of other reactions, such as isomerizations, it appears that the analogous data, for example, on a solvent viscosity frequency dependence q ( w ) , or on a dynamic Stokes fluorescence shift may presently be absent. Its absence probably provides one main source of the differences in opinion [5, 40-433 on solvent dynamics treatments of isomerization.

SOLVENT DYNAMICS AND RRKM THEORY OF CLUSTERS

395

In the next section we summarize two treatments of microcanonical systems [2, 441, one of the steady-state Kramers’ one-coordinate type and one including vibrational assistance. An earlier approach to the problem was given by Troe [45].

II. MICROCANONICAL SOLVENT DYNAMICS MODIFIED RRKM THEORY A. One-Coordinate Qpe Treatment The Kramers-type equation corresponding to Eq.( 1 . 1 ) and adapted in Ref. 2 to the microcanonical case for a system with coordinate q and its conjugate momentum p is

where S,(q) is the local vibrational entropy at q. T,[=l/(aS,/aE,)] is the local microcanonical vibrational temperature and is a function of q, and P is the probability distribution in (q, p) space. The adaptation was such as to pennit the equilibrium microcanonical phase-space distribution function to satisfy the equation identically [2]. In the steady-state approximation, solving Eq. (2.1) for the rate constant at the given total energy E yields [ 2 ]

where a t denotes T,d2S,/dq2 at q = qt and is an effective barrier frequency. The ~ R R K Mis the RRKM (microcanonical)rate constant at the given energy E.

B. Vibrational Assistance Treatment A second procedure, based on the vibrational assistance model for calculating the solvent-dynamics-modified rate, is given in Ref. 44. The reaction4iffusion equation, adapted from &. (1.2), is, for the case where the back reaction is neglected, given by (2.3).The more complete treatment, where the back reaction (recrossings) is included, is given in Ref. 44:

R. A. MARCUS

396

The adaptation is such as to pennit the equilibrium microcanonical distribution for the slow coordinate X to be a solution (2.3) when k(X) = 0. The S,(X) in EQ. (2.3) is the vibrational entropy change needed to reach X from

x=o:

(2.4)

[the constant does not affect the aS,/aX appearing in Eq. (2.3)J; p is the density (i.e., the number per unit energy) of quantum states of the reactant, and p(X) is that density at X per unit X. It is given by [44] 2

Px

=

[2(E - U(X)- Em)]”*

(2.5)

rn

where Px is the momentum conjugate to X. Throughout we use a massweighted unit for X for notational brevity. Here, Em is the energy of the mth quantum state of the reactant for all coordinates but X and U(X)is the potential energy at X at the local equilibrium value of the remaining coordinates. It is readily verified [44]that (2.5) satisfies jp(X)dX = 1, where the integral is over the reactant’s region of X space. We give in Fig. 2 a schematic plot showing contours on which a vibrational entropy S , ( q , X ) is constant, q is a fast coordinate, and the line C is the transition state in the (X, q ) space. This S , ( q , X ) can be defined as in an equation similar to (2.4) in terms of a k g In p(q,X), p ( q , X ) being the local density of study, that is, the number per unit energy per unit q and per unit X. While this plot is not used in the derivation, it can be visually helpful. The k(X)in Eq. (2.3) is given by an RRKM-like expression for the-given X [44]:

where N ( X ) is the local number of quantum states (per unit X) for the given X, along the transition state, with energy equal to or less than E and is given by (2.7), and x is a coordinate that is a projection (defined in Ref. 44) on

SOLVENT DYNAMICS AND RRKM THEORY OF CLUSTERS

397

Product Region

Slow coordinate X Reactant Region

-

I

r

Figure 2. Schematic entropic surface as a function of a slow coordinate X and a fast coordinate q. Here, S is a saddle point, the line C is the transition state in this (X,q ) space, and X, lies at its intersection with the X axis.

the transition state space. We have

Here, En is the energy of the nth quantum state of the transition state for all coordinates but X and Q,the reaction coordinate, V , ( X ) is the potential energy in the transition state at the point X and at the position of minimum potential energy with respect to all other coordinates in the transition state, and px is the momentum conjugate to n. From the expressions for k(X) and N ( X ) it can be shown that the usual RRKM expression obtains when the diffusion along X is rapid, that is, when P(X)has its equilibrium value Peq(X)[=p ( X ) / p ] : We have

398

R. A. MARCUS

and so from (2.6) and (2.7)

where the limits on the integral in (2.9) are at the end-points p x = 0. [The system is now near the X where U,(X)is a minimum and so x has become a vibration.] The second equality in (2.9) arises because the integral there can be written as an integral over one cycle of the x motion and so equals $ p xdx/h. The latter is a constant, which we write as I + Semiclassically, 1 is an integer for any x quantum state. We now have in (2.9) what can be shown [44]to be a sum over the quantum states (for all coordinates but the reaction coordinate) with energy equal to or less than E. The sum is denoted by N ( E ) , and one obtains the usual RRKh4 expression. It remains now to solve Eq. (2.3). Here, there are various approaches, depending on the conditions. When a non-steady-state solution is required, one can introduce the decoupling approximation of Sumi and Marcus, if there is the difference in time scales mentioned earlier. Or one can integrate Eq. (2.3) numerically. For the steady-state approximation either Ekj. (2.3) can again be solved numerically or some additional analytical approximation can be introduced. For example, one introduced elsewhere [44]is to consider the case that most of the reacting systems cross the transition state in some narrow window (XI,XI f +A), narrow compared with the X region of the reactant [e.g., the interval (O,X,) in Fig. 21. In that case the k(X) can be replaced by a delta function, R(XI)A6(X - XI). Equation (2.3) is then readily integrated and the point XI is obtained as the X that maximizes the rate expression. The A is obtained from the width of the distribution of rates in that system [44]. A simple expression for the rate constant results:

i.

where X,,, is the X that maximizes krate(X)and hence minimizes the reaction time l/krate(X). That time appears in the final expression in Ref. 44 as (2.1 1)

SOLVENT DYNAMICS AND RRKM THEORY OF CLUSTERS

399

where the diffusion-controlled and activation-controlled rate constants are given by

and (2.13) [When Xma,occurs at the end-point X,, there is a minor change in procedure, and the system now crosses the transition state in the interval (X,,X,- A), Equations (2.11H2.13) are again obtained.] The result in (2.11) that the reaction time is the sum of two other times is fairly common in the general reaction4ffusion literature, in which a steadystate approximation is used and there is a diffusion toward a sink followed B -+ C, with by reaction at that sink. For example, in the scheme A forward and reverse rate constants kl and k:! for the first step (equilibrium constant K = k,/k2) and rate constant k j for the last step, a steady-state l/kl), which has the same funcapproximation for B yields 1/krate(l/k3K)+( tional form as (2.11). The more complete expression, which allows for the back reaction (recrossings), has a slightly more complicated structure [44].

111. DISCUSSION At present the body of data on reactions in clusters is insufficient to test the above two microcanonical approaches. For electron transfers in solution it seems clear that the vibrational assistance approach, stemming from Eq. (1.2), with its extensions mentioned earlier, is the one that has been the most successful [27-301. For slow isomerizations Sumi and Asano have pointed out that an analysis based on Eq. (1.2) was again needed [40]. An approach based on FQ. (1.1) or on its extension to include a frequency-dependent friction, they noted, led to unphysical correlation times [40]. In investigations of fast isomerizations the most commonly studied system has been the photoexcited trans-stilbene [5,41-43,46]. Difficulties encountered by a one-coordinate treatment for that system have been reported [4, 81. Indeed, coherence results for photoexcited cis-stilbene have shown a coupling of a phenyl torsional mode to the torsional mode about the C=C bond [42, 471. Other investigatorshave used systems that are more apt to represent a one-

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coordinate-like behavior, for example, the isomerization of “stiff-stilbene,” where the phenyl groups are tethered further to the respective carbon atoms of the double bond [48], and the conformational change of binaphthyl [lo]. In the last study a microviscosity was introduced into the Kramers formula to obtain the friction coefficient instead of using the bulk viscosity of the solvent, and was inferred from rotational relaxation or translational diffusion coefficient (f = ksT/D in mass-weighted units). In that case the expected Kramers function of q (via the Stokes-Einstein equation) was obeyed for the one-coordinate model. Use of the bulk viscosity led, instead, to an observed fractional dependence on q [ 101. A microfriction inferred from a rotational or translational relaxation time has been similarly successfully used by various other investigators [41, 43, 49, 501. An approximate molecular expression (“ballpark”) for a t for a cluster was given in Ref. 2. Many questions in the analysis of solvent dynamics effects for isomerizations in solution have arisen, such as (1) when is a frequency-dependent friction needed; (2) when does a change of solvent, of pressure, or of temperature change the barrier height (i.e., the threshold energy), and (3) when is the vibrational assistance model needed, instead of one based on Q. (I. 1) or its extensions? In the case of electron transfers in solution there appears to be a greater cohesiveness of views, and the need for vibrational assistance is well established for reactions accompanied by vibrational changes (e.g., changes in bond lengths). A detailed analysis of the experiments could be made because of the existence of independent data, which include X-ray crystallography, EXAFS, resonance Raman spectra, time-dependent fluorescence Stokes shifts, among others. One may inquire as to what this experience with solutions suggests for the study of reactions in clusters. In the case of electron transfers supplementary information, such as time-dependent fluorescence Stokes shift in clusters, would again be helpful. Equation (2.3) can be modified to include a D(t), as in the isothermal case, if needed from the results of such data. For isomerizations, also, it would be useful to have, for solutions or clusters, detailed analogous data such as the above Stokes shift. However, because of the low intensity of such a fluorescence in this case, such data appear to be absent or scarce. The questions that have arisen in regard to isomerizations in solution also apply to isomerizations in clusters. One of these questions, which has now been addressed, is the threshold energy and its variation with size of the cluster. Data on the threshold energies were obtained in the microcanonical study by Heikal et al. [l]. New questions, however, also arise for clusters: How rapidly is the energy transferred between the solute and the solvent molecules in the cluster? Outside the threshold region a reduction in k,,,(E) with

r,

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increasing cluster size was observed experimentally 11. A principal question is whether this reduction is due to the increase, with increased cluster size, in the number of coordinates which can share the excess energy or whether it is due to increased frictional effects by the solvent molecules, or both. If the solvent molecules outside the first solvent layer in the cluster have little effect on the frictional forces, then this question can be addressed by comparing the reaction rates with those clusters that contain more than one solvent layer. Again, if instead of using stilbene one replaces one of the phenyl groups attached to the C=C double bond, the solvent viscosity effects should be less and the energy-sharing role of the extra coordinates more readily discerned. Yet again, microcanonical studies with molecules deliberately chosen, as in the solution case, to favor a one-coordinate approach would also be of particular interest. It is clear that the study of solvent dynamics in solution has proved to be a rich field. A number of questions remain to be resolved, and the study of clusters can open new avenues.

Acknowledgments It is a pleasure to acknowledge the support of this research by the National Science Foundation and the Office of Naval Research.

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Klosek-Dygas, B. M. Hoffman, B. J. Matkowsky, A. Nitzan, M. A. Ratner, and Z. Schuss, J . Chem. Phys. 90, 1141 (1989); E. Pollak, J . Chem. Phys. 85, 865 (1986); H. Grabert, Phys. Rev. Lett. 61,1683 (1988); E. Pollak, H. Grabert, and P. Hhggi, J. Chem. Phys. 91, 4073 (1989); T. Fonseca, H. J. Kim, and J. T. Hynes, J. Photochem. Phorobiol. A: Chem. 82, 67 (1 994); R. Graham, J. Sfat. Phys. 60, 675 (1 990); R. F. Grote and J. T. Hynes, J. Chem. Phys. 74,4465 (1981). 17. M. Berkowitz, J. D. Morgan, and J. A. McCammon, J. Chem. Phys. 79,5563 (1983); S. H. Northrup and J. A. McCammon, J. Chem. Phys. 78,987 (1983). 18. S. Tucker and E. Pollak, J. Star. Phys. 66, 975 (1992); A. M. Frishman, A. M. Berezhkovskii, and E. Pollak, Phys. Rev. E 49, 1216 (1994); A. M. Berezhkovskii, A. M. Frishman, and E. Pollak, J. Chem. Phys. 101,4778 (1994); E. Pollak, Ref. 4, p. 5. 19. N. Agmon and J. J. Hopfield, J. Chem. Phys. 78,6947; ibid., 80,592 (1984) (E); 79,2042

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31. M. Cho, S.J. Rosenthal, N. F. Scherer, L. D. Ziegler, and G. R. Fleming, J. Chem. Phys. 96, 5033 (1993); G. van der Zwan and J. T. Hynes, J . Phys. Chem. 89,4181 (1985); R. Jiminez, G. R. Fleming, P. V. Kumar, and M. Maroncelli, Narure 369, 471 (1994); S. J. Rosenthal, X. Xie, M. Du, and G. R. Fleming, J. Chem. Phys. 95, 4715 (1991). 32. T. Fonseca, J. Chem. Phys. 91, 2869 (1989). 33. D.Chandler, J. Stat. Phys. 42,49 (1986); D. C. Chandler, J. Chem. Phys. 68,2959 (1978); R. A, Kuharski, 1. S. Bader, D. Chandler, M. Sprik, M. L. Klein, and R. W. Impey, J. Chem. Phys. 89,3248 (1988); I. S. Bader and D. Chandler, Chem. Phys. Lett. 157,501 (1989); J. S. Bader, R. A. Kuharski, and D. Chandler, J. Chem. Phys. 93,230 (1990). 34. M. Maroncelli, J. Maclnnis, and G. R. Fleming, Science 243, 1674 (1989); M. Maroncelli, J. Chem. Phys. 94,2084 (1991); S. Roy and B. Bagchi, J. Chem. Phys. 99, 1310 (1993); S. J. Rosenthal, R. Jiminez, G. R. Fleming, P. V. Kumar. and M. Maroncelli, J. Mol. Liq. 60, 25 (1994); M. Maroncelli and G. R. Fleming, J. Chem. Phys. 86, 6221 (1987); J. Gardecki, M. L. Horng, A. Papazyan, and M.Maroncelli, J. Mol. Liq. 65/66,49 (1995). 35. Y. J. Yan, M. Sparpaglione, and S. Mukamel, J. Phys. Chem. 92,4842 (1988). 36. K. Tominga, D. A. V. Kliner. A. E. Johnson, N. E. Levinger, and P. F. Barbara, J. Chem. Phys. 98, 1228 (1993). 37. Y. J. Yan and S. Mukamel, Phys. Rev. A 41,6485 (1990); L. E. Fried and S. Mukamel, J. Chem. Phys. 93,932 (1990). 38. Y. J. Chang and E. W. Castner, Jr., J. Chem. Phys. 99,7289 (1993); and refs cited therein; 39.

M.Cho et al., Ref. 31. N. G. Baklshiev, Opf. Specfr. 16,446 (1969); Yu. T.Mazurenko and V. S . Udaltsov, Opt.

Spect,: 44,417 (1978). 40. H. Sumi and T. Asano, Chem. Phys. Left. 240, 125, 1995; J. Chem. Phys. 102, 9565 (1995); H. Sumi, J. Phys. Chem. 95,3334 (1991); T. Asano, H. Furuta, and H. Sumi, J. Am. Chem. SOC. 116, 5545 (1994); H. Sumi, J. Mol. Liq. 65/66, 65 (1995). 41. Y.-P. Sun, J. Saltiel, N. S. Park, E. A. Hoburg, and D. H. Waldeck, J. Phys. Chem. 95, 10336 (1991); J. Saltiel and Y.-P. Sun, J. Phys. Chem. 93,6246, 8310 (1989). 42. R. J. Sension, S. T. Repinec, A. Z. Szarka, and R. M. Hochstrasser, J. Chem. Phys. 98, 6291 (1993); A. Z. Szarka, N. Pugliano, D. K. Palit, and R. M. Hochstrasser, Chem. Phys. Lett. 240,25 (1995).

43. D. M. Zeglinski and D. H. Waldeck, J. Phys. Chem. 92, 692 (1988); D. H. Waldeck, Chem. Rev. 91,415 (1991). 44. R. A. Marcus, J. Chem. Phys., 105,5446 (1996). 45. I. Schroeder and I. Troe, Ref. 4, p. 206, see refs. 1, 2, and 5 in that article. 46. D. Raftery, R. J. Sension, and R. M. Hochstrasser, Ref. 4, p. 163. 47. S. Pedersen, L. Banares, and A.

H.Zewail, J. Chern. Phys. 97,8801 (1992).

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DISCUSSION ON THE REPORT BY R. A. MARCUS Chairman: E. Pollak J. Tree: In our own work measuring energy-specific excited stilbene lifetimes in stilbene-hexane clusters, we found for larger clusters that there is no isomerization. We interpreted this as evidence for “boiling off’ of the cluster partners, removing energy from stilbene and thus suppressing the isomerization. This is an alternative to increasing the effective size of the reacting molecule. Which interpretation do you favor? We also clearly showed that the barrier for isomerization is decreased by the first cluster partners, as evidenced in our studies of the thermal reaction between gas and liquid phases. R. A. Marcus: I understand that the observed effect of a decrease in rate constant k(E) with increasing cluster size at a fixed E was not due to a boiling off of the hexane molecules (and hence to a reduced E), but I refer to my colleague, Ahmed Zewail, for an answer to your question. A. H. Zewail: My answer to Prof. Troe is that, in our experiment, already the cluster with one solvent shows the shift in Eo. As for the boiling off of solvent molecules in larger clusters this is a nontrivial problem that we have considered in our paper. Based on the analysis of the translational energy and the kinetics, we concluded that the exponential decays (rates) are determined by the isomerization [see Chem. Phys. Lett. 242, 380 (199511. In any event, only one solvent molecule (at most) can be evaporated for the available energy studied experimentally; recall that the binding energy of hexane is relatively large. H. Hamaguchi: I would like to comment on the stilbene photoisomerization in solution. We recently found an interesting linear relationship between the dephasing time of the central double-bond stretch vibration of S1 trans-stilbene, which was measured by time-resolved Raman spectroscopy, and the rate of isomerization in various solutions. Although the linear relationship has not been established in an extensive range of the isomerization rate, I can point out that the vibrational dephasing time measured by Raman spectroscopy is an important source of information on the solvent-inducedvibrational dynamics relevant to the reaction dynamics in solution. R. A. Marcus: It is good to hear about that; certainly one needs all types of information to be incorporated. D. M. Neumark Prof. Marcus, your theoretical treatment was motivated by experimental studies of isomerization in clusters with

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very few solvent molecules (n = I , 2). How appropriate is your theory to these small clusters? In particular, can one discuss concepts such as viscosity and solvent friction in small clusters?

R. A. Marcus: The experiments involved hexane rather than argon and went from n = 0 to n = 5 hexane molecules. In Chem. Phys. Lett.

244, 10 (1993, I considered two limiting cases for the energy sharing of the trans-stilbene with the modes of the solvent molecules. Experiments comparing results for one and two shells of solvent molecules in the cluster may provide information on which limiting model might be the more appropriate. Previous experiments on trans-stilbene in solvents suggest a rapid energy sharing. In the above article I also gave a rough expression relating the “viscosity” or friction for the cluster to molecular properties, but I am sure it can be improved upon.

B. Hess: The structure and function of solvent components constituting the active site of enzymic reactions represent an exciting puzzle in protein chemistry. In an active pocket, a restricted cluster number is given by the small set of amino acid residues, mostly hydrophobic, in the nearest neighborhood to the ligand and its reaction partners. As Prof. Marcus pointed out, Frauenfelder and his collaborators studied experimentally the effect of solvent viscosity on protein dynamics. In case of CO myoglobin they could show that over a wide range in viscosity the transition rates in heme-CO are inversely proportional to the solvent viscosity and can consequently be described by the Kramers equation [I]. A complementary study was carried out to explore the effect of viscosity on the photocycle of bacteriorhodopsin. Here again the Kramers equation in a modified form was found to be useful [ 2 ] . Most recently, the photodissociation of carbon monoxide myoglobin was studied in crystals at liquid helium temperatures by two different groups [3]. Schlichting et al. [4]could show that CO dissociation leads to tilting of the proximal histidine and a decompression and motion of the F-helix toward its junction with the E-helix. These conformational changes are linked to an increase of the enthalpic barrier decreasing the association rate coefficient. It was speculated that the energy stored in this conformation of the residue and its neighbors is released during structural fluctuations associated with ligand escape. These and other observations (see also Ref. 5) illustrate the necessity to extend the theoretical approach of Prof. Marcus’s theory to the domain of intramolecular interactions in protein dynamics. I would be glad if he could comment on this development. 1. D.Beece, L. Eisenstein, H. Frauenfelder, D. Good, M. C. Morgan, L. Reinisch, A. H. Reynolds, L. B. Sorensen, and K. T. Yue, Biochemistry 18, 3421 (1979).

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2. D. Beece, S. F. Bowne, J. Czege, L. Eisenstein, H. Frauenfelder, D. Good, M. C. Marden, J. Marque, P. Ormos, L. Reinisch, and K. T. Yue, Phorochem. PhorobioZ. 43, 171 (1981). 3. G. Petsko, Nature 371, 740 (1994). 4. I. Schlichting, J. Berendzen, G. N. Phillips, Jr., and R. M. Swwet, Nature 371, 8008 ( 1994). 5. H. Akiyama, T. Kakitani, Y. Imamoto, Y. Shichida, and Y. Hatano, J. Phys. Chem. 99,7147 (1995).

R. A. Marcus: Concerning the issue raised by Prof. Benno Hess, a number of treatments of the “solvent dynamics” of chemical reactions in proteins or liquid solvents assume one slow (X)and one or more fast coordinates. The theories then differ in the nature of how a rate constant k(X)depends on X. Agmon and Hopfield, for example, used a k(X)specific for the ligand-heme dissociation process they were considering. Sumi and I used a k(X) specific for the electron transfer reactions we were considering [ 11. In today’s talk, which concerns an isomerization, I used for k ( X ) the analog of an RRKM rate constant appropriate to it. These various treatments have in common the same differential equation for the probability distribution P(X)along X.They differ in the process considered and in the nature of solution. There is still much to be done, particularly for systems that deviate from single time-exponential behavior, to use some of the existing numerical solutions, to test various analytical approximations, and to develop new analytical approximations. It was interesting to hear from Prof. Hess about the new and detailed structural information becoming available for the protein systems, and extending the theory will be an interesting problem. 1. H. Sumi and R. A. Marcus, J. Chem. Phys. 84,4894 (1986).

W. Hebel: I have a rather general question for Prof. Marcus. You are discussing the complicated solvent dynamics of molecular clusters. Does this also include large biomolecules such as proteins in aqueous solutions? Could you perhaps comment on how far research has gone in analyzing and understanding the interaction of biopolymers in aqueous solvents? R. A. Marcus: Even though solvents and solvent-solute interactions or interactions with a protein can be very complicated and the resulting motion can be highly anharmonic, under a particular condition there can be a great simplification because of the many coordinates (perhaps analogous to the central-limit theorem in probability theory).

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The condition is that of linear response of the solvent or protein, for example, that the change in dielectric polarization of the solvent be proportional to the change in charge of a solute. With this condition fluctuations give rise to a quadratic expression for free-energy changes. This simplification ultimately led, in the case of electron transfer reactions in appropriate atoms or group transfer, to new predicted relationships among rate constants of different reactions. The linear response approximation for the electron transfer systems was subsequently also tested by various investigators by computer simulations of solvent and of proteins. J. Troe: Professor Marcus, you were mentioning the 2D SumiMarcus model with two coordinates, an intra- and an intermolecular coordinate, which can provide “saddle-point avoidance.” I would like to mention that we have proposed multidimensional intramolecular Kramers-Smoluchowski approaches that operate with highly nonparabolic saddles of potential-energy surface [Ch. Gehrke, J. Schroeder, D. Schwarzer, J. Troe, and F. Voss, J. Chem. Phys. 92, 4805 (1990)]; these models also produce saddle-point avoidances, but of an intramolecular nature; the consequence of this behavior is strongly nonArrhenius temperature dependences of isomerization rates such as we have observed in the photoisomerization of diphenyl butadiene. R. A. Marcus: I used the words saddle-point avoidance, incidentally, to conform with current terminology in the literature. More generally, one could have said, instead, avoidance of the usual (quasi-equilibrium) transition-state region (ie., the most probable region if viscosity effects were absent). E. Pollak: In relation to the point discussed by Profs. Troe and Marcus, we have shown that those cases considered as saddle-point avoidance are consistent with variational transition-state theory (VTST). If one includes solvent modes in the VTST, one finds that the variational transition state moves away from the saddle point; the bottleneck is simply no longer at the saddle point. A. H. Zewail: I have a question for Prof. Marcus concerning the fact that, in the bulk solvation problem, there are two regimes for the description of solvation, the continuum model and the detailed molecular dynamics. Do you expect that in clusters the friction model will change as the number of solvent molecules changes from small to large? R. A. Marcus: In Chem. Phys. Lett. 244, 10 (1995), a very rough approximate hard-sphere model used for liquids was mentioned to relate the frictional coefJicient to the pair distribution function in the cluster.

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D. J. Tannor: One would think that as one adds more and more layers of solvent one is introducing irreversible decay of the correlation function of the solute-solvent coupling. The main physical content of the Grote-Hynes expression for the rate constant is that contributions from this correlation function that are slow compared with the time scale for reaction do not really contribute to the reaction rate. This suggests that by starting with a description of only the first solvent shell and introducing shorter and shorter solvent memory, one will see a transition that resembles that of adding more and more solvent shells. R. A. Marcus: About the problem raised by Prof. Tannor, there are a number of questions to be resolved, such as energy migration to the solvent in the cluster, the detailed dynamical effect of successive layers of solvent in larger clusters, and comparing with cluster experiments of other suitably chosen reactants, for example, RlRzC = CR3R4, where R3 and R4 are so small that the molecule has no frictional effect in solution. Such an isomerization has previously been studied in liquids. In the present chapter several experiments are suggested to disentangle the various factors influencing the energy-dependent rate contants. H. Hamaguchi: What information do you have concerning the structure and dynamics of the hexane-dressed stilbene molecule? How flexible or how rigid is the structure? R. A. Marcus: I personally do not have the information. A. H. Zewail: To provide a partial answer to the question of Prof. Hamaguchi, the structure of the 1:1 stilbene-hexane species was determined with the help of rotational coherence spectroscopy. For higher clusters we used atom-atom model potentials and deduced structures.

HIGH-RESOLUTION SPECTROSCOPY AND INTRAMOLECULAR DYNAMICS H. J. NEUSSER* and R. NEUHAUSER Znstitut f i r Physikalische und Theoretische Chemie Technische llniversitat Miinchen Garching, Germuny

CONTENTS I. Introduction 11. Intramolecular Dynamics in Electronically Excited S1 State of Benzene A. Mechanism of Intramolecular Dynamics in Polyatomic Molecular System B. Intramolecular Dynamics in Benzene 1. States at Low Excess Energy 2. Dynamic Behavior of States at Intermediate Vibrational Excess Energy C. Influence of Van der Waals Bonded Noble-Gas Atoms on Intramolecular Dynamics 111. Laser-Driven Population Dynamics and Coherent Ion Dip Spectroscopy A. Introduction B. Incoherent Population Dynamics C. Coherent Population Dynamics D. Coherent Population Dynamics for Special Pulse Sequences E. Coherent Ion Dip Pulse Sequence F. Experimental Results 1. Experimental Setup 2. Experimental Procedure of Coherent Ion Dip Spectroscopy 3. spectra IV. Intramolecular Dynamics of High Rydberg States in Polyatomic Molecules A. General Remarks B. Experimental C. Experimental Results V. Conclusion *Report presented by H. J. Neusser Advances in Chemical Physics, Volume 101: Chemical Reactions and Their Control on the Femtosecond lime Scale, XXth Solvay Conference on Chemistry. Edited by Pierre Gaspard, Irene Burghardt, I. Prigogine, and Stuart A. Rice. ISBN 0-471-18048-3 0 1997 John Wiley & Sons, Inc.

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I. INTRODUCTION In this work we discuss the intramolecular dynamics initiated by the interaction of light with isolated polyatomic molecules and van der Waals complexes. As is well known, in many polyatomic systems the intramolecular dynamics induced in this way precedes the chemical reaction and leads to an energy randomization within the numerous internal degrees of freedom of the system. This justifies the application of statistical theories that have been successfully applied to describe the unimolecular decay behavior of these systems [l, 21. In order to find precise information on the coupling mechanism and the coupling strength, experiments are required that lead to a sufficient selection of individual molecular states. Thus the experiments discussed in this work are performed with nanosecond light pulses providing the highest resolution of several tens of megahertz possible for their duration, that is, Fourier transform limited and thus coherent light pulses. We will present results on the intramolecular dynamics in different energy regimes, that is, the electronically excited state (Sl), the electronic ground state (SO), and in high Rydberg states with various kinds of couplings and a resulting different intramolecular dynamic behavior. Even in a molecule the size of benzene the resolution achieved in this way is sufficient to investigate the dynamic behavior of individual rotational states. For this it is necessary to eliminate the Doppler broadening of the rovibronic transitions. Two methods have been applied: (i) the elimination of Doppler broadening in a Doppler-free two-photon-transition and (ii) the reduction of Doppler broadening in a molecular beam. Measurements of the dynamic behavior have been performed in the frequency [3] and time domain [4].We will briefly summarize the results from high-resolution measurements and discuss the conclusions on the intramolecular decay mechanism. Then it will be discussed how the intramolecular dynamics is influenced by the attachment of an Ar or Kr atom to the benzene molecule, leading to a weakly bound van der Waals complex. If intense coherent light pulses interact with a molecular system, a special population dynamics in a three-level system results that is important and helpful for the spectroscopy of high vibrational levels in the electronic ground state of molecules [5, 61. In this work we present the features of a coherent high resolution technique leading to the sensitive spectroscopy of ground-state levels in molecules and van der Waals complexes. Exploiting the characteristics of the coherent excitation, we are able to detect intramolecular couplings not only in the excited S1 state but also in the electronic ground state. First examples are presented for benzene and applications to van der Waals complexes are discussed.

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41 1

II. INTRAMOLECULAR DYNAMICS IN

ELECTRONICALLY EXCITED S1 STATE OF BENZENE In this section we present experimental results for the lifetime of individual rovibronic states of benzene at different excess energies in the S, electronic state. In this way the dependence of the lifetime of the states on their excess energy and their rotational quantum number is studied. A general model for the underlying coupling mechanism is presented, and the influence of a van der Waals bound noble-gas atom on the intramolecular dynamics is investigated.

A. Mechanism of Intramokcular Dynamics in Polyatomic Molecular System Absorption of light in a molecule via electric dipole interaction normally leads to the excitation of a zero-order state, typically a rovibrational Bom-Oppenheimer (BO) level within the first excited singlet SIstate, These levels are usually not eigenstates of the molecular system since they are coupled by the kinetic-energy operator to the ground-state electronic potential surface, by the spin-orbit coupling operator to the lowest triplet state (interstate coupling), and/or by anharmonic coupling or Coriolis coupling operators to other rovibrational states within the same electronic potential surface (intrastate coupling) [7, 81. At modest vibrational energies of the electronic ground state SOthe interstate coupling does not exist and intrastate coupling is prevailing. On the other hand, in the energy range of high Rydberg states (see Section IV) the density of electronic states is much higher than the density of vibrational and even rotational states and interstate coupling is the dominating mechanism. In the S, state both inter- and intrastate coupling takes placed. In a polyatomic molecule consisting of more than 10 atoms, the density of coupled states p within the SO ground electronic state, and sometimes the T I triplet state, is so high that coupling (with matrix element V) to a large number of states may occur ( p >> V - I ) . This condition approaches the statistical limit for the density of coupled states. Hence, coherent excitation leading to a defined phase of the complete set of eigenstates that contribute to a given BO state results in the initial production of this zero-order state. This initial character is consequently lost by dephasing of the prepared packet of eigenstates. Since the density of states is so high, no recurrence occurs on the time scale of the experiment and a single exponential decay of the prepared population will be observed (large-molecule limit [9]). These conditions produce a Lorentzian line shape for a continuous-wave (CW) spectroscopic experiment that uses high spectral resolution and a small coherence width. By contrast, the density of coupled states within the electronically excited SI state

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H. J. NEUSSER AND R. NEUHAUSER

(or the ground state SO) of a polyatomic molecule depends strongly on the vibrational energy. At low vibrational excess energy and for weak interstate coupling, only one proper state might be located within the range of the intrastate coupling matrix element, and coupling to only a single rovibronic state occurs. The conditions produce a repulsion of the two eigenstates, a situation frequently observed in the spectra of small molecules (small-molecule limit [lo]). The coherent excitation of the two coupled eigenstates results in an oscillatory behavior of the two populations (quantum beats) Ill, 121. Depending on the magnitude of the coupling matrix element, nanosecond pulses or even subnanosecond pulses are required for this coherent excitation. For higher excess energies with increasing density of states a more complicated time behavior is expected.

B. Intramolecular Dynamics in Benzene As an example we consider the prototype molecule benzene (C6€&)consisting of 12 atoms and possessing 30 vibrational degrees of freedom. A fluorescence quantum yield of SIbenzene smaller than unity (0.2) was observed under collision-free conditions on the time scale of fluorescence [ 13-15], and it was concluded that a nonradiative electronic decay takes place that is irreversible on the time scale of fluorescence and occurs in the statistical limit [16, 171. As zero-order €30states we take rovibronic states in the lowest singlet and triplet electronic states. Each state may be coupled to others by the different coupling mechanisms mentioned above. There is, however, a hierarchy of the coupling strengths with interstate coupling being weaker than intrastate coupling in benzene and many other molecules. At the energy of the benzene S, state (38086 cm-') the density of vibrational states in SO is so high (=1015 l/cm-') that coupling to the electronic ground state is in the statistical limit that results in irreversible internal conversion. The situation is not so clear for the coupling to the triplet state. There the density of states is considerably smaller (=lo6 l/cm-') due to the smaller excess energy (SI-TI energy gap: 8464 cm-I) and it is not a priori certain that coupling to triplet states really occurs in the statistical limit. Finally, the situation for coupling within the S1 potential surface is completely different. For excess energies less than 2500 cm-I, the density of states is sufficiently low that the coupling is expected to be in the small-molecule limit; that is, spectral lines involving coupling states are expected to be split rather than broadened. At very high excess energies, where a corresponding high density of states occurs, coherent excitation is expected to lead to a fast irreversible dephasing of the vibrational states within Sg and a significant broadening of spectral lines [ 181.

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1. States at Low Excess Energy

The 14’ state of C6H6 at 1571 cm-’ vibrational energy above the zero point of S1 can be excited through two-photon absorption. Doppler-free CW spectra with thousands of well-separated rovibronic lines of 15 MHz width were recorded at room temperature. Individual transitions were assigned up to J’ = K’ = 10 [19] and the frequencies of 90% of these lines could be well fitted by a semirigid symmetric top model. The remaining lines are shifted from the expected position or split into two components separated by some gigahertz [20].The appearance of these isolated perturbations is due to the fact that the “light” 14’ state is coupled locally to “dark” states within SI through higher order rotation-vibration couplings [3]. From emission spectra of the single eigenstates we were able to determine the identity of the three dark states involved [21]. Pulsed excitation allowed the decay of single rotational states of the 14’ vibronic state to be measured. The decay was found to be singly exponential in all cases investigated [4, 211. For states that are not perturbed by a coupling within SI(unperturbed states) the decay time was found to be independent of the molecular rotation, whereas for perturbed states the decay times are shorter due to the admixture of the dark states, which decay faster. Most likely these dark states are combination states containing quanta of low-frequency out-of-plane vibrations that are strongly coupled to the So ground or T I triplet state and undergo a faster electronic nonradiative relaxation process than the excited “bright” states. The rotationally independent decay of the “unperturbed” states points to an electronic nonradiative relaxation process to T I in the statistical limit. To summarize, the rotationally resolved two-photon spectra at low vibrational excess energy show isolated perturbations that can be understood as rovibrational couplings in the strong limit to discrete background states in SI.This coupling causes a mixing of the vibrational character of the resulting quasi-eigenstates and therefore a rotational dependence of the nonradiative decay, which otherwise would not depend on the rotation of the molecule. The behavior of states excited through one-photon transition (i.e., differing symmetry and parity) is found to be qualitatively the same. Since Dopplerfree one-photon experiments were performed in a molecular beam at low rotational temperatures of 2 K only low J’, K’ values are observed [22]. In this narrow energy range a coupling to dark background states is not very likely. No perturbation was observed in the 6’ band at the low excess energy of 522 cm-I . The lifetime measured for various rotational states up to J‘ = 7, K’ = 6 was found to be independent of the rotational quantum number excited. This demonstrates that the excited rovibronic states do not couple to background states, as is expected for the low J’, K’ quantum num-

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bers and the small excess energy with its low density of background states in S,. The decay is faster than that of the 14’ state at more than twice the excess energy. This has been explained by the energy gap model and different Franck-Condon factors for the Inter Systems Crossing (ISC) process

P31.

2. Dynamic Behavior of States at Intermediate Vibrational Excess Energy

For higher excess energies the density of background vibrational states increases. This causes a different dynamic behavior of the excited light zeroorder states. As an example we briefly describe the results for the 14’ l 2 state of C& at an excess energy of 3412 cm-’ [24].In the blue part (low rotational energy) of the qQ-branch of the 14’ band only (unshifted) K’ = 0 lines were seen in the fluorescence excitation spectrum, whereas further to the red (higher rotational energy) the only observed structure seemed to be due to lines with K’ = J’. This finding was interpreted as due to predominant coupling in the weak limit of 14’l 2 to a strongly diffusely broadened background state that allows only a minority of uncoupled (or weakly coupled) states to survive in the fluorescence excitation spectrum. The measured strong dependence of the collisionless linewidth on the J quantum number shown in Fig. 1 was supposed to be due to the rotational dependence of the coupling matrix element, that is, parallel Coriolis coupling for the blue part and perpendicular Coriolis coupling for the red part of the band [3]. A detailed theoretical description in this and other bands of benzene has been successfully performed using an artificial intelligence model for the description of the coupling pathways [25, 261. It was also possible to measure the decay times and the homogeneous collisionless widths of the K’ = 0 lines in the blue part of the 14’ l2 band. The decays were all found to be singly exponential within the experimental accuracy. The values of the decay rates determined from the time-resolved and frequency-resolved experiments agreed quite well, indicating that in our experiment single quantum states were observed. This was one of the first examples of a combined measurement of the dynamic behavior of a large molecular system in both the time and frequency domains. Similar results were found €or the one-photon band 6Ali band at 3287 cm-’ excess energy. This corroborates the model of a rotationally dependent intramolecular dynamics due to intrastate coupling [3, 4, 271. C. Infiuence of Van der Waals Bonded Noble-Gas Atoms on Intramolecular Dynamics When we attach an Ar atom to the benzene molecular plane, a weakly bound van der Waals complex is formed. Experiments with high-resolution rotation-

HIGH-RESOLUTION SPECTROSCOPY AND DYNAMICS

. .

L

..5

-

* .

JK= 20

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-100

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12.7 MHz

-:

.

I

415

I

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l

1 :

I

l

0

I

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I

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+loo

A u tMHzl

Figure 1. Linewidths of different rotational transitions in the 14A1; vibronic band of benzene measured with Doppler-free two-photon absorption. The observed strong dependence on the quantum number J of the rotational angular momentum is evidence for a rotationally dependent intramolecular coupling process. (Taken from Ref. 3.)

ally resolved ultraviolet (UV) spectroscopy in supersonic cooled molecular beams clearly show that the Ar atom is located on the Cg axis of benzene at a distance of 3.58 A from the molecular plane [28, 291. A decrease of the average van der Waals distance to 3.52 A after excitation to the electronically excited SIstate is found. In benzene-Ar three low-frequency intermolecular

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H. J. NEUSSER AND R. NEUHAUSER

modes (two bending, 31.16 cm-' ; one stretching vibration, 40.1 cm-I [30, 311) are added to the 30 vibrational degrees of freedom of benzene. The binding energy of the benzene-fir van der Waals complex in the SIstate is about 360 cm-', as has been found from mass-selective pulsed field threshold ionization measurements [32, 331. This is in reasonable agreement with recent results from ab initio calculatioy [34-361. For benzenes4= an average van der Waals distance of 3.68 A in SI[37] and a binding energy of less than 435 cm-I was measured [32, 331. When the benzene-Ar complex is excited to the low lying vibrational state 6l, we find a small red shift of 21 cm-' of this band from the respective transition in bare benzene that is mainly caused by the red shift of the zeropoint energy. Even for this low-lying vibrational state the excess energy of 522 cm-' is larger than the dissociation energy of 360 cm-' of the complex. Thus, in principle, coupling to the van der Waals modes and dissociation is a new relaxation channel in the complex. It is interesting to investigate whether the intermolecular channel can compete with the intramolecular relaxation processes in the bare molecule at this excess energy. In Fig. 2 the highresolution spectra of the 6; band in benzene-& leading to the 6' state at an excess energy of 522 cm-I and the 6Al; band leading to an excess energy of 1444 cm-' are shown. They were measured for benzene-Ar complexes produced in a supersonic molecular beam of benzene seeded in Ar gas under high pressure. Supersonic expansion leads to a low rotational temperature of 2 K. The observed linewidth of 120 MHz corresponds to the experimental linewidth given by the laser bandwidth and the residual Doppler broadening in the skimmed molecular beam. There is no additional broadening due to dynamic processes faster than nanoseconds. Even for an excess energy of 1444 em-', which is more than four times the binding energy, the complex is stable on the nanosecond time scale and the intramolecular dynamics is not affected drastically. Furthermore, no perturbations are observed in these bands, which would point to additional coupling paths in the van der Waals complexes due to the increased density of states. To discover smaller specific effects on the intramolecular dynamics after attachment of an Ar atom to the benzene molecule, we performed lifetime measurements of single rovibronic states in the 6; band of the benzene-Ar and the benzene-84 Kr complex. No dependence of the lifetime on the quantum number within one vibronic band was found [38]. This is in line with the results in the bare molecule and points to a nonradiative process in the statistical limit produced by a coupling to a quasi-continuum, for example, the triplet manifold. In Fig. 3 the decay curves of benzene and benzene-Ar are shown after excitation of the same individual rotational state in the 6' vibrational state. Though the rotational quantum numbers . I ; , = 44 and the vibrational quan-

a

I "

LY

-

Figure 2. Rotationally resolved REMPI spectra of the 6; and 6h1b vibronic bands of the benzene-Ar complex. No broadening of the lines in the 6AlA band is observed showing that even for an excess energy of 1444 cm-I, which is more than four times the binding energy, the complex is stable on the nanosecond time scale. (Taken from Ref. 29.)

L

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H. J. NEUSSER AND R. NEUHAUSER I

Time [ns] Figure 3. Comparison of the fluorescence decay curves after selective excitation of the 6 ‘ , J’ = K’ = 4 rovibronic state of benzene and benzene-Ar. Note the shorter lifetime of the same rovibronic state in the complex.

tum numbers are identical in both cases, a pronounced shortening of the lifetime from 86 ns in benzene to 53 ns in benzene-& is found. The measurement of the lifetime of the J ; , = 22 state in benzene-& leads to an even shorter lifetime of less than 10 ns. Most likely the lifetime is in the range of a few nanoseconds since the exciting laser pulse has a pulse duration of 7 ns and no broadening of the lines is seen in the spectra [37]. We can exclude a predissociation process [39] responsible for the decrease of the lifetime for three reasons. (i) Dispersed emission spectra did not show any indication of emission from the fragment monomer [40]. Thus no dissociation occurs on the time scale of the fluorescence emission. (ii) The additional excitation of the van der Waals stretching vibration in benzene-Ar does not lead to a further decrease of the lifetime. (iii) The stronger decrease of the lifetime of the 6l state in benzene-Kr would not be expeced for a predissociation process since the benzene-& complex is more strongly bound and has only a slightly higher density of states since the frequencies of the three van der Wads modes do not differ very much from that of benzene-Ar 1411.

After exclusion of a predissociation process responsible for the lifetime shortening in complexes of benzene with noble gases, we consider the external heavy-atom effect on the intersystems crossing rate as the origin of the lifetime shortening [421. The strong decrease of the lifetime in the

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benzene-= complex supports this explanation. An ambient heavy atom leads to stronger spin-orbit coupling in the aromatic molecule. External heavy-atom effects have been studied in the condensed phase. Experiments have been performed on benzene molecules in a noble-gas matrix at low temperatures [43,441 and on tetracene-noble gas complexes [45]. Here we present a first example where the external heavy-atom effect is studied for a specific rovibronic state in a van der Waals system with defined position of the heavy atom and known van der Waals vibrations. In conclusion, we have found that the intramolecular dynamics in the benzene molecule at low excess energy is not strongly influenced by the additional three vibrational degrees of freedom of the benzene-Ar complex. The coupling of the excited intramolecular modes to the low-frequency intermolecular modes is weak. The observed 40% decrease of the lifetime of the 6' state does not depend on the individual excited rotation and points to an external heavy-atom effect as the source of the lifetime shortening observed for the same selectively excited rovibronic state.

JII. LASER-DRIVEN POPULATION DYNAMICS AND COHERENT ION DIP SPECTROSCOPY

A. Introduction As pointed out in the previous sections, a major goal of this work is the investigation of intramoleculardynamic processes leading to an energy redistribution after excitation of defined quantum states of the molecule. Of great importance for this is the population dynamics that is introduced by the laser light itself, especially by the use of intense Fourier-transform-limited laser pulses. Here the laser light modifies the quantum mechanical system in such a way that not only the molecular system but the whole quantum mechanical system containing the molecular Hamiltonian as well as the electric field interaction Hamiltonian has to be considered [46]. This combined system can be modified by changing the interaction conditions, which is not the case for excitation with weak laser pulses leaving the molecular system unperturbed. This offers the possibility to prepare a system in a special state leading to an effective control of the intramolecular process [4749]. Here we will show that the laser-driven population dynamics makes feasible new spectroscopic techniques for the investigation of high-lying vibrational states in the electronic ground state of polyatomic molecules. We present a brief discussion of the population dynamics encountered in the molecular excitation process using intense coherent nanosecond laser pulses. The typical population dynamics is shown to lead to the new spectroscopic technique of coherent ion dip spectroscopy (CIS), which is useful for the investigation of the

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intramolecular dynamics in molecules and van der Waals complexes [5,6, 301.

B. Incoherent Population Dynamics In most experiments in the frequency domain as well as in the time domain the population dynamics can be described in terms of rate equations based on transition probabilities depending on the coupling strength and levels of different density [50]. Neglecting the polarization properties of the laser light, the transition probability can be written as a product of the square of the transition matrix element times the laser light field strength and the level density of the final level. In this way absorption cross sections and stimulated and spontaneous emission coefficients have been deduced from quantum mechanical calculations. In Fig. 4 a typical lambda-type three-level system is shown that is realized, for example, in stimulated emission pumping

12>

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13a>

Decay

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dvv===

II> Figure 4. Level scheme of a lambda-type double-resonance eperiment. The pump laser pulse couples the initial level 1 to a single rovibronic intermediate level 2 in the electronically excited S1 state. The intermediate level 2 is coupled by the dump laser pulse to the vibrationally excited level 3 in the electronic ground state So. Ionization from level 2 is possible by absorption of an additional photon from the intense dump laser pulse. The coupling of the final level 3 to a single dark state (level 3 4 is indicated and discussed in the text.

HIGH-RESOLUTION SPECTROSCOPY AND DYNAMICS

42 1

(SEP) experiments (for a review see Ref. 51). This excitation scheme has been used for the excitation of high-lying vibrational staes in the electronic ground state of a series of molecules to investigate the intramolecular redistribution processes occumng at high internal energies [52]. Population in level 1 is transfered to level 2 by absorption of a photon provided by the pump laser while the dump laser laser light can cause stimulated emission into level 3, which is going to be investigated. The possibility of a fast decay of level 3 [e.g., by intramolecularenergy redistribution (IVR) in the statistical limit] or coupling of level 3 to single dark states (e.g., IVR in the intermediate case) is neglected for a moment. The rate equations for this three-level system read

where ni denotes the population in level i, 1p.d the intensity of the pump-anddump laser, and Bij, A, the Einstein coefficients. No decay in this three-level system is considered. The time-dependent incoherent population dynamics can be obtained in the usual way by solving the rate equations as a system of coupled linear differential equations with special initial conditions. For CW laser fields and for laser pulses that are long compared to the typical relaxation rates, (quasi)stationary solutions can be found. For short pulses a time-dependent solution of the system of differential equations [Eq. (l)] is necessary. In Fig. 5 the typical population dynamics in a three-level system is shown for temporally slightly overlapping intense laser pulses and 100% population in level 1 before laser interaction.The intense laser light field used in this calculation saturates all transitions. First the pump laser pulse causes a nearly equal distribution of population in levels 1 and 2. Then the dump laser pulse applied after the pump laser pulses causes stimulated emission into level 3, leading to nearly 25% population in levels 2 and 3 and about 50% population in level 1. If only the pump laser frequency is in resonance with the 2 c 1 transition and the dump laser frequency is not in resonance with the transition connecting levels 2 and 3, the population is distributed in equal parts between levels 1 and 2 and 50% of the population will be found in level

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H. J. NEUSSER AND R. NEUHAUSER

dump pulse

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Figure 5. Typical incoherent population dynamics within the level scheme of Fig. 4 for fully saturated resonant transitions calculated from the rate equations (1).

2. This behavior is exploited in SEP experiments [SI] where the lowering of the population of level 2 for double-resonance conditions is probed by laser-induced fluorescence (LIF) or ion detection (ion dip experiments) by ionizing the molecules in level 2 with a third laser pulse. It is obvious from the rate equations that no dip depth larger than 50% of the maximum offresonant signal can be obtained as long as no fast decays of the final levels must be considered. (However, for fast-decaying final levels deeper dips can be expected and the dip depth has been used for an estimate of the decay rate [53].) In the following we want to show how this 50% limit can be overcome by the use of intense Fourier-transform-limitednanosecond laser pulses. In this case the rate equations do not describe the population dynamics correctly, and a solution of the density matrix equation of the system is necessary. The resulting population dynamics allows the detection of weak transitions resulting from the coupling of the optically accessible bright states to dark states and the investigation of the coupling strength and energetic position of the unperturbed transitions.

C. Coherent Population Dynamics

When a molecular system interacts with intense Fourier-transform-limited

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laser pulses, the full Hamiltonian of the molecular system and the field interaction must be considered and solved as a whole. The interaction Hamiltonian Hinteraction must not be treated by time-dependent perturbation theory but has to be included into the complete Hamiltonian of the system:

After transformation into the interaction picture and application of the rotating-wave approximation [46, 50, 541 the population dynamics can be calculated numerically by solving the time-dependent three-level Schriidinger equation or (if phenomenological relaxation rates are considered) by solving the density matrix equation (3) for the molecular system. The density matrix equation is given by

Here p is the density matrix for all molecular states in the three-level system depicted in Fig. 4, and all incoherent relaxation terms caused, for example, by collisions, spontaneous emission, or decay in a (quasi)continuum are incorporated in the relaxation matrix rrelax. For a three-level system the Hamiltonian in the interaction picture H i in Rotating Wave Approximation is given in matrix representation by

The Rabi frequencies 61p.d are given by

where pP,d is the transition matrix element for the pump- and dump-transition and Ep,d the amplitude of the laser field strength of the pump- and dumplaser pulse. For simplicity different polarizations of the laser light fields are not included in this discussion, although interesting applications have made use of the polarization dependence [55-591. Solution of Eq.(3) yields values for the diagonal matrix elements of the density matrix p and thus the timedependent population in every level. In a general approach it is possible to show that a coherent treatment

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H. J. NEUSSER AND R. NEUHAUSER

of the population dynamics is necessary if the Rabi frequency exceeds all intramolecular decay rates and typical inverse-phase relaxation times of the laser light [50].

D. Coherent Population Dynamics for Special Pulse Sequences Figures 6a-c show the population dynamics encountered in a three-level system (see Fig. 4) interacting resonantly with two Fourier-transform-limited laser pulses with three different delay times between the two pulses. The calculation was done assuming that the chosen Rabi frequencies fulfill the relation ( Q p , d > l/pulse duration) in all three cases. This relation ensures that the typical time for a Rabi oscillation of the population in an isolated two-level system is shorter than the pulse duration. Ionization from level 2 was introduced as a fast laser intensity-dependent decay of level 2 [6, 601, and resonant laser frequencies were assumed. In the upper parts of Figs. 6a-c the time-dependent Rabi frequencies of both laser pulses are shown for different delays. In all cases the dump laser pulse has a higher Rabi frequency than the pump laser pulse and twice its duration. Note that the Rabi frequency is proportional to the laser field strength and therefore to the square root of the pulse intensity. In the lower part the population dynamics for the three different pulse sequences is shown. The part of population transferred to the ionization continuum is indicated by a strong line.

6 3

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0.6

$ 0.4

n 0.2

0.o -20.0 0.0 20.0 Time [ns]

-20.0 0.0 20.0 Time [ns]

Time Ins]

Figure 6. Coherent population dynamics calculated using the density matrix equation (3) for different delays (a+) of the laser pulses. Upper part: Time evolution of the Rabi frequencies of both laser pulses. Lower part: Calculated time evolution of the level populations for three different delays.

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(a) Pump Laser Pulse before Dump Laser Pulse. This pulse sequence is equivalent to the pulse sequence used in most SEP experiments. The damped Rabi oscillation of the population between levels 1 and 2 clearly indicates the coherent character of the interaction. Most of the population is ionized during the first laser pulse because of the high laser intensities. Although the dump laser is in resonance, ionization is very efficient and no prominent ion dip will be observed in this case.

(b) Dump Laser Pulse Overlapping with Pump Laser Pulse. In this situation the oscillatory behavior vanishes and nearly no change in level population is observed. The ionization signal vanishes for resonance conditions and a nearly 100% ion dip is expected. This pulse sequence is used in the CIS experiments described below. (c) Pump Laser Pulse after Dump Laser Pulse. With this pulse sequence an effective population transfer to the final level 3 is possible over a wide range of delay times and Rabi frequencies. This behavior was exploited in population transfer experiments using the stimulated Raman rapid adiabatic passage technique (STIRAP), which has been analyzed in detail analytically and numerically [60-64].

E. Coherent Ion Dip Pulse Sequence As shown in Fig. 6b, for a dump laser pulse overlapping with the pump laser pulse no net population transfer occurs. It is particularly interesting that the intermediate level 2 is not significantly populated at any time although level 3 is weakly populated during the interaction. This surprising population dynamics can be exploited to check whether the dump laser frequency is in resonance with the 2 -+ 3 transition and thus the double-resonance condition is fulfilled: As in SEP experiments it is possible to monitor the population of level 2 either by fluorescence from level 2 or by ionization after absorption of an additional photon (see Fig. 4). In a simple model the ionization process from level 2 can be introduced by a time-dependent decay rate of level 2 [6, 601 that is proportional to the intensity of the laser pulses, whereas the fluorescence is only proportional to the population in level 2 after interaction with both laser pulses [54]. For an off-resonant dump laser frequency and a resonant pump laser frequency a high ion current is observed, whereas for double-resonance condition no ion current can be measured due to the negligible population of level 2. In Fig. 7a the calculated ion current and fluorescence signal is shown using the parameters of our recent work [6].The pump laser frequency was kept in resonance with the transition 2 t 1 and the dump laser frequency was scanned across the 2 -+ 3 transition. (It should be mentioned that for

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H. J. NEUSSER AND R. NEUHAUSER

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Figure 7. ( a ) Three-level system: Calculated ion dip and fluorescence dip spectra with 1 transition. Ionization is treated as pump laser frequency tuned to resonance with the 2 an intensity-dependent decay of level 2. The sharp dip with nearly 100% depth indicates the coherent character of the excitation. (b) Four-level system: Different from (a) level 3 is now coupled to a dark state, yielding a splitting of level 3 in two states separated by 0.016 cm-' that are well resolved in the calculated ion dip spectrum.

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the calculation the more realistic model of Ref. 6 for the interaction process was used. Here not only the time dependence of the Rabi frequencies and ionization decay rates but also a radial Gaussian distribution of the laser intensities and the dependence of the transition moments on the rn-quantum number of the orientation of the molecule in a laboratory fixed axis were included.) The calculated spectrum shows two interesting aspects. (i) For the given Rabi frequencies the fluorescence dip is less pronounced than the ionization dip. This behavior is due to the fact that ionization takes place mainly in regions with high Rabi frequencies because of the quadratic dependence of the ionization rate on the laser field strength, whereas the fluorescence signal originates from all spatial regions interacting with the laser pulses. (ii) While the linewidth of the dips [full width at half maximum (FWHM)] is large (several gigahertz), they are very sharp at the minimum. This allows us to determine the exact frequency position of the 2 --+ 3 transition. The linewidth of the dip depends strongly on the Rabi frequencies and ionization rates. High ionization rates and low Rabi frequencies yield small but narrow dips. When we apply the coherent technique to study intramolecular processes occurring in the final state 3, it is important to investigate how an intramolecular coupling or a dynamic process of level 3 affects the population dynamics and whether small spiittings can be observed. In Fig. 7b the calculated ionization signal for the same laser intensities and time delays as used in Fig. 7a is shown. Now, however, level 3 is coupled to a dark state 3a as depicted in Fig. 4. The state 3a is assumed to be dark since the transition moment 2 ---c 3 is very small due to small Franck-Condon factors or for symmetry reasons. The coupling is assumed to lead to two mixed eigenstates with an energetic separation of 0.016 cm-' that is larger than the laser bandwidth. As demonstrated by the calculated CIS spectrum of Fig. 7b, both states can be resolved even for intensity conditions leading to a broad CIS signal. Finally, we want to emphasize an interesting result of the numerical calculation that has been proven experimentally. As shown in Fig. 6, there exist two pulse sequences (b, c) leading to a small population of level 2. In case (c) most of the population is transferred to level 3 while in case (b) nearly all of the population remains in the initial level. In coherent ion dip experiments case (b) is used as it provides deeper dips due to the more effective suppression of ionization. Using higher laser intensities would allow us to achieve nearly 100% ion dips also in case (c); however, for off-resonant conditions the ion current would be smaller by an order of magnitude than in the pulse sequence of case (b) and dips are more difficult to detect.

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F. Experimental Results In the following section a brief survey of our recent experimental results obtained with CIS is presented. 1. Experimental Setup

The experimental setup (Fig. 8) is similar to the one used in previous CIS experiments [6]. Both laser pulses are provided by two pulsed amplified CW ring lasers operating with Coumarin 102 and Fthodamine 110 dye, respectively. Amplification of the CW light in two three-stage amplifier systems and frequency doubling of the nearly Fourier-transform-limited visible laser light pulses yield UV pulses with energies of 400 pJ, pulse durations of 15 ns (FWHM), and a bandwidth of 70 MHz. The two counterpropagating laser beams are focused down to a common focus of less than 0.5 mm diameter. The first laser pulse exciting the molecule or the van der Waals complex from the electronic ground state to a single rovibronic state in the electronically excited Sl state is attenuated by a factor of 50 to reduce direct photoionization by absorption of two photons of this pulse. The counterpropagating narrow-band light pulses interact with the molecules in the center of a cooled molecular beam expanding from a reservoir with 2% benzene seeded in Ar at a backing pressure of 2 bars through a nozzle with a 300-pm orifice [22]. A conical skimmer (1.5 mm diameter) collimates the beam and reduces the Doppler width below the laser linewidth, and an experimental saturation broadened resolution of 200 MHz is achieved in a resonance-enhanced two-photon ionization (REMPI) experiment with the frequency of the second pulse not in resonance with the 2 3 transition. The produced ions are mass analyzed in a time-of-flight (TOW mass spectrometer and detected with multichannel plates.

-.

2. Experimental Procedure of Coherent Ion Dip Spectroscopy

In Fig. 9 the ion current calculated as described above is shown as a function of the detunings of the pump-and-dump laser frequency in a two-dimensional plot. A cut along the pump laser frequency axis for a nonresonant dump laser frequency yields REMPI spectra. The same cut for a resonant dump laser frequency show a “splitting” of the 2 1 transition that can be explained in the intuitive picture of a dynamic Stark effect. (For a closer discussion of this point, see Refs. 6 and 30). A cut along the dump laser frequency

-

Ar ton Laser

Kr ton Laser

I

Beam

Molecular

Figure 8. Scheme of the experimental setup of the CIS experiment. Two Fourier-transform-limited nanosecond laser pulses with different frequencies are interacting with cold molecules or van der Waals complexes in a skimmed supersonic molecular beam.

cw - Dye Laser

cw - Dye Laser

I

1

TOF~

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H. J. NEUSSER AND R. NEUHAUSER

Ion Current

Figure 9. Calculated ion current spectra for different detunings of the pump laser and the dump laser frequency. The two-dimensional plot includes REMPI spectra as well as CIS spectra.

axis with resonant pump laser frequency yields CIS spectra with ion dips. The process of measuring CIS spectra is indicated by the solid arrow in Fig. 9: In a first step the dump laser frequency is tuned out of resonance with the downward transition 2 3 and the pump laser frequency is scanned, yielding a rotationally resolved REMPI spectrum of the investigated vibronic band. In a second step the pump laser frequency is fixed on top of a single rotational line in the selected vibronic band and the dump laser frequency is scanned. (In Fig. 9, this means that the pump laser frequency is fixed on top of the ridge and the dump laser frequency is scanned along the ridge, as indicated by the arrow.) For double-resonance conditions, when the dump laser frequency is in resonance with the 2 -+ 3 transition, the ion current vanishes. In the two-dimensional plot this leads to the valley crossing the ridge in the middle of the plot.

--.

3. Spectra

Coherent ion dip spectroscopy has been shown to be a versatile tool for the investigation of high-lying intramolecular vibrations in the ground state of molecules and of intermolecular vibrations of van der Waals complexes.

-

(a) Intramolecular Coupling in Benzene Molecule. Incoherent SEP and CIS spectra have been measured for benzene (C6H6 and C6D6). Pre62 viously, we investigated the down transitions 6' + 12 and 6'

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431

in C6H6 in an incoherent SEP experiment with high-resolution pump but low-resolution dump laser [65]. Some rotational resolution was achieved in this way. Applying the high-resolution CIS technique to the 6' --c 62 transition leads to a complete resolution of all close-lying rotational transitions. For the 6' -+ 12 transition additional lines were observed in our previous SEP experiments. These were explained by a coupling of the 12 level through a Darling-Dennison [66] resonance to the close-lying 52 state. The CIS spectrum of the same transitions shows the well-resolved Darling-Dennison resonance but also a hitherto unknown splitting of the higher energetic component whose origin has not been explained yet. The measured spectrum resembles the theoretical spectrum in Fig. 7 b and demonstrates the feasibility of CIS for the sensitive highly resolved detection of intramolecular couplings. (b) CIS of Intermolecular Vibrations of Benzene-Ar Complex. The CIS technique has been used to detect the weak transitions to high-lying intermolecular van der Waals vibrational states in the benzene-Ar complex with rotational resolution. In Fig. 10, as an example the CIS spectra of the SI,6's' -SO, 12s' and SI,6's' +SO, l2b2 transitions in the c6D6-A~complex are shown. Here the pump laser frequency was fixed on top of the R44 transition of the S1 state of the benzene-Ar complex and the dump laser frequency scanned in the region where allowed rotationally resolved downward transitions are expected. (Here s denotes the totally symmetric intermolecular stretching mode, b the degenerate bending mode, and l b in Fig. 10 the vibrational angular momentum of the degenerate bending mode). Four prominent dips appear in the ion dip spectra that can be assigned to the four allowed rotational transitions in each case. Additional CIS spectra have been measured via different van der Waals vibrational states in the SI electronic state to a series of van der Waals vibrational states in the electronic ground state SOof the complex. In Fig. 11 an overview of all investigated transitions in the benzene (C6&)-Ar complex is given demonstrating the high sensitivity of CIS. Seven hitherto unknown transitions up to a van der Waals excess energy of 130 cm-' have been observed [30], and their rotationless frequency positions have been determined with a high accuracy of 0.03 cm-' . Recent experiments concentrate on the investigation of strongly perturbed bands like the 6's' band in c6D6-A~ using two-dimensional CIS, scanning both lasers independently as depicted in the calculation shown in Fig. 9. A further goal is to determine the rotational constants of the different van der Waals states. The effective rotational constants represent a value averaging over the vibrational motion and thus information about the vibra-

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H. J. NEUSSER AND R. NEUHAUSER

100 Oh-

50 % -

0 Yo-

61b2(Ib-O), (J'=K'=4,+/) 1-

+

6,b2(/':=0)

100 %-

50 % -

0 %-

Dump Frequency

-

I )

Figure 10. Experimental CIS spectra of the 6's1(J' = K' = 4,+1)-62sl (upper trace) 62b2(lL = 0) transition (lower trace) in transition and the 6'b2(1; = 0).(J' = K' = 4, +1) benkene-Ar ( C 6 b .Ar). Four dips in each spectrum are shown indicating single rovibrational transitions with different Hoed London factors. The iodine spectrum on top of each spectrum was synchronously recorded for absolute frequency calibration.

tional wave function. Theory predicts a strong mixing of the stretching and bending vibrations for higher states in benzene-Ar [67,681. Experimental evidence for mixing of the intermolecular bending and stretching modes has been found in the S1 state of para-difluorobenzene-Ar by an analysis of the

HIGH-RESOLUTION SPECTROSCOPY AND DYNAMICS

Van der Waals internal energy 0 20 40 60 80

100

433

[CHI]

m. Figure 11. Investigated van der waals transitions in the benzene-t\r (CgH6 . Ar) complex. The high sensitivity of the CIS allows the detection of weak transitions to hitherto unknown high-lying van der Waals states in the electronic ground state yielding precise information about their frequency.

rotational constants obtained from rotationally resolved REMPI spectra of different van der Waals states [69, 701.

IV. INTRAMOLECULAR DYNAMICS OF HIGH RYDBERG STATES IN POLYATOMIC MOLECULES A. General Remarks Sections I1 and I11 focused on the intramolecular dynamics occumng in low electronic potential surfaces (SO,SI, T I ) .For higher electronic valence states and low Rydberg states a drastic shortening of lifetimes due to fast dynamic processes has been observed. Qpical time constants in these energy regions are found to be in the subpicosecond range [71, 721. At very high excitation energies, close to the ionization continuum, there exist electronic states resembling the Rydberg states of hydrogen that are expected to be long-lived from the scaling laws [73]. The Rydberg series consist of electronic states with a single electron excited into a quasi-classical orbit around the positively charged molecular core with high Rydberg quantum number n. In this case the energetic separation of neighbored electronic states (i.e., Rydberg states with different hydrogen quantum numbers n, 1) becomes smaller than the spacing of vibrational or even rotational levels of the molecular core. This situation is depicted in the level scheme of Fig. 12 with three Ryd-

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H. J. NEUSSER AND R. NEUHAUSER

Figure 12. Level scheme of the rotationally resolved high-n Rydberg experiment. A first narrow-band laser pulse excites the molecule from the electronic ground state So into a single rotational state in the electronically excited SI state. The frequency of the second laser pulse is scanned to obtain the rotationally resolved Rydberg spectrum shown in Fig. 13.

berg series converging to different rotational energy levels of the molecular core. The term N + denotes the rotational angular momentum of the molecular core. (Note that the other relevant rotational quantum numbers, such as the projection K + of the core angular momentum N + on the symmetry axis of a symmetric top molecule, are omitted for simplification.) In this section we address the question of whether high Rydberg states can be selectively excited in a polyatomic molecule by high-resolution techniques to investigate their dynamic behavior, High Rydberg states play an important role in spectroscopic techniques utilizing pulsed-field ionization 1731for the investigation of the threshold ionization processes. Recent developments include the detection of electrons with zero kinetic energy (ZEKE) [74, 751 and the mass-analyzed detection of threshold ions (MATI) [76, 771 after pulsed-field ionization of high long-lived Rydberg states. While these techniques have been successfully applied to a variety of molecules and clusters leading to spectroscopic and dynamic information on the ionic ground state, little is known about the nature of the high Rydberg states of these systems involved in the excitation process. A major point of interest has been the discrepancy between the lifetimes found from extrapolation of the linewidth of low Rydberg levels and results for high Rydberg levels deduced from unresolved pulsed-field ionization experiments [78-801. This discrepancy has initiated the discussion of different mechanisms leading to a lengthening of the lifetime of high-n Rydberg states [81-881.

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B. Experimental

The experimental setup used for the investigation of high-n Rydberg states is similar to that used in our coherent ion dip experiments (see Fig. 8). The main difference is that no external field is applied during excitation with the two overlapping laser pulses. Typically 100 ns after excitation of the molecule by the two laser pulses a pulsed electric field of 100 V/cm is turned on for 10 ps to field ionize the high-lying Rydberg levels. The produced ions are mass analyzed in a TOF mass spectrometer and detected with multichannel plates. Because of the lower laser power and smaller transition strength to the Rydberg series as compared to transitions investigated in CIS experiments (see Section III), coherent effects can be neglected. In Fig. 12 a typical doubleresonance excitation scheme is shown. In a first step the molecule is excited to a single rotational state in the electronically excited S1 state by absorption of a photon with energy hv 1 provided by one of the pulsed amplified single6’, J’ = K’ = 1(-1) mode CW laser systems shown in Fig. 8 (e.g., into the SI, state of benzene (C6D6). The symbols J’ and K’ denote the symmetric top quantum numbers and - I the lower lying level of the degenerate 6’ vibration of benzene split by Coriolis coupling [22]). By absorption of a photon with energy hv2 from the second pulsed amplified narrow-band laser system, the molecule is excited into individual states in the Rydberg series converging to different limits above the lowest ionization energy (IE) with N + = 0. Keeping the first laser frequency V I in resonance with a single rovibronic transition frequency to the S1 electronic state and scanning the frequency vq of the second laser yield the Rydberg spectrum presented in Fig. 13. The doubleresonance excitation scheme with sub-Doppler resolution in both excitation steps ensures that Rydberg series excited from different intermediate rovibronic levels do not overlap in the high-n region.

C. Experimental Results In Fig. 13 the Rydberg spectrum of benzene (C6D6) pumped via the lower component of the Coriolis-split SI, 6 ’ ,J’ = K’ = 1, ( - I ) intermediate state is shown demonstrating the resolution of Rydberg series for n > 100. A more detailed investigation of the spectrum reveals several weak series overlapping with the main series. All series can be identified throughout the complete measured spectra [89] and fit very accurately to the Rydberg formula

with a quantum defect p c 0.01. In Eq. (6) IE denotes the series limit and Rknzenethe mass-corrected Rydberg constant for benzene.

H. J. NEUSSER AND R. NEUHAUSER

C,D,

I d n=89

Pumped via 610,P11

74555.0

74565.0

74575.0

Total Energy [cm-’1 Figure 13. Rydberg spectrum of Q D 6 obtained by pumping via the P I1 transition to the SI,6 ’ , .I’= K’ = 1 , ( - I ) rovibronic intermediate state. Note that Rydberg states up to n > 100 are resolved.

In Fig. 14 the magnified central part of the spectrum shown in Fig. 13 is presented together with the positions of the Rydberg peaks of two series calculated using Eq. (6). The assignment of the series limits leads to accurate values of the lowest adiabatic ionization energy of the benzene (C6D6) molecule and the rotational constants of its cation 1891. The width of the Rydberg peaks is about 1 GHz and larger by a factor of 10 than expected from the laser resolution. It is caused by Stark splitting of the n levels induced by small permanent residual stray fields present in our apparatus. Individual Rydberg peaks can no longer be resolved when the Stark splitting becomes larger than the distance of two neighbored Rydberg peaks of a series. In a simple model the Stark splitting increases with n2, and the distance of the Rydberg peaks decreases with l/n3. For the highest resolvable n = 110 we find from l/n3 = 3Fn2 an upper limit for the residual field of 50 mV/cm. When the pulsed-field ionization signal of a single excited Rydberg peak is measured as a function of the delay time between the extraction field pulse

437

HIGH-RESOLUTION SPECTROSCOPY AND DYNAMICS

,

CI

C

Q)

f

0

C

0

:

- 74563.0

74565.0

L

I

74567.0

74569.0

Total Energy [cm-’1

Figure 14. Magnified central part of the Rydberg spectrum of C& shown in Fig. 13. The filled circles indicate the positions for two Rydberg series converging to different core rotational states of the ion.

and the laser pulse, a decay of the signal is found that can be interpreted as a “lifetime” of the Rydberg level. Varying the delay in the 30-1000 ns range we found a multiexponential decay behavior with a fast decay component of some 10 ns and very long lived components. In recent experiments we were able to show that even for n = 58 there are Rydberg states that survive 45 ps delay time. This decay time is longer by a factor of 40 than extrapolated from the 500-fs lifetime [72] of low Rydberg states with n = 5 using the scaling law. Several models have been proposed to explain this lengthening of the Rydberg lifetime [82-881. All models include external effects such as I-mixing through the Stark effect in an electric field or rn-mixing through collisions. In these models fast optically accessible light zero-order states with small I-quantum number are coupled to dark zero-order states with larger Z-quantum number similar to the intrastate coupling scheme of the S, state discussed in Section Il. However, while in the S1 state the optically accessible light state is long-lived and coupling occurs to short-lived dark background states, in the Rydberg region the light states are short-lived and coupling occurs to long-lived dark states. Thus a lengthening of the lifetime is expected for Rydberg states rather than a shortening as in the SIstate.

438

H. J. NEUSSER AND R. NEUHAUSER

This coupling leads to another interesting aspect concerning the measured decay time of the pulsed-field ionization signal: The rotationally resolved Rydberg states presented in Figs. 13 and 14 are states with a single n but consist of a manifold of Stark states that can be expressed as linear combinations of zero-order Rydberg states with the same molecular core rotational quantum numbers but different angular momentum quantum numbers 1 of the Rydberg electron. These zero-order Rydberg states may exhibit different optical accessibilities and different decay times. As the observed Rydberg peaks in the spectra are caused by the overlap of different Stark states with different decay times and optical accessibilities, this can result in a nonexponential decay behavior, as investigated theoretically in Ref. 88, and is expected to lead to changes of the “line shape” of the Rydberg peak measured for different delay times of the pulsed field as demonstrated in a recent high resolution experiment [89].

V. CONCLUSION We have shown that the investigation of the state-selected intramolecular dynamics of polyatomic molecules and their van der Waals complexes is now feasible in different energy regimes up to the ionization energy. At low excess energies (522 em-’) in SIthe intramolecular dynamics in the benzene molecule can be described by intrastate coupling of pairs of rovibronic levels in S1,leading to quasi-eigenstates. The zero-order levels are coupled to a quasi-continuum of levels in the T I or SOstate (interstate coupling), leading to an exponential decay. As the dark zero-order states have a shorter lifetime than the optically accessible light zero-order states, the intrastate coupling that is found to be induced by Coriolis forces results in a shortening of the lifetime of the quasi-eigenstates. When the intrastate couplings to short-lived dark background states become more frequent, this leads to a further shortening of the lifetime and a rapid decrease of the fluorescence quantum yield at higher excess energy. The intramolecular dynamics at high excess energies is not substantially affected in the benzene-& complex by the attachment of the Ar atom. Predissociation of the complex does not play a major role, though internal energy exceeds the dissociation energy. Evidence is found for an enhancement of the intersystem crossing rate by the external heavy-atom effect. We have presented a new technique for the investigation of intramolecular couplings in the electronic ground state SO. The new technique of CIS is based on the special population dynamics induced by the coherent excitation of a three-level system with two narrow-band Fourier-transform-limited laser pulses. It allows the investigation of high-lying intermolecular vibrational states in the electronic ground state of van der Waals complexes. These

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439

states are found to be strongly mixed by a coupling between the bending and stretching van der Waals modes. In this way, in the low-energy region basic information on the relevant coupling mechanisms is obtained, leading to an energy flow within and between the subsets of intramolecular and intermolecular modes in a weakly bound van der Waals complex. In high Rydberg states (n > lOO), which were resolved for a large polyatomic molecule in the present work, the intramolecular dynamics is governed by interstate coupling between a manifold of close-lying electronic levels. The intramolecular dynamics in this energy range is influenced by external effects: Small electric stray fields present in every apparatus utilizing field ionization for Rydberg state detection lead to an externally induced mixing of many Rydberg states with identical principal quantum number n but different 1 angular momentum quantum numbers. This results in a lengthening of the lifetime as compared to the value expected for the zero-order optically accessible light state with low 1 quantum number and is an important characteristicof pulsed-field ionization experiments. Internal intramolecular couplings due to the frequent crossing of close-lying Rydberg series in polyatomic molecules may be observable after suppression of external fieldinduced effects. In conclusion in this work we have presented basic information on the nature of the coupling processes leading to the intramolecular dynamics in isolated molecules. This information is useful for the understanding of the origin and mechanism of the fast femtosecond energy flow in high valence and low Rydberg states.

Acknowledgments The authors thank R. Sussmann for helpful discussions. Financial support from the Deutsche Forschungsgemeinschaft is gratefully acknowledged.

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50. V. S. Letokhov and V. P. Chebotayev, Nonlinear Laser Spectroscopy, Springer, Berlin, 1977.

5 1. Special issue, Molecular Spectroscopy and Dynamics by Stimulated-Emission Pumping, (J. Opt. SOC.Am. B 7, 1990). 52. C. E. Hamilton, J. L. Kinsey, and R. W. Field, Ann. Rev. Phys. Chem. 37,493 (1986). 53. T. Ebata, M. Furukawa, T. Suzuki, and M. Ito, J. Opt. SOC.Am. B 7 , 1890 (1990). 54. B. W. Shore, The Theory of Coherent Atomic Excitation, Wiley, New York, 1990. 55. W. L. Meerts, I. Ozier, and J. T. Hougen, J. Chem. Phys. 90,4681 (1989). 56. H. G. Rubahn, E. Konz, S. Schiemann, and K. Bergmann, Z Phys. D 22,401 (1991). 57. Y.B. Band and P. S. Julienne, J. Chem. Phys. %, 3339 (1992). 58. A. F. Linskens, N. Dam, J. Reuss, and B. Sartakov, J. Chem. Phys. 101,9384 (1994). 59. R. Neuhauser and H. 1. Neusser, J. Chem. Phys. 103,5362 (1995). 60. G. W. Coulston and K. Bergmann, J. Chem. Phys. 96,3467 (1992). 61. F. T. Him, Phys. Lett. 99A, 150 (1983). 62. U. Gaubatz, P. Rudecki, S. Schiemann, and K. Bergmann, J. Chem. Phys. 92, 5362 (1990). 63. A. Kuhn, G. W. Coulston, G. 2.He, S. Schiemann, and K. Bergmann, J. Chem. Phys. 96, 4215 (1992). 64. Y. B. Band and 0. Magnes, J. Chem. Phys. 101,7528 (1994). 65. Th. Weber, E. Riedle, and H. J. Neusser, J. Opt. SOC.Am. B 7 , 1875 (1990). 66. B. T. Darling and D. M. Dennison, Phys. Rev. 57, 128 (1940). 67. Ad van der Avoird, J. Chem. Phys. 98, 5327 (1993). 68. E. Riedle and Ad van der Avoird. J. Chem. Phys., 104, 882 (1996). 69. R. Sussmann, R. Neuhauser, and H. J. Neusser, Chem. Phys. Lert. 229, 13 (1994). 70. R. Sussmann and H. J. Neusser, J. Chem. Phys. 102,3055 (1995). 71. J. M. Wiesenfeld and B. I. Greene, Phys. Rev. Lett. 51, 1745 (1983). 72. R. L. Whetten, S. G. Grubb, C. E. Otis, A. C. Albrecht, and E. R. Grant, J. Chem. Phys. 82, 1I15 (1985). 73. F. B. Dunning, in Rydberg Stares of Atoms and Molecules, R. F. Stebbings and F. B. Dunning, Eds., Cambridge University Press, 1983. 74. G. Reiser. W. Habenicht, K. Muller-Dethlefs, and E. W. Schlag, Chem. Phys. Lett. 152, 119 (1988). 75. F. Merkt and T. P. Softley, Phys. Rev. A 46, 302 (1992). 76. L. Zhu and P. M. Johnson, J. Chem. Phys. 94, 5769 (1991). 77. H. Krause and H. J. Neusser, J. Chem. Phys. 97,5923 (1992). 78. U. Even, R. D. Levine, and R. Bersohn, J. Phys. Chem. 98,3472 (1994). 79. F. Merkt, J. Phys. Chem. 100, 2623 (1994). 80. S. T. Pratt, J. Chem. Phys. 98, 9241 (1993). 81. C.Bordas, P.F. Brevet, M. Broyer, J. Chevaleyre, P. Labastie, and J. P.Petto, Phys. Rev. Lett. 60, 917 (1988). 82. W. A. Chupka, J. Chem. Phys. 98,4520 (1993). 83. D. Bahatt, U. Even, and R. D. kvine, J. Chem. Phys. 98, 1744 (1993). 84. W.A. Chupka, J. Chem. Phys. 99,5800 (1993).

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85. E. Rabani, L. Y. Baranov, R. D. Levine, and U. Even, Chem. Phys. Lett. 221,473 (1994); F. Remacle and R. D. Levine, Chem. Phys. Letters 257, 1 1 1 (1996). 86. M.J. J. Vrakking and Y. T. Lee, Phys. Rev. A 51, R894 (1994). 87. M.J. J. Vrakking and Y. T. Lee, J. Chem. Phys. 102, 8818 (1995). 88. M.Bixon and J . Jortner, L Phys. Chem. 99,7466 (1995). 89. R. Neuhauser, K. Siglow, H. J. Neusser, J. Chem. Phys. 106,896 (1997).

DISCUSSION ON THE REPORT BY H. J. NEUSSER Chairman: E. Pollak

J. Manz: Prof. H. J. Neusser has presented to us beautiful highresolution spectra of medium-size molecules and clusters such as benzene and C6H6- At (see current chapter). The individual lines have been assigned to individual rovibronic eigenstates of the systems, and their widths have been interpreted in terms of various intramolecular processes between zero-order states (e.g., Coriolis coupling, anharmonic couplings between bright and dark states, and so on). Now let me be the advocate of the molecule, which does, in fact, not know anything about zero-order states. Instead, it has molecular eigenstates, at least in the ideal case of decoupling from the radiation field. Ultrahigh resolution spectroscopyof these eigenstates should yield spectra with zero linewidth, in contrast with the experimental results. I would like to ask, therefore, for an interpretation of the width of the observed spectral lines, not in terms of zero-order states, but in terms of molecular eigenstates. My guess is that the observed spectral width of molecular eigenstates can only be due either to couplings to the radiation field or to continuum states due to (pre-) dissociation but not to Coriolis coupling or anharmonic couplings of vibrational states. H. J. Neusser: As an approximation I separated intrastate coupling within the S1 state from the interstate coupling to the TI or So state. Eigenstates resulting from intrastate coupling are eigenstates with respect to the S1 state (“quasi-eigenstate”). From the J , K independent exponential decay it is concluded that the interstate coupling in benzene is in the statistical limit and thus leads to a Lorentzian line shape. In a completely isolated molecule with a finite density of triplet states, the individual eigenstates could be resolved (i) if the spontaneous lifetime is very long so that the zero-order states are not broadened and do not overlap and (ii) for a very high resolution of the exciting laser light. In real systems, like benzene in our experiment, the spontaneous lifetime

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and collisions (and the limited spectral resolution) lead to a broadening of lines and make the observation of real eigenstates impossible. M. Quack: Relating to the discussion by Neusser and Manz, I should like to point out that in highly excited states of polyatomic molecules (such as benzene) the true spectrum is a continuum, and it thus makes little sense to talk about “true” discrete molecular eigenstates. This is rather a useful, simplified picture. The continuum nature arises from spontaneous emission [important in the IR, when the average distance between levels is smaller than the spontaneous emission linewidth (about cm-’ in the IR emission)]; there is furthermore a continuum due to possible dissociation channels in most polyatomic molecules. Finally, there is always coupling to the outside world by collisions and ultimately gravitation. Thus, while there are several useful levels of approximate descriptions of polyatomic molecules, the ultimately correct one would be a time-dependent state in the continuum, not an “eigenstate” of the energy. I have given a related discussion concerning IVR (intramolecular vibrational redistribution) in 1981

PI.

1. M. Quack, Faraday Disc. Chem. SOC. 71, 359 (1981).

L. Woste: Prof. Neusser, you are able to select specific rotational states at very high specific Rydberg state quantum numbers n. So your orbiting electron becomes very slow, even slower than the molecular rotation. Do you see any chance to bring both into phase, causing stroboscopic effects? H.J. Neusser: In the energy regime this condition is fulfilled for a close spacing of the two Rydberg series converging to different rotational quanta of the cation. I have shown in one of my slides that a frequent crossing of Rydberg series is expected for a large molecule with small rotational constants. At the crossing point the Rydberg lines from two series can be close to each other. This leads to a coupling of the two series. In particular, this crossing of two rotational Rydberg series with small An becomes possible. R. W. Field: Concerning the question by Prof. Woste about the possibility of efficient energy transfer between ion-core rotation and the Rydberg electron, Prof Neusser mentioned following two Rydberg series that converge to two different rotational levels of the ion. He also noted the location of the An = 1,2,3,. . . series of level crossings between the two series. In fact, these level crossings correspond to the situation where the Rydberg orbit period is equal to the rotational period, twice the rotational period, three times, and so on. The

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H. J. NEUSSER AND R. NEUHAUSER

classical period corresponds to h/AE, where AE is the quantum level spacing. V. Engel: Prof. Neusser, you mentioned the technique of Stimulated Raman Rapid Adiabatic Passage STIRAP, which allows for the coherent transfer of vibrational population selectively. Is the “technique” not another very efficient and experimentally verified scheme of coherent control? H. J. Neusser: Choosing a special pulse sequence of the dump and the pump laser pulse leads to a complete blocking of the population transfer in the CIS experiment or else makes it very efficient. We can say that a special channel is open or closed, that is, controlled by the experimental parameter. This is similar to STIRAP experiments. However, it was shown by Band and Magnes [l] that the adiabatic passage population transfer in STIRAP experiments does not represent a solution of an optimal control problem. 1. Y. B. Band and 0. Magnes, J. Chem. Phys. 101,7528 (1994).

T. Kobayashi: Let me ask two questions of Prof. Neusser: 1. Is it possible to observe a shift in coherent Raman scattering in the three-level system with A-type coupling? We have done an experiment to obtain a femtosecond Raman gain spectrum in polydiacetylenes. The Raman spectrum is shifted to the red under increased , amplification peak signal is pump ( w , ) intensity. By changing ~ 2 the to be shifted to lower frequency. If the optical Stark effect is observed, then, in principle, it should be possible to observe the effect of a high field on the coherent Raman spectrum (see Fig. 1).

a1

7

02

II 01

w2

__

amplification OfW

weak field

I

1 1

/ -

01

---_ Figure 1.

9

01

amplificationofq ’~mmgainsignd

strong field

-

2. Why is the dump frequency exactly resonant with the (2)

13)

HIGH-RESOLUTION SPECTROSCOPY AND DYNAMICS

445

transition (see Fig. 2)? Why were Mowllow triplets not observed? In other words, why is only the splitting observed, without evidence of the triplet?

Figure 2.

H. J. Neusser 1. Concerning the first question of Prof. Kobayashi, I think that shifts of Raman lines by interaction with other levels than the investigated ones should be observable in the Raman process. A pronounced shift is expected, for example, if one of the laser frequencies is close to a resonant multiphoton transition to another rovibronic state. 2. Refemng to the second question, let me point out that in a CIS experiment we do not monitor the fluorescence from the split level 2 to the split level 3, which would lead to an equally spaced triplet in the fluorescence spectrum.

The CIS “trapping experiment” monitors the population in level 2 by an incoherent ionization step. In the weak-field limit of the pump laser this can be compared to an experiment detecting the fluorescence from level 2 to a spectator state (here level 1) at the field-free resonance frequency between levels 1 and 2. In such an experiment no fluorescence is expected at this frequency due to the strong shift of level 2 for resonant dump laser frequency.

K. Yamanouchi: I would like to make the comment that, in the Rydberg series of a molecule, as the principal quantum number n increases, the energy spacings between adjacent Rydberg states become close to those between vibrational levels. In such a situation, an effective energy exchange could occur between the motion of a Rydberg electron and the vibrational motion of the ion core. If the total energy is above the ionization threshold, vibrational excitation would cause autoionization in an efficient way, such that the vibrational quantum number decreases in a stepwise manner and the vibrational energy is transferred to the Rydberg electron, resulting in a relatively large nega-

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H. J. NEUSSER AND R. NEUHAUSER

tive change in u, that is, Au = Uion - my& where URyd and Uion represent vibrational quantum numbers of the Rydberg state and of the ion produced after the autoionization, respectively. Recently, we investigated the Rydberg states of a HgNe van der Waals dimer by an optical-optical double-resonance method and found autoionization channels associated with Av = -3, -2 besides Av = - 1 in the energy region just above the ionization threshold. This observation may be interpreted in terms of an efficient energy exchange between a Rydberg electron and an ion-core vibration. This type of energy exchange in an autoionization process may correspond with the behavior of a kicked rotator in classical mechanics, which is known to exhibit chaos. It would be worthwhile to consider an autoionization process of a simple diatomic molecule in its Rydberg states to understand experimentally the essential dynamics of a quantum system, whose classical counterpart exhibits chaos. H. J. Newer: In relation to the comment by Prof. Yamanouchi, we should notice that an efficient interaction of the Rydberg electron with vibrations of the core is expected for small vibrational frequencies. Benzene as a rigid molecule has relatively large vibrational frequencies of more than 300 cm-' . An efficient coupling is expected for van der Waals complexes (e.g., the benzene-Ar complex) with low van der Waals vibrational frequencies of about 30 cm- I . T. P. Softley: The energy flow in high Rydberg states between electronic and nuclear motions is of active interest with regard to lifetime lengthening. Do you see any evidence in your high Rydberg spectra of benzene for coupling between Rydberg series converging to different rotational states of the ion, either under the lowest field conditions or when higher fields are applied? H. J. Neusser: For a selected intermediate J;, state we observe a couple of Rydberg series; for example, for J k , = l I we can identify two series under minimum residual field conditions. When we apply a stationary electric field of 300 mV/cm, additional series appear that are coupled by the electric field. All series have different limits representing different rotational states of the benzene cation. At present we cannot say whether the coupling observed under minimum residual field conditions is induced by the small stray field or by field-free intramolecular coupling. M.S. Child: The first part of the talk by Prof. Neusser concerned intramolecular interaction between the SI and T I states of benzene, and the second part referred to the high Rydberg spectrum. Each member of this singlet series must have a corresponding triplet, such that

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447

the singlet-triplet energy separation is predicted on general grounds to decrease as l/n3. Is there any possibility of observing intersystem crossing at relatively low n values? If so, variation of n would provide a method for tuning through the vibrational manifold of the triplet. H. 3. Neusser: In this regard, I should say that, up to now, the assignment of Rydberg triplet states has not been possible. In our highresolution spectra only Rydberg series with a small quantum defect have been observed. It appears to be questionable whether triplet series become visible through spin-orbit coupling, which is expected to be small in a molecule like benzene. Furthermore, we expect that intersystem crossing plays a minor role in the nonradiative relaxation of high Rydberg states in benzene.

Advances In Chemical Physics, Volume101 Edited by I. Prigogine, Stuart A. Rice Copyright © 1997 by John Wiley & Sons, Inc.

GENERAL DISCUSSION ON INTRAMOLECULAR DYNAMICS Chainnun:E. Pollak

V. S. Letokhov: Let me make two comments. My first comment is about terminology and the second about the possibility to laser control the intramolecular vibrational distribution (IVR) rate. 1. Prof. H. Neusser introduced the term “coherent ionization dip.” Talking about the coherence in an ionization process we should distinguish two different schemes. One scheme is based on two-frequency coherent control of a two-level A scheme (see Fig. l), which was discussed by Prof. H.Neusser.

Here coherence occurs only for three low-lying quantum states. The 449

450

GENERAL DISCUSSION

effect of blocking of the population of excited level 3 is related to a well-known effect in laser optical physics: “coherent population trapping.” Two coherent fields of frequencies w 1 and w 2 create a coherent superposition of atomic states 1 and 2. Under proper conditions the atom in such a superposition state cannot absorb radiation and will stay in the low-lying states. The ionization channel has no coherence. Ionization serves as a probing of the population of excited state 3 only. Another scheme exploits rhe coherence in ionization. Let us consider the following scheme (see Fig. 2).

Figure 2.

Suppose we have two close intermediate states, 2 and 3. In a two-€requency (a1 , w2) laser field the atom has two nondistinguishable channels of photoionization. Depending on the phase relationship, the quantum interference of the two channels can be “constructive” or “destructive.” In the latter case, the yield of photoionization can be greatly reduced. This effect is related to the well-known “Fano resonances” in atomic photoionization spectra. Also the interference will generate

P

VI c

Nonresonant modes

Stochastitation onset

I

\ ?'

1

I

1

-

Broadening 7 of absorption band

Frequency shift 6 due to intermode anharmonicity

001

IR Absorption spectrum

iI

I1

'

\ I

I

0

Frequency shift due to intramode anharmonicity

0

j!+l

$

Figure 3. Model of IR MP E/D of plyatomic molecules.

Resonant mode

0-

Resonant 4= modeselective interaction with discrete vibrational-rotational transit ions

*--->

*----+

Ferrni Range resonances, of high-density transition to vibrational stochastization

Quasi-resonant noncoherent mode-nonselective interaction with systems of coupled vibrational modes

Dissociation limit

Range of transient vibrational overexcitation

Dissociation of weakest bond

452

GENERAL DISCUSSION

12

2

13,800 20,700 Vibrational energy, E (crn-l) Figure 4. Theoretical dependence of rate of IVR as a function of vibrational 6900

energy of model molecule CFC12Br with given anharmonicity constant Xi,k (from Ref. 1).

quantum beats of the ionization probability, with the frequency of the energy splitting of levels 2 and 3. 2. Let me raise the following question: Can we control or modify the IVR rate by intense femtosecond laser pulses? For the laser coherent control of a chemical reaction we need ultrashort laser pulses with duration T~ << T 2 , T ~ v R where , T2 is the phase relaxation time and 71VR is the time of intramolecular vibrational relaxation. It would be nice to extend 7 1 ~Let ~ us . consider the case of IR multiple-photon excitation and dissociation (IR MP E/D) as applied to polyatomic molecules (Fig. 3). The molecule in a low-lying level interacts coherently with a strong resonant IR field. At a certain level of vibrational energy the density of Fermi resonances of modes coherently driven with the rest of vibrational modes becomes very large. As a result the molecule “penetrates” into the vibrational quasi-continuum (“vibrational heat bath”) where noncoherent vibrational heating by strong IR field occurs. The process of vibrational stochastization has a certain threshold (E,), as you can see from the results of theoretical considerations [ 11 (Fig. 4). It will be quite interesting to control the position of E, by an intense IR

INTRAMOLECULAR DYNAMICS

453

field. Perhaps the strong Rabi splitting of vibrational levels can modify the Fermi resonances and induce some “regularization” of vibrational levels. It is an interesting subject for future research. 1. A. A. Stuchebrukhov, M. V. Kuzmin, V. N. Bagratashvili, and V. S. Letokhov, Chem. Phys. 107,429 (1986).

M. Quack: In the discussion by Prof. Letokhov, the question was raised to what extent IVR [l] can be “controlled or influenced by laser radiation, in particular in relation to changing the mechanism of IR multiphoton excitation in its various regimes. I should like to point out that this “laser field control” is in principle contained in the formulation of the theory of multiphoton excitation in terms of “spectroscopic states” [2]. Whereas the consequences of the general treatment are not always immediately transparent, a simplified mechanistic picture arises by considering the switching over from the “case C” to “case B” of multiphoton excitation as a function of level density, intramolecular couplings, and, in this context important, laser field intensity as a function of time [24]. While this exists already as a quantitative theory, making more detailed use of this approach depends heavily upon our detailed understanding of intramolecular anharmonic and rovibronic couplings. This problem can be well illustrated with the example of CF31, also mentioned by Vladilen Letokhov. We have been working for more than a decade to understand the spectrum of resonance structures related to the COz laser pumped v I (CF3 symmetric stretching) fundamental [MI.Beyond the easily visible anharmonic resonances of v I with 2v5 and v 2 + v3, mentioned also by V. S. Letokhov, we have identified further resonances with u3 + 3V6, also 4V6 and so on. Our present understanding suggests that all Zow-frequency vibrational modes are already heavily connected by rovibrational resonances in the fundamental range V I of CF31. The situation is very complex indeed [MI.While I believe that radiative control of intramolecular rovibrational redistribution will be an important topic, a good understanding of the underlying high-resolution spectra is crucial. Some progress has been made using advanced experimental and theoretical techniques (for reviews see Refs. 7 and 8), but obviously more work is needed. 1. M. Quack and

W.Kutzelnigg, Ber. Bunsenges. Physik Chem. 99,231 (1995).

2. M. Quack, J. Chem. Phys. 69, 1282 (1978). 3. M. Quack, Adv. Chem. Phys. 50, 395 (1982).

4. M. Quack, Infrared Phys. Technol. 36, 365 (1995). 5. H. Burger, K. Burczyk, H. Hollenstein, and M. Quack, Mol. Phys. 55, 255 (1985). 6. H. Hollenstein, M. Quack, and E. Richard, Chem. Phys. k t t . 222, 176 (1994).

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7. M. Quack, Ann. Rev. Phys. Chem. 41, 839 (1990). 8. M. Quack, J. Mol. Struct. 292, 171 (1993); ibid., 347, 245 (1995).

V. S. Letokhov: My answer to Prof. Quack is that it is indeed difficult to predict theoretically the effect of intense femtosecond IR pulses on the IVR rate of polyatomic molecules, which is important for the transfer of vibrationally excited molecules from low-lying states to the vibrational quasi-continuum. We are developing the relevant theoretical mechanisms of IR MP E/D of polyatomics since the discovery of this effect for isotopic molecules BC13 and SF6 in 1974-1975. I hope that it will become more realistic to study experimentally the influence of intense IR pulses on IVR due to the great progress of femtosecond laser technology. R. A. Marcus: It might be interesting to apply the idea of Prof. Letokhov of using intense short lasers to study the time evolution of the acetylenic CH excitation in the molecules (CH3)3, SiCECH, and (CH3)3CC=CH, whose high-resolution IR spectrum in the CH region was studied by Scoles and Lehmann and indicated considerable localization: You recall that for the Si case this IR line was very narrow, in spite of the greater density of vibrational states of the Si compound. The reason, Stuchebrukhov and I found, was due to the sparsity of nearresonant zero-order vibrational states in the first few tiers (i.e., nearly resonant with the excited CH vibrational state) of anharmonically coupled zero-order vibrational states in the Si compound, in contrast with the situation for the C compound. This tier structure for “superexchange” energy transfer to the background dark states is described in Ref. 1. 1. A. A. Stuchebrukhov, A. Mehta, and R. A. Marcus, J. Phys. Chem. 97,12491 (1993); A. A. Stuchebrukhov and R. A. Marcus, J. Chem. Phys. 98, 8443 (1993).

M. Quack: I think I can give an answer to the question raised by Rudy Marcus in relation to the problems mentioned by V. S. Letokhov. Briefly the question is: What is the dynamics of the acetylenic CH stretching motion in substituted polyatomic acetylene molecules of the type (XY3)3 C-CSC-H (or generally R-CEC-H) under strong (femtosecond) coherent IR excitation and considering the coupling of the CH vibration with the frame modes. As we have shown some time ago [l] and as has been confirmed in much detail more recently [2], the coupling of CH stretching to the frame in acetylenes is relatively weak. With coherent femtosecond IR excitation one thus obtains relatively simple one-dimensional anharmonic oscillator motion, and wavepacket results for this case have been published [3].

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The IVR becomes important on the 10-ps to I-ns time scale, according to current wisdom. For some molecules our knowledge is sufficient to consider this an experimental (spectroscopic) resuk rather than a prediction [l]. 1. K. von Puttkamer, H. R. Diibal, and M. Quack, Chem. Phys. Lett. 95, 358 (1983); Faraday Disc. Chem. SOC. 75, 197 (1983); M. Quack, Jerusalem Symp. 24, 47 (1991); Mode Selecrive Chemistry, J. Jortner, R. D. Levine, and B. Pullman, Eds., D. Reidel, Dordrecht. 2. K. Lehmann, G. Scoles, and B. H. Pate, Annu. Rev. Phys. Chem. 45,241 (1994). 3. M. Quack and J. Stohner, J. Phys. Chem. 97, 12574 (1993).

R. A. Marcus:

Then it would be good to do the experiment. L. Woste: Infrared multiphoton excitation schemes have theoretically and experimentally well been studied over the past 15 years. If now we repeat this at femtosecond time scales and intensities, should not potential energy deformations, as proposed by Prof. Manz, be considered, or are they inherently incorporated into the consideration? V. S. Letokhov: In the case of very intense IR fields we should take into consideration the effects of both (1) rearrangement of quantum vibrational levels (e.g., Rabi splittings) and (2) distortion of potential molecular curves. But with such strong IR femtosecond pulses we perhaps cannot ignore the excited electronic states. I believe that a simple molecule can be excited to some higher vibrational levels and subsequently can be involved in a multiphoton jump to excited electronic states. As a result, it will be difficult to observe the effect of distortion of molecular potential curves.

M. Quack 1. In relation to the questions raised by Prof. Woste on the “timedependent, moving effective potentials” shown by Prof. Manz, I would like to add a clarification that might help to avoid misunderstanding. The description with these time-dependent potentials in a laser field shown by Prof. Manz in his movie is very imaginative, beautiful, and perhaps didactically useful. It is, however, not necessary to use such a description. One can do the full dynamical treatment in the laser field, for instance on the basis of “spectroscopic states” or “molecular eigenstates,” without any reference to effective moving potentials (I assume in fact that the calculations by Prof. Manz were done this way and the moving potentials were only calculated for illustration). On real problems in IR multiphoton excitation, there are even some advantages in using the basis of “spectroscopic states” [1-31. 2. Concerning the question of Prof. Letokhov on the high intensi-

456

GENERAL DISCUSSION

ties used in the simulations causing possible problems in experiments, I might point out that in 1979 I made a simple model prediction that ionization at high intensities will always become the fastest process (compared to dissociation or isomerization reaction) [4,5]. In this early work we considered the possible mechanisms of vibrational preionization, perhaps field assisted, as well as direct electronic excitation and field ionization [4, 51. Infrared laser ionization with C02 lasers was recently shown to occur in diatomic molecules by Dietrich and Corkum [6], and Professor Letokhov has right now informed me about previous work in his group on this problem (experiments in the 1980s). We have also some evidence from recent experiments in our group that ionization with C0.r lasers may occur more easily than usually anticipated. 1 . M. Quack, J. Chem. Phys. 69, 1282 (1978). 2. M. Quack, Adv. Chem. Phys. 50, 395 (1982). 3. M. Quack, Infrared Phys. 29, 441 (1989). 4. M. Quack, Ber. Bunsenges. Physik. Chem. 83,757 (1979). 5. M. Quack, Infrared Phys. Technol. 36, 365 (1995). 6. P. Dietrich and P. Corkum, J. Chem. Phys. W,3187 (1992).

J. ’ h e : I would like to comment on the role of the solvent in the photoisomerization of trans-stilbene, as discussed by Prof. Marcus. From our extensive studies in series of nonpolar and polar solvents in compressed gases and in the liquid phase, a very detailed picture arises; we now can distinguish specifically different types of “solvation”: first, there is strong interaction with polar solvents; second, we can distinguish two kinds of interaction with nonpolar solvents, one site specific of only mildly polarizable small alkanes that possibly can squeeze in between the two phenyl groups in stilbene and one non-site specific of more polarizable large alkanes that can only solvate around the outer periphery of stilbene. These different types of “solvation” result in characteristically different solvent effects on the kinetics. E. Pollak The RRKM theory and Kramers theory and its later generalizations by Grote, Hynes, and other are two sides of the same coin. In the spatial diffusion limit, one can show that Kramers’s rate expression is identical in form to the RRKM expression, that is, a ratio of equilibrium unidirectional flux and density of reactants. The difficult problem in the application of RRKM theory to the stilbene molecule with a few attached benzenes is whether the equilibration of energy occurs fully on the time scale of the isomerization. One should also

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keep in mind that addition of cluster molecules changes potentials of mean force, not only fluctuational properties. Both Prof. Marcus’s talk and the experimental results of Troe, Zewail, and Hamaguchi should stimulate the theorists to undertake a thorough molecular dynamics study of this very interesting problem. R. A. Marcus: With regard to a comment made during this discussion, it is certainly true that in the limit of large systems the microcanonical expression gives the same result as a canonical one. However, a typical molecule is not large enough. Landau’s little known “thermodynamic” theory of unimolecular reactions [11used that approximation. Qualitatively, it was appropriate, but quantitatively it can make a huge error for typical molecules. For example, consider a molecule with an energy barrier for reaction Eo,an energy E, and an effective number of classical harmonic oscillators s. The RRKM expression for the energy-dependent (microcanonical) unimolecular rate constant is A[(E - E o ) / E ] S - ’ , where A is some constant. The corresponding canonical expression, obtained by taking the limit as s - 1 -+ at fixed E / ( s - 1) is A exp(-E/kT), where ‘kT’ = E / ( s - 1). Now, consider a system with EO = 5 kcal/mol, (electronically excited t-stilbene has an even smaller Eo! ) s - 1 = 10 and k T = 300 K, that is, E = 6 kcal/mol. Then, the ratio of the canonical to the microcanonical expressions is 1.5 x lo4!

-

I . L. Landau, Phys. 2. Sowjet. 10, 67 (1936).

R. D. Levine: As emphasized by the speakers on femtosecond pumping schemes, an important point is that the initial excitation is localized within the Franck-Condon regime. The question is whether the sheer localization can be used to advantage to induce laser-selective chemistry [K.L. Kompa and R. D. Levine, Acc. Chem. Res. 27, 91 (1994)l. As we understand better the topography of potential-energy surfaces for polyatomic molecules, it may be possible to launch the system with such initial conditions that it will, of its own accord, proceed to cross a particular transition state and so exit toward a particular set of products. G. R. Fleming: In relation to the comment by Prof. Levine, Jonas has proposed that ultrafast spectroscopy can be used to study thermal reactions by placing a wavepacket on the ground-state surface at a specific location and with a specific momentum. In think such experiments are quite feasible. J. Manz: I would like to suggest that one should go even beyond Prof. R. D. Levine’s suggestion to put the nuclear wavepacket on spe-

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cific domains of the potential-energy surface, specifically close to the transition state, and then let it find its own way toward new molecular configurations, in an exploratory “scout”-type manner. Instead, I would like to support the vision of Prof. S. A. Rice; that is, one should try and use the options of laser control in order to “impose one’s own will” on the molecule’s time evolution (S. A. Rice, “Perspectives on the Control of Quantum Many Body Dynamics: Application to Chemical Reactions,” this volume; see also Ref. 1). For example, one may use a laser pulse such that the molecular wavepacket is put on the transitionstate region with the proper momenta so that it is driven selectively toward one specific product. Alternatively, one may extend Neumark’s photoelectron detachment spectroscopy of the transition state [2] to laser control of product branching ratios. Essentially, a laser pulse may be used to prepare one or several near-degenerate selective resonance states close to the transition state, such that the resonance will evolve toward the desired product channel, The control of chemical reactions by preparation of selective resonances at the transition state has been predicted first in Ref. 3. This method allows one to make products that would not be obtained easily by traditional thennochemistry or chemical kinetics. 1. R. Kosloff, S . A. Rice, P. Gaspard, S. Tersigni, and D. J. Tannor, Chem. Phys. 139,

201 (1989). 2. D. M. Neumark, Acc. Chem. Res. 26,33 (1993). 3. R. H. Bisseling, P. L. Gertitschke, R. Kosloff, and J. Manz, J. Chem. Phys. 88,6191 (1988).

R. D. Levine: In the context discussed by Prof. Neusser, I want to comment on the relevance of the experiments on the high Rydberg states to photoselective excitation of molecules. The point is that in the energy range of the experiment the density of states is extremely high. The optical excitation of a high Rydberg state prepares a state in which almost all the energy is in the orbital motion of the electron, which moves in a Bohr-Sommerfeld orbit of high n about a nearly groundstate core. Such a state is typically directly coupled to the continuum which corresponds to an ionized electron and a colder core. It is also coupled to very many other isoenergetic states in which the core has gained more energy at the expense of the electron. In other words, there is a very large bound phase space while the experiment prepares states that are at the very bottleneck to the continuum (cf. Fig. 7 in my chapter in this volume, “Separation of Time Scales in the Dynamics of High Molecular Rydberg States”). The competition between the two alternative fates of the electron is of interest in connection with the

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question of whether one can optically prepare states that by virtue of the preparation will go one way rather than the other. L. Wkte: Professor Neusser, with respect to your slowly orbiting Rydberg electron, do you envisage the possibility of replacing it by a negatively charged (atomic) ion, for example C1-, so the Rydberg particle becomes an ionically bonded planetary-type particle with an incredibly long bond distance? (See the comment by J. Manz in the General Discussion on Molecular Rydberg States and ZEKE Spectroscopy: Part I). H.J. Neusser: In reply to Prof. Woste let me mention that a situation similar to the high Rydberg states I discussed has been proposed for a dipolar bound electron in molecular anions. Here large distances of the electron and small binding energies are expected. First experimental indications were found. Ion pair states represent another interesting example. M. Quack: The situation Prof. Woste refers to corresponds to excited rotational states of highly excited ion pair (electronic) states, which are well known for diatomic molecules, for example. T.P. Softley: With regard to the question by Prof. Woste, I would like to add that ion pair states of many small molecules are known experimentally. For example H:! has been excited to H’ . .. H- states very close to the dissociation limit by S. Pratt and co-workers [ 11, and then these states are dissociated by small pulsed electric fields. The difficulty in all such experiments is in accessing the very large internuclear distances necessary to get close to the “transition state.”

1. S. T.Pratt, E. F. McCormack, J. L. Dehmer, and P. M. Dehmer, Phys. Rev. Let?. 68, 584 (1992).

T. Kobayashi: In the same context, I would like to point out that, in polymer systems with degenerate ground states such as truns-polyacetylene, it may occur that a fraction of the excited species (exciton) is split into a soliton and an antisoliton with opposite charges. These can be bound by Coulombic interaction and also by steric hindrance.

REGULAR AND IRREGULAR FEATURES IN UNIMOLECULAR SPECTRA AND DYNAMICS

INTRAMOLECULAR DYNAMICS IN THE FREQUENCY DOMAIN R. W.FIELD*, J. P. O'BRIEN, M.P. JACOBSON, S. A. B. SOLINA, W. F. POLIKt, AND H. ISHIKAWAS Department of Chemistry Massachusetts Institute of Technology Cambridge, Massachusetts

CONTENTS I. Introduction 11. Dispersed Fluorescence Spectrum of Acetylene 111. From Spectrum to Potential to Dynamics IV. A Change in Resonance Structure V. Summary References

I. INTRODUCTION Three topics will be discussed here. The first topic is a description of dispersed fluorescence (DF) spectra of acetylene, recorded via the 'Au-k'q electronic transition. These spectra systematically sample regions of the SO potential-energy surface that are far from equilibrium geometry. By basing an analysis of these spectra on feature states rather than on eigenstates, the systematic sampling of the potential is local region by local region as opposed *Report presented by R. W Field b e s e n t address: Dept. of Chemistry, Hope College, Holland, MI 49423. $Present address: Dept. of Chemistry, Graduate School of Science, Tohoku University, Aoba-ku, Sendi: 980-77, Japan. Advances in Chemical Physics, Volume 101: Chemical Reactions and Their Control on the Femtosecond 'lime Scale, XXth Solvay Conference on Chemistry, Edited by Pierre Gaspard, Irene Burghardt, I. Prigogine, and Stuart A. Rice. ISBN 0-471-18048-3 0 1997 John Wiley & Sons, Inc.

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to being simultaneously sensitive to the global potential. The sampling is local because the spectra are low resolution; thus the dynamics is restricted to the early-time evolution of a p e ~ e c t l yknown initial state. The second topic is a simplified scheme for going from a spectrum, Z(w), to a potential surface, V(Q),to dynamics, q ( Q , t ) . The essential feature of this scheme is embodied i‘ii two complemenm- reduced-dimension pictures: dynamics in configuration space and dynamics in state space. Long progressions of feature states in the two Franck-Condon active vibrational modes (CC stretch and trans-bend) contain information about wavepacket dynamics in a two dimensional conjiguration space. Each feature state actually corresponds to a polyad, which is specified by three approximately conserved vibrational quantum numbers (the polyad quantum numbers nstretch,nresonance, and [total, Ins,nres, El), and every symmetry accessible polyad is initially “illuminated” by exactZy one a priori known Franck-Condon “bright” state. The finer structure within each feature state corresponds to the dynamics of the Franck-Condon bright state within a four-dimensional state space. This dynamics in state space is controlled by the set of all known anharmonic resonances. The state space is four dimensional because, of the seven vibrational degrees of freedom of a linear four-atom molecule, three are described by approximately conserved constants of motion (the polyad quantum numbers); thus 7 - 3 = 4. The spectra being discussed here are CzH2A-2 DF spectra, and the initial state created on the SOsurface corresponds to a perfectly known vibrational eigenstate of the 2 state (S1) surface transferred onto the 2 state (SO)surface. However, any conceivable initial state could be expressed as a superposition of independently evolving polyads, each initially illuminated via one or more a priori known bright zero-order states. In order to extract dynamical information from a low-resolution (DF) spectrum, it is important to have a procedure for separating overlapping polyads. By comparing two DF spectra recorded from different intermediate states, spectrally overlapping polyads may be “unzipped” from each other. This unzipping pennits a systematic survey of the rates and initial directions of energy flow in state space. In particular, IVR patterns in successive members of a progression of polyads may be compared against vibrational quantum-number-scaling predictions of the spectroscopic effective Hamiltonian model, H$:,n,es,fj,for each polyad. This leads to the third and final main topic, the use of Heffmodel u-scaling predictions to detect the changes in the w 1 :0 2 :...~3~ - 6 resonance structure that occur near chemically important topographic features of a potential surface. Such features include an isomerization saddle point or a sharp bend in the minimum-energy isomerization path. The key feature of this matrix model is

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that the off-diagonal elements obey harmonic oscillator vibrational selection and scaling rules, which allow quantitative predictions of dynamics for any Rns, polyad polyad and any initially prepared state. The u scaling of the model provides a description of an otherwiseindescribably complex pattern in t). The change the spectrum or in any dynamical quantity calculablefrom 4(Q, in resonance structure would be manifest as a turning on of aquantifiable discrepancy between the polyad predictions and actual observations. Many of the topics discussed in this chapter have been dealt with in detail elsewhere [ 1-51, in particular, Jonas et al., “Intramolecular Vibrational Relaxation in the SEP Spectrum of Acetylene” [l]; Field et al., “Pure Sequence Vibrational Spectra of Small Polyatomic Molecules: Feature State Assignments, SequentialDynamics, Energy and Time Scaling, and the Bag-of-Atoms Limit” [2]; Solina et al., “The Acetylene SO Surface: From Dispersed Fluorescence Spectra to Polyads to Dynamics” [3]; Solina et al., “Dispersed Fluorescence Spectra of Acetylene Excited via the A ‘A, - % E; Origin Band: Recognition of Polyads and Test of the Multiresonant Effective Hamiltonian Model for the %-State” [4];and Ishikawa et al., “Stimulated Emission Pumping of HCP Near the Isomerization Barrier: E V ~ SI25,3 15cm-’” [5]. In the interest of brevity, the reader is referred to those papers for additional figures and detailed description, discussion, and analysis.

q:,

’

11. DISPERSED FLXJORESCENCE SPECTRUM OF ACETYLENE Stimulated emission pumping (SEP) and DF spectra provide respectively high- and low-resolution samples of the structure and dynamics on the electronic ground-state potential-energy surface, especially in vibrationally highly excited regions of the Franck-Condon active normal modes. For acetylene, the Franck-Condon active normal modes are CC stretch (Y;’) and trans-bend (Y:) owing to the large changes in internal coordinates AWC = 0.18 A and A&CH = 60”,between the trans-bent A ‘A, (a* ?i) SIstate and the linear 2 ‘qSO state. The DF spectrum of C2H2, excited via the KL = 1 levels of the origin (0”), 2u; (Y; is the trans-bend in the Si state), and 3 4 intermediate vibrational states, contains long progressions in overtones and I”, u”l:: 1” ) combinations of the Franck-Condon bright states (uy, vy, vy ,u”: (0, n, 0, 2moy2,Oo)o*2. These zero-order bright states (ZOBSs) are not eigenstates, because they are coupled to other zero-order states (“dark” states) by several anharmonic resonances known from the analysis of infrared [6, 71, Raman, and high overtone spectra [8].Consequently, each ZOBS appears in the spectrum as a broadened and/or fractionated “feature state” rather than a single sharp vibrational band. As the result of a fortuitous coincidence for the acetylene A-2 system, each feature state in the DF spectrum accessed via

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cold bands of the form VGK; [V is the symbol for the trans-bending normal mode, which has different mode number in the S1 (mode 3) vs. So (mode 4) states and K is the projection of angular momentum along the near-symmetry axis (K = Ki in Sl,K = 1” in SO)]belongs to a different polyad. More precisely, each g symmetry, I = 0+,2 polyad is illuminated by exactly one a priori known ZOBS. What is a polyad? A polyad is a subset of the zero-order states within a specifiable region of E V I (typically ~ a few hundred reciprocal centimeters) that are strongly coupled by anharmonic resonances to each other and negligibly coupled to all other nearby zero-order states. If approximate constants of motion of the exact vibration-rotation Hamiltonian exist, then the exact H can be (approximately) block diagonalized. Each subblock of H corresponds to one polyad and is labeled by a set of polyud quantum numbers. For the C2H2 SO state, a procedure proposed by Kellman [9, 101 identifies the three polyad quantum numbers %retch = vs = VI Uresonance = U ,

+ u2 + v3

= 5LJl + 3U2

ltotal= 1 = l4 + 15

+ 5U3 + U4 + LJ5

as the three linearly independent (but not orthogonal) vectors in seven-dimensional quantum number space that are orthogonal to all of the directions specified by the resonance vectors (which summarizethe vibrational selection rules for off-diagonal matrix elements) correspondingto all known anharmonic resonances. (See Table I for a summary of all h o w n anharmonic resonances for the SO skate of C2H2.) For example, the stretch Darling-Dennison resonance, caused by the kll33QiQ; anharmonicity term, has off-diagonalmatrix elements (in the harmonic oscillator product basis set) AUI= -Au3 = f2,Au2 = Auq = Au5 = A14 = A15 = 0, which correspond to the resonance vector (Avl,A v ~Au3, . Av4. Al4, A u ~AZ5) , = (*2,0, T2,0,0,0,0). Since the factorization of H into HF~~,n,s,rl effective Hamiltonian blocks is approximate,each polyad H$:, nres, will describe the low-resolution spectrum of each polyad and the early-time intrapolyad dynamics. Since the acetylene DF spectrumhas the special property that each g, 1 = 0’ , 2 polyad is illuminated by exactly one known ZOBS, for example, (0, n, 0, 2m092,Oo)o*2,the polyad is specified by the ZOBS quantum numbers Ins,nres, 21 = In, 2m + 3n, 0 or 21

or, vice versa, the ZOBS quantum numbers are specified by the polyad quantum numbers (0, n,, 0, nres - 3ns, Oo)oor2. The eigenvalues of H$:,nes,rlgive the locations of all energy levels within the polyad or feature state, and the

a

8

TABLE I

AUI= -b3= 2

’

Stretch-Only Interactions

c

i= I

in Signed4 Basis

Scaling Matrix Element. H;,,

Reduced Coupling Constants (cm- ) for acetylene42

Au3 = - A w = -Au4 = -Au5 = 1, Al4 = -A15 = f1 AUI = - A u ~= - 2 Au4 = 1 Aul = - A m = -:As = 1 AUl = -Au3 = Au4 = -AQ = 1, A14 = -A15 = f l

A u =~ - A u ~= 2 Au5 = -Au4 = 2, Al4 = -A15 = 72 A14 = -A15 = T2

eff

Note: Matrix elements are given in the signed-l basis. Since the laser prepares and probes acetylene states with a defined parity, an orthogonal transformation is performed on the signed-1-basis states to transform them to a parity basis. See Refs. 1, 3, 6, and 8.

DD stretch resonance

1,244 resonance 1,255 resonance 14.35 resonance

3,245 resonance

vibrational-l resonance

DD bend resonance I DD bend resonance I1

A

Anharmonic Resonances used in H , i -eff H;j = (u/ IH I(u; + Au;)“”

Scalable Resonances in Global Treatment of IR, Raman, Overtone, and SEP Data for Standard Normal-Mode Hamiltonian

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fractional ZOBS character in each eigenvector gives the relative intensity of each single-eigenstatecomponent within the polyad (i.e., a complete description of the fractionation pattern observable in the frequency-domain spectrum). The eigenvalues and eigenstates give a complete picture of the earlytime intrapolyad dynamics in state space that would be observed in the time domain i f the ZOBS (or any other zero-order state) were prepared at t = 0 by a sufJiciently short S1- So DUMP pulse. The most serious obstacle to extracting useful information from the lowresolution DF spectrum is the progressively more severe spectral overlap ~ This is due to two effects: between separate polyad features as E V Iincreases. the density of Franck-Condon bright ZOBSs increases (approximately as E V I B [ ~ 2 ~ 4 ] - 1and / 2 ) the width of each polyad increases (owing to the = uq scaling of the dominant anharmonic resonance experienced by the ZOBS). The worsening overlap between polyads, however, is not an insurmountable obstacle. We have developed an “unzipping” procedure whereby the fractionated patterns associated with different polyads can be disentangled from each other. This unzipping procedure is based on the fact that each acetylene polyad is illuminated by exactly one ZOBS. Thus, if the ZOBS is accessible from different intermediate vibrational levels (e.g., Oo, 2 4 , and 3v;), then the pattern of splittings and relative intensities within each polyad is exactly reproduced, but the relative intensities of different polyads (controlled by known Franck-Condon factors) will change by large and predictable amounts. Thus, by comparing pairs of low-resolution DF spectra recorded via different intermediate states, the resolvable subfeaturescan be apportioned into their respective polyads. For the energy region E V ~5B 16,OOOcm-’, this unzipping process can be done by eye. Recently we have developed a statistically rigorous “extended cross-con-elation” (XCC) procedure that has proven capable of unzipping polyad subfeatures with intensities comparable to baseline noise and, to a limited extent, unzipping individual overlapped lines [111. Two of the valuable consequencesof being able to unzip the C2H2 DF spectrum into separate polyads are (i) the ability to measure “deperturbed” ZOBS energy levels and Franck-Condon factors and (ii) the ability to survey and explain trends in IVR as the u2 and u4 quantum numbers are increased. This is the kind of information that can be used to extract local and systematicinformation about V(Q) from the frequency-domain spectrum,Z(o), without the necessity of assignrng and fitting eigenstates against a global V(Q). -

HI. FROM SPECTRUM TO POTENTIAL TO DYNAMICS A rotation-vibration eigenstate is typically sensitive to the global potentialenergy surface. In time-dependent language, in order to resolve individual

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eigenstates, it is necessary to allow the initially prepared wavepacket (for a DF spectrum this is the vibrational wave function of the intermediate S1 state, which is not an eigenstate on the SOsurface) to evolve for a time longer than 1 / P V t B . During such long-time evolution, the wavepacket explores much of the energy and symmetry accessible region of V(Q). On the other hand, the low-resolution DF spectrum samples severely resscted, a priori known, yet systematically selectable regions of the V ( Q ) . Low resolution means early time, and the system does not have time t6explore V ( Q ) far from its t = 0 location in the Franck-Condon dark normal coordinatesyThe spectrum is sensitive only to the most important anharmonic effects. Extracting information from an eigenstate spectrum is an all-or-nothing affair. Every observed eigenstate (a small fraction of the total number of eigenstates in the explored spectral region) must be correctly related to the corresponding eigenstate of a model H matrix. When all of the parameters in H are at their “true” values, it is not difficult to make this correspondence between observed and calculated eigenstates. However, when the model parameters are not yet refined to their “true” values, a huge number of eigenstate correspondence decisions must be made correctly, or else any attempt at least-squares adjustment of the model parameters will fail. The difficulty of making these correct correspondences can only be appreciated by someone who has attempted to fit a perturbed spectrum! This is “the least-squares fitting bottleneck.” The use of relative intensities computed from the eigenvectors of H can be very helpful in making the correct model c)observed eigenstate correspondences, but at high p v l ~the “leastsquares fitting bottleneck” is usually insurmountable. This hypersensitivity of the eigenstates to details of the potential surface should not be surprising, especially in view of the chaotic nature of classical trajectories at chemically significant E v ~ eon most polyatomic molecular V(Q)’s. Extracting information from a low-resolution DF spectrum is straightforward, logical, and systematic. At lowest resolution, the DF spectrum conveys information about the motion of the center of the initial wavepacket in the configuration_space defined by the Franck-Condon active normal modes. For the A-X transition of CzHz, this restricted subspace is either two or three dimensional, consisting of the CC stretch and the trans-bend (initially in the plane defined by the trans-bent state and, somewhat later, perpendicular to this initially selected plane). The spectrum first fractionates into polyad features. This fractionation reflects the first few partial recurrences of the autocorrelation function (*(?)(\k(O)) as the wavepacket travels primarily along the Franck-Condon active coordinates. This is “dynamics in configuration space.” The spectrum then fractionates into finer subfeatures: the internal structure of independent polyads. This reflects the essentially independent evolution of Franck-Condon bright ZOBSs within non-

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communicating polyads. The intrapolyad ZOBS evolution is “dynamics in state space.” The DF spectrum consists of a series of features, each associated with a single, known ZOBS. The intensity weighted average energy of the lower state of the feature transition is the “deperturbed” energy of the named ZOBS. Fractionation of the Franck-Condon bright state spreads the intensity over many of)the eigenstates, but the center of gravity (intensity-weighted average E V ~ B of the zero-order state (@;@\kp = feature is the zero-order energy EpSp).No multistate deperturbation model is required (nor any of the associated assumptions about resonance mechanisms) to extract the zero-order energy of the Franck-Condon bright state. The shortcomingof this simple relationship between feature state and zero-order energy is that only the energy of the single Franck-Condon bright state in each polyad is determined. Of course, it is possible to obtain the deperturbed energies of different sets of ZOBSs by using a different intermediate state that illuminates a given polyad via a different Franck-Condon bright state. So several zero-order energy levels could be obtained sequentially from the same polyad, and this can be accomplished without any model. Thus progressions of energies for the two or three Franck-Condon active normal coordinates can be obtained or refined from DF spectra, permitting systematic refinement of a family of cuts through the potential-energy surface. The integrated intensity of each feature state (polyad) is proportional to the deperturbed Franck-Condon factor for the known intermediate state-bright state vibrational transition (see Figs. 1 and 2). As long as there is only one Franck-Condon bright state per polyad, and interpolyad interactions can be neglected, the deperturbed Franck-Condon factors can be used to test and refine the Franck-Condon active region of the SoV(Q). Polyatomic molecule Franck-Condon factors have always been problematic, largely due to the difficulty of separating contributions to the intensity of a given eigenstate transition into unknown (and unknowable without a refined model) bright-state amplitudes in each eigenstate and the individual brightstate Franck-Condon overlap integrals. By combining a successful poly ad unzipping procedure with the fact that each polyad in the CzH2 DF spectrum contains a single ZOBS, one obtains long progressions of deperturbed Franck-Condon factors, extending to chemically significant excitation energies. Such Franck-Condon factors provide information complementary to that obtained from the deperturbed bright-state energies. In addition, they provide information about possible vibronic coupling mechanisms whereby transition intensity is borrowed from other electronic states with qualitatively different equilibrium geometries (i.e., neither D..h nor C 2 h ) . The fractionation patterns within each polyad provide a different kind of deperturbed information that is useful for refining the global V(Q). - The frac-

(e)

INTRAMOLECULAR DYNAMICS IN THE W Q U E N C Y DOMAIN

4

6

8 10 12 14 16 Quanta of trans-bend (V,)

v4=4

6

8

18

47 1

20

v,= 4 v,= 3 v,= 2

v,= 1 v,= 0 10

12

I4

16

18

20

Figure 1. Unzipped polyads (bottom) from the C2H2 A 0’ DF spectrum as progressions in has constant LQ (CC stretch) and each column has constant u4. Integrated intensity (deperturbed Franck-Condon factor) for each polyad in the Oo DF spectrum, arranged as progressions in u4 (top). Shading indicates the value of u2 for the progression. u4 (rruns bend). Each row

tionation describes dynamics in a reduced-dimension state space, in contrast to the dynamics in (Q2, Q4) configuration space as described by the deperturbed bright-state energies and Franck-Condonfactors. By “dynamics in state space” we refer to the time evolution of a ZOBS, which is restricted by the specified polyad quantum numbers [n,,rims, I ] , where *(O) is the pre-

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cc

Stretch

4

6

8 10 12 14 16 Quanta of trans-bend (V,)

18

20

v2=4 v2=3

v2=2

v,= 1 v2=0 v4= 4

6

8

10

12

14

16

18

20

Figure 2. Same as Fig. 1, except spectra originate from the 2"; level. Note the nodes in the progressions, originating from the two internal nodes in the 2v; state, near u4 = 12, 18.

cisely known ZOBS. Since a linear four-atom molecule has seven vibrational degrees of freedom, and there are three approximately conserved polyad quantum numbers on the C2H2S0 surface, the state space accessed by the ZOBS has 7 - 3 = 4 generalized dimensions. What we see in a low-resolution DF spectrum is the early-time dynamics of the ZOBS. In other words, what region of V (-Q ) does the ZOBS explore

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first and how quickly does it get there? Thus, by systematically selecting a variety of ZOBSs, one can map out portions of the non-Franck-Condon active normal coordinate regions of V(Q). The fractionation patterns exhibited 6y successive members of a progression of polyads (along u2, CC stretch, or along u4, trans-bend) provide a surveyor’s map of IVR.One can look at the IVR trends and see whether the multiresonance model expressed in the Hf,:,,res,llpolyads provides a qualitative or quantitative representation of the fractionation patterns. The dynamics of even a four-atom molecule is so complicated that, unless one knows what to look for, one can neither identify nor “explain” trends in the dynamics versus u2 or u4 or EVIB.Moreover, by defining the pattern of the IVR polyad and how this pattern should scale with u2, u4, or EVIB,the l$:,nEs,,l model may make it possible to detect a disruption of the pattern. Such disruptions could be due to a change in the resonance structure of the exact H near some chemically interesting topographic feature of the V(Q), - such as an isomerization saddle point. Figure 3 shows that fractionation increases with u4 at constant u2, whereas it decreases with 9 at constant u4. The reason for this is displayed in Figs. 4-6, where the “mixing angle” H;/(Ep - E;) is plotted versus u4 at constant u2 and versus u2 at constant u4. The polyad model contains explicit u2, u4 scalings of the anharmonic coupling matrix elements, H’., and anharmonic detunings ( x 2 2 , x ~ , x 2 4 ) of the energy denominator AE$= EP - EP. The IVR increases as u4 increases because AEE 0 at %nd = u4 + us = 16. The IVR decreases along u2 because the X 2 4 and X25 anharmonicity constants cause the dominant first-tier bend Darling-Dennison resonance to detune as u2 increases:

-

where IAE:.I increases more rapidly than H; as u2 increases. The unzipping procedure reveals the diagnostically significant trends in fractionation widths and patterns illustrated by Figure 3. These trends can lead to qualitative insights into IVR mechanisms and can suggest optimal schemes for external control over intramolecular dynamics. The unzipped polyads can also yield quantitative least-squares refinements of anharmonic coupling constants, from which any dynamical quantity based on may be calculated. Each PolYad q:,nm,ll matrix is based on the same set of anharmonic coupling strengths. Although, as the dimension of each matrix increases rapidly with EVIB,one should not be misled by an illusion of increasing complexity. If one is interested in a particular region of Ev~B, it

*(elf)

qt,nes,Il

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.iii

v,=4

v2= 3 v2= 2 v2= 1 v,= a v4=4

6

8

12

10

14

16

18

20

Figure 3. Unzipped polyads from the Oo DF spectrum, arranged as in Fig. 1. Note that IVR, manifest as fractionation, increases as u4 increases but decreases as u2 increases.

0.01

2 0

5

10

IS

20

25

30

Quanta of trans-bend( V4) +A

-tB

+C

Figure 4. Within a bend Darling-Dennison stack of zero-order levels, Hi, increases and

AE; decreases as u4 increases. The spike in Hij/AE; occurs where the zero-order energies “crash near Z&nd = 16.

INTRAMOLECULAR DYNAMICS IN THE FREQUENCY DOMAIN 5 4.5 4

475

A 1

0

I

2

3

4

Quanta of CC Strctch ( V,)

Figure 5. Within a bend Darling-Dennison stack, H;,increases but AEo. increases more /' rapidly as ~2 increases, resulting in a gradually decreasing mixing angle, Hij/AE$

:I f

.' ..

.... ....

'a

.*

0

....... ".'"

I' I

2

3

4

5

Quanta of CC Stretch ( V,) Figure 6. The entire bend Darling-Dennison stack, which contains the ZOBS, pulls away from the other Darling-Dennison stacks as ~2 increases. The different stacks are connected by the 3, 245 anharmonic resonance. Although the inter-stack Hij increase as ~2 increases, the Hij increase more slowly than AE$ thus the mixing angle decreases slowly as u2 increases.

R. W. FIELD et al.

476

is a simple matter to identify all relevant [n,,nres,I] polyads and to instruct a computer to set up and diagonalize all of the relevant H~,$res,ll matrices. The eigenvalues and eigenvectors of these matrices, Ins. nres. 11, +f". nres, 4

{Ei

1

contain all of the information needed to compute the intrapolyad dynamics associated with any conceivabie initial state [2]. We can choose *(O) to be the ith zero-order state of the [n,,nres,I] polyad, in which case

where q5 and # are respectively zero-order state and eigenstate, and the cij's are mixing coefficients (obtained from the ith row of the U matrix that diagonalizes ~ ~ , n r c s , I Once l). the *(O) state is expressed in terms of eigenstates, it is trivial to express the time dependence of this initial state,

The time-dependent probability of finding the system in the kth zero-order state belonging to the [n,, nres,I] polyad is p ik ( t ) = i(4f"s.nres. ,

where

nres-

( t ) )12

INTRAMOLECULAR DYNAMICS IN THE FREQUENCY DOMAIN

477

Here, )cijl2and ]ckjl2 are, respectively, the fractional character of the zeroorder bright ( i ) and dark (k) states in thejth eigenstate. If is a real matrix, then [ c ~ ~ c $ ' c ; cis~ ~real P ] (because all c's are real), and the sin wj,* term vanishes,

J$i!,n,,,Il

For the special case of i = k, Pii(t) is the survival probability,

Note that, although the survival probability of the ZOBS is derivable from the inverse Fourier transform of the spectrum [ I ( @ ) ] ,here Pii(t)and all other conceivable dynamical quantities are obtained from the parameters in H"ff that are derived from a fit to Z(w). Parameters derived from a fit [ns.rims, 11 to one polyad, illuminated by a particular ZOBS,should adequately describe the dynamics of any polyad illuminated by any ZOBS!Whenever Z(o) has both many wiggles and several approximate constants of motion (e.g., n,, nres,l ) , Z(w) provides far more easily interpretable information about dynamics than any comparably difficult time-domain experiment. However, a little humility may be in order here. Small changes in anharmonicity parameters (especially a change in sign of an off-diagonal matrix element) can result in qualitative changes in some dynamical quantities. Although we have so far detected no evidence of interpolyad interactions in HCCH, these interactions certainly exist. Moreover, although we have insisted on using harmonic oscillator v-scaling relationships for off-diagonal matrix elements, we have included anhnnonic correction terms along the diagonal of our HFi:,n,s, matrices. It is interesting to note that, if one takes only the piece of *(t) associated with one polyad, then the intrapolyad dynamics includes no motion of Q - or f, where

Since the selection rule for nonzero Qiand Pi matrix elements in the harmonic oscillator basis is Av = f l , and since the definition of a polyad is such that all pairs of states differing by only one vibrational quantum number

478

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and

belong to difSerent polyads, all intrapolyad matrix elements of all Qi and all Pi are zero. Thus

, =0

(f>t= 0

for any intrupolyud *(t). This reinforces our earlier statement that the two reduced-dimension pictures, dynamics in the Franck-Condon active configuration space and dynamics in intrapolyad state space, are complementary. The former is appropriately described in terms of real ball-and-spring motions; the latter is probably only describable in terms of abstract motion in state space. Figure 7a illustrates some of the dynamics in the state space of the [n,= 1,nres= 11,Z = 01 polyad illuminated at t = 0 via the (0, 1,0, 8'. ')'0 ZOBS [2].The four panels of this figure depict the survival probability of the ZOBS (upper left), the probability of transfer to the two most important "first-tier'' states [coupled by the bend Darling-Dennison interaction: upper right (0, 1, 0, 6+', 2-2)0 and lower left (0, 1, 0, 6', 2'), and the transfer probability to one of the dynamically most remote dark states in the polyad (1, 0, 0, Oo, 6')' (lower right), which occurs via a minimum of four anharmonic coupling steps (k1,244followed by k44,55 three times). In a sense, the dynamics depicted in Fig. 7a are almost indescribably complex. The complexity reflects the presence of the N(N- 1)/2 beat frequencies that arise from N eigenstates in a polyad. However, several crucial qualitative features are evident. The ZOBS has P(0) = 1 and all of the dark states have P(0) = 0. For the bright state P(t) oscillates about -0.8 with a peak-to-peak amplitude of -0.4 and a shortest dominant period of -200 fs (although the oscillation contains at least two dominant Fourier components). For one of the dark states P(t) oscillates about -0.15 with a peak-to-peak amplitude of -0.30 and a period of -200 fs (also containing at least two Fourier components). Thus (0, 1, 0, 6+', 2-')O is the dominant partner of the ZOBS. Most of the 0.1 oscillatory probability not accounted for by the (0, 1, 0, So, O')', (0, 1, 0, 6+', 2-2)0 pair is found in the (0, 1, 0, 6', 2O)O dark state but with a period of -400 fs. The remote dark state shows a very small, spiky P(t), with a stochastic appearance.

2

time (ps)

2

A

8:

(a)

0

2

t

0

n -.-n

0.5

1.o

1.5

2.0

.- 2.5

3.5

0

n -.-n

0.2

1

1

2

time (ps)

2

time (ps)

1 I

3

3

Figure 7. Survival (PiJ and transfer (Po)probabilities for the C2H2 [I, 1 I, 01 polyad. These Pi,(t) and Pij(t) curves were computed using the eigenvectors and eigenvalues of the H&l,Lo, polyad matrix. The ZOBS is (0,1, 0, 8O,' )'0 and the four panels correspond to (0,1.0, 8', O0)O upper left, the first-tier zero-order states (0,1.0, 6+2,2-*)O upper right and (0,1,0,6O, 2O)O lower left, and a far-edge dark state (1.0.0, Oo,')'6 lower right. The four panels are repeated, at time scales increasing in factor of 10 or 100 steps: ( a )0-4ps; (b)0-40 ps; ( c ) W ps; (4 040ns.

4

f-

-

-

0.2 0.0 0

-

-

g0.4

1

-

8:

5

r,

-

4

9 0.6

3

3

-

time (ps)

-

1

0.8

0

0.2

(0,l ,0,6tL,2'L)

4

4

4

0 P

m 0

u)

09

-

Nln

.-E

0

r

0

480

h

0

'9 9 9

0

9

v Y

48 1

r-zl-

*

0

0

0 0

0

m

n

0.5 N

.-E

.-

0

2

0

0

M

iz

482

INTRAMOLECULAR DYNAMICS IN THE FREQUENCY DOMAIN

483

Figures 7a-d illustrate the intrapolyad dynamics of the [ 1, 11, 01 polyad on several time scales, increasing in steps of 10, 10, and 100 from Fig. 7a (4 ps) to Fig. 7d (40 ns). So much dynamic information is contained in \k(t) that each factor of 10 in time scale of Pi&) and Pij(t) reveals new features. Lacking a trick to accomplish “unzipping” in the time domain, the dynamics uncovered in a time-domain experiment would appear far more compli-

cated. Even if unzipping were possible, it would be necessary to record the

intrapolyad dynamics of several zero-order states to obtain sufficient information to characterize a polyad Heff. Without such a characterization, it would be impossible to make even qualitative predictions of the dynamics in another region of EVIBor of another class of probe-able zero-order state. As an illustration of the EvlB-scaling and u-scaling power of the polyad model, Fig. 8 shows the intrapolyad dynamics in the [ 1,23,0] polyad (which lies at roughly twice the E v l ~for the [ 1, 11, 01 polyad), illuminated via the (0, 1, 0, 20°, O0)O ZOBS. There is a profound qualitative difference between Figs. 7a and 8. In Fig. 8, P ( t ) for the ZOBS decays more rapidly and then oscillates (without any apparent periodicity) with smaller amplitude (0.25 peak to peak) about a lower value (-0.12). The first-tier dark states oscillate stochastically about much lower values of P ( t ) (50.3) after a few strong (-0.2) initial recurrences. For the most remote dark state P(t) also appears is considerably larger spiky and stochastic, but its average value (-2 x than that for the corresponding state in the [I, 11, 01 polyad (-1.5 x despite the fact that (1,0,0, Oo, 18’)’ is at least 10 anharmonic steps removed from the ZOBS [vs. 4 steps for (1, 0, 0, Oo, 6O)OI. Most of the differences between the [I, 11, 01 and [l, 23, 01 polyads is related to the vastly larger dimension of the latter (91 vs. 19 states), the scaling up of all off-diagonal matrix elements (by a factor of - 20/8), and an increased spread of the eigenenergies (1850 vs. 700 cm-I). However, the most important difference between [ 1, 11, 01 and [ 1, 23, 01 is that the transand cis-bend modes “crash” near h n d = u4+u5 = 16. The w4, wg,x44,x55 and x45 constants lead to a near-perfect collapse of the entire (u4 = h n d , US = 01, (u4 = &nd - 2, u5 = 2), . . . , (u4 = 0, u5 = %nd) bend manifold near &end = 16. It is clear that the polyad Heff model provides a useful framework for describing and extrapolating the intramolecular dynamics to higher EVIBor to different ZOBSs. When and how the extrapolation fails can provide a sensitive means of detecting and describing a change in the W I:w2 :. ..~ 3 ~ resonance structure. Such a change in resonance structure could be associated with access to a chemically significant topographic feature of V(Q).Although we have some tentative evidence for deviations from polyad Heffpredictions for C2H2 at E V ~ B = 16,000crn-l, the predicted energy of the acetylene t) vinylidene barrier, we must postpone discussion of these EVIB2 l6,000cm-’ I

6

R. W. FIELD et al.

484

"

'.or-(0,i ,0,18+2,2'2)

(OBI ,0.20° ,0°) O

I

1

0.q.

0.8

$0.8

igi

0.4

0.2 0.0

3

2

1

lime @s)

4

time (ps)

( 1 ,o,o,oo,18O)

(0.1.O. la0,2') 5x10.'

l

'

O

time (ps)

I

I

1

I

I

time (ps)

Figure 8. Same as Fig. 7A, except this is the [I, 23, 01 polyad and the ZOBS is (0,I , 0,200, ooy.

DF spectra until the optimized unzipping algorithm [ll] has been carefully tested. Instead, SEP spectra of HCP will be discussed in which the WHX :wcp :W k n d resonance structure changes (HX is the H stretch against the CP center of mass) from 5 :2 : 1 near HCP equilibrium to 3 :2 : 1 near the HPC saddle point.

IV. A CHANGE IN RESONANCE STRUCTURE

Stimulated emission pumping spectra of HCP 2 C+ have been recorded via the 'A" [ 121 and 2. 'A' states [5]. These 0.05-cm-' resolution spectra sample eigenstates rather than feature states, extending to EVIB= 25,315 cm-', above the 2 state zero-point level, about 1300 cm-' above the ab initio predicted linear HPC saddle point [5, 131. The SEP spectra sample a large set of completely assignable vibrational eigenstates as well as some partially assignable interlopers, which are also vibrational levels of the HCP 2 state, made observable either by perturba-

INTRAMOLECULAR DYNAMICS IN THE FREQUENCY DOMAIN

485

tions of the Franck-Condon bright states or by an axis-switching mechanism [14]. The assignable states are all high overtones of the bend (0, u2, 0) 26 Iu2 5 42 [5], combination bands (0, u2, 1) u2 = 24, 26 [12] or members of polyads, P I u2 + 2u3 = 30, 38, based on the 1 :2 resonance between w2 (bend) and w3 (CP stetch) [5]. The bend overtone progression (0, u2, 0) exhibits surprisingly regular Morse-like vibrational spacings [5, 12, 191, described by AzG(u2)= G ( u +~ 1) - G ( u -~ 1) = 2wiu2 + ~

2 2 4

Figure 9 shows that A ~ G ( u+~6). 2 6 = ~ const ~ for 1 I u2 I 4 1 (mi = 670 cm-', x22 = -6.26 cm-I). However, the regularity of A2G(u2) is in marked contrast to the other molecular constants and to the appearance of the spectrum! As is expected for the B value of a linear triatomic monohydride, the rotational constant increases very slowly with u2 at low u2 [15]. Then, owing to strong 2 : 1 anharmonic interaction with the CP stretch (for which B is expected and observed to decrease with u3 at low ug [IS]), B o , ~ ,begins o to decrease slowly and continues to decrease slowly and monotonically to u2 = 34 (Bo = 0.6663 cm-', B34 = 0.6454 cm-I). Suddenly B o , ~ ,reverses o 1350 1300 ?-

1250


1200

i) v

g1150

1100 1050

0

10

20 Ua

'

40

Figure 9. Vibrational level spacing for HCP 2 C,AzG(u2) = C(0,u2 + l,O)G(O, u2 1,O) plotted (open circles) vs. the bending quantum number m.Also plotted l(open squares) is the reduced A2C. A2G(9) - 2 x 2 2 ~ 2 which , would be a horizontal line for a Morse oscillator.

R. W. FIELD et al.

486 0.70

.

0

d

10

20

30

40

ua

Figure 10. E ( o , ~ ~ , vs. o)

u2

for HCP. Values plotted are partially deperturbed [S].

direction and increases by 10% to B o , ~ , =o 0.705 cm-’ and then reverses direction again to B0,42,0 = 0.695 cm-’ (see Fig. 10). Although these changes in B may seem small to a nonspectroscopist, one should keep in mind that complete removal of the H at constant rcp would cause B to increase by only 18%! The rapid change in B o , ~ ~is, ?probably due to neglect of Coriolis interaction of (0, u2, 0)I = 0 levels with (1, u2 - 3, 0)E = 1 levels. Near the eH(Cp) bond angle of the HCP H HPC saddle point, WHX decreases from - 5uhnd to slightly less than 3 W k n d and then increases to slightly more than 3 W k n d [16]. When the WHX : @bend ratio crosses through 3 : 1, the Coriolis effects on Fffreach their maximum value and then change sign, exactly as is observed in the spectrum. This Coriolis hypothesis is borne out by the u2 dependence of all of the other “vibrational fine structure” constants (42,g22, yli) [5]. All of these constants are nearly independent of u2 up to u2 = 36, where they suddenly begin to change rapidly. The q 2 constant, which controls the A1 = 2 interaction between 1 = 0 and 1 = 2 components of the same (u, ,u2, u g )vibrational state, is affected by the same A1 = +1 Coriolis interaction that caused the rapid . 10 and 11 show the nearly identical u:! depenvariation of B o , ” ~ , oFigures ,~. 12, which is a plot of q2(u2) versus B o , ~ , o . dence of q2 and B O , , , ~Figure shows that despite the erratic u2 dependences of both q2 and B, the q 2 versus B plot is linear. This indicates that the changes in both constants originate in the same Coriolis interaction! The onset of sudden variations in vibrational fine structure is one of the most sensitive indicators of a change in resonance structure. The magnitudes of fine-structure parameters are determined by second-order perturbation theory (a Van Vleck or “contact” transformation) [17]. The energy denominators in these second-order sums over states are approximately independent of EVIBas long as the o 1 :w2 : .. .W 3 N - 6 resonance structure is conserved.

INTRAMOLECULAR DYNAMICS IN THE FREQUENCY DOMAIN

487

2.2,

1.2L

0

-

... . .<..-. 1 . .

20

10

30

..,. .. ,,,.I 40

v2

Figure 11. Off-diagonal interaction parameter 42 between 1 = Oe and 1 = 2e levels of (0,9,0) is plotted vs. 9.Note that the actual Oe - 2e interaction matrix element is proportional to - (u2 + I)q2 (see eqs. 4-6 of Ref. 5). Although 42 is nearly constant from u = 2, ... , 34, the actual interaction increases by a factor of -35/3.

When one of the vibrational frequencies starts to change rapidly [because the curvature of V ( Q ) along that coordinate reverses sign, as it must at a barrier or saddle pointr all of the energy denominators start to change much more rapidly than the u scaling of the matrix elements squared in the numerator. The HCP case exhibits one additional signal of the onset of rapid change in vibrational resonance structure. This is the sudden onset of vibrational perturbations at (0, 32, 0) [ S ] . Local perturbations, where one rotational term curve crosses another, are manifest as level shifts, extra lines, and intensity anomalies [18]. Such perturbations are typically rare at low E v l ~and

1.2 0.64

0.65

0.66

0.67

B, (cm-')

0.60

0.69

0.70

Figure 12. A plot of 42 vs. B ( O , ~ , Ofor ) HCP.The nearly linear relationship indicates that B and 42 are both affected by the same A1 = f l Coriolis interaction.

R. W. FIELD et al.

488

increase monotonically in number and strength as E V ~ increases. B Since perturbation matrix elements typically scale along a progression linearly with the progression-forming quantum number up and anharmonicity-based differences between polyad members also typically scale linearly with up, the mixing angle Hi d/(E:p, - E , - 2,d + 1) for a family of perturbations is roughly indepen&nt of up. Thus it is very unusual for perturbations to appear suddenly at up = 32! But this abrupt turning on of local perturbations is precisely what occurs for HCP [5]. Only one weak perturbation was observed below u2 = 32 in the entire SEP data set [12]. This is another manifestation of a change in resonance structure. As WHX decreases from - 5 ~ to 2 = 4W2, WHX passes through three new, lower order anharmonic resonances:

.

Here, WHX also passes through two new, lower order A1 = f l Coriolis resonances near WHX = 3Wbend:

Since the perturbations turn on at slightly lower u2 than the onset of changes in fine-structure parameters [5], it seems likely that the perturbations in (0,32, 0)are due mostly to new WHX = h k n d anharmonic resonances and the rapid changes in fine-structure parameters at (0,36,O) due mostly to new WHX = 3Wbend Coriolis resonances.

V. SUMMARY Examples from the DF spectrum of C2H2 [3, 4) and the SEP spectrum of HCP are discussed [S, 121. in both cases V(Q) is systematically sampled along long progressions in two Franck-Conaon active modes, a bend and a heavy atom stretch. Information about intramolecular dynamics (especially bond-breaking isomerization) and chemically interesting topographic features of the potential-energy surface (especially an isomerization saddle point) is obtained from the vibration-rotation spectrum in unusual ways. The combination of low-resolution and spectral unzipping into noninteracting polyads enables systematic, model-free surveys of deperturbed Franck-Condon factors, deperturbed zero-order energy levels, and trends in intramolecular vibrational redistribution (IVR) rates and pathways [3]. The He" tnf,nres,Ilpolyad model permits extraction of the most important resonance strengths directly from fits to a few polyads [6-81. Once these anharmonic

INTRAMOLECULAR DYNAMICS IN THE FREQUENCY DOMAIN

489

coupling parameters are known, a complete and scalable picture of IVR is obtained from the eigenvectors of each H$:,,,es, This leads to insights into optimal control schemes and facilitates detection of the onset of changes in the w 1 :w2 :. . . 0 3 ~ 6- resonance structure. In a smaller molecule (HCP),these diagnostically important changes in vibrational resonance structure are manifest in several ways: (i) the onset of rapid changes in molecular constants, especially B values and secondorder vibrational fine-structure parameters associated with a doubly degenerate bending mode; (ii) the abrupt onset of anharmonic and Coriolis spectroscopic perturbations; and (iii) the breakup of a persistent polyad structure

PI.

In both the HCCH and HCP examples, the study of long progressions provides information about V(Q), not only along the progression-forming Franck-Condon bright normal coordinates, but also for those coordinates strongly coupled to the Franck-Condon active coordinates. The V ( Q )is sampled and refined locally, strip by strip, rather than globally. The IVR mechanisms are described, extrapolated, and surveyed, and this can generate more insight than direct time-domain measurements of a few well-chosen (but not unzipped) survival probabilities. ' h o complementary reduced-dimension representations of dynamics are described and distinguished. The earliest time dynamics in Franck-Condon active configuration space involves real ball-and-spring motions. A wavepacket corresponding to the vibrational eigenstate on the electronically excited surface is transferred at t = 0 onto the electronic ground-state surface. After this moving wavepacket splits up into independently evolving Franck-Condon bright states, it becomes useful to consider the evolution of each of these ZOBSs within their respective polyad. This is dynamics in abstract state space. There is no motion of (Q) or (P).The neglect of interactions between polyads is based on the exstence-of generalized quantum numbers that are simultaneously orthogonal to the directions in (3N - 6)dimensional state space defined by the resonance vectors corresponding to all important anharmonic resonances [9, 101. We expect that soon a new unzipping procedure, based on extended cross-correlation of many DF spectra 1111, will provide a quantitative measure of the hitherto neglected interactions between overlapping polyads.

Acknowledgments This research has been supported by grants from the Department of Energy (DE-FG0287ER13571) and the Air Force Office of Scientific Research (F49620-94-1-0068). One of us (H. I.) thanks the Japan Society for the Promotion of Science for Japanese Junior Scientists for a fellowship. M. P. J. thanks the Department of Defense for a predoctoral fellowship.

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References 1. D. M. Jonas, S. A. B. Solina, B. Rajaram, S. J. Cohen, R. J. Silbey, R. W. Field, K. Yamanouchi, and S. Tsuchiya, J. Chem. Phys. 99,7350 (1993).

2. R. W. Field, S. L. Coy, and S. A. B. Solina, Prog. Theoret. Phys. Suppl. 116, 143 (1994). 3. S. A. B. Solina, J. P. O’Brien, R. W. Field, and W. F. Polik, Ber. Bunsenges. Phys. Chem. 99, 555 (1995). 4. S. A. B. Solina, J. P. O’Brien, R. W. Field, and W. F. Polik, J. Phys. Chem. 100, 7797 ( 1996). 5. H. Ishikawa, Y.-T. Chen, Y. Ohshima, B. Rajaram, J. Wang, and R. W. Field, J. Chem. Phys. 105,7383 (1996). 6. J. Pliva, J. Mol. Spectrosc. 44, 165 (1972). 7. Q. Kuo, G. Guelachvili, M. Abbouti Temsamani, and M.Herman, Can. J. Phys. 72,1241 (1994); A. Campargue, A. Abbouti Temsamani, and M. Herman, Chem. Phys. Lett. 242, 101 (1995); M. Herman, M.Abbouti Temsamani, D. Lemaitre, and J. vander Auwera, Chem. Phys. Lett. (1996). 8. B. C. Smith and I. S. Winn. J. Chem. Phys. 89, 4638 (1988); ibid., 94,4120 (1991). 9. M. E. Kellman, J. Chem. Phys. 93,6630 (1990). 10. M. E. Kellman and G. Chen, J. Chem. Phys. 95,8671 (1991). I 1. M. Jacobson and S. L. Coy, MIT, unpublished. 12. Y.-T. Chen, D. M.Watt, R. W. Field, and K. I<. Lehmann, J. Chem Phys. 93,2149 (1990). 13. S. C. Farantos, H.-M. Keller, R. Schinke, K. Yamashita, and K. Morokuma, J. Chem. Phys. 104, 10055 (1996). 14. J. T. Hougen and J. K. G . Watson, Can. J. Phys. 43,298 (1965). 15. A. Cabaiia, Y. Doucet, J.-M. Gameau, C. Pipin, and P. Puget, J. Mol. Spectrusc. 96,342 (1982). 16. M. Stumpf and R. Schinke, Max Planck Institut fur Stromungsforschung,Gottingen, twodimensional adiabatic surfaces, computed from the ab initio potential of Ref. 13, private communication, 1995. 17. G. Amat, H. H. Nielsen, and G. Tarrago, Rotation-Vibration of Polyatornic Molecules, Marcel Dekker, New York, 1982. 18. H. Lefebvre-Brion and R. W. Field, Perturbations in rhe Spectra of Diatomic Molecules, Academic, Orlando, E,1986. 19. K. K. Lehmann, S. C. Ross, and L. L. Lohr, J. Chem Phys. 82,4460 (1985).

EMERGENCE OF CLASSICAL PERIODIC ORBITS AND CHAOS IN INTRAMOLECULAR

AND DISSOCIATION DYNAMICS P. GASPARD* and 1. BURGHARDT' Service de Chimie Physique and Center for Nonlinear Phenomena and Complex Systems Universite' Libre de Bruxelles Brussels, Belgium

CONTENTS I. Introduction 11. Semiclassical Quantization around Equilibrium Points and Periodic Orbits A. Time Evolution in Quantum Mechanics and Trace Formulas B. Quantization around Isolated Equilibrium Points c. Gutzwiller Trace Formula for fsolated Periodic Orbits D. Zeta Function and Interferences between Isolated Periodic Orbits E. Periodic-Orbit Expression for Eigenfunction Averages F. Berry-Tabor Trace Formula and Nonisolated Periodic Orbits G. Bifurcating Periodic Orbits H. Semiclassical Scattering: Scattering Orbits versus Trapped Orbits 1. Emergence of Rate and Relaxation Behaviors: Quasiclassical Regime 111. Bounded Systems A. Energy Spectrum and Its Different Scales 1. Average Level Density 2. Periodic-Orbit Structures 3. Energy Scale below Mean Spacing B. Statistics of Level Curvature and Other Parametric Properties

'Report presenred by P: Gaspard h e s e n t address: Institut fur Physikalische und Theoretische Chemie der Universitat Bonn, Bonn, Germany Advances in Chemical Physics, Volume 101: Chemical Reactions and Their Control on the Femtosecond Time Scale, XXth Solvuy Conference on Chemistry, Edited by Pierre Gaspard, Irene Burghardt, 1. Prigogine, and Stuart A. Rice. ISBN 0-471-18048-3 0 1997 John Wiley & Sons, Inc.

49 1

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P. GASPARD AND I. BURGHARDT

C. Time Domain 1. Beyond Heisenberg Time 2. Emergent Ciassical Orbits and Vibrograms D. Diatomic Molecules 1. Morse-Qpe Model for 12(i E) 2. Experimental Vibrogram of NaI by Zewail and Co-workers E. Triatomic -Molecules 1. CSz(X_'Eg+) 2. N02(X 2AI -A 2B2) F. Tetra-atomic Molecules 1. '2C2HD(X'C+) 2. '*C2H2 (5 G. Synthesis IV. Open Systems A. Energy and Time Domains B. Unimolecular Dissociation Rates: RRKM Theory and Distribution of Resonances C. Dissociation on Potentials with a Saddle: Classical Properties 1. Classical Dynamics: The Repeller 2. Bifurcation Scenario Associated with Transition to Chaos 3. Fully Chaotic Regime: Smale Horseshoes D. Dissociation on Potentials with a Saddle: Semiclassical Quantization I. Quantization in Periodic Regime 2. Quantization in Transition Regime 3. Periodic-Orbit Quantization in Fully Chaotic Regime E. Ultrashort-Lived Resonances in Triatomic Molecules 1. HgIz 2. c02 3. H3

'

4.

03

5 . H20 6. Comparison of Lifetimes

V. Conclusions References

I. INTRODUCTION

Understanding chemical reactions has been a major preoccupation since the historical origins of chemistry. A main difficulty is to reconcile the macroscopic description in which reactions are rate processes ruling the time evolution of populations of chemical species with the microscopic Hamiltonian dynamics governing the motion of the translational, vibrational, and rotational degrees of freedom of the reacting molecules. With the development of shorter and shorter laser pulses, the inter- and intramolecular relaxation processes in the nano- and picosecond windows have been progressively resolved in "real time" down to the vibrational and reactive dynamics in the femtosecond window [I]. These observations have

EMERGENCE OF CLASSICAL PERIODIC ORBITS AND CHAOS

493

highlighted the fact that quantum Hamiltonian dynamics is not always the stage of oscillatory motions associated with a discrete and regular energy spectrum but may also show more complex behaviors determined by discrete but irregular energy spectra or by continuous energy spectra with a superimposed spectrum of discrete resonances [2]. In particular, irregular vibrational spectra with Wignerian level spacing statistics have been observed this last decade for a number of highly excited molecules [3-71. On the other hand, many recent works have characterized the reactive dynamics in terms of quantum resonances, which allows a rigorous definition of metastable states with finite lifetimes and hence of dissociation rates [4, 8-10]. Moreover, emerging classical orbits have recently been discovered in Rydberg atoms and highly excited molecules by high-resolution spectroscopy 1111. By the notion of emerging classical orbits, we imply that a highly excited quanta1 wavepacket shows time recurrences around the periods of corresponding classical orbits. Since time and energy are always subject to Heisenberg uncertainty relations, AE At - A, the emerging classical orbits remain fuzzy objects. However, the classical actions in play are often one or two orders of magnitude larger than the Planck constant, under which conditions the classical orbits emerge with sharpness, together with their nonlinear properties of stability, bifurcation, and chaos. Intramolecular dynamics and chemical reactions have been studied for a long time in terms of classical models. However, many of the early studies were restricted by the complexities resulting from classical chaos. The application of the new dynamical systems theory to classical models of reactions has very recently revealed the existence of general bifurcation scenarios at the origin of chaos. Moreover, it can be shown that the infinite number of classical periodic orbits characteristic of chaos are topological combinations of a finite number of fundamental periodic orbits as determined by a symbolic dynamics. These properties appear to be very general and characteristic of typical classical reaction dynamics. Moreover, new semiclassical methods have been developed that are based on the Gutzwiller and Berry-Tabor trace formulas [12, 131. These methods allow the calculation of energy levels or quantum resonances in systems with many interfering periodic orbits, as is the case for chaotic dynamics. The purpose of this chapter is to review the recent results obtained in this context over the last decade and, in particular, in our group. The report is organized as follows. In Section 11, we summarize the relevant quantummechanical principles and the Gutzwiller and Berry-Tabor trace formulas. Section I11 is devoted to bounded systems. The semiclassical properties relevant for short time scales are discussed and illustrated with so-called vibrograms ,of 12,NaI, CS2, CzHD, and C2H2. Section III is also concerned

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with properties related to irregular spectra and Wignerian level repulsion, which appear on long time scales beyond the Heisenberg time. We distinguish two well-established mechanisms by which the level spacing statistics may become Wignerian, that is, by overlap between two electronic surfaces, or by accumulation of several vibrational anharmonic resonances on a single electronic surface. Further mechanisms involving rotational-vibrational interactions or molecular Rydberg states are discussed. Section IV is devoted to open reactive systems. Here, we distinguish between potential surfaces with a single saddle point and potential surfaces with quasibound regions separated from the exit channels by a potential barrier. The latter type of systems typically feature long lifetimes and Wignerian repulsion between quasidiscrete energy levels. For direct, ultrafast dissociation processes on surfaces with a saddle point, we present classical-dynamical properties that have recently been discovered and the results of semiclassical quantization using the Gutzwiller trace formula [lo]. This part is illustrated by the dissociation of Hg12 and C02 in particular.

II. SEMICLASSICAL QUANTIZATION AROUND EQUILIBRIUM POINTS AND PERIODIC ORBITS A. Time Evolution in Quantum Mechanics and 'Ikace Formulas In quantum systems, the time evolution is determined by the Hamiltonian operator according to the Schrodinger equation

Since the equation is linear, th: propagation is fully determined by the evolution operator, or propagator K(t),

The propagator can be obtained as a Fourier integral of the resolvent operator 1

L1

G ( E )=

~

E-il

(2.3)

which is defined for complex energies E. The singularities of the resolvent correspond to the energy spectrum of the Hamiltonian. These singularities may be poles at real or complex energies or branch cuts. The poles are asso-

EMERGENCE OF CLASSICAL PERIODIC ORBITS AND CHAOS

495

ciated respectively to bound states that evolve in time like exp(-iE,t/h) or to resonant states that decay like exp(-ie,t/h) exp(-rnt/2A) for forwardtime propagation. The lifetime of the resonance is related to the half-width according to 7, = h/rn.The branch cuts exist at the energy thresholds where new dissociation channels are opened, and they are associated with algebraic decays [14]. The energy spectrum is obtained from the spectral determinant of the Hamiltonian det@ - E) = 0

(2.4)

if this determinant is well defined; otherwise alternative forms have been proposed. The logarithmic derivative of the spectral determinant is equal to the trace of the resolvent tr

1

E-H

= &ln det@ - E)

On the other hand, the level density is also related to the trace of the resolvent by

where we have used the distribution identity l/(x + i0) = P(l/x) - i d ( x ) . In scattering systems, it is the logarithmic derivative of the S matrix that is related to the trace of the resolvent by [15]

I ( E ) = -ih &ln det i ( E ) = 2 ~ tr[G(E h - I?) - 6(E - H0>1

(2.7)

where HOis the free-particle Hamiltonian. Equation (2.7)defines the average Wigner time delay, which is the time difference between the scattered wave and the free wave at energy E. This equation shows that the singularities of the S matrix can be obtained by considering the difference between the level densities corresponding to the scattering Hamiltonian and the free Hamiltonian and identifying the singularities of the analytical continuation for this difference. The interrelationsbetween the propagator, the resolvent, and the level density will be central to our discussion. In particular, the trace formulas referred to in Section I represent semiclassical approximations to the quantities (2.6) or (2.7)and turn out to involve the periodic orbits of the classical dynamics.

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P. GASPARD AND I. BURGHARDT

The highly excited and reactive dynamics, the details of which have been made accessible by recently developed experimental techniques, are characterized by transitions between classically regular and chaotic regimes. Now molecular spectroscopy has traditionally relied on perturbation expansions to characterize molecular energy spectra, but such expansions may not be valid if the corresponding classical dynamics turns out to be chaotic. This leads us to a reconsideration of such perturbation techniques and provides the starting point for our discussion. From there, we will proceed to discuss the Gutzwiller trace formula, which provides a semiclassical description of classically chaotic systems. B. Quantization around Isolated Equilibrium Points

One of the early methods to obtain molecular energy eigenvalues is the contact transformation technique by Van Vleck [16, 171. This method has originally been developed to obtain the vibrational energy levels of bound molecules, but recent works, in particular, by Miller and co-workers, have shown how the method can be extended to obtain the vibrational energy levels of metastable species [18]. In this way, the energy levels have been obtained in the form of the Dunham expansion, which we quote here disregarding the rotational degrees of freedom,

E,

= Eep+ AE,

+.

* *

where v and 1 are the vibrational quantum numbers and d the degeneracies of the oscillators. This type of expansion is carried out around an equilibrium point of the potential energy surface, where a,,V(eP) = 0, and which is

located at the energy Eep.Within the harmonic approximation, the calculation is based on the normal modes and their associated frequencies o.These frequencies are all real at minima of the potential but may be imaginary for other types of equilibrium points, in particular, saddle points. In the latter case, the imaginary part of the relevant frequencies must be taken negative, Im w j = - IIm w j I, in order to define the quantum resonances associated with the forward semigroup that corresponds to propagation for positive times. The frequencies determine the stability of the equilibrium point and, in particular, the Lyapunov exponents [19], which are given by

EMERGENCE OF CLASSICAL PERIODIC ORBITS AND CHAOS

497

Within the harmonic approximation, we observe that the lifetimes of the dominant resonances (with uj = 0 for the unstable modes j ) are therefore directly related to the sum of positive Lyapunov exponents, (2.10) The anharmonicities of the potential contribute by the terms involving the constants x, g, y, ... as well as the energy shifts AE, = O(A2), .. . and the frequency shifts aW, = O(h2), . . , . These anharmonic constants can be calculated by the Van Vleck contact transformations [20] as well as by a semiclassical method based on an h expansion around the equilibrium point [14], which confirms that the Dunham expansion (2.8) is a series in powers of A. Systematic methods have been developed to carry out the Van Vleck contact transformations, as in the algebraic quantization technique by Ezra and Fried [21]. It should be noted that the constants x and g can also be obtained from the classical-mechanical Birkhoff normal forms [22]. The energy shifts AE,, .. . may be considered as anharmonic corrections to the harmonic zero-point energy associated with the constants d. Recent works have shown that the energy shift AE, is far from negligible and may be of the order of 100 cm-’ for a harmonic zero-point energy of the order of loo0 cm-’ [20]. If imaginary frequencies are involved, some of the constants x, y , . .. also turn out to be imaginary. Since the imaginary part of the eigenenergies determines the lifetimes of the resonances, such anharmonicities cause variations in the lifetimes. Rotational degrees of freedom can also be included in this scheme; that is, we may thus study the effect of rotation on the molecular resonances [23]. Let us comment on the domain of validity of the Dunham expansion (2.8). As discussed by Miller and Seideman [24], the Dunham expansion implies a regular distribution of the energy levels or resonances. The anharmonicities have the effect of nonlinearly distorting the spectrum of energy levels with respect to the harmonic situation, without however preventing an assignment of the levels in terms of the quantum numbers v and 1. Therefore, such expansions should hold for regular spectra. To confirm this conclusion, let us note the fact that the quantum numbers v and 1 are constants of motion (defined in terms of the original position and momenta operators through the sequence of Van Vleck contact transformations) under the condition that the series generated by these transformations converges. If this is the case, the dynamics appears as effectively integrable in spite of the anharmonicities.

498

P. GASPARD AND I. BURGHARDT

In many systems, though, the constants x, y , ... of the Dunham expansion may turn out to be large because of small denominators, which are due to resonance conditions between the frequencies. These resonances may be of Fermi or Darling-Dennison types or of vibrational or rotational Z-doubling types. In such cases, it is necessary to remove the resonant terms of the Hamiltonian before carrying out the Van Vleck contact transformation in order to avoid the occurrence of the critical small denominators. The Hamiltonian must be separated into a diagonal part given by a Dunham expansion like (2.8) but with different anharmonic constants involving only partially the anharmonicities and a nondiagonal part containing the resonances [17]. The nondiagonal part induces interactions between the energy levels of the diagonal part so that the assignment in terms of the quantum numbers v and 1 has to be modified. In the presence of a single resonance, alternative assignments have been proposed which involve the definition of local modes [25]. In such cases, an assignment is still possible because the corresponding single-resonance Hamiltonian remains integrable. For this type of Hamiltonian, local modes may appear through bifurcations, such as the period-doubling bifurcation in the Fermi resonance or the pitchfork bifurcation in the Darling-Dennison resonance [26]. However, at high energies the number of resonances that have to be taken into account increases since the anharmonicities play a growing role. In such regimes, more and more intruder lines prevent the assignment in terms of a complete set of good quantum numbers. A theorem by Kellman [27] shows that the number of constants of motion decreases. The spectrum is thus expected to become irregular and dominated by statistical Wignerian repulsion so that the Dunham expansion loses its validity. However, regularities may still persist on intermediate energy scales, even if the spectrum appears irregular on small energy scales, as recent works have shown. Such regularities appear when many energy levels accumulate, and they are explained semiclassically in terms of the classical periodic orbits sustained by the potential. Because the periodic orbits extend over global regions in phase space, they can take into account anharmonic features that are too remote from the equilibrium point to be faithfully represented by the Dunham expansion. In such regimes, a semiclassical quantization is possible based on the periodic orbits according to the trace formulas by Gutzwiller and by Berry and Tabor, to be discussed below.

C. Gutzwiller Trace Formula for Isolated Periodic Orbits Let us turn back to the results of Section 1I.A. The level density (2.6),which involves the trace of the resolvent, may be evaluated with semiclassical methods. The early works by Wigner, Weyl, Thomas, and Fermi already showed how to obtain the average level density and, in general, the average values of

EMERGENCE OF CLASSICAL PERIODIC ORBITS AND CHAOS

499

such traces as we introduced above. These main contributions are obtained by using the Weyl-Wigner transforms of the quantum operators into functions over a mock phase space defined in terms of position and momentum variables (q,p) [28]. However, this quasiclassical method smooths out the energy quantization. In the 1970s, Gutzwiller showed that further contributions must be added to the trace formulas, which involve the classical periodic orbits [121. Accordingly, the semiclassical expression for the trace becomes

with the average contribution [28]

x

- H,I)

+ 0(h4)

(2.12)

where Hcl is the classical Hamiltonian function of positions and momenta. The leading term is the integral of the microcanonical density over phase space. The following terms are corrections in powers of the Planck constant, which have been first derived by Wigner. The contribution of the periodic orbits is 112, 293

=

p

2 r=l

T pcos[(r/h)Sp - r(7r/2)pp+ AC,(rT,) + h3C3(rTp)+ . . .I ahldet(m;; - 1)1*/2 exp[A2C2(rTp)+ h4C4(rTp)+ . --I

(2.13)

where the first sum extends over all the prime (i.e., nonrepeating) periodic orbits p , while the second sum takes into account the repetitions of each

500

P. GASPARD AND I. BURGHARDT

periodic orbit p . The quantity T, denotes the prime period that is the shortest among the periods; (2.14)

is the reduced action; and p, is the so-called Maslov index, which can be defined topologically as the winding number of invariant manifolds around the periodic orbit [30]. m, is the linearized Poincari mapping in a surface of section transverse to the periodic orbit. The eigenvalues A$) of mpdetermine the linear stability of the periodic orbit. In particular, the Lyapunov exponents of the periodic orbit are given by (2.15)

so that the amplitude of the rth repetition of the periodic orbit p decreases as

A,,

- exp

[

--T~

C A): hp>O

]

(2.16)

The more unstable the orbit is the smaller is the amplitude of its contribution. This result confirms the intuitive reasoning that the amplitude of an orbit is essentially the square root of the probability for nearby trajectories to remain close. This probability decreases exponentially with time t = rTp along each direction that is linearly unstable. The amplitude of the periodic orbits is therefore determined by the linear stability with respect to perturbations transverse to the orbit. In this sense, the leading term in expression (2.13), obtained by setting C, = 0, treats the dynamics transverse to the orbit at the level of the harmonic approximation. The nonlinear stability properties appear thus as anharmonic corrections to the dynamics transverse to the orbit. These anharmonicities contribute to the trace formula by corrections given in terms of series in powers of the Planck constant involving the coefficients C,, which can be obtained as Feynman diagrams 114, 311. For periodic orbits that undergo a bifurcation, some Lyapunov exponents may vanish so that the orbit becomes of neutral linear stability in the critical directions [32]. In such cases, the dynamics transverse to the periodic orbit

EMERGENCE OF CLASSICAL PERIODIC ORBITS AND CHAOS

501

becomes controlled by the nonlinear stability in the critical directions. The anharmonicities transverse to the orbit then provide the dominant contribution to the amplitude. The periodic-orbit contribution derived by Gutzwiller is general and applies to different kinds of periodic orbits. However, the applicability of (2.13) rests on the property that the periodic orbits are isolated, that is, they do not belong to a continuous family. This is the case in hyperbolic dynamical systems where all the periodic orbits are linearly unstable. We should emphasize that the Gutzwiller trace formula may apply both to bounded and scattering systems. If the hyperbolic dynamical system is moreover chaotic, the periodic orbits are known to proliferate exponentially as [19] (2.17)

where htopis the topological entropy per unit time of the system. The periodic orbits are labeled by sequences of symbols o = 01 02 . - - ON taken from some alphabet w i E A = (0, 1,2,. ..,M - 1). There may be constraints on the way the symbols follow each other in these words, but the exponential proliferation essentially stems from the fact that the words are formed by concatenation of the symbols E193. In this case, the sum over the prime periodic orbits converges in absolute values only in the half plane [33] 1

- Im E 2 P ( i ) i O(A)

n

(2.18)

where P(P) is the so-called Ruelle topological pressure, here evaluated at fl and bounded by [14, 341

=

502

P. GASPARD AND I. BURGHARDT

(2.19)

These bounds on the pressure are based on two different kinds of averages. The average is taken over the invariant measure that gives equal probability weight uniformly to each orbit independent of its stability. This invariant measure ktopappears in the evaluation of the topological entropy, hence its name. The other average involves the natural invariant measure, which is evaluated by the time average in bounded systems [35]. Besides, ~~1 denotes the classical escape rate [33]. These bounds will be used below in discussing the semiclassical properties of the energy spectrum.

(.k,,

D. Zeta Function and Interference between Isolated Periodic Orbits

In this section, we arrive at the quantization condition expressed in terms of periodic orbits. The periodic-orbit contribution to the trace formula can be written as the logarithmic derivative of a so-called zeta function, (2.20) which is defined as [36]

m l , ...,mu=O n l , ...,ns = O

p

(2.21) where lAik)1> 1 are the stability eigenvalues of the unstable directions while exp(+_i27rpf) are those of the stable directions. Because of the logarithmic derivative, the poles of the resolvent appear at the zeros of the zeta functions so that we obtain the quantization condition

EMERGENCE OF CLASSICAL PERIODIC ORBITS AND CHAOS

503 (2.22)

Z(E) = 0

That this condition is a generalization of the standard WKB (WentzelKramers-Brillouin) condition can be seen by considering a one-degree-offreedom system, where we have

[a

"1

Z ( E ) = 1 - exp - S p ( E )- i - p p = 0 2

(2.23)

so that the Bohr-Sommerfeld quantization condition is recovered: (1/2") f p 4 = + clp/4). The zeta function takes into account the interferences between the different periodic orbits. Indeed, when the system features an infinity of periodic orbits, the zeta function can be expanded as [37]

(2.24) The terms that occur high in the series have a small amplitude and contribute little. Therefore, the zeros of the zeta function will be determined essentially by the first few terms associated with the least unstable periodic orbits. Contrary to systems with one degree of freedom, no factorization is possible so that the different periodic orbits have additive contributions that interfere. The distribution of zeros will therefore have the tendency to become irregular, contrary to classically integrable systems. The quantization condition (2.24) is suitable for scattering systems as well as for bounded systems. However, in the latter case, the zeros of the zeta function are not guaranteed to be real, in contrast to the simple case of (2.23), where we have only one periodic orbit. To remedy this difficulty for chaotic bounded systems, several authors have suggested to consider a quantization condition that incorporates the quantum unitarity [38]. This quantization condition is the object of active current studies, especially in the context of understanding the transition between the large energy scales ruled by the periodic orbits and the small energy scales ruled by random matrix properties. Such a transition is obtained in the analytic continuation across the entropy barrier given by Eqs. (2.18) and (2.19). Let us remark that the ability of periodic-orbit methods to calculate eigenenergies even in classically chaotic systems is related to the validity

504

P. GASPARD AND I. BURGHARDT

of the semiclassical method in the time domain much beyond the time scale of dynamic instability (l/A) In(l/tt). This last result by Heller and Tomsovid. suggests the possibility of using semiclassical methods such as the so-called cellular dynamics for the time propagation of quantum amplitudes [39].

E. Periodic-Orbit Expression for Eigenfunction Averages Certain semiclassical properties involving the eigenfunctions can also be calculated with periodic-orbit theory. Con%ideringthe Wigner functions conesponding to the energy eigenfunctions %& = En& [281,

we have at a given energy

This expression can be used to calculate averages of the form [14,401

{or example, f y the photo!bsorptio? cross section at zero temperature, with A = ( 1 r w / 3 c ~ o > d- ~(&Id, ~ ) where d is the electric dipole moment operator. We will obtain as a first term a quasiclassical average over the microcanonical invariant density. This quasiclassical average is corrected by a Weyl series in powers of the Planck constant, which reproduces the smooth background of the absorption cross section. To these terms are then added the contributions from the periodic orbits, which reproduce oscillating structures in the photoabsorption cross section.

EMERGENCE OF CLASSICAL PERIODIC ORBITS AND CHAOS

505

We obzerve pat the level density is recovered if the observable is the identity, A = I , since 4 dt = T,,. The above formula may be generalized to situations where tKe observable itself depends on time, for example, = kexp(+ifit/A)iexp(-ikt/A), where k and i are other observables, as is the case in the Green-KubWYamamoto-Zwanzig formulas for the transport and reaction rate coefficients. The expression (2.26) shows that the eigenfunctions are distributed on average, at the leading approximation, as predicted by the microcanonical invariant density. This observation, which was made by several authors including Berry [41], Voros [42], Shapiro et al. [43], and others, leads to a number of predictions, one of which is that the nodal structure of the eigenfunctions, although irregular, shows correlations obeying

with = 1x1 42rn[E - V(q)]/h. On these grounds, Berry conjectured that the eigenfunctions are Gaussian random variables, which leads to the same result as (2.28). Berry’s conjecture generalizes the early assumption by Porter and Thomas 1441 that the matrix elements of the observables evaluated between eigenfunctions follow Gaussian distributions. Classical ergodicity is expected to hold when the high-lying eigenfunctions behave according to Eq. (2.28) [42], which has been rigorously justified for several model systems [45]. Moreover, Srednicki showed that Berry’s conjecture implies a Maxwellian probability distribution in many-body systems at high temperatures as well as the Fermi-Dim and Bose-Einstein distributions at lower temperatures under dilute gas conditions [46]. However, Heller [47] has drawn attention to the fact that the eigenfunctions of classically chaotic systems are not structureless at high excitations but present so-called scars. Scars are fluctuations in the Wigner or Husimi transforms of the eigenfunctions, which have the form of a cross in (p,q ) space, near unstable periodic orbits and extending along their stable and unstable manifolds. In position space, these structures form lines that follow the projection of certain periodic orbits and thus create anisotropic correlations in the oscillations of the wavefunction. The role of periodic orbits in scarring phenomena can be explained by the presence of the last term in Eq. (2.26) and will therefore be of higher order in A than the leading isotropic correlations of (2.28). Accordingly, the scarring phenomena are expected to fade away as the number of degrees of freedom increases ( F - =).

506

P. GASPARD AND I. BURGHARDT

F. Berry-Tabor 'Jkace Formula and Nonisolated Periodic Orbits

In integrable systems, the periodic orbits are not isolated but form continuous families, which are associated with so-called resonant tori. In action-angle variables, the Hamiltonian depends only on the action variables, similar to the Dunham expansion,*

H,,(J) = E,,

+w J +X *

: J 2 + y : J3 + Z

J4 +

. * *

(2.29)

For such systems, the trajectories are of the form

O ( t ) = Cnt + e(0)(mod 2 4 with Cn =

aHcl -

aJ

(2.30)

The periodic orbits are obtained by solving the following equations, which express that the full period T is reached after ni oscillations in the ith degree of freedom:

(2.31)

The solutions of these equations are the actions of the periodic orbits J = J n ~= Jp as well as the prime period T p , which is the smallest period such that T = TnE= rTp (see Fig. 1). Here, r is the repetition number of the prime period that is obtained as the largest integer such that the n,, = n/r remain integer. In contrast to chaotic systems, the periodic orbits are thus labeled by the F integers n. As a consequence, the periodic orbits proliferate only algebraically, Number{n : T n < t } - t F

(2.32)

Moreover, the periodic orbits are here of neutral stability. Berry and Tabor [ 131 have derived a trace formula for the level density of integrable systems,

*It should be noted that anharmonic h corrections, in particular to the zero-point energy, are ignored in this section.

507

EMERGENCE OF CLASSICAL PERIODIC ORBITS AND CHAOS

Figure 1. Schematic view of an energy shell in the space of actions of F = 3 classically integrable system. Bulk periodic orbits are depicted by circles (f = F = 3); edge periodic orbits withf = 2 by stars; and edge periodic orbits withf = 1 by squares.

(2.33) where the average level density is given as before by (2.6) while the periodicorbit contribution is now [13]

.c r= 1

1

-cos

[

2rmnp

(+ F)+ -

T

a.1

with np refemng to the smallest integers n labeling the same prime periodic orbit p: n = rnp. Each resonant torus is characterized by the actions J,,, the frequencies ~ ) J Hthe ~ , stability properties according to aJ'H,, as well as the individual Maslov indices pp and another index ap.The derivative of the period with respect to energy can be expressed in terms of derivatives of the Hamiltonian as (2.35)

P.GASPARD AND I. BURGHARDT

508

Again, an expression can be derived for the averages of observables, namely,

tr h ( E -

B)=

I

dFJ dFO Aw(8, J)Aw(O, J; E )

where J, = h(v + f p ) . Equation (2.36) expresses the fact that the Wigner transforms of the eigenfunctions are concentrated on the quantized tori of action J,. Via Fourier transforms using the Poisson summation formula, we obtain a sum over the resonant tori support of the periodic orbits, the actions Jp of which are determined by Eqs. (2.31):

The concentration of the Wigner transforms of the eigenfunctions on the quantized ton provides the counterpart in classically integrable systems to the scarring phenomena associated with unstable periodic orbits. The periodic orbits (2.31) are referred to as bulk periodic orbits in the sense that all the F actions are nonvanishing. Therefore, all the F degrees of freedom are excited in this periodic motion. On the other hand, there exist edge periodic orbits in the subsystems in which one or several action variables vanish (see Fig. 1). These subsystems have a lower number of excited degrees of freedom, but their periodic orbits also contribute to the trace formula. However, they have smaller amplitudes, related to the amplitude of the bulk periodic orbits as

EMERGENCE OF CLASSICAL PERIODIC ORBITS AND CHAOS

for h - 0

509

(2.38)

since F If I1. In general, the amplitudes decrease with the repetition number as power laws, in contrast to unstable periodic orbits where the decrease is exponential according to (2.16). The larger the number of degrees of freedom is, the more rapid is the decrease with repetition number. Therefore, we should expect to observe only the first few repetitions of periods. However, for one-degree-of-freedom subsystems (f = l), the amplitudes do not decrease with the repetition number. The amplitude of the periodic orbits is much larger in the Berry-Tabor formula than in the Gutzwiller trace formula for isolated periodic orbits. This can be concluded from the behavior of the amplitudes when we formally take 0. The leading behavior comes from the average level density, the limit h which increases as tivF. The next largest contributions are those of the neutrally stable periodic orbits of resonant tori, the amplitudes of which go with h-(F+ 'M2. Finally, the isolated unstable periodic orbits contribute with amplitudes in tt-' . This order can be understood from the fact that the amplitudes depend on the stability of the orbits. If a periodic orbit undergoes a bifurcation and becomes unstable, its amplitude decreases accordingly. Another implication is that the importance of the periodic-orbit contributions relative to the average level density decreases as the number of degrees of freedom increases 1481. Indeed, the number of powers of the Planck constant that separates the average level density from the amplitudes of the neutrally stable periodic orbits is equal to i(F - 1). An equal number of powers of h separates the amplitudes of the neutrally stable periodic orbits from those of the unstable ones. Therefore, the periodic-orbit contributions disappear as F+=.

-.

G. Bifurcating Periodic Orbits Due to the nonlinearities of the classical Hamiltonian, the periodic orbits undergo bifurcations at critical energies. At these bifurcations, the stability of the orbit changes and extra periodic orbits are created or existing ones annihilated [19]. These bifurcations have dramatic effects on the semiclassical amplitudes of the periodic orbits [49]. In particular, the comparison between the amplitudes of neutrally stable and unstable periodic orbits shows that the amplitude is expected to be globally lowered after a destabilization. In the vicinity of a bifurcation, the corresponding amplitude in the

5 10

P. GASPARD AND I. BURGHARDT

Gutzwiller trace formula should present a peak. At the leading order, the amplitude is predicted to have a divergence because some stability eigenvalues pass through A$) = 1 so that the denominator vanishes in Eq.(2.13). Consequently, uniform semiclassical approximations are required in the vicinity of bifurcations, which show that the amplitude is strongly peaked but still remains finite. Recent results show that even after the annihilation of periodic orbits at a bifurcation there remain traces of the annihilated periodic orbits in the quantum amplitudes [SO]. These so-called ghost periodic orbits have been interpreted in terms of the complex periodic orbits created when the corresponding real periodic orbits are annihilated. Such phenomena have been experimentally observed for Rydberg atoms in a magnetic field [11, 5 11. We shall show below that such phenomena are also important in intramolecular and dissociation dynamics. Note that complex periodic orbits generally account for “non-classical” effects. The most conspicuous example is provided by the coupling between electronic surfaces beyond the Born-Oppenheimer approximation, at an avoided crossing or conical intersection 1141. The analysis of such problems has recently seen remarkable advances due to the discovery of the gauge structure induced by the separation of the quantum system into the subsets of electronic and nuclear degrees of freedom [52-541.

H. Semiclassical Scattering: Scattering Orbits versus ’Jkapped Orbits Semiclassical theories of the scattering amplitudes have been developed, in particular, by Miller for reactive problems [55] and by Child for predissociation [25]. In this context, we should notice that the Gutzwiller trace formula. extends to scattering systems. As we mentioned above, the Gutzwiller trace formula is of use to determine the energy spectrum of the system, which is constituted by the complex-energy resonances in the case of scattering systems. The scattering resonances are semiclassically calculated from the periodic orbits trapped in the scattering region. However, there exist many problems that require the evaluation of scattering amplitudes instead of the scattering resonances. In the case of long-lived resonances, the scattering amplitudes are strongly affected by the resonances but they may be largely independent of the resonances under certain conditions. This independency is related to the fact that the semiclassical calculation of the scattering amplitudes involves classical orbits belonging to an invariant set that is complementary to the set of trapped orbits in phase space 1561. The trapped orbits form the so-called repeller in systems where all the orbits are unstable of saddle type. The scattering orbits, by contrast, stay for a finite time in the scattering region. Even though the scattering orbits are controlled

EMERGENCE OF CLASSICAL PERIODIC ORBITS AND CHAOS

5 11

to some extent by the stable and unstable manifolds of the trapped orbits, they will visit regions that may be remote from the repeller. Therefore, they are in general expected to determine different behaviors than the trapped orbits of the repeller. For the scattering orbits, the action (2.14) should be replaced by [55]

s=-Iq*dp

(2.39)

which is finite for scattering orbits. Together with energy conservation, the action s can be used to generate the classical motion using a variational principle. Recently, Smilansky and co-workers [57] have introduced a canonical transformation for scattering systems, which maps the ingoing trajectories onto the outgoing ones. For scattering problems, this mapping plays a similar role as the Poincar; mapping for the repeller. Trace formulas involving the S matrix can be derived in this way as sums over scattering orbits, which are periodic for this new mapping. In particular, Smilansky and co-workers have obtained a trace formula for the eigenphases of the S matrix [57]. The eigenphase distribution may become irregular even when the classical motion on the repeller is nonchaotic because chaos is here induced only on the set complementary to the repeller. Wignerian repulsion appears between the eigenphases under such circumstances. We expect that this scattering mapping could be fruitfully applied to the semiclassical evaluation of reactive amplitudes and probabilities.

I. Emergence of Rate and Relaxation Behaviors: Quasiclassical Regime

-

As emphasized by Berry and others [58], the formal limit A 0 presents an essential singularity in the mathematical expressions involving the quantum phases exp(iSp/A) such that the analyticity of the relevant quantities is lost near A = 0. For this reason, we may expect a variety of different behaviors as this limit is approached, depending on the type of systems considered and, especially, on the type of observables. Motivated by this remark, we would like to point out the existence of a regime different from the semiclassical one, in which the quantization of the energy levels is not the dominant feature. In fact, the quantized energy levels can only be detected if the time evolution of the detecting observable features quantum beats. This requires the observable to have a spectral decomposition that is concentrated on a limited number of energy-levels. An example is given by the time autocorrelation of an observable D:

5 12

P. CASPARD AND I. BURGHARDT

Ht)

(2.40)

where $ is the density matrix at the initial time. If this initial state is a pure state (e.g., the projection on the ground state of the Hamiltonian j3 = i&)(&I), the autocorrelatio? function is given by the survival amplitude of the quantum state IrC.0) = Dl&), (2.41) Now, the eigenenergies of the Hamiltonian can be detected directly if the time dependence of the above average exhibits quantum beats. This will be the case if the spectrum is not too dense and the linewidths are smaller than the level spacings. From a Fourier transform of the autocorrelation function, we then obtain an expression of the form (2.26)-(2.27), which can be evaluated semiclassically in terms of periodic orbits and their quantum phases. However, if the initial stateAisa thermal state, such as the canonical density matrix $ = (l/Z)exp(-PH), the autocorrelation is no longer given by a single quantum amplitude but becomes a sum of quantum amplitudes in which quantum phases are randomized. In the classical limit A ---t 0, the leading expression becomes the purely classical autocorrelationfunction with the dynamics being ruled by the classical Liouvillian operator ,?,I = {H,,, [28, 591 a}

CDD(= ~ ) dFq dFp f(q, p)Dw(q, ~ ) [ ~ ' ' ' ~ wp)1+ ( q , O(A)

(2.42)

which can be derived by Weyl-Wigner transforms. The decay of scattering systems can be considered along similar lines. If the detecting observable has a Weyl transform that extends over a large region in mock phase space and if the initial state has a similar Wigner function, the time evolution is likely to be of Liouvillian type rather than of Hamiltonian type. The classical Liouvillian operator ,?,I, which is the classical limit of the Landau-von Neumann superoperator in Wigner representation, can also be analyzed in terms of a spectral decomposition, such as to obtain its eigenvalues or resonances. Recent works have been devoted to this problem that show that the classical Liouvillian resonances can be obtained as the zeros of another kind of zeta function, which is of classical type. The resolvent of the classical Liouvillian can then be obtained as [60, 611

EMERGENCE OF CLASSICAL PERIODIC ORBITS AND CHAOS

5 13

where 1 0 . ~ 1 is a reference Liouvillian defined in order for the trace to be well defined. The trace TrE refers to a trace over functions defined on the energy shell. Equation (2.43) gives in fact the leading term in an expansion in powers of A. On the right-hand side of Eq. (2.43), we introduced the classical zeta function [60,611,

for periodic orbits unstable in all directions. The classical zeta function is similar to the semiclassical zeta function (2.21) except that we are here concerned by the probability itself rather than by the probability amplitude, hence the power one of the stability eigenvalue in the denominator. Actually, the classical zeta function concerns the time evolution of statistical ensembles while the semiclassical zeta function concerns the time evolution of quantum waves, which correspond to two different levels of description. In this regard, we should notice that the time evolution of a quantum system is ruled by two different types of eigenvalues corresponding to the wave function and the statistical descriptions. On the one hand, we have the eigenenergies of the Hamiltonian within the wave function description. On the other hand, we have the eigenvalues of the Landau-von Neumann superoperator in the Liouville formulation of quantum mechanics. These quantum Liouvillian eigenvalues s are related to the Bohr frequencies according to S

1 Ih

= - (Em

-

1 1 = - Re(E, - E,,) + - Im(Em+ En) 1A

A

(2.45)

For bounded systems, the imaginary parts of the energies vanish. For scattering systems, Eq. (2.45) gives the Liouvillian eigenvalues of the forward semigroup defined with factorizable boundary conditions. Nonfactorizable boundary conditions have been considered by Frensley [62], which may lead to other kinds of Liouvillian eigenstates. Let us emphasize that the Bohr frequencies form a continuous spectrum in scattering systems and that Liouvillian eigenvalues like (2.45) then correspond to a discrete spectrum of decay rates obtained by analytic continuation from real to complex Bohr frequencies.

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In the classical limit h -+ 0, the spectrum of the Landau-von Neumann superoperator tends to the spectrum of the classical Liouvillian operator. If the classical system is mixing, the classical Liouvillian spectrum is always continuous so that we may envisage an analytic continuation to define a discrete spectrum of classical resonances. It has been shown that such classical resonances are given by the zeros of the classical zeta function (2.44) and are called the Pollicott-Ruelle resonances sn(E) [63]. These classical Liouvillian resonances characterize exponential decay and relaxation processes in the statistical description of classical systems. The leading Pollicott-Ruelle resonance defines the so-called escape rate of the system, (2.46) which is the rate at which the probability of being in the scattering region decays under the classical time evolution [33,61]. For bounded systems, the escape rate vanishes and the resonance so(E) = 0 coincides with the Liouvillian eigenvalue associated with the microcanonical invariant probability. As indicated above, the leading resonance also corresponds to the Ruelle topological pressure evaluated at the exponent /3 = 1 (cf. Refs. [14, 331). The zeta function methods have proved to be extremely powerful to obtain the resonances of classical scattering systems, which give the quasiclassical reaction rates [6 13. In transport processes, the classical resonances give the dispersion relations that characterize the relaxation of hydrodynamic modes 1641. These results bring about a new understanding of the problem of irreversibility at the classical level, as discussed elsewhere [a]. The previous discussion shows that the relaxation processes emerge from the quantum dynamics under appropriate circumstances leading to the formation of time-dependent quasiclassical parts in the observable quantities. Let us add that quasiclassical and semiclassical methods have been recently applied to the optical response of quantum systems in several works [65,66] where the relation to the Liouville formulation of quantum mechanics has been discussed, without however pointing out the existence of Liouvillian resonances as we discussed here above. The connection between the property of chaos and n-time correlation functions or the nth-order response of a system in multiple-pulse experiments has also been discussed [67, 681.

III. BOUNDED SYSTEMS A. Energy Spectrum and Its Different Scales Bounded quantum systems have in general a discrete energy spectrum formed by a sequence of energy eigenvalues {En}. Discrete vibrational spec-

EMERGENCE OF CLASSICAL PERIODIC ORBITS AND CHAOS

5 15

tra extend usually up to the dissociation threshold, which lies at several eV above the equilibrium point for typical molecules [69]. In a general way, the discreteness of bounded quantum spectra may be viewed as a consequence of the fact that phase-space volumes smaller that the elementary cells of size h” are “inactive” for quantum dynamics, as opposed to classical dynamics. This quantum property is at the root of the third law of thermodynamics,

which states that quantum systems have an absolute entropy per unit vol-

ume, in contrast to classical systems, which have only a relative entropy or €-entropy per unit volume [70].We assume here that the coupling to the electromagnetic field is neglected; otherwise, the energy levels acquire a natural radiative linewidth. In model systems like billiards or for potentials that grow indefinitely, the discreteness of the spectrum extends up to infinite energy, which allows different statistical quantities to be properly defined on the infinite ensembles of energy eigenvalues in the limit E 00. In the following, we will discuss such quantities, which account for different features that appear in the spectrum. We will show that the different spectral features may be classified in terms of the quasiclassical domain, the semiclassical domain, and the domain below the mean level spacing. The quasiclassical domain has already been introduced above, in the context of the average level density first derived by Wigner, Weyl, Thomas, and Fermi [cf. (2.12)] as well as in the context of time-dependent observables such as correlation functions [cf. (2.42)]. We will define below the characteristic energy scale of the semiclassical contribution, which has to be distinguished from the still smaller scale below the mean level spacing. Let us proceed now to a more detailed description of the different regimes.

-

I . Average Level Density The quantities that provide the most “coarse” way of characterizing the spectrum are the average staircase function Na,(E) and its derivative, which is the average level density given by (2.12) up to terms of order A2. The average staircase function is a smooth and monotonously increasing function that counts the number of levels below a given energy E. It captures the structures of the energy spectrum on the largest energy scales, since the average quantities are associated with the largest contribution, of the order of A-“, in the trace formula. This contribution defines the quasiclassical domain. For bounded molecules, which may often be conveniently modeled in terms of harmonic oscillators, it is well known that [71, 721

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P.GASPARD AND I. BURGHARDT

where the dots refer to anharmonic corrections as well as to h corrections. The anharmonic corrections can be calculated by including the full classical Hamiltonian with its anharmonicities in the leading term of (2.12). The next-to-leading terms of (2.12) give h corrections that may turn out to be important. Equation (3.1) shows that the level density increases extremely fast in systems with a large number of degrees of freedom. At sufficiently high level density, the spectrum may appear as a quasicontinuum, or even as a true continuum if the mean level spacing becomes comparable with the natural linewidth. 2. Periodic-Orbit Structures

Between the quasiclassical domain and the energy scale characterized by the mean spacing between the levels lie spectral features that are related to the shortest among the periodic orbits. These oscillations in the energy spectrum are described by the Berry-Tabor or Gutzwiller periodic-orbit terms in the trace formulas. Since the Berry-Tabor periodic-orbit amplitudes are of the I)/* in the level density while the Gutzwiller amplitudes are order of trcF+ only of the order h-', the periodic-orbit fluctuations are much more important in regular systems than in hyperbolic and chaotic systems, as has been confirmed by many observations. The range in energy of the fluctuations is of the order of the inverse of the period AE, = 27rh/Tp, which defines the semiclassical domain. 3. Energy Scale below Mean Spacing

The smallest energy scale is the one of the individual energy levels; this scale refers to all spectral details below the mean level spacing. On this finest energy scale, the energy spectra have been classified into regular or irregular spectra according to the type of statistics and, in particular, of the spacing distributions. In early works [73, 741, two kinds of spacing distributions were surmised: the Poisson distribution, which supposes that the energy levels are independent of each other, and the Wigner distribution, which takes into account the fact that energy levels repel each other. The latter surmise was confirmed by random matrix theories, where an exact and universal spacing distribution was derived by Mehta, Gaudin, and Dyson for ensembles of infinite random matrices [73,741. Recent works have shown that the situation is more complicated. It appears that a Poisson distribution is too strong an assumption in finite systems, where slight deviations with respect to the Poisson distribution are expected [751. Such regular systems as coupled harmonic oscillators have complicated spacing distributions that depend critically on the commensurability between the frequencies [76], suggesting that the situation is not simple.

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5 17

Although many numerical works have provided evidence of a relationship between classical chaos and the existence of a Wignerian spacing distribution [77], here, too, several counterexamples have shown that the relationship appears to rest on a delicate set of assumptions. Classical models referring to certain geodesic flows on surfaces with negative curvatures, which are chaotic in the sense that they have positive Lyapunov exponents and Kolmogorov-Sinai entropy per unit time, turn out to have a Poisson-like distribution [78]. This is explained by the existence of an infinite set of quantum (Hecke) operators commuting with the Hamiltonian [78]. This pathology occurs for particular geodesic flows defined on domains of high geometric regularity, which have a high degeneracy in the spectrum of periodic orbits. On the other hand, models have been found that are classically pseudointegrable, such as a rectangular billiard with a pointlike scatterer [79] or a pointlike magnetic flux [80], but have a Wignerian spacing distribution. In this case, repulsion is explained by diffractive effects on the pointlike singularity, which causes interferences between different diffractive rather than classical paths. These counterexamples point to the suggestion that Wignerian repulsion should arise because of interference effects between paths of quite different lengths, in particular, when the spectrum of classical periods is itself irregular without too many degeneracies. The role of dynamical instability characterized by the Lyapunov exponents is not direct. It appears essentially on the level of the bounds (2.19) on the absolute convergence of the zeta function and of the trace formula. In bounded systems, where the classical escape rate vanishes, ycl = 0, the bounds given in the bottom row of (2.19) show that the topological pressure P( is positive and of the order of one-half of the sum of positive Lyapunov exponents. This bound corresponds to a time scale of the order of the inverse of the sum of positive Lyapunov exponents, or the inverse of the topological entropy per unit time. The term entropy burrier has been introduced for this bound since it prevents us from reaching the energy levels directly with the trace formula [81]. An analytic continuation is in general required, after which the properties of the energy spectrum may turn out to differ considerably from the ones suggested by the trace formula. Indeed, if we believe the many numerical observations, in particular on the Riemann zeta function [82], this analytic continuation is at the origin of the Wignerian repulsion by a mechanism that is still mysterious. In the domain (2.18), the zeta function or the trace formula may be assimilated to a quasiperiodic function of the energy that is completely determined by the periodic orbits, but there is no energy level in this region [33, 831. When the entropy barrier is crossed, the zeta function falls in a different class of functions, with properties that allow the energy levels to show the universal Wignerian statistics. The conclusion is that dynamical instability and the Lyapunov time scale does not play a

i)

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F? GASPARD AND I. BURGHARDT

direct role in discrete energy spectra, while the spectrum of the classical periods is of primary importance. Other statistical quantities have been proposed and applied that can probe the intermediate scales from below the mean spacing up to the semiclassical scales. Among these are the two- and multilevel cluster functions, the related number variance C2, and the spectral rigidity A3, as well as the twolevel form factor b(k),which is the Fourier transform of the two-level cluster function Y ~ ( E[73, ) 74, 76, 841. These statistical quantities are based on a semilocal rescaling of the energy spectrum by the average level density so that they lose any reference to the intrinsic energy or time scales. The C2 and A3 statistics are used to complement the spacing statistics in testing the irregularity of the energy spectrum and have been applied to a number of molecules, notably to NO2 by Jost and co-workers [6]. The two-level form factor b(k) is known to present a minimum at short rescaled times in systems with Wignerian repulsion, which has been called a correlation hole [85]. However, minima similar to the correlation hole may also appear in the form factor b(k) of classically integrable systems, as discussed by Wilkie and Brumer [86]. Therefore, the correlation hole does not provide a clear evidence of random-matrix behavior if it is not used in conjunction with other statistical criteria. Recently, a distinction between classically integrable and chaotic systems has been observed in the fluctuation properties of the oscillatory part of the staircase function No&) = N ( E ) - N,,(E). In classically integrable systems, this oscillatory part is proved to be an almost-periodic function of energy with non-Gaussian fluctuations [75]. In contrast, it is conjectured that No&) has Gaussian fluctuations in classically chaotic systems, as numerically observed even in the aforementioned classically chaotic geodesic systems showing a Poisson-like spacing distribution [87]. More recently, new methods have been developed which connect more closely classically chaotic systems to random-matrix theories [881. The irregularity of the spectrum has consequences on the properties of the matrix elements of observables like the electric dipole moment and, thus, on the radiative transition probabilities. For radiative transitions, a single channel is open and the statistics of the intensities follow a Porter-Thomas or x 2 distribution with parameter u = 1, as observed in NO2 [5,61. The preceding considerations are essentially based on the model of random-matrix ensembles proposed by Dyson and others in the 1960s. Recent works, in particular by Casati and co-workers [89], have focused on band random matrices. Such matrices naturally arise in quantum systems with subspaces coupled only to next-neighboring subspaces such as for electronic states in a chain of atoms or in the kicked rotator. In such systems, localized states are observed that present a level statistics intenne-

EMERGENCE OF CLASSICAL PERIODIC ORBITS AND CHAOS

5 19

diate between Dyson’s universal statistics. The level repulsion is numerically observed to depend on the degree of localization of the eigenfunctions [89]. These results open broad perspectives in our understanding of irregular spectra.

B. Statistics of Level Curvature and Other Parametric Properties Recently, there has been a growing interest in parametric properties of the energy levels [90]. For example, when the quantum system is interacting with an external field, the energy spectrum as well as the matrix elements depend on the magnitude of the external fields X. Several statistical quantities have been proposed to test the sensitivity of the spectrum to external perturbations: the parametric number variance [91], the parametric velocity correlation function [92], the curvature distribution [93-951, as well as Fourier transform quantities such as parametric spectral correlations [96]. The level curvature, defined as K, = d2E,/dh2, has been considered in early works on irregular spectra [93, 941. Recently, the tail of the curvature distribution at large curvatures has been shown to have universal properties inherited from the spacing distribution [95]. In parallel with the dependencies of the type P , ( S ) - S’ of the spacing distribution for the three universality classes introduced by Dyson (v = 1 for orthogonal ensembles, v = 2 for unitary ensernbles, v = 4 for symplectic ensembles), the tail of the curvature distribution behaves as [95] 1

- (KJY+2 where

K

for 1.1

-+=

= P K / ( v m , , ) is the semilocally rescaled curvature with

P-’

=

((k, - (kn))2) and En = dE,/dX. This universal behavior has been numeri-

cally confirmed for several systems [97](e.g., the hydrogen atom in a magnetic field [98]) and experimentally observed in microwave cavities [99]. Zakrzewski and Delande have proposed global density functions for the curvature distributions that incorporate the universal tail [98]. Von Oppen [ 1001 and Fyodorov and Sommers [1011proved that the Zakrzewski-Delande functions are actually the exact curvature distributions in the Gaussian ensembles of infinite random matrices. The universality* of the Zakrzewski-Delande curvature distributions is still being debated. The behavior (3.2) also appears in parametric variations of the line intensities [102]. Furthermore, the parametric sensitivity of eigenfunctions has been considered, as well as general*Universality holds if a distribution applies not only to the Gaussian ensembles but also to the other ensembles based on the different orthogonal polynomials, such as the Legendre ensembles, within each of the three Dyson universality classes: OE, UE, and SE [73].

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I? GASPARD AND I. BURGHARDT

izations to parametric motion of the energy levels with respect to more than a single external parameter [103]. Some paramettic properties have also been investigated in molecular systems [104]. C. Time Domain I . Beyond Heisenberg i7me

The different scales in the energy spectrum are in correspondence with different time scales in the evolution of the quantum system. The small energy scale is in correspondence with the long time scale and vice versa. The time evolution immediately follows from the energy spectrum. Take as an example the time evolution of some observable X, which may be, for iptance, the position or the momentum pertaining to a degree of freedom (X = or

b):

Other examples are the time autocorrelation functions of the form (2.40). The long-time evolution is determined by the structure of the energy spectrum on the smallest energy scale. Functions like (3.3) are known to be almost periodic in time, that is, functions that present time recurrences for arbitrarily long times [ 1051. This almost-periodic behavior occurs when a sufficiently long time has elapsed for the individual levels to be resolved by the dynamics, that is, essentially beyond the Heisenberg time, which is determined by the average level density according to (3.4)

Even if recurrences are known to occur, they are nevertheless rare: The larger the amplitude of the recurrence, the longer the time we have to wait before the recurrence. The long-time regime is of essentially a quantum-mechanical nature. As the number of degrees of freedom increases, the almost-periodic regime is delayed to longer time scales because of the increase of the level density with the number of degrees of freedom. 2. Emergent Classical Orbits and Vibrograms

Classical behavior is expected to occur on time scales that are shorter than the Heisenberg time (3.4). As the energy of the initial wavepacket increases, the quasiclassical and semiclassical regimes will extend over longer time intervals if the level density increases, as is usually the case in bounded

EMERGENCE OF CLASSICAL PERIODIC ORBITS AND CHAOS

521

systems. If the Wigner function corresponding to the initial wavepacket is localized around some positions and momenta in mock phase space, we may expect that the average (3.3) will follow closely the classical trajectory for a while before the Heisenberg time is reached, according to Ehrenfest’s theorem. For autocorrelation functions (2.40), the classical behavior (2.42) is expected on the same time scale under similar quasiclassical conditions. However, the autocorrelation function will soon become almost-periodic even if the classical system is mixing; thus transport properties may not exist in bounded quantum systems since the Green-Kubo integrals would be divergent. For these reasons, classical properties may be considered to emerge from quantum dynamics only on short time scales. This limits the observation of the periodic orbits to those with a period shorter than the Heisenberg time. In Coulomb systems, it turns out that the Heisenberg time can become arbitrarily long as the ionization threshold is approached with an appropriate rescaling. Taking Fourier transforms of rescaled quantities has therefore been proposed to display the periodic-orbit spectrum of Coulomb systems like the hydrogen atom in a magnetic field [ 1061. In such rescaled Fourier transforms, the periodic orbits emerge in principle with an arbitrary precision since an infinite sequence of energy levels is available. However, this is not the case in molecular systems, where the relevant potentials are of Morse type, with a finite number of energy levels. To circumvent the difficulty due to the absence of appropriate rescaling for such potentials, it has been proposed to consider windowed Fourier transforms of the energy spectrum. In an early work, Kinsey and Johnson studied the continuous spectrum of the Hartley band of ozone using Fourier transforms with a scanning Blackman-Harris window function [107]. Recently, Hirai et al. [I081 have defined so-called vibrograms, which result from applying Gaussian window functions in the Fourier transform of the energy spectrum. The vibrogram allows us to unfold the energy spectrum into a diagram displaying time-energy features of the intramolecular dynamics. In the semiclassical limit, the vibrogram compares with the classical period-energy diagram, or (E, T)plot, as recently denoted in an independent work by Baranger et al. [109]. Let us also mention that a similar technique has been recently used by Rouben and Ezra [110] in the study of bifurcations of periodic orbits, where the unfolding in terms of an energy-time diagram is essential. The vibrogram technique has been applied to different bounded molecules-N0~ [ 1111, C2HD I1121, CSZ 114, 1131, C2Hz [108, 1141-and dissociating molecules-0~ [107], C02, H2S, and D2S [14]. By selecting windows in energy and time, this theoretical approach turns out to be related to the experimental “pump-dump” technique, that is, pumping of molecules with femtosecond laser pulses followed by the observation with a dump pulse. In this context, we will Consider below exper-

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iments by Zewail and co-workers that allowed the observation of penodic recurrences in NaI [ 1151. The windowed Fourier transform is defined by [lo81

S,(f,E)=

I"

n(E)exp

(E - E)2

i

- - Ef] dE A

(3.5)

where E is the width of the window and n(E) is the level density defined by (2.6). This windowed Fourier transform can be viewed as a Bargmann, Gabor, or Husimi transform of the level density or, vice versa, of the trace of the evolution operator. This method is known in acoustics as a sonogram [ 1081. The function (3.5) can be interpreted as the survival amplitude corresponding to initial wavepackets with a Gaussian distribution in energy,

where anare arbitrary phases. The vibrogram is given as a contour or density plot of the real function [lo81

(3.7) with a normalization such that the function (3.7) becomes a constant independent of the average level density at long times beyond the Heisenberg time, as suggested by Pique. As we discussed in Section I1 in relation to (2.41), a survival amplitude has a semiclassical behavior that is directly related to the periodic orbits by the Gutzwiller or the Berry-Tabor trace formulas, in contrast to the quasiclassical quantities (2.42) or (3.3). Therefore, we may expect the function (3.7) to present peaks on the intermediate time scale that are related to the classical periodic orbits. For such peaks to be located at the periodic orbits' periods, we have to assume that the level density is well approximated as a sum over periodic orbits whose periods T p = a& and amplitudes vary slowly over the energy window [E - ;E, E + $1. A further assumption is that the energy window contains a sufficient number of energy levels. At short times, the semiclassical theory allows us to obtain

EMERGENCE OF CLASSICAL PERIODIC ORBITS AND CHAOS

523

which shows that the vibrogram is a superposition of Gaussian peaks centered on the classical periods and of time width A t = h / e h The height of each peak is determined by the quantum amplitude of the corresponding periodic orbit, that is, by its stability properties. The spectrum of the periodic orbits can be resolved as long as the spacing between the periods is larger than the time width At. Due to the proliferation laws (2.17) and (2.32), the period spectrum soon becomes so dense that the peaks start to overlap, which turns out to occur even before the Heisenberg time. Therefore, only a finite number of periods can be resolved. In this sense, the periodic orbits are only emergent properties of the quantum systems; they are fuzzy objects limited by Heisenberg uncertainty relations. Because of its long-time character, the proliferation of periodic orbits remains outside the domain where we see classical properties emerge from wave-mechanical behavior. Nevertheless, the semiclassical theory provides us with a unique interpretation of the short-time recurrences in the vibrogram and of related survival amplitudes. Quantum-mechanically,wavepacket recurrences are obtained by a Fourier transform of the energy spectrum; that is, the Fourier transform establishes the connection between the complementary time and energy variables. We may hence proceed to an assignment of the complementary structures in the energy and time domains. Thus, in the energy domain, we have the structure given by the energy levels, which are assigned in terms of their quantum numbers. In the time domain, the complementary structure is given by the time recurrences which can be assigned in terms of the symbols or integers labeling the periodic orbits, as discussed in Section 11. In the semiclassical regime, the periodic orbits thus provide the connection between the time and energy structures, with the time recurrences corresponding to classical periods. This assignment will be illustrated below with the vibrogram analysis of several bounded molecules. Let us remark here that the classical correspondence we use below is established as usual with

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for non-degenerate normal modes and with

(3.10)

for doubly degenerate normal modes, where (6i,J i ) and (xi, Li) are canonically conjugate variables such that -Ji I Li I+Ji.

D. Diatomic Molecules

The vibrational motion of diatomic molecules is one dimensional so that the classical motion is integrable, as expected for two-body problems. The classical motion is periodic on each energy shell, the period being given by

T = rTp(E)= r

d 2 m dq jq< &=a q>

(3.11)

where r = 1, 2, .. , is the repetition number of the prime period. As a function of energy, the period is constant near the minimum of the potential and increases as the energy approaches the dissociation threshold, with the potential progressively broadening. At the threshold, the periodic orbit turns into the separatrix between bounded and scattering trajectories, which is known to have an infinite period [14]. Separatrices also exist if the potential has equilibrium points other than a minimum. If the potential has a maximum, the corresponding separatrix is at the origin of a so-called homoclinic bifurcation of the periodic orbit. The effect of the separatrices can be observed in the vibrogram representation. In one-dimensional systems, the amplitude Apr of the periodic orbits is constant with respect to the repetition number r, as shown by (2.34), when F = 1, so that recurrences due to repetitions of the prime orbit are observable.

a.

1. Morse-Type Model for I z ( 2 'C)

Figure 2 depicts the vibrogram corresponding to the dynamics on the ground state of iodine, modeled by a Morse potential with the equilibrium distance r = 2.67 A and the dissociation energy D = 12,542 cm-' [14, 1081. The periodic orbit and its repetitions clearly appear in the vibrogram. The classical

EMERGENCE OF CLASSICAL PERIODIC ORBITS AND CHAOS

525

1345

0

0

E [cm-'1

12600

Figure 2. Vibrogram of a Morse model of 1 2 ( i 'C) calculated with the function (3.7) with c = loo0 em-'. The solid lines are the classical periods given by (3.11).

periods obtained from (3.11) have been indicated and show a nice agreement. The period of the time recurrences increases near the dissociation threshold as expected; note, however, that, due to the finite number of energy levels, the classical recurrences tend to disappear near the threshold as a consequence of overlap between periods and of the almost-periodic fluctuations. 2. Experimental Vibrogram of NaZ by Zewail and Co-workers [I151 As a second example, consider the experimental observation by Zewail and co-workers of time recurrences in NaI, as recorded in a pump-dump experiment [ 1151. In such experiments, the pump pulse creates a wavepacket in the potential with a variable mean energy, similar to the wavepacket (3.6) except that the distribution is not generally Gaussian. In the semiclassical limit, the time recurrences occur at the periods of the emerging classical orbits. The corresponding vibrogram analysis has been carried out in Ref. 115, and Fig. 3 shows the resulting experimental vibrogram, where the lengthening of the period can be observed. Here, the potential is composed of two surfaces with an avoided crossing. Moreover, one of the surfaces extends above the dissociation threshold, which affects the amplitudes of the periodic orbits since a coupling to the continuum occurs by predissociation.

E. Triatomic Molecules

More complicated behaviors are expected for triatomic molecules (i.e., for three-body problems). In general, the analysis is facilitated by the fact that

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P. GASPARD AND I. BURGHARDT

5000

2500

0

, . 1 4ooo 8000 loo00

2000

6OOo

E [cm-'1

Figure 3. Experimental vibrogram of NaI obtained by Zewail and co-workers [115]. The triangles give the prime periods and the diamonds their repetitions. The solid line is a fit of the experimental points. (Experimental data from Ref. 115.)

Born-Oppenheimer potentials associated with bound electronic states feature minima around which the surface is nearly harmonic. Therefore, the motion is expected to be regular at low energies, as for coupled harmonic oscillators. Anharmonicities start to play a role as the energy increases. The anharmonicities cause the nearby energy levels to repel each other due to resonances of Fermi or Darling-Dennison type. A single such resonance may create local modes of vibration besides the normal modes without affecting the classical integrability, as we discussed earlier. Such phenomena can be pinpointed in the vibrogram from the analysis of the periodic orbits [14,110]. In triatomic species, the number of vibrational degrees of freedom is 4 for a linear molecule like CS2 and 3 for a nonlinear molecule like N02. Obviously, the search of periodic orbits is significantly more complicated than in diatomic molecules. Nevertheless, tools have been developed to obtain the shortest among the periodic orbits, involving in particular the identification of elliptic islands in Poincart5 surfaces of section, the use of integrability at low energies, and the search for commensurabilities among the frequencies. Let us discuss some details of the analysis for two particular molecules.

1. CS&?'C,') This molecule is currently being studied in the laboratory of Pique in Grenoble [ 116, 1171. In the ground state, the molecule of carbon disulfide is collinear and presents a Fermi resonance like carbon dioxide.

EMERGENCE OF CLASSICAL PERIODIC ORBITS AND CHAOS

527

The vibrogram of the experimental data has been interpreted by Gaspard et al. [14], who used an effective model Hamiltonian proposed by Pique et al. [ 1171 on the basis of a Dunham expansion including the Fermi resonance between the symmetric stretching mode and the degenerate bending mode. The classical counterpart of the model Hamiltonian is integrable. Therefore, the Berry-Tabor trace formula applies and the periodic orbits can be labeled by integers (nl,ny2,n3), which give the numbers of periods of the individual modes in the full period. In the analysis of the bulk periodic orbits, a simplification occurs for the bending oscillations. Because the Hamiltonian of a linear molecule depends quadratically on the angular momentum variable L2, the time derivative of vanishes with L2, in contrast to the conjugated angle x2 given by x 2 = &&l the time derivatives of the other angle variables, which are essentially equal to bi = oi. Therefore, the subsystem L2 = 0 always contains bulk periodic orbits that are labeled by (nl,n;, n3). In the analysis of the periodic orbits, it is convenient to first study the edge periodic orbits of relevant 1F , 2F, or 3F subsystems rather than to start with the bulk periodic orbits for which all the F = 4 degrees of freedom are active. The amplitudes of the edge orbits are significantly smaller than those of the bulk periodic orbits, as explained in Section 11. Nevertheless, they help to localize the bulk periodic orbits because the edge periodic orbits form a skeleton for the bulk phase-space dynamics in integrable systems. The periods of the three modes of CS2 taken individually are respectively equal to TY = 2?r/wl = 50 fs for the symmetric stretch, T i = 27r/w2 = 83 fs for the bend, and 7'; = = 21 fs for the asymmetric stretch. Since the asymmetric stretching mode 3 is of high frequency, we may expect that many bulk periodic orbits (nl,n:,n3) have periods close to the period of each edge periodic orbit (n~,ni,-)with n3 = (w3/wl)nl f: ( ~ 3 / 0 2 ) n 2 . Since w3 is large with respect to both the other frequencies, a number of closely neighboring commensurabilities are possible. Following this argument, we may conclude that an important role is played by the subsystem in which only the symmetric stretching and the bending modes are active, as studied in Ref. 14. This study [ 141 has shown that a period-doubling bifurcation associated with the Fermi resonance occurs in this subsystem at the energy E = 3061.3 cm-' (with Eep= 0).Below the Fermi bifurcation, there exist edge periodic orbits of normal type, which are labeled by (nl,nT2,-)normal. At the Fermi bifurcation, a new periodic orbit of type (2, lo, -)Fermi appears by period doubling around a period of 2T1 = 100 fs. This orbit is surrounded by an elliptic island that forms a region of local modes in phase space. Therefore, another family of edge periodic orbits of local type are born after the F e d bifurcation that may be labeled by the integers (nl,ni,-)lWa1. They are distinct

528

P. GASPARD AND I. BURGHARDT

from the periodic orbits of normal type because a homoclinic orbit (also called separatrix) separates the normal from the local orbits. The experimental vibrogram shows an important recurrence around 160 fs, which may be assigned to the edge periodic orbit (3,2*, Recently, the vibrogram analysis has been carried out by Michaille et al. [113] on the basis of another model proposed by Joyeux [ 1181 as well as on an ab initio potential fitted to the experimental data of Pique [1191. Essentially the same classical periodic orbits appear in the different models at low energies. In the same context, let us add that Joyeux has recently applied the Berry-Tabor trace formula to a 2F Fermi-resonance Hamiltonian model of CS2 [120] and carried out a classical analysis of several related resonance Hamiltonians [121]. At higher energies, a transition around 13,000 cm-I above the first vibrational level has been observed when the asymmetric stretching mode starts to interact with the other modes 171. Further anharmonic resonances may be expected in this region, which are the object of current studies.

This nonlinear molecule has been studied by several groups [5J and, recently, by Jost and co-workers 16, 1111. The ground electronic state of the molecule 2 2A1 appears very harmonic and may be modeled by a Dunham expansion. The excited electronic state 2B2 has its minimum about 10,000 cm-' above the minimum of the ground electronic surface. In the region above this minimum, the vibrational states of both electronic surfaces are intknnixed by nonadiabatic couplings beyond the Bom-oppenheimer leading approximation [52-541, which induce Wignerian repulsions between the vibrational levels. The spacing and intensity statistics have been obtained by Jost and co-workers [6] that obey respectively the Wigner and Porter-Thomas distributions, as predicted by random-matrix theories. The overlap between both electronic surfaces is therefore a mechanism of formation of irregular spectra that is extremely strong. The intermixing between both electronic surfaces explains the quenching of radiative transition rates, as discussed by Koppel et al. [53]. The vibrogram has also been obtained by Jost and co-workers in the region of 12,000-15,000 cm-*, where the vibrational spectrum is irregular [ 11I]. The main recurrence occurs at 47 fs and multiples thereof and corresponds to the bending frequency of about 714 cm-'. Thus, the spectrum is dominated by the bending motion in this regime of N02. A conical intersection is expected above 10,000 cm-' that has not yet been rigorously identified. This conical intersection also creates important anharmonicities in the fully diagonalized effective Harniltonians of Weigert

A

EMERGENCE OF CLASSICAL PERIODIC ORBITS AND CHAOS

529

and Littlejohn [54] that should destroy the near harmonicity of the lower surface and give rise to classically chaotic behavior.

F. Tetra-atomic Molecules

In tetra-atomic molecules that are linear, the number of degrees of freedom is F = 7,which again considerably complicates the analysis. Nevertheless, the dynamics of such systems can turn out to be tractable if the anharmonicities may be considered as perturbations with respect to the harmonic zero-order Hamiltonian. In such cases, the regular classical motions remain dominant in phase space as compared with the chaotic zones, and the edge periodic orbits of subsystems again form a skeleton for the bulk periodic orbits. Recent works by Herman et al. and Field et al. have focused on molecules of the family of acetylene, in particular CzHD [112) and C2H2 (see Refs. 122 and 123 and Field et al., “Intramolecular Dynamics in the Frequency Domain,” this volume). These linear molecules have three stretching modes, 1, 2, and 3, and two doubly degenerate bending modes, trans 4 and cis 5. Isotopic effects appear particularly striking in the vibrational dynamics, as shown in the comparative study of the dynamics of the above isotopomers. In the energy range 0-16,000 cm-’, the vibrational Hamiltonian of this molecule can be modeled by a Dunham expansion without anharmonic resonances of the classical form [ 1121

(3.12)

The classical periodic orbits of this completely integrable system can be obtained by solving Eqs. (2.31). The action space can be systematically scanned to search for approximate commensurabilities between the frequencies. The harmonic periods of the individual modes are respectively TY = 10.09 fs, T i = 2 ~ / 0 2 = 18.19 fs, T ! = 2 ~ / 0 3 = 13.07 fs, T t = ~?F/WI 2?r/w4 = 65.79 fs, and Tg = ~ ? F / W S= 50.48 fs. In this integrable system, the bulk periodic orbits can be labeled by the integers (n, n2, n3, n?, ny ), as explained in Section 11. As in CS2 [ 141, the frequencies of the angles x4 and xs of bending defined by (3.12) are proportional to L4 and L5 so that periodic orbits always exist

P. GASPARD AND I. BURGHARDT

530

SA

-

.Y(t 200

100

0

0

4000

8000

12000

E [cm-11 Figure 4. Vibrogram of C2HD calculated with 6 = 2000 cm-* from all the vibrational energy levels predicted by the Dunham expansion corresponding to the Hamiltonian (3.12) obtained by Herman and co-workers by fitting to high-resolution spectra [I 121. The periods of the bulk periodic orbits of Table I obtained numerically for the classical Hamiltonian (3.12) are superimposed as circles. On the right-hand side, the main labels ( n 4 , n 5 ) of the periodic orbits are given.

for which m4 = m5 = 0. Because of their low frequency, the bending modes determine the main recurrences of the vibrogram depicted in Fig. 4. These main recurrences at 200,250,400, and 450 fs correspond to the commensurabilities (n4,n5) = (3,4), (4,5), (6,8), (7,9), respectively (see Fig. 4). Table I shows a set of bulk periodic orbits associated with these recurrences. Here, again, several bulk periodic orbits correspond to the same commensurabilities specified only by (n4,n5) because the stretching modes are much faster than the bending modes. It is also important to notice that the periodic orbit (20, 11, 15, 3 O , 4') at about 200 fs is the shortest of the bulk periodic orbits and that all the shorter recurrences are due to edge periodic orbits in subsystems withf < F = 7. According to the Berry-Tabor formula, the amplitudes of these edge periodic orbits are necessarily smaller than the ones of the bulk periodic orbits [cf. (2.38)], which is in agreement with the amplitudesobserved in the vibrogram. This effect can be qualitatively described as a stroboscopic effect of Lissajous type [ 108,116,1171. The Berry-Tabor formula offers a systematic explanation of this effect in terms of the periodic orbits. 2.

12C2H2 (X'C')

An effective Hamiltonian for this system has been obtained recently by Herman and co-workers based on the high-resolution Fourier spectroscopy of

EMERGENCE OF CLASSICAL PERIODIC ORBITS AND CHAOS

53 1

Table I Recurrences of Vibrogram of C2HD and Their Periodic-Orbit Assignment [ 1121

-200

(20, 11, 15, 3'. 4')

-250

(25, 14, 19, 4', 5') (25, 14, 20, 4', 5')

-400

(40, 22, 30,6', (40, 22, 31, 6', (41, 22, 31, 6', (45, 25, 34, 7', (45, 25, 34, 7', (45, 25, 35, 7'. (45, 25, 36,.'7

-450

8') 8') 8') 9') 9') 9') 9')

IOOO-

5000250010002000350050002000800500-

the spectrum up to 12,000 cm-' [123]. Contrary to its isotopomer C*HD, acetylene presents important anharmonic resonances, due to the fact that the harmonic frequencies oi allow closely neighboring commensurabilities, which causes many perturbations among the vibrational levels. Actually, this molecule has been the object of an early study that showed the presence of Wignerian repulsion at high energies around 25,000 cm-' [3]. The effective Hamiltonian by Abbouti Temsamani and Herman [123] is composed of a diagonal part given by a Dunham expansion with all the x's and g's as well as y z ~ and , of a nondiagonal part including the following resonances [ 1231: 0 0

0

vibrational Z-doubling 45/45; bending Darling-Dennison interaction 44/55; stretch-bend anharmonic resonances 3/245, 1/244, 1/255, 14/35, 33/1244; and stretching Darling-Dennison interaction 11/33.

The work of Kellman [27] shows that the Hamiltonian of Ref. 123 preserves the constants of motion:

with ii; = 5;+ i d i , which are the remaining good quantum numbers of the vibrational dynamics. These quantum numbers label superpolyads in which

P. GASPARD AND I. BURGHARDT

532

the vibrational levels are intermixed so that (u1, vz, u3, 1155) are no longer good quantum numbers. The vibrogram of acetylene is depicted in Fig. 5, where we observe important differences with respect to the vibrogram of CzHD. The main recurrences of the vibrogram can be interpreted by the commensurabilities between the harmonic periods, which are here TY = 27r/w1 = 9.53 fs, 7': = . 453.65 fs, and T i = 2 ~ / ~16.57 2 fs, T! = 2 1 ~ / ~=39.76 fs, T: = 2 ~ / ~ = 27r/ws = 44.67 fs. In particular, the very strong recurrences at 50 fs and multiples thereof are due to commensurabilities between the stretching modes l and 3 and either of the bending modes 4 or 5. The longer recurrences are also due to commensurabilities among the trans and cis bending modes 4 and 5. The recurrences at 275, 320, and 380 fs can be assigned to commensurabilities ( q , n 5 ) = (5, 6), (6, 7), (7, a), respectively. We notice that the bending recurrences of C2HD and C2H2 are different. The only common recurrence (4, 5 ) corresponds to a real periodic orbit in C2HD but to a complex periodic orbit in the 2F bending subsystem of C2Hz (see Fig. 5). Another difference is

&, 1

500 400

300 200 100

1

0 1

0

I

I I

'

*

.

- !

4000

E [cm']

aooo

12000

Figure 5. Vibrogram of C2H2 calculated with e = 2000 cm-' from all the vibrational energy levels predicted by the effective Hamiltonian of Abbouti-Temsamaniand Herman [ 1231 obtained by fitting to high-resolution spectra. The period-energy diagram of some periodic orbits of the correspondingclassical Hamiltonian, given by (3.14) for the bending subsystem, is superimposed. The labels (n4,n5)refer to the edge periodic orbits (-, -, -,n!,n:) represented by solid lines when real and by short dashed lines when complex. The transition between real and complex periodic orbits is depicted by long dashed lines. The vertical dashed line marks the energy of the bifurcation as explained in the text. The bottom solid line around 10 fs is the edge periodic orbit (1, 1, -, -) while the next one around 50 fs is the edge periodic orbit (5. -, 5 , lo, -).

-.

EMERGENCE OF CLASSICAL PERIODIC ORBITS AND CHAOS

533

that the bending recurrences' amplitudes decrease around 7000-8000 cm- I above the minimum. In order to give a more quantitative interpretation, we analyzed the classical motion. The classical Hamiltonian is a function of the following actionangle variables: HCl(Jfl,J 2 , J 3 , J4, J s , L 4 , LS.81 - 0 3 , 81- 8 2 - 2 8 4 , 0 4 - 0 5 , x 4 - X S ) , so that (3.13) are also classical constants of motion. As another consequence, the dynamics can be reduced to a 4F subsystem that determines the motion of the 7F system. This 4F subsystem contains itself many subsystems with a smaller number of active degrees of freedom. In order to assign the recurrences around 50 and 100 fs, we considered the 3F subsystem J 2 = 4 = J S = 0, which is completely integrable. The Hamiltonian of this subsystem is of the form H c , ( J 1 , J 3 , J 4 , 0 1 - 0 3 ) . The stretching Darling-Dennison interaction is particularly intense with k l 1 / 3 3 = -102.816 cm-' so that the bifurcation leading to local modes occurs at very low energies. The edge periodic orbit (5, -, 5, lo, -) has been calculated numerically for comparison with the corresponding recurrence of the vibrogram. We observe in Fig. 5 a very good agreement. The integrability of this subsystem explains that the recurrence persists up to higher energies. The bending recurrences have been analyzed by focusing on the 4F bending subsystem J1 = J 2 = J 3 = 0, which we suppose forms a skeleton for the bulk periodic orbits of the full system. Certainly, this assumption has the consequence that the action variables J 4 and J S are forced to take larger Values at fixed energy than would be the case if the energy was distributed on all the degrees of freedom. Accordingly, the anharmonicities are stronger in the subsystem J I = J2 = J 3 = 0 than in the full system at similar energies. We should keep in mind these differences between edge and bulk orbits in the following discussion. The Hamiltonian of the subsystem has the form

which preserves the constants of motion, P,I = J 4 + J5 and &,,I = 4 + L5. so that the motion of this 4F subsystem is driven by a 2F subsystem, which allows us to represent the motion by two-dimensional Poincar6 mappings. With the further constraint that L 4 = L5 = 0 and x 4 = x 5 initially, the action-

534

P. GASPARD AND I. BURGHARDT

angle variables L4, L5, ~ 4 and , x 5 remain constant in time. Under these conditions, the dynamics reduces to the 1F subsystem in the variables [04 - 05, ~ ( Js J5)], which drives the angle O4 + 05. Periodic orbits corresponding to (n4,n5) can thus be obtained by integration of the one-degree-of-freedom subsystem. The bifurcation diagram of these periodic orbits is depicted in Fig. 5, where the analogy with CS;! appears clearly (cf. Fig. 52 of Ref. 14). The periodic orbits are generated in bifurcations along lower and upper borders defined by the periods of the 1F subsystems J4 = 0 and J5 = 0, respectively. A pitchfork bifurcation occurs at E = 4348.6 cm-' where local modes arise from J5 = 0. The progression of periodic orbits [-, -, -,n:,(nq + l)'] with n4 = 4, 5, 6, 7, 8, ... appears in nice agreement with the recurrences of the vibrogram, which confirms that these recurrences are due to bending. Moreover, the Poincari mappings of (3.14) at values of PCl fixed by the existence of (5,6) and L4 + L5 = 0 show the presence of classically chaotic motions with a bifurcation at E = 6900 cm-* (see Fig. 6). At this bifurcation, the periodic orbit (5,6) becomes unstable because one of its Lyapunov exponents turns positive, as shown in Fig. 7. The periodic orbit (6.7) destabilizes by a similar scenario around E = 7200 cm-I. These results show that the interaction between the bending modes leads to classically chaotic behaviors that destabilize successively the periodic orbits. For the bulk peri-

()

t.-'~ -

---.. ...... .

.... ... . .... ... . . . . . . . .. .. . . .. . . ......- .... . . .._- ._

-

.

-:.:.d

, ... , .._. . . ...., , .. _. .... . ... --.

Figure 6. Phase portraits of (3.14) in the Poincark surface of section x4 - x5 = 0 in the bending subs stem R - Lz = 0 in which the periodic orbit (n4,ng)= (5, 6) exists: ( a ) E = 5022.15 cm-', P = 6.97; (b)E = 7040,92 cm-'. P = 9.84; (c) E = 8060,88 cm-*, P = 11.31. (From Ref. 114.)

S‘E 0

2

W

UI

r

r

h

Y

11-

S’E-

S’E 0 Y

S’E-

odic orbits, for which the bending anharmonicities are weaker as explained above, we may expect that this chaotic destabilization occurs at higher energies. The transition to classical chaos can explain why the amplitudes of the main recurrences decrease, as observed in the vibrogram. The presence of a transition is further confirmed by Wignerian repulsions leading to irregular spectra in the superpolyads corresponding to the highest bending excitations [114]. This destabilization of the bending recurrences is due to the combined anharmonic resonances of vibrational l-doubling and of bending Darling-Dennison types, which arise because the frequencies of the normal

536

0.02

I

I

‘

-

-

s

n

x

0.0 I

-

0

.

4000

*

I

6000

E [ern-*]

8000

lo000

Figure 7. Lyapunov exponents of the periodic orbits (n4, n5) = ( 5 , 6 ) , (6,7)in the bending fs-’. (From Ref. 114.) subsystem of Hamiltonian (3.14). The numerical error is -5 x

modes are closer to the corresponding resonance conditions in C2H2 than in C*HD.* The vibrogram analysis [ 1081 based on the dispersion fluorescence spectrum of acetylene by Solina et al. [125] reveals a recurrence around 50 fs from 0 to 12,000 cm-I, which is similar to the one in the previous analysis. However, an important recurrence appears at 70 fs at higher energies from 4000 to 16,000 cm-’, which is caused either by anharmonic period lengthening or by a transition to a slower regime at higher energies.

G. Synthesis The vibrogram reveals the emerging periodic orbits of the vibrational dynamics. The periodic-orbit contribution to the spectrum concerns an energy scale larger than the mean spacing but small enough to “see” quantization effects. The recurrences of the vibrogram characterize the intramolecular dynamics in the time domain. We may attribute the presence of time recurrences to an inefficient vibrational energy exchange in the system, which is in agreement with the fact that the amplitudes of the recurrences are most prominent in classically integrable systems. Conversely, when the dynamics becomes classically chaotic, the recurrences tend to disappear, which provides evidence of a rapid energy exchange between the different vibrational degrees *Let us note that the energy range studied here is well below the vinylidene barrier at 17,300 cm-I or even the vinylidene minimum at 15,400 cm-I [124].

EMERGENCE OF CLASSICAL PERIODIC ORBITS AND CHAOS

537

of freedom. The role of strong anharmonic resonances in the increase of intramolecular vibrational relaxation has been discussed by Quack [ 1261, in particular. Here, we obtain a more detailed evidence and understanding of such mechanisms in terms of the instability of the emerging periodic orbits of the vibrational dynamics. We may speculate that the time recurrences predicted by the theoretical vibrograms of acetylene could be observed in femtosecond pump-dump experiments. In this regard, let us remark that, very recently, energy-time plots similar to the ones we describe here have been experimentally obtained for rubidium Rydberg wavepackets [1271. Concerning the structures of the energy spectrum below the mean spacing, the above discussion shows the existence of two mechanisms by which the energy spectra may become irregular:

1. On a single electronic surface, by accumulation of anharmonic resonances that cause the intermixing of the vibrational levels within the superpolyads. 2. On two overlapping electronic surfaces, by the intermixing of the vibrational levels due to the nonadiabatic coupling beyond the leading Bom-oppenheimer approximation. Near the minimum of the ground electronic surface, the anharmonicities generally play the role of perturbations so that the spectra are regular and the first mechanism is weak. It should become more important when the potential surface deviates significantly from the parabolic shape. The second mechanism, by contrast, may have a very marked effect, as illustrated by the example of NO2 [5, 61. The formation of irregular spectra by the two preceding mechanisms has implications for intramolecular vibrational and electronic relaxation. In the work by Jortner et al. [128], electronic relaxation processes are modeled by the coupling of an isolated level to a dense quasicontinuum of levels, which leads to approximately exponential time evolution. Here, in NOz, for instance, the coupling occurs between two sets of levels that are sparsely distributed, a scenario that was also envisaged by Jortner [128]. The new results obtained since 1970 show that the coupling between sparsely distributed levels leads to Wignerian statistical repulsion and a Porter-Thomas distribution of intensities, as modeled by random-matrix theories. In sparse irregular spectra, the time evolution of the intramolecular relaxation may be more complicated than an approximate exponential, as evidenced by the recurrences in the vibrograms. Against this background, we may envisage that irregular spectra may also appear due to the intermixing between the rotational levels of several overlapping vibrational bands. Such a mechanism would involve the

538

P. GASPARD AND I. BURGHARDT

rotational-vibrational interactions, which are expected to be strong, especially in the case of floppy molecules. In view of the recent results on irregular spectra observed in Rydberg atoms, we may also expect that similar phenomena exist in Rydberg molecules, as already suggested a few years ago by Lombardi et al. [129].

IV. OPEN SYSTEMS A. Energy and Time Domains If the quantum system is open (e.g., of scattering type), its energy spectrum will be continuous. Far from being featureless, the continuous spectrum shows structures that are interpreted in terms of the analytic singularities of the S matrix, that is, poles and branch cuts. The poles define resonances that correspond to metastable states characterized by a quantized energy but a finite lifetime, which is given by the inverse of the resonance half-width. Molecular resonances have been much studied recently and have been systematically used to interpret the dissociation dynamics of unimolecular reactions [9]. In open systems the discrete spectrum of the resonances plays a role similar to the spectrum of energy levels in bound systems. The global structures appearing in the spectrum provide a characterization of the dynamics. In particular, regular sequences of equally spaced resonances may be interpreted in terms of a recurrence period in the time domain, as shown by Heller [130], Pack [131], and others. On the other hand, irregular sequences arise in systems with several interfering recurrence periods, as we will discuss below. Thus, we can distinguish between regular and irregular spectra of resonances as well as between sparse and dense spectra of resonances. Since the resonances are distributed in a complex energy surface, we find spectra that extend not only along the real direction but also along the imaginary direction, as revealed in numerical studies, for example, for the inverted harmonic potential and the Eckart potential 19, 24, 1321. The resonance spectrum reflects the openness of the potential. We may distinguish between different types of potentials:

1. Weakly open potentials with a quasi-bounded region that is separated from the exit channels by potential barriers. This class of systems can be studied, for instance, with the R-matrix theory of Wigner [133] according to which the quasi-bounded region is first treated as a closed system with arbitrary boundary conditions at the bottlenecks, where the wave functions are thereafter matched with those of the exit channels. In such systems, we expect the resonances to have a number of properties in common with discrete energy levels, while the lifetimes are relatively long. The lifetimes are determined, on the one hand, by tunneling below the barrier and, on the

EMERGENCE OF CLASSICAL PERIODIC ORBITS AND CHAOS

539

other hand, by the number of open channels above the barrier. In this case, we expect a dense spectrum of resonances that is either regular or irregular depending on whether the potential gives rise to anharmonicities (cf. our previous discussion). 2. Strongly open potentials that have no energy minimum but instead a saddle point, a maximum, or possibly no equilibrium point. In such potentials, the dissociation process is direct and very fast so that the lifetimes are very short. If there is no equilibrium point, it is possible that no resonance exists. We may expect sparse spectra that are regular, or irregular depending on the spectrum of interfering periodic orbits. Apart from this classification, there are systems like Rydberg molecules where it is essential to take into account surface hopping between several coupled potential surfaces. As the work of Heller and co-workers [ 1301 has shown, the energy spectrum is complementary to the time autocorrelation function by a Fourier transform relation. The continuous nature of the energy spectrum implies the decay of the autocorrelation function. At intermediate times, the decay is given in terms of a superposition of exponentials determined by the lifetimes of the resonances. The real parts of the energies determine the periods of recurrences in the autocorrelation function. In the semiclassical method, these recurrences can be interpreted as emerging periodic orbits, as discussed below. At long times, the decay is controlled by the fine structures in the energy spectrum and, in particular, by the behavior of the spectrum near its thresholds. Near these energies, where the kinetic energy is minimal, the spectrum is associated with the slowest translational motions of the dissociation fragments in the exit channels. Hence, the decay obeys power laws that are of quasiclassical character.

B. Unimolecular Dissociation Rates: RRKM Theory and Distribution of Resonances

Around a fixed energy E, the average reaction rate is given by the famous RRKM formula, which can be derived from both quasiclassical and quanta1 considerations [71, 721. In the context of the Wigner R-matrix theory [133], the rate is given by the sum of the half-widths of all the open channels. The rate is thus the product of the number v ( E ) of open channels and the rate per = l/hn,,(E), where h is the Planck constant. The average channel zCchannel(E) reaction rate is obtained as [ 134, 1351

P. GASPARD AND I. BURGHARDT

540

in the case of potentials with nearly harmonic minima, where E$ is the minimum energy at the bottleneck, Eo IE* is the minimum energy of the potential in the quasi-bounded region, and F is the number of internal degrees of freedom. This formula is very powerful and can even explain the general behavior of the resonances in systems with other types of potentials. For Fdimensional well potentials we have %(E)- (E - E* ) ( F - ')/2/(E- E o ) ( ~2)/2 since the number of open channels increases as v ( E ) (E- E* ) ( F - ')I2 while n,,(E) - (E - E o ) ( ~ - ~In) /billiards, ~. the minimum energies are equal, E* = Eo, so that the decay rate is equal to %(E)- h independently of the number of degrees of freedom. This behavior has been numerically observed, in particular, for disk scattering systems. Formally, the formula (4.1) applies not only to deep potentials of type 1 but also to potentials of type 2 without minimum. In such a case, both extrema again coincide at the same energy, E* = Eo, but the average dissociation rate is constant, as observed for potentials with a near-harmonic saddle. Miller has shown that the number of open channels v ( E ) should be replaced by the sum of tunneling probabilities associated with the modes of the transition state when the energy approaches the barrier E = E$ 1136). Besides the average behavior of the rate, recent works have focused on the fluctuations in the distribution of resonances around the average rate. Random-matrix theories have been used to explain these fluctuations in the dense spectra of quasi-bounded systems. Wignerian repulsions are predicted and observed along the real energy axis. Along the imaginary axis, the Gaussian model by Porter and Thomas predicts chi-square probability distributions with parameter v for the reduced half-widths*:

-

d

Prob{c < x } = dx

y/2 - 1 2~/2r(~/2)

which is derived for v open channels contributing equally to the total halfwidth (1371. Let us emphasize that the Porter-Thomas distribution is here applied to the resonances of the molecular Hamiltonian in the absence of a radiation field. In the case of NO2 mentioned in Section 111, the same distribution with v = 1 was applied, by contrast, to the radiative linewidths of the molecular Hamiltonian [5, 61. Miller and co-workers performed a systematic analysis of the distribution of decay rates for the SO electronic state, as well as of the S1 - SO cou*The reduced half-widths are defined as the half-widths divided by the velocity associated with the resonance, I$ = r,,/u,,.

EMERGENCE OF CLASSICAL PERIODIC ORBITS AND CHAOS

541

pling matrix elements in deuterated formaldehyde, D2CO [4,1381. In this molecule, the dissociation occurs slightly above the barrier of the SO state, which has been studied by Stark level-crossing spectroscopy between the SO and the S, states by Polik et al. [138]. This analysis revealed that the coupling between the electronic states Si and SOinvolves only a small number of open channels, which is almost insensitive to the electric field: Y = 1.6 and dv/d!E = -0.02 (kV/cm)-' [138]. On the other hand, the effective number of open channels responsible for the decay of formaldehyde in the SO state is observed to be equal to Y = 3.8 as obtained by extrapolation to zero electric field at an energy of about 0.5 kcal/mol above the barrier located at 80.6 kcal/mol. This effective number depends on the electric field according to d v / d z = +0.28 (kV/cm)-' [138]. Recently, Miller and co-workers have obtained a generalized form of the distribution of unimolecular decay rates for the case of coupled open channels contributing with unequal partial half-widths [139].Further results have also recently been obtained in the statistical theory of reactions where the possibility of algebraic decay besides the RRKM exponential decay has been discussed [140].* If the number of open channels increases with energy, the Porter-Thomas distribution (4.2)shows that the imaginary parts of the resonance energies no longer accumulate near the real axis like in the case v = 1 and, to a lesser extent, in the case v = 2. For v >> 1, the resonances tend to move below the real axis, leaving an empty region. This general behavior is confirmed by semiclassical theories based on the inequalities (2.18H2.19)that actually predict the formation of such a gap empty of resonances below the real axis [14].This gap exists in strongly open systems of type 2, which we shall focus on below.

C. Dissociation on Potentials with a Saddle: Classical Properties The prototype potential surface invoked in chemical kinetics is a two-dimensional surface with a saddle equilibrium point and two exit channels at lower energies. The classical and quantal dynamics of such surfaces has been the object of many studies since the pioneering works by Wigner and Polanyi. Recent advances in nonlinear dynamical systems theory have provided powerful tools, such as the concepts of bifurcations and chaos, to investigate the classical dynamics from a new point of view and to perform the semiclassical *In this context, it should be pointed out that an algebraic decay has also been numerically observed in classical Coulomb-type models of atomic autoionization processes by Bliimel [141]. This might turn out to be relevant for Rydberg molecules. which also represent Coulomb-type systems. For the recent observation of algebraic decays in Rydberg atoms, see Ref. 142.

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quantization of the dissociation process. These new results provide a unifying scheme to interpret a broad set of observations on ultrafast processes and to predict the distribution of resonances in the nonseparable systems under investigation. We will start by describing the relevant aspects of the classical dynamics, introducing the concepts of the invariant set and the repeller, the Smale horseshoe and its symbolic dynamics, and the bifurcations at their origin [191. We then turn to the semiclassical quantization based on the Gutzwiller trace formula and the zeta functions. We proceed to show how this new theoretical framework allows us to explain the distribution of resonances in several molecules like HgI,, CO;?,and H3, to calculate their lifetimes, and to provide a synthesis with respect to previous work. We notice that the theory described here is issued from a previous work by Gaspard and Rice in which disk scattering systems were used as a vehicle for the study of unimolecular fragmentation 1331. The recent results obtained by Burghardt and Gaspard show the remarkable generality of these considerations in the context of ultrashort dissociation processes [lo, 141. I.

Classical Dynamics: The Repeller

Let us consider a triatomic molecule on an antibonding Bom-Oppenheimer potential surface with a saddle equilibrium point. Normal-mode analysis shows that the equilibrium point is characterized by real and imaginary frequencies. If the molecule is linear, the frequencies of the (nondegenerate) symmetric stretching and the (doubly degenerate) bending modes are typically real while the frequency of the (nondegenerate) asymmetric stretching mode (which corresponds to the reaction coordinate) is imaginary. The imaginary part is given by the positive Lyapunov exponent of the equilibrium point at the threshold energy [cf. (2.9) and (2.10)]. In nonlinear molecules, we typically have two instead of three real frequencies [23]. We focus on the nonlinear dynamics for the collinear configuration which we treat as part of the full multidimensional system. This is to be contrasted with two-degree-of-freedom models where the molecule is assumed to be frozen in some angular configuration, such that the bending degree of freedom is excluded from the dynamics. In our analysis, bending is taken into account in terms of linearized dynamics, which allows us to extend the results for the collinear situation to the full three-dimensional system. The restriction we must be aware of is that the three-dimensional system may have periodic orbits that are not of collinear type. For symmetric molecules XYX in the collinear configuration, the Hamiltonian is of the form

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where rl and r;?are the two distances between the central nucleus Y and the outer nuclei X's, while pxy = (mi'+ m;')-I is the reduced mass of the XY system. We will focus on such symmetric species but indicate throughout how the analysis extends to nonsymmetric XYZ-type molecules. At energies slightly above the saddle energy, there exists a single unstable classical periodic orbit. This periodic orbit corresponds in general to symmetric stretching motion (or an equivalent mode in XYZ-type molecules). The Lyapunov exponent of this periodic orbit tends to the one of the equilibrium point as the threshold energy is reached from above. Although unstable, this periodic orbit is an example of classical motion which leaves the molecule bounded. Other periodic and nonperiodic trajectories of this kind may exist at higher energies. The set of all the trajectories of a given energy shell that do not lead to dissociation under either forwardor backward-time propagation is invariant under the classical flow. When all trajectories belonging to this invariant set are unstable, the set is called the repeller R [19, 33, 35, 481. There also exist trajectories that approach the repeller in the future but dissociate in the past, which form the stable manifolds of the repeller Ws(R). Reciprocally, the trajectories that approach the repeller in the past and dissociate in the future form the unstable manifolds W , ( R ) (see Fig. 8). All other trajectories spend only a finite time in the scattering region. They are referred to as scattering orbits, and they constitute the vast majority

P

9 Figure 8. Schematic representation of a chaotic repeller and its stable W, and unstable W, manifolds in some Poincari surface of section ( q , p ) together with one-dimensional slices along the line L of typical escape time function I+.

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of trajectories. The phase-space volume occupied by the repeller is actually equal to zero so that a typical trajectory has a vanishing probability to be on the repeller or on its stable or unstable manifolds. Therefore, the trajectories have probability 1 to be of scattering type. Nevertheless, the repeller and its stable and unstable manifolds play a CNcia1 role in guiding and delaying the scattering trajectories. Indeed, evidence of the repeller can be obtained by comparing the time delay associated with the escape of different scattering trajectories from the transition-state region. To this end, we construct numerically the escape-time function, which gives the time taken by a trajectory to escape from a certain region 'B surrounding the scattering region [35]: T+(q,p) = max{T > 0 : @k(q, p) E B

for t

E

[O, T [ }

(4.4)

where @; denotes the flow on the energy shell H(q,p) = E and (q,p) is some initial condition on this energy shell, usually taken in a two-dimensional Poincar6 surface of section. Equation (4.4)is a function that becomes arbitrarily large when the initial condition comes close to the stable manifolds of the repeller. Indeed, on the stable manifold, the time to escape will be infinite. The singularities of the forward escape-time function (4.4) can thus be used to reveal the stable manifolds of the repeller. Conversely, the unstable manifolds are uncovered by the backward escape-time function obtained by reversing time [35],

which has its singularities on the unstable manifolds. Since the trajectories of the repeller are located at the intersections of the stable and unstable manifolds we can construct the repeller from the escape-time function: We take the sum of the absolute values of the forward and backward escape-time functions and identify the intersections of the singularities of this sum [lo],

By this method, we have been able to study the repeller, in particular for the systems HgI, and CO,. Let us add that the stable and unstable manifolds play the very important role of separatnces between reacting and nonreacting trajectories [25]. The invariant set undergoes bifurcation sequences in the course of which its topology and its stability are modified. These bifurcations are responsible

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for the transition from periodicity at low energies to chaos at higher energies. The following bifurcation scenarios are typically observed in saddletype potential surfaces. 2. Bifircation Scenarios Associated with Transition to Chaos As we said, the repeller is composed of only a single unstable periodic orbit at energies slightly above the saddle energy. For dissociative electronic surfaces of molecules like HgI,, COz, and H3, several bifurcation scenarios have been observed, which appear to be generic in the case of triatomic species with nuclei of similar mass, or of the light-heavy-light type. One famous scenariohas been first observed by Pechukas et al. in H3 [1431. These authors discussed the bifurcations in the context of reaction rate theory where the bottleneck of the reaction is identified with a dividing surface of minimal flux, that is, which should separate as well as possible the reacting from the nonreacting scattering orbits. A special role has been assigned to several periodic orbits of the invariant set of trapped trajectories, denoted periodicorbit dividing surfaces (PODSs). Each of these special periodic orbits has then been used to calculate reaction rates according to the variational reaction rate theory. Now in the context of our above discussion on the repeller, these special periodic orbits appear as part of the invariant set. This suggests that one might reconsider the general method of calculating variational reaction rates by taking into account a priori all the periodic orbits on the same footing and finding dynamical criteria to distinguish between them. One of the criteria is the stability of the periodic orbits. Another criterion, of topological nature, will be given below, which shows that in fully chaotic regimes the PODSs, as the shortest periodic orbits, generate all trajectories of the repeller by topological combination. As we show below, the PODSs turn out to be the most important periodic orbits in the bifurcation scenarios leading to chaos. Thus dynamical systems theory provides a reinterpretation of the role of these periodic orbits, which allows a new approach to the analysis of dissociation. We have shown elsewhere that the different bifurcation scenarios can be conveniently discussed in terns of area-preserving mappings generated by the action function [ 101

and which are of the form [14]

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qn+l =

as --- 4 n + Pn+ I aPn+I

(4.8)

Such Hamiltonian mappings are generated by a Poincark surface of section transverse to the orbits of the flow. Thus, v(4) plays the role of a potential function for the motion perpendicular to the periodic orbit. Note that the mapping takes into account the nonseparability of the dynamics. The theory of bifurcations shows that the different types of bifurcations can be described in terms of normal forms, which represent local expansions of the dynamics around the bifurcating periodic orbit [ 19, 32, 491. The purpose of the above mapping is to describe the successive bifurcations of the symmetric-stretch periodic orbit, starting from low energies above the saddle point. Appropriate truncation of the Taylor series of the potential u(q)around q = 0, which corresponds to the location of the symmetric-stretch orbit, provides us with the normal forms of the bifurcations [la]. The bifurcations relevant for the dissociation dynamics under discussion can be described by truncating at the sixth order in q, (4.9)

Let us remark that the mapping should be invariant under the reflection q -+ -q for symmetric molecules so that K = v = u = 0 in this case. (a) Supercritical or Direct Antipitchfork Bifurcation in Symmetric Molecules XYX (See Fig. 9). In one of the simplest scenarios, as observed

in HgI,, the unstable periodic orbit of symmetric-stretch type undergoes a supercritical antipitchfork bifurcation. The orbit becomes neutrally stable of elliptic type at a critical energy E,. Just above this critical energy, the symmetric-stretch periodic orbit is embedded in an elliptic island. The pitchfork bifurcation gives birth to two unstable periodic orbits that exist above the bifurcation. These two periodic orbits are bordering the elliptic island and are the PODSs identified by Pechukas and Pollak [143]. They correspond to nonsymmetric vibrational motion of the molecule with one bond being stretched out further than the other. Since the molecule is symmetric, two distinct such periodic orbits must exist. We shall denote as 1 and 2 these new periodic orbits of shortest period born in the antipitchfork bifurcation, while the symmetric-stretch periodic orbit is called 0 [lo]. The periodic orbits l and 2 inherit the instability lost by the symmetric-stretch orbit

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0

1

547

I\

Ed

Figure 9. Supercritical antipitchfork bifurcation scenario for symmetric XYX molecules; on the left, bifurcation diagram in the plane of energy versus position; in the center, typical phase portraits in some Poincari section in the different regimes; on the right, the fundamental periodic orbits in position space.

0 in the bifurcation, that is, they are hyperbolic (without reflection).* Their stable and unstable manifolds extend into the exit channels and form separatrices between nonreacting trajectories, which are immediately repelled by 1 or 2 back into the channels, and other trajectories that pass beyond the PODSs 1 and 2. The fate of these trajectories is still uncertain because of the homoclinic tangle [191between the inner branches of the stable and unstable manifolds of 1 and 2, which constitute partial separatrices between scattering orbits and orbits of the elliptic island. This homoclinic tangle forms a small chaotic zone surrounding the elliptic island. This chaotic zone as well as the interior of the elliptic island contain many periodic and nonperiodic orbits, but they are in general of longer periods than the periodic orbits 0, 1, and 2. The fact that the antipitchfork bifurcation is supercritical implies that all these new trapped orbits of the invariant set are born above the bifurcation *The cases of hyperbolic-withaut-reAectionand hyperbolic-with-reflection stability have to be distinguished. In both cases, the trajectories in the neighborhood of the periodic orbit trace out hyperbolic paths in the Poincak section, but if the stability is hyperbolic with reflection, the trajectories cross over between the branches of the hyperbola on each iteration.

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so that the periodic orbit 0 is the only orbit of the repeller below the bifurcation (unless other global bifurcations with successive sub- and supercntical bifurcations have taken place below the antipitchfork bifurcation we are considering). At still higher energies, the elliptic island undergoes a typical cascade of bifurcations in which subsidiary elliptic islands of periods 6, 5, 4,3 are successively created, which leads to the global destruction of the main elliptic island to the benefit of the surrounding chaotic zone. The cascade ends with a period-doubling bifurcation at Ed, above which the periodic orbit 0 is hyperbolic with reflection, and the main elliptic island has disappeared [ 1451. Subsidiary elliptic islands of very small area continue to exist until a last homoclinic tangency occurs at Ehr, above which all the trapped orbits of the invariant set are unstable of saddle type. The system is then fully chaotic. According to this scenario, the invariant set may contain quasiperiodic motion for energies E, < E < Ehr, while the main elliptic island exists only for E, c E < Ed < Eh,. The interval Ehr - E, turns out to be small as compared with the energy interval above Eh,, where full chaos has set in and the invariant set is a repeller. The above scenario is accounted for by the normal form (4.9)truncated at fourth order in q with K = v = u = p = 0 and x c 0, taking p as the bifurcation parameter, which increases with energy ( p thus plays a similar role as the total energy in the actual Hamiltonian dynamics). The antipitchfork bifurcation occurs at pa = 0. The fixed points of the mapping (4.8)are given by p = 0 and du/dq = 0. Since the potential is quartic, there are either one or three fixed points that correspond to the shortest periodic orbits 0, 1, and 2 of the flow. (b) Subcritical Antipitchfork Bifurcation in Symmetric Molecules XYX (See Fig. 10). A different scenario is observed in H3 [143] as well as COz [146], where the unstable periodic orbit 0 stabilizes in a subcritical antipitchfork bifurcation at E, that is preceded by the birth of the periodic orbits 1 and 2 in two symmetric tangent bifurcations at E,, < E,. (Let us notice that a subcritical antipitchfork bifurcation is an inverted pitchfork bifurcation.) Therefore, the unstable symmetric-stretch orbit 0 alone constitutes the repeller only before the tangent bifurcations giving birth to 1 and 2. These tangent bifurcations occur at some distance from the periodic orbit 0. At each of the tangent bifurcations two new periodic orbits are generated, one of which is neutrally stable and embedded in a small elliptic island while the other is hyperbolic (without reflection) and can be identified with either 1 or 2. Therefore, slightly above the tangent bifurcation, there exist five periodic orbits of shortest period, as first observed by Pechukas and Pollak [143]: 1 and 2, which are hyperbolic, and 1’ and 2’, which are elliptic immedi-

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P

Figure 10. Same as Fig. 9 for the subcritical antipitchfork bifurcation scenario for symmetric XYX.

ately above the tangent bifurcation. These orbits are ordered in space as 2 c 2 ' < 0 < 1'< 1. As the energy increases in the interval Ell < E < E,, the orbits I' and 2' progressively shift toward the symmetric-stretch orbit 0 and merge at the subcritical antipitchfork bifurcation. Just below this bifurcation, 1' and 2' are elliptic while 0 is still hyperbolic (without reflection). Between Err and E,, the periodic orbits 1' and 2' may either remain of elliptic type or become hyperbolic in the energy interval [Ed&,Ed&'] such that E,, C Ed8 c Ed&* C Ell. Above the subcritical antipitchfork bifurcation E, c E, the symmetricstretch periodic orbit 0 is elliptic and surrounded by a main elliptic island. From there onward, the bifurcation scenario is similar to the previous case as energy increases. The main elliptic island undergoes a cascade of bifurcations ending with a period doubling at Ed above which the periodic orbit 0 is hyperbolic (with reflection). At higher energies, there is a last homoclinic tangency Eh, leading to complete chaos. In this scenario, quasiperiodic motions exist for energies E,, c E c Ehr with a main elliptic island being present for E,, < E, < E c Ed < Eht. Here, again, the interval Eh, - Ell turns

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out to be small as compared with the energy interval above E h t , in which the chaos is complete. This scenario is described by the normal form (4.9) with K = v = u = 0, x > 0, and p < 0. The antipitchfork bifurcation occurs at pa = 0, but it is preceded by a pair of tangent bifurcations at prr= x2/4p < 0.For prr< p < pa = 0, five fixed points exist that correspond to the five PODSs 2, 2’, 0, l’, and 1 of the flow. (c) Tangent Bifurcation in Nonsymmetric Molecules XYZ (See Fig. 11). If the molecule is not symmetric, as is the case for the transition complexes of FH2 and He12 studied by Poll& et al. 11431,the flow is not symmetric under reflection through the central nucleus Y, which is no longer compatible with the pitchfork bifurcation scenario. In this case, one of the simplest scenarios is given by (4.9) with Y = u = p = 0 but K 9 0, which breaks the reflection symmetry under q -+ -4, In this case, the antipitchfork bifurcation is replaced by a tangent bifurcation. Below the tangent bifurcation, there exists the unstable periodic orbit of hyperbolic type that is the analogue of the symmetric-stretch periodic orbit 0 in the previous scenarios. This orbit may continue to exist without changing its stability through the

2

0”

1

I

I

Figure 11. Same as Fig. 9 for the tangent bifurcation scenario for nonsymmetric XYZ.

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55 I

tangent bifurcation: It is simply shifted to the right- or left-hand side, where it turns into the analogue of either 1 or 2, depending on the sign of K . At some distance from this unstable periodic orbit, a tangent bifurcation occurs in which a pair of periodic orbits is born, one of which is elliptic and the analogue of 0 above the bifurcation, while the other is hyperbolic (without reflection) and the analogue of either 2 or 1. Above this main tangent bifurcation, the three shortest periodic orbits 2 < 0 c 1 exist, and the elliptic periodic orbit 0 is embedded in a main elliptic island that undergoes a cascade of bifurcations essentially similar to the previous ones. Therefore, the major difference between the nonsymmetric and the symmetric (supercritical) cases is the smooth exchange of roles between the central and one of the bordering shortest periodic orbits. In the mapping (4.9) with v = u = p = 0 and x c 0, this main tangent bifurcation, which replaces the antipitchfork bifurcation, occurs at p, = ~ ( - x K 2/4)'/3, which is shifted from zero to positive values when K does not vanish. Other more complicated scenarios may occur if x is close to zero and changes its sign, in which case higher order terms should be included to describe the bifurcation scenario. To conclude, the normal form (4.9) allows us to classify the different types of bifurcations leading to the emergence of a chaotic repeller, to the description of which we now turn. As soon as more than one unstable periodic orbit exist in the invariant set, the stable and unstable manifolds of these periodic orbits may form homoclinic intersections. According to a theorem by Birkhoff, chaotic behaviors may exist in the flow under these circumstances [19]. However, as long as not all of the stable and unstable manifolds intersect transversally, but certain manifolds present homoclinic tangencies, other theorems, in particular by Shil'nikov and Gavrilov [147], show that elliptic periodic orbits exist, This means that tiny elliptic islands sustaining quasiperiodic motions persist. The dynamics is then a mixture of chaotic and regular motions, which requires a special treatment both classically and quantum mechanically. Now a very remarkable property of the systems we are considering here is that all the stable and unstable manifolds do intersect transversally at sufficiently high energies above the transition region, that is, above the last homoclinic tangency. Thus the energy windows where the motion is fully chaotic turn out to be very broad. This allows us to apply without further approximation the Gutzwiller trace formula and the zeta function quantization. In the fully chaotic regime, the repeller, which is highly unstable, can be constructed as explained above in terms of the sum of absolute values of the forward and backward escape-time functions, which displays the folding

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structures formed by the stable and unstable manifolds. In this way, it is possible to confirm the transversality of the intersections between the stable and unstable manifolds, which guarantees that the system is purely hyperbolic.

3. Fully Chaotic Regime: Smale Horseshoes The key concept in understanding the fully chaotic regimes of classically chaotic systems has been introduced by Smale [148]and is a repeller with the shape of a horseshoe. The dynamics on the Smale horseshoe is described by a global Poincark mapping of the type (4.7)-(4.9), for instance with x = u = p = 0, which defines the area-preserving Hknon map. Horseshoe structures are formed by a mechanism of stretching and folding in phase space: Due to the nonlinear classical motion, a given phase-space region stretches out and then folds back onto the initial domain. This mechanism lets an infinite number of trapped orbits appear (see Fig. 12). In the Smale horseshoe and its variants, the repeller is composed of an infinite set of periodic and nonperiodic orbits indefinitely trapped in the region defining the transition complex. Ail the orbits are unstable of saddle type. The repeller occupies a vanishing volume in phase space and is typically a fractal object. Its construction is based on strict topological rules. All the periodic and nonperiodic orbits turn out to be topological combinations of a finite number of periodic orbits called the fundamental periodic orbits. Symbols are assigned to these fundamental periodic orbits that form an alphabet A = {0,1,2,. . . ,M - 1 }. The other orbits of the repeller are in one-to-one correspondencewith symbolic sequences constructed by concatenation of the symbols,

In particular, the periodic orbits are in correspondence with finite sequences such as wlw2...wp of period p. The periodicity occurring in the symbol sequences translates into the periodicity of the corresponding trajectory crossing the Poincar6 surface of section. The concatenation is performed without constraint on the successive symbols so that the motion on the repeller corresponds to a Bernoulli random process. The regions around the fundamental periodic orbits are successively visited in a random fashion without memory of the previous fundamental periodic orbit visited. As a consequence, the periodic orbits proliferate exponentially with their period, as described by (2.17). The topological entropy per symbol is equal to hrop= In M. In the horseshoe that has two branches, the number of fundamental peri-

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Figure 12. Formation of a Smale horseshoe with (a)two branches, (b) three branches.

odic orbits is M = 2. Smale has also proposed variants of the horseshoe in which the number of branches is higher. In our context, Burghardt and Gaspard have discovered that horseshoes with three branches describe the repeller at high energies in collinear models of HgI, [lo]. The three fundamental orbits are the symmetric-stretch orbit 0 together with the off-diagonal periodic orbits 1 and 2, which are born in the

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bifurcations described above. In the fully chaotic regime, the periodic orbit 0 is hyperbolic with reflection, while 1 and 2 are hyperbolic without reflection. All the other periodic and nonperiodic orbits of the repeller are topological combinations of these three fundamental ones according to the rule (4.10). Our recent works have shown that this result is of broad generality and should be expected for a general class of symmetric triatomic molecules with atoms of similar mass, or of light-heavy-light type. In particular, we have recently observed the same three-branch horseshoe in a collinear model of C02 [146], and the comparison with the results by Pechukas, Pollak, and Child [I431 strongly suggests that the repeller of dissociating H3 is also a three-branch horseshoe. The formation of three-branch horseshoes in scattering systems can be modeled by a global mapping like (4.7) and (4.8) with the potential [lo]

-

a model we proposed elsewhere with 6 = = 0. If p = S = 0, the poten-4,as applies to symmetric XYX molecules; tial is symmetric under q in the general case, (4.11) models nonsymmetric XYZ species. We should emphasize here, that the occurrence of the Smale horseshoe is not restricted to symmetric molecules but may extend to nonsymmetric molecules like F H 2 or HC12 [143], in which case the horseshoe has a nonsymmetric form. Other classically chaotic scattering systems have been shown to have repellers described by a symbolic dynamics similar to (4.10). One of them is the three-disk scatterer in which a point particle undergoes elastic collisions on three hard disks located at the vertices of an equilateral triangle. In this case, the symbolic dynamics is dyadic (it4 = 2) after reduction according to C3v symmetry. Another example is the four-disk scatterer in which the four disks form a square. The C,, symmetry can be used to reduce the symbolic dynamics to a triadic one based on the symbols {0,1,2}, which correspond to the three fundamental periodic orbits described above [ 141. In the presence of reflection symmetry with respect to the diagonal of the potential-energy surface, as in symmetric molecules or in the four-disk scatterer, Burghardt and Gaspard have shown that a further symmetry reduction can be performed in which the symbolic dynamics still contains three symbols A’ = {O, +, - } (101. The orbit 0 is the symmetric-stretch periodic orbit as before. The orbit + is one of the off-diagonal orbits 1 or 2 while - represents a half-period of the asymmetric-stretch orbit 12. Note that the latter has also been denoted the hyperspherical periodic orbit in the literature. The central feature of the symbolic dynamics is that it provides a complete classification of the periodic orbits: Once the symbolic dynamics has been

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established on the basis of the phase-space structure, no periodic orbit can be missed. It is remarkable that this scheme indeed appears to provide the key to the dynamics of a whole class of dissociative species over a broad energy range.

D. Dissociation on Potentials with a Saddle: Semiclassical Quantization 1. Quantization in Periodic Regime Just above the saddle energy, the quantization can be performed by the usual perturbation theory applied to scattering systems as described by Miller and Seideman [24]. This equilibrium poirit quantization uses Dunham expansions of the form (2.8) with imaginary coefficients. This method is valid for relatively low-lying resonances above the saddle, up to the point where anharmonicities become so important that the Dunham expansion is no longer applicable (see the discussion in Section 1I.B). The regime of validity of the Dunham expansion, that is, the regular or periodic regime, coincides with the energy range where the repeller consists of a single unstable periodic orbit. Indeed, it is the onset of the bifurcations we described above (which go along with the resonance conditions mentioned in Section II.B) that causes the divergence of the coefficients in the Dunham series. Thus, for the regular regime, equilibrium point quantization can be compared with periodic-orbit quantization for the symmetric-stretch periodic orbit. For the latter, we are applying the quantization condition given in terms of the zeta function (2.21), which is limited to the leading semiclassical approximation. In the presence of a single periodic orbit for the collinear dynamics of a triatomic molecule, the total zeta function factorizes into zeta functions for states of given bending excitation (u, 1 ) with u = 0, 1, 2, . .. and 1 = u, u - 2, ... , -u and of given quantum number rn = 0, 1,2, . . . [lo, 1491 leading to as many quantization conditions, ZvdE) = 1 -

exp[(i/A)S(E) - i(7r/2)p - 2 ~ i p ( E ) ( + u l)] =O lA(E)I '/2A(E)'"

(4.12)

where S ( E ) is the action of the symmetric-stretch periodic orbit at energy E. Besides its action, the unstable periodic orbit is characterized by its stability eigenvalue A ( E ) under collinear infinitesimal perturbations transverse to the direction of the orbit as well as by the doubly degenerate stability eigenvalue exp[i2xp(E)] under infinitesimal perturbations of bending type. Here, we assume that the periodic orbit is unstable with respect to asymmetric stretching perturbations lA(E)I > 1 and neutrally stable with respect to bending perturbations characterized by the rotation number p(E). Moreover,

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the motion of the asymmetric stretching perturbations is also characterized by the so-called Maslov index, which is the winding number of the stable or unstable manifolds during a prime period of the periodic orbit 1301. (The Maslov index is constant with respect to all classically continuous parameters such as E as long as the topology of the periodic orbit does not change.) All these quantities can be obtained as functions of energy E by numerical integration of the classical Hamiltonian equations. A polynomial fitting of the numerical data for S(E), A(E), and p ( E ) can then be carried out to obtain the analytic dependence on E. Rotation can in principle be included in this scheme by calculating the periodic orbits for a rotating molecule. The resonances are then obtained by searching for the complex zeros of the zeta functions (4.12) in the complex surface of the energy. Assuming that the action is approximately linear, S ( E , J ) = T(E - E$ ), while the stability eigenvalues are approximately constant near the saddle energy ES , the quantization condition (4.12) gives the resonances [101 Enm"l ES

+T

2rAp

(u + 1) - i

ti ln(lA)'/2A") T

(4.13)

These zeros are thus labeled in terms of the expected quantum numbers for a triatomic molecule: n is the principal quantum number corresponding to the progression in symmetric stretching excitation; u and I are the quantum numbers for the bending excitations. They appear in (4.12) in the harmonic approximation for the bending excitations. We further have the quantum number m associated with the unstable asymmetric stretching excitations. The result (4.13) shows that the lifetime of the resonances decreases with increasing value of the Lyapunov exponent X = ( 1 / T ) In IAl as well as with the quantum number m of asymmetric stretch. The corresponding resonant states have n nodes along the symmetric stretching direction and rn nodes along the asymmetric stretching direction. The result (4.13) suggests that the excitation of an m = 1 state instead of an m = 0 state from vibrational states with corresponding symmetries on a lower electronic surface should lead to a drastic reduction of the lifetime, approximately by a factor 3. The derivation of (4.13) shows that the equilibrium point quantization and the periodic-orbit quantization can be compared term by term. This comparison shows that the periodic-orbit quantization is able to take into account the anhannonicities in the direction of symmetric stretch. However, the anharmonicities are neglected in the other directions transverse to the periodic orbit. Their full treatment requires the calculation of A corrections to the Gutzwiller trace formula, as shown elsewhere [14]. It should be noted that this periodic-orbit quantization is no longer valid in

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the immediate neighborhood of the first bifurcation where A approaches the value 1 for the bifurcating orbit, and a uniform approximation is required to take into account the nonlinear stability properties of the periodic orbit [49, 501. Indeed, equilibrium point quantization turns out to have a better behavior than periodic-orbit quantization as the region of the bifurcation is approached.

2. Quantization in Transition Regime For molecules composed of atoms with similar mass, as well as light-heavy-light species, the transition region, which covers the bifurcations leading up to the fully chaotic regime, turns out to have a very small extension in energy, which might even be smaller than the spacing between two resonances of successive principal quantum number n. In this case, the resonant states extend in mock phase space over a volume that is larger than the volume of the elliptic island around the symmetric-stretch periodic orbit. Under such circumstances, the properties of the resonant states are essentially determined semiclassically by the classical dynamics outside the elliptic island in regions that are far from the regions of validity of linearized stability analysis. The dynamics in these regions affects the local dynamics only at the antipitchfork bifurcation, which requires the knowledge of the nonlinear stability properties of the periodic orbit in order to obtain the semiclassical quantum amplitudes, as discussed in Section 1I.G. By contrast, in heavy-light-heavy molecules such as HMuH, ClHCI, or IHI, a very extended elliptic island exists in the classical phase space [150]. In such cases, the elliptic island may be the support of several metastable states that can be obtained by Bohr-Sommerfeld quantization. Their lifetime is determined by dynamical tunneling from inside the elliptic island to the outside regions. 3. Periodic-Orbit Quantization in Fully Chaotic Regime

Periodic-orbit theory provides the unique semiclassical quantization scheme for nonseparable systems with a fully chaotic and fractal repeller. AS we mentioned in Section 11, the different periodic orbits of the repeller have quantum amplitudes weighted by the stability eigenvalues, and the periodicorbit amplitudes interfere among each other as described by the zeta function. The more unstable the periodic orbit is, the less it contributes in (2.24). Therefore, only the least unstable periodic orbits play a dominant role. Let us consider here the case of a repeller that is a Smale horseshoe with three branches as described above. The periodic orbits are in one-to-one correspondence with the bi-infinite sequences constructed from the symbols {0,1,2} associated with the three fundamental periodic orbits. A complete list of periodic orbits can be established: { p} = {0,1,2,01,02,12, . ..}. Here,

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the zeta function also factorizes into different zeta functions corresponding to the approximately good quantum numbers (i,e., associated with bending in our case), but each zeta function remains a product over all the periodic orbits

(4.14) where S,, Ap, pp, and pp are the action, the asymmetric stretching stability eigenvalue, the Maslov index, and the rotation number of bending, respectively, which are required for each period orbit. The product over the periodic orbits is expanded as in (2.24) into the so-called cycle expansion [lo, 141

P

= 1 - to - t l - t2 - (tol - t o t l ) - (rO2 - t 0 t 2 ) - (tI2 - t l t 2 ) - ... - \ * fundamental po

=O

(4.15)

The first three terms beyond 1 are the semiclassical quantum amplitudes of the three fundamental periodic orbits. The other terms appear as corrections that may be neglected in a first approximation. Indeed, each of these terms represents a difference between the amplitude of some periodic orbit o 1 w2 - - .w p minus the amplitudes of the shorter periodic orbits entering into its topological composition. In many cases, it turns out that the amplitude of the periodic orbit is very well approximated by the product of amplitudes of its topological parts so that an approximate cancellation holds within each of these terms [lo]. Of course, better approximations can be obtained by including more and more of these terms. In the case of symmetric molecules, we have that tl = t 2 so that our approximate quantization condition becomes [lo]

EMERGENCE OF CLASSICAL PERIODIC ORBITS AND CHAOS

559

0 = 1 - to - 2tl

In the three-branch horseshoe, the periodic orbit 0 is hyperbolic with reflection and has a Maslov index equal to po = 3 while the off-diagonal orbits 1 and 2 are hyperbolic without reflection with the Maslov index p1 = 2 [ 101. Fitting of numerical actions, stability eigenvalues, and rotation numbers to polynomial functions in E can then be used to reproduce the analytical dependence on E. The resonance spectrum is obtained in terms of the zeros of (4.16) in the complex energy surface. We remark that an expression like (4.13) can no longer be derived in general because of the interference between the two amplitude terms in (4.16). This is the general feature of a nonseparable regime where the spectrum of resonances loses its regularity. When there exist two fundamental periodic orbits, we may expect that the spectrum of resonances still displays quasiperiodic regularities, as is the case for the three-disk scatterer [33]. However, for repellers with more than two fundamental periodic orbits, the regularities are even further destroyed and the spectrum becomes irregular. By inspection of (4.16) we can infer that the spectrum of the periods plays a crucial role in the distribution of the resonances. If the fundamental periodic orbits have very similar periods &So = (nearly degenerate period spectrum), we may expect that the resonance spectrum is still quite regular, as in the periodic case (4.12), but with an overall amplitude given by the sum of interfering amplitudes. On the other hand, the resonance spectrum will be irregular if the periods of the fundamental orbits are very different. A very important role is also played by the stability eigenvalues with respect to the unstable asymmetric stretching perturbations. If among the stability eigenvalues for two given periodic orbits one is much larger than the other, the corresponding amplitude may be neglected so that the condition (4.16) reduces to the periodic case (4.12) for the less unstable of both fundamental periodic orbits. For instance, if IAol >> (A, I > I, the resonance spectrum is determined by the off-diagonal periodic orbits 1 and 2 and should present a regular progression. Similar considerations apply to the eigenfunctions associated with the resonances. According to Eq.(2.26), the contribution of a periodic orbit to the eigenfunctions is also given in terms of the stability eigenvalue A,,. Therefore, the eigenfunction is localized essentially around the least unstable periodic orbits. This result shows that the

560

P. GASPARD AND I. BURGHARDT

resonant states are localized around the off-diagonal periodic orbits 1 and 2 if the periodic orbit 0 is markedly more unstable than the off-diagonal ones. Since the off-diagonal orbits 1 and 2 are composing the periodic orbit 12, the resonant states would hence be localized around the hyperspherical periodic orbit 12. This coincides with an interpretation found in the literature according to which the orbit 12 represents the resonant periodic orbit (RF'O) [143]. Another remark is that the resonances exist only below the line defined by (2.18) and (2.19) so that there is a gap between the real energy axis and the resonance spectrum. This is the feature of a strongly open scattering system in which the decay process is ultrafast. This gap is given in terms of the topological pressure that is the leading zero, so(E) = P ( p ; E ) , of the inverse Ruelle zeta function, (4.17)

When p = {, the leading zero gives thus an approximate value of the gap (2.18) in the spectrum of scattering resonances. If the repeller has only one unstable periodic orbit, the topological entropy vanishes in Eq. (2.19) so that the gap is completely determined by the sum of positive Lyapunov exponents as described by (4.13). In comparison with the periodic case, the gap is reduced when the repeller becomes chaotic, which is a direct effect of emerging classical chaos on a quantum property. This lifetime lengthening which has been studied in detail by Gaspard and Rice [33] can be regarded as the precursor of the fluctuations of the quantum decay rates around the average RRKM rate, which appears as the degree of openness of the potential decreases from strongly open to weakly open. This lifetime lengthening studied by Gaspard and Rice also appears when the longest quantum lifetimes are compared with the average classical lifetime [33]. In Section 11, we made the distinction between quantal quantities such as survival amplitudes and quasiclassical quantities such as averages or correlation functions. The decay of the quantal quantities is determined by the scattering resonances while the decay of the quasiclassical quantities is determined by the classical escape rate. According to (2.46), the escape rate can be calculated as the leading zero of the classical zeta functions (2.44),which takes here the approximate form (4.17) with = 1. Gaspard and Rice have pointed out that the average classical lifetime is shorter than the longest quantum lifetimes for chaotic repellers while they coincide for periodic repellers [33]. A similar phenomenon exists in weakly open potentials where the quantum resonances are distributed around the average RRKM

EMERGENCE OF CLASSICAL PERIODIC ORBITS AND CHAOS

561

quasiclassical rate, as described, in particular, by the Porter-Thomas distribution. Therefore, here also the quantum resonances may have a lifetime longer than the quasiclassical lifetime, which may also be considered as an effect of classical chaos. It should be emphasized that this argument is based on the distribution of the eigenenergies in their imaginary parts but not in their real parts. The periodic-orbit quantization can be used to calculate not only the resonances but also the full shape of the photoabsorption cross section using (2.26) and (2.27). This semiclassical formula for the cross section separates in a natural way the smooth background from the oscillating structures due to the periodic orbits. In this way, the observation of emerging periodic orbits by the Fourier transform of the vibrational structures on top of the continuous absorption bands can be explained. To conclude this section, let us add that the formulas (4.15H4.17) can be generalized in a straightforward way to repellers with more than three fundamental periodic orbits. In the following, these tools are applied to several dissociating molecules.

E. Ultrashort-Lived Resonances in Watomic Molecules 1. Hg12

Zewail and co-workers have reported femtosecond laser experiments on the photodissociation of mercury(I1) iodide (HgI,) [ 1511:

h~ + HgI,(X ' C i )+ [I * . Hg

1

*

I]'

HgI(X 2Ct)+ I(,P3/2)

---c

(4.18)

Motivated by this experiment, Burghardt and Gaspard have carried out a detailed analysis of the resonances of the transition complex on the semiempirical potential surface proposed by Zewail and co-workers [151], that is, a damped Morse potential for the two degrees of freedom of collinear motion. Bending motion is neglected in this model so that the bending rotation number is assumed to be zero. The results reported below must therefore be shifted by the bending zero-point energy, and also have to be corrected if the rotation number varies significantly with energy. The potential surface for the stretching dynamics is typical of strongly open systems, with a single saddle equilibrium point and two exit channels, i.e., we have a typical example to which the previous analysis applies. If the saddle is at the origin of energy, the total dissociation threshold is at an energy of 1800 cm-I, while the bottom of the exit channels lies at - 100 cm-' . The classical dynamics follows the first bifurcation scenario we discussed above, i.e., involving a supercritical antipitchfork bifurcation.

562

P. GASPARD AND I. BURGHARDT

(a) Periodic Repeller for 0 cm-' c E < E, = 523 cm-' . In this regime, the repeller contains the single periodic orbit 0 of symmetric stretching motion that is unstable (hyperbolic without reflection). Equilibrium point and periodic-orbit quantization have been applied to this regime. Both semiclassical quantization schemes show that six resonances with m = 0, u = 0, and n = 0, ..., 5 exist in this regime, which are regularly spaced with a spacing corresponding to a period of 363 fs. The first resonance has a lifetime of 1 1 1 fs. The lifetimes become longer as energy increases. In Fig. 13, we observe that the resonances lie on the border of the spectral gap, which coincides with the curve corresponding to the escape rate, that is, to the Lyapunov exponent of the symmetric-stretch periodic orbit 0, because the repeller is here periodic: Im E = -(A/2)Xo(ReE) = hP(1/2; Re E) = (A/2)P(1;Re E ) . Differences between the lifetimes obtained from equilibrium point quantization and periodic-orbit quantization appear as the bifurcation is approached. The lifetimes are underestimated by equilibrium point quantization but overestimated by periodic-orbit quantization. The reason for the upward deviation in the case of periodic-orbit quantization is that the Lyapunov exponent vanishes as the bifurcation is approached. The quantum eigenfunctions,however, are not characterized by the local linearized dynamics but extend over larger distances that are of more unstable character.

0

500

lo00

1500

Re E [cm"]

Figure 13. Scattering resonances of a two-degree-of-freedom coilinear model of the dissociation of Hg12 [lo]. The filled dots are obtained by wavepacket propagation, the crosses by equilibrium point quantization with (2.8), the dotted circles from periodic-orbit quantization with (4.16). The solid lines ace the curves corresponding to the Lyapunov exponents, Im E = -(h/2)Xp(Re E ) , of the fundamental periodic orbits p = 0, 1, 2. The dashed line is the spectral gap, Im E = hP( 1/2; Re E). The long-short dashed line is the curve corresponding to the escape rate, Im E = (h/2)P(I;ReE).

EMERGENCE OF CLASSICAL PERIODIC ORBITS AND CHAOS

563

The periodic repeller exists up to the antipitchfork bifurcation. (b) Transition Region for E, = 523 cm-' < E < Ehr = 575 cm-'. Above the antipitchfork bifurcation, the symmetric-stretch periodic orbit becomes elliptic in the interval 523 cm-' < E < 548 cm-' . It is embedded in a main elliptic island of phase-space area much smaller than 2&, which cannot sustain a long-lived resonant state. Actually a wavepacket calculation shows that a single short-lived resonance with m = u = 0 and n = 6 at E = 55 1 cm-I covers the transition region. The elliptic island is delimited by the two off-diagonal periodic orbits 1 and 2 which are born at the antipitchfork bifurcation. From there up to above 1500 cm-' ,the periodic orbits 1 and 2 are hyperbolic (without reflection). At E d = 548 cm-' , the elliptic island is destroyed by a period-doubling bifurcation at which the periodic orbit 0 becomes hyperbolic (with reflection) up to high energies. A last homoclinic tangency between the stable and unstable manifolds of the three periodic orbits 0, 1, and 2 occurs around Ehr = 575 f 5 cm-I. For E d = 548 cm-' < E c Ehr = 575 cm-', the main elliptic island no longer exists but small subsidiary elliptic islands of high periods still persist up to the last homoclinic tangency. At the antipitchfork bifurcation E,, the periods of the new periodic orbits 1 and 2 bifurcate supercritically from the period of 0, as seen in Fig. 14.

(c) Fully Chaotic Repeller for Eh, = 575 cm-' C E --t 1500 cm-'. In this region, the repeller is a Smale horseshoe with three branches (see Fig. 15). The orbits of the repeller are in one-to-one correspondence with sequences built from symbols (0, 1,2} associated with the three fundamental periodic orbits. All the orbits are unstable of saddle type. The periodic-orbit quantization based on the zeta function quantization

..

.. :: *.. ... ,...* .

I * .I

I .

0

*

0

500

I

E [cm-'1

lo00

.

.

.

.

1500

Figure 14. Period-energy diagram of the fundamental periodic orbits 0, 1, and 2 in HgI2 with the bifurcations of the transition region marked by dashed lines.

564

P. GASPARD AND I. BURGHARDT

P

Q

Figure 15. Three-branch h a l e horseshoe in the 2F collinear model of Hg12 dissociation at the energy E = 600cm-' above the saddle in a planar Poincari surface of section transverse to the symmetric-stretch periodic orbit. The Smale horseshoe is here traced out in a density plot of the cumulated escape-time function (4.6).

condition (4.16) compares favorably with the resonances obtained from wavepacket propagation. In this region, 15 resonances with m = u = 0 and n = 7, 8, . .. have been identified numerically, The relative regularity of the spectrum is explained by the similarity of the fundamental periods, as seen in Fig. 14. The agreement between the periodic-orbit quantization and the wavepacket calculation is very good as far as the energies of the resonances are concerned. At low energies near the transition region, the lifetimes are again overestimated because the Lyapunov exponents approach zero, while the nonlocal features of the dynamics are not taken into account. At higher

EMERGENCE OF CLASSICAL PERIODIC ORBITS AND CHAOS

565

energies, the lifetimes are well reproduced by the approximation (4.16). Their values range between 160 and 240 fs, in agreement with the results of the femtosecond laser experiments. Variations in the lifetimes are observed that we interpret as the effect of the interference between the three fundamental periodic orbits. This interference causes a lengthening of the lifetimes as compared with the lifetimes of the individual periodic orbits. In Fig. 13, we thus observe that the resonance spectrum extends below the spectral gap, Im E = trP(1/2;ReE). On the other hand, the curve corresponding to the escape rate, Im E = (A/2)P( 1; Re E), lies here below the spectral gap but above the curves corresponding to the Lyapunov exponents, Im E = -(R/2)Ap(ReE) with p = 0, 1, 2. This is evidence for the Gaspard-Rice lengthening of lifetimes due to classical chaos, which was previously observed in the three-disk scatterer [33]. The eigenfunctions associated with the resonances have been obtained via wavepacket propagation. They appear to be localized along the symmetricstretch periodic orbit 0, with a number of nodes equal to n and even under the exchange of iodine nuclei. Due to the relative stability of the symmetricstretch orbit, we have thus here a system where the hypothesis of the orbit 12 representing the W O ,that is, resonant periodic orbit, does not hold. 2. c02

Much work has been devoted to the photodissociation of C02 in its most intense electronic band between 85,000 and 90,000 cm-' [152-1551. This band, which corresponds to the C;: electronic state, shows irregular vibrational structures that have proved difficult to assign. The photodissociation on this electronic surface has been studied in the classical work by Kulander and Light, who used a collinear model based on a semiempirical LEPS (London-Eyring-Polanyi-Sato)surface [ 1521. Their calculation revealed irregular vibrational structures that have been more recently interpreted by Schinke and Engel [153], Kulander et al. [154] and Zobay and Alber [155] in terms of periodic orbits. The periodic orbits that have been identified in these works correspond to the orbits 0, 01, 02, 12, 011, 022, 012, and 0102, according to the symbolic dynamics we introduced. These results raise the question whether a Smale horseshoe indeed exists, by analogy with the case of HgI,. We have recently been able to show that the collinear periodic orbits in this model of C02 are in fact controlled by such a three-branch Smale horseshoe. Below we give a detailed description of the classical-dynamical regimes leading up to-and beyond-the regime characterized by the three-branch horseshoe and its symbolic dynamics. As the LEPS surface comprises all vibrational degrees of freedom, it is possible to include the effect of bending in the analysis. To this end, we have carried out the equilibrium point quantization for this system at energies just

'

566

P. GASPARD AND I. BURGHARDT Table I1 Resonance Constants of Several Dissociative Molecules

363 -103 108

TYf TY

7y 7,

111

h

Echannel Ediss.

47.8 -lo00 2800

AE*

-0.1

en

91.9 -30 -i49.1 -I .23

0 1

02 03 x1 I x12

+i1.7

XI3

x22 x23 x33 g22

-1.2

30 112 3.2 3 f l

35 244 6.2 6.4 590 -10,006 80,650 -23.4 952.4 136.9 - i 856.3 -2.8 21.7 +i53.9 1.23 -i2.3 -43.0

18 f 2

10f2

6 f 3

32.9 3863 -3193

21,300 1108 i 25 298 f 12 - i 1650 k i300 -9 f 3

1850 f 100 -80 -I 10

-0

-0

1.O ~

-

~~

~

~

Nore: Frequencies and energies in reciprocal centimeters and times in femtoseconds. aCollinear two-degree-of-freedom model by &wail et al. [15 I ] by quantization around the equilibrium point of linear geometry (bending ignored). bFour-degree-of-freedom model of I2CI602 with LEPS surface [ 1521 by quantization around the equilibrium point of linear geometry (bending and rotation included). ‘Dunham expansion fitted to experimental data of the Hartley band by Joens [158] (nonlinear equilibrium geometry). dThree-degree-of-freedommodel fitted to experimental data for the overall band structure by Schinke et al. [I591 (nonlinear equilibrium geometry). Tollinear two-degree-of-freedom model with the Karplus-Porter surface by Pechukas et al. [ 1431 (linear equilibrium geometry). f T! = 2h/wj with tc= 5308.84 fs/cm. g7:

= h/lw31.

hEnergy and lifetime of the first available resonance in the previous analyses.

above the unique saddle point of the LEPS surface [146]. Let us note that, for this surface, both the total dissociation threshold and the bottom of the exit channels are at a much larger separation in energy from the saddle than in the case of HgI,. Table I1 gives the relevant data. In the following, we will characterize the different dynamical regimes for COz on the LEPS surface. The scenario in a general way resembles the one for HgI,. However, the initial bifurcations are of the second type described in Section IV.C.2; that is, the symmetric-stretch orbit undergoes a subcritical

567

EMERGENCE OF CLASSICAL PERIODIC ORBITS AND CHAOS

antipitchfork bifurcation, which is preceded by a pair of tangent bifurcations. We have the following scenario: (a) Periodic Repeller for 0 cm-’ < E < 2637cm-' . Below the pair of tangent bifurcations, the repeller is composed of the symmetric-stretch periodic orbit that is hyperbolic (without reflection). Equilibrium point quantization (including bending) shows that three resonances n = 0, 1, 2 with rn = u = 0 exist below the bifurcation (see Fig. 16). The period of the symmetric stretching motion at the saddle is equal to 35 fs. The lifetimes become longer as energy increases, as in HgI,. The lifetime extrapolated to the saddle turns out to be equal to 6.2 fs, which is very short. (b) Transition Region for Ett = 2637 cm-’ < E < E h t . The transition region is extremely narrow in this model. The pair of tangent bifurcations occur at Ett = 2637 cm-’, where the off-diagonal periodic orbits 1 and 2, of hyperbolic-without-reflection stability, as well as the elliptic periodic orbits 1’ and 2’ are born. The subcritical antipitchfork bifurcation occurs at E, = 2672 cm-’, where the orbits 1‘ and 2’ merge with the symmetric-stretch periodic orbit 0, which changes its stability from hyperbolic (without reflection) to elliptic. The orbit 0 remains elliptic up to a period doubling at E d 0

5

-200

E

-400

Y

w Y

-600 L 0

1

1000

I

I

3000 Re E [cm“]

2000

1

4000

I

5000

Figure 16. Scatteringresonances of the full rotational-vibrational Hamiltonian describing the dissociation of COz on a LEPS surface obtained by equilibrium point quantization with (2.8). The resonances with J = 0, ... 10 are given by dots. Their close vicinity explains the formation of “hyphens”, i.e., unresolved sequences of dots. Note that rotation is very slightly destabilizing in the present model. The successive hyphens are the bending progressions with uz = 0, . . . 5 . The solid line is given by the Lyapunov exponent of the symmetric-stretch periodic orbit 0 expressed as an imaginary energy.

.

568

P. GASPARD AND I. BURGHARDT

= 2707 cm-', above which it is hyperbolic (with reflection) up to E = 54,000

cm-' (see below). The last homoclinic tangency is expected at a not much higher energy than E d . According to the equilibrium point quantization, there is no resonance in this tiny transition region, which remains much below the wave-mechanical phase-space resolution. We notice that the transition region is even narrower than in the HgI, system. However, the successive tangent and subcritical bifurcations have important consequences for the period of the orbit 0 relative to the one of 1 and 2. As shown in Fig. 17, the period of 0 has a slightly increasing monotonous behavior while the period of 1 or 2 rapidly decreases, which is in contrast with the Hgl, system (see Fig. 14). This behavior is due to the tangent bifurcation at which the curve of the period of 1 or 2 versus energy must have a vertical slope so that the branch of the curve corresponding to 1' or 2' joins the curve of 0 at the subcritical bifurcation from lower energies. This has the effect of forcing the period of 1 or 2 to be significantly different from the period of 0. (c) Fully Chaotic Repeller for Eh, c E -+ 54,000 cm-'. In this region, the repeller is a three-branch Smale horseshoe associated with the three fundamental periodic orbits 0, 1, and 2 (see Fig. 18). The Smale horseshoe has been constructed with the escape-time function as described above, as for the Hgl, system. All the orbits are unstable of saddle type. This result allows us to establish a complete list of periodic orbits in terms of the symbols {0,1,2}. A semiclassical quantization is thus possible over this large energy interval via the zeta function formalism. The shortest periodic orbits

60

40 20

0

0

5000

10000

E

15000

[cm-'1

Figure 17. Period+nergy diagram of the fundamental periodic orbits 0, 1, 2, 01, 02, 12 of the C02 system with the bifurcations of the transition region marked by dashed lines.

EMERGENCE OF CLASSICAL PERIODIC ORBITS AND CHAOS

569

P

9

Figure 18. Three-branch Smale horseshoe in the collinear model of C02 dissociation at

the energy E = 10,759.4 cm-' above the saddle in a hyperbolic Poincari surface of section transverse to the symmetric-stretchperiodic orbit. The figure represents a density plot of the cumulated escape-time function (4.6), as in Fig. 15.

are thus (0,1,2,12,01,02,. . .}. Figure 17 shows that the composite orbits have periods close to the sum of periods of the fundamental orbits. As energy increases, the symmetric-stretch orbit 0 becomes very unstable while the offdiagonal orbits 1 and 2 have a Lyapunov exponent that decreases toward very small values. From this behavior, we may expect that the off-diagonal periodic orbits play the dominant role in the quantization and the resonant states will be localized around them, that is, around the hyperspherical periodic orbit 12, which is topologically composed of 1 and 2. This is what has been numerically observed by Kulander et al. [154]. On the other hand, the large

570

P. GASPARD AND I. BURGHARDT

difference of periods between 0 and 1 or 2 provides an explanation for the irregularity of the resonance spectrum in this system. A detailed analysis in terms of a zeta function quantization according to (4.16) should give a clue as to what is the role of the different periodic orbits in the spectrum, especially taking into account their widely different stabilities. We will report on such an analysis in a future publication [146]. The semiclassical formula (2.26) and (2.27) for the photoabsorption cross section shows that the Franck-Condon region may be identified with the region around the symmetric-stretch periodic orbit 0. From this we may expect that all the periodic orbits with the symbol 0 contribute significantly to the photoabsorption cross section, which explains why the periods of 0, 0 1, or 02 appear in the Fourier transform of the band, while the periods of 1 or 2 and 12 do not. In accordance with this observation, Schinke and Engel [153] first suggested that the periodic orbits 0,012 and 0102, according to our symbolic dynamics, should be relevant to the analysis of the photoabsorption cross section. Zobay and Alber [ 1551 later carried out a semiclassical calculation of the absorption cross section, taking into account the periodic orbits 01,02,011 and 022 in addition to those mentioned above. Their analysis is based on the semiclassical expression for the absorption cross section given in Eq. (2.27) (see also Ref. [40]) and yields very good agreement with the exact quantum-mechanical result for the collinear model. (d) Transition to Another Fully Chaotic Repeller for 54,140cm-' < E. At E = 54,140 cm-', the symmetric-stretch periodic orbit 0 changes its stability from hyperbolic (with reflection) to elliptic in an inverse period-doubling bifurcation, which is followed over a very narrow energy interval by a pitchfork-type bifurcation. By the latter, the periodic orbit 0 becomes hyperbolic without reflection. This sequence of events occurs over less than 1 cm-'. It is preceded by bifurcations of other periodic orbits of the horseshoe, in particular of the orbit 01. During these bifurcations, two new periodic orbits are born in the system, and we conjecture that a new repeller with five fundamental unstable periodic orbits exists above 54,200 cm-' .Its trajectories should be in one-to-one correspondence with sequences built on the alphabet {0,1,2,3,4}. Its quantization is possible by a straightforward generalization of (4.14H4.16). Note, however, that this regime is well above the experimentally accessible region. Let us compare these results with the experimental data on C02. A photoabsorption spectrum of COz has been recorded in 1965 by Nakata et al. [156]. The vibrogram as well as direct Fourier transforms of the intense band under discussion reveal main time recurrences around 50 and 60 fs [14]. On the other hand, the above model shows periods corresponding to 0 and 01 or 02 around 40 and 60 fs, respectively. Although there is a qualitative agree-

EMERGENCE OF CLASSICAL PERIODIC ORBITS AND CHAOS

57 1

ment, the differences suggest the need for a quantitative analysis based on ab initio surfaces. 3. H 3

This dissociative system, which represents the prototype system for chemical reaction dynamics, has been the object of many studies. Child et al. [143] have carried out a detailed analysis of the classical dynamics in a collinear model based on the Karplus-Porter surface. These authors have introduced the concept of PODS and first observed the subcritical antipitchfork bifurcation scenario in this system. The bottom of the exit channels is at -3 194 cm-' if the origin corresponds to the saddle of the Karplus-Porter surface. The pair of tangent bifurcations occur at E,, = 1670 cm-', which is followed by the subcritical antipitchfork bifurcation at E , = 2633 cm-' . The bifurcation scenario is thus similar to the CO2 system, and we may expect a three-branch Smale horseshoe in this system as well. The quanta1 character is even more prominent than in CO:!and the first reported resonances occur at 3863 and 7453 cm-', respectively, above the transition region (see Fig. 19). Here the zeta function semiclassical quantization with (4.16) should apply. The reported lifetimes are of 33 fs, which is

0

Q

-100

E

-200

u

w

U

-300

a

0

loo00

a

20000 Re E [cm-'1

3

m

Figure 19. Scattering resonances of 2F collinear models of the dissociation of H3 obtained by Manz et al. [ 1431 on the Karplus-Porter surface and by Sadeghi and Skodje on a DMBE surface [132]. The energy is defined with respect to the saddle point. The dashed lines mark the bifurcations of the transition to chaos: 1, 5, and 3 are the numbers of shortest periodic orbits, or PODS, in each region.

572

P. GASPARD AND I. BURGHARDT

longer than the symmetric stretching period of about 10 fs, in contrast with both the HgI, and COz systems. In a recent work, Sadeghi and Skodje have calculated the resonances according to a different collinear model of H3, based on a doubly-manybody-expansion (DMBE) surface [132]. The energies of the two first resonances are very close to the ones on the Karplus-Porter surface, but the lifetimes are here given by 17 and 25 fs, respectively (see Fig. 19). The similarity with the COz system suggests that the periodic orbit 0 becomes more and more unstable as energy increases, while the periodic orbits 1 and 2 become less and less unstable. This behavior is in accordance with the localization of the resonant eigenfunctions around the off-diagonal periodic orbits 1 and 2, and with the lengthening of the lifetimes at high energies (see Fig. 19). The periodic-orbit quantization of the DMBE model has recently been carried out [157]. 4.

03

The Hartley band of ozone has been the object of systematic investigations. The vibrational structures on top of the band have motivated studies of the dissociation dynamics on the corresponding potential surface [ 1071. Johnson and Kinsey have interpreted the time recurrences in Fourier transforms of the band in terms of the periodic orbits of a two-degree-of-freedom model with a frozen bending angle [1071. The structure of the periodic orbits of this model turns out to be very similar to the periodic orbits of the collinear models described above. Drawing on this similarity, we may identify the periodic orbits observed by Johnson and Kinsey with the periodic orbits 0,Ol or 02,011 or 022, and 0111 or 0222 of a three-branch horseshoe, with periods of 43,70, 103, and 131 fs, respectively. Let us remark here that one may in principle carry out a periodic-orbit analysis that goes beyond the one for a two-degree-of-freedom model with frozen bending configuration. In particular, the periodic orbits of the three-degree-of-freedom model can be numerically obtained from the ones of a two-degree-of-freedom model by relaxing the constraint on the bend angle and letting it evolve according to the full dynamics. In a more recent work, Joens [158] has assigned the structures of the Hartley band using a Dunham expansion, that is equilibrium point quantization. The lifetime predicted by his analysis is extremely short, equal to 3.2 fs, while the symmetric stretching period is of 30 fs. Recall, however, that the interpretations in terms of equilibrium point expansions and in terms of periodic orbits are strictly complementary only for regular regimes. 5. H20

The first UV band of water has been analyzed by Schinke and co-workers,

EMERGENCE OF CLASSICAL PERIODIC ORBITS AND CHAOS

573

who obtained a nice agreement between the experimental and theoretical shapes [159]. The vibrational structures on top of the band are very weak but regular. The corresponding period and lifetime of the time recurrences are respectively of 18 fs and about 6 fs. Recently, Hiipper et al. El601 have carried out a periodic-orbit quantization of the model of Ref. [159], as well as of another model [161] which predicts a shorter lifetime of about 3 fs instead of 6 fs.

6. Comparison of Lifetimes Table 11 shows a compilation of some of the data we collected for the above systems, which provide the lifetimes extrapolated to the saddle equilibrium point. As expected, a heavy molecule like HgI, has long period and lifetime as compared to the lighter species. In general, the period of the symmetric stretching oscillations behaves as expected, becoming shorter as the species becomes lighter. Concerning the lifetimes, Table I1 shows that they are usually three to four times shorter than the period such that the observable wavepacket recurrences are expected to be rather weak. For the comparison of periods and lifetimes, note here that the period is defined as the period of rotation of a trigonometric function like cos(2rf/T), while the lifetime is defined by the decay of an exponential function like exp(-t/.r), which shows that they differ by definition by a factor 2r. Apart from the general trend, a surprisingly short lifetime appears for 0 3 and a surprisingly long one for H3. This might be partially due to the different character of the potential surfaces for the different species, which will affect the dynamical instability. We hope that future works will shed more light on these observations.

V. CONCLUSIONS In this chapter we have summarized recent results obtained by applying semiclassical methods to the study of vibrational and dissociative dynamics on the femtosecond scale. In a first part, we described the methods of equilibrium point quantization and periodic-orbit quantization. The former method represents a perturbation treatment that is valid as long as an assignment in terms of good quantum numbers is possible. Beyond such regular regimes, we encounter irregular spectra related to nonlocal dynamics. Periodic-orbit theory provides an avenue to the understanding of such spectra on an intermediate energy scale. Periodic-orbit methods have been formulated on the one hand for classically integrable systems and on the other hand for classically hyperbolic systems: The former are described by the Berry-Tabor trace formula and the latter by the Gutzwiller trace formula. The Gutzwiller trace formula at its leading order is equivalent to a quantization condition expressed in terms

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of a zeta function, which takes into account the interference between periodic orbits, that is characteristic of chaotic systems. Even though the periodic orbits are of infinite number, the analysis becomes manageable due to the fact that the shortest and least unstable periodic orbits provide the dominant contributions. For discrete spectra, we discussed the relationship between classical chaos and random-matrix properties, which concern the smallest spectral energy scale, below the mean level spacing. Note that dynamical instability, which is the signature of classically chaotic dynamics, directly affects the imaginary parts of the system’s eigenenergies, which is of immediate relevance to the lifetimes in scattering systems but does not play a direct role for bounded systems. Irregular spectra in bounded systems rather seem determined by the distribution of the periods of the periodic orbits or, more generally, of the interfering quantum paths. In the context of irregular spectra, we further summarized new results concerning their parametric properties. Turning back to the role of the periodic orbits emerging from the wavemechanical dynamics, we have shown that the new vibrogram technique may be used to provide evidence for such periodic orbits in the intramolecular vibrational dynamics. The vibrogram represents the period-energy diagram of the emergent periodic orbits. The periodic orbits that may thus be observed are necessarily limited to the shortest ones, since the accumulation of periodic orbits soon exhausts the limits of resolution of the technique. Nevertheless, several examples show clear evidence of the periodic-orbit signature. Of particular interest are changes in the stability of the periodic orbits, which can be observed, for example, in C2H2, due to a transition to chaos in the classical bending dynamics. This is in contrast to the isotopomer CzHD, whose dynamics can be modeled via a classically integrable effective Hamiltonian. Periodic orbits also play a key role in the semiclassical calculation of the transition-state resonances that characterize ultrafast dissociation processes. In this context, we have considered the dissociation of several triatomic molecules on potential-energy surfaces with a single saddle point. From a detailed analysis of the classical collinear dynamics for the symmetric species HgI, and C02, a general scheme may be inferred: Over a broad energy range, we have a classically chaotic regime where all periodic orbits are part of the global phase-space structure of a Smale horseshoe with three branches. This result paves the way for a periodic-orbit quantization based on the zeta function, which provides an understanding of the spectrum in terms of the periods and stabilities of a few fundamental periodic orbits. This theoretical scheme offers a unifying viewpoint with respect to a number of results obtained in previous research. On this background, we discussed possible dynamical scenarios and lifetimes for several molecules. Ultrafast dissociation, and reactive processes in general, can thus be

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understood in terms of the dynamics of open systems and turn out to be markedly susceptible to the emerging regular or chaotic classical dynamics. In open systems, as opposed to bounded systems, a more direct connection exists between classical chaos and the properties of the resonance spectrum. We already mentioned above that the dynamical instability has direct consequences for the lifetimes of the metastable states. On the other hand, irregularities in the energy spectrum appear to be primarily related to the distribution of the periods. We thus believe that the explanations offered by the theoretical framework presented here will add to the interpretation of many experimental observations on elementary dynamical processes.

Acknowledgments We express our gratitude to G . Nicolis, I. Prigogine. and S. A. Rice for support in this research. We also thank M. Herman, R. Jost, J. Lievin, 1. Mills, J.-P. Pique, and R. Schinke for fruitful discussions as well as P. van Ede van der Pals for assistance in the preparation of this report. Financial support was granted by the National Fund for Scientific Research (FNRS Belgium) for P. G. and by an EEC doctoral fellowship, No.ERBCHBICFl30857 for I. B. The research is further financially supported by the Quantum Keys for Reactivity ARC Project of the CommunautC Franqaise de Belgique, and by the Banque Nationale de Belgique.

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143. E. Pollak and P. Pechukas, J. Chem. Phys. 69, 1218 (1978); E. Pollak, M. Child, and P. Pechukas, J. Chem. Phys. 72, 1669 (1980); E. Pollak and M. S . Child, J. Chem. Phys. 73, 4373 (1980); E. Pollak and M. S . Child, Chem. Phys. 60, 23 (1981); E. Pollak, J. Chem. Phys. 74,5586 (1981); J. Manz, E. Pollak, and J. Romelt, Chem. fhys. Lett. 86,26 (1982); E. Pollak, J. Chem. Phys. 76,5843 (1982): E. Pollak, in The Theory of Chemical Reaction Dynamics, D. C. Clary, Ed., Reidel, Dordrecht, 1986, p. 135. 144. M. A. M. De Aguiar, C. P. Malta, M. Baranger, and K. T. R. Davies, Ann. fhys. 180, 167 (1987). 145. S. Wiggins, Global Bifurcarions and Chaos, Springer, New York, 1988; S. Wiggins, Chaotic Transport in Dynamical Systems, Springer, New York, 1992; E. Ott, Chaos in Dynamical Systems, Cambridge University Press, Cambridge, 1993. 146. I. Burghardt and P. Gaspard, in preparation. 147. N. K. Gavrilov and L. P. Shil'nikov, Math. USSR Sbornik 17,467 (1972); ibid., 19, 139 ( 1973). 148. S . Smale, The Mathematics of lime, Springer, New York, 1980. 149. P. Gaspard and S . A. Rice, Phys. Rev. A 48,54 (1993). 150. 0. Hahn, J. M. Gomez-Llorente, and H. S. Taylor, J. Chem. Phys. 94, 2608 (1991); R. T. Skodje and M. J. Davis, J. Chem. Phys. 88, 2429 (1988). 151. A. H. Zewail, Faraday Discuss. Chem. Soc. 91, 207 (1991); M. Dantus, R. M. Bowman, M. Gruebele, and A. H.Zewail, J. Chem. fhys. 91,7437 (1989); M. Gruebele, G. Roberts, and A. H. Zewail, Phil. Trans. Roy. SOC.Lond. A 332, 223 (1990). 152. K.C. Kulander and J. C. Light, J. Chem. Phys. 73,4337 (1980). 153. R. Schinke and V. Engel, J. Chem. Phys. 93,3252 (1990). 154. K. C. Kulander, C. Cerjan, and A. E. Orel, J. Chem. Phys. 94,2571 (1991). 155. 0. Zobay and G. Alber, J. fhys. B; Ar. Mol. Opt. fhys. 26, L539 (1993). 156. R. S . Nakata, K. Watanabe, and F. M. Matsunaga, Sci. Light 14, 54 (1965). 157. B. Hupper, Ph.D. Thesis, Carl von Ossietzky Universitat Oldenburg (1997). 158. J. A. Joens, J. Chem. Phys. 100,3407 (1994). 159. V. Engel, R. Schinke, and V. Staemmler, J. Chem. fhys. 88, 129 (1988); M. von Dirke and R. Schinke, Chem. Phys. Lett. 1%, 51 (1992). 160. B. Hupper, B. Eckhardt, and V. Engel, Semiclassical photodissociation cross section for H20, preprint (1996). 161. M. L. Doublet, G. J. Kroes, and E. J. Baerends, J. Chem. fhys. 103,2538 (1995).

Advances In Chemical Physics, Volume101 Edited by I. Prigogine, Stuart A. Rice Copyright © 1997 by John Wiley & Sons, Inc.

GENERAL DISCUSSION ON REGULAR AND IRREGULAR FEATURES IN

UNIMOLECULAR SPECTRA AND DYNAMICS Chairman: M. Herman G. Casati: The change in some of our basic ideas about classical mechanics, enforced by dynamical chaos, is so deep that quantum mechanics may hardly ignore it. On the other hand, the investigation of the properties of quantum systems that become chaotic in the classical limit is bringing to light a qualitative picture of their quantum motion that could hardly be suspected at times when the qualitative understanding of quantum dynamics was modeled just after integrable cases and perturbation theory. For example, in the problem of excitation of Rydberg atoms under microwave fields a somewhat unexpected result is that the most efficient ionizing process is not a single-photon but a multiphoton one [l]. A remarkable fact is that, in spite of all fluctuations and fractal properties exhibited by quantum motion, strong empirical evidence has been obtained that the quantum evolution is very stable, in sharp contrast to the extreme sensitivity to initial conditions that is the very essence of classical chaos [2]. Some 30 years ago, a breakthrough in the understanding of the qualitative behavior of classical systems undergoing the so-called stochastic transition was made possible by the analysis of the “standard map” or “kicked-rotator” model. The main consequence of this transition is that a statistical picture of the Fokker-Planck type applies, leading to an energy absorption linear in time, in a diffusive-like fashion. One of the main problems is to what extent quantum mechanics mimics this behavior. Surprisingly enough, it turned out [3] that, typically, the quantum excitation follows the classical pattern only up to a finite time ts, after which the quantum rotator appears to enter a stationary regime and the energy absorption comes to an end. This quantum-mechanical suppression of the classical diffusive behavior is of a quite general nature and it has been confirmed, theoretically and experimentally, in the problem of excitation and ionization of hydrogen atoms under 583

584

GENERAL DISCUSSION

microwave fields [4,5]. The Hamiltonian of this model is given by (in atomic units)

where e and w are the intensity and frequency of the linearly polarized microwave field. In the classical case, when the field intensity exceeds a threshold value, the electron’s motion becomes chaotic and strong excitation and ionization takes place. In the quantum case instead, interference effects may lead to the suppression of the classical chaotic diffusion, the socalled quantum dynamical localization, and in order to ionize the atom, a larger field intensity is required (see Fig. 1). Another interesting example is the chaotic autoionization of molecular Rydberg states caused by the interaction of the electron with the degrees of freedom of the core. We consider the model in which the core consists of a positive Coulomb charge plus a rotating dipole that lies in the same plane of the electron orbit (m = 1). The Hamiltonian reads (atomic units)

L2 1 xcos++ycos+ H = !j(p: + py’)+ - - - + d 21 r r3 where d , L, I are the dipole moment, the orbital momentum, and the moment of inertia of the core, respectively. The analysis of the classical dynamics shows a transition to chaotic motion leading to diffusion and ionization [63. In the quantum case, interference effects lead to Iocaiization and the quantum distribution reaches a steady state that is exponentially localized (in the number of photons) around the initially excited state. As a consequence, ionization will take place only when the localization length is large enough to exceed the number of photons necessary to reach the continuum. In conclusion, quantum dynamical localization plays an important role in the excitation and ionization process of atoms and molecules. A question that remains open in connection with the previous talk is whether quantization via periodic orbits can account for this phenomenon.

1 . G. Casati, B. V. Chirikov, I. Guameri, and D. L. Shepelyansky, Phys. Rev. h f f . 57, 823 (1986); Phys. Rep. 154,77 (1987). 2. G. Casati, B. V.Chirikov, I. Guameri, and D. L. Shepelyansky, Phys. Rev. Leir. 56, 2437 (1986).

585

REGULAR AND IRREGULAR FEATURES

0.5

1.O

1.5

2.0

oo=on:

2.5

3.0

Figure 1. Comparison at identical parameter values of experimental and quantummechanical values for the microwave field strength for 10% ionization probability as a function of microwave frequency. The field and frequency are classically scaled, wo = niw and €0= rite, where no is the initially excited state. Ionization includes excitation to states with n above n,. The theoretical points are shown as solid triangles. The dashed curve is drawn through the entire experimental data set. Values of no, n, are 64, 114 (filled circles); 68, 114 (crosses); 76, 114 (filled squares); 80, 120 (open squares); 86, 130'(triangles); 94, 130 (pluses); and 98, 130 (diamonds). Multiple theoretical values at the same w o are for different compensating experimental choices of no and a.The dotted curve is the classical chaos border. The solid line is the quantum 10% threshold according to localization theory for the present experimental conditions. 3. G. Casati, B. V. Chirikov, J. Ford, and F. M.Izrailev, Lecrures Notes in Physics (Springer) 93,334 (1979);G. Casati and B. V. Chirikov, Quantum Chaos, Cambridge University Press, Cambridge, 1995. 4. G . Casati, I. Guameri, and D. L. Shepelyansky,IEEE J. Quantum Electron. 24,1420 (1988).

5. I. Bayfield. G. Casati, I. Guameri, and D. W.Sokol, Phys. Rev. Lett. 63,364 (1989).

6. F. Benvenuto, G. Casati, and D.

L. S. Cederbaum

L. Shepelyansky, Phys. Rev. Lett. 72, 1818 (1994).

1. Dr. Gaspard, you discussed periodic orbits of a number of triatomic molecules. 1 would like to know how many degrees of freedom have been included in the analysis? 2. To make the periodic orbit approach a competitive method of analysis, it would be relevant to extend it to more than two dimen-

586

GENERAL DISCUSSION

sions. As is well known, it is nontrivial to generalize the periodic-orbit approach to more dimensions. Could you discuss the advances made in this direction?

P. Gaspard

1. The theoretical analysis of HgI, by Burghardt and Gaspard has been carried out for the two-degree-of-freedom collinear dynamics. 2. The bending degrees of freedom can be taken into account as perturbations with respect to the collinear motion, thereby obtaining a complete description of the vibrational dynamics, as we discussed (Gaspard and Burghardt, “Emergence of Classical Periodic Orbits and Chaos in Intramolecular and Dissociation Dynamics,” in this volume). One of the possibilities provided by the ti-expansion theory is to include also nonlinear perturbations transverse to each periodic orbit. We think that periodic orbits for the full 3F dynamics can also be numerically obtained, for instance, in bent molecules. Approximate periodic orbits pertaining to a restricted 2F dynamics can be calculated in a first stage by freezing the bending degree of freedom. This constraint can thereafter be progressively removed while numerically tracking the periodic orbits in order to obtain the 3F periodic orbits.

A. H. Zewail: Dr. Gaspard has shown that for HgI, the period increases with energy. I assume that this reflects the “opening” of the potential in the coordinate perpendicular to the reaction coordinate, above the saddle point. P. Gaspard Yes. This is due to the damped Morse potential of your model of HgI,. There is thus a lengthening of the periods as the threebody dissociation is approached.

M. Quack I agree with most of what Prof. Field said. In fact, I have agreed with

this type of analysis for a long time. However, I have two remarks and a question. 1. The first remark concerns the abbreviation IVR. Prof. Field said that people even did not agree with what the three letters stand for, and in his lecture he actually never spelled out the words. He is, indeed, right that there are a variety of usages in reading words into IVR, the two most frequent ones being (a) “intramolecular vibrational relaxation” and (b) “intramolecular vibrational redistribution.” (There are also others, such as “intramolecular vibrationai randomization” and further, quite unrelated ones, such as “initial-value representation”). Personally, I strongly prefer the usage of “redistribution,” because this lan-

REGULAR AND IRREGULAR FEATURES

587

guage can be used both for the time-independent and the time-dependent pictures of the underlying physical phenomenon. It also allows for very general types of time dependence, whereas “relaxation” has an overly strong connotation of irreversibility, which may not always apply. There is also IVRR (“intramolecular vibrational-rotational redistribution”; see also Refs. 1 and 2). The basic physics of IVR(R) corresponds to a Hamiltonian being nonseparable (with a strictly separable vibrational molecular Hamiltonian there is no IVR), and the consequences of this can be seen both in stationary states and in time-dependent states. 2. The second remark concerns the question of the relation between the effective (algebraic) spectroscopic Hamiltonian and the “true” molecular Hamiltonian in coordinate and momentum space. The spectroscopic analysis Prof. Field has used gives an effective Hamiltonian, which is a matrix representation of the molecular Hamiltonian in some basis, which is not known and which in general cannot be written as the simple set of product functions in some set of coordinates as it was written down by Prof. Field. In the traditional spectroscopic treatment of such Hamiltonians going back to the work of Nielsen, Amat, Tarrago, Mills, and others decades ago, one had assumed that adequate basis functions and potential constants can be derived from such an analysis by perturbation theory. This was certainly the generally accepted wisdom until the 1980s (very widely assumed by many even today). If this were true to a good approximation, Prof. Field’s analysis would be fully justified. I do not want to exclude here the possibility that this works accidentally for the special case of the acetylene spectra discussed by him. However, when we introduced the polyad concept into the treatment of IR multiphoton excitation and IVR [l], we found that there were possible ambiguities and failures of the accepted treatment by perturbation theory. In the years following 1984 we have shown beyond any doubt that, indeed, the standard spectroscopic perturbation theory fails badly for a large class of Fermi resonance and Darling-Dennison resonance-type systems in the alkylic CH stretching-bending dynamics [3, 41. We have also less complete evidence on other systems. To illustrate the failure more quantitatively, I might quote the errors of anharmonic potential constants derived from the effective Hamiltonian analysis, which can be wrong by factors of 2-5 in individual cases, and the nature and structure of the basis states of H,ff, which can be quite different in reality from the simple analysis based on writing product functions CP I (Ql )@p2(Q2) * * .

588

GENERAL DISCUSSION

Thus, in order to derive quantitatively meaningful wavepacket dynamics of molecular motion (and also quantitative stationary-state wave functions) from the effective Hamiltonian, it is necessary to cany out afirther step in the analysis, which is highly nontrivial. We have canied out such analyses for a number of systems and I think an adequate understanding of the problem does exist nowadays and is accepted by the subcommunity most interested in this question. I like to illustrate this in the scheme from “molecular spectra” to “molecular motion” [5] (see Scheme 1).

High-resolutionmolecular spectroscopy Fourier transform spectroscopy Laser spectroscopy MOLECULAR SPECTRA

L t

1

Effective Hamiltonians Electronic Schriidin er equation L ? Rovibrational Schrodinger equation Molecular Hamiltonian +- Ab initio potential hypersurfaces

L

Time evolution operator (matrix) L MOLECULAR MOTION Molecular rate processes and statistical mechanics Scheme 1. Molecular spectra and molecular motion.

While the derivation of Heff is an important and useful step in this scheme (and I fully agree with Prof. Field that it is a most important exercise i,n spectroscopic “pattern recognition” [6]), the further step from H,ff to Hmol,tme must not be omitted in quantitative work, in general. The nature of the failure in the dynamics derived from the simplified Heft. analysis can be seen from the highly oversimplified sketch in two coordinates Ql, Q2 in Fig. 1. The error on the effective Hamiltonian basis (initial) state may be sufficiently large to introduce very large errors on the time propagated state. For quantitative results on some systems I can refer to Ref. 7, for example, where one can find also some interesting analytical considerations. Over the years we have assembled definitive experimental evidence for several molecular examples that the wavepacket evolution using incorrect (perturbation theory) initial states differs significantly from the evolution using the proper initial states. There remains no reasonable doubt about this, although this fact may not yet be widely recognized.

REGULAR AND IRREGULAR FEATURES

I

589

Q2

Figure 1. Sketch of probability densities as function of time in two coordinates: (1) correct initial state (from He# with transformation); (2) initial (basis) state of Heft (by perturbation theory); (3) true state at time t ; (4) state derived at time t (incorrect) if (2) is taken as initial state. 1. M.Quack, J. Chem.Phys. 69, 1282 (1978); Faraday Discuss Chem. SOC. 71, 309, 325,359 (1981); H.R. Dubal and M. Quack, J. Chem. Phys. 81,3379 (1984); Mol. Phys. 53,257 (1984). 2. M.Quack and W. Kutzelnigg, Be,: Bunsenges. Physik. Chem. 99, 231 (1995). 3. M.Lewerenz and M. Quack, J. Chem. Phys. 88,5408 (1988). 4. M. Quack, Ann. Rev. Phys. Chem. 41,839 (1990). 5 . M.Quack, Jerusalem Symp. 24,47 (1991); Mode Selective Chemistry, J. Jortner, R.

D. Levine. and B. Pullman, Eds., Reidel, Dordrecht (1991). 6. M. Quack, J. Chem SOC. Faraday Trans. 2,84, 1577 (1988). 7. R. Marquardt and M. Quack, J. Chem. Phys. 95,4854 (1991).

R. W. Field: I never claimed that molecular vibrations are even remotely close to being described by products of harmonic oscillators. I said that spectra are mysteriously more regular than they have any right to be. We can only adopt a simple representation for simple patterns. Harmonic oscillators are convenient because of their Au selection rules and matrix element u-scaling rules. Many of the inaccuracies of harmonic oscillators are irrelevant to dynamics, because the weak Au selection-rule-violating matrix elements between energetically well-separated basis states are much less important at early time than stronger Au-obeying interactionsbetween near-degenerate basis states. Of course it is possible to refine the model for the individual oscillators. This is especially important if the parameters determined from the analysis of a spectrum are going to be used to build an accurate and molecule-tomolecule transferablemodel for generic anharmonic couplings.I am sure you agree that one should start simple and build in complexity only as the specific class of problem at hand demands it. I continue to believe that one can construct reasonably accurate pictures of dynamics in configuration space (for the Franck-Condon

590

GENERAL DISCUSSION

active normal coordinates) and in state space (for the other modes strongly anharmonically coupled to the Franck-Condon active modes, restricted by approximate constants of motion) directly from intensity and frequency spacing information derived from feature-state progressions and from the width and complexity of each assigned feature state (polyad). The picture you drew on the board (Fig. 1 of the preceding comment by M. Quack), suggesting that a normal-mode picture would predict wavepacket motion in the opposite direction from reality, is ar best hyperbole. For large displacements, harmonic oscillators give an excellent description of the gradient of the potential in the Franck-Condon region.

M. E. Kellman: I believe the relations described by Prof. Quack among all the aspects of the effective spectroscopic Hamiltonian-its zero-order modes and the nature of these in coordinate space; the coupling terms in the spectroscopic Hamiltonian; and its connection to the potential-energy surface in coordinate space-are not completely understood, but there has been recent progress. Fried and Ezra [L. E. Fried and G. S . Ezra, J. Chem. Phys. 86,6270 (1987)] and McCoy and Sibert [A. B. McCoy and E. L. Sibert 111,J. Chem. Phys. 95,3476 (1991)] have shown how to obtain a spectroscopic Hamiltonian from a potential-energy surface in coordinate space. I have discussed [M. E. Kellman, Annu. Rev. Phys. Chem. 46,395 (1995)] the raising and lowering operators in the spectroscopic Hamiltonian in terms of actions defined among anharmonic modes, rather than as matrix elements of position and momentum operators for harmonic modes. So Prof. Field’s interpretation of the Hamiltonian makes a good deal of sense to me, though I agree with Prof. Quack that there are still difficulties in understanding just what all of us are doing when we use these Hamiltonians. In particular, the nature of the zero-order modes in this Hamiltonian, and what they look like in the ordinary molecular position and momenta coordinates, is something that, to my knowledge, is not yet understood.

M. Quack: In answer to the question by Prof. Kellman on exact analytical treatments of anharmonic resonance Hamiltonians, I might point out that to the best of my knowledge no fully satisfactory result beyond perturbation theory is known. Interesting efforts concern very recent perturbation theories by Sibert and co-workers and by Duncan and co-workers as well as by ourselves using as starting point internal coordinate Hamiltonians, normal coordinate Hamiltonians, and perhaps best, “Fermi modes” [I]. Of course, Michael Kellman himself has contributed substantial work on this question. Although all the available analytical results are still rather rough approximations, one can always

REGULAR AND IRREGULAR FEATURES

591

solve the problem “exactly” by numerical means, and we have done so in numerous cases. In that sense there is not a real problem left. We understand the theoretical essence of the problem and can carry out quantitatively valid calculations, even though there are no simple formulas around, which would be exactly valid. 1.

R. Marquardt and M.Quack, J. Chem. Phys. 95,4854 (1991).

M. E. Kellman: Prof. Field showed that there is a lot of order in the chaotic region of the C2H2 spectrum, by “unzipping” the spectrum into polyads defined by approximate quantum numbers. This gives a partial assignment of the spectrum, but not yet a complete assignment, because there are fewer polyad numbers than the N degrees of freedom of the system. This raises the difficult question, is there any way in these molecules to give a complete assignment with N quantum numbers? Another way of putting it is to ask if the spectrum can be further unzipped into sequences and progressions defined by a complete set of N approximate quantum numbers. We have been working at this problem and have recently made progress. We have used bifurcation theory to determine the anharmonic normal modes of the H20 spectroscopic Hamiltonian in the chaotic region, including the modes “born” from the original normal modes in bifurcations. This was used in a Husimi phase-space analysis [l] of the wave functions to assign the spectrum with quantum numbers appropriate for the bifurcated normal modes. These were used to “reconstruct” sequences and progressions. An example is shown in Fig. 1 for polyad P = 8 of the spectroscopic Hamiltonian. The sequences resemble those of the normal-mode region, but there are marked deviations as well. The figure shows as lines the levels calculated with both the 1 : 1 Darling-Dennison stretch-stretch resonance and the 2 : 1 stretch-bend Fenni resonance and compares these with levels shown as dots calculated by including just the 1 : 1 resonance. Particularly in the chaotic region, interaction of the F e d and Darling-Dennison resonances causes pronounced reorganization of the spectrum. In an attempt to avoid having to perform the difficult bifurcation analysis that leads to this assignment, John Rose and I devised a simpler diabatic correlation diagram approach that leads to the same results. We have applied this very recently to CzHz, where the spectral reorganization, due to the many couplings spoken of by Prof. Field, is much more pronounced than in H20. Nonetheless, the correlation diagram approach works very well. Figure 2 shows the correlation diagram for the polyad with 16 quanta of bend, “unzipped” into subpolyads defined by the effective Z-resonance quantum number obtained

592

GENERAL DISCUSSION

E

(cm

-'

27500

-[lAO -[26]0

-

[OW [16]2

-.

(52)'2mm -

27000

P =n,+n,+n,/2=8 -[06]4

-72414

26500

A

(71)+-0

-

26000

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(60)'-4M'" -

25500

(7O)'2"Om -c

- (50)+'6 0

25000

(41)'s

A

-

11318 (31)+8

(80)+-0

I

I

I

1

I

C

2

4

6

8

Figure 1.

by our procedure. Figure 3 shows the energy levels unzipped into two types of sequences: 2-resonance subpolyads and subpolyads of the Darling-Dennison bend-bend coupling.

593

REGULAR AND IRREGULAR FEATURES

C2H2 P = 16 bend polyad I - resonance subpolyads 7200

cm-' 7100

7000

6900

6800

6700

6800

6500

6400

6300

I

I

I

full coupling Ho+ V Figure 2.

I

I

GENERAL DISCUSSION

594

C2H, P = 16 subpolyads Darling-Dennison"

l-resonance" 9800

Energy (cm-')

i

-

"

_

j

9600

9200

9000

-

In regard to control of energy flow, one can think of defining and trying to prepare time-dependent states closely related to these subpolyads (which contain time-independent levels of the spectroscopic Hamiltonian). If the proper time-dependent states could be prepared, energy could be made to flow with confinement to the chosen subpolyad. Now, 1 would like to pose a question to Prof. Field and Quack and to the entire conference. How do we use spectroscopic information to devise effective schemes to control chemical reactions? Should we try to force the molecule to follow our will? Or should we try to make use of what we learn from spectroscopy about what the molecule wants to do and use this knowledge to get the molecule to do what we actually want? 1. 2. Lu and M.E. Kellman, Chem. Phys. Leit. 247, 195 (1995).

R. W.Field: It is an extremely good idea to ask what the molecule "wants to do." Control schemes based on such knowledge are likely to be both more effective and easier to implement, because less frequent and forceful outside interventions will be required. A simplified picture of molecular dynamics might be very helpful to an experimentalist in designing a control strategy. It is very difficult to visualize the motion of an N-atom molecule in a full (3N - 6)-dimensional configuration space. A reduced dimension picture would serve as a stepping stone to insight.

REGULAR AND IRREGULAR FEATURES

595

The polyad model for acetylene is an example of a hybrid scheme, combining ball-and-spring motion in a two-dimensional configuration space [the two Franck-Condon active modes, the C-C stretch ( Q 2 ) and the trans-bend (Q4)] with abstract motion in a state space defined by the three approximate constants of motion (the polyad quantum numbers). This state space is four dimensional; the three polyad quantum numbers reduce the accessible dimensionality of state space from the seven internal vibrational degrees of freedom of a linear four-atom molecule to 7 - 3 = 4. M. Quack: Prof. Kellman has very nicely phrased the questions concerning reaction control: (1) Should we rather impose our will on molecules to have them do what we wish them to do? Or (2) should we not rather let the molecules do what they would like to do (but using our understanding in generating a good initial state)? He has addressed this question to everybody but did not get an answer. My answer would be, using an analogy between molecules and human beings, that it is neither nice nor possibly easy to use brute force on the molecules. However, often the molecules may be in a state where they do not really know what they want to do. Then we might use some very mild means to seduce them to do what we would wish them to do. As an early example for such mild seduction I might quote the theoretical scheme for potentially mode selective infrared laser chemistry of ozone [ l , 23, which predates some of the more widely publicized subsequent schemes using excited electronic states. 1. M.Quack and E. Sutcliffe, Chem. Phys. Lett. 99, 167 (1983). 2. M. Quack and E. Sutcliffe, Chem. Phys. Lett. 105, 147 (1984)

R A. Marcus: Concerning the report by Dr. Gaspard, I wonder if it might be useful to extract from the data on vibrational resonances the “natural motions” of the system, namely by looking at the typical periodic orbits (or the short-time wavepacket analogues), together with the relevant potential-energy contour that bounds the orbit. The resulting picture would be topological, that is, basis set free and coordinate system free. It could be a useful visual representation of the results and could be extended from pictures in two to then in three dimensions. To be sure, other than codifying the spectra, which the present algebraic scheme already does, it may not have other benefits unless some regularities appear for various classes of molecules. P. Gaspard: Concerning the issue raised by Prof. Marcus, I should remark that information on the shape of the periodic orbits in the original coordinates is lost at the level of a spectroscopic effective Hamil-

596

GENERAL DISCUSSION

tonian as given by a Dunham expansion. Such an information could be partially obtained if the electric dipole moment operator is also known. Further information can be extracted from observables depending directly on the position coordinates through the Stark and Zeeman effects. On the other hand, we should mention that, at the level of classical mechanics, periodic-orbit analysis provides a topological characterization of the system in terms of a symbolic dynamics, which appears as a common feature for a given class of systems. R. A. Marcus: I have a question for Dr. Gaspard concerning the following point. For some systems (e.g., H6non-Heiles), high-order perturbation theory provided a reasonable picture of the Poincar6 surface of section, if not performed to too high an order. It did not contain the irregularities of the actual motion, but nevertheless it provided a relatively good semiclassical quantization. Are there any analogous results in the systems you are studying, systems of more relevance to chemists, than Hinon-Heiles? P. Gaspard: To answer the question by Prof. Marcus, let me say that we have observed, in particular in HgI,, that higher order perturbation theory around the saddle equilibrium point of the transition state may indeed be used to predict with a good accuracy the resonances just above the saddle. However, deviations appear for higher resonances and periodic-orbit quantization then turns out to be in better agreement than equilibrium point quantization. Concerning the Poincar6 surface of section, it should be noticed that a sort of quantum surface of section can be constructed by intersection of the Wigner or Husimi transform of the eigenfunctions expressed in the quantum action-angle variables of the effective Hamiltonian, which can provide a comparison with the classical Poincark surfaces of section (e.g., in acetylene). M. S. Child: Prof. Marcus asks whether there is any way to identify “natural motions” of different species. We have a partial answer for the stretching motions of symmetrical AB2 species [l], based on the simple observation that the bonds of real molecules can break and that the dominant mechanism for interbond coupling in nonhydrides is via the kinetic energy. The old model of Wilson and Thiele [2] embodies just these characteristics by using Morse bond potentials and a kineticenergy coupling term dependent on atomic masses and the interbond angle. Furthermore the classical Hamiltonian may be scaled to a twoparameter form, in which energies are expressed as fractions of the dissociation energy and different molecules are identified by a coupling

REGULAR AND IRREGULAR FEATURES

597

parameter X = MB/(MA + MB)COS 8 , or equivalently by the ratio of small-amplitude antisymmetric and symmetric vibrational frequencies (w,/w,) = [(I - X)/(1 + X)]'I2.

0.5

0.4

$0.1

0

2

4

::0.2

0.1

frequency ratio

Figure 1. Symmetric-stretchclassical stability diagram. The energy is scaled by twice the bond dissociation constant.

The study that we have made was stimulated by the results of Pique et al. [3] on the dispersed fluorescence spectrum of CSz, which shows an assignable progression of Fermi resonance polyads between symmetric-stretch and bending motions, up to roughly half the dissociation energy, followed by a sharp transition to a softly chaotic regime. Moreover, the number of observed states rises as E 2 in the lower energy region and as E3 above the transition energy, indicating a transition from two-mode to three-mode coupling. Our suggestion is that this transition is associated with a change in character of the symmetricstretch periodic orbit from stable to unstable, which leads naturally to excitation of the antisymmetric stretch. Our model deals of course only with coupling between the two stretching motions. The method of analysis is an extension of Hill's method [4], which

GENERAL DISCUSSION

598

reduces to periodically forced motion of linearized displacements perpendicular to the periodic orbit. The extension is to allow for nonharmonic motion along the orbit, followed by a uniform asymptotic analysis rather than reduction to the Mathieu equation [2]. The resulting symmetric-stretch stability diagram is shown in Fig. 1, which shows a sequence of 1 : 1, 3 :2, 2 : 1, 5 :2, ... resonances on passing from the left to the right of the figure. The locations of different molecules in this phase plane are also indicated. Thus, for example, the facile transition from normal-mode to local-mode behavior in H 2 0 is seen to arise from the transition from 1 : 1 stability to instability. CS2, on the other hand, has a much larger q+.db ratio of approximately 5 :2 at low energies, rising to 2 : 1 when the symmetric stretch becomes unstable. It is also noticeable that this instability arises at roughly E/D = 0.5, which coincides well with the transition point in the experimental spectrum

PI.

As a crude two-mode simulation of the three-mode quantummechanical spectrum we have performed a spectral decomposition of two wavepackets initiated at the inner turning point of the symmetricstretch periodic orbit, in other words, with only zero-point excitation of the antisymmetric stretch. Figure 2a shows that a low-energy wavepacket contains only a simple progression in the symmetric stretch. However, increasing the mean energy of the wavepacket so that it spans the symmetric-stretch instability leads to the more complicated spectrum in Fig. 2b. Moreover, a similar break is found to occur at the same energy regardless of the magnitude of Planck’s constant, or equivalently of the actual masses of the atoms involved. The classical mechanics scales according to the relative atomic masses and fractional energy toward dissociation, and the qualitative appearance of the spectrum follows this scaling. Only the quantitative density of lines depends on the actual masses. To answer Prof. Marcus’s question, we may therefore conclude that the natural motions of the system are the short-time periodic orbits. Those that arise from the symmetric-stretch bifurcations depend on the frequency ratio: local modes in the 1 : 1 case, y-shaped orbits at the 3 :2 instability, horseshoes at the 2 : 1 resonances, and so on. 1. T. Weston and M. S. Child, in preparation. 2. E. Thiele and D. J. Wilson, J . Chem. Phys. 35, 1256 (1961). 3. G. Sitja and J.-P. Pique, Phys. Rev. Lett. 73, 232 (1994). 4. W. Magnus and S. Winkler, Hills Equation, Wiley-Interscience, New York, 1966.

M.S. Child: I have moreover a question for Prof. Field. Given that the Darling-Dennison resonance between the symmetric and antisym-

I-

599

REGULAR AND IRREGULAR FEATURES Packet energy = 0.143 I

0.04

0.03

0

0.05

0.1

0.15

0.2

0.25

0.3

Scaled energy

0.35

0.4

0.45

0.5

s (a)

Packet energy = 0.286

0'03 0.025

-

0.02

-

0.015

-

0.01

-

0.005

-

-

0

II

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

Scaled energy (b) Figure 2. Spectral decomposition of two symmetric-stretch wavepackets at (ua/us)= 2.5, which is appropriate for CSz; ( a ) at reduced energy 0.143 and (b) at 0.286.

600

GENERAL DISCUSSION

metric stretch is a local stretch, is the Darling-Dennison resonance of the acetylene bends a local bend? If so, is it a step toward the vinylidene transition state?

R. W. Field: Each acetylene polyad contains zero-order states that are easily accessible via plausible direct or multiple-resonance 'A, -2 ' C i Franck-Condon pumping schemes. Each Evib 2 16,000 cm-' polyad also contains zero-order states that have good overlap with the half-linear acetylene t) vinylidene saddle-point structure. However, the eigenstutes that have detectable intensity in a Franck-Condon spectrum (-1 96 the intensity of the strongest transition within that polyad) typically have negligible (c fractional character of the best saddle-point overlapping state, and vice versa. It seems plausible that the bend Darling-Dennison resonance is capable of creating local-bender states by strongly mixing the cisand trans-bending normal-mode states. Such local-bender states would have excellent overlap with the acetylene t)vinylidene transition state. However, I do not know whether the increase in strength of the bend Darling-Dennison resonance as h n d E u4 + u5 increases will be overwhelmed by a Franck-Condon dilution effect owing to an even more rapid increase in the dimension of the polyad. However, I remain optimistic that something about the change in the w I :w I :w2 :- . - :w5 resonance structure that occurs at the energy of the acetylene c)vinylidene saddle point will be detectable in the dispersed fluorescence or stimulated emission pumping spectrum. E. Pollak: There is a unifying theme in the talks of Profs. Field and Gaspard. A huge body of work by people such as Child, Taylor, Farantos, Schinke, Tennyson, Schlier, and others has demonstrated the utility of the concept of periodic-orbit normal modes. At the bottom of the potential-energy surface, a periodic orbit corresponds to each normal mode. As one goes up in energy, bifurcations take place, and Child has beautifully analyzed the normal to local transition in terms of localmode periodic orbits. We have seen in H3+ how periodic orbits can be used to assign coarse-grained spectra in correspondence to short-time localizations. Taylor, Gomez Llorente, and co-workers have demonstrated the same for light-atom transfer systems, Na3, and more. Farantos has demonstrated how one can follow up in energy periodic-orbit bifurcation diagrams in three dimensions. The bottom line: Periodicorbit normal modes should become the language of high-energy molecular spectroscopy. Finally a question for Prof. Gaspard. You have shown the beautiful structure of periodic orbits in H3.Why, beyond a preliminary work of

REGULAR AND IRREGULAR FEATURES

601

Stine and Marcus, has no one given a good semiclassical analysis of the reaction probability in collinear H3? P. Gaspard: To answer the question by Prof. Pollak, we expect from our present knowledge that the periodic-orbit quantization of the H + H2 dissociative dynamics on the Karplus-Porter surface can be performed with the same theory as applied to HgI,. E. Pollak: The computation of Stine and Marcus for H3 was preliminary. I do not think that ghost orbits need be invoked but rather it is high time for a detailed and careful computation. B. A. Hess: Dr. Gaspard has introduced the vibrogram as a tool to extract periodic orbits from the spectrum by means of a windowed Fourier transform. This raises the question whether other recent techniques of signal analysis like multiresolution analysis or wavelet transforms of the spectrum could be used to separate the time scales and thus to disentangle the information pertinent to the quasiclassical, semiclassical, and long-time regimes. Has this ever been tried? P. Gaspard: As far as I know, the wavelet analysis of spectra has not yet been done and would be very interesting to develop. A remark is that the vibrogram also depends on the width E of the Gaussian window, which may be varied to construct another kind of plot. V. Engel: You showed the plot of the Nal molecule, which is a system with a curve crossing. The treatment of classical mechanics in such a system is not well defined. What can be done to treat the nonadiabatic effects if you calculate periodic orbits that are then used for interpretation? P. Gaspard To answer the question by Prof. Engel about curvecrossing problems, we should mention the work by Littlejohn and Weigert on matrix Hamiltonians [R. G. Littlejohn and S.Weigert, Phys. Rev. A 48,924 (1993)l. These authors have shown how to derive systematically the adiabatic Hamiltonians describing the classical motion on each potential surface. It turns out that surface hopping between surfaces requires one to use complex periodic orbits. J. Manz: The theoretical method of Prof. Field (See Field et al., “Intramolecular Dynamics in the Frequency Domain,” this volume.) evaluates the fluorescence dispersion spectra of HCCH in terms of the Fourier transform of the autocorrelation function,

where

602

GENERAL DISCUSSION

-

denotes the normalized (factor N ) initial (i) vibrational (v) eigenstate lClevi(q)prepared in the electronic excited state (e), multiplied by the g transition transition dipole functio! pge(q)for the electronic e and propagate9 [exp(- iHet/h)] on the electronic ground state ( 8 ) with Hamjltonian H e , as suggested by E. J. Heller [l]. I would like to ask Prof. Field about his model for pge(q).In principle, the transition dipole function pge(q) will affect all the subsequent quantitative results, in particular if pse(q)varies strongly along the vibrational coordinates q in the domain of +evi(q).In fact, one should expect substantial variations of pge(q)as the linear equilibrium structure of acetylene is transformed into vinylidene. 1. E. J. Heller, J. Chem. Phys. 68, 3891 (1978); Acc. Chem. Res. 14, 368 (1981).

R.W. Field: I must apologize for not being sufficiently clear about

the excitation scheme we use for our acetylene experiments. Although the initial and final states are both on the acetylene 2 'EB surface, the final state we prepare is the result of two electronic transitions +-i followed by --t i )rather than one vibrational-rotational infrared or Raman transition. There is a profound difference between the knowledge of the excitation function needed to describe electronic versus vibrational processes. The acetylene tj 2 electronic transition is a bent tj linear transition that would be electronically forbidden ('E;-'Ei) at the linear structure. The usual approximation is to ignore the possibility that the electronic part of the transition moment depends on nuclear configuration and to calculate the relative strengths of vibrational bands as the square of the vibrational overlap between the initial and final vibrational states (Franck-Condon factor). A slightly more accurate picture would be to express the electronic transition moment as a linear function of Qy (the trans-bending normal coordinate on the linear 2 'Xi state); in such a treatment, the transition moment would be zero at the linear structure and the vibrational overlap factors would be replaced by matrix elements of @(. Nevertheless, as long as one makes use of low vibrational levels of the state, neglect of the nuclear coordinate dependence of the electronic excitation function is unlikely to affect the predicted dynamics or to complicate any proposed control scheme. R. Jost: Prof. Field has shown three dispersed fluorescence spectra

A

(A

REGULAR AND IRREGULAR FEATURES

603

that contain information on ground-state vibrational energy levels of C2H2 up to 27,000cm-' ,but your analysis in terms of feature states and polyad quantum numbers breaks down at about 13,000 cm-'. In order to see if new feature states (i.e., new periodic orbits) appear at higher energy, I suggest to perform a windowed Fourier transform analysis (i.e., to get vibrograms as explained by P. Gaspard). It is tempting to think that the presence of the vinylidene minimum is responsible for the disappearance of some of the polyad quantum numbers, but new feature state@)[periodic orbit(s)] may appear above the vinylidene threshold. Prof. Field, did you try such a Fourier transform analysis? R. W.Field: No, I have not done it so far.

MOLECULAR RYDBERG STATES AND ZEKE SPECTROSCOPY

ZEKE SPECTROSCOPY E. W. SCHLAG Znstitut f i r Physikalische und Theoretische Chemie Technische Universitat Miinchen Garching, Germany CONTENTS I. Introduction 11. ZEKE Spectroscopy 111. Conclusion References

I. INTRODUCTION Zero electron kinetic energy (ZEKE) spectroscopy is a new form of spectroscopy with photoelectrons that, however, differs from photoelectron spectroscopy (PES) in some substantive ways. Basically it employs scanning photon sources much as in the photoionization efficiency (PIE) technique of Watanabe [ l ] (Fig. 1). The photoelectron spectroscopy of Siegbahn, Turner, and Terenin, on the other hand, employs a fixed wavelength in the vacuum ultraviolet (VUV)region of the spectrum and then analyzes the kinetic energy of the various photoelectrons emitted. This difference is more than one of procedure. Albeit it has proven to be nearly impossible to obtain electron energy peaks at a resolution of much better than some 100 cm-' in a standard experiment, ZEKE peaks are performed today at a resolution of some 0.1 cm-' and better. Such an increase in spectral resolution alone has made ZEKE spectroscopy a full-fledged spectroscopic technique. In addition, however, ZEKE spectroscopy is characterized by new selection rules that enable access to large number of levels, even normally forbidden levels. Hence, whereas in ZEKE spectroscopy nearly all levels are accessible

Advances in Chemical Physics, Volume 101: Chemical Reactions and Their Control on the Femtosecond Time Scale, Xxth Solvay Conference on Chemistry, Edited by Pierre Gaspard, Irene Burghardt, I. Prigogine, and Stuart A. Rice. ISBN 0-471-18048-3 0 1997 John Wiley & Sons, Inc.

607

E. W. SCHLAG

608

(1954)

PIE

Photo Ionization Efficiency Watanabe

CambridgeMass.

PES

Photo Electron Spectroscopy Siegbahn Turner Terenin Kirnura

Uppsala Oxford St. Petersburg Okazaki

TPES

Threshold Photoelectron Spectroscopy Peatman

Zero Electron Kinetic Energy

-

(1984)

Munich

Anion ZEKE Neumark Drechsler

(1969)

Evanston

ZEKE

Muller-Dethlefs

(1961)

(1989) Berkeley Munich

Figure 1. Historical development leading to ZEKE spectroscopy.

if the energy is right, PES is a vertical spectroscopy that totally depends on the Franck-Condon principle in direct transitions. This is most clearly illustrated in Fig. 2a, where all vibronic states of nitric oxide are seen, including states in the Franck-Condon gap, whereas PES observes only the first five vibronic bands (Fig. 2). The scanning of a laser across the ionization threshold produces an almost continuous increase in the photocurrent with excess energy, as seen in Fig. 3a. Putting this same signal through a ZEKE “filter,” one produces the signal in Fig. 3b. In other words, the ZEKE signal that represents the ionic states of Nz+is seen here as a sharp signal that can be analyzed in terms of the species N2+ alone. This signal is buried but contained in Fig. 3a. It can be extracted by separating the direct ions produced above in the ionization energy, waiting some time until only high-lying Rydberg states survive, and now ioniz-

609

ZEKE SPECTROSCOPY

90,000 95.000

85,000

80.000

75,000

CM-I

105;OOO . 115;OOO CM-I

'

125,000

-P 10,000 2 -

m OD c

._

2

50,000

I

I

I

'

1

'

I

' '

I

7

I

' '

I

'

I

l~I'"'I""I""I""1"~'1"''~"

75,000 80,000 85,000 90,000 95,000 105,000 115,000 125,000 CH-I

I

10

CM-1

11 12 Binding energy (eV)

I

13

Figure 2. Comparison of the NO+ spectra with equal energy scales of (a) ZEKE spectrum,showing ion states up to d = 26; (b) total ion spectrum;and (c) photoelectron spectrum from Turner's book [2].

ing these with an extraction pulse. Hence, simply waiting produces the sharp ZEKE spectrum. This is due to the fact that it has been discovered that just below each possible state of an ion there reside high-n Rydberg states with an enormous lifetime of some 20-100 ps. This band of high Rydbergs is very narrow and only just below each ion state. Hence, separating these ZEKE

610

E. W. SCHLAG

a) photdonlzatlon

ZEKE filter b) ZEKE

W.vcnumk1 (an.')

Figure 3. Comparison of PIE spectrum of N2+ with the corresponding ZEKE spectrum: (a) photoionization spectrum in the region of the first IP [3]; (b) ZEKE photoelectron spectrum [41.

states out of the sea of ions gives a signature of all possible ionic states. This is totally different, in principle, from PES. Another example is the silver dimer Ag,, as shown in Fig. 4 [ 5 ] . Here the signature of the total spectrum of direct ions is seen in the top of Figure 4. In the same experiment destroying these with a spoiling field and analyzing the remaining ZEKE states with a delayed pulsed field give the correct vibronic spectrum of Ag2+, as shown in the bottom of Fig. 4. In the top spectrum there are clearly many excited ionic states produced directly, but these are autoionization resonances at energies other than the vibronic threshold of the E K E delayed pulsed-field ionization (PFI)spectrum. This technique has now been applied to many molecular systems, as shown in Fig. 5 . It can also been applied to van der Waals (vdW) complexes, where often all six intermolecular vibrational modes are observed. Another important application is to anions, where here the electron detachment produces ground-state neutral systems (Fig. 6). The anion state can be

61 1

ZEKE SPECTROSCOPY

Ag2 : Delayed PFI versus Direct Ions

-

X2Zi

B'II,(V-4)

+

X'X; ( v " - 0 )

Delayed PFI

v'-0

I

I

61800

'

i

'

l

"

62000

'

l

"

-'

'

l

62200

'

62400

M

Figure 4. Comparison of PIE spectrum of Ag2+ with the corresponding ZEKE spectrum

PI.

mass selected, and thus the neutral produced from this anion is also mass selected. Hence this provides a means for mass-selected neutral ZEKE spectroscopy. Since anions can be made of various metastable species, so can the neutral spectrum be measured for this metastable species. One example is the high-resolution neutral ground-state spectrum of a radical, here OH (Fig. 7). The various types of archetypal classes of systems that have been studied is shown in Fig. 8.

II. ZEKE SPECTROSCOPY Threshold electron spectroscopy, or its newest variant ZEKE spectroscopy, represents a new approach to these problems that has already afforded a broad set of new applications, particularly for soft bonds and metastable species, such as is characteristic for metastable reactive intermediates.

612

E. W. SCHLAG

Small Molecules

nitric oxide ammonia hydrogen sulfide carbon disulfide methyl iodide, ethyl iodide

Aromatic Molecules

benzene para-difluorobenzene phenol toluene

Van der Waals Complexes

benzene-argon benzene-krypton para-difluorobenzene-argon

H-Bonded Complexes

phenol-water phenol-methanol phenol-ethanol phenol-dimethylether phenol dimer

Mixed Complexes

phenol-water-argon

Metal Complexes

silver dimer

Molecular Complexes

nitric oxide dimer

Anions

hydroxyl radical iron oxide iron dicarbide

Figure 5. Molecular systems studied with ZEKE spectroscopy.

In the late 1960s steradiancy techniques were initiated that measured highly accurate drift-free thresholds (TPS), albeit with a modest increase in resolution. This work was started by Peatman et al. [6-81. This abandons all measurements of electron energies but instead measures the resonance wavelength via the steradiancy of the emitted electrons, a unique property at threshold. Only at threshold is it possible to produce electrons with no kinetic energy, and hence constitutes a source that lacks any angular divergence. A primitive optics, in fact a straight pipe, will, in the limit of extreme length, only transmit these threshold electrons. Coincident timing of these electrons also suppresses the remaining straight-through component 191. The threshold is ips0 fact0 exact (as exact as the light source) and cannot be affected by surface effects, for example (except in intensity, not energy). This became a new state selector of molecular ions since coincident with these threshold electrons are ions in a defined molecular state. The kinetics

613

ZEKE SPECTROSCOPY

;

f

=

Figure 6. Spectroscopy of neutral ground state via anion ZEKE spectroscopy. Rotationally resolved anion-ZEKE of OH

-m .u) op. W

Y N W

-

U(0) EA

EE Ii 1.5 em-1

14695 14700 14705

Figure 7. High-resolution spectroscopy of the ground state of the neutral OH radical.

E. W. SCHLAG

614

Molecular Catlons Fragment Cations

CH;

,C=C-C+

Catlon Complexes

van der Wsds Energies F-

Molecular Anions Fragment Anions

SH

I-H-l -

Transient Anions sholt lived

Au 2-

-

Fe C-C

Neutrals in Ground State IR. Ramen. MC.

6

Fragments in Ground State

SH

Transients in Ground State

I-H-l Fe-C-C

Reactive Intermediates

Au

Slate md Mess selected Spectra h Ground State

Surfaces - Clusters Figure 8. Types of systems studied with ZEKE spectroscopy.

of such state-selected ions has made important contributions to the understanding of the theory of unimolecular reactions [101 employing coincidence techniques (PIPECO). This TPS technique gives the exact location of states of molecular ions at a reasonable resolution of some 10-20 cm-' . The differences are demonstrated in Fig. 9, where the left-hand side shows threshold measurements. At the threshold, and only here, is the energy of the emitted electron near zero. Employing a detector for such zero kinetic energy electrons will show a peak only at the exact position of the molecular energy level (Fig. 10). The undifferentiated total electron current would, however, in contrast have kept increasing, since the excitation energy merely leads to an increasing velocity of the two charged particles produced. A very simple example (atomic argon) demonstrates this principle (Fig. 11). The total current simply reflects the onset of the ionization potential (IP) at the 2P3p state and the Rydberg series leading to the next level of the ionized atom, the 2 P , p state, this, however, not being directly visible. In contrast, the threshold spectrum (Fig. 11) clearly gives a peak for both states as well as suppresses the Rydberg spectrum, which generally leads to hot electrons.

615

ZEKE SPECTROSCOPY

t h r es ho Id spec t ros c opy

photo e l ec t ron spec f r o scop y

Figure 9. Comparison of threshold spectroscopy and photoelectron spectroscopy.

High Rydberg states were found, however, to have a very important property [Ill. Rydberg states with quantum numbers n = 100, ..., 200, just 1-2

'.' ;Y;. [k-

cm-' below the respective level of the free ions, have recently been discovered to be surprisingly long-lived, typically some 50 ps [11, 121, whereas some 10 ns would have been anticipated [13]. If now the laser excitation

... ..._.... .......[... I

..,. . ... . , , , .

_j.

.

.

M ("+,Nil

. . . .. . ... ... ., , , , . M Iv+, N I;

_......._............................ M (v: NTl

total Ion Signal

Threshold

w tuned

Signal

M(v.NI

Figure 10. Comparison of total ion spectrum with threshold spectrum.

E. W. SCHLAG

616 12 s

13 s

us

-

Ar : ION YIELD

+ E

U

ZEKE

127 000

128 000

L Icni’l

Figure 11. Argon spectrum.

occurs in a field-free environment, such Rydberg states will persist for some 50 ps. If an electric pulse is now applied after some 1-5 ps, only these highlying states will be observed and all other states will have disappeared. Since these high-lying states directly reflect the onset of each ionic level to which they converge within some 1-2 cm-’ ,this is a direct measure of the molecular ion spectrum displaced by some 1-2 cm-I , a number that can be calibrated or extrapolated away. This is the basis of modem ZEKEi spectroscopy of molecular cations. The resolution of ion spectroscopy now becomes astonishing in that it increases to near laser resolution and thus ion spectroscopy emerges as being on an equal footing with traditional high-resolution conventional spectroscopies. Figure 12 demonstrates the marked increase in resolution; we compare the historical first ZEKE spectrum (below) to the corresponding photoelectron spectrum [2, 14-16]. In fact, such a ZEKE spectrum will continue to display a rich set of energy levels in the ionic ladder up to u = 26 (top of Fig. 2), whereas the trace of the total electron current (middle photoionization efficiency trace) as a function of energy displays intially the traditional staircase function, which then becomes obscured by

617

ZEKE SPECTROSCOPY

Ebind lev1 9

11

10

12

-0 C

.-(31

m

W Y

W N I

I

0

I

10.0 E int Icm-’l

I

20.0

Figure 12. Comparison of the photoelectron spectrum of NO with the original ZEKE spectrum of the first band, with demonswation of rotational resolution of the ion state.

many autoionization and other resonances. Interestingly the photoelectron spectrum breaks off at u = 4 or u = 5 (bottom of Fig. 2). More recent experiments on nitric oxide are displayed in Fig. 13 (left side) [17], showing typical alternations due to parity but also showing large changes in the orbital quantum number of the outgoing electron. This was quite puzzling, until detailed theoretical calculations by McKoy et al. (right side Fig. 13) [ 181 confirmed this to be a new property of this spectroscopy in conformity to large-scale theoretical calculations. Figure 14 demonstrates a photoelectron spectrum (top) for benzene [19] compared with a current ZEKE spectrum (bottom) of some 0.2 cm-’ resolution [20], even demonstrating the expected splitting of the vibration due

618

E. W. SCHLAG Cdculstd

N, = O ).(

3

1

3

Ekprimantal

NA='

(b)

Exprimeold

5 Figure 13. Comparison of experimental ZEKE spectra of NO' [17] with ab initio calculated spectra [I81 N A reflects the initial rotational state of the transitions in the SIintermediate state. 0

1

200

100 I

I

E i ntl m e V 1

0

P

y1

W

a

0

P

y1

W

Y

W N

36288

-

36308

E [ cm"]

36328

Figure 14. Benzene spectra: ( a ) photoelectron spectrum; (b) ZEKE spectrum of a single band in (a).

619

%EKESPECTROSCOPY

U

6l

62

u -

O0

7L000

3

"\,

4:

75000

w,

0

w2

2l

lcrn-ll

76000

Figure 15. Spectrum of para-ditluorobenzene: ( a ) high-resolution photoelectron spectrum [19]; (b) ZEKE spectrum in the same energy range [21].

to the Jahn-Teller effect showing rotational resolution. Figure 15 shows a high-resolution photoelectron spectrum for para-difluorobenzene compared to the ZEKE spectrum below [21]. Figure 16 demonstrates the results for a cluster phenol-water [22]. The top of Fig. 16 is the total current [23], repeated in our laboratory (middle), and the ZEKE spectrum below demonstrates the direct observation of the new soft modes derived from intennolecular vibrations and their many overtones and combinations for the soft-modes phenol-methanol clusters [24]. Such normal vibrations are pictorially displayed in Fig. 18. An interesting demonstration for a chemical application to neutral ground states is in a mixture that represents a study of Fe complexed to a hydrocarbon, in this case the complex FeC4H2. This complex can be uniquely identified in the anion mass spectrum of an iron target in a hydrocarbon stream (top of Fig. 19). Threshold photodetaching this mass-selected species in a very high resolution ZEKE experiment then directly produces the unequivocal spectrum of the ground state of this interesting reactive intermediate, demonstrating that the structure must have linear C-C-Fe bonding. This can have many interesting application to the chemistry of such intermediates. Figure 8 gives a short summary of the types of systems that are amenable to this new form of high-resolution photoelectron spectroscopy for extreme

620

E. W. SCHLAG

phenol water

65 000

6L 500

Icm-’l

6 L 000

total energy / cm-1

Figure 16. Spectrum of the phenol-water complex: (a) total ion signal [23]; (b) total ion signal [22]; ( c ) ZEKE spectrum [22].

metastables (even transition states [25]) and for reactive intermediates, thus opening up a new spectroscopic look at metastable states and soft modes. III. CONCLUSION

Zero electron kinetic energy spectroscopy provides a new tool in the study of chemical systems. In particular, it is applicable to molecules that are not “in a bottle” (metastable species) which nevertheless are of fundamental

62 1

ZEKE SPECTROSCOPY

27600

28ooo

Ionising Laser Energy / cm-'

28500

Figure 17. Phenol-methanol ZEKE spectrum via the vibrationless state of S1. n

(J

(24Ocrn.')

Z (2%:257 cm-')

V

1 y ' (328 cm")

p ' (84 cm-')

y

(261 cm")

p"(67cm-')

Figure 18. Phenol-water complex: intermolecular normal modes.

622

E. W. SCHLAG

I

1

I

‘98

i

1

I

100

I

1

l

I

1

t

102

l

I

I

1

1

l

I

104

I

.

106

I

I

I

I

I

I

amu

I

108

I

I

I

I

I

110

I

I

I

I

I

I

112

.

I

I

*

I

I

I

I

I

114

Figure 19. Neutral spectrum as mass-selected anion spectrum.

importance to chemistry as intermediates in chemical reactions. The struc-

ture of these systems is important for an understanding of the mechanism of chemical reactions. This applies not only to cations and anions as objects of study but also to free radicals, complexes, hydrogen bonded or just vdW complexes, and other unstable species. In spite of their substantial instability, these can be mass selected as anions and then studied as a spectrum of the mass-selected neutral ground state of these species. Considering the great interest in such species, this should become one of the major areas of application of this new spectroscopy.

ZEKE SPECTROSCOPY

623

References I . K. Watanabe, (a) J. Chern. Phys. 22, 1564 (1954); (b) ibid., 26, 542 (1957). 2. D. W. Turner, C. Baker, A. D. Baker, and C. R. Brundle, Molecular Photoelectron Spectroscopy, Wiley, London, 1970. 3. 3. Berkovitz and B. Ruscic, J. Chem. Phys. 93, 1741 (1990). 4. F. Merkt and T. P. Softley, Phys. Rev. A 46, 302 (1992). 5. Ch. Yeretzian, R. H. Hermann, H. Ungar, H. L. Selzle, E. W. Schlag, and S. H. Lin, Chem. Phys. Lett. 239, 61 (1995). 6. W. B. Peatman, Ph.D. Thesis, Northwestern University, Evanston, 1969. 7. W. B. Peatman, T. B. Borne, and E. W. Schlag, Chem. Phys. Lett. 3,492 (1969). 8. T. Baer, W. B. Peatman, and E. W. Schlag, Chem. Phys. Lett. 4, 243 (1969). 9. P. M. Guyon, T. Baer, L. F. A. Ferreira, 1. Nenner, A. Tabche-Fouhaile, R. Botter, and T. R. Govers, J. Phys. B 11, L141 (1978). 10. T. Baer, in Gas Phase Ion Chemistry, M. T. Bowers, Ed., Academic New York. 1979, p. 153. 11. G. Reiser, W. Habenicht, K. Muller-Dethlefs, and E. W. Schlag, Chem. Phys. Left. 152, 119 (1988). 12. W. G. Scherzer, H. L. Selzle, and E. W. Schlag, Z. Naturforsch 48% 1256 (1993). 13. E. W. Schlag and R. D. Levine, J. Phys. Chem. 96, 10608 (1992). 14. K. Muller-Dethlefs, M. Sander, and E. W. Schlag, Chem. Phys. Lett. 112, 291 (1984). 15. K. Muller-Dethlefs and E. W. Schlag, Ann. Rev. Phys. Chem. 42, 109 (1991). 16. K. Muller-Dethlefs. E. R. Grant, K. Wang, V. B. McKoy, and E. W. Schlag, Adv. Chem. Phys., in press. 17. M. Sander, L. A. Chewter, K. Miiller-Dethlefs, and E. W. Schlag, Phys. Rev. A 36,4543 (1987). 18. H. Rudolph, V. McKoy, and S. N.Dixit, J. Chem. Phys. 90,2570 (1989). 19. E. Sekreta, K. S. Viswanathan, and J. P. Reilly, J. Chem. Phys. 90,5349 (1989). 20. R. Lindner, H. Sekyia, and K. Muller-Dethlefs, Angew. Chem. 105, 1384 (1993). 21. D. Rieger, G. Reiser, K. Muller-Dethlefs, and E. W. Schlag, J. Phys. Chem. 96, 12 (1992). 22. G. Reiser, 0. Dopfer, R. Lindner, G . Henri, K. Muller-Dethlefs, E. W. Schlag, and S. D. Colson, Chem. Phys. Lett. 181, 1 (1991). 23. R. J. Lipert and S. D. Colson, J. Chem. Phys. 89,4579 (1988). 24. T. G. Wright, E. Cordes, 0. Dopfer, and K. Muller-Dethlefs, J. Chem. Soc. Faraday Trans. 89, 1609 (1993). 25. G . Drechsler, C. Bksrnann, U. Boesl, and E. W. Schlag, J. Mof. Srruct. 348,337 (1995). Note: cf ZEKE homepage: http://www.chemie.tu-muenchen.de/zeke

DISCUSSION ON THE REPORT BY E. W.SCHLAG Chairman: G. Casati

G. Gerber: Prof. Schlag, could you shortly explain what is an inverse Born-Oppenheimer (BO) situation?

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E. W. SCHLAG

E. W. Schlag: It simply means that the electron is the slowest particle compared to the more rapid nuclear motions. For this reason, each rotation has its own Rydberg series. This is referred to typically as an inverse BO representation, similar to a Hund’s case d. S. A. Rice: Have you studied any excited ion states with degenerate vibrations to ascertain the coupling of the vibrational angular momentum to the electron angular momentum? E. W. Schlag: Yes, we do see splitting of the ion states. In particular, we see the Jahn-Teller splitting for the first time for benzene ions and in great detail. A. H. Zewail: Are we supposed to think, according to the discussion by Prof. Schlag, that for high-n Rydberg orbitals we have an electronic rnanifoEd within each vibrational state, the inverse of conventional molecular description? E. W. Schlag: Each rotational or vibrational level has its own private Rydberg series that are strongly coupled to other series and are driven by entropy to the highest n state of all accessible manifolds. M. Shapiro: The cuckoo effect: How do we know where the ZEKE electron comes from? If A is present among B, the ZEKE electron can come not from B, influenced by A, but from the minority species A itself. E. W. Schlag: The pickup is from a sea of ions of one species that picks up the E K E electron of the other species. Thus it shows up at the wrong energy. When the sea of ions is shut off, the signal disappears since it is at the wrong mass. This demonstrates that ZEKE states have an existence of their own and need not be formed necessarily from optically excited low4 Rydberg states.

SEPARATION OF TIME SCALES IN THE DYNAMICS OF HIGH MOLECULAR RYDBERG STATES R. D.LEVINE The Fritz Haber Research Center for Molecular Dynamics The Hebrew Universily Jerusalem, Israel and Department of Chemistry and Biochemistry University of California Los Angeles Los Angeles, California

CONTENTS I. Background

11. Preliminaries

111. Dynamics A. Effective Hamiltonian B. Trapping Versus Dilution Iv. Concluding Remarks References

1. BACKGROUND There are many motivations for the study of the unusual dynamics of high Rydberg states of molecules. The two that most capture my imagination are the exceptionally wide range of time scales involved (Fig. 1) and the unusual limiting situation of a very slow electron being perturbed by the faster motion of the nuclei in the core about which it revolves. What this means is that, as one varies the hydrogenic principal quantum number n, it is possible Advances in Chemical Physics, Volume 101: Chemical Reactions and Their Control on the Femtosecond Time Scafe. XXfh Solvay Conference on Chemistry, Edited by Pierre Gaspard, Irene Burghardt, I. Prigogine, and Stuart A. Rice. ISBN 0471-18048-3 0 1997 John Wiley & Sons, Inc.

625

R. D.LEVINE

626

/-.time resolved

'

-I

Stark modulatio

1

10 100 principal quantum number/n

Figure 1. Orbital period of an electron moving in a Coulomb field, the time scales of some internal and external perturbations [3a-3d], and the observed (shorter, see below) lifetime for the polyatomic molecule known as BBC [4]. Note that at the highermost values of n the decay lifetime begins to shorten; cf. Fig. 4.

to cover the entire range from the Born-Oppenheimer limit to its inverse (Fig. 2). Zero electron kinetic energy (ZEKE) spectroscopy [6a, 71 has opened up the study of high Rydberg states and, in particular, has drawn attention to the many time scales that are involved. Extremely long living (tens of microsecond) states (which can be detected by their ionization by a pulsed weak electrical field) have been observed [ l , 6b, 7-17]. However, shorter living (submicrosecond scale) states can also be seen [4, 18, 191, and the preliminary experimental evidence is that the shorter living states are more intense whereas the extremely stable states constitute about 5-20% of the ZEKE signal [19]. There is also some experimental indication and much theoretical evidence for a prompt (i.e., comparable to an orbital period) decay. Our interest in the study of the lifetime of high Rydberg states was motivated by the phenomena of delayed ionization [20]. Multiphoton (and also single-photon excitation of large molecules [21,221 and of clusters [24-321) has revealed that such molecules do not necessarily promptly ionize, even though they contain enough energy to do so without the assistance of an electrical field or any other external perturbation. There is also no experimental doubt that they decay also by electron emission (since one can detect the electrons and even measure their kinetic energy distribution [33]). There can, however, be other decay mechanisms, in particular dissociation of the

627

DYNAMICS OF HIGH MOLECULAR RYDBERG STATES

-1"

-20

1 " hverse '

"

"

0

'

"

'

Born-Oppenheimer (~=200) "

0.2

"

'

"

"1

"

0.4

0.6

0.8

U nz

I

Figure 2. Effect of the frequency w of the perturbation by the core on an electron moving in a Bohrdommerfeld orbit of high eccentricity (low angular momentum). Plotted vs. the angle u, which varies by 2r over one orbit. Note that the perturbation is localized near the core. In the inverse Born-Oppenheimer limit o( >> 1) the perturbation oscillates many times during one orbit of the electron. (For further details and the formalism that describes the motion at high x as diffusive-like (see Refs. 3c and 5.) For higher angular momentum 1 the effective adiabaticity parameter is x( I - e) = x12/2, where e is the eccentricity of the Bohr-Sommerfeld orbit. States of high 1 are thus effectively decoupled from the core.

energy-rich molecule (or cluster) into neutral fragments. We have indeed proposed [21] that the energy-rich molecule stores the energy in the very many degrees of freedom of the core and that it is the slow, diffusivelike transfer of the energy back to the outgoing electron that is rate determining. This point of view does account for the size dependence [21] and also provides definite predictions for the dependence on the total energy (Fig. 3).

,

0

0.5

I

1.5

2

Energy I Ionization potential

Figure 3. Transition from the small- to the large-molecule limit. (The large molecule is twice as large as the middle one, and both are far larger than the small one, which undergoes prompt ionization.) A schematic plot of the yield of ionization vs. the available energy scaled by the ionization potential.

R. D. LEVINE

628

The study [ 181 of time-resolved ZEKE spectroscopy was undertaken to obtain a better and detailed understanding of the phenomena of delayed ionization. The earlier work by Even and co-workers at TeI Aviv University had several limitations. First, there was a delay time between excitation and the time at which detection could begin of about 100 ns. In the range of states of interest (say, n > 100) the experiment was therefore blind to what happens during the first 100 orbits or more (cf. Fig. 1). In the decade of 100-1OOO ns the signal decayed roughly exponentially but with a long-time component of roughly 5-20% of the total intensity (depending on the molecule, the wavelength, etc.). These results were obtained for a number of (jet-cold) aromatic molecules, all of which exhibited a “turnover” in that the (shorter) lifetime increased with increasing principal quantum number, as does the orbital period (Fig. 1) but then started to decrease (Fig. 4). Another limitation of the early Tel Aviv experiments was that the noise level was such that one could not examine the long-time (microsecond-scale) component at a good signal-to-noise level. Furthermore, experiments on Hg atoms suggested that for times longer than 1 ps the atoms were subjected to external perturbations. A third limitation was that at the time one did not have a precise enough value for the ionization potential, and therefore one did not know whether the optically prepared state was above or below the threshold for ionization. This has since been rectified [4] by observation of many members of Rydberg series that allowed an independent determination of the ionization potential. Thereby an assignment of n can be made and hence provided the lifetime as a function of n, as shown in Fig. 1. In the experimental arrangement used by Schlag and co-workers at the

DCA

.

* -800 0

.i700 s“ 6 0 0 c

500”

-8

I

I

I

I

-6 -4 -2 0 Energy below threshold /cm*’

Figure 4. The measured (shorter) decay lifetime (in nanoseconds) of dichloroanthracene (DCA) vs. the energy below the threshold to ionization. The curve is a fit for a diffusional motion of the electron about an anisotropic core in the presence of a weak (stray) DC field. For the derivation of the kinetic description from a Hamiltonian see Ref. 3c.

DYNAMICS OF HIGH MOLECULAR RYDBERG STATES

629

Technical University Munich, the delay time before detection can begin is about 10 ps (or down to 1 ps in the recent experiments by Muller-Dethlefs and co-workers [6a]). Most of the time-resolved experiments were, so far, carried out for benzene. The question of whether the longer decay observed (but not well characterized) in the Tel Aviv experiment is the same as the long-time (microsecond scale) decay observed for benzene in a variety of Munich experiments [6b, 7, 9, 10, 13, 35, 361 is still open. It is suggestive that it is, but the definite experiments are not yet done. At the same time, interest in ZEKE spectroscopy focused attention on a rather different aspect, namely the extreme sensitivity of the high Rydberg states to external perturbations [2, 3a, 6, 8, 11, 12, 15, 17, 37401. What is clear is that the presence of an electrical and/or magnetic field or external charges can elongate the lifetime of states into the microsecond range. From a practical point of view this is very beneficial since it gives rise to a larger ZEKE signal that is useful in both the spectroscopic and the analytical applications. On the other hand, it clouds the issue of what effects are inherent to the isolated molecule and what is due to external perturbations. Of course, there is the other side of the coin. The external perturbations provide probes with the right time scales for the study of high Rydberg states; cf. Fig. 1. Detailed considerations (Fig. 5 ) show that the effect of external perturbations is not invariably in the direction of elongating the lifetime. Indeed, the dynamical computations also shown in Fig. 5 support the interpretation that the decrease in the lifetime with increasing electrical field is due to the lowering of the threshold for ionization [41]. This, we think is the same effect

v)

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-Computed

.\.

0.1 01 ” ” 0 0.2 0.4 0.6 0.8

1

o

.j ’

’

1.2 1.4 1.6

DC field I (V/cm)

Figure 5. Experimental (shorter) lifetimes of DABCO in the presence of a DC field (in addition to a stray field of 0.1 V/cm) compared to a computation using classical mechanics [ I ] for an electron revolving about a quadrupolar anisotropic core. The smooth line is a fit to (see Ref. 1 for more details). an exponential dependence on

630

R. D. LEVINE

that is due to the shortening of the lifetime as one approaches threshold, as seen in Fig. 4. In terns of the picture of delayed ionization, this effect corresponds to reducing the value of the ionization potential and so, at a given energy, is equivalent to moving to the left on the abscissa of Fig. 3. 11. PRELIMINARIES

We consider that the essential physical nature of the decay of high Rydberg states (and of delayed ionization in general) is that there is a very congested set of bound states coupled to a sparse continuum. In the language of transition-state theory, the number of states of the transition state is far smaller than the number of states of the energy-rich molecule to which they are coupled. This is also the case for an ordinary unimolecular dissociation, and there are indeed many useful analogies between the two types of time evolution. This makes the study of Rydberg states of relevance to reaction dynamics in general. Indeed, the two very actively debated questions of mode-selective chemistry and of control of chemical reactions can be usefully studied through their close analogies in the Rydberg case. There are two sources of congestion of states. One is orbital. There is the n2 degeneracy of a hydrogenic state of principal quantum number n. Moreover, since the spacing of adjacent hydrogenic states is (in atomic units) K 3 , it follows that there are n5 distinct quantum states per atomic energy unit or n5/2 Ry states per wavenumber, where Ry = 105cm-' is the Rydberg constant. For the high n's of interest, this orbital degeneracy is by itself quite high. Of course, these states will remain bunched unless the spherical degeneracy of the Coulombic field is lifted, for example, by an anisotropic molecular core about which the Rydberg electron revolves and/or by an external DC electrical field. To completely resolve the degeneracy, one needs to also break the cylindrical symmetry, and this can be done by a variety of ways including external perturbations (a magnetic field, other molecules, etc.). Moreover, as has been well documented for atoms [42], when the central field is not strictly Coulombic, additional perturbations such as a DC field can mix states of different n. The other source of congestion is due to the molecular core. It is most readily discussed using the inverse Born-Oppenheimer point of view to define the zero-order quantum numbers. Here each state of the ionic core has its very own series of high Rydberg states. The physical reality of this approximation is the observation [36,43] of the long-time stable ZEKE states not just below the lowest ionization threshold but also just below the threshold of ionization processes that leave an excited ionic core. Indeed, it is for this very reason that ZEKE spectroscopy is useful for the spectroscopy of ions (or for such neutrals that are produced by ionization of negative ions

DYNAMICS OF HIGH MOLECULAR RYDBERG STATES

63 1

[44]). The high value of the total energy (comparable or larger than the ionization potential) means that many states of the ionic core are available, up to and including neutral fragmentation channels. (Of course, the higher the energy of the core, the less energy is in the electronic excitation.) As in the case of the orbital degeneracy, the availability of isoenergetic states does not necessarily mean that they are coupled to one another. Indeed, the very use of the inverse Bom-oppenheimer approximation implies a weak coupling between these zero-order states. Here, too, the coupling is often limited by selection rules and one role of external perturbers is to relax these rules. Under the circumstances that the number of bound states is high and ionization requires the most of the available energy is localized in the electron, the dynamics can exhibit two or more time scales [45] (Fig. 6). The prompt decay is a minority route. In this mode, the optically excited electron departs without having sampled much (or even any) of the bound phase space that is available. All other decay channels are delayed. Figure 6 is a schematic drawn for an intramolecular coupling of uniform strength between all the bound states. The actual phase space has bottlenecks (cf. Fig. 7), and these cause a spread in the magnitude of the decay rates. Even so, the typical result of computational studies in which the Hamiltonian is diagonalized is that the rates tend to cluster into groups of quite different magnitudes. A quantum mechanical proof of the bifurcation of the decay modes is

prompt

1o-'q- . -

I

2

,

0

,

1

2

, \ 4

1

6

log (density of qbound states) Figure 6. The average decay width for a given uniform coupling strength V to the continuum (as indicated) vs. the density of (quasi)bound states (logarithmic scale). When the bound phase space is congested, the individual state rates bifurcate into a few (equal in number to the number of dissociation channels) promptly decaying states and many "trapped states whose decay is far slower. (See Section 1II.B.) In the real molecule there are considerable variations in the magnitude of the intramolecular couplings. Therefore, the magnitudes of the rates are scattered but fall within two (or more) groups.

632

R. D.LEVINE

simple and will be given shortly. It must however be emphasized that this is not an inherently quanta1 result. The delay due to trapping in a congested bound phase space is essentially the point made by Lindemann in interpreting the delayed dissociation of energy-rich polyatomic molecules. It was further elaborated and quantified by RRKh4 (Rice, Ramsperger, Kassel, and Marcus). Their initial (RRK) result for the dissociation rate can be cast in the present terms as

where v is the frequency of motion along the bottleneck, K is the number of dissociative continua (the number of states with enough energy along the reaction coordinate for dissociation to take place), and N is the number of (quasi) bound states. When K << N it follows that the rate of unimolecular dissociation is delayed (k << v). That the effect is classical in nature can also be seen from the old-fashioned way of catching lobsters by lowering into the water a large container with an opening wide enough for a lobster to crawl in. The exit of the lobster will be much delayed due to the time necessary to sample the inside of the container. There can be a difference between the dissociation of polyatomic molecules and delayed ionization in the nature of the initial excitation. In ZEKE spectroscopy the state that is optically accessed (typically via an intermediate resonantly excited state) is a high Rydberg state, that is a state where most of the available energy is electronic excitation. Such a state is typically directly coupled to the continuum and can promptly ionize, unlike the typical preparation process in a unimolecular dissociation where the state initially accessed does not have much of its energy already along the reaction coordinate. It is quite possible however to observe delayed ionization in molecules that have acquired their energy by other means so that the difference, while certainly important is not one of principle. The reason external perturbers often elongate the lifetime can thus be understood on general grounds. They break the symmetry and thereby significantly increase the number of bound states that determine the available phase space where the system can linger prior to dissociation. Figure 7 is a schematic illustration of the available phase space. The following points about this figure should be noted: (i) The optically excited Rydberg state has a low angular momentum 1 and is typically directly coupled to the ionization continuum. This is shown in the figure by drawing the initial state near the bottleneck of the bound phase space.

DYNAMICS OF HIGH MOLECULAR RYDBERG STATES

633

Figure 7. A schematic outline of the available bound phase space. The shaded area is the region that is optically accessed. It is near the exit to the continuum. The ordinate refers to the angular quantum numbers of the electron. These can be changed primarib (but not only) by the external perturbations. The bottleneck for such changes is shown as a dashed line. The abscissa refers to the principal quantum number of the electron or to the rotational quantum number of the core. These two change in opposite directions due to the coupling to the core.

(ii) There are bottlenecks for the intramolecular dynamics within the bound phase space. One often noted (partial) barrier is for the angular momentum Z of the electron to increase. It can increase due to angular momentum exchange with the anisotropic core. An external electrical field can readily change I (“Stark splitting”). A magnetic field (or external charges) can also change rnl and this has the advantage that 1, which has to be larger than rnr, stays larger. An electron of high I moves in a nearly circular orbit and is therefore effectively decoupled from the core. For this reason states of high I are shown in Fig. 7 as being far from the exit to the continuum. The role of I can be quantitatively stated by noting that the effective adiabaticity parameter is x(1 - E) = xZ2/2, where € is the eccentricity of the Bohr-Sommerfeld orbit, (cf Fig. 2). (iii) The exchange of angular momentum between the Rydberg electron and the core results either in ionization if n goes up (while t h e j of the core goes down) or in n going down while the rotational energy of the core goes up. In lighter diatomics where the rotational constant is high and the energy spacings are wide (x, cf. figure 2, is higher or, equivalently, the rotational period is shorter), this is not an efficient process. For molecules where the rotational spacings are low, this is a viable route for the sampling of the bound phase space. Section III provides a more technical discussion of the dynamics as schematically summarized in Fig. 7.

634

R. D. LEVINE

III. DYNAMICS We have used quantum [45a, fl, semiclassical [5],and classical [ 1-31 dynamics to study the detailed time evolution of Rydberg states. The same Hamiltonian was used, that of an electron moving in the field of an ionic core, where the potential of the core was not purely Coulombic but included anisotropic terms (dipolar and/or quadrupolar) that could also depend on the vibration of the core; see Eq. (3.2). Starting with this Hamiltonian we have also derived a mapping [3, 51 for the change in n per revolution around the core and from the mapping derived a kinetic description (valid for a time resolution coarser than an orbital period) in which the electron undergoes a diffusive motion. The initial ad hoc use of a diffusion equation [20, 211 could thereby be validated from first principles, with the important advantage that the map specifies the relevant coupling constants. There are primarily two, one of which is x = an3, where w is the frequency of the perturbation (e.g., the vibration or rotation frequency of the core. The term x determines the coupling regime (cf. Fig. 2). It was also possible [5] to use the Hamiltonian, which includes a quantum defect, to derive the equation for the width of autoionizing lower Rydberg states [46]. The same generic Hamiltonian was previously used to describe, both qualitatively and quantitatively, the dynamics of lower Rydberg states [47-501. It is worthwhile to examine the Hamiltonian in some detail because it enables one to discuss both intramolecular and intermolecular perturbations from the same point of view. To do so, we start from a zero-order Hamiltonian that contains just the spherical part of the field due to the core (which need not be Coulombic as it includes also the quantum defect [42]) and add two perturbations: U due to external effects and V due to the structure of the core. Here, U contains both the effect of external fields (electrical and, if any, magnetic [11) and the role of other charges that may be nearby [8, 1 1 , 12, 171. The technical point is that both the effect of other charges and the effect of the core not being a point charge are accounted for by writing the Coulomb interaction between two charges, at points rl and r2, respectively, as

-= >

+P/((cos 0)

(3.1)

Here r< and r, are the smaller and larger rl and r2,respectively, and 8 is the angle between the two vectors. If one is dealing with the effect of the core, then r, is the distance r of the electron from the core and (3.1) becomes just

DYNAMICS OF HIGH MOLECULAR RYDBERG STATES

635

the usual multipole expansion for the potential about an anisotropic charge distribution. Hence, apart from the Coulombic term included in Ho, c1 Q v=P , ( ~ O S e)+ - P~(COS e)+ r2 r3

. ..

(3.2)

where j.4 is the dipole moment of the core, 8 is the angle between the vector to the electron and the axis of the dipole, and the higher order terms in V are quadrupolar. The multiple moments of the core can, of course, depend on the vibrational coordinates and thereby couple the Rydberg electron also to the vibration. As the derivation shows, such a Hamiltonian is valid for a Rydberg electron that does not penetrate the electronic core, as otherwise one cannot take r, to be the distance r of the electron from the core. The higher is the orbital angular momentum 2 of the Rydberg electron the better is this assumption because at higher n's the distance of closest approach is

Z(Z + 1)/2.

On the other hand, as long as the external ions are few and hence typically not inside the Rydberg orbit, r, is the distance of the ion from the center of the core. Hence the contribution of an external ion goes the other way,

u.

'On

r2 - r p , ( c ~ se) + - P~(COSe) -

R2

R3

+ .. .

(3.3)

where R is the distance of the ion to the core (where, by assumption, R > r ) and 8 is the angle between the vectors to the electron and the ion, respectively. Now (r) = n2 and (r-*) = n-3 so that the relative importance of the core vs. the external dipolar contributions scales as (c1/n)(R2/(r)*)and it is possible for the ion to overwhelm the importance of the anisotropy of the core. Both the terms in (3.3) and the external fields affect the electron when it is primarily far away from the core. In terms of Fig. 2, this is the midrange where u = ?r. The influence of the core is primarily near the inner turning point where u = 0, 2a. Even at high n's one needs to follow the system for many orbital periods if one is to mimic the experimental results. The difficulty is compounded if one measures the time in units of periods of the core motion. This suggests that the time evolution be characterized using the stationary states of the Hamiltonian rather than propagating the initial state. We have done so, but our experience is that in the presence of DC fields of experimental magnitude (which means that Stark manifolds of adjacent n values overlap), and certainly so in the presence of other ions that break the cylindrical symmetry and hence mix the m1 values, the size of the basis required for convergence is near the limit of current computers. In our experience, truncating the quan-

636

R. D. LEVINE

tum mechanical basis cannot be trusted without extensive convergence tests. (There is a theoretical reason for this, which is discussed below.) Hence we tend to regard classical mechanics as providing a viable alternative (particularly so in view of the already mentioned) very high density of states. Indeed, for the times in the microsecond range the mapping [3c, 51 is the most realistic tool. To discuss the separation of time scales, we begin with the argument that a system that reaches the continuum via a narrow bottleneck can exhibit more than one time scale [45a,b,f, 511. Particular attention will be given to the question of when this will be the case. The argument begins by considering the time evolution in the bound subspace. As is well known [52,53,54], one can confine attention to the bound levels by the introduction of an effective Hamiltonian H in which the coupling to the continuum is accounted for by a rate operator I’: In general, I’ contains both a Hermitian and an anti-Hermitian part. The latter causes a “level shift,” and if it contributes, we include it in H so that r is Hermitian. The usual discussion of dynamics in a congested bound-level structure begins with the dynamics induced by H. In other words, H is diagonalized first and only then is the role of I’ examined. Here we shall first diagonalize r and only then discuss how off-diagonal terms in H modify the results. The essential point is that if there are K independent decay channels, then the N x N matrix r is necessarily a matrix of rank K. If K < N,then I’ has N - K (or more) zero eigenvalues. The N - K associated eigenvectors have a zero width. If the time evolution is determined by r, then these states are trapped forever. The limiting situation when this is the case is the “trapping limit.” In the genera1 case H has nondiagonal elements between the different eigenvectors of I?. The time evolution will mix the different eigenvectors of I’, and the coupling strength to the continuum will be spread over all those states that are coupled by H. The reason for the emphasis is that this is where the selection rules on which states can be coupled by H are of key importance. The more states are coupled by H, the more is the given coupling to the continuum spread over more states. The opposite limit to trapping is when the mixing induced by H is dominant. We proceed to a detailed derivation of these results.

A. Effective Hamiltonian

To describe the dynamics in a congested bound phase space above the threshold for dissociation, it is convenient to use an effective Hamiltonian, which

DYNAMICS OF HIGH MOLECULAR RYDBERG STATES

637

is an operator confined to the subspace of bound states: !Jf = QHoQ + Q(V + U)Q + Q(V + U)P(E+- PHp)-'P(V + U ) Q

(3.5)

Here, Q is the projector on the bound subspace and P projects onto the open, continuum channels. The intramolecular coupling is written as V + U so that, as before, U is any additional coupling brough about by external perturbations. The equation H = Ho + V+ U,where HOis the zero-order Hamiltonian of the Rydberg electron and so includes only the central part of the potential due to the core plus the motion (vibration, rotation) of the core, uncoupled to the electron. The perturbations V + U can act within the bound subspace, as the operator Q(V + U)Q is not necessarily diagonal and is the cause of any intramolecular dynamics even in the absence of coupling to the continuum. The intramolecular terms can also couple the bound and dissociative states. The coupling to the continuum is implicitly contained in the last term in (3.5). The decay is due to the imaginary part of the Green's function P(E+ - PHP)-'P. In what follows we assume: (i) The spectrum of PHP is in principle known. Typically, what is readily known is the spectrum of PHoP and the additional coupling terms can give rise to so-called final-state interactions (discussed, e.g., in Ref. 55). (ii) The contribution of the real part, the so-called level shift [53], due to the coupling to the continuum can be neglected. This is justified if the coupling terms QHP are only weakly energy dependent [561. If this is not the case, then the effective Hamiltonian contains an additional Hermitian term. Otherwise, the last term in (3.5) is a purely anti-Hermitian operator and hence can be written as iQrQ, where r is Hermitian:

In (3.6) the matrices are in the bound (Q) subspace and I' is the Hermitian rate operator. The essential point of this chapter is that the physics of the problem we wish to address dictates that the rank of the matrix I' is small compared to the dimension N of the bound subspace, where N is the rank of H or of Q. The proof that I' is of a low rank begins with the explicit form

r = ?~QHPG(E - PNPIPHQ or

(3.7)

R. D. LEVINE

638

where the states ) k ) are states of the continuum that are isoenergetic with the zero-order bound states In). The summation over the index k in (3.7) is over the number of linearly independent isoenergetic states of PHP. Physically these are dissociative states where the intramolecular coupling is still “on”, that is, they are what one loosely refers to as transition states. In the RRK terminology these are the states with enough energy along the reaction coordinate. The range K of the index k is therefore the number of such states. In the notation of transition-state theory, K = N f ( E- Eo), where EO is the threshold energy for dissociation. For any but a diatomic molecule, most ways of partitioning the energy E among the internal degrees of freedom results in that there is not enough energy in the reaction coordinate so that K is small compared to the range N of values that the index n can assume. To prove that the rank of the matrix I’ is K, (3.7) is written as

k= I

The K vectors Vk are linearly independent and have N components, where if N is the number of bound states,

n= I

Some, or many, matrix elements in (3.9) may vanish because of selection rules. It is convenient to take the vectors Vk as eigenvectors of the N x N Hermitian matrix r, N n= 1

N

K

n=1 k=l

The first K such eigenvectors can be normalized by

DYNAMICS OF HIGH MOLECULAR RYDBERG STATES

639

Note that since the matrix r is positive (semi)definite, the first K eigenvalues rk, k = 1, ... , K, are all nonnegative. The other N - K eigenvectors of r all have zero as an eigenvalue. (Note that the nonvanishin eigenvalues of r must equal those of the K x K Hermitian matrix 42 a V V.) The time evolution is determined by the full effective Hamiltonian H and not by the rate matrix F alone. One cannot therefore discuss the time evolution without reference to the matrix H.Say, however, H and r commute, [H, I"] = 0. A simple condition that ensures this result is that the bound states are strictly degenerate. If H and r commute, the eigenvectors of r evolve in time independently of one another. In the basis of states defined by the N eigenvectors of r there will be K states that will decay by direct coupling to the continuum and N - K states that are trapped forever. An arbitrary initial state is a linear combination of the N eigenvectors of I' and hence can have a trapped component. The K eigenstates that are directly coupled to the continuum decay promptly because they carry the entire coupling strength

f

K

TrI'=x

rk

(3.12)

k= I

so that, when K << N,on the average, rk >> N-'Tr r. The Rydberg state which is optically prepared in a typical ZEKE experiment is usually directly coupled to the continuum [45c, 571. Other considerations being absent, it should decay promptly, possibly with a stable, trapped component. The point is that the initially prepared state is also directly coupled to many other states, due both to external perturbations 1371 and to intramolecular coupling [3b]. The conclusion that the initial state has two components, one that decays promptly and one that is trapped, is thus only valid in zero order (so-called golden rule limit). One needs to allow for the coupling terms represented by V and U.

B. Trapping Versus Dilution

In the golden rule approximation, the decay rates are the eigenvalues of the matrix I'. The result that when the bound-state subspace is dense, I' is of a rank smaller than the number of bound states means that, in the golden rule approximation, some of these bound states will remain bound. That is, they are trapped states and do not decay. Of course, the golden rule is just

640

'

R. D.

LEVINE

the first-order approximation. But in first order it follows that states can be strictly classified as either decaying or as stable states. This is just as in the simple RRK picture, where states either have enough energy along the reaction coordinate or they do not and only the former do dissociate. Analytical considerations applied to the full dynamics (where it is rather than r that determines the time evolution) and computational results [45Q suggest that even when the coupling is allowed for, remnants of the zero-order dichotomy between promptly decaying and stable states do survive, that K states can either decay promptly, without much sampling of the bound space of N states, and that N - K states have a delayed decay. The larger is the density of bound states, the slower is the decay of the states that, in first order, are trapped. Figure 6 is the schematic case while Fig. 8 shows explicit computational results.

1 0.8

0.6 0.4

0.2

s

0.8

0.6 0.4

0.2

n., -3

-2

-1

0

1

log (t /ns)

2

3

Figure 8. Time evolution, obtained via a numerical diagonalization of the effective Hamiltonian H (adapted from Ref. 45) when N = K and N > K for two different initial states. Shown is the survival probability of the initial state (dashed line) and the probability to remain bound (full line) vs. time. The reason for the difference between these two is due to the system sampling the rest of the bound phase space. At higher number N of bound states, for a given value of K, the delayed decay would be shifted to even longer times while the survival probability will remain essentially unchanged, showing that the delay is due to sampling of the bound phase space.

DYNAMICS OF HIGH MOLECULAR RYDBERG STATES

To discuss the role of are uncoupled

641

H,consider first the limit where the bound states

31 = I&,- ir

[&,r]= 0

(trapping limit)

(3.13)

where I&,is diagonal in the basis of states that diagonalize r. Then the eigenvectors of H are the eigenvalues of r and the eigenvalues have a real part, which is the energy (i.e., the eigenvalue of €I,-,), and an imaginary part, which is the eigenvalue of:'l

H

* I = (El - rl)q

I = I , . ..,N

rr= 0

for I > K

(3.14)

An arbitrary initial state cp can be expanded using the eigenvectors of I' as a basis where, since these states are orthogonal [cf. Eq. (3.1 l)], N

(3.15) I=l

It follows that in the trapping limit an arbitrary initial state will have a component that is trapped and does not decay:

(3.16)

The trapped component is the component of the initial state in the subspace of the N - K trapped eigenvectors of I".

A special case of these results, discussed in Ref. 51, is when all the states are degenerate, €I,-, = El, where I is the identity matrix. This special form is sufficient to show the trapping phenomena but it is not necessary. Strict trapping is possible even when the states are not degenerate. The necessary condition is that the Hamiltonian H in the bound space commutes with the

642

R.D.LEVINE

matrix I?. Here we discussed a more general case where the two matrices commute, namely H = &, where 6 is diagonal. The general case corresponds to allowing for H being nondiagonal, so that (3.13) is replaced by

H=&+H1-ir

(3.17)

As in (3.13) we take the matrices in this equation to be defined in the basis that diagonalizes I‘ so that, in the absence of HI, the effective Hamiltonian is diagonal and the dynamics manifests trapping. Computing the time evolution is equivalent to diagonalizing the effective Hamiltonian. One can proceed by using perturbation theory and thereby eliminate the effect of the intramolecular coupling HI in successive orders. This route is particularly useful if the perturbation HI itself has terms of different orders, as is often the case for real molecules. In this way one can discuss the sequential sampling of phase space [45b, 58-61] with the modification that dissociation is possible. Another route is to exactly diagonalize, which can be done numerically (as, e.g., in Fig. 8). Either way, the coupling mixes the trapped and the promptly decaying states. If the mixing is so complete that one cannot uniquely correlate the eigenstates of H with the eigenstates of r (or of &),then the distinction between trapped and promptly decaying states is lost upon diagonaIization of H . The eigenstates cannot be said to be of one kind or the other. Since the intramolecular coupling causes a “repulsion” of the energy levels of H, the region of dominant coupling is when the spacings of the levels of H exceeds the magnitude of the eigenvalues of r. In the language of resonance scattering theory [52,53,54,62] the “resonances” are far apart in energy and so are isolated. In such a coupling regime it is more reasonable to first diagonalize H and then to regard r as a perturbation. In the language used below, in this range the coupling to the continuum has been effectively democratically diluted over all the states. This is the limit of a low density of states, and the merging of the two branches is clearly seen in Fig. 6. Also seen therein is that when the coupling is to the continuum is weaker, a higher density of states before the bifurcation into two types of decay is evident. For a high density of states it will typically be the case that dilution is incomplete. What this means is that when an eigenstate w, of H is expanded in the basis that diagonlize r, K

N

I= I

l=K+l

(3.18)

DYNAMICS OF HIGH MOLECULAR RYDBERG STATES

643

one or the other of the two terms dominate. As an eigenstate of 31 the state w, decays independently of all others and its rate of decay is

Those eigenstates of 31 that are made up predominately of prompt states will decay promptly and vice versa, the extent of dilution being measured by the expansion coefficients lc;tI2, I 5 K . The dramatic effect of the dilution of the coupling strength to the continuum is to endow the trapped states with a finite (but long) lifetime. We consider that this is the origin of the extremely long decay times observed in ZEKE spectroscopy. Dilution also tends to extend the lifetime for the prompt decay. For a discussion of dilution from a time-dependent point of view (the so-called time stretch), see Refs. 3a and 3d. The overall effect of dilution is to cause a more uniform distribution of the lifetimes, so that the shorter lifetimes are longer and the very long lifetimes are shorter. In other words, dilution and trapping, which are due to different terms in the effective Hamiltonian, have an essentially opposite effect. When dilution is the dominant effect, the distribution of lifetimes is unimodal and narrower. (Increasing dilution means moving to the left in Fig. 6.) When trapping dominates, the distribution of lifetimes is at least bimodal. The long lifetimes corresponding to states that would be fully trapped in the absence of coupling and the short lifetimes are the prompt decays. It is possible for the distribution of lifetimes to have more than two modes because the coupling need not be a simple process but can manifest a sequential character [45b, 58, 60,611. A structure in the matrix H1 can give rise to additional bifurcations in the distribution of lifetimes. The distinction between prompt decay and trapped states has been made here for the basis that diagonalizes the r matrix. When the Hamiltonian H that governs the evolution in the bound space is not diagonal in that basis, the distinction between the two types of states is valid only to lowest order and the actual dynamics mixes the two types of states. It is possible to define trapped states even in the general case [45d] by diagonalizing the Hermitian operator €I-'I', and this is equivalent to a simultaneous diagonalization of H and r. The resulting generalized eigenvectors can be strictly classified according to whether their eigenvalues does or does not have an imaginary part and there are K of the former. However, the required transformation is not unitary and so the generalized eigenvectors do not diagonalize functions

644

R. D. LEVINE

of H so that they are mixed by the time evolution (except, as in the above, to first order). Finally we explicitly discuss external perturbations as a source of dilution. When the molecule is coupled to its environment, strictly speaking, one should use a Hamiltonian description for the combined system. Just as we eliminated the explicit coupling to the continuum by the use of an effective Hamiltonian, one can do the same for the perturbation due to the surroundings. In the lowest order, when such perturbations are weak and essentially static, the effect is equivalent to an additional coupling mechanism, denoted as U in Eq.(3.5). States that otherwise would be trapped can acquire a finite width when U is included in H. Since such a description is valid only when U is a weak perturbation, one might think that the effect will be minor. The reason that this is not necessarily so is the high degeneracies typical of highly excited Rydberg states. The optical excitation of the Rydberg state typically accesses only very low angular momentum states. The breaking of the spatial symmetry of the isolated atom by an external anisotropy (as in Ref. 12) or by a magnetic field (Ref. 1) couples in a very large number of states, many more states than are coupled when only an external electric field is present since such a DC field retains the cylindrical symmetry of the problem. One point about the perturbation of the Rydberg electron is therefore the number of states that are coupled by H. The other is the dispersion in the eigenvalues of H. The extent of dilution depends on the failure of H and r to commute. Larger energy spacings favor dilution, where larger is measured in respect to W ' T r r. Since the typical long decay time is orders of magnitude longer than an orbital period of a Rydberg electron, it follows that N-lTr r is very much smaller than the spacing of (unperturbed) Rydberg states (which at high n's are fractions of a wavenumber). The long time evolution governed by the small spacings in the eigenstates of H is therefore sensitive to unusually small spacings, of the order of Ry/n5, in reciprocal centimeters. In computational studies of quantum dynamics this implies that truncating a basis set can lead to severe truncation errors in the long-term time evolution. In experimental studies it means that slight disturbances can have observable effects, particularly so at longer times.

IV. CONCLUDING REMARKS Prompt and delayed ionization is familiar for very energy rich molecules. The special feature of high Rydberg states is the initial state that is optically prepared, a state directly coupled to the continuum on the one hand and to a very dense bound manifold on the other. The dynamical theory necessary to describe such states has been reviewed, with special reference to the extremely long-time decay. It is suggested that this resilience to decay is due

DYNAMICS OF HIGH MOLECULAR RYDBERG STATES

645

to admixture of states that are, in zero order, trapped. These can be mixed in due to both intramolecular and external perturbations.

Acknowledgments This work would not have been possible without the close collaboration with the experimental groups of E. W. Schlag (Munich) and of U. Even (Tel Aviv). I am very grateful to them and to their co-workers, H. L. Selzle and K. Muller-Dethlefs in particular. It is also a pleasure to thank my co-workers Eran Rabani, FranGoise Remade, and Leonid Baranov, whose contribution is reflected through the references to their papers. The work was supported by the Volkswagen Foundation and by SFB 377.

References I . A. Muhlpfordt, U. Even, et al.. Phys. Rev. A 51, 3922 (1995). 2. E. Rabani and R. D. Levine, J. Chem. Phys. 104, 1937 (1995).

3. (a) E. Rabani, L. Y. Baranov, et al., Chem. Phys. Lett. 221, 473 (1994); (b) E. Rabani, R. D. Levine, et al., J. Phys. Chem. 98, 8834 (1994); (c) E. Rabani, R. D. Levine, et al., Ber. Bunsenges. Phys. Chem. 99,310 (1995); (d) E. Rabani, R. D. Levine, et al., J. Chem. Phys. 102, 1619 (1995). 4. U. Even, R. D. Levine, et al., 1.Phys. Chem. 98, 3472 (1994). 5. E. Rabani and R. D. Levine, J. Chem. Phys. (1996). 6. (a) K. Muller-Dethlefs and E. W. Schlag, Ann. Rev. Phys. Chem. 42, 109 (1991); (b) G. 1. Nemeth, H.L. Selzle, et al., Chem. Phys. Lett. 215, 151 (1993). 7. E. W. Schlag, W. B. Peatman, et at., J. Elect. Spectry. 66, 139 (1993). 8. X.Zhang, J. M. Smith, et al.. J. Chem. Phys. 99, 3133 (1993). 9. C. E. Alt, W. G. Scherzer, et al., Chem. Phys. Len. 224, 366 (1994). 10. I. Fischer, R. Lindner, et al.. J. Chem. Soc. Faraday Trans. 90, 2425 (1994). I 1. F. Merkt, Chem. Phys. 100, 2623 (I994). 12. F. Merkt and R. N. Zare, J. Chem. Phys. 101, 3495 (1994). 13. G. I. Nemeth, H. Ungar, et al., Chem. Phys. Left. 228, (1994). 14. W. G. Sherzer, H. L. Selzle, et al., Phys. Rev. Lett. 72, 1435 (1994). 15. C. E. Alt, W. G. Scherzer, et al., J. Phys. Chem. 99, 1660 (1995). 16. A. Muhlpfordt and U. Even, J. Chem. Phys. 103,4427 (1995). 17. (a) M. J. J. Vrakking and Y.T. Lee, Phys. Rev. A 51, 894 (1995); (b) M. J. J. Vrakking and Y. T. Lee, J. Chem. Phys. 102, 8818 (1995). 18. D. Bahatt, U. Even, et al.. J. Chem. Phys. 98, 1744 (1993). 19. U. Even, M. Ben-Nun, et al., Chem. Phys. Lett. 210,416 (1993). 20. E. W. Schlag and R. D. Levine, J. Phys. Chem. 96, 10668 (1992). 21. E. W. Schlag, J. Grotemeyer, et al., Chem. Phys. Left. 190, 521 (1992). 22. R. Weinkauf. P. Aicher, et al., J. Phys. Chem. 98, 8381 (1994). 23. E. E. B. Campbell, G. Ulmer, et al.. Z. Phys. D 24,81 (1992). 24. P. Wurz and K. R. Lykke, J. Phys. Chem. 96, 10129 (1992). 25. B. A. Collings, A. H. Amrein, et al., J. Chem. Phys. 99, 4174 (1993). 26. M. Foltin, M. Lezius, et al., J. Chem. Phys. 98, 9624 (1993).

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27. T. Leisner, K. Athanassenas, et al.. J. Chem. Phys. 99,9670 (1993). 28. K.Toglhofer, F. Aumayr, et al., Europhys. Len. 22, 597 (1993). 29. C. Yeretzian, K. Hansen, et al., Science 260,652 (1993). 30. H. Hohmann, C. Callegari, et al., Phys. Rev. Len. 73, 1919 (1994). 31. H. Hohmann, R. Ehlich, et al., Z. Phys. D 33, 143 (1995). 32. B. D. May, S. F. Cartier, et al., Chem. Phys. Lett. 242, 265 (1995). 33. H. Weidele, D. Kreisle, et al., Chem. Phys. Len. 237, 425 (1995). 34. D. Bahatt, 0. Cheshnovsky, et al., 2.f: Phys. Chem. 184,253 (1994). 35. H.-J. Dietrich, R. Lindner, et al., 1.Chem. Phys. 101, 3399 (1994). 36. W. G. Schemer, H. L. Selzle, et al., Phys. Rev. Lerr. 72, 1435 (1994). 37. W. A. Chupka, J. Chem. Phys. 98,4520 (1993). 38. S. T.Pratt, J. Chem. Phys. 98, 9241 (1993). 39. M. Bixon and J. Jortner, J. Chem. Phys. 103,4431 (1995). 40. F. Merkt, S. R. Mackenzie, et al., J. Chem. Phys. 103, 4509 (1995). 41. L. Y. Baranov, Y. Kris, et al., J. Chem. Phys. 100, 186 (1994). 42. M. Born, Mechanics of the A r m , Blackie, London, 1951. 43. H. Krause and H. J. Neusser, J. Chem. Phys. 99, 6278 (1993). 44. D. M. Neumark, Ann. Rev. Phys. Chem. 43, 153 (1992). 45. (a) F. Remacle and R. D. Levine, Phys. Leu. A 173, 284 (1993); (b) F. Remacle and R. D. Levine, J. Chem. Phys. 98,2144 (1993); (c) F. Remacle and R. D. Levine, J. Chem. Phys. 104, 1399 (1996); (d) F. Remacle and R. D. Levine, Mol. Phys. 87,899 (1995); (e) F. Remacle and R. D. Levine, J. Phys. Chem. 100, 7962 (1995); (9 F. Rernacle and R. D. Levine, J. Chem. Phys. 105,4949 (1996). 46. G. Herzberg and C. Jungen, J. Mul. Specrrosc. 41, 425 (1972). 47. R. S. Berry, J. Chem. Phys. 45, 1228 (1966). 48. A. Russek, M. R. Patterson, et al., Phys. Rev. 167, 17 (1968). 49. E. E. Eyler, Phys. Rev. A 34, 2881 (1986). 50. R. D. Gilbert and M. S. Child, Chem. Phys. Leu. 287, 153 (1991). 51. F. Remacle, M. Munster, et al., Phys. Len. A. 145, 265 (1990). 52. H. Feshbach, Ann. Rev. 19, 287 (1962). 53. R. D. Levine, Quantum Mechanics of Molecular Rate Processes, Oxford, Clarendon, 1969. 54. S. Nordholm and S . A. Rice, J. Chem. Phys. 62, 157 (1975). 55. J. A. Beswick and J. Jortner, J. Chem. Phys. 68, 2277 (1978). 56. M. Desouter-Lecomte, J. Lievin, et al., J. Chem. Phys. 103,4524 (1995). 57. Merkt and Softley, lnr’l. Rev. Phys. Chem. 12, 503 (1993). 58. J. P. Pique, Y.Chen, et al., Phys. Rev. Lert. 58,475 (1987). 59. D. E. Logan and P. G. Wolynes, J. Chem. Phys. 93,4994 (1990). 60. K. Yamanouchi, N. Ikeda, et al., J. Chem. Phys. 95,6330 (1991). 61. M. J. Davis, J. Chem. Phys. 98,2614 (1993). 62. M. Bixon, J. Jortner, et al., Mol. Phys. 17, 109 (1969).

Advances In Chemical Physics, Volume101 Edited by I. Prigogine, Stuart A. Rice Copyright © 1997 by John Wiley & Sons, Inc.

GENERAL DISCUSSION ON MOLECULAR RYDBERG STATES AND ZEKE SPECTROSCOPY: PART I Chairman: G. Casati

J.4. Lorquet: The energy resolution of Prof. Schlag’s spectra is indeed impressive. But would there be an advantage in recording the spectra at a low resolution, just good enough to resolve the vibrational structure, in order to get a better appreciation of the intensity pattern? Could then that pattern be interpreted as a superposition of several progressions? Also, can one rationalize any observed dependence of the intensities on experimental parameters of the ion source? E. W. Schlag: The intensities can in many cases be calculated from the Franck-Condon principle, but in some cases there are important exceptions that are most clearly interpreted in the inverse Born-Oppenheimer (BO) framework. Here one clearly sees the superposition of Rydberg stacks from individual rotations or vibrations. R. D.Levine: To answer the question of Prof. Lorquet, let me say that the peaks in the ZEKE spectra correspond to the different energy states of the ion. From the beginning one was able to resolve vibrational states, and nowadays individual rotational states of polyatomics have also been resolved. The ZEKE spectrum is obtained by a (weak) electrical-field-induced ionization of a high Rydberg electron moving about the ion. The very structure of the spectrum appears to me to point to the appropriate zero-order description of the states before ionization as definite rovibrational states of the ionic core, each of which has its own Rydberg series. Such a zero-order description is inverse to the one we use at far lower energies where each electronic state has its own set of distinct rovibrational states, known as the Born-Oppenheimer limit. D. Gauyacq: I have a short comment on Prof. Schlag’s remark on the multichannel quantum defect theory (MQDT) approach to ZEKE spectra: The first ZEKE spectrum of NO was actually interpreted by using MQDT as early as 1987 [ 13. In this work, a full calculation of the ZEIE peak intensities is carried out by the MQDT approach, which 647

648

GENERAL DISCUSSION

takes into account the relative population of the intermediate levels (see p. 859 and Fig. 14 of Ref. 1). 1. S. Fredin, D.Gauyacq, M. Horani, Ch. Jungen, G . Lefevre, and F. Masnou-Seeuws, Mol. Phys. 60, 825 (1987).

E. W. Schlag: Thank you for pointing out this reference to me which totally agrees with our originally measured spectrum. R. Schinke: Prof. Levine, I have two questions: 1. Did you and your co-workers compare the results of classical and quantum mechanical calculations? 2. For an electronic system, how do you define a transition state necessary for calculating an RRKM rate constant?

R. D. Levine: Our classical and quantal simulations use the same Hamiltonian. However, we have not, so far, performed the dynamical computations on the same system by both classical and quantum mechanical methods. We have not done so because, as I discussed, one needs a rather large basis set if the quantal computation is to converge. In particular, computation on the role of the electrical field using a limited basis set (using not enough 1 values) give completely spunous results, such as a lengthening of the lifetime. These results change markedly when a basis that properly spans the Stark manifold is used. So we have focused the quantal effort on a few topics. Encouraged by your interest we will certainly try a direct quantalclassical comparison. Of course, we have a clear idea that classical mechanics will be at some fault if the spacings of the states of the ionic core are wider than the strength of the coupling of such states due to the electron. (And for this reason we prefer to treat the vibrations by semiclassical or quantal means.) I agree with you that it will be useful to have a direct quantitative comparison and we will work on providing it. J.-C. Lorquet: Leaving Rydberg states aside for a moment and concentrating on unimolecular reactions, I wonder whether a relationship exists between the trapping mechanism described in your present communication and the healing process you introduced many years ago? As a second question, how does the efficiency of both of these processes vary as one increases the internal energy? I imagine that the numbers N and K increase with energy in a different way. One might imagine that the higher the internal energy, the greater the tendency of exploding rapidly. R. D. Levine: An important motivation for the detailed understand-

RYDBERG STATES AND ZEKE SPECTROSCOPY I

649

ing of the dynamics of high molecular Rydberg states is indeed that they provide a wonderful laboratory for examining many of the issues we want to explore for unimolecular reactions of energy-rich polyatomic molecules. Two recent papers [F. Remacle and R. D. Levine, J. Chem. Phys. 104,1399 (1996) and J. Phys. Chem. 100,7962 (1996)l. A particular illustration of this cross-cultural influence is the question of whether one can pump states directly coupled to the transition state. In the Rydberg problem the answer is very much “yes” because these are often the states that are optically excited. They are directly coupled to the continuum but are also coupled to other Rydberg series, where n is lower and the core is more excited. Healing is said to occur when a bond that is already broken, so that the products begin to recede from one another, is “healed”; the products approach again and recross the transition state back to the bound region of phase space. When there are many (quasi)bound states, healing can occur in an off-diagonal manner in that the (quasibound) state that they come back to need not be the same as the (quasibound) state that they dissociated from. Healing thus provides for an enhanced mixing of the bound phase space. Healing thus provides for a longer lifetime. In formal theory healing is due to the rate operator part of the effective Hamiltonian being nondiagonal in the zero-order basis. The rate operator is a nonnegative definite operator and can be diagonalized. You and your co-workers have argued that under the circumstances that we are discussing (a dense bound phase space coupled to a sparse continuum) there will necessarily be many “trapped” eigenvectors of zero eigenvalue for the rate operator. However, the dynamics is described by the full effective Hamiltonian, and this will mix in the eigenvectors of the rate operator. A trapped eigenstate will thus not be stable but may have a very long decay time. In other words, healing can provide for a bottleneck to dissociation. J.-C. Lorquet: That’s a very good point. In order words, resurrection is an extreme case of healing, isn’t it? R. D. Levine: I am coming from Jerusalem. Ch. Jungen: I would like to make a comment on the concept of “inverse Born-Oppenheimer approximation,” which has been invoked by both speakers this morning. Here is a quotation from a paper published by R. S . Mulliken [J. Am. Chem. Soc. 86, 3183 (1964)l more than 30 years ago: “In most discussions on molecular wave functions,

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GENERAL DISCUSSION

the validity of the Born-Oppenheimer approximation is assumed. This approximation is most nearly accurate when the frequencies of motion, which can be gauged by energy level spacings, are much larger for the electronic than for the nuclear motions. In a Rydberg state series, as n increases, the frequencies of the Rydberg electron become smaller and smaller relative to those of nuclear vibration and rotation. This leads to more or less radical changes in coupling relations.” Mulliken then goes on to denote the two limiting cases as Rydberg-coupled and Rydberguncoupled wave functions. I do not think we need a new word here for something that has been in use for a long time.

R. D. Levine: As I am sure you will agree, it is not possible to

categorically state that one zero-order basis set is “better” than another. At best one can state that a particular basis is more convenient. If one intends to actually diagonalize the Hamiltonian, then the convenience is a matter of technicalities: Which basis offers a better computational route? If, however, one wants to use the zero-order basis to discuss physics, then, as we all agree, one would like the basis to diagonalize as much of the problem as is possible. My point is that the Rydberg states span an enormous coupling range and what can be a useful zero-order basis in one range of n need not necessarily be so in a quite different range. This is in part due to the n3 dependence of the orbital period of the electron (cf. Fig. 1 of the chapter by R. D. Levine, “Separation of Time Scales in the Dynamics of High Molecular Rydberg states,” this volume). There is also the question of which zero-order basis is more “natural” for the experiment. Of course, here, too, simplicity is in the eye of the beholder, but as I already argued, to my mind, ZEKE spectroscopy points out the advantage of the inverse Born-Oppenheimer zero-order basis. I recognize that MQDT is based on the fact that near the core the separation inherent to the inverse Born-Oppenheimer basis is no longer valid and the MQDT procedure switches basis. It does so because it seeks to diagonalize the Hamiltonian in an efficient manner. We recognize the coupling by labeling the inverse Born-Oppenheimer states as providing a zero-order basis. It is precisely the nature and role of this coupling that we wish to elucidate. For a derivation of the Herzberg-Jungen formula starting with our point of view see the appendix of E. Rabani and R. D. Levine, J. Chem. Phys. 104, 1937 (1996).

From a purely operational point of view, please note that the individual peaks in what one calls the ZEKE spectrum (E. W. Schlag,

RYDBERG STATES AND ZEKE SPECTROSCOPY I

65 1

“ZEKE Spectroscopy,” this volume) are obtained by ionizing the Rydberg series which is built on a particular state of the ionic core. The inverse Bom-Oppenheimer picture is thus natural when one begins the discussion with what is actually the raw observation. M. Shapiro: Concerning the inverse Bom-oppenheimer regime, I wonder if we are not simply in a decoupling regime. The relevant parameters for the Bom-Oppenheimer approximation are not so much the relative frequencies of the electronic and nuclear motions (were this the case in the continuum, when the spacing between the electronic levels is zero, we would always be in the “inverse Bom-Oppenheimer” regime, which is clearly not the case) as the magnitude of the “nonBom-Oppenheimer” coupling terms and the level separations. There are systems where, in fact, as we increase the size of the system (which is what we do when we go to a high Rydberg state), the non-Bom-Oppenheimer terms go to zero faster than the separation between levels, and the Bom-Oppenheimer approximation stays valid. Therefore I wonder if you actually checked the magnitude of these terms? R. D. Levine: The case of the Rydberg electron orbiting a rotating ionic core is formally analogous to that of a very low energy rare-gas atom bound by van der Waals forces to a rotating diatomic. In both cases the system can dissociate by rotational energy of the core being made available to the translational motion. What is new in the Rydberg problem is that because of the great depth of the Coulomb well, the already bound electron can also lose energy by going down in that well while the core is further rotationally excited. In both cases, the same zero-order basis set, in which the faster motion (namely the rotation) adapts itself to the slower one, is useful [e.g., M. Shapiro, R. D. Levine, and B. R. Johnson, J. Chem. Phys. 52, 1755 (1970)l.In the Rydberg case this basis set is the one in which the electron is solved for in the field of the core. This is the zero-order basis that we (and others) use. In both cases what the basis set provides is only a zero-order description because in both cases one knows that the system can exit to the continuum. A. H.Zewail: I have two questions for Profs. Schlag and Levine: 1. For lower n we observe femtosecond dynamics [l]. Is it reasonable to think that because of the low n (low angular momentum I ) the probability for penetrating the core is much larger, leading to femtosecond instead of microsecond dynamics?

652

GENERAL DISCUSSION

2. When the level structure is that dense for high n, you have a distribution of lifetimes, and in the language of intramolecular dynamics you would expect a fast component and a much slower average decay. 1. M. H. M.Janssen et al., Chem. Phys. Lett. 214, 281 (1993); H. Guo and A. H. Zewail, Can. J. Chem. 72,947 (1994).

E. W. Schlag 1. Yes, indeed, coupling to the core is stronger for lower n, leading to shortened lifetimes for low n. 2. For complex lifetimes at high n, there are expected to be a series of couplings leading to short and long lifetimes.

R. D. Levine: The latest results for benzene as reported by Prof. Neusser suggest that benzene is the molecule we should study at all time scales. In other aromatic molecules (and also in other molecules [U.Even et al., J. Phys. Chem. 98, 3472 (1994)l) you have reported that there are at least two times scales for the decay of high Rydberg states. A faster (hundreds-of-nanosecond) scale and a slower one (microsecond range). Even the faster time scale is two to three orders of magnitude slower decay than one would expect from an extrapolation of the decay measured at low or intermediate n Rydberg states. Theory [F.Remacle and R. D. Levine, Phys. Lett. A 173,284 (1993)] and simulations suggest that there should also be a prompt decay corresponding to a yet shorter scale. Your statement that you can now begin the time-resolved ZEKE spectroscopy at much earlier times than before makes me suggest that benzene be one of the first molecules that you choose for experimental study. A. H. Zewail: I wanted to emphasize in my comment that the dynamics is rich and the definition of a “lifetime” is only possible if the entire temporal behavior is unraveled. E. W. Schlag: It is very hard to do the ideal experiment of perfect state selection in a field-free environment. The perfect experiment here may well not be the ideal ZEKE experiment since some field appears to be needed for state mixing. D. M. Newark: I would like to make a comment to Prof. Schlag. One expects an anion ZEKE spectrum to have the same overall intensity profile as the corresponding photoelectron spectrum only if direct detachment is the only process that occurs. However, FeO- has several dipole-bound and valence-excited states near the detachment threshold.

RYDBERG STATES AND ZEKE SPECTROSCOPY 1

653

So it is not too surprising that the FeO- ZEKE spectrum is very different from the photoelectron spectrum. E. W.Schlag: That is a good point, although there are some other very profound intensity changes in anion-ZEKE spectra. D. M.Neumark: We have recently investigated resonant multiphoton detachment in C4, C,, and C,. Photoelectron spectroscopy followed by multiphoton excitation strongly suggests that “thennionic emission” occurs in these small clusters, and this is supported by our observation of delayed electron emission from C, and C,. The latter results yield the emission rate as a function of total energy. We can compare these results to RRKM calculations and find the calculated rates to be an order of magnitude too high. Do you have any insights into possible nonstatistical effects in this electron emission process that might explain thk discrepancy? (See Figs. 1-5 .)

1

21200

.

t

22200

.

n

23200

.

m

24200

.

I

25200

P

Q m

Figure 1. One-color resonant multiphoton detachment spectra of C,, C 6 . and C,.

654

GENERAL DISCUSSION

One-Photon

hu4.66eV

-z

.a v)

c

e

B

Multi-Photon

0.m

0.20

0.40

0.60

0.80

1.m I.P

Electron Kinetic Energy (eV)

Figure 2. Time-of-flight photoelectron spectra of C, . Top: one-photon direct detachment with photon energy 4.66 eV. Bottom: resonant multiphoton detachment with photon energy 2.04 ev. The electron affinity of linear c6 is 4.18 ev.

Time (nanoseconds)

Figure 3. Electron emission time profiles from resonant multiphoton detachment of C; at various photon energies. The time width at full width at half maximum (FWHM)of the laser pulse is 30 ns.

RYDBERG STATES AND ZEKE SPECTROSCOPY I

655

c; x%,

c; x*n,

-

Figure 4. Schematic drawing of C, resonant multiphoton detachment mechanism X2n, transition (IC, internal confor excitation at the band origin of the C2n, version; TE, thermionic emission).

5.0 47wo

48000

51m

53OOo

Total Photon Energy (m')

Figure 5. Comparison of experimental and calculated electron emission rate constants for C, and C,. Experimental results are shown in solid circles connected by a dashed line. Rates are calculated using microcanonical statistical model for thermionic emission. Solid lines are results when ab initio vibratiohal frequencies are used. Dashed line (C6 only) is calculation in which the value of the lowest vibrational frequency in C, was reduced by 30%.

656

GENERAL DISCUSSION

R. D. Levine: Prof. Neumark, your very detailed results on the excess energy dependence of the rate of delayed detachment of the electron from small carbon clusters should help test an ongoing discussion. One possibility is to view the process as that of thermionic emission. Now thermionic emission is the oldest version of transition-state theory known to me. What one assumes is that the system is in thermal equilibrium and the rate is given by the rate at which thermal electrons will cross a (hypothetical) surface surrounding the cluster, where the barrier height is the work function. In other words, the theory takes the rate of crossing of the transition state to be rate determining. It is not possible for me to fit your interesting data “by eye,” but very superficially I would say that your measured rate is too slow as compared to the predictions of the theory based on thermionic emission. Sometime ago we [E. W. Schlag and R. D. Levine, J. Phys. Chem. 96, 10608 (1992)l discussed the possibility that crossing the barrier is not rate determining. Rather, the slow process is the exchange of energy between the electron and the nuclear degrees of freedom (electron-phonon coupling is the technical term). This gives rise to a diffusive-like motion of the electron. Typical results are shown in Fig. 3 of the chapter by myself (R. D. Levine, “Separation of Time Scales in the Dynamics of High Molecular Rydberg States,” this volume) where your clusters qualitatively correspond to the rightmost curve shown therein. The situation is analogous to the so-called energy diffusion regime in the Kramers picture of reactions in solution except that here “the molecule acts as its own solvent” [E. W. Schlag, J. Grotemeyer, and R. D. Levine, Chem. Phys. Lett. 190, 52 1 (1992)l.

D. M. Neumark: I talked to Prof. Marcus, and he mentioned some-

thing similar.

V. Engel: Let me come back to the distribution of lifetimes of the ZEKE Rydberg states. I wonder if there is a simple picture behind. Con-

sider a much simpler molecule, namely the NaI molecule Prof. Zewail told us about. There you have a bound state coupled to a continuum. It can be shown that in such a system the lifetimes of the quasibound states oscillate as a function of energy. In fact, Prof. Child showed with the help of semiclassical methods that there are lifetimes ranging from almost infinity to zero [l]. That can be understood by the two series (neglecting rotation) of vibrational levels obtained from the adiabatic and diabatic picture. If two energy levels of different series are degen-

RYDBERG STATES AND ZEKE SPECTROSCOPY 1

657

erate, an infinite lifetime is obtained; in the case the excitation energy is between two levels, a shorter lifetime occurs. For a Rydberg electron in a molecule there are many series, but the picture behind might be the same. 1 . S. Chapman and M. S . Child, J. Phys. Chern. 95,578 (1991).

J. Manz 1. The classical trajectory simulations of Rydberg molecular states carried out by Levine (“Separation of Time Scales in the Dynamics of High Molecular Rydberg States,” this volume) remind me of the related question asked yesterday by Prof. Woste (see Berry et al., “SizeDependent Ultrafast Relaxation Phenomena in Metal Clusters,” this volume). Here I wish to add that similar classical trajectory studies of ionic model clusters of the type A; ’B; have been carried out by 0. Knospe and R. Schmidt [I]. Here the two charged clusters A; and B; rotate around each other, similar to the rotation of the Rydberg electron e- around its cationic center M+ in the Rydberg state M+ . e- . Several properties of the trajectories for A; - B; and M+ - e- are found to be analogous, except for the different masses and effects of internal motions in the fragments A; and B; [l]. 1. 0. Knospe and R. Schmidt, Z. Phys. D 37, 85 (1996); 0. Knospe and R. Schmidt, Phys. Rev. A, 54, 1154 (1 996).

2. Prof. Schlag (“ZEKE Spectroscopy,” this volume) has introduced a new sequential technique of ZEKE spectroscopy: In the first step, a negative ion M- is photoionized, yielding the neutral core M of the excited Rydberg state of the anion M-*. In the second step, M is further photoionized, yielding the cationic core M+ of the excited Rydberg state of the neutral molecule M*. The overall sequence is thus M- -M-*

-M

+ e- -M* + e- -+M++ 2e-

and this sequential scheme is (at least formally) similar to the new NeNePo method which has been introduced into femtosecond chemistry by L. Woste and co-workers (see Ref. 1 and Berry et al., “SizeDependent Ultrafast Relaxation Phenomena in Metal Clusters,” this volume). I would therefore like to ask Profs. Schlag and Woste to comment on the relations between their new techniques. I . S . Wolf, G. Sommerer, S. RUU, E. Schreiber, T. Leisner, L. Woste. and R. S. Berry, Phys. Rev. Lett. 74,4177 (1995).

3. Prof. Levine (“Separation of Time Scales in the Dynamics of

658

GENERAL DISCUSSION

High Molecular Rydberg States,” this volume) has pointed to the fascinating situation that exists in Rydberg molecules in electric fields for energies E close to the ionization threshold Ethr,that is, the number of may be even larger (near-) degenerate (quasi)bound states &,,,nd(E) than the number of near-degenerate dissociative states Ndiss(E). I would like to ask Prof. Levine about the energy dependence of the ratio

I would expect that R ( E ) increases as E approaches Ethr, due to the enormous increase in N b o u n d ( E ) relative to Ndi,,(E). This conjecture may have important consequences for the stability of Rydberg molecules; cf. the experimental results of Prof. Schlag, which show that highly excited molecular Rydberg states may have exceedingly long lifetimes (see Schlag, ‘‘ZEKEi Spectroscopy,” this volume). Usually the stability of Rydberg molecules is attributed to dynamical effects; that is, any ionization processes are limited to the rare situations where the Rydberg electron interacts with the molecular core (see Levine, “Separation of Time Scales in the Dynamics of High Molecular Rydberg States,” this volume). On the other hand, the stability of Rydberg molecules could also be interpreted in terms of an “entropic effect”; that is, the formation of bound states rather than ionization states could be due to the increasing ratio R ( E ) as defined above.

-

E. W. Schlag: On the way up from M- -+M M+ it would be often helpful to state select M via ZEKE spectroscopy. It would also be useful for its overtones. L. Woste: In stationary spectroscopy ZEKE certainly provides spectroscopic results at an impressive resolution. Using femtosecond pulses one can certainly not excite specific states as compared to ZEKE. The Fourier transform of the wavepacket evolution, however, exhibits also spectral resolution that easily reaches and even exceeds what we see in ZEKE spectra. For this reason, I do not see any disadvantage in using “femtosecond NeNePo” to probe states of a prepared molecule. E. W. Schlag: I would suggest that an interesting experiment is ZEKE selection on the way to further decomposition. This would directly answer the question. R. D. Levine: The suggestion of Prof. Manz about the dynamics of extremely high Rydberg states is a very interesting one to discuss.

RYDBERG STATES AND ZEKE SPECTROSCOPY I

659

Unfortunately it is not going to be easy to test experimentally or even to simulate on the computer. The reason is the extreme sensitivity of states of high n to external perturbations. In the laboratory, stray electrical fields, which cannot be completely avoided (or black-body radiation) will cause ionization of these states. Even on the computer, numerical roundoff errors will act as external noise. M.S. Child: Perhaps I can add a comment about time scales. For small nonpolar species such as H2 and N2 the dominant interaction between the Rydberg electron and the nuclear vibrational and rotational motion occurs within a small radius around the ionic core, which is traversed in a fraction of a femtosecond. This short encounter justifies the “sudden” treatment of vibration and rotation in MQDT theory, while also permitting Born-Oppenheimer estimates of the necessary quantum defect functions. It is also central to the n-3 scaling law because the core transit time is almost energy independent, while the Rydberg orbit time increases as n3. In dipolar situations the electron will continue to rotate with the core out to a radius such that the ion-dipole anisotropy is small compared with the relevant rotational energy separation for the ion core. Again since the switch-off distance is insensitive to energy, the dipole transit time will also be roughly the same for all n and one again expects an n-3 scaling law, but with a different coefficient. If this reasoning is correct, it is hard to see how the presence of a dipole can substantially enhance the lifetimes of ZEKE states. E. W. Schlag: The paper by Akulin et al. [l] explicitly calculates this model. 1.

V. M. Akulin, G . Reiser, and E. W. Schlag, Chem. Phys. Lett. 195, 383 (1992).

R. D. Levine: I completely agree with the picture of Prof. Child, and indeed this is how we calculate the effect of coupling to the vibrations [E. Rabani and R. D. Levine, J. Chern. Phys. 104, 1937 (1996)]. As to the special case of dipolar anisotropy, see L. Ya. Baranov, F. Remacle and R. D. Levine, Phys. Rev. A 54,4789 (1996). U. Even: In a recent series of papers [M. Bixon and J. Jortner], using a model Hamiltonian quantum treatment, it is shown that all multipole contributions to 1 mixing are negligible when compared with 1 mixing by low external fields. Thus the long lifetimes associated with ZEKE states are attributed (in atoms and in molecules) to the external fields alone. M. Herman: There is a striking, obvious and fascinating comparison between overtones and Rydberg states. Both cases present a high

GENERAL DISCUSSION

660

density of levels and exhibit scaling factors (referring to the acetylene case, to be extended to smaller and possibly larger species). How far can one pursue the comparison? What are the chances that a model developed in one case could help elucidate the other case?

V. S. Letokhov: In his exciting and dynamical report Prof. Schlag mentioned the historical development leading to ZEKE spectroscopy. Let me comment on this point. Laser-induced WMPI and ZEKE spectroscopies belong to the rich family of laser ionization spectroscopy techniques (Fig. 1) [l]. In resonance photoionization of a neutral particle (atom or molecule), the absorption by the particle of a few laser photons gives rise to an easily detectable pair of charged particles-a photoion and a photoelectron. In most experiments, the number of charged particles

r Mt

/

Photoion mass spectra

M*

+ eK

'Photoelectron spectrum

k

Photoions Photons

Je

Photoelectrons 'Resonant ionization optical spectrum

(b)

(a1 Figure 1. Versions of photoionization spectroscopy wherein not only the dependence of the multiphoton ionization efficiency on the laser wavelength is subject to measurement, but also the mass spectrum of photons and energy spectrum of photoelectrons: (a) energy-level diagram; (6)collision of a neutral particle with laser photons.

RYDBERG STATES AND ZEKE SPECTROSCOPY I

66 1

(i.e., the photoionization efficiency) as a function of the laser wavelength at one or, in principle, several resonance excitation steps is subject to measurement. This simplest version of photoionization spectroscopy is referred to as resonance photoionization spectroscopy (Fig. 1). The technique provides fairly accurate information on the structure of the quantum transitions of atoms and molecules with extremely high sensitivity. An excited molecule can be ionized by several pathways, charged fragments of various masses being formed in the process. A mixture of charged particles of various masses is also formed in the course of photoionization of a mixture of isotope atoms or molecules or else of molecules with close absorption lines. To identify more accurately the charged particles formed, we can in addition analyze the mass spectrum, that is, change over to photoionization muss spectrometry, as is mentioned in Ref. 2. This spectroscopic technique is being widely used in experiments with molecules and mixtures of isotopic atoms [I, 31. In my papers in 1975-1976 I considered various applications of laser resonance photoionization for the detection of single atoms and molecules, particularly by means of laser optical mass spectrometry [3, 41. The first steps in the realization of this idea were the successful experiments on two-color photoionization of polyatomic molecules in the gas cell [3, 51 and in the mass spectrometer chamber [6, 71. It seems to me that these works are a first demonstration of the REMPI technique. Independent experiments of Prof. Schlag and Prof. Bemstem have been cited by Prof. Schlag. In this respect I should mention also Prof. Mamyrin’s work at the Ioffe Institute, who realized the mass reflectron in 1966. Now this is a most useful type of TOF mass spectrometer for experiments with laser pulses. Finally, the photoelectron formed as a result of the decay of the ionic state of a particle acquires some excess energy, equal to the difference between the initial energy of the particle and the ion energy. This energy excess is transferred to electron translation; that is, it is converted to kinetic energy. If it populates different energy levels, there occurs a whole spectrum of energies of the photoelectrons produced in the course of ionization. Measurement of the photoelectron energy spectrum provides additional information about the particle being ionized and lies at the root of one or more photoionization spectroscopy versions-photoelectron spectroscopy of excited states with resonance ionization. Zero electron kinetic energy spectroscopy is based on the detection of low-kinetic-energy electrons and tuning of the laser wavelength XZ

662

GENERAL DISCUSSION

in Fig. 1. This technique allows to obtain a high spectral resolution from the optical channel for the ionizing state (0.1 cm-' and better), as compared with 100 cm-' in standard photoelectron spectroscopy, exploiting the spectral resolution of the electron channel. As concerning the development of ZEKE spectroscopy, I should say that we are using this technique for experiments with atoms routinely since perhaps 1978 (see relevant references in Ref. 1). In the case of multistep ionization of atoms we excite high-lying states below (Rydberg states) and above (autoionization states) the ionization limit. In both cases we exploit pulsed electric time-delayed ionization and (or) detection of ions or electrons. This technique allows us to detect charged particles with low kinetic energy. In the case of photoion detection we can denote this technique as ZIKE (zero ion kinetic energy) spectroscopy. This technique is very powerful for the study of narrow metastable autoionization (AI) states (above the ionization limit). In our experiments we have discovered the very narrow (-0.01 cm-') autoionization state in gadolinium above the ionization limit with nanosecond lifetime IS]. Such A1 states are very important for the practical realization of laser isotope separation of uranium by the resonant multistep ionization technique. Many other AI states of rare-earth elements have been studied by the ZIKE technique (see Ref. 1). The nature of the narrow autoionization long-lived atomic and molecular states may be quite different. (See also Ref. 9.) Finally I am very impressed how efficiently Prof. Schlag has used the ZEKE technique, which was already well known for atoms, in the study of molecular states. 1. V. S. Letokhov, Laser Photoionization Spectroscopy, Academic, Orlando, 1987. 2. R. V. Ambartsumian and V. S. Letokhov, Appl. Opt. 11(2), 354 (1972). 3. V. S. Letokhov, In Tunable Lasers and Their Applications, A. Mooradian, T. Jaeger, and P. Stokseth, Eds., Springer-Verlag.Berlin, 1976, p. 122. 4. V. S. Letokhov, Physics Today 30(5), 23 (1977). 5. S. V. Andreev, V. S. Antonov, I. N. Knyazev, and V. S. Letokhov, Chem. Phys. 45(1), 166 (1977). 6. V. S. Antonov, I. N. Knyazev, V. S. Letokhov, V. M. Matyuk, V. G. Movshev, and V. K. Potapov, Pis'ma ul. Tekhn. Fiz. 3(23), 1287 (1977) (in Russian). 7. V. S. Antonov, I. N. Knyazev, V. S. Letokhov, V. M. Matyuk, V. G. Movshev, and V. K. Potapov, Opt. Lett. 3(2), 37 (1978). 8. G . I. Bekov, V. S. Letokhov, 0. I. Matveyev, and V. I. Mishin, Pis'ma 2.Eksp. Teor. Fiz. 28, 308 (1978). 9. G. I. Bekov, E. P. Vidolova-Angelova, L. N. Ivanov, V. S. Letokhov. and V. I. Mishin, Optics Commun. 35, 194 (1988); Zh. Eksp. Teor. Fiz. 80, 866 (1981).

E. W. Schlag: Prof. Letokhov, the interesting development here is

RYDBERG STATES AND ZEKE SPECTROSCOPY I

663

that there are many molecular states above the IP that have anomalous lifetimes of some 100 ps. This is the required signature for this spectroscopy. This was not expected and is the basis of ZEKE since 1984. This gives a new rich spectroscopy of ions. Some atoms have a few states also above the IP but due to a different mechanism. Technically pulsed-field ionization detection has been used for both atoms and molecules, but generally speaking for atoms below the IP, whereas for molecules these are detecting imbedded long-lived states that are ZEKE states in the ionization continuum.

V. S. Letokhov: As concerning the autoionization states of atoms, for example Gd, the lifetime is in the nanosecond range. But according to our calculations some atoms, for example Yb, have very long-lived (microsecond range) lifetimes (see Refs. 1 and 2). You have observed long-lived Rydberg autoionization states for molecules. It seems to me that such states have already been well known in atomic physics. I also have a comment about the historical development of laser resonance ionization spectroscopy with mass spectroscopy (REMPI + MS) leading to ZEKE. Zero electron kinetic energy spectroscopy is just one version of this technique, which is quite known in atomic physics. As concerning the observed long-lived Rydberg autoionization states of molecules I should say that by the same technique we observed long-lived autoionization states for atoms about 20 years ago. Such states can live even on the microsecond time scale [2]. Moreover, other researchers have observed long-lived autoionization Rydberg states in atoms, which have the same origin as in molecules. The long lifetime of these states is explained by weak coupling with the continuum. I . V. S. Letokhov, Laser Photoionization Spectroscopy, Academic, Orlando, 1987. 2. G. I. Bekov, E. P. Vidolova-Angelova, L. N. Ivanov, V. S. Letokhov, and V. I. Mishin, Opt. Commun. 35, 194 (1988); Zh.Eksp. Teor. Fiz. 80, 866 (1981).

E. W. Schlag: The term zero-kinetic-energy electrons was used already by us in Chem. Phys. Lett. 4, 243 (1969). The new mechanism of redistribution of low-l into high-l molecular states is the basis of ZEKE states. T. P.Softley: The aims of ZEKE spectroscopy are conceptually different from the atomic pulsed-field ionization experiments that predate ZEKE. In the latter, the aim is always to observe and study the individual Rydberg states. In ZEKE spectroscopy the aim is to detect small batches of Rydberg states lying below successive vibration-rotation thresholds of the ion without specific interest in the individual Ryd-

664

GENERAL DISCUSSION

berg spectra. The experimental methodology is the same, but the aims differ.

M. Chergui: Let me invoke a rather exotic behavior of Rydberg states that may or may not relate to the lifetime lengthenings discussed in the previous talk. From a nonexpert point of view, my impression is that the lifetime lengthenings here reported are due to external perturbations, and Prof. Levine has shown us the cases of the electric and magnetic fields. But there exists another type of external perturbation, and this is the case of a condensed-phase environment. We have measured the lifetime of the low-n (actually n = 3) Rydberg states of NO in Ne, Ar, Kr, and Xe matrices [l]. We have observed a lifetime lengthening of up to two orders of magnitude (see Fig. l), as compared to the gas phase value, in going from low-polarizability matrices (Ne) to high-polarizability ones (Xe). In order to lift any ambiguities and to stress that the environment specifically affects the Rydberg electron, I would like to add the following points: (a) The effect cannot be due to a modification of the ground-state wave function as we have checked this point against valence transitions, which show no lifetime modification.

1.4 10'

-

1.2 10'

h

I 'A4

2000 0

1

2

3

, A

4

atomic polarizability (lO%d) Figure 1. Lifetime of the n = 3 Rydberg states of NO in Ne, Ar,Kr, and Xe matrices vs atomic polarizability.

RYDBERG STATES AND ZEKE SPECTROSCOPY I

665

(b) The effect is not caused by mixing of the Rydberg state with near-resonant quadruplet states of infinite lifetime.

(c) For a given matrix, the Rydberg lifetime changes dramatically with the trapping site of the molecule. 1. F. Vigliotti, G. Zerza,and M. Chergui, in Femtochemistry, Ultrafmt Chemical and Physical Processes in Molecular Systems, M.Chergui, Ed., World Scientific, Singapore, 1996.

njl

-3

’

nj2

.,

- 4 - 3 - 2 - 1

nj4

0

nj8

1

log 1o(dmo)

nil6

2

732

3

11164

4

I 5

Figure 1. Stability borders for the system Hamiltonian at fixed w = 0.001 and m = 0.4. The lower curve is the usual chaos border € 0 = 1/50Wk’3 = ( m ~ / w ~ ) ’ / 4 / 5 0 ( ~ m 3 )For L/L2. small 00 this border approaches the usual static border eo = 0.13. The “magic mountain” of stability is delimited from below by the stabilization border € 0 1200 mg and from above by the destabilization border € 0 = (l6L/r)(wo/rn0)~ with L = In[ 2eo/(er)/(womo)].The dashed lines co = (em/w)(wo/mg) are drawn at constant e: ( a ) e = 0.0025; ( b ) e = 0.05;( c ) e = 1. The border (3) below which the Kepler map description is valid is given, in the present case, with fixed m and w , by the line €0 = O.Z(oo/mo) (not drawn in the figure). The present picture is drawn at fixed w and m. If instead we keep no fixed, then the system will always be stable in the region to the right of the dotted vertical line given by worn: = 3.

$-1

666

GENERAL DISCUSSION

G. Casati: I have the following general comment. When a Rydberg atom, prepared in some initial state, interacts for a certain time with a radiation field of a given intensity and frequency, one would naively expect the ionization probability to be a monotonically increasing function of the field intensity. This is however not the case. Instead, it may happen that on increasing the field strength above a critical value the atom becomes increasingly stable against the field-induced ionization (see Fig. 1). This phenomenon is a property of the classical motion and qualitatively can be understood, as mentioned by Levine in his talk, by the fact that by increasing the field strength, the electron is kept far from the nucleus, and this prevents ionization.

FROM RYDBERG STATE DYNAMICS TO ION-MOLECULE REACTIONS USING ZEKE SPECTROSCOPY T.P. SOmLEY,* S. R. MACKENZIE, F. MERKT, and D. ROLLAND Physical and Theoretical Chemistry Laboratory Oxford, United Kingdom

CONTENTS I. Introduction 11. State-Selected Ion-Molecule Reactions A. Principles of State Selection B. Experimental 111. Examples of Preparation of State-Selected Ions A. Hydrogen, H; B. Carbon monoxide, CO+ C. Nitrogen, N$ D. Nitric oxide, NO+ IV. Studies of Ion-Molecule Reactions A. H; + H2 H; + H B. Collision Energy Resolution C. Transmission Effects D. Rydberg State Perturbation by Collision V. Rydberg State Lifetimes VI. Experimental Measurements of Rydberg Lifetimes VII. MQDT Calculations of Spectra of Autoionizing Rydberg States A. Method Employed in the Calculations B. Calculations for Argon C. Calculations for Nitrogen VIII. Conclusions References

-

*Report presented by I: l? Sofley Advances in Chemical Physics, Volume 101: Chemical Reactions and Their Control on rhe Femtosecond lime Scale, XXth Solvay Conference on Chemistry, Edited by Pierre Gaspard,

Irene Burghardt, I. Prigogine, and Stuart A. Rice. ISBN 0-471-18048-3 0 1997 John Wiley & Sons, Inc.

667

668

T. P. SOFTLEY, S. R, MACKENZIE, F.MERKT, AND D. ROLLAND

1. INTRODUCTION

The delayed pulsed-field ionization (PFI) of Rydberg states now lies at the heart of an expanding range of experiments in gas-phase spectroscopy and dynamics. Foremost are the techniques of zero-kinetic-energy(ZEKE) photoelectron spectroscopy [l-31 and mass-analyzed threshold ionization (MATI) spectroscopy (41. Over 100 molecules have been studied using these techniques ranging from H2 to large aromatics, resulting in a wealth of spectroscopic data on molecular cations and an improved understanding of photoionization dynamics [l] and Rydberg channel couplings [2]. The ZEKE spectra of anions [5] and clusters [6] have also been recorded, and ZEKE or MATI has been used as a state-specific probe in studies of intramolecular dynamics in the femtosecond (see Ref. 7 and Baumert et al., “Coherent Control with Femtosecond Laser Pulses,” this volume) and picosecond [8] time domains. In a separate development Rydberg tagging time-of-flight spectroscopy has proved to be a valuable means for determining product velocity distributions in photofragmentation [9] and bimolecular collisions [lo]. Recent research has concentrated both on broadening the range of applications and on improving the understanding of the physical mechanisms underlying these PFI techniques. In the latter category, there has been considerable interest in explaining the unexpected long lifetimes of high Rydberg states of a wide range of molecules [ll-181. The study of the dynamics of molecular Rydberg states has now become a topic of importance in its own right, with particular interest in the significance of intramolecular couplings and external electric and magnetic field effects. This work is a natural follow-up to two decades of measurements of properties of atomic Rydberg states [191. In this chapter, results of importance to both categories are reported. In the first part a new application of PFI to the study of state-selected ion-molecule reactions is described, and in the second part some novel experiments and calculations relating to Rydberg state dynamics are presented. The goal of the ion-molecule reaction studies is to select ions in unique vibration-rotation quantum states and to study the effects of selectivity on reaction cross sections. The additional possibility of controlling the spatial alignment of the ionic angular momenta using the polarization properties of lasers is also of interest and is under investigation. Rotational effects on reaction cross sections act as a probe of anisotropy in the reaction potential-energy surface, which might arise from long-range forces (see Ref. 20 and Troe, “Recent Advances in Statistical Adiabatic Channel Calculations of State-SpecificDissociation Dynamics,” this volume) or shorter range valence interactions. The effects of ionic rotation have received very little attention either theoretically or experimentally.

FROM RYDBERG STATE DYNAMICS TO ION-MOLECULE REACTIONS 669

11. STATE-SELECTED ION-MOLECULE REACTIONS A number of techniques have been used previously for the study of stateselected ion-molecule reactions. In particular, the use of resonance-enhanced multiphoton ionization (REMPI) [21] and threshold photoelectron photoion coincidence (TPEPICO) [22] has allowed the detailed study of effects of vibrational state selection of ions on reaction cross sections. Neither of these methods, however, are intrinsically capable of complete selection of the rotutional states of the molecular ions. The TPEPKO technique or related methods do not have sufficient electron energy resolution to achieve this, while REMPI methods are dependent on the selection rules for angular momentum transfer when a well-selected intermediate rotational state is ionized; in the most favorable cases only a partial selection of a few ionic rotational states is achieved [23]. There can also be problems in REMPI state-selective experiments with vibrational contamination, because the vibrational selectivity is dependent on a combination of energetic restrictions and Franck-Condon factors. In this chapter, the design and instigation of a new experimental method aimed at achieving an improved level of ionic state selectivity is described, which allows, in small molecules, the selection of individual rotational states. The technique, developed from ZEKE spectroscopy, makes use of the existence of a pseudocontinuum of high-n Rydberg stakes of the neutral molecule located energetically below each ionic threshold (corresponding to each vibration-rotation state of the ion), which in the range of principal quantum number n 2 100 show metastability with respect to decay processes such as fluorescence, predissociation, and autoionization. Experiments have shown that even the Rydberg states of complex molecules with many internal degrees of freedom, or Rydberg molecules with electronically excited core states, can live for tens of microseconds [17,24-263 (see also Section V). In order to exhibit such stability, the Rydberg electron must have virtually no interaction with the ionic core of the molecule, which itself may be considered to exist in a well-defined internal quantum state. The Rydberg electron is easily ionized by the application of a small pulsed electric field. The ion core is unaffected in this process (but see next paragraph), and therefore the resultant ion is left in the same well-defined state. Thus, the P H of Rydberg states provides the basis for a method of ionic state selection. However, the formation of “prompt ions” by direct photoionization at the same time as Rydberg state excitation is generally unavoidable, and therefore such nonselected ions must be separated from those arising from field ionization. The possibility of making such a separation was first demonstrated by Zhu and Johnson [4] in the MAT1 experiment. An important consideration for the purposes of ionic state selection is the

670

T. P. SOlTLEY, S. R. MACKENZIE, F. MERKT, AND D. ROLLAND

possibility that the ionizing pulsed electric field actually induces a change of rotational state of the ion core by enhancing the interchannel couplings 127, 281. In ZEKE or MATI spectroscopy this question is disregarded because one only collects all the ions or electrons produced by delayed PFI in a given energy range, and the actual states of the ions formed do not matter. Nevertheless, a Rydberg state (v+N+nl)could be coupled by the pulsed field to the (u+N+- x d ’ ) continuum in a forced rotational autoionization, leaving the ion in rotational state N + - x instead of the selected N+.Two arguments offer strong evidence that such processes do not occur. First, there is no reason why forced rotational autoionization, if it is happening, should be confined to occur within the field ionization range; it could also occur below this range and would give rise to long red tails on every ZEKE or MAT1 peak. The general observance of well-defined field ionization behavior in ZEKE spectroscopy points to the nonoccurrence of forced autoionization. Second, in order to survive the several-microseconddelay before field ionization, the Rydberg electron must almost certainly have found its way into a high-m,, high-Z state (see Section V) in which the electron is noninteracting with the core. In this state the forced autoionization, which requires core penetration, cannot occur because the pulsed electric field will preserve the high-mr value of the Rydberg electron and hence the nonpenetrating character.

B. Experimental The state-selected ion-molecule reaction experiment is illustrated schematically in Fig. 1 and has been described in more detail elsewhere [29].

I EXCLTATION AND DISCRIMINATfoN 2 FIELD 1ONlZATlON AND ACCELERATION

AND PRODUCT DETECTION

STATESELECTED /////#

BEAM

ALS

Figure 1. Schematic diagram of the state-selected ion-molecule experiment.

FROM RYDBERG STATE DYNAMICS TO ION-MOLECULE REACTIONS

67 1

Molecules are excited in a doubly skimmed pulsed supersonic beam using a two-color resonance-enhanced multiphoton excitation process, to Rydberg states with the desired ionic core quantum state. Prompt ions are deflected out of the beam by applying a small perpendicular field (typically Fd = 0.1 V/cm) or, alternatively, are retarded using a field parallel to the beam axis. The surviving Rydberg molecules continue unaffected by the field into the extraction region (with the exception of the very highest states, which are field ionized) and then, after a suitable time delay, they are field ionized by a pulsed field (-4 V/cm) to produce state-selected ions. The metastability of Rydberg states is crucial for the selectivity because it is generally necessary to allow a period of a few microseconds between excitation and field ionization to give adequate separation between Rydberg molecules and prompt ions. The extraction pulse serves a second important purpose in that it accelerates (or decelerates if the polarity is reversed) the ions in the neutral beam direction, and the degree of acceleration is determined by the duration and amplitude of the pulse. This causes a slippage in the beam between the ions and the neutrals and hence a much enhanced ion-neutral collision rate. Reactive collisions can therefore occur over a length, which in the present setup is variable in the range 5-15 cm. A quadrupole mass filter with resolution below 1 amu is placed at the end of this reaction zone and is tuned to transmit either the parent ions or the product ions, which are detected by a multichannel plate (MCP) detector. We choose to vary the collision energy by changing the duration (20 ns-20 ps) rather than the amplitude of the extraction pulse, because changing the duration does not affect the field ionization probability (and hence the spectroscopic discrimination between Rydberg states belonging to different series). In preliminary experiments using the new apparatus we have produced rovibrationally state-selected Ht, CO+, Nt, NH;, and NO+ and have begun Hi + H to study simple bimolecular reactions, for example, H$ + H2 at low collision energy ( 4 . 5 eV) [29]. For studying the H;/H2 reaction, where the molecular ions react with the species that itself is the precursor neutral molecule, the beam consists of neat H2 only, backing pressure of 2 bars. However, for other reactions such as CO+ + H2, the ion precursor and reactant gases are coexpanded from the same nozzle. The translational temperature has been measured (see Section 1V.B) to be of order 2 K, and the rotational temperatures, determined from REMPI spectra, are typically in the range 5-20 K (except H2). The laser system consists of two dye lasers (Spectra Physics PDL 3) pumped by a single Nd-YAG laser (Quanta Ray GCR 270). Dye Laser 2, pumped at 355 nm, operates in these experiments in the range 370-500 nm (5-15 mllpulse), whereas Dye Laser 1, pumped at 532 nm, operates at 660-600 nm and is frequency tripled using KD*P and BBO crystals

-

672

T. P. SOFTLEY, S. R. MACKENZIE, F. MERKT, AND D. ROLLAND

in an INRAD autotracking system to produce light at 220-200 nm (0.3-1 mJ/pulse). The two beams, which are temporally overlapped, are combined on a dichroic mirror and are then focused into the vacuum chamber using a 15-cm lens to intersect the molecular beam perpendicularly. The beam waste at the focal point is approximately 0.4 mm.

III. EXAMPLES OF PREPARATION OF STATE-SELECTEDIONS In all cases a two-color multiphoton process is used to excite the molecules to the high-n Rydberg states. In the remainder of this text we use a primed notation to refer to the transitions from the ground state to the intermediate state (e.g., Q‘ branch, 0’ branch, etc.), whereas an unprimed notation is used for transitions from the intermediate state J’ to the Rydberg states with core rotation quantum number N + ;for example, S(2) implies the transition J‘ = 2 - + N + = 4, whereas S’(2) implies J” = 2 J‘ = 4.

-.

A. Hydrogen, H i

The hydrogen molecular ion, H,+ is prepared by 2 + 1’ multiphoton excitation via the E, F ’ C i , u’ = 0 state, with the frequency tripled Laser 1 tuned to around 202 nm (0.5 &/pulse) and Laser 2 to ~ 3 9 6nm (5-10 mJ/pulse). The transitions Q’(O), Q’(l),Q‘(2), and Q’(3) are well resolved in the transition to the intermediate state (rotational temperature =180 K), and Q-type transitions to the high Rydberg pseudocontinuum are used to produce ions in u+ = 0, Nt= 0, 1,2,3. Approximately 1000 state-selected ions are detected per pulse after transmission through the mass filter. The S-type transitions, which in principle could allow population of ionic states up to N + = 5, are too weak to be observed. Using a DC discrimination field of 0.1 V/cm and a pulsed extraction field of 3.7 V/cm, a resolution of 5 cm-’ is obtained in the PFI peaks (however, see Section IV.D and Fig. 9). Full details of the state-selective preparation of Hi are given elsewhere [29]. An alternative procedure demonstrated earlier [30] was to use a tunable extreme ultraviolet source, based on four-wave mixing of pulsed ultraviolet laser beams, to excite directly from the ground state to the high Rydberg states. On balance it appears that the multistep excitation is a more efficient use of the dye laser energy and is less demanding experimentally. Moreover, for all ions except H;, the enhanced selectivity gained by exciting via an intermediate well-defined rotational level is a major advantage.

B. Carbon Monoxide, CO+ The CO+ ions are prepared by 2 + 1’ excitation via the E ’n,u’ = 0 state; Laser 1 is frequency tripled to produce light of 215 nm (0.5 ml/pulse) and

FROM RYDBERG STATE DYNAMICS TO ION-MOLECULE REACTIONS

.. .

673

. . .. .. . . . . ... .. . .*

' p , , ,

z.-

M

e v)

a

-

.

CI

S'(3) &

Figure 2. The 2 + 1 REMPI spectrum of the E 'H-X 'CC transition of CO (0-0 vibrational band).

Laser 2 is at 4 9 6 nrn (5 ml). The 2 + 1 REMPI spectrum of the E 'II- X 'C+transition is shown in Fig. 2, from which a rotational temperature of 6 K may be deduced. From the point of view of optimizing selectivity, this transition, which shows well-resolved P,R,and S' branches, is most likely to be a preferable starting point compared to the B-X or C-X bands of this molecule; the two-photon transition intensity for the latter bands would be completely concentrated in the Q'-branch lines, which would not be well resolved. Figure 3a shows the PFI spectrum obtained by selecting the intermediate state J' = 2 with Laser 1 tuned to the S'(0) transition. Six peaks are observed corresponding to N + = 0 to 5. The DC/pulsed fields used were 0.2 and 3

113010.0

113030.0

113050.0

~ o t atmn l value / cm-'

113070.0

113090.0

Figure 3. Pulsed-field ionization spectra from the E 'II (u' = 0) state of CO with the initial level J' = 2 pumped via (a)S'(0) and (b) R'(1).

674

T. P. SOFTLEY, S. R. MACKENZIE, F. MERKT, AND D. ROLLAND

V/cm, respectively, giving a resolution of 4cm-’ , sufficient to observe the individual rotational states. The peaks correspond to 10-50 ions per pulse. Figure 3b shows the PFI spectrum probed via the same intermediate state J‘ = 2 but with the R’(1) transition as the initial excitation process. The two spectra show quite different rotational line intensities; the Q-type transition to N + = 2 is the strongest line when the R’(1) transition is pumped (Fig. 3b), but in the S’(0) spectrum (Fig. 3a) it is weaker than the P- and R-type transitions. Also the T(2) peak is observed in the spectrum via S’(0) but not via R’(1). The observed differences are explained in part by parity selection rules. The S’(0) transition populates only the e component (II+) of the J’ = 2 state According whereas the R‘( 1) transition populates only thef component (W). to Xie and Zare [31], the selection rule applying is N + - J’ + p+ + I = odd where 1 is the angular momentum of the departing electron and the Kronig symmetry indices are p+ = 0 for the ionic 2C+ state and p’ = 1, 0 for the intermediate II- and II+ states, respectively. For the present case this implies N + - J ’ = f l , + - 3 ,... forleven,n+(e) or N + - J’ = 0, f 2 , k4,. . . for 1 even, n-(f) or

forZodd,W(f) for 1 odd, II+(e)

Fujii et al. observed a strong propensity to the even-l selection rule [32] when using laser-induced fluorescence to determine the ionic state distribution following direct ionization from the same intermediate state, observing no Q-type transitions following S‘-branch excitation. The even4 propensity is to be expected because the electron is ionized from a nearly pure p orbital and hence I should equal 0 or 2. In the PF’I spectrum via S‘(O), however (Fig. 3a), the Q(2) line gains some intensity by rotational channel coupling with Rydberg states converging to the N ++ 1, limit; that is, the Q-type transition is gaining intensity from an R-type transition. This is shown more clearly in the Q(3) line following S’(1) excitation in Fig. 4, where the apparently noisy structure observed on the Q-type transition is resolved into individual Rydberg states converging to the N + = 4 limit. The intensity reduces to zero between each Rydberg state, suggesting that there is no intrinsic intensity for direct excitation to the N + = 3 pseudocontinuum from the J’ = 3(e) level: All the intensity is borrowed. Nevertheless, the position and overall width of this feature are determined by the field ionization of the pseudocontinuum. The coupling that gives rise to the intensity borrowing must occur between states of the same overall parity. Therefore, the even-l Rydberg states populated via the R’-type transition must couple to Rydberg states with odd-2 character associated with the N + = J’ limit, mediated by the core dipole. The PFI spectrum of CO illustrates nicely the interplay between pure photoionization dynamics and Rydberg channel couplings, as is often observed

il

FROM RYDBERG STATE DYNAMICS TO ION-MOLECULE REACTIONS 675

1

1 13040.0

nd4

68

113042.0

76

70

113044.0

113046.0

Total term value I cm-'

78

80

113048.0

113050.0

Figure 4. The Q(3)transition in the PFI spectrum of CO following S'(1) excitation showing Rydberg states converging to N + = 4. The total term value is given with respect to CO X 'E+,v" = 0, J" = 0.

in ZEKE spectra [2]. This interplay is important in controlling our ability to produce specifically chosen states effectively. In many cases it is possible to produce a wider range of ionic quantum states using PFI than in direct ionization. Vibrational intensities do not necessarily follow Franck-Condon factors, although most of the known examples of non-Franck-Condon behavior occur in small molecules.

C. Nitrogen, NZ

The lowest accessible two-photon transition of N2 is the a" 'E; - X 'c;'g system, for which the 0-0 vibrational band is at =202 nm. The intensity is concentrated in the strong Q' branch, with S' and 0' branches weaker by two orders of magnitude. The S' branch transitions in the REMPI spectrum allow an estimate of the rotational temperature of 7 K. implying a significant population of J" = 0 , . . .,4. The rotational lines of the Q' branch are not resolved in the present study, and in order to obtain rotational selectivity, it is preferable to use the well-resolved S' and 0' branch transitions in the two-photon step under high-power conditions [33]. Figure 5 shows the PFI spectrum from J' = 0,2,3,4 leading to population of the N; ions in N + = 0,. .., 6 . Again the spectroscopic resolution is =4 cm-' . Approximately 100 selected ions per laser pulse are produced in this way, compared to the 600 ions per pulse at lower laser power, but with only partial state selection, when exciting via the Q/Q'-type sequence. The 0 branch line profiles

676

T. P. SOFTLEY, s. R. MACKENZIE, F. MERKT, AND D. ROLLAND

N+=

0' ! 'z

m

I

3

I

4

I

5

I

6

J'=4

125640.0

125660.0

125680.0

125700.0

125720.0

125740.0

125760.0

TOUIterm value / cm-' Figure 5. The P H spectra from J' = 0,2,3,4 of the a" 'Z; state of N2 leading to population of the Nt ions in N + = 0,. ..,6. The total term values are given relative to N2 X 'C;, V" = 0,J" = 0.

in the PFI spectrum show a superimposed structure associated with the coupling of Rydberg states = n = 60,. . .,80, converging on a higher rotational state of the ion, with the field-ionized pseudocontinuum in question. This gives an overall factor-of-2 enhancement to the intensities of the 0 branch lines compared to the S branch. The Q branch transitions are in any case the strongest.

D. Nitric Oxide, NO+ The optimum method for production of NO+ is to use the 1 + 1' excitation via theA 2C+ state (226 + 337 nm). The ZEKE spectrum is well documented [34] and the resolution achieved to date in our experiments is sufficient to obtain good state selectivity. We have also investigated the 1 + 1' ionization via the B *IIstate, primarily because of the convenience of wavelengths used (202 + 387 nm,very close to those used for N2 and H2).The B 211 state of NO has a potential minimum at larger internucleardistance compared to the ground state of either NO or NO', and therefore good Franck-Condon overlap can only be obtained via 'v > 0. The most convenient level to access in this work was u' = 4, and the 1 + 1REMPI spectrum is shown in Fig. 6. The PFI spectrumobtained via the P;,(5.5) transition (J' = 4.5) is shown in Fig. 7.The spectrum is weak and corresponds to the production of only a few ions per pulse but shows that a very wide range of rotational states can be populated, apparently rather more than in the excitation from the A 2C+ state [34].

-

FROM RYDBERG STATE DYNAMICS TO ION-MOLECULE REACTIONS 677

-m . .-8 :

R

:

6.5

-4;(

v)

I + -

t- i

5!

9:. 5;.

-

,

i4.5

55

-

5

3.5 2 5 15

-

-

zs 1.50.5

35

1

I

45

3.5

I

-

L

d

4

Figure 6. The 1 + 1 REMPI spectrum of the B 2111p - X 2111/2 (4-0) transition of NO.

The reason for the weakness of the PFI spectrum is of interest in itself, in that the one-photon ionization from the B 'II state to the ground state of the ion is formally forbidden as it involves a nominal two-electron change:

-

N 0 + ( 3 u ) * ( l ~ ) ~ ;'E+ X

N 0 ( 3 0 ) ~ ( 1 ~ ) ~ ( 2Ba2) n~ ;

It is therefore surprising that this process can be observed at all, and the intensity can be ascribed to a configuration mixing of the B 'II state with ]. the rotational intenthe C 211 state [configuration ( 3 ~ )1~~() ~ ( 3 p ) 'Indeed, sity distribution is similar to that shown in the ZEKE spectrum via the C state

-

N+=b

'1 5

I

4

I

I

6

5

I

7

-

d

el

.+ % v1

*

E74700.0

-

74750.0

74800.0

TOM~term value / cm-I

Figure 7. The PFl spectrum of NO from the J' = 4.5 level of the B leading to population of the NO+ ions in N + = 0 , . .. ,7.

74850.0

(d= 4) state

678

T. P. SOFIZEY, S. R. MACKENZIE, E MERKT, AND D. ROLLAND

[35]. The configuration mixing is much stronger above u' = 7 of the B state

[36], and one would expect a much stronger spectrum in excitation from higher levels. The use of the u' = 4 level of the €3 state as a suitable intermediate for ion preparation is not to be recommended, although it is potentially a valuable starting point for the one-electron excitation to Rydberg states converging to the u 3C+ state of the ion [configuration ( 3 0 ) * ( 1 ~ ) ~ ( 2 ~ ) 'which ], would however require the second laser frequency at =78,000 cm-' .

-

IV. STUDIES OF ION-MOLECULE REACTIONS A. HZ

+ H2

HS

+H

The Hi + H2 reaction is studied using pure normal hydrogen in the molecular beam with the Hi prepared as described in Section IILA. The 1000-shot averaged Hi and H; signals are recorded by tuning the quadrupole mass fiiter to masses 2.0 and 3.0, respectively. Although the Hi parent ion peak is -100 times greater in intensity, there is no remnant signal at mass 3.0 due to H;. The H; signal corresponds to a few ions per pulse. Figure 8 shows the variation of the measured H;-H,+ signal ratio as a function of center-of-mass collision energy E,, for Hi selected in u+ = 0, N + = 1. The data are uncorrected for various transmission losses and dynamical factors (see below), but the figure illustrates the scope and accuracy of the measurements. After applying instrumental corrections, the collision energy dependence of the signal follows that predicted using the Langevin model. We have measured

I

0.0

100.0 200.0 Center of mass collision energy / meV

300.0

Figure 8. Variation of the measured H;-H$ signal ratio as a function of center-of-mass collision energy for Hi selected in u+ = 0, N + = 1.

FROM RYDBERG STATE DYNAMICS TO ION-MOLECULE REACTIONS 679

similar profiles for other rotational states N + = 0,. . .3, but any differences from the curve shown in Fig. 8 were within the error bars of the measurements. Hence, at the present time we are unable to make any authoritative statements about the magnitude of the rotational effects on the reaction cross section, except to say that the variation between N + = 0 and N + = 3 does not appear to be any greater than the value of 10% suggested by Chupka et al. [37]. Unfortunately the largest effects of rotation would be expected at the lowest collision energies, but this is currently where the error bars are largest. Nevertheless, the results shown demonstrate the ability of this apparatus to study reactions, not just state selection, and work is in progress to improve the sensitivity at low collision energy. In evaluating the viability of a new experimental technique for studying ion-molecule reactions, a number of factors must be considered. Ultimately our aims are to measure relative cross sections for reactions as a function of both the internal energy of the ion and the collision energy. It is important that the collision energy can be varied down to =10 meV where the rotational energy may be comparable with the translational energy.

B. Collision Energy Resolution The velocity distribution of the H2 reactants in the direction parallel to the molecular beam has been measured by exciting the molecules to stable Rydberg states around n = 80 (v+ = 0, N + = 0) and then allowing them to continue from the excitation region all the way to the MCP detector, a distance of about 20 cm, under field-free conditions. The width of the time-offlight distribution shows that the translational temperature is =2 K.The ions produced by field ionization in the extraction region are all given an equal acceleration by the pulsed field, assuming the field is perfectly homogeneous, and therefore the laboratory velocity spread for the ions will be identical to that of the neutral molecules. If these velocity-spreading effects dominate the collision energy distribution, then we expect a f 3 meV collision energy spread at E,, = 10 meV and +I0 meV spread at E,, = 100 meV for the H2 + Hl reaction. The angular divergence of the molecular beam is difficult to determine but is likely to be a less significant factor. Ideally we would like all reactions to occur after the field ionization pulse is switched off and prior to the reactants entering the quadrupok, that is, at a nominal constant collision energy. It is important to ensure therefore that the ionization pulse is as short as possible to minimize collisions during acceleration. Given that the acceleration is controlled by both the magnitude and duration of the pulse, there is some scope to test whether this problem is significant and to limit its effect. For a pulse of length 100 ns and amplitude 3.7 V/cm, the probability of collision is approximately 2% of that along the remainder of the flight. Concerning the collisions within the quadruple

680

T. P. SOFIZEY, S. R. MACKENZIE, F. MERKT, AND D. ROLLAND

filter, it is unlikely that the resultant product ions will have the correct trajectory to be transmitted to the detector, and moreover, the parent HZ ions will penetrate the quadrupole very little when it is tuned to mass 3. In any case the lower beam density in this region further downstream reduces the probability of collision. It may therefore be concluded that the collision energy is well controlled for >95% of the collisions.

C. Transmission Effects

+ H2 reaction is that it is highly exothermic and the product H; ions are light. The average center-of-mass frame product ion velocity is of order 3500 ms-I ,compared to the center-of-mass velocity of, for example, 4850 ms- at a collision energy of 0.1 eV. This restricts the detection angle in the center-of-mass frame (i.e., the products scattered in the forward direction are preferentially detected) with the limits depending on exactly where the collision takes place and on the center-of-mass velocity. The HS + H2 reaction shows a strong backward-forward peaking, and hence the collection efficiency is still high; nevertheless, the integral cross section cannot be determined in this case without detailed knowledge of the product angular distribution. This transmission factor will be less of a problem for reactions with heavier product ions and lower energy release. An improvement to the apparatus might be to carry out the PFI within a guiding octupole. The transmission probability is also affected by the angular divergence of the neutral beam and by divergence of the ions due to space charge effects. The excitation of the neutral molecules takes place at the focal points of two pulsed lasers, and therefore a small bunch of Rydberg states are formed in a volume of approximately 0.1 mm3. There is some natural spreading of the group of high Rydberg molecules due to the molecular beam velocity distribution, as the neutrals drift into the extraction region, where we estimate that the volume occupied by Rydberg states is approximately 1 mm3.Great care has to be taken to ensure that we do not produce too many ions, so as to avoid space charge effects. One method to reduce the ion density without a corresponding reduction in the total number of ions is to use a cylindrical focusing arrangement producing the initial bunch of Rydberg molecules in a line focus 4 x 0.1 mm. Such an arrangement allows one to take full advantage of the laser energy available, especially in the case where the two-photon transition is saturated, while avoiding space charge effects.

A significantproblem with the Hi

D. Rydberg State Perturbation by Collision The presence of a large number of ions in the excitation volume, which are mainly caused by one-color direct ionization at the same time as Rydberg state excitation, can lead to additional problems as illustrated in Fig. 9. This

FROM RYDBERG STATE DYNAMICS TO ION-MOLECULE REACTIONS

124000.0

124100.0

124200.0

124300.0

68 1

124400.0

TO^ term value / cm-'

Figure 9. The Q(1) rotational line in the PFI spectrum of H2 recorded with increasingly tight laser focus from (a) to ( d ) .

shows the Q(1) rotational line in the H2 PFI spectrum using a field ionization pulse of just 10 V/cm with Figs. 9a-d resulting from increasingly tight focusing of Laser 1 and Laser 2. The spectra in Figs. 9b-d show that ions are produced by field ionization, not only when the neutrals are excited to Rydberg states within the calculated field ionization range (n > 90), but also when they are excited initially to levels well below this range, even to n = 20. Our interpretation is that the presence of high densities of ions leads to n-changing collisions, or charge exchange, in which the low-n states are upconverted to high-n states that are then field ionized. The observed spectra reflect the initial excitation rather than the final states ionized. In practice, it is possible to suppress this effect by lowering the laser power or defocusing the beams, as shown in Fig. 9a. Again the use of a cylindrical lens is helpful in this respect.

V. RYDBERG STATE LIFETIMES A factor of key importance in these studies is the metastability of the longlived Rydberg states, that are field ionized to produce state-selected ions. This metastability, in which the states exist for tens of microseconds, has been the subject of much discussionrecently [ 11-18] and is not expected a priori. A simple extrapolation of the known decay behavior of low-n Rydberg states using the conventional n3 scaling law leads to predictions of lifetimes for the highn states that are too short by several orders of magnitude. For example, in N:! the u+ = 0, N + = 2 Rydberg states with n 5 200 are predicted to decay by rotational autoionizationin less than 20 ns, whereas the experiments show that such Rydberg molecules survive for many microseconds (see Section VI).

682

T. P. SOFTLEY, S. R. MACKENZIE, F. MERKT, AND D. ROLLAND

Chupka [Ill, in a landmark paper, pointed out the importance of electric fields in enhancing the stability of the Rydberg molecules. The transition intensity for the initial excitation process canies the molecules into low4 states, which have core-penetrating orbits. These states are unstable with respect to decay processes, because of the energy exchange when the electron is close to the core. However, the effect of the electric field (stray fields of order 20-50 mV/cm are almost unavoidable in most ZEKE experiments) is to mix the low4 states with the degenerate sets of nonpenetrating high-I states (hydrogenic manifolds), and these latter states have no Rydberg electron+ore interaction and hence are long lived. The sharing out of lifetime and transition intensity results in a set of populated states whose lifetime is greater than the zero-order low4 states by a factor of approximately n. The existence of inhomogeneous electric fields, which could for example be created by the presence of high densities of ions near the Rydberg molecules, leads to rnr mixing (or more strictly M J mixing in most systems) and this would lead to further lifetime dilution, by a factor approaching n. Thus in the limit of complete Z and ml mixing, the lifetimes would scale as n5 rather than n3. Recent experiments have demonstrated that the presence of background ions can lead to enhancement of ZEKE signals [7], presumably through lifetime lengthening effects, although other experiments appear to demonstrate the lack of such effects [38]. Levine and co-workers, in their theoretical studies of high Rydberg lifetimes [14], have concentrated more on the importance of intramolecular processes, utilizing a classical trajectory approach. They point to the importance of rotational-electronic energy exchange between the ion core and the Rydberg electron. In this picture the Rydberg electron drifts from one orbital angular momentum state to another, exchanging angular momentum and energy with the rotating core each time it comes into the core region. It is probable that such effects are more important in larger molecules when the rotational energy spacing and Rydberg energy spacing become comparable in magnitude, with the additional requirement for strong coupling (and hence a highly anisotropic ion core). Recently they have shown that the presence of an external field has important effects in modifying this “diffusive motion,” leading to less frequent collisions with the core [14]. They have also adopted a quantum mechanical approach based on diagonalization of a complex Hamiltonian (similar to the method of Bixon and Jortner [13], as described in Section VI1.B) to demonstrate a “trapping effect,” which occurs when the density of coupled bound states exceeds the density of continuum states (see Ref. 18 and Levine, “Separation of Time Scales in the Dynamics of High Molecular Rydberg States,” this volume). Such an effect is more likely to be prominent in molecules than atoms, because of the large number

FROM RYDBERG STATE DYNAMICS TO ION-MOLECULE REACTIONS

683

of interacting Rydberg series. It has been shown that this situation leads to a bifurcation of lifetimes, particularly when the mean level spacing is less than the decay width. The ideas of Chupka and those of Levine are not incompatible. A general framework for discussing Rydberg state lifetimes must embrace both intramolecularand external effects. Two limiting cases may be envisaged for any atom or molecule; the low-external-field regime, in which intramolecular effects dominate the dynamics, and the high-field limit, where the nature of the molecule becomes almost irrelevant and the external effects dominate. In the intermediate regime the intramolecular and external-field effects may show a complex interaction, with the field modifying and perhaps facilitating the intramolecular couplings. Ultimately, as the field is increased, intramolecular couplings can become effectively switched off (see Section VII.C), and lifetimes are determined by the dilution of the zero-field bound-continuum couplings. In this sense the general framework we refer to is analogous to the well-known transition in atoms from the Zeeman effect at low magnetic field to the Paschen-Back effect at high magnetic field. The matter that is not clearly established is whether the transition to the high-field regime occurs under rather similar external conditions for all molecules or whether it is highly dependent on the rotational constant, ion-core anisotropy, or possibly the density of vibrational states. In the context of designing experiments such as the one described in Sections 11-IV, three fundamental questions should be addressed: Are the weak homogeneous fields present in most experiments sufficient to cause the observed lifetime lengthening, and if not, what is the relative importance of homogenous and inhomogeneous field effects? Is there a universal stabilization mechanism valid for the complete range of systems from simple atoms to complex molecules or are intramolecular effects more dominant in some cases than others? Is the stabilization highly variable from one experimental setup to another, and what can be done to control it? In the following sections some new experimental and theoretical results are presented that shed further light upon the answers to these questions.

VI. EXPERIMENTAL MEASUREMENTS OF RYDBERG LIFETIMES Three basic methods for determining Rydberg lifetimes may be envisaged. 1. High-resolution measurement of linewidths of transitions to individual

eigenstates.

684

T. P. SOFILEY, S. R. MACKENZIE, F. MERKT, AND D. ROLLAND

2. Measurement of the survival probabiiity of Rydberg molecules as a function of delay time for pulsed-field ionization. 3. Direct observation of the appearance of decay products in real time.

It should be noted with respect to method 1 that in most Rydberg excitation experiments to date pu1sed lasers have been used even to obtain the highest resolution spectra, and this naturally precludes the measurement of lifetimes longer than -10 ns. We have carried out careful measurements, in the apparatus described in Section 11, of lifetimes as a function of principal quantum number for Rydberg states of nitrogen converging to the N + = 2 , 3 ionic thresholds [24] using methods 2 and 3. These states can in principle undergo rotational autoionization to the continua associated with N + = 0, 1, respectively. The molecules are excited from the J’ = 2, 3 intermediate levels of the a 1qstate (see 3 parallel to the Section 1II.C) in the presence of a small DC field ~ 0 . V/cm, molecular beam direction, which is used to retard the prompt ions relative to the neutral Rydberg states. A pulsed field is applied after a time delay of typically 7 ps to ionize the Rydberg molecules, and all ions are accelerated by this pulse to the detector. The ion time-of-flight profile shows two peaks in general, as shown in Fig. 10, the early peak due to field-ionized Rydbergs and the later peak due to the retarded prompt ions formed immediately on

0.10

.g

0.09

A

0.08

g om

h

--2

3

a!

0.06

0.05

.gJ 0.04

.g .$

0.03

0

d

0.02

4

0.01 0.00 -0.01 27.0

27.5

28.0

Ion timc of flight / ps

28.5

29.0

Figure 10. The Nt ion time-of-flight profile following Rydberg state excitation around n = 150 (N+= 2) in the presence of a 0.3-V/cm retarding field. The dotted line shows the one-color direct ionization signal, which is subtracted from the total signal in the analysis of the results.

FROM RYDBERG STATE DYNAMICS TO ION-MOLECULE REACTIONS 685

excitation. A signal also appears at times intermediate between the field ionization and prompt ionization peaks, that can be ascribed to ions formed by autoionization after a short time delay with respect to excitation. An exponential decay curve due to autoionization is obtained as a shoulder on the prompt ionization peak, and as shown in Fig. 10, the time of flight can be converted into a real time measurement of the time of ionization (method 3). A series of measurements of the time-of-flight profiles at different excitation energies has been carried out [24], from which the ionization dynamics as a function of n can be extracted. Experiments of type 2 have also been carried out in which the pulse delay is varied over the range 1-26 ps and the remnant population detected by PFI. For the np Rydberg states converging to the N + = 2 limit in N2 there are three J levels ( J = 3,2, l), arising from the coupling of N+ and I , and these are optically prepared in approximately equal proportions. Only the J = 1 set can couple to the N + = Onp(J = 1) continuum in the absence of an electric field. For n = 80,. . . ,100 it is found experimentally that roughly one-third of the states decay by fast rotational autoionization with 7 c 500 ns (presumably the J = 1 set), while two-thirds are not observed either by delayed-field ionization or by fast autoionization at all. Note that the total excitation probability is assumed to be constant across the threshold region and is determined from the magnitude of the direct ionization signal above the N + = 2 threshold. For the nonautoionizing J = 2, 3 states there must be some other decay process, presumed to be predissociation, competing on a submicrosecond time scale. Hence virtually none of the populated n = 80,. . . ,100 states survive the 7-ps delay before field ionization. The 0.3V/cm field is apparently having very little stabilizing effect in this range of n. For n = 100,. ..,140 the lifetimes with respect to autoionization gradually increase to = 2ps at n = 140 (a longer tail is observed on the prompt-ion timeof-flight peak) while the fraction of molecules undergoing predissociation on the experimental time scale decreases, thereby increasing the fraction of states that survive to be field ionized. The increased lifetimes are in part due to the DC field inducing the mixing of the short-lived np, J = 1, states with the longer lived high4 states and with the J = 2,3 states. The effects of 1 mixing and J mixing should only be observed when the np Rydberg states are submerged into the hydrogenic high-2 manifolds, predicted to occur at n > 100, and this explains why the increased lifetimes are observed in the range n = 100,. ..,140. The decreased importance of predissociation is also caused by I mixing. Surprisingly the fraction of states undergoing autoionization within 7 p s does not increase, despite the fact that when 1 and J mixing occur, all states should have comparable autoionization lifetimes. It is probable that there is a subtle interplay between predissociation, autoionization, and rnr mixing in this range of n.

686

T. P. SOFTLEY, S. R. MACKENZIE, F. MERKT, AND D. ROLLAND

For n = 160 a decay time of approximately 3 ps is measured for 25% of the Rydberg states and a very long lifetime >30 ps for the remaining 75%. The lifetimes >30 ps cannot be explained purely in terms of a democratic I and J mixing of all states in this range, caused by the homogeneous field. The multichannel quantum defect theory (MQDT) calculations discussed in the next section suggest an average lifetime of -5 ps in a homogeneous field, which is quite close to that observed for the fast-decay component. It therefore appears that other processes make an important contribution to the stabilization for a large fraction of the Rydberg states with n > 150. The inhomogeneous field created by ions produced within the laser excitation volume is likely to be a significant factor [12] in causing m mixing and hence further stabilization.

VII. MQDT CALCULATIONS OF SPECTRA OF AUTOIONIZING RYDBERG STATES The MQDT picture [39, 401 of a Rydberg molecule in an electric field [41, 421 reveals much about the esentid physics of the Rydberg stabilization problem. The space of the electron is divided primarily into three regions. The effects of the external field are only felt in the outermost region III, where the total potential for the electron is a sum of Coulombic and external field terms (in ax.) 1 r

V ( r )= --

+ Fz + - -

where Fz is the homogeneous field term, but further terms may be added for an inhomogeneous field. The effects of ion-core anisotropy are assumed to be negligible in this region. If the field is purely homogeneous, the hydrogenic Stark wave functions (the solutions of the Schrodinger equation in region 111) are best described in a parabolic coordinate system. In the intermediate region 11 the electron experiences forces identical to those in a field-free hydrogen atom; that is, the ion core may be considered as a point charge and the field effects are negligible in comparison to the Coulombic force. The wave function must be a linear combination of field-free hydrogenic solutions to the Schrodinger equation, most conveniently expressed in spherical polar coordinates. In the short-range region, molecule-specific interactions of the Rydberg electron with the ion core occur, and the hydrogenic functions are no longer true eigenfunctions of the system; there is interaction between the long-range channels, including mixing of channels with different ioncore states. It should be stressed that although the Schriidinger equation is

FROM RYDBERG STATE DYNAMICS TO ION-MOLECULE REACTIONS

687

molecule independent in the external regions I1 and Ill, the need to make a smooth join to the wave function in region I leads to a molecule-specific combination of the hydrogenic functions as a valid solution. This may be generally written as

where & is the wave function for the core state with quantum numbers y and is a hydrogenic function for the Rydberg (or free) electron characterized by quantum numbers i. However, as the field and/or effective principal quantum number increases, the importance of couplings in region I decreases, the electron spends a greater fraction of its time in regions I1 and 111, and the total solutions converge toward a molecule-independent hydrogenic solution. In the execution of the MQDT calculations described below, we take a handful of input parameters-the energy-independent quantum defects and transition moments for the molecule or atom of interest-derived from a spectroscopic analysis of a limited energy region, or ab initio calculation, and these can then be used to calculate spectroscopic and dynamical properties over wide energy ranges without further adjustment of parameters. A photoionization spectrum for a given field is simulated by determining the linear combination of hydrogenic functions, as in Eq. (2), as a function of energy and then using it to calculate the transition probability. In order to extract dynamical information regarding the lifetimes 7 of the autoionization features in the spectrum, the simplest procedure is to use the formula +iy

_T -S

5 . 0 1 4 ~lo-'* r/cm-'

(3)

where r is the calculated linewidth [full width at half maximum (FWHM)]. More correctly the lines should be fitted to a Fano line shape to extract the true width. However, both procedures cause difficulties when overlapping complex resonances occur and then a time-dependentcalculation of the decay becomes desirable.

A. Method Employed in the Calculations The method used in these calculations is based on the theoretical developments of Sakimoto [42] and is also an extension of our previous work on the Stark spectrum of spin-orbit autoionized Rydberg states of argon [43] and of the vibrationally autoionized states of hydrogen [44, 451. Only the homoge-

688

T. P. SOFTLEY,S. R. MACKENZIE, F. MERKT, AND D. ROLLAND

neow component of the electric field is included in these calculations; hence MJ-mixing processes are not described. A fundamental assertion is that the open-channel components of the wave function are unchanged by the electric field, and therefore these can still be represented in region 111by the quantum numbers E, ml, and y [ I , m l = i in Eq. (Z)]. The closed channels, however, are mixed by the field and are defined by the quantum numbers y, mi and the parameter p, which is the separation constant (value, 0 .. . , 1) in the solution of the Schriidinger equation in parabolic coordinates; takes INT( Y - m) values at a given effective principal quantum number v and corresponds to the INT(v - m) pairs of values for the parabolic quantum numbers nl and n2. Tunneling effects are ignored in the calculations. The /3 values need to be determined for each field and v value [43].The total photoionization cross section, proportional to the oscillator strength, is given by a summation of partial photoionization cross sections over all the open channels (yo,I , m); only the open channels with non-zero quantum defects need be included.

where dr;s..m,, and dkom are transition moments to closed-channel final states y~plrm”and open-channel final states yJm, respectively:

with A;:,,,,,, representing the admixture of bound-channel $p“rn“ into the continuum wave function y,lm. In Eq. ( 5 ) the Stark scattering matrix x F is related to the K matrix (reactance matrix) via

x F = (1 + iKF)(l - iKF)-’

(6)

The KF matrix is related to the conventional K matrix of MQDT by a frame transformation from parabolic to spherical co-ordinates: the K matrix is then related by a further frame transformation to the quantum defects [43, 451. The first term in Eq. (4)gives a contribution to the photoionization intensity borrowed from the “bound-state” spectrum. The dkomterm represents direct photoionization, and the overall expression allows Fano-type interference between these terms. In Eq. ( 5 ) A is a phase shift in the parabolic rep-

FROM RYDBERG STATE DYNAMICS TO ION-MOLECULE REACTIONS

689

resentation of the closed-channel components and is calculated as described in Ref. [43]. The theory is common to both system (N2and Ar) for which calculations are presented here and the differences occur in the detailed determination of the KF and dF matrices in the two cases.

B. Calculations for Argon

The first system chosen is the autoionizing Rydberg states of argon converging to the 2P1/2 ionization limit. In a previous paper [43] the results of similar calculations for the Rydberg states in the range n = 15, ... ,20 were reported and compared with the single-photon excitation spectrum obtained by the same authors. At low field the spectrum is dominated by transitions to the ns’ and nd’ states, but as the field increases, the np’ states and the high-2 manifolds appear. The agreement obtained between experiment and theory was generally very good in terms of linewidths and positions as a function of applied field. In the present case the aim is to extend the calculations to the high-n Rydberg states n = 80,. ..,100; the extension has been made possible primarily by improvements in efficiency of the computer code. Of particular interest is a comparison with the experimental results of Merkt [46], who obtained the ZEKE-PFI spectrum of Ar in the presence of three different DC fields ( 4 . 1 , 1 and 2 V/cm) with a pulsed-field delay time of 200 ns. At 0.1 V/cm the PFI spectrum shows a constant signal over the range n = 90,. ..,200 and then drops off at lower n due to autoionization. This suggests that the states with 90 < n < 200 have lifetimes 2 200 ns, implying a stability more than 50 times greater than predicted using an n3 scaling law for the optically accessible ns’ series and lo00 times greater than for the more intense nd‘ series. The effect of increasing the electric field was found experimentally to lead to significant shortening in the lifetime of the high-n Rydberg states and hence loss of signal in the delayed PFI. The method used in the calculations follows that explained in detail in Ref. [43]. The input quantum defects pa defined with respect to the Russell Saunders coupling scheme, which is the appropriate short-range basis, are given in Ref. 43. At energies corresponding to u = 100 the total number of open and closed channels in the final KF matrix is 414. Figure 11 shows the calculated photoionization spectra in the region of n = 90 for various fields in the range 0.001-2 V/cm. A step size of cm-’ was used in the calculations. At 0.001 V/cm (Fig. 1la) the 92s (sharp) and 90d (broad) Rydberg states appear strongly, and from the linewidths we ~ At 0.1 infer that the lifetimes are approximately 3.8 and 0 . 2 respectively. V/cm the n = 90 manifold appears superimposed on the tail of the 90d resonance. The manifold levels are sharper than the s and d resonances and

690

T. P. SOnLEY, S. R. MACKENZIE, F. MERKT, AND D. ROLLAND

I

128528.10

I

128528.15

I

128528.20

128528.25

Wavenumber /cm -’ Figure 11. Multichannel quantum defect theory simulations of the photoionization cross section of Ar versus excitation wavenumber, in the presence of a DC field of magnitude ( a ) 0.001 V/cm, (b) 0.1 V/cm, (c ) 0.2 V/cm, ( d ) 0.3 V/cm and (e) 2.0 V/cm.

have an average width corresponding to approximately 25 ns. These lifetimes are much shorter than the experimental observations of Merkt, who found lifetimes > 200 ns [46]. It is noticeablethat as the field increases to 0.2 V/cm (Fig. 1Ic), the 90d resonance is strongly shifted in position and narrows quite dramatically, whereas the 92s resonance is barely changed in intensity and actually becomes slightly broader. It is only at around 0.3 V/cm (Fig. 1Id) that the 92s resonance starts to become absorbed into the high-l manifold. The main reason for this delayed mixing is that the s states can only interact with the manifold via a chain of A1 = 1 coupling through the np and nd Rydberg states. It is only at 0.3 V/cm that the np states become immersed in the manifold. At 2 V/cm (Fig. Ile) the spectrum is dominated by hydrogenic manifold structure and there is strong overlap between many manifolds. Other calculations (not shown) indicate that there is no major qualitative change in average lifetimes between 1 and 5 V/cm, although the resonances become

FROM RYDBERG STATE DYNAMICS TO ION-MOLECULE REACTIONS 691

less isolated at increased field, due to greater overlap of manifolds. A notable feature is that there is quite a wide range of widths predicted at 2 V/cm, if anything more so than at 0.3 V/cm. The main intensity occurs in lines with widths corresponding to lifetimes in the range 3-50 ns. This raises an important question as to whether ZEKE experiments measure the entire population initially excited in the pseudocontinuum or whether only a small fraction is observed that survives the initial decay. These calculations may also illustrate the bifurcation of lifetimes suggested in the model of Remacle and Levine [18] (see also Levine, “Separation of Time Scales in the Dynamics of High Molecular Rydberg States,” this volume). In recent work [13], Bixon and Jortner have used a matrix diagonalization approach to calculate lifetimes for autoionizing Rydberg states in argon for individual manifolds of a given n (where n = 100,. ..,280) in the presence of an electric field. Their method involved the use of a complex matrix of dimension =2n of the effective Hamiltonian H = Ho + eFz - $’, although the off-diagonal matrix elements of ir were omitted, making this method equivalent to earlier calculations on Xe [47]. They concluded that there were two distinguishable field ranges, as defined by the value of the reduced field,

F=

’

(F/V cm- )a5 3.4 x 109[p(mod l)]

(7)

where p is the quantum defect for the initially populated state. In range A (0.7 1 P 1 2), the onset of coupling between the low-l and high4 manifolds occurs, and the states show a bimodal distribution of lifetimes: There is a long-lived component whose transition intensity increases and whose lifetime decreases with increasing F, while there is a short-lived component showing the opposite trends. This range corresponds to Figs. l l b and l l c (F = 0.96, 1.92, respectively). In range B (F 2 3 corresponding to Figs. lld and 1le), there is a democratic sharing of lifetime between all coupled states with the resultant mean lifetime

The sum in the denominator represents the total coupling width of all the low4 states to the continuum (or continua). The predictions of Bixon and Jortner are in semiquantitative agreement with the MQDT results presented here; for example they would predict a mean lifetime of approximately 18 ns for the long-lived states in the region of n = 90 in the high-field limit.

692

T. P. SOFIZEY, S. R. MACKENZIE, F. MERKT, AND D. ROLLAND

The matrix diagonalization method suffers two disadvantages that are not intrinsic to the MQDT calculations presented in this chapter. First, in order to account for the effects of overlap between adjacent manifolds, a very large matrix would need to be diagonalized, particularly in the high-n range, and convergence may be difficult. The second disadvantage comes in the requirement for a knowledge of the matrix elements of I?; the diagonal elements can only be obtained from measurements of experimental linewidths for low-n autoionizing states, while there is no straightforward method to obtain the off-diagonal elements other than by calculation [17]. The zero-field quantum defects are also required to calculate (Ho).For some of the basis states of interest in argon, the diagonal matrix elements of r are unknown experimentally, and educated guesses need to be made. In the MQDT treatment, the overlapping manifolds are automatically included in the calculation because the principal quantum number n is no longer a parameter in the formalism and is effectively replaced by the energy as a continuous variable. A channel wave function can describe the properties of an infinite range of n values for given angular momentum quantum numbers. The contributions to the wave function at a given energy from states of all n values are automatically included. Second, the quantum defects are used in MQDT to characterize not only the line positions but also the bound-continuum interactions, and there is no need to determine interaction widths independently. The interactions corresponding to the off-diagonal matrix elements of I' are included automatically. The disadvantage of the MQDT method for the Stark effect is that fairly large complex matrices ( 4 0 0 x 500) must still be handled and inverted at each point on a very fine energy grid. Although these operations may be more feasible than diagonalizing complex matrices with dimensions of thousands, the method is still very computer intensive. In this work we have demonstrated that the MQDT method can be applied to obtain meaningful results in the high-n region. Although it is particularly interesting to compare our results with those of Bixon and Jortner for argon, and satisfying to find agreement, we are also interested in determining whether there are molecule-specific effects that cannot be determined by simply calculating properties of atomic systems. Of particular interest is to determine the effects of interactions between the many Rydberg series converging on different vibration-rotation states of the ion, which is equivalent to the rotational electronic couplings in the Levine model [14].

C. Calculations for Nitrogen For nitrogen we would ideally like to calculate the excitation spectra of the high-n N+ = 2 , 3 Rydberg states discussed in Section VI in the presence of a 0.3-V/cm field, including as many as possible of the interacting Rydberg series

FROM RYDBERG STATE DYNAMICS TO ION-MOLECULE REACTIONS

693

converging on different ionic rotational thresholds. However, there is a need to compromise between including sufficient rotational channels and calculating for high enough n. We are restricted by the dimensionsof the matrices involved (which on the particular computer system being used means 400 x 500). Two types of calculationshave been perfomed in the first type only N + = 0 (open) and N + = 2 (closed) channels are included, which can be carried out for up to n = 120. In the second type the N + = 4 , 6 channels are also included but the calculationsare carried out at lower n (516).The complicationin the latter case is that the N* = 0 channel only becomes open at an energy corresponding to -n = 80 of the N + = 2 channel, and therefore rotational autoionization cannot occur for n c 80. However, in this work the N + = 0 limit is falsely lowered to below n = 16 of N + = 2. Although this is a physically artificial calculation, it does have validity within the general framework of MQDT, which treats bound states and continua uniformly and assumes an energy independence of the quantum defects. Furthermore, it is expected that the results obtained at a given value of P will mimic those obtained at a corresponding value of P for higher n Rydberg states. The u+ > 0 channels are ignored in these calculations. We assume R-independent quantum defects using the values given in Ref. 48 for thep channels, and in the absence of further information,the calculated values for s and d Rydberg channels belonging to the A 211uand B 2qion-core states from Ref. 49. Redissociation is not included in the calculations,although in principle it could be. The appropriateframe transformationsfor the diatomic molecule case are discussed in detail in Ref. 45, although the integration over vibrational wave functions need not be included for R-independent quantum defects. Figure 12a shows the calculated spectra (0.3 V/cm field) in the region of the 70p Rydberg states with the three J components showing exactly the behavior described in Section VI; that is, two long-lived components and one short-lived component are predicted, with linewidths corresponding to lifetimes of 200 ns for J = 2, 3 and 1 ns for J = 1. In zero field the autoionization lifetimes of the J = 2 , 3 components should be infinite, and therefore the calculated lifetimes are an indication of the extent of J mixing induced by the field. However, as shown in Fig. 126, at n = 80 the effect of the field is apparently already sufficient to mix completely these states such that they have a comparable lifetime. In the experiments, however, there is no evidence for such J mixing occurring in the range n = 80,. ..,100; that is, only one-third of the states autoionize rapidly, suggesting that predissociation is competing on a time scale of a few nanoseconds and is still causing decay of two-thirds of the populated states. At n = 95 in the calculations, only high4 manifold structure is apparent with a mean lifetime of 689 ns. An extrapolation of this lifetime to n = 160 using an n4 scaling law would give a maximum lifetime of 5.0 ps, which is reasonably consistent with the fast

694

T. P. SOFTLEY, S. R. MACKENZIE, F. MERKT, AND D. ROLLAND

r

1

I

I ' J=2

I

I J=3

k (b)n=80

125661.10

125661.12

125661.14

1 25661.16

125661.18

Wavenumber /cm-'

Figure 12. Multichannel quantum defect theory simulations of the photoionization cross section of N2, field = 0.3 V/cm (a) near n = 70 and (b) near n = 80, including N + = 0, 2, M J = 1 channels only, with excitation from J' = 2, M; = 1.

decay observed experimentally for ~ 2 5 % of the states but totally incompatible with the long-lived component (>30 ~ s )It. must be stressed once again that only the effects of homogeneous fields are included in these calculations at the present time. Figures 13a and 13b show a comparison of spectra calculated at n 16 with and without the N + = 4, 6 channels included. At 100 V/cm (F = 0.5) there is considerable mixing between the N + = 2 Rydberg states and the N + = 4 hydrogenic manifold, which is primarily induced by the electric field and does not exist at much lower field. This leads to significant differences from the case where rotational couplings with the higher N + channels are ignored. On the other hand, at 200 V/cm (F = 1.5), where the hydrogenic manifolds completely dominate the spectrum (Fig. 14), the effect of the rotational channel mixing is somewhat less, and the inclusion of the N + = 4, 6 channels leads only to a few additional lines that do not perturb the N + = 2 transitions very strongly, Figure 14 shows a small part of the N + = 2 n = 16 manifold structure, each group of lines being one Stark component split by fine-structure interactions into three or four components.

-

FROM RYDBERG STATE DYNAMICS TO ION-MOLECULE REACTIONS

695

1000 V/cm

(a) N=0.2 only

L (b) N=0.2,4,6

125212.0

125

125216.0

0

12

18.0

Wavenumber /cml

Figure 13. Multichannel quantum defect theory simulations of the photoionization cross section of N2, near n = 16, field = lo00 V/cm; ( a ) including only N + = 0 . 2 and (b)including N + = 0. 2, 4, 6, with excitation from J' = 2, M; = 1 .

2000 V/cm

(a) N=0,2 only

1

d I

L 125225.0

(b)N=0.2,4,6

1

125 30.0

Wavenumber /cm"

Figure 14. Multichannel quantum defect theory simulations of the photoionization cross section of Nz,near n = 16, field = 2000 V/cm: ( a ) including only N + = 0, 2 and (b) including N + = 0, 2, 4, 6, with excitation from J' = 2, M; = 1.

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T. P. SORLEY, S. R. MACKENZIE, F. MERKT, AND D. ROLLAND

The most interesting conclusion at this point is that the mixing between 1 components belonging to a particular rotational channel is so strong at high fields that the coupling with other rotational channels is effectively switched off. The reason for this is that the high-Z states do not undergo rotational channel couplings, and therefore when every state has a large admixture of high4 components, there is very little interchannel coupling. This point reiterates the comments made in Section V concerning the transition to the high-externalfield limit. Analogous behavior has been observed in calculations for HZ[50].

VIII. CONCLUSIONS In Sections 11-IV it is demonstrated that rotationally state-selected ions can be produced in sufficient quantities to study reactions. Product ions for lowenergy collisions (Ecoll values of 0.04-0.24 eV) have only been detected to date in the reaction of H2+ + Ha, but it is likely that this method should be reasonably general. The importance of Rydberg state lifetime enhancements has been discussed in depth in Sections V-VII. The comparison of MQDT calculations with experimental results shows clearly that the inclusion of homogeneous field effects only (1 mixing, no m mixing) is insufficient to explain the observed lifetimes in the experiments, pointing to the importance of inhomogeneous fields. A question raised by the argon calculations is whether the majority of PFI experiments measure only a small fraction of the total excitation cross section to the pseudocontinuum of high Rydbergs, because of the wide range of lifetimes of these states, even in the high-field limit. The N2 calculations show that internal rotational-electronic couplings can be initially enhanced as a homogeneous electric field is increased, but the couplings are effectively switched off at sufficiently high field. In the future further investigation of the effects of electronic-rotational coupling must be carried out to determine under what conditions the transition between the regime dominated by internal couplings and that dominated by external-field effects occurs in various molecular systems.

Acknowledgments We are grateful to the EPSRC for an equipment grant and to the EU Human Capital and Mobility Program (no. CHRX-CT93-0150) and the British Council for their additional support. S. R. M. gratefully acknowledges the EPSRC for his Studentship and F. M. thanks St John’s College, Oxford, for his Research Fellowship.

References 1. K. Miiller-Dethlefs. E. W. Schlag, E. Grant, K. Wang and V. McKoy, Adv. Chem. f h y s . 90, 1 (1995).

FROM RYDBERG STATE DYNAMICS TO ION-MOLECULE REACTIONS 697 2. F. Merkt and T.P. Softley, Inf. Rev. Phys. Chem. 12, 205 (1993). 3. 1. Powis, T.Baer, and C. Y. Ng, Eds.. High Resolution Laser Photoionization and Photoelectron Studies, Wiley, Chichester, 1995. 4. L. Zhu and P. Johnson, J. Chem. Phys. 94, 5769 (1991). 5 . C. C. Arnold, Y. Zhao, T.N. Kitsopoulos, and D. M. Neumark, J. Chem. Phys. 97,6121 ( 1992). 6. 0. Dopfer, G. Reiser, K. Muller-Dethlefs, E. W. Schlag, and S. D. Colson, J. Chem. Phys. 101,974 (1994). 7. M. J. J. Vrakking. I. Fischer, D. M. Villeneuve, and A. Stolow, J. Chem. Phys. 103,4538 (1995). 8. X. Zhang, J. M. Smith, and J. L. Knee, J. Chem. Phys. 100, 2429 (1994). 9. M.N. R. Ashfold, 1. R. Lambert, D. H. Mordaunt, G. P. Morley, and C. M. Western, J. Phys. Chem. 96,2938 (1992). 10. F. Merkt, H. Xu and R. N. Zare, J. Chem. Phys. 104,950 (1996). 11. W. A. Chupka, J. Chem. Phys. 98,4520 (1 993). 12. F. Merkt and R. N. Zare, J. Chem. Phys. 101, 3495 (1994). 13. M. Bixon and J. Jortner, J. Chem. Phys. 103,4431 (1995). 14. E. Rabani, R. D. Levine, A. Muhlpfordt, and U. Even, J. Chem. Phys. 102, 1619 (1995). 15. M.J. J. Vrakking and Y.T. Lee, J. Chem. Phys. 102,8818 (1995). 16. S. T. Pratt, J. Chem. Phys. 98, 9241 (1993). 17. X.Zhang, J . M. Smith, and J. L. Knee, J. Chem. Phys. 99,3133 (1993). 18. F. Remacle and R. D. Levine, J. Chem. Phys., 104, 1399 (1996). 19. T. F. Gallagher, Rydberg Aroms, Cambridge University Press, Cambridge 1994. 20. D. C. Clary, Annu. Rev. Phys. Chem. 41,61 (1990). 21. S. L. Anderson, Adv. Chem. Phys. 82, 177 (1992). 22. P. M. Guyon and C. Alcaraz, Proc. SPIE 1858, 398 (1993). 23. H. K. Park and R. N. Zare, J. Chem. Phys. 99,6537 (1993). 24. F. Merkt, S. R. Mackenzie, and T. P. Softley, J. Chem. Phys. 103,4509 (1995). 25. H. Krause and H.J. Neusser, J. Chem. Phys. 99,6278 (1993). 26. W. Kong, D. Rodgers, and J. W. Hepbum, J. Chern. Phys. 99,8571 (1993). 27. C. R. Mahon, G. R. Janik, and T. F. Gallagher Phys. Rev. A 41, 3746 (1990). 28. F. Merkt, H.H. Fielding, and T. P. Softley, Chem. Phys. Left. 202, 153 (1993). 29. S. R. Mackenzie and T. P. Softley, J. Chem. Phys. 101, 10609 (1994). 30. F. Merkt, S. R. Mackenzie, and T. P. Softley, J. Chem. Phys. 99,4213 (1993). 31. J. Xie and R. N. Zare, J. Chem. Phys. 93,3033 (1991). 32. A. Fujii, T. Ebata, and M. Ito, Chem. Phys. Lett 161, 93 (1989). 33. S. R. Mackenzie, F. Merkt, E. J. Halse, and T. P.Softley, Molec. Phys. 86, 1283 (1995). 34. G. Reiser and K. Muller-Dethlefs, J. Phys. Chem. 96,9 (1992). 35. K.Muller-Dethlefs, M. Sander, and E. W. Schlag, Chem. Phys. Left. 112, 291 (1984). 36. M. Raoult, J. Chem. Phys. 87,4756 (1987). 37. W. A. Chupka, M. E. Russell, and K.Refaey, J. Chem. Phys. 48, 1518 (1968). 38. C. Alt, W. G. Schener, H. L. Selzle, and E.W. Schlag. Chem. Phys. Left.240,457 (1995).

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39. M. I. Seaton, Rep. Prog. Phys. 46, 167 (1983). 40. U. Fano, Phys. Rev. A 2, 353 (1970). 41. D. A, Harmin, Phys. Rev. A 24,2491 (1981). 42. K. Sakimoto, J. Phys. B 22,2727 (1989). 43. H. H. Fielding and T. P. Softley, J. Phys. B 25, 4125 (1992). 44. H. H. Fielding and T. P. Softley, Chem. Phys. Lett. 185, 199 (1991).

45. H. H. Fielding and T. P.Softley, Phys. Rev. A 49, 969 (1994). 46. F. Merkt, J. Chem. Phys. 100, 2623 (1994). 47. W. E. Emst, T. P. Softley, and R. N. Zare, Phys. Rev. A. 37,4172, (1988). 48. K. P. Huber and Ch. Jungen, J. Chem Phys. 92, 850 (1990).

49. W. M. Kosman and S. Wallace, J. Chem. Phys. 82, 1385 (1985). 50. T. P. Softley, A. J. Hudson and R. Watson, J. Chem. Phys. 106, 1041, (1997).

DISCUSSION ON THE REPORT BY T. P. SOFTLEY Chairman: M. S. Child D. M. Neumark: Prof. Softley, what is the kinetic-energy resolution of your ion-molecule experiment? I am particularly concerned about collisions that occur while the ions are being accelerated by the pulsed field. T.P.Softley: At 10 meV collision energy for the Hi + H2 reaction the collision energy resolution is approximately +3 meV. This is due to the translational temperature of the beam and angular divergence primarily. The acceleration pulses are short, confining collisions during acceleration to less than 5%. R. D. Levine: The rate of an ion-molecule reaction can be governed either by the (physical) polarization potential in the reactant's region or by a possible barrier of chemical origin along the reaction coordinate, Similarly, the stereochemistry of ion-molecule reactions is of particular interest because it can be due to either the anisotropy of the potential in the entrance valley [R. D. Levine and R. B. Bernstein, J. Phys. Chem. 92,6954 (1988)J or to the angle dependence of the chemical barrier [E. Rabani, D. M. Charutz, and R. D. Levine, J. Phys. Chern. 95, 10551 (1991)l. Rotational state selection is one way to probe the steric requirement €32 of reactions. Your preliminary result that the reaction rate for Hi iis only weakly dependent on the rotational state will thus attract much attention. This is particularly so since the system is simple enough from

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a quantum chemistry point of view that ab initio potential-energy surfaces of realistic accuracy can be computed. The study of the dynamics of diatom4iatom ion-molecule reactions is only beginning, and there are already a number of puzzling issues. To my mind, not the least of which is the rather high rate of such reactions. In view of much recent work on the N; + H2 reaction and the possible role of the charge transfer channel, N2 + H;, it might be of interest to look at this reaction directly. T. P. Softley: At the present time, we do not say that there are no rotational effects on the cross section of the H; + H2 reaction but that our signal-to-noise levels at the lowest collision energies (where these effects might occur) are not sufficiently good to draw any conclusions (see the current chapter). D. M. Neumark: Didn’t Chupka investigate rotational effects in the H; + H2 reaction many years ago? T. P. Softley: Chupka concluded that the cross section for H; + H2 did not vary by more than 10% over the range of rotational levels studied, although in common with our studies at present, they were not able to increase the accuracy beyond this level [W. A. Chupka, M.E. Russell, and K.Refaey, J. Chem. Phys. 48, 1518 (1968)l. B. Kohler: I have a question for T. Softley: Experimentally, how do you distinguish predissociation of the initially populated high Rydberg states from other decay mechanisms? T. P. Softley: The predissociation yield is determined indirectly. The total number of molecules excited is known, by comparison with above-threshold ionization yields. The number of autoionizing Rydbergs and the number of field-ionized Rydbergs is also known. The fraction of dissociating Rydbergs is determined by subtracting the total ionization signal from the total excitation signal (presumed to be equal to that above threshold).

QUANTUM DEFECT THEORY OF THE DYNAMICS OF

MOLECULAR RYDBERG STATES CH.JUNGEN Laboratoire Aimk Cotton du CNRS Universitk de Paris-Sud Orsay, France

CONTENTS I. Introduction

11. Frame Transformations and Bound States

[II. High Orbital Angular Momentum States IV. States in the Electronic Continuum V. Determination of Quantum Defects from Experiment VI. Conclusion References

I. INTRODUCTION With the advent of the technique of zero-kinetic-energy (ZEKE) photoelectron spectroscopy there has been renewed interest in the physics of highly excited atomic and molecular Rydberg states and, in particular, the breakdown of the Born-Oppenheimer approximation in those states. The recent discussion of the long-lived so-called ZEKE states [ 11, that is, Rydberg states with principal quantum numbers n L 100, has focused on the mixing of Rydberg channels characterized by different values of the core quantum numbers u+, N + (u+ vibration, N + rotation) as well as different values of the electron orbital angular momentum 1. Rydberg states with n 1 100 correspond to Advances in Chemical Physics, Volume 101: Chemical Reactions and Their Control on the Femfosecond Time Scale, XXth Solvny Conference on Chemistry, Edited by Pierre Gaspard, Irene Burghardt. I. Prigogine, and Stuart A. Rice. ISBN 0-471-18048-3 0 1997 John Wiley & Sons, Inc.

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orbits of almost macroscopic dimension (of the order of for n = 100) so that weak external perturbations such as electric stray fields become effective in mixing the unperturbed Rydberg channels [2]. This mixing “dilutes” the short-lived low4 states among the manifold of long-lived high-Z states and hence leads to a lifetime lengthening. On the other hand, a molecular Rydberg electron is also strongly attracted by the positively charged core, and collisions with the latter also lead to mixing of Rydberg channels. It has been surmised [3] that this intramolecular Rydberg dynamics also plays a role in the lifetime lengthening of the ZEKE states. This mechanism is related to the breakdown of the Born-Oppenheimer approximation and as such has been studied for a number of molecules, mainly diatomic systems, in the framework of multichannel quantum defect theory. The purpose of this contribution is to provide a brief review of the work that has been carried out along these lines in the author’s group and where the role of the vibrational and rotational degrees of freedom has specifically been considered. The reader is also referred to a recent review by Lefebvre-Brion [4] on rotationally resolved autoionization of molecular Rydberg states. 11. FRAME TRANSFORMATIONS AND BOUND STATES

A central feature of molecular quantum defect theory is the use of frame transformations. These provide an elegant way of treating the breakdown of the Born-Oppenheimer approximation that occurs systematically once an electron is excited into a high Rydberg state. The realization that this breakdown occurs is quite old. Mulliken [ 5 ] , in the first of his papers on molecular Rydberg states, wrote in 1964 (p. 3 189). “In most discussions on molecular wave functions, the validity of the Born-Oppenheimer approximation is assumed. This approximation is most nearly accurate when the frequencies of motion, which can be gauged by energy level spacings, are much larger for the electronic than for the nuclear motions. In a Rydberg state series, as n increases, the frequencies for the Rydberg electron become smaller and smaller relative to those of nuclear vibration and rotation. This leads to more or less radical changes in coupling relations. . . . In the B.O. approximation, the wave function for the lowestn Rydberg states of a diatomic molecula with a closed-shell core takes the

form

where the operator A in Eq. (1) makes % antisymmetric in all the electrons. As n increases and the inner loops of the Rydberg molecular orbital become

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less and less important relative to the outer loops, the Rydberg electron withdraws more and more from influencing the rotational and vibrational motions of the nuclei and for large n the wavefunction takes the form

Here J / R , , ~ is no longer included within the electronic factor of the B.O. approximation." Mulliken referred to Eqs. (1) and (2) as Rydberg-coupled and Rydberg-uncoupled wave functions, respectively. As far as rotational motion is concerned, the transition between Eqs. (1) and (2) is equivalent to transitions between Hund's coupling cases (a) and (b) (uncoupling of the electron spin) or between cases (b) and (d) (uncoupling of the electron orbital angular momentum I). The corresponding recoupling transformation is implied in Van Vleck's work [6] published in 195 1. The connection with multichannel quantum defect theory was established by Fano [7] in 1970. By combining the concepts of Seaton's atomic quantum defect theory [8] with the frame transformation approach used in electron-molecule scattering [9], he showed how multichannel quantum defect theory (MQDT) can account for rotation4ectron coupling (1 uncoupling) in H 2 . A comprehensive account of the theory of rotation-electron coupling as well as of vibration-electron coupling in diatomic molecules was given by Jungen and Atabek [lo] in 1977. These authors based their application to the ungerude Rydberg states of H2 on ab initio theory, and they showed that MQDT couple with frame transformation approach accounts for nonadiabatic level shifts quite accurately without requiring the knowledge of the usual momentum coupling functions, which are necessary in the customary coupled equations calculations. The key quantity in their approach is a rovibronic reaction matrix of the form

Kv+N+, ,+"+'

=

where K is the body-fixed reaction matrix K"(R) = tan 7rp"(R) that depends on molecular symmetry (electronic angular momentum component A) and nuclear configuration (internuclear distance R). Here, p is the quantum defect in the usual sense. The terms J and M are the total angular momentum of the molecule and its space-fixed component, which are conserved for an isolated molecule in field-free space. The terms )u+N+)denote the uncoupled

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or asymptotic Rydberg channels whereas IRA) denote the coupled or shortrange channels. The transformation between the two limits is embedded in the bra-kets J ’ ( ~ + N +IRA)JMfor which explicit expressions can be found in the relevant papers. Equation (3) is written for a given electronic state of the core and a given value 1 of the electron orbital angular momentum. The paper by Jungen and Atabek [lo] also contains a sketch of the extension of the theory to include additional purely electronic interactions, such as the mixing of different partial waves 1 by the nonspherical field of the molecular core and/or interactions with core-excited Rydberg channels. This was not implemented at the time, however. This gap has been filled only recently in a series of papers by Ross et al. dealing with the gerade Rydberg states of H2 [ll]. Equation (3) is now generalized to take the form

where K$(R) = tan 7rp$(R) is a nondiagonal quantum defect matrix with i or j specifying a given electronic core state and associated Rydberg orbital angular momentum 1. In the application to gerade H2 the manifold of Born-Oppenheimer channels was represented by a 3 x 3 electronic matrix including sog and dug electrons associated with the * ground state of H$ and a po, electron associated with the excited core. All other molecular symmetries were represented by a single matrix element. The connection with ab initio theory here and in Eq. (3) above is the following: The clamped-nuclei reaction matrix K$Z) is determined such that when entered into a multichannel quantum defect calculation for fixed R and A, it yields precisely the corresponding known ab initio clamped-nuclei electronic energies U k ( R ) of the molecule. This approach has been used to calculate 382 observed gerude singlet and triplet excited-state levels of HZ with total angular momentum values (exclusive spin) between N = 0 and N = 5, spanning a range of 23,000 cm-I and associated with 12 electronic states. The mean deviation observed (calculated) is only 5.8 cm-’ . By comparison, the errors of the ab initio input potential-energy curves [12] depend on the electronic state and vary with R, but a reasonable mean value is probably about 2 cm-’. In turn, the nonadiabatic level shifts themselves amount up to several hundreds of wavenumber units.

’

*

xi

QUANTUM DEFECT THEORY OF MOLECULAR RYDBERG STATES

705

111. HIGH ORBITAL ANGULAR MOMENTUM STATES As the orbital angular momentum 1 of the Rydberg electron increases, the centrifugal potential term +&I+ 1)/2r2 becomes more and more effective in keeping the Rydberg electron outside the core region. For a threshold electron (E = 0) the inner turning point for classical motion in a Coulomb field is ro = !1(1+ 1). For 1 = 3 one thus has ro = 6 a.u., which is larger than the core size of a small molecule. On the other hand, any molecule possesses electric multipoles as well as polarization fields that extend beyond the core region. These contribute to the body-frame quantum defects p*(R) of all Rydberg channels, but for 1 1 3 their contribution dominates. This means that the body-frame quantum defects can be evaluated in terms of the ioncore multipole moments and polarizabilities. For example, for a symmetrical molecule (no core dipole moment) the body-frame quantum defect for an f electron interacting with a C diatomic core is given by [13] A 1 p 3 , 3 ( ~ , R=) -(1 + 4 ~ ) a ( R-) 630

Here E is the electron energy in Rydbergs; ;[2all(R) + a l ( R ) ] = a(R) and i[all(R)- a l ( R ) ] are, respectively, the isotropic and anisotropic dipole polarizabilities of the core that are nuclear-coordinate dependent, and Qzz is its quadrupole moment. Expressions analogous to Eq. ( 5 ) for I 2 3 are readily derived. Equation ( 5 ) takes account of the long-range field components proportional to r-k with k up to 4 . For higher accuracy higher multipole moments and hyperpolarizabilities must also be included. In the case of HZ all these quantities are known from ab initio theory and are available for a wide range of R values. High4 Rydberg states have been observed for this molecule for 1 = 3 , 4 , 5 by Fourier transform spectroscopy [14-161 and for 1 = 5,6 by microwave spectroscopy [17-201, and they have also been calculated [14-161 from first principles by multichannel quantum defect theory by using formulas of the type of Eq. ( 5 ) in conjunction with the frame transformation approach of EQ. (3) (i.e., neglecting I mixing). In the calculations terms up to r-6 were taken into account. It has been found that the calculations yield the observed rovibronic Rydberg levels to within about 0.5 cm-' for 1 = 3 and to within about 0.04 cm-' for I = 5, showing that quantum defect theory provides an adequate description of high-1 molecular states just as for the more familiar low4 states. A problem that apparently has not

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yet been tackled and may be of interest for studying the physics of high-Z ZEKE states is the calculation of channel couplings off-diagonal in I based on the long-range field model, followed by use of Eq. (4).

IV. STATES XN THE ELECTRONIC CONTINUUM Some of the earliest applications of MQDT dealt with vibrational and rotational autoionization in H2 [21-251. One concept that emerged from these studies is that of complex resonances [26), which are characterized by a broad resonant distribution of photoionization intensity with an associated rather sharp fine structure. These complex resonances cannot be characterized by a single decay width; they are the typical result of a multichannel situation where several closed and open channels are mutually coupled. The photoionization spectrum of H2 affords a considerable number of such complex resonances. The H2 molecule is a system for which quite recently it has been possible to measure in unprecedented detail state-selected vibrationally and rotationally resolved photoionization cross sections in the presence of autoionization [27-291. The technique employed has been resonantly enhanced multiphoton ionization. The theoretical approach sketched above has been used to calculate these experiments from first principles [30], and it has thus been possible to give a purely theoretical account of a process involving a chemical transformation in a situation where a considerable number of bound levels is embedded in an ensemble of continua that are also coupled to one another. The agreement between experiment and theory is quite good, with regard to both the relative magnitudes of the partial cross sections and the spectral profiles, which are quite different depending on the final vibrational rotational state of the ion. V. DETERMINATION OF QUANTUM DEFECTS FROM EXPERIMENT

The key quantities in the traditional Born-Oppenheimer theory of molecules are the coordinate-dependent electronic energies. They supply the potentials for nuclear motion from which the level fine structure can be predicted. These curves or surfaces need not necessarily be obtained from ab initio theory. The inverse approach is followed in most spectroscopic work in that the potentialenergy surfaces or sections thereof are extracted from experiment. Indeed, the structural information contained in the electronic energies provides the most commonly used interface for the comparison between ab initio theory and experiment. Without this key feature of the theory, molecular physics could never have progressed as it has in the past decades.

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An analogous situation occurs in multichannel quantum defect theory where the coordinate-dependent quantum defect matrices of Eiq. (4) above play a similar role. In a number of instances the inverse procedure has indeed been followed here too. Thus, for Liz, a 2 x 2 quantum defect matrix involving su and du channels has been determined [3 11 from mixed Rydberg series at the equilibrium internuclear distance Re as well as the derivative of each of its elements with respect to R. Similarly, for N2,2 x 2 matrices representing the p andf series and their interactions have been determined for R = Re from the highly resolved vacuum ultraviolet (VUV) absorption spectrum [131. The NO molecule is another example where quantum defect matrices have been extracted from experiment and have been compared directly to corresponding quantities calculated from first principles [32, 331. Work on the Rydberg states near n* = 14 is in progress for the very strongly dipolar molecule CaF. Here it has been possible to extract from the observed fine structure of the super-1-complex the full nondiagonal quantum defect matrices including the partial-wave components 1 = 0,. .., 3 for A = 0,. . . ,3, that is, in all 20 matrix elements [34].

VI. CONCLUSION It appears in conclusion that molecular multichannel quantum defect theory is a method of choice for treating channel couplings in high Rydberg states of molecules. This theory is based on coordinate- and symmetry-dependent quantum defect matrices, which in tum are related to the clamped-nuclei Bom-Oppenheimer potential-energy surfaces or equivalent electron-core scattering phase shifts. Nonadiabatic effects are taken into account through geometric frame transformations,while the customary radial momentum coupling or Coriolis coupling functions are not required. Interference effects between various channel interaction paths are fully taken into account, as is exemplified by the calculations of complex resonances discussed above. We expect that such complex resonances are also important in the physics of very high Rydberg states.This approach can also be extended to account for external electric fields [35] and Rydberg electron wavepacket dynamics [36]. References 1. G . Reiser, W. Habenicht, K. Muller-Dethlefs, and E. W. Schlag, Chem. Phys. Lett. 152, 119 (1988). 2. W. A. Chupka, J. Chem. Phys. 98,4520 (1993); 99,5800 (1993). 3. D. Bahatt, U. Even, and R. D. Levine, J. Chem. Phys. 98, 1744 (1993). 4. H. Lefebvre-Brion, in High Resolution Laser Photoionization and Photoelectron Studies, I. Powis, T. Baer, and C. Y. Ng, F A , Wiley Series in Ion Chemistry and Physics, Wiley, Chichester, 1995. p. 171.

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5. R. S. Mulliken, J. Am. Chem. Soc. 86,3183 (1964); 88, 1849 (1966); 91,4615 (1969). 6. J. H. Van Vleck, Rev. Mod. Phys. 23,213 (1951). 7. U. Fano, Phys. Rev. A 2, 353 (1970). 8. M. J. Seaton, Rep. Prog. Phys. 46, 167 (1983). 9. N. F. Lane, Rev. Mod. Phys. S2,29 (1980). 10. Ch. Jungen and 0. Atabek, J. Chem. Phys. 66,5584 (1977). 11. S. C. Ross and Ch. Jungen, Phys. Rev. A 49, 4353 (1994); 49, 4364 (1994); 50, 4618 (1994). 12. S.Yu and K. Dressler, J. Chem. Phys. 101,7692 (1994). 13. K. P. Huber, Ch. Jungen, K. Yoshino, K. Ito, and G. Stark, J. Chem. Phys. 100, 7957 (1994). 14. Ch. Jungen, I. Dabrowski, G. Herzberg, and D. J. W. Kendall, J. Chem. Phys. 91, 3926 (1989). 15. Ch. Jungen, I. Dabrowski, G. Herzberg, and M. Vervloet, J. Chem. Phys. 93,2289 (1990). 16. Ch. Jungen, I. Dabrowski, G. Herzberg, and M. Vervloet, J. Mol. Spectrosc. 153,ll (1992). 17. W. G. Sturms, P. E. Sobol, and S. R. Lundeen, Phys. Rev. Lerr. 54,792 (1985). 18. W.G. Sturms, E. A. Hessels, and S. R. Lundeen, Phys. Rev. Lett. 57, 1863 (1986). 19. W. G. Sturms, E. A. Hessels, P.W. Arcuni, and S. R. Lundeen, Phys. Rev. Lett. 61,2320 (1988).

20. W. G. Sturms, E. A. Hessels, P. W. Arcuni, and S. R. Lundeen. Phys. Rev. A 38, 135 (1988). 21. G. Herzberg and Ch. Jungen, J. Mol. Specrmsc. 41,425 (1972). 22. Ch. Jungen and Dan Dill. J. Chem. Phys. 73, 3338 (1980). 23. M. Raoult and Ch. Jungen, J. Chem. Phys. 74, 3388 (1981). 24. N. Y. Du and C. H. Greene, J. Chem. Phys. 85,5430 (1986). 25. J. A. Stephens and C. H. Greene, J. Chem. Phys. 100,7135 (1994). 26. Ch. Jungen and M. Raoult, Faraday Discuss. Chem. Soc. 71,253 (1981). 27. M.A. OHalloran, F! M. Dehmer, F. S. Tomkins, S. T. Pratt, and J. L. Dehmer, J. Chem. Phys. 89, 75 (1988). 28. M. A. OHalloran, S. T. Pratt, E S. Tomkins, J. L. Dehmer, and P. M. Dehmer, Chem. Phys. Lerr. 146,291 (1988). 29. J. L. Dehmer, P. M. Dehmer, S. T. Pratt, F. S. Tomkins. and M.A. O’Halloran, J. Chem. Phys. 90,6243 (1989). 30. Ch. Jungen, S. T. Pratt, and S. C. Ross, J. Phys. Chem. 99, 1700 (1995). 31. A. L. Roche and Ch. Jungen, J . Chem. Phys. 98,3637 (1993). 32. S. Fredin, D. Gauyacq, M. Horani, Ch. Jungen, G. Lefivre, and F. Masnou-Seeuws, Mol. Phys. 60, 825 (1987). 33. M. Raoult, J. Chem. Phys. 87,4736 (1987). 34. Ch. Jungen, N. A. Harris, and R. W. Field. (in press). 35. H. H. Fielding and T. P. Softley, Phys. Rev. A 49, 969 (1994). 36. H. H. Fielding, J. Phys. B: At. Mol. Opt. Phys. 27, 5883 (1994).

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DISCUSSION ON THE REPORT BY CH. JUNGEN Chairman: M. S. Child U. Even 1. Prof. Jungen, at what distances did you calculate these quantum defect matrix elements? 2. How sensitive is the calculation to the division of space into the three regions?

Ch. Jungen 1. I should have mentioned that in this example only one vibrational channel is included in the calculations, that is, only rotation-electron coupling is taken into account. Vibration-electron coupling will be discussed in the second part of my talk. 2. The switch-over point does not have to be specified since this is a “sudden approximation.’’All that matters is that a value r, exists for which (i) the field due to the core has vanished except for the Coulomb component and (ii) the Born-Oppenheimer approximation is still valid.

SUBPICOSECOND STUDY OF BUBBLE FORMATION UPON RYDBERG STATE EXCITATION IN CONDENSED RARE GASES M.-T.PORTELLA-OBERLI, C.JEANNIN, and M. CHERGUI* Znstitut de Physique Expirimentale Universite‘ de Lausanne Lausanne-Dorigny, Switzerland

The fate of Rydberg states in the condensed phase is still a matter of debate from both a spectroscopic and a dynamical point of view. As far as dynamics is concerned, it is now well established that following absorption of a photon by low-n Rydberg states, a “bubble” is formed around the excited atom or molecule due to the repulsion of the surrounding closed-shell atoms by the Rydberg electron [ 1, 21. In conventional spectroscopy, this is inferred from the strong blue shifts of absorption bands and the strong absorption-emission Stokes shifts of several hundreds of millielectron-volts, which have been observed for pure and doped rare-gas solids or liquids [l-31. Recently, molecular dynamics (MD)simulations have confirmed this picture [4, 51. The aim of this work is (a) to determine the time scales for bubble fonnation and (b) to investigate the dynamics of the cage relaxation process and if any, in particular, look for coherences of the cage vibrations and their disappearance. It has been suggested that bubble formation upon Rydberg state excitation corresponds to an increase of the cage radius by up to 20% following photoexcitation [3-51, and it would be of interest to investigate the dynamics response to the crystal driven out of equilibrium by an ultrashort laser pulse. *Communication presented by M. Chergui Advances in Chemical Physics, Volume 101: Chemical Reactions and Their Control on the Femrosecond Time Scale, XXth Solvay Conference on Chemistry, Edited by Pierre Gaspard, Irene Burghardt, I. Prigogine, and Stuart A. Rice. ISBN 0-471-18048-3 0 1997 John Wiley & Sons, Inc.

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M.-T. PORTELLA-OBERLI, C. JEANNIN, AND M. CHERGUI

The system we have singled out for this study is NO in Ar matrices. The reasons are as follows: (a) The NO molecule has low-lying Rydberg states due to its low ionization potential. This is an advantage over rare-gas atoms, which need vacuum ultraviolet (VUV) photons for excitation [11. (b) The spectroscopy of its Rydberg states in condensed rare gases is now well established [3,6]. This is an advantage over metal atoms (in particular alkali atoms), whose spectroscopy in the condensed phase is still a matter of debate [7]. (c) It is, to our knowledge, the sole molecule to exhibit a fluorescencethat has also been well characterized by time- and energy-resolved studies [3, 81. This fluorescence stems from the lowest lying A (u = 0) state (vibrationless), and it turns out to be of great use for probing the time evolution of the Rydberg state, as we will see later.

* c'

In order to carry out the study some prerequisites are necessary: (i) Observables need to be established that would allow us to probe the cage relaxation process in real time. For that matter a preliminary nanosecond experiment was carried out that consisted in pumping the A(v = 0) level by a first laser and then probing transitions from this level to higher lying Rydberg states by means of a second tunable VIS-XR laser after a time delay of -20 ns. The detected signal is the depletion of the A(u = 0) fluorescence. The results of this study are published elsewhere [9] and show several Rydberg-Rydberg transitions from the A(u = 0) level around 1.1,0.8, and 0.6 pm,which can be used to probe the dynamics going on the A(u = 0) state by transient absorption or fluorescence depletion. Of particular interest are the bands near 0.8 pm which fall close to the fundamental of our Ti-Sa laser. It should be stressed that these transitions correspond to the relaxed configuration of the cage around the A(u = 0) state. (ii) A scheme had to be found to pump the A(v = 0) level that has its maximum absorption at 6.33 eV in Ar matrices (FWHM = 100 meV) and probe its evolution in time on one of the Rydberg-Rydberg transition bands. This was done by taking the fundamental (around 800 nm) of the Nd-Y1F pumped Ti-Sa regenerative amplifier, tripling it and mixing the fundamental and the frequency tripled pulses in a &barium borate (BBO) crystal (100 pm thick) to generate the hard UV pump beam around 200 nm. Part of the fundamental beam was used as probe beam after passing through a delay line. The zero delay between pump and probe pulses and their cross-correlation were done by crossing the pump and probe beams in a second BBO crystal

BUBBLE FORMATION IN CONDENSED RARE GASES

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in place of the sample and detecting the difference frequency signal around 266 nm. The samples were prepared as described in Ref. 9. The experiment therefore consists in probing depletion of A(u = 0) fluorescence as a function of delay time between the pump and the probe pulses. The difference between the undepleted fluorescence minus the depleted fluorescence is plotted in Fig. 1 as a function of pumpprobe delays for excitation of the A(u = 0) absorption band at 195 nm. The cross-correlation of the pump and probe beams is also shown in Fig. 1 and shows a UV pulse width of -200 fs. It is found that the signal rises with the cross-correlation of pumpprobe pulses and then shows a decay on a time scale of a few picoseconds followed by stabilization at a level some 4040% lower than the maximum. Pursuing the pumpprobe scan up to -120 ps does not yield any additional structures, and the level reached after -2-5 ps (depending on hpump) remains constant throughout. It is tempting to interpret the trace in Fig. 1 as a loss of population in the first few picoseconds following excitation, followed by stabilization of the population. However, in steady-state measurements with synchrotron radiation, it was clearly established that excitation of the A(u = 0) Rydberg state of NO leads first to a cage relaxation followed by a nonradiative relaxation to near-resonant valence levels on a nanosecond time scale, which can compete with the nanosecond time scale of radiative decay to the ground state [lo, 111. Rather, we believe that our signals reflect the evolution in time of an absorption coefficient between the A state and the upper state (or states) to which the probe wavelength is tuned. The point that remains to be clarified is: Does the evolution of the absorption coefficient for the probe transition reflect that of the cage? Recently, Jortner and co-workers [4, 51 carried out simulations on the bubble formation in Arn (n = 12,. ..,200) clusters following Rydberg excitation of a Xe impurity to the 6s state. The interesting feature in relation with our system is that NO-Ar and Xe-Ar have virtually the same difference potential between their ground and first excited Rydberg state [12], and therefore the forces involved therein should be very similar. According to these MD simulations for large clusters (n = 146) at 10 K, following excitation at the band maximum of the Xe (6s) absorption, there is a sudden expansion of the cage with an increase by some 10% of the cage radius, in about 200 fs after the pump pulse. This is followed by an oscillation of the cage for some 2-2.5 ps at a period of 400-500 fs around the radius of the expanded cage. A further expansion of the cage takes place on a long time scale between 2 and 5 ps where a more complex oscillatory pattern is exhibited. For times larger than 5 ps, the cage has reached its final equilibrium configuration, which corresponds to an increase by -20% of the ground-state cage radius. The time

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M.-T. PORTELLA-OBERLI, C. JEANNIN, AND M. CHERGUI

1.0 : 0.8

.-ro

-

6P

0.6 0.4

g 0.2 v1

u

0.0

-:-0.2 -0.4 h

v

-0.6

B 8u1 -0.8

5

-1.0

-1.2 -10000

-5000

0

5000

lo000

15000

20000

25000

Time delay (fs)

Figure 1. Pumpprobe signal for the A(u = 0) Rydberg level of NO in Ar matrices at 4 K. The plotted signal is the fluorescence in the presence of the pump only minus the fluorescence in the presence of pump and probe pulses, as a function of the time delay between them: pump 195 nm, probe 784 nm.The upper curve is the cross-correlation of the pump and probe pulses.

scales for complete cage relaxation inferred from these simulations are very close to the ones obtained from our pumpprobe experiments, and the latter reflect the structural modifications of the cage. However, the trajectories obtained by Jortner and co-workers [4,51 could not explain our pump-probe scans as it would mean that the signal stabilizes after the first 200-400 fs and changes again only after 2 to 5 ps. Our results would be more in line with the adiabatic bubble expansion suggested for free electrons in liquid He [13]. A complete characterization of the bubble expansion mechanism is underway, including other media such as solid Ne and solid hydrogen, and molecular dynamics simulations for the case of NO in Ar matrices are being carried out. Finally, a more extensive account of the methodology of this experiment can be found in ref. 14.

References E. E. Koch, Electronic Excitations in Condensed Rare Gases, Springer Tracts in Modem Physics, Springer, Berlin, 1985. 2. I. Ya. Fugol, Adv. Phys. 27, 1 (1978). 3. M. Chergui, N. Schwentner, and V. Chandrasekharan, J. Chem. Phys. 89, 1277 (1988). I. N. Schwentner, J. Jortner, and

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4. D. Scharf, J. Jortner, and U. Landman, J. Chem. Phys. 88,4273 (1988). 5. A. Goldberg and J. Jortner, in Femtochemistry, Physics and Chemistry of Ultrafast Processes in Molecular Systems, M. Chergui, Ed., World Scientific, Singapore, 1996, p. 15. 6. M. Chergui, N. Schwentner, and W. Bohmer, J. Chem. Phys. 85, 2472 (1986). 7. N. Schwentner and M. Chergui, J. Chem. Phys. 85, 3458 (1986).

8. F. Vigliotti, G . Zerza, and M. Chergui, in Femtochemistry, Physics and Chemistry of Ultrafast Processes in Molecular Systems, M. Chergui, Ed.,World Scientific, Singapore, 1996, p. 654. 9. G. Zerza, F. Vigliotti, A. Sassara, M. Chergui, and V. Stepanenko, Chem. Phys. Lett., 256, 63 (1996). 10. M. Chergui, R. Schriever, and N. Schwentner, J. Chem. Phys. 89, 7083 (1988). 11. M. Chergui and N. Schwentner, J. Chem. Phys. 91, 5993 (1989). 12. R. Reinginger, private communication; 1996 A. Kohler, Dissertation, Hamburg, 1986. 13. M. Rosenblit and J. Jortner, Phys. Rev. Lett. 75,4079 (1995). 14. M.-T. Portella-Oberli. C. Jeannin and M. Chergui, Chem. Phys. Lett. 259,475 (1996).

DISCUSSION ON THE COMMUNICATION BY M. CHERGUI Chairmun: M. S. Child

L. Woste: Prof. Chergui, the bubble created by your Rydberg electron is very exciting. It should have a remarkable dimension. Do you think that your NO molecule can rotate inside the bubble, just to give more evidence to that bubble? M. Chergui: Indeed, we estimated the increase of the cage radius to be of the order of 5-10% [see Chergui et al., J. Chem. Phys. 89, 1277 (1988)], as compared to the ground state. These are dramatic local modifications comparable to those of large-scale character that could be induced by a hydrostatic pressure. To answer your question, we have carried out preliminary polarization measurements in which the polarization of the Rydberg fluorescence was seen to remain unchanged as compared to that of the pump laser. This would suggest that the molecule does not rotate inside the bubble. K. Yamanouchi: Recently, we investigated the interatomic potential V R , ~ ( Rof ) the Rydberg states of a HgNe van der Waals dimer by optical-optical double-resonance spectroscopy. It was demonstrated that VRyd(R) sensitively varies as a function of the principal quantum number n [ J . Chem. Phys., 98, 2675 (1993); ibid., 101, 7290 (1995); ibid., 102, 1129 (1995)], and in the lowest Rydberg states of Hg(7 3S~)Neand Hg(7 'So)Ne, the interatomic potentials exhibit a distinct barrier at around R 4 A. The existence of the barrier was interpreted in terms of a repulsive interaction caused by the 7s Rydberg

-

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M.-T.PORTELLA-OBERLI, C. JEANNIN, AND M. CHERGUI

electron and the attached rare-gas atom on the basis of the quantum defect orbital approach. As n increases, it was found that the barrier disappears in the Franck-Condon region and VRyd(R) converges rapidly to the interatomic potential of the ion core, Vion(R), of HgNe+. We also determined the ion-core potential Vion(R) as a convergence limit of VRyd(R). By assuming VRyd(R) to be represented by a simple sum of the ioncore potential and the repulsive exchange interaction, Vex(K), that is, VR~~(R = )Vion(R) + Vex(R), the repulsive contribution Vex(R) for the lowest Rydberg state (n = 7) was extracted. The shape of Vex(R), which has a barrier at K - 3 A,was very similar to that of the probability distribution of the Rydberg electron for n = 7 evaluated using its quantum defect orbital wave function. Therefore, it was concluded that the barriers in the interatomic potentials for Hg(7 3S1)Ne and Hg(7 'So)Ne are caused by the exchange repulsive interaction between the 7s Rydberg electron and the rare-gas atom. The bubble of the Rydberg electron of NO in the rare-gas matrix observed by Dr. Chergui can be interpreted by a similar model, which we proposed for the Rydberg states of a free HgNe van der Waals molecule. The driving force for the structural relaxation in the cage of rare-gas atoms could originate from the repulsive exchange interaction between the Rydberg electron of the NO moiety and the surrounding rare-gas atoms. M. Chergui: This goes along with our interpretation. The A *C+ state has n = 3 and a quantum defect of -1.1; that is, it shields the ionic core quite efficiently and, in the ground-state configuration of the molecule-matrix system, the Vion(R) you mentioned should be negligible. This does not mean that at much larger NO-Ar distances a weak van der Waals attraction does not appear for the A state and indeed it has been observed in molecular beam studies [see Quaid et al., Chem. Phys. Lerf. 227,54 (1994); Tsuji et al., 1.Chern. Phys. 100,5441 (199411. Concerning higher Rydberg states of NO in rare-gas matrices which we probe from the A(v = 0) level [Zerza et al., Chem. Phys. Left., 256, 63 (1996)], we also invoke a balance between Vion(R) and Vex(R) for increasing n in the interpretation of our results. B. A. Hess: My naive view of Rydberg states in the condensed phase was that they are simply quenched. Do I have to change this picture on account of your experiments? M. Chergui: Yes, in the past the first experiments on the absorption of Rydberg states in condensed rare gases showed that either they disappeared from the spectrum or only the lowest ones, bearing

BUBBLE FORMATION IN CONDENSED RARE GASES

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a parentage with the atomic or molecular states, could be observed while the higher ones were interpreted as Wannier excitons (see, e.g., Schwentner, Koch, and Jortner, in Electronic Excitations in Condensed Rare Gases, Springer Tracts in Modem Physics, Berlin, 1985). More recently, we showed that higher states of NO in rare-gas matrices still bear a molecular character [Chergui et al., J. Chem. Phys. 85, 2472 (1986); Zerza et al., Chem. Phys. Lett., 256, 63 (1996)l. Now concerning the fluorescence, while rare-gas atoms in rare-gas matrices do exhibit fluorescence (see the above reference of Schwentner et a].), NO is, to our knowledge, the sole molecule to exhibit Rydberg fluorescence in the condensed phase. That and the fact that NO has low-lying Rydberg states are the reasons we singled it out as our model system in this study.

T. Kobayashi 1. Prof. Chergui, how about doing experiments on the polarization dependence? 2. If the delay time is made very long, an oscillatory structure may be observed in case the sensitivity is high enough. M. Chergui 1. As mentioned in my reply to Prof. Woste (see above), such experiments are underway. 2. We have scanned up to -120 ps and seen no oscillatory structure whatsoever.

M. Herman: Do you see any evidence of NO dimers? Could they possibly interfere? M. Chergui: Dimers absorb at -207 nm in rare-gas matrices and excitation of this band does not yield any fluorescence [Chergui et al., Chem. Phys. Lett. 201,187 (1993)l. Furthermore, our detection is based on the fact that we record the depletion of the fluorescence of one of the A(0, u”) bands due to NO monomers. There is therefore no possibility that NO dimers could interfere with our measurements. A. H. Zewail: Prof. Chergui’s observations are very interesting. As discussed in this conference, coherent wavepacket motion has been observed in condensed phases in many laboratories. In Prof. Chergui’s experiment it is now possible to study the time scale for “bubble” formation. Molecular dynamics should tell us the nature of forces which maintain any coherence in such systems.

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L. Woste: A speculative remark Could the Rydberg molecule even form a bubble in a gas cell (scattering environment)? M. Chergui: “Bubble” formation is a consequence of the change in molecule-environment interactions. If the density of the gas is such that you probe, from the ground state, a distribution of distances for which this interaction is repulsive in the excited state, then you will form a “bubble.” The relevant regime of densities need not be very high since Rydberg states are extremely sensitive to the environment [for the case of NO, see Miladi et al., J. MoZ. Spect. 55,81 (1975)and 69,260 (1978); Morikawa et al., J. Chem. Phys. 89,2729 (1988); and Chergui et al., Chem. Phys. Let?. 216, 34 (1993)l. W. E. Schlag: In the future we will find these long lived states in other media as well, I think. Bound states on surfaces may become one such example.

Advances In Chemical Physics, Volume101 Edited by I. Prigogine, Stuart A. Rice Copyright © 1997 by John Wiley & Sons, Inc.

GENERAL DISCUSSION ON MOLECULAR RYDBERG STATES

AND ZEKE SPECTROSCOPY PART 11 Chairman: M. S. Child

R. D. Levine: Prof. Jungen, your results for the low Rydberg states of molecules with exceptionally high dipole moment prompt me to ask if you can extend such multichannel quantum defect theory (MQDT) computations to the region of high n’s. The motivation is that the coupling of the zero-order inverse Bom-oppenheimer states is due to the electrical anisotropy of the core. With a large dipole moment this coupling will be quite strong and, because of the rather long range of the electron-dipole potential, can be quite effective. Ch. Jungen 1. What I would do is the following: (a) For low 1 take the quantum defect matrix elements as they result from the fittings. (b) For high I evaluate them in elliptic coordinates assuming no penetration (the effects of the dipole field are then fully included); in this way a full calculation with an arbitrary number of 1 components can be carried out. 2. Notice that in our calculation we have both closed and open ionization channels. This means that the quantum defect/frame transformation approach appears to work very well both below and above the “critical region” for which n - 100,. . . ,1OOO.I would find it surprising if the approach failed in that region.

W. H. Miller: I believe that the reason the multichannel quantum defect theory (MCQDT) works well is that it assumes the ordinary Bom-oppenheimer approximation (i.e., that the electron follows the molecular vibrational and rotational motion adiabatically) in the region close to the molecule, but not so in the region far from the molecule (where the electron moves more slowly than molecular vibration and rotation). The “frame transformation” provides the transition between 719

720

GENERAL DISCUSSION

the body-fixed basis function in the interior region and the space-fixed basis functions in the exterior region.

Ch. Jungen: All I can say at this point is that the quantum d$fect/frame transformation approach appears to work for CaF around n = 14. We have chosen CaF, which is so highly polar, in order to ascertain this, and this is also the reason why we have made ab initio calculations in order to compare experiment and theory. I suppose that the dipole field is averaged out by the rotational motion, and thus one can get away with the customary frame transformation approach. L. S. Cederbaum: Prof. Jungen, you mentioned that MCQDT takes into account nonadiabatic effects. I would like to point out that this approach only considers those nonadiabatic effects that arise due to the motion of the Rydberg electron. The nonadiabatic effects in the ion core are not considered. These effects can often be substantial. The MCQDT could probably be extended to include long-range potentials other than Coulomb, for instance, dipole potentials.

Ch. Jungen: You are quite right. The Renner-Teller effect in the water ion is an example of this kind. The strong vibronic coupling in the core leads to nonadiabatic effects in the Rydberg states of the neutral species, which are of course not accounted for by the coordinate dependence of the quantum defect and have to be taken into account separately. With regard to your second remark I would like to say this: The calculations I showed for CaF and BaF Rydberg states are examples where the core field is a Coulomb field but where a very strong dipole field is superimposed. Here we have used the generalized version of quantum defect theory, which takes account of this modified long-range field.

B. A. Hess: Prof. Jungen, in your talk you emphasized that you don’t have to calculate matrix elements of a/aQ or Coriolis coupling. My impression is that this is due to your most appropriate choice of a diabatic basis, which is generally what ab initio quantum chemists do when they want to avoid singularities in the adiabatic basis. On the other hand, the absence of explicit Coriolis coupling matrix elements is due to the transformation to a space-fixed coordinate system.

RYDBERG STATES AND ZEKE SPECTROSCOPY I1

72 1

Ch. Jungen: The vibronic coupling is included through the R dependence of the diagonal and off-diagonal quantum defect matrix elements. The effective principal quantum number, or more precisely the quantum defect, gives a handle on the electronic wave function. The variation with R then contains the information concerning the derivative with respect to R of the electronic wave function.

T.P. Softley: I would like to ask Prof. Jungen if there is any experimental evidence for the need to include singlet-triplet interaction in a quantum defect description of H2? Ch. Jungen: Oka has recently obtained Fourier transform infrared (FTIR) emission spectra of the 5g4f transition in H2. These spectra are currently being analyzed. They show singlet-triplet splittings of the order of 0.04 cm-', which thus are comparable in magnitude to the hyperfine splittings of Hi. (This had actually already been observed by Jungen, Dabrowski, Herzberg and Vervloet [J. Chem. Phys. 93,2289 (1990)l.) The new aspect in Oka's work is that satellite transitions are observed that demonstrate that a recoupling of angular momenta occurs when the singlet-triplet splitting becomes very small.

R. W. Field: I'd like to make a comment and ask two questions. My comment is the following. The internuclear distance (R) dependence of the quantum defect ( p ) expresses one of the most important mechanisms for the exchange of energy between the Rydberg electron and the vibrating nuclei. In a sense, the quantum defect function p ( R ) is a generalization of the Born-oppenheimer potential-energy curves. I am concerned with convenient experimental methods for determining d p / d R . There appear to be at least three ways to determine this crucial coupling constant, d p / d R . The first involves analysis of Au = -An* = 1 perturbations that occur at relatively low n* when the Rydberg orbit period is equal to the vibrational period. For example, for we = 500 * * cm-', the n , u = 0 - n - 1, u = 1 perturbation occurs at n* = 7.6. If the quantum defect is expanded about R:, the equilibrium internuclear distance of the ion core,

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GENERAL DISCUSSION

the nonzero part of the perturbation matrix element (eu)p(R)le’u’)= (e(dp/dRle’)(ul(R - R:)(u f 1) provides an experimental value of d p / d R . A second way of sampling d p / d R is through the rate of vibrational autoionization. The In*, u+ = 1) state is autoionized by the ionization continuum of u+ = 0.For 0: = 500 cm-‘, all u+ = 1 levels with n* 2 14.8 lie above the u+ = 0 ionization limit. Once again, the (ul(R - R:)lu f 1) vibrational matrix element picks out the d p / d R coupling constant. A measurement of the vibrational autoionization rate gives an independent measure of d p / d R . The third method seems very different from the first two. However, it most elegantly illustrates how an entire Rydberg series forms a single structural unit. Suppose the potential curves for the ion, V+(R),and for a low-n* member of a Rydberg series, V,,(R), are both known. Then the R dependence of the vertical ionization energy determines d p / d R at R:,

p ( R ) = n - 4 1 f 2 [ V + ( R) Vn*(R)]-’/’ where d p / d R at R: can be evaluated directly or expressed as a power series in w t -we,,* and R t - Re,,*. This last method is ideally suited for quantum chemical calculations, because the problem of diffuse orbitals is absent for the ion core and minimized for a low-n* Rydberg state. (It also implies useful n * scaling laws for u: - w ~ , and ~ + - Re,n* at the low-n* end of a series where the Born-oppenheimer approximation is sufficiently well obeyed to permit construction of potential-energy curves.) At last, here are the two questions: 1. Have there been examples where the intrachannel d p / d R has been independently determined by the perturbation matrix element, autoionization rate, and V + ( R )- V,,(R) methods? 2. Do the values of d p / d R determined by the various methods typically agree? If not, which method do you believe is most reliable?

Ch. Jungen: Yes, I would agree with Prof. Field that autoionization widths in the continuum, spectral perturbations at high n, and the

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distortion of the vibrational frequency at low n with respect to that of the ion in principle provide the same information on vibration-electron coupling. This has been demonstrated in the case of H2.

B. Kohler: My question to T. Softley ties in to one of the major themes of the meeting, namely coherence. In your presentation you

briefly mentioned that it may be important to consider an initially coherent superposition of states in the preparation step of experiments on highly excited Rydberg states. Several groups have now prepared coherent electronic wavepackets using picosecond (and shorter) pulses. Would this kind of an initial state be useful for any of the classes of pulsed-field ionization experiments that you have described?

T.P. Softley: There is little doubt that in most ZEKE experiments using nanosecond lasers the Rydberg level structure is so dense that a coherent superposition of levels is populated initially, and the correct description of the dynamics should be a time-dependent one. It is possible that some control over the dynamics could be achieved using some of the methods described earlier in the conference, for example, simultaneous excitation through three-photon and one-photon transitions, using third-harmonic generation. The MQDT calculations we have performed yield a solution to the time-independent Schriidinger equation; in principle the time-dependent information can be obtained by Fourier transformation, possibly using the vibrogram method described by Gaspard. M. S. Child: I would like to ask to T. Softley what limitations apply to vibrational states of H2+ that can be prepared by the mass-analyzed threshold ionization (MATI) technique. In particular it would be interesting to known whether the states observed in Carrington’s H3+predissociation excitation experiment [ 11 could be more selectively prepared by the reaction H;(u)

+ H2 -+H$(u’) + H

with known initial u values. 1. A. Carrington and R. A. Kennedy, J. Chern. Phys. 81, 91 (1984).

T. P. Softley: It has been demonstrated by Hepburn et al. that H; can be formed in u = 13 by a pulsed-field ionization technique from a

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GENERAL DISCUSSION

low vibrational level of the E, F state of Hz.It would be feasible for us to study reactions of Hi (v = 13) in our experiments.

S. R. Jain: I have two questions for Prof. Levine:

1. With respect to the debate on Born-Oppenheimer vs. inverse Born-Oppenheimer regimes, we know that adiabaticity leading to Born-Oppenheimer description is associated with a geometric phase. Similarly, it is not difficult to envisage a geometric phase with an inverse Born-Oppenheimer description. As we have seen in the talk of Prof. Levine, in situations where there are time-varying fields, does it not give us an opportunity to settle (to some extent) this debate by observing (or not observing) this phase? 2. What guarantees the area-preserving nature of your classical map PI?

1. E. Rabani, R. D. Levine, and U. Even, Bet Bunsenges. Phys. Chern. 99, 310 (1995).

R. D. Levine

1. The discussion of the geometric phase requires that the zero-order basis be a good approximation. If there is mixing, then the concept of a phase factor is replaced by a unitary matrix. Indeed, one can think of the frame transformation of MQDT as just such a matrix. If there is no coupling, then the geometric phase is, in essence, what gives rise to the quantum defect of Rydberg states. 2. As to your other question, a generating function for our map is given by

where k is the index of the orbit and E is the energy of the Rydberg state, E = - 1/2n2. The map is given by

W. H. Miller: In treating electronically nonadiabatic processes one often introduces (usually on physical grounds) a diabatic model, which has a nondiagonal electronic potential matrix, and then neglects any remaining derivative coupling. The total (vibronic) wave function is

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then a multi-(electronic) state expansion, and no “geometric phase” needs to ever be introduced (for it is implicitly there!). The geometric phase could arise if one diagonalized the diabatic Hamiltonian, that is, going to the adiabatic representation.

B. A. Hess: The notion of a geometric phase generally requires an adiabatic situation, where an adiabatic connection and adiabatic transport can be defined. In the presence of many nearby avoided crossings, in a highly nonadiabatic situation (as in the case of the inverse Bom-oppenheimer regime), the notion of a geometric phase is ill defined. S. R. Jain: After the initial work of Berry, it is now well-known that cyclicity or adiabaticity is not necessary to associate geometric phase from the work of J. Anandan and Y.Aharonov [Phys. Rev. Left. 58, 1863 (1989)l.

M.Lombardi: What is not needed is the validity of the adiabatic approximation, that is, that there is no transition between adiabatic states. But the geometric phase is defined by following states along a path in parameter space (here nuclear coordinates) with some continuity condition. In the diabatic representation, there is no change of basis at all and thus the geometric phase is identically zero. Do not confuse adiabatic basis (which is required) and adiabatic approximation (which may not be valid). Consider the example of the Jahn-Teller interaction at a conical intersection of two electronic surfaces of energy, the pre-Berry example of geometric phase studied by Longuet-Higgins. A doubly degenerate electronic state at a highly symmetric reference configuration (e.g., an E state of an X3 molecule at equilateral configuration) is split by a doubly degenerate vibration [an e vibration described by a radial ( r )and an angular (4) parameter]: One of the two degenerate states is lowered in energy such that the minimum occurs for a lower symmetry (isosceles instead of equilateral) for a given value of the distortion r = ro and any phase 4. The vibrational energy levels of that molecule can be (and usually are) computed by diagonalizing a Hamiltonian matrix set up in a diabatic basis, that is, a fixed basis independent of r and 4. Then if the vibronic interaction is strong, the lower energy levels computed that way follow a law El = BZ(Z + 1) with half-odd integer I , in a system with absolutely no spin. In the diabatic basis, this is a nonunderstandable peculiarity that pops out of the black box that effects the diagonalization of the matrix. However, as pointed out by

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GENERAL DISCUSSION

Longuet-Higgins,one can also set up the problem in an adiabatic basis, obtained by rediagonalizing the electronic potential at every value of r and 4. One obtains in that way the two adiabatic electronic surfaces, and the lower energy levels are confined to the lower surface, avoiding the center of the cone, that is, the reference equilateral configuration. Longuet-Higgins rationalized this half-odd integer value of I by noticing that, when one follows by continuity the adiabatic basis along a full turn around the center, the two electronic states change sign, exactly as does a spin after a full turn, and this change is the geometric phase. (But the total wave function, composed of electronic and nuclear parts, is unchanged, the change of sign of the electronic state being compensated for by a change of sign of the nuclear wave function, so that there is no contradiction with unicity of the total wave function.) So this halfodd integer value of I is an observable consequence of geometric phase, connected with properties of adiabatically following a basis in parameter space. This property is evidenced only if the vibronic interaction is strong, causing enough levels in the lower adiabatic surface to recognize this Ef law; otherwise there are only weak perturbations between pairs of degenerate E states. Nevertheless, nothing forbids making all computations in the diabatic basis, which has no geometric phase. The preceding discussion pertained to dynamical situations in intermediate cases, in which it is necessary to take into account interactions between adiabatic states, enabling transitions between them.

M. S. Child: With regard to the question about Berry’s geometric phase, the most common molecular manifestation is the half-odd angular momentum associated with the Jahn-Teller effect. Two cases may be distinguished in the context of Rydberg spectroscopy, according to whether the electronic degeneracy involves the excited Rydberg orbitals or states of the positive ion core. Examples of the former type are well established for NH3 [l] and H3 121 and a quantum defect treatment has been given for the latter species by Stephens and Greene [3]. The theoretical situation is more complicated if the ion core is Jahn-Teller active. Mueller-Dethlefs [4] has given a very complete picture of the Jahn-Teller dynamics of C&j+, but the consequences for interaction with benzene Rydberg electrons remain to be worked out. 1. 2. 3. 4.

M. N. R. Ashfold, R. N. Dixon, and R. J. Stickland, Chem. Phys. 88 463 (1984). G. Herzberg, Faraday Discuss. Chem. Soc. 71, 165 (1981). J. A. Stephens and C. H.Greene, Phys. Rev. Lett. 72, 1624 (1994). I. Fischer, R. Linder, and K. Muelier-Dethlefs. J . Chem. SOC. Faruday Trans. 90, 2425 (1994).

TRANSITION-STATE SPECTROSCOPY AND PHOTODISSOCIATION

PHOTODISSOCIATION SPECTROSCOPY AND DYNAMICS OF THE VINOXY (CH2CHO) RADICAL D. L. OSBORN, H. CHOI, and D. M. MUMARK* Department of Chemistry University of California Berkeley, California and Chemical Sciences Division Lawrence Berkeley Laboratory Berkeley, California

CONTENTS I. Introduction 11. Experimental 111. Results IV. Discussion A. CH3 + CO Channel B. D + CD2CO Channel V. Conclusions References

I. INTRODUCTION Photodissociation experiments have become one of the most valuable tools in chemical physics for the purpose of understanding how excited electronic states couple to the dissociation continuum. These experiments, and the the*Report presented by D. M.Neumark Advances in Chemical Physics, Volume 101: Chemical Reactions and Their Control on the Femtosecond Eme Scale, Mth Solvay Conference on Chemistry. Edited by Pierre Gaspard, Irene Burghardt, I. Prigogine, and Stuart A. Rice. ISBN 0-471-18048-3 0 1997 John Wiley & Sons, Inc.

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D. L. OSBORN, H. CHOI, AND D. M. NEUMARK

ory developed to explain them, have yielded considerable insight into the variety of dynamical processes that occur subsequent to electronic excitation [l]. From these studies, one hopes to obtain bond dissociation energies, characterize the symmetry of the excited state, measure the product branching ratios, and determine if the excited state undergoes direct dissociation on an excited-state surface, predissociation via another excited state, or internal conversion to the ground state followed by “statistical” decay to products. There are two clear recent trends in photodissociation experiments. One is to perform extremely detailed mesurements on systems where the basic photodissociation dynamics are reasonably well understood. As an example, experiments have been reported on CD31 in which the parent molecule is state selected and oriented prior to photodissociation [2]; multiphoton ionization of the CD3 fragment combined with imaging detection yields angular and kinetic-energy distributions for each product rotational state. An alternate direction is to apply well-developed photodissociation techniques to larger and/or more complex species, with the goal of seeing what new phenomena appear that are absent in the more commonly studied small-model systems. The work of Butler [3] on acetyl halides is a nice example of this; these studies have probed subtle nonadiabatic effects that dramatically affect the product branching ratio. The approach taken in our laboratory combinesboth of these trends. Specifically, we have developed a new experiment that allows us to study, for the first time, the photodissociationspectroscopyand dynamics of an importantclass of molecules: reactive free radicals. This work is motivated in part by the desire to obtain accurate bond dissociation energies for radicals, in order to better determine their possible role in complex chemical mechanisms such as typically occur in combustion or atmospheric chemistry. Moreover, since radicals are open-shell species, one expects many more low-lying electronic states than in closed-shell molecules of similar size and composition. Thus, the spectroscopy and dissociation dynamics of these excited states should, in many cases, be qualitatively different from that of closed-shell species. While one might expect that the techniques developed for photodissociation studies of closed-shell molecules would be readily adaptable to free radicals, this is not the case. A successfulphotodissociation experiment often requires a very clean source for the radical of interest in order to minimize background problems associated with photodissociating other species in the experiment. Thus, molecular beam photofragment translation spectroscopy, which has been applied to innumerable closed-shell species, has been used successfully on only a handful of free radicals. With this problem in mind, we have developed a conceptuallydifferent experiment [4] in which a fast beam of radicals is generated by laser photodetachment of mass-selected negative ions. The radicals are photodissociated with a second laser, and the fragments are detected in coinci-

PHOTODISSOCIATION OF THE VINOXY RADICAL

73 1

dence with a position- and time-sensing detector. This detection scheme yields high-resolution photofragment energy and angular distributions for each product channel. The negative ion production scheme ensures that only the desired radicals are produced. We have studied several radicals with this experiment during the last few years. In this chapter we report new results on the photodissociation spectroscopy and dynamics of the vinoxy radical, CH2CHO. The vinoxy radical is an important intermediate in hydrocarbon combustion; it is one of the primary products of ethylene oxidation. The electronic absorption spectrum was obtained by Hunziker [5],who found two bands with origins at 8000 and 28,760 cm-' . On the basis of ab initio calculations by Dupuis [6], these were assigned to transitions from the X 2A" ground state to the A 2A' and B *A" states, respectively. The higher energy B 6 X band extends at least to 35,700 cm-I . In Hunziker's experiment, this band shows some partially resolved features near the origin, but little structure remains above 3 1,200cm-I .Inoue [7] and Miller [8] have measured laser-induced fluorescence (LIF) spectra of the B +- X band, and the appearance is very different. These spectra consist of only a few sharp features between the origin and 30,200 cm-I ;no fluorescence signal is observed for excitation to the blue of this. The comparison of the absorption and LIF spectra implies that rapid predissociation occurs beyond 30,200 cm- ,thereby quenching any fluorescence. This is consistent with a recent hold-burning study by Gejo et al. [9]. Hence, this is an attractive speciesfrom the perspective of our photodissociation experiment, since we can examine the competition between predissociation and fluorescence in more detail. The primary photochemistry of the vinoxy radical is also of interest. Several relatively low-lying dissociation channels are available [ 10, 111:

'

--

CH2CHO CH3 + CO CH2CHO H + CHzCO CH2CHO ---t CH2 + HCO

A E = 0.05 eV AE = 1.45 eV AE = 4.1 eV

(1) (2) (3)

Channel (l), the lowest energy channel, requires a hydrogen shift, while (2) and (3) are simple band fission channels. Channels (1) and (2) are accessible from all levels of the B state, whereas channel (3) can be accessed only at the blue edge of the B c X band. Our experiment reveals that all levels of the B state in CH2CHO predissociate. We observe dissociation to channels (1) and (2). Kinetic energy distributions at several dissociation energies indicate that CH3 + CO production takes place following internal conversion to the ground-state potential-energy surface. These distributions are indicative of a barrier of 1-2 eV along the reaction coordinate to dissociation on the ground-state surface. A comparison of our results to photodissociation studies of the acetyl (CH3CO) radical

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D. L. OSBORN, H.CHOI, AND D. M. NEUMARK

[12, 131 suggests that this barrier corresponds to the isomerization barrier between vinoxy and acetyl on the ground-state surface.

II. EXPERIMENTAL The fast-beam photodissociation instrument has been described in detail previously [4].Briefly, a beam of vinoxide (CH2CHO-) anions is generated by bubbling O2 at 3 atm through acetaldehyde at -78°C. The deuterated species (CD2CDO-) is made in a similar fashion. The mixture is introduced to the spectrometer through a pulsed molecular beam valve. Ions are generated by means of a pulsed discharge assembly attached to the faceplate of the valve through which the gas pulse passes. By firing the discharge just after the valve is open, one forms a variety of ions that cool significantly (-50 K) in the resulting free jet expansion. The pulsed beam passes through a skimmer, and negative ions in the beam are accelerated up to 8 keV and mass separated by time of flight. The vinoxide ions are then photodetected with an excimer-pumped dye laser. The CH2CHO- and CD2CDO- were photodetached at 663 nm (1.870 eV) and 667 nm (1.859 eV), respectively. These energies are just above the electron affinities [14] of CH2CHO (1.824 eV) and CD2CDO (1.818 eV). Remaining ions are deflected out of the beam, leaving a fast pulse of mass-selected vinoxy radicals. These are photodissociated with a second excimer-pumped dye laser. The photofragments are detected with a microchannel plate detector that lies on a beam axis 1 m downstream from the dissociation region. A blocking strip across the center of the detector prevents parent radicals from reaching the detector, whereas photofragments with sufficient recoil energy miss the beam block and strike the detector. These fragments are generally detected with high efficiency (up to 50%) due to their high laboratory kinetic energy. Three types of measurements were performed in this study. First, photodissociation cross sections were measured, in which the total photofragment yield was measured as a function of dissociation photon energy. In these experiments, the electron signal generated by the microchannel plates is collected with a flat metal anode, so that only the total charge per laser pulse is measured. The beam block is 3 mm wide for these measurements. To perform photodissociation dynamics experiments, the dissociation laser is tuned to a transition where dissociation occurs. Most of the dynamics studies described here employ a time-and-position sensing photofragment coincidence detection scheme [ 151 in which a dual wedge-and-strip anode [4] is used to collect the electron signal from the microchannel plates. For each dissociation event, we measure the distance R between the two fragments on the detector, the time delay T between their arrival, and the individual displacements of the two fragments, r1 and r2, from the detector center. From this we obtain the center-of-mass recoil energy ET, the scattering angle with

PHOTODISSOCIATION OF THE VINOXY RADICAL

733

respect to laser polarization, 8, and the two photofragment masses ml and m2 via

e =tan-'

($)

(4)

Here EO and uo are the ion beam energy and velocity, respectively and 1 is the flight length from the photodissociation region to the detector. An 8-mmwide beam block is used for these measurements; a narrower block results in crosstalk between the two halves of the anode. The coincidence scheme works very well so long as m1/m2 I -5 and is thus quite suitable for channel (1). However, for channel (2), the fragment mass discrepancy is too large to perform this type of measurement. The recoil velocity of the heavy fragment is too small to clear the beam block, and the laboratory energy of the H (or D) atom is so low that their detection efficiency drops considerably. Since either of these effects makes coincidence measurements difficult, if not impossible, we performed a somewhat less complex experiment to detect and analyze channel (2). The flat-anode configuration of the detector (with the 3-mm beam block) was used, and we simply measure the time-of-flight distribution of all fragments at the detector. This was used previously to measure the kinetic-energy release in NCO photodissociation [161. For the present study, we only investigated CD2CDO since the fragment mass ratio for channel (2) is smaller than for CH2CHO. As will be seen below, our ability to measure the kinetic-energy distribution for channel (1) via the coincidence scheme allows us to subtract the contribution of this Channel from the time-of-flight measurement. This enables us to isolate the contribution of channel (2).

HI. RESULTS Photodissociation cross sections for CH2CHO and CD2CDO are shown in Fig. 1. Both spectra consist of sharp, extended vibrational progressions indicative of predissociation. A comparison of Fig. 1 with the LIF and absorption spectra [5, 7, 81 shows that we observe predissociation all the way to the origin of the B X band; this is labeled peak 1 in Fig. 1. We observe photodissociation over the entire range of the absorption band. However, peaks 1-7 are the only peaks seen in the LIF spectrum. Peaks 1,2,5, and 6 are particularly prominent in the LIF spectrum; the latter three peaks are assigned

.-

734

29000

D. L. OSBORN. H.CHOI, AND D. M. NEUMARK

3oooo

31000

32000

33000

Photon Energy (cm-') Figure 1. Photodissociation cross section of CHzCHO (top) and CDzCDO (bottom). Peaks 1-7 are the only features seen in the laser-induced fluorescence spectrum of CH2CHO (Ref. 8).

by Yamaguchi [17]to the 9;, 8;, and 7h transitions involving the C-C-0 bend, C-C stretch, and CH2 rock, respectively. While peak 1 is the most intense peak in the LIF spectrum, it is the weakest of the four in Fig. 1 . This shows that the competition between LIF and predissociation tilts sharply in favor of the latter even over the small energy range spanned by peaks 1 4 . Photofragment coincidence data were taken at several of the peaks in Fig. 1. Mass analysis of the fragments showed that only coincidences corresponding to channel (l), CH3 + CO (or CD3 + CO), were seen at all dissociation wavelengths examined. As discussed previously, the time and position data yield a coupled translational energy and angular distribution P(ET,e), which can be written as

Here P(ET) is the angle-independent translational energy distribution, and ~ ( E Tis) the (energy-dependent) anisotropy parameter, with - 1 5 P I 2. Fig. 2 shows the P(ET) distributions for CH2CHO at the four disso-

PHOTODISSOCIATION OF THE VINOXY RADICAL

735

1

Translational Energy (eV) Figure 2. Translational energy distributionsP ( E T ) for CH2CHO four dissociation energies corresponding to peaks A-D in Fig. I .

+

CH3

+ CO taken at

ciation energies indicated by peaks A-D in Fig. 1. The distributions all peak at nonzero translational energy. The most striking feature of Fig. 2 is the insensitivity of the P(&) distributions to photon energy; only a very small shift toward higher ET is seen over the entire energy range that was probed. The P ( E T ) distributions for CD2CDO photodissociation are similar and again show little variation with photon energy. Figure 3 shows the average anisotropy parameter at each photon energy for CH2CHO photodissociation. This shows that the angular distributions are isotropic (fl 0) at B state excitation energies below 1000 cm-’, but for higher energies we find fl in the range of 0.4-0.5. Note that the angular distribution becomes anisotropic in the energy range where fluorescence is quenched. Finally, the photofragment time-of-flight distribution for CD2CDO photodissociation at 31,980 cm-I is shown in Fig. 4. This will be analyzed in detail in the next section, but for now it suffices to point out that the wings in the distribution are from D atoms, indicating that photodissociation channel (2) to D + CD2C0 is indeed occurring.

736

D. L. OSBORN, H. CHOI, AND D. M. NEUMARK

Figure 3. Anisotropy parameter 0 for CH2CHO energies.

3.4

3.5

3.6

3.7

-.

CH3

3.8

+ CO at several dissociation

3.9

4.0

Time of Flight (ps) Figure 4. Photofragment time-of-flight spectrum for CD2CDO excited at 3 1,980 cm- I , including experimental results, contributions from D + CD2CO and CD3 + CO channels, and total simulated spectrum. The CD3 + CO contribution is obtained from an independent measurement of the energy and angular distribution at this energy using the photofragment coincidence detection scheme. The inset shows the translational energy distribution for the D + CD2CO channel (with 0 = 1.2) used to simulate the contribution of this channel to the timeof-flight spectrum.

PHOTODISSOCIATION OF THE VINOXY RADICAL

737

IV. DISCUSSION

A. CH3 + CO Channel The shape of the P(Er) distributions for channel (1) and their insensitivity to excitation energy is characteristic of statistical decomposition over a barrier. Distributions of this type are often seen in infrared multiphoton dissociation of molecules in which there is a barrier to product formation on the groundstate potential-energy surface [ 181 and also for electronic excitation in which internal conversion to the ground state occurs prior to dissociation [19]. The rationale for these distributions is that dissociation is statistical up until the top of the barrier. At this point, the most likely trajectories have nearly zero translational energy since this maximizes the number of vibrational states perpendicular to the reaction coordinate that can be populated. This is true regardless of the total excitation energy, provided dissociation is sufficiently slow so that energy randomization can occur. However, once the barrier is traversed, the photofragments move apart too quickly for the newly available energy to be fully randomized, so that the translational energy distribution peaks at some fraction of the barrier height, typically 4040%. It therefore appears that channel (1) occurs via internal conversion from the initially excited B (2A”) state to the X (2A”) state and that the peaking of the P(&) distributions around 1 eV translational energy is caused by a barrier between 1.2 and 2.4 eV on the ground-state potential-energy surface. We now consider the location of this barrier along the reaction coordinate. Internal conversion from the B state will result in highly vibrationally excited CH2CHO. In order to dissociate to CH3 + CO, this species must first isomerize to the acetyl radical, CH3C0, and then break the C-C bond in this radical. One certainly expects a barrier to be associated with isomerization, and the photodissociation experiments by North et al. [121 indicate that there is a barrier associated with breaking the C-C bond in the acetyl radical. Hence, there should be two barriers along the reaction coordinate, as shown in Fig. 5 . Given this reaction coordinate, we now want to consider which barrier is responsible for the maximum in the P(&) distributions. In North’s experiment, acetyl chloride (CH3COCl) is photodissociated at 248 nm to yield C1 + CH3CO. Time-of-flight analysis of the photofragments shows that the CH3C0 radical undergoes secondary dissociation when it contains more than 17 f 1 kcal/mol of internal energy, indicating that this is the barrier height for the reaction CH3CO + CH3 + CO. This reaction is endothermic by only 11 kcal/mol, however, so the barrier is 6 kcal/mol (0.26 eV) above CH3 + CO products. These energetics are consistent with recent ab initio calculations by Deshmukh et al. [13]. Thus, according to our model for the dissociation, this barrier cannot be responsible for the peak

738

D. L. OSBORN, H. CHOI, AND D. M. NEUMARK

Figure 5. Energetics for CH2CHO dissociation, including qualitative picture of the reaction coordinate for CH3 + CO production on ground-state potential-energy surface. See text for discussion of barrier heights.

at 1 eV in the P(&) distributions, since one generally expects only a fraction of the barrier height to manifest itself in this manner. This suggests that the peak in the translational energy distributions reflects the isomerization barrier in Fig. 5. If this is so, the barrier should lie 1-2 eV above the products, and preliminary ab initio calculations in our group indicate that this is a reasonable range of values. Our interpretation implies that once isomerization occurs, the resulting energized CH3CO falls apart very rapidly, before energy randomization can happen. Otherwise, the smaller barrier would determine the product translational energy distributions. This is what happens in the infrared multiphoton dissociation (IRMPD) of CH3N02, where the CH30 + NO product channel is formed by isomerization to CH30NO followed by dissociation with no exit barrier [18]. The translational energy distribution peaks at zero, consistent with the dynamics being determined by the absence of an exit barrier for dissociation rather than the isomerization barrier, which lies about 0.6 eV above the products. However, the CH30N0 well lies 1.8 eV below CH30 + NO, whereas the CH3CO well is only 0.7 eV deep relative to the exit barrier to dissociation. In addition, the ratio of the excitation energy to the well depth is much higher in our experiment than in the IRMPD study (>5 vs. 4).Thus, once isomerization to CH3CO occurs, dissociation may be so rapid that the well does not noticeably affect the final state dynamics. Alternatively, concerted elimination to CH3 + CO may occur at the top of the isomerization barrier, in

PHOTODISSOCIATION OF THE VINOXY RADICAL

739

which case the resulting reaction path would not pass through the CH3CO well (and over the second barrier) at all. Clearly, the detailed dissociation mechanism on the ground-state surface will have to be examined in more detail once a reasonable potential-energy surface is developed. At excitation energies close to the band origin, fluorescence still competes effectively with dissociation. We interpret this to mean that the initial internal conversion to the ground state is sufficiently slow at these energies that fluorescence is competitive. However, it appears that as the vibrational energy in the B state increases, internal conversion becomes so rapid that fluorescence is completely quenched. This is consistent with the angular distributions. These are isotropic (p 0) for those transitions that exhibit fluorescence, but anisotropic distributions occur at higher excitation energies where the fluorescence is quenched, indicating the lifetime with respect to dissociation has dropped substantially. Thus, it is the early-time dynamics following excitation that determine the outcome of the competition between fluorescence and dissociation. The rather abrupt change in the relative quantum yields for fluorescence vs. dissociation is of considerable interest. A possible mechanism has been proposed by Yamaguchi [20] based on ab initio calculations of the various CH2CHO excited-state energies as a function of the C-C torsional angle. He finds that energy of the B(2A”) state rises only slightly (3300 cm-’) over the entire torsion angular range. In contrast, the A(’A’) state, which is well separated in the planar geometry, approaches to within 2200 cm-’ of the B state at a geometry where the planes CH2 and CHO groups are perpendicular. The implication here is that, with minor adjustments, one can imagine an intersection between the two states at relatively low excitation energy of the B state, and this is what promotes the abrupt increase in the internal conversion rate with energy. Our results are certainly consistent with this if one views internal conversion to the A state as the rate-limiting component in a two-step internal conversion process that finishes on the ground electronic state.

B. D + CDzCO Channel

The time-of-flight distribution in Fig. 4 has contributions from channels (1) and (2). However, we know the detailed form of the energy and angular distribution for channel (1) from our photofragment coincidence measurements. The contribution from the CO3 and CO fragments to Fig. 4 can then be determined by using a Monte Car10 simulation to calculate the fragment time-of-flight distribution from channel (1) based on the known energy and angular distribution; this procedure, which averages over all relevant experimental parameters, is described in more detail in Ref. 16. The result is shown as the solid line in Fig. 4. The comparison of the simulation with the data shows that channel (2) is responsible for the far wings of the distribution (from the D atoms) and the sharp peak at the center.

740

D. L. OSBORN, H. CHOI, AND D. M. NEUMARK

This latter feature must come from CD2C0, which barely recoils out of the beam and therefore misses the smaller beam block used in this detector configuration. The inset to Fig. 4 shows a center-of-mass translational energy distribution for channel (2) that, when iun through our Monte Carlo program, generates the time-of-flight distribution shown by the dashed lines in Fig. 4. This distribution, when added to that for C03 + CO, adequately reproduces the total experimental time-of-flight distribution. We can therefore learn about the dynamics of channel (2), although not in as much detail as channel (1). The translational energy distribution for channel (2) also peaks away from zero, implying that there is an exit channel barrier to hydrogen atom loss. This is not unusual for the ground-state dissociation of a radical to a radical (D) + closed-shell species (ketene). Although we do not have data for channel (2) at the whole series of excitation energies as we do for channel (l), it is reasonable to expect that channel (2) also proceeds by internal conversion to the ground state followed by statistical decomposition over a barrier. It is clearly of interest to know the branching ratios between the two channels. Unfortunately, this is complicated by two factors. First, we only observe CDzCO fragments near the high-energy limit of the translational energy distribution; slower fragments strike the beam block and are not detected. This effect is taken into account in the Monte Carlo generation of a time-of-flight spectrum, but it does mean that the low-energy portion of the translational energy distribution is poorly determined. Second, the detection efficiency of the D atoms is considerably lower than that of the heavy fragments, but we do not know exactly how low. We estimate the detection efficiency to be between 1 and 10%;the true branching ratio clearly depends on this value. It does appear that channel (2) is a major channel, contributing at least 50% to the total fragmentation. A more detailed analysis is currently underway to better quantify this channel.

V. CONCLUSIONS This work represents the first study of the photodissociation dynamics of the vinoxy radical. We observe predissociation over the entire B 2A'' +-X *A" band of CHzCHO, including the origin at 28,760 cm-' . Two dissociation channels are observed: CH3 + CO and H + CH2CO. Translational energy distributions for the CH3 + CO channel are largely independent of excitation energy, indicating that this channel is likely due to statistical decomposition on the vinoxy ground-state potential-energy surface. The translational energy distributions all peak near ET = 1 eV, and we believe this is indicative of the isomerization barrier for conversion of vinoxy (CHzCHO) to the acetyl radical (CH3CO) prior to dissociation. A comparison of our results

PHOTODISSOCIATION OF THE VINOXY RADICAL

74 1

with previous laser-induced fluorescence work on CH2CHO shows that predissociation dominates over fluorescence at excitation energies > 1000 cm-' above the band origin. This indicates a greatly increased internal conversion rate above this energy, possibly due to another excited state of vinoxy intersecting the B 2Att state. The results presented here show that our instrument can also be used to investigate dissociation channels in which the mass disparity of the two fragments is very large, namely the H + CH2CO channel; the study of this channel was facilitated by using CDzCDO. Although the dynamics of this channel cannot be elucidated at the same level of detail as the CH3 + CO channel, our ability to study it at all represents an important extension of the capabilities of the instrument.

Acknowledgments This research is supported by the Director, Office of Energy Research, Office of Basic Energy Sciences, Chemical Sciences Division, of the U.S. Department of Energy under Contract No. DE-AC03-76SF00098.

References I . R. Schinke, Photodissociation Dynamics, Cambridge University Press, Cambridge, 1993. 2. J. W. G. Mastenbroek, C. A. Taatjes. K. Mauta, M. H. M. Janssen. and S. Stoke, J. Phys. Chem. 99,4360 ( I 995). 3. M. D. Person, P. W. Kash, and L. J. Butler, J. Chem. Phys. 97, 355 (1992). 4. R. E. Continetti, D. R. Cyr, D. L. Osborn, D. J. Leahy, and D. M. Neumark, J. Chem.

Phys. 99, 2616 (1993); D. J. Leahy, D. L. Osborn,D. R. Cyr, and D. M. Neumark, J. Chem. Phys. 103,2495 (1995). 5. H. E. Hunziker. H. Kneppe, and H. R. Wendt, J. Photochem. 17,377 (1981). 6. M. Dupuis, J. J. Wendoloski, and W. A. Lester, Jr., J. Chem. Phys. 76, 488 (1982). 7. G. Inoue and H. Akimoto, J. Chem. Phys. 74, 425 (1981). 8. L. F. DiMauro, M. Heaven, and T. A. Miller, J. Chem. Phys. 81,2339 (1984). 9. T. Gejo, M. Takayanagi, T. Kono, and I. Hanazaki, Chem. Lett. 2065 (1993). 10. S. G. Lias, J. E. Bartmess, J. F.Liebman, J. L. Holmes, R. D. Levin, and W. G. Mallard, J. Phys. Chem. Ref- Data 17 (Suppl. 1) (1988). 11. C. W. Bauschlicher, Jr., J. Phys. Chem. 98, 2564 (1994). 12. S. North, D. A. Blank, and Y. T. Lee, Chem. Phys. Lett. 224, 38 (1994). 13. S. Deshmukh, J. D. Myers, S. S. Xantheas, and W. P. Hess, J. Phys. Chem. 98, 12535 ( 1994).

14. R. D. Mead, K. R. Lykke, W. C. Lineberger, J. Marks, and J. I. Brauman. J. Chem. Phys. 81,4883 (1984). 15. D. P. DeBruijn and J. Los, Rev. Sci. Instrum. 53, 1020 (1982). 16. D. R. Cyr, R. E. Continetti, R. B. Metz, D. L. Osborn, and D. M. Neumark, J. Chem. Phys. 97,4937 (1992). 17. M. Yamaguchi, T. Momose, and T. Shida, J. Chem. Phys. 93,4211 (1990). 18. A. M. Wodtke, E. J. Hintsa, and Y.T. Lee, J. Phys. Chem. 90, 3549 (1986).

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19. X. Zhao, R. E. Continetti, A. Yokoyama, E. J. Hintsa, and Y. T. Lee, J. Chem. Phys. 91, 4118 (1989).

20. M. Yamaguchi, Chem. Phys. Lett. 221,531 (1994).

DISCUSSION ON THE REPORT BY D. M. NEUMARK Chairman: J. Manz

M. Shapiro: Prof. Neumark, in general, I was struck by the similarity between your data on the CH3O radical and the photodissociation of CH31, which we analyzed many years ago. In particular, it is interesting to note that the degree of excitation of the umbrella mode is similar and appears to increase with increasing excitation of the parent molecule. Obviously the exit channel dynamics dominate here. The other comment I would like to make is that the positive value of the P parameter you observe is due to a quantum interference effect. A simple mixing of the ground state with the excited state in the final continuum state hardly affects the directional properties of the dipole matrix elements per se because the transition dipole matrix elements between different states within the ground state are very small. Namely, if Pex=

+

then

because

which represents a high overtone transition.

D. M. Neumark: We indeed take the translational energy distribution from CH30 dissociation to be evidence for exit channel interactions on a repulsive potential-energy surface. This is in contrast to photodissociation of the vinoxy radical, for which very little variation of the CH3 + CO translational energy distribution occurs over a 0.5-eV range of excitation energy.

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743

Regarding your comment on the photofragment angular distributions, the 3;66:, transition in the 2A1-* 2E band of methoxy radical (involving single excitation of the degenerate CH3 rocking mode) is nominally allowed because of Jahn-Teller coupling in the 2E ground state. However, you are correct in pointing out that this does not explain the positive value of 0 observed for this perpendicular electronic transition. The most likely explanation is that the upper level is vibronically mixed with an a1 vibrational level of a nearby electronic state of E symmetry; this argument has also been proposed by Terry Miller (Ohio State University) to explain new spectroscopic features seen in the electronic spectrum of CH3O. J. The: Prof. Neumark, how well under control do you have energy and angular momentum of the dissociating excited species? D. M. Neumark: In cases where we can measure it (02, for example), the rotational temperature is 30-50 K. T.P. Softley: Would you comment on the vibrational temperature of the photodissociated parent molecules in the beam. For 0 2 you clearly observe u” = 5 6 , but is this also the case for methoxy? D. M. Neumark We make no effort to produce vibrationally cold 02, since the B +-X transitions to predissociating upper state levels are rotationally resolved and completely understood. In the case of CH30, we detach the CH3O- just above the detachment threshold so that we do not produce vibrationally excited CH3O. K. Yamanouchi: There are many kinds of small polyatomic molecules that emit detectable fluorescence in the energy region where they predissociate. Acetylene, S02, and CS2 are examples of such molecules. The two processes, fluorescence emission and dissociation, are in general competing processes. In order to discuss predissociation dynamics, it is also important to derive a state-specific rate constant based on the measurements of absorption and dissociation cross sections. D. M. Neumark: The competition between fluorescence and predissociation is an interesting and complex problem, as evidenced by the observation that it is quite different in the vinoxy and methoxy radicals. Unfortunately, high-resolution absorption spectra are not available for these radicals so it is not so straightforward to compare absorption spectra with LIF or predissociation spectra. R. A. Marcus: The large amount of energy liberated as translational energy in the CH2CHO -+ CH3CO -+ CH3 + CO reaction was very interesting. Do you have information on the energy distribution among the other coordinates of the products: If the first step, the transfer of

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an H from the CHO group to the CH:! group is vibrationally adiabatic, that is, no significant change in this H-stretching vibrational quantum number, and if the C=O and other CH fragments are largely spectators, then all of the excess energy of the highly exothermic process will go into the C-C vibration and hence into translational energy of separation of the CH3 and CO fragments. The actual distribution will depend on how short-lived the hot CH30 is. D. M. Neumark: Once the vinoxy radical surmounts the isomerization barrier, we don’t really know if it passes through the acetyl minimum before dissociating to CH3 + CO, or if instead CO elimination is concerted. This would clearly be an interesting theoretical problem to pursue. K. Yamashita: Prof, Neumark, you talked about the experiment in which anions are photodetached to the bound states of neutral molecules, and these bound states are excited to dissociate by a second photon. However, you can also excite anions to the transition state of neutral reactions instead of bound states and can reach the resonance states, as you do for several systems. My question is then, is it possible to excite these resonance states directly by your second photon? If it is possible, you would be able to control photodissociation product channels, because there should be cases where some resonance states localize on one side of the transition state and other resonance states localize on another side. D. M. Newnark: In our current experiment we use pulsed nanosecond lasers, and the lifetimes of even the longest lived transition-state resonances are too short to excite with a second photon. A related and very interesting experiment would be to measure the product energy distributions resulting from decay of these resonances. For example, one could photodetach IHI- and measure the I + HI kinetic-energy distributions as a function of electron kinetic energy. This requires a “triple-coincidence” experiment that cannot be done easily in our laboratory. However, experiments of this type are currently being carried out by Prof. Robert Continetti at the University of California at San

Diego.

RESONANCES IN UNIMOLECULAR DISSOCIATION: FROM MODE-SPECIFIC TO STATISTICAL BEHAVIOR R. SCHINKE,* H.-M. KELLER, H. FLOTHMANN,

M.STUMPF, C. BECK, D. H. MORDAIRVT, and A. J. DOBBYN Max-Planck-Institut jiir Stromungsforschung Gottingen, Germany

CONTENTS 1. 11. 111. IV. V. VI. VII.

Introduction Potential-Energy Surfaces Quantum Mechanical Calculations HCO: A Textbook Example of Regular Dynamics K O : A Spectroscopic Challenge HNO: A Mixed Reguhr-Irreguhr System HO2: Classical Chaos Reflected in Dissociation Rates and Product-State Distributions VIII. Resume and Outlook References

I. INTRODUCTION Resonances in open systems, that is, systems whose total energy is higher than the first dissociation threshold, are an old theme of scattering dynamics. For very elementary introductions the reader is referred to, for example, Messiah (Ref. 1, Chapter III), Cohen-Tannoudji et al. (Ref. 2, Chapter XIII), Satchler (Ref. 3, Chapter 4), and Schinke (Ref. 4, Chapter 7). They *Report presented by R. Schinke Advances in Chemical Physics, Volume 101: Chemical Reactions and Their Control on the Femtosecond Time Scale. XXth Solvay Conference on Chemistry, Edited by Pierre Gaspard, Irene Burghardt, I. Prigogine, and Stuart A. Rice. ISBN 0-471-18048-3 0 1997 John Wiley & Sons, Inc.

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are induced by the temporary trapping of some part of the available energy in internal modes (collectively represented by r in what follows) that are “orthogonal” to the fragmentation coordinate R. The storage of energy in these internal degrees of freedom can significantly delay the rupture of the bond between the two (or more) entities because first sufficient energy must be transferred to the dissociation coordinate. Resonances are well-known features in all kinds of scattering [5-71, for example, nuclear reactions, electron collisions, scattering of atoms and molecules from solid surfaces, photodissociation, and molecular spectroscopy. The latter topic is particularly enlightening, because with modem experimental and theoretical tools all facets of resonances can be studied in great detail [8]. Recent progress in the minute investigation of resonance states in polyatomic molecules will be highlighted in the present contribution for the Twentienth Solvay conference in chemistry. A typical example of a molecule ABC in its electronic ground state 2 is depicted in Fig. 1, showing a one-dimensional cut through a multidimensional potential-energy surface (PES). The quantum states below the AB(X, 0) + C threshold are true bound states with discrete energies. When the total energy is increased above the first threshold (E = 0). it becomes a continuous variable. However, this does not necessarily imply that the progression of bound states immediately dies out when E > 0. On the contrary, it can persist into the continuum, even to high energies above the threshold, with one important distinction: Unlike true bound states, resonance states have a finite energetic width. In view of the time-energy uncertainty relation, resonances have a finite lifetime whereas true bound states live forever (if we ignore spontaneous emission). The quantities that specify a resonance are the energy E$\, the width Ak“), where k(’) is the decay rate of the resonance, and last but not least the final distribution of the particular quantum states /3 of the products, P(i)(/3). The energy of a resonance is essentially determined by the PES in the inner (i.e., bound) region. Its width (or lifetime #) = l/k(i) if the resonance has a Lorentzian line shape) depends on the coupling between the inner region and the exit channel, that is, the efficiency of internal vibrational energy redistribution (IVR) between the dissociation coordinate R on one hand and the internal coordinates r on the other and therefore on the shape of the potential near the transition state (TS). The lifetime of the temporary excited complex can range from a few tens of a femtosecond for a rather fast process to microseconds, or even longer, for a weakly bound van der Waals complex. Finally, the product-state distributions reflect both the wave function at the TS and the dynamics (i.e., the energy exchange) in the exit channel [4, 9, lo]. In order to fully understand the fragmentation dynamics of a system, it is, in principle, necessary to consider all three of these observables for as

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+

dissociation coordinate R

Figure 1. Schematic representation of resonances in the continuum of a polyatomic molecule ABC(X) dissociating into products AB(X, 0) and C. The left-hand side shows an absorption-type cross section O&&) with a rich resonance pattern. The term p ( E ) is the density of states at the energy E and N & ( E ) is the number of states at the TS, orthogonal to the dissociation path, that are accessible at energy E. Several experimental schemes for a spectroscopic analysis of resonances are also indicated. (Reprinted, with permission of the Royal Society of Chemistry, from Ref. 34.)

many resonances as possible. The possibility of measuring and calculating resonance widths and final-state distributions makes photodissociation, from the dynamical point of view, more interesting than bound-state spectroscopy; in the latter case only the excitation energies can be analyzed. Experimentally one can investigate resonances by various spectroscopic schemes, as indicated in Fig. 1: by direct overtone pumping [ l l ] from the ground vibrational state, by vibrationally mediated photodissociation [121 using an excited vibrational level as an intermediate, or by stimulated emission pumping (SEP)[13-151 from an excited electronic state. In all cases it is possible to scan over a resonance and thereby determine its position E Z and its width hkc').A schematic illustration of an absorption or emission spectrum is depicted on the left-hand side of Fig. 1; all of the more or less sharp structures at energies above threshold are resonances. Figure 2 shows an overview SEP spectrum measured for DCO [16]. It consists of

8PL ODEEL

OOBEL

WtEL

OOZEL

OOOEL

OOSZl

OOBZL

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Excitation Energy / (cm-') Figure 2. Overview SEP spectrum for DCO(2) as measured by Stiick et al. [16]. The energy is measured with respect to the vibrational ground state (0, 0.0). Each vibrational band consists of four different rotational lines. For a detailed discussion of the assignment the reader is referred to the theoretical analysis of Keller et al. [17]. (Reprinted with permission of the American Institute of Physics, from Ref. 16).

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a long series of vibrational bands each being composed of four different rotational lines. The bands below 5500 cm-' are true bound states while all bands above this energy are resonances. (The highest bound state has an energy of 5471 cm-' above the ground vibrational state [17].) A finer scan of this spectrum bears out that the resonance widths do not depend much on the particular rotational state, within the same vibrational band, but depend sensitively on the vibrational excitation. Finally, the distribution of quantum states of the product molecule, f i i ) ( n , j ) ,can be monitored with a third laser [18]. (In what follows the quantum numbers n a n d j specify the vibrational and rotational state of the fragment molecule AB.) As examples we show in Fig. 3 two measured rotational distributions of CO in the fragmentation of a particular vibrational resonance of HCO [19], in comparison with the results

0.2

n

;3

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oexperiment

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i Figure 3. Comparison of measyed [19] and calculated [20] rotational state distributions following the dissociation of HCO(X) for two different initial total angular momentum states.

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of an ab initio calculation [20); the total angular momentum states are 303 and 312.The particular shapes of these distributions reflect, in a qualitative sense, the angular dependence of the wave function at the TS, as we will elucidate below. Although each resonance has its own individual width and final-state distribution (quantum state speciJicify), one can broadly distinguish between two limits: mode selectivity and statistical behavior [21, 221. In the first case, both the widths and the product-state distributions depend in a predictable way on the kind and the degree of excitation of the molecule, that is, on the quantum numbers with which the resonance can be labeled; mode specificity obviously requires some kind of unique assignment. In the second case, a systematic dependence on the particular resonance is not possible, simply because a meaningful assignment of the resonance states does not exist. Because of the lack of a clear-cut relationship between the resonance on the one hand and k(’)and P(’)(P)on the other, a statistical analysis might be more meaningful. The widths and the final-state distributions can be thought of as reflections of the nodal pattern of the underlying resonance wave function and therefore, we argue that mode specificity is confined to systems with mainly regular motion whereas statistical behavior is normally found in irregular, that is, classically chaotic systems. Mode-specific behavior is expected when the density of states is relatively small and/or when the coupling between the modes is weak. In contrast, statistical behavior is believed to occur when the density of states is large and/or the internal coupling is strong. In the first case exact quantum mechanical (or possibly semiclassical) calculations on a complete and accurate PES are indispensable if one wants to compare theoretical predictions with detailed experimental data. In the statistical case it is hoped that much simpler calculations based on statistical assumptions, for example, the Rampsberger-Rice-Kassel-Marcus (RRKM)[23] theory or the statistical adiabatic channel model (SACM) [24,25], can be used to describe the average values of decay rates and final-state distributions. These theories do not require detailed knowledge of the complete PES, but only a small portion of it in the region of the TS. The fragmentation of a molecule in its ground electronic state is commonly known as unimolecular dissociation [26-281. [For a recent review see Ref. 29 and the Faraday Discussion of the Chemical Society, vol. 102 (1995).1 Because of its importance in several areas of physical chemistry, such as combustion or atmospheric kinetics, there is a high demand of accurate unimolecular dissociation rates. On the other hand, however, the calculation of reliable dissociation rates by dynamical methods (i-e., the solution of the classical or quantum mechanical equations of motion) is, for obvious technical problems, prohibited for all but a few simple molecules. For

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this reason one is forced to use very simple models for predicting dissociation rates, the accuracy of which is not a priori known. The most widely employed method is based on the assumption that prior to dissociation all (zero-order) states, or the classical phase space in classical mechanics, are statistically populated [23, 30, 311. The dissociation rate as a function of energy E is then given by

where N l ( E ) is the number of accessible states at the transition state “orthogonal” to the fragmentation path, p ( E ) is the density of states, and h is Planck’s constant. The various versions such as RRKM, variational RRKM, or SACM differ by how the numerator is calculated. By definition, Eq. (1) can only be expected to yield a reasonable approximation for the average quantum mechanical rate, while fluctuations, which normally are caused by quantum interference, cannot be reproduced. In the last two years or so our research group has analyzed in detail the dissociations of HCO [32, 331 DCO [16, 171, HNO [34], and H02 [35-371 on their ground-state PESs using quantum mechanical methods. In the case of HCO we concluded that it is an essentially regular system with mostly assignable wave functions and that it illustrates mode specificity. On the other extreme, analyses of the bound-state wave functions as well as the energy spectrum of the bound states showed that the motion of H02, at energies near the H + 02 threshold, is mainly irregular, that is, classically chaotic [36]. The dissociation rates and intensities in an absorption-type spectrum can be well modeled by statistical theories [37]. Most interestingly, the dissociation of HNO embraces both regular and irregular motion; it thus represents an intermediate case between HCO and HO2. The purpose of this chapter is a detailed comparison of these systems and the elucidation of the transition from regular to irregular dynamics or from mode-specific to statistical behavior. The main focus will be the intimate relationship between the multidimensional PES on one hand and observables like dissociation rate and final-state distributions on the other. Another important question is the rigorous test of statistical methods for these systems, in comparison to quantum mechanical as well as classical calculations. The chapter is organized in the following way: The three potential-energy surfaces and the quantum mechanical dynamics calculations are briefly described in Sections I1 and 111, respectively. The results for HCO, DCO, HNO, and H02 are discussed in Sections IV-VII, and the overview ends with a short summary in Section VIII.

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II. POTENTIAL-ENERGY SURFACES The calculations presented here include all three internal degrees of freedom. Since we consider the decay of an unstable complex into two (or in principle more) fragments (“half collision”), it is necessary to use coordinates appropriate for scattering calculations rather than normal coordinates, for example [4]. Furthermore, since the fragmentation into a single product channel is investigated, the usual Jacobi coordinates can be used: R, the distance from the atom C to the center of mass of the diatomic fragment AB; r, the vibrational coordinate of the AB entity; and y , the angle between R and r. The HCO PES has been determined by extensive ab initio calculations at the multi-reference configuration interaction (MRCI) level using the MOLPRO program package [32]; the almost one thousand original energy points were then fit to an elaborate analytical expression [33]. In order to compensate for a small underestimation of the CO stretching frequency by the ab initio calculations, the PES was slightly modified. The PES for HNO has also been constructed by ab initio calculations, again making use of MOLPRO [38]. About 1200 points have been calculated and a three-dimensional spline interpolation scheme was employed to calculate the potential between the grid points. In this case, too, slight modifications have been applied in order to make the theoretical term energies agree better with the experimental ones. The PES for HO2 is the double many-body expansion (DMBE) IV potential of Pastrana et al., which has been constructed by using both ab initio and experimental data [39]. Contour plots of all three PESs are depicted in Fig. 4. The left-hand side shows the potentials as functions of R and r for fixed y and the right-hand side demonstrates the (R, y) dependence for fixed r. In order to present in the clearest way the similarities and differences, the scales of the axes are the same for each molecule. The HCO PES has the shallowest well (D, = 0.834 eV) and a small barrier separates the inner region from the exit channel. The values of 2.14 potential wells for HNO and HO2 are considerably deeper (0, and 2.37 eV, respectively) and no potential barriers hinder the bond rupture. Let us first discuss the (R,r ) behavior. In all three cases the bonds of the diatomic entities, CO, NO, and 0 2 , significantly change when the hydrogen atom is attached. In the well the anharmonicity along r is much larger than in the free molecule. This effect is least pronounced for HCO and most dramatic for HO2; in the latter case the opening of the 0 + OH reaction channel is clearly seen at large 02 separations. Since the fundamental frequency w r associated with r is smaller in the case of H02 than for HNO, and since the anharmonicity is so strong in this coordinate for H02, the density of states is substantially larger for HO2, despite the fact that the dissociation

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Figure 4. Contour plots of the potential-energy surfaces for HCO, HNO. and H02.The left-hand side shows the (R, r ) dependence, with the angle y being fixed at the equilibrium in the well. The right-hand side highlights the (R, y ) dependence, with r fixed at the equilibrium. The spacing of the contours is 0.25 eV and the lowest contour is 0.25 eV above the minimum. Energy normalization is so that E = 0 corresponds to H + XO(r,). The Jacobi coordinates R, r, and y are as described in the text, with y = 180” corresponding to H-X-0. (Reprinted, with permission of the Royal Society of Chemistry, from Ref. 34.)

energies are similar. The number of bound states is merely 15 for HCO, 215 for HNO, and 361 (365) for the odd (even) parity of H02. In all three cases the coupling between R and r, qualitatively indicated by the angle between the two “reaction channels,” is relatively weak, and therefore high excitation in r is, roughly speaking, the “carrier” for the many resonances above the threshold observed for all three molecules [32]. In each case more or less “pure” vibrational states with excitation only in r are found up to very high energies. The decoupling of r from the other two modes seems to be most effective for HCO, less so for HNO and H02. The coupling between the angle y and the dissociation coordinate R is always large if Jacobi coordinates are used. At low energies deep inside the well, this coupling is linear and normal coordinates are usually better suited for interpretation and assignment than are Jacobi coordinates. However, if the molecular dynamics above the dissociation threshold is studied, the normalmode picture breaks down and scattering coordinates have to be employed.

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The coupling between R and y is reflected by the curvature of the minimum energy path inside the well, which seems to be roughly equal in all three cases. The PES for HCO has a very shallow second minimum at small angles that, however, is not seen in the figure because it is too high in energy. For HNO this second minimum occurs at lower energies and is clearly seen in Fig. 4. The PES for HO2 is symmetric with respect to y = 90” and the two wells are obviously equally deep. The PES of HCO has a barrier and therefore the TS is clearly defined. When the molecule leaves the well region, it has to “squeeze” through this bottleneck, and that is clearly shown by the stationary wave functions [32, 401. HCO is an example of a “tight” transition state. In the case of HO2 there is no potential barrier, but nevertheless a distinct bottleneck is still seen where the wide HO2 well (wide with respect to r ) quite abruptly narrows and turns into the relatively cramped exit channel. Here, the 0 2 force constant changes rapidly with R. Thus, as we will discuss below, H 0 2 can still be considered as an example of a tight transition state. The situation is different, however, for HNO: No potential barrier hinders the dissociation and the NO force constant changes very smoothly with the distance from the H atom. If we look from the free-product side, the narrow exit channel gradually opens into the wide HNO well. HNO exemplifies a system with a “loose” transition state. The consequences will become apparent below. The general behavior of the potentials in the asymptotic channel can be qualitatively understood in simple chemical terms [341. It is not difficult to surmise the existence of a long series of resonances in each case. They are basically the result of high overtone excitation in the vibrational coordinate r. the mode that for all three molecules is rather weakly coupled to the dissociation coordinate. In order to enable the system to dissociate, first a sufficient amount of energy has to be transferred from motion in r to motion along R, which, depending on the coupling strength, can be very time consuming. Simultaneous excitation in the angular degree of freedom generally accelerates the fragmentation because y and R are usually rather strongly coupled. Finally, direct excitation of the dissociation mode normally leads to rather short-lived complexes. This is, of course, only a qualitative picture. Pronounced differences among the systems considered will be emphasized below. At the end of this section, it is worthwhile to point out that resonances in quantum mechanics are intimately related to the existence of trapped classical trajectories. The smaller the classical forces between r and y on one hand and R on the other, the longer is the lifetime and vice versa. In this sense it might be helpful for understanding the complex quantum dynamics by imagining the trajectories of a classical billiard ball moving on multidimensional potential-energy surfaces (see, e.g., Chapter 5 of Ref. 4).

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111. QUANTUM MECHANICAL CALCULATIONS

Continuum wave functions are required to fulfill specific boundary conditions in the exit channel, that is, in the limit R-+-, and therefore they are much more difficult to Calculate than bound-state wave functions. In our applications we use a modification of the log-derivative version of Kohn’s variational principle [37]. The essential steps can be summarized as follows: (i) The coordinate space is divided in an inner (R < R,) and an outer (I7 > R,) region, where R, is chosen so that the coupling between the vibrational-rotational channels of the diatom is negligibly small. (ii) In the inner region an appropriate set of energy-independent basis functions is constructed from a large primitive basis set by a general contraction-truncation scheme, and the corresponding Hamiltonian matrix is diagonalized; this yields, as a by-product, all the bound-state energies and wave functions. (iii) In the outer region the appropriate coupled-channel equations are solved approximately for each value of the energy E. (iv) Matching the solutions in the inner and the outer regions at the boundary R, leads to algebraic equations from which the wave functions for each energy and all open channels P of the diatom, *$), are obtained. The diagonalization of the Hamiltonian matrix is time consuming, but as this step is independent of E, it has to be done only once. Varying the energy is very efficient and normally several thousand energies are included in one scan of the spectrum. Having the partial wave functions *$’, we usually calculate the overlap with an “initial” wave function xo and determine an absorption-type cross section according to [4].

where the sum runs over all open product channels /3. If we were to calculate an absorption spectrum, we would take xo as a particular vibrational state in the ground electronic state; if, on the other hand, our goal would be the calculation of a SEP spectrum, the initial wave function must represent a vibrational state in an upper electronic manifold. In any case, because we calculate directly the wave functions (rather than discrete energies in the complex plane), we can determine any observable and compare it to its experimental counterpart. Plotting the stationary wave functions of the system facilitates the assignment of the resonances in terms of vibrational quantum numbers (if there is any) and illustrates the overall dissociation path. For this reason we usually consider the so-called total wavefunction * E (Chapter 2 of Ref. 4), that is, a

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superposition of all partial wave functions Of’for a given energy, weighted by the overlap factor (xo IOf’).Figure 5 depicts several examples of resonance wave functions for HCO. The left-hand side shows the (R, r) dependence for fixed angle y while the right-hand side features the (R, y) dependence for fixed r. Although these wave functions look like bound-state wave functions, they are, nevertheless, real continuum wave functions; except for state (3, 0, 0), the asymptotic tails are too small, relative to the amplitudes in the inner region, to be seen on this scale. The assignment in terms of the three quantum numbers u1 (R, H-C stretch), u2 (r, C-O stretch), and u3 (7, H-C-O bending) is rather obvious, except for the wave function displayed in the lowest panel. In the next section we will elucidate the relationship between the shape of these wave functions on one hand and the dissociation rates and the final rotational state distributions of CO on the other. Resonances are features inherent in the Hamiltonian and show up as more or less sharp structures in all quantities that contain the wave functions, for example, the absorption cross section aab,(E).The absorption spectrum obviously depends on the initial-state wave function x o , and therefore the various resonances are weighted with different Franck-Condon factors, which complicates the analysis of the spectrum. A numerically more convenient quantity is

where the norm 11.11 is calculated only in the inner region R < R,. Division by the cross section compensates for the weighting with the overlap matrix element in the definition of the total wave function. For an isolated resonance with hrentzian shape it can be shown that 7(E,,) evaluated at the center of the resonance is the lifetime of the resonance, and therefore we will call ; ( E ) the lifetime function in what follows. The lifetime function is largely independent of the particular initial state, and in our applications we found that in general it gives a better resolution of the resonance structures, especially in the overlapping regime. Figure 6 shows ?(E) for all three systems as a function of the total energy available for the fragments, E-Eo, on a logarithmic scale. The energy regime is chosen to be the same in all three cases in order to demonstrate the different densities of (resonance) states. The spectrum of HCO has been calculated to much higher energies, about 1.7 eV above threshold, and even at these high energies there still exist many very narrow resonances [32]. The spectrum for HNO and H02 have been determined up to 0.7 and 0.4 eV, respectively. Most of the resonances for HCO can be unambiguously

RESONANCES IN UNIMOLECULAR DISSOCIATION

3.0

1801

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Figure 5. Selected resonance wave functions for HCO. The left-hand panel shows (R,r) cuts for fixed angle y and the right-hand panel illustrates the (R, y ) behavior for fixed value of r. The distances are given in a0 and the angle is given in degrees. Plotted is the modulus square of the wave function. Except for the lowest panel, all wave functions are easily assigned by the quantum numbers U I (H-CO stretch), u2 (C-O stretch), and u3 (H-C-Obend).

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0.1

0.2

E-Eo / e V

0.3

0.4

Figure 6. Lifetime function ?(E), defined in F q (3), is shown on a logarithmic scale as a function of the available energy E - Eo above threshold. Atomic time units are used (1 a.u. = 2.42 lo-” s). The numbers on the vertical axis indicate powers of 10. (Reprinted, with permission of the Royal Society of Chemistry, from Ref. 34.)

assigned to the three quantum numbers (UI,u2, ug) mentioned above. Only in some cases the accidentally strong coupling between two energetically close states may disturb the clear picture. In the case of HNO the majority of states have an irregular nodal structure, although a fair number of resonances still can be assigned; the assignable states are mainly associated with excitation in the NO mode with none, one, or two quanta in the bending mode. Finally, for HOz an assignment is hopeless in view of the fact that the bound states close to threshold are already irregular [36]; going into the continuum does not make the dynamics more regular [37]. Figure 6 clearly bears out that for each of the three molecules the lifetimes fluctuate a great deal across the entire

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energy regime shown. The “spectra” for HNO and HO:! distinctly illustrate how the resonances gradually begin to overlap as the energy increases. Extracting decay rates k(’)from the spectra is simple if the resonances are isolated. In that case the lifetime function, for example, is to a good approximation given by

where hk is the full width at half maximum of a resonance. With increasing energy the average spacing between the resonances decreases, and at the same time the average width increases so that the resonances begin to overlap. This general behavior makes the extraction of rates gradually more tedious and at some point it is even impossible [41]. This problem, together with the fact that the larger rates are obscured in the background, means that the rates that we can extract from the spectra with reasonable confidence are limited to values smaller than about 1013 s-’. The dissociation rates extracted from the three spectra in Fig. 6 are depicted as a function of the energy above the threshold, E - Eo, in Fig. 7. Before we discuss details of the three systems, we mention only that in all three cases the rates fluctuate a great deal about an average and that this average gradually increases with energy, as generally predicted by statistical theories. Whether or not the statistical models correctly predict this average is a question at the heart of unimolecular dissociation. Although the fluctuations of the rates appear to be similar, we will stress in the following sections that their origins are rather different for, for example, HCO and HOz: Modespecific variation of the wave functions for HCO and irregular behavior in the latter case.

IV. HCO. A TEXTBOOK EXAMPLE OF REGULAR DYNAMICS The dissociation of the formyl radical HCO into H and CO is an illuminating example for studying in great detail the breaking of a bond in a triatomic molecule. It is a textbook example of resonances in polyatornic systems. Because of its importance in combustion, the fragmentation of HCO has been intensively studied both experimentally and theoretically. For a recent overview covering the literature up to 1994 see the review by Neyer and Houston [ 181. The most detailed spectroscopic investigation was published less than half a year ago by Tobiason et al. [42], who used stimulated emission pumping to resolve about 60 resonances. On the theoretical side, many research groups have exploited the dissociation of HCO as

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0

0.1

0.2

E-E,

IeV

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0.4

Figure 7. Dissociation rates k as extracted from the quantum mechanical calculations (open circles). The statistical rates are represented by the step functions and the filled circles represent the classical rate constants as obtained from elaborate classical trajectory calculations. (Reprinted, with permission of the Royal Society of Chemistry, from Ref. 34.)

a test case of developing time-independent [43-46] as well as time-dependent [40, 471 quantum mechanical methods. Most of these studies used the Bowman-Bitman-Harding (BBH) PES, which was constructed by-at that time extensive-ab initio electronic structure calculations about 10 years ago [48]. Although this PES provides a generally realistic representation of the dissociation process, the number of details, which became available with the

RESONANCES IN UNIMOLECULAR DISSOCIATION

76 1

most recent experimental data for HCO [42] and DCO [16, 491, demanded a new, hopefully more accurate PES. Simultaneous with the latest round of SEP experiments, a new global ab initio PES has been constructed on the MRCI level [32] that is the basis for all our dynamics calculations. After a slight adjustment of the potential in order to correct for an underestimation of the CO stretching frequency by about 25 cm-' , the measured term energies of the 15 bound states and the about 60 resonances are reproduced with a root-mean-square (rms) deviation of about 20 cm-I, compared to 50 cm-l obtained with the BBH potential [42]. The deviation of our calculations is more or less uniform over the entire energy range considered while the deviation obtained with the BBH PES increases rather drastically with E. There are many interesting facets of this elementary fragmentation process. In this presentation we emphasize primarily the mode specificity, that is, the more or less sytematic dependence of the resonance widths and the final-state distributions on the excited mode and the number of quanta in each mode. (For more complete analyses the reader is referred to Refs. 32 and 33 and some future publications.) Due to the low density of (resonance) states and the weakness of the intramolecular coupling, especially between the dissociation coordinate R and the CO stretching mode r, the vibrational dynamics of HCO is mainly regular, even at energies high above the dissociation threshold. The wave functions in the upper four panels of Fig. 5 intriguingly reflect this regularity: They show a clear-cut nodal pattern that in most cases allows a straightforward assignment (u,, u2, u j ) (the three modes have been defined above). Only if two resonances are accidentally very close in energy can they mix with the result that the wave functions do not have a simple behavior, thus complicating the assignment. The lower panel of Fig. 5 shows an example; in this case it is more sensible to make the assignment in view of the energy of the state rather than the wave function pattern. The rate constants k extracted from the resonance widths are depicted as a function of the access energy E - EO in the upper part of Fig. 7. In order to emphasize the comparison with the other two systems, only a small portion above the threshold is shown. The rates show the generic behavior typical for a system with a barrier in the exit channel: An exponential increase reflecting tunneling through the barrier is followed by a leveling off as the energy exceeds the adiabatic potential barrier, that is, the potential barrier plus the zero-point energies corresponding to the modes orthogonal to the dissociation coordinate [50]. Superimposed on this secular behavior are pronounced fluctuations that encompass about three orders (!) of magnitude. These fluctuations are typical for a mode-specific system [21, 51-53] and basically reflect the different types of wave functions of the various resonance states.

762

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Resonances with pure excitation of the CO stretching mode (0, u2, 0) (an example with u2 = 8 is shown in Fig. 5) have the smallest rate and therefore the longest lifetime; energy transfer from r to R is rather inefficient, and therefore the system needs a long time before enough energy is accumulated in the dissociation coordinate to permit dissociation. On the other extreme, direct excitation in R allows a rather rapid bond rupture, and therefore the resonances (u1, 0, 0) have the shortest lifetimes. Excitation of the bending mode (0, 0, u3) leads to lifetimes that are between C-O excitation as the lower limit and H-C excitation as the upper extreme. This mode specificity is further elucidated in Fig. 8, where we show the widths for several

2

6

I0

12

"2

Figure 8. Resonance widths (in cm-l) for the dissociation of HCO as a function of the CO stretching quantum number y for several series as indicated. The terms vl and u3 are the H-CO stretch and H-C-0 bending quantum numbers, respectively. (Reprinted, with permission of IOP Publishing, from Ref. 8.)

763

RESONANCES IN UNIMOLECULAR DISSOCIATION

resonance series as a function of u2. Of course, mode specificity requires an assignment, whether it be in normal modes, local modes, or whatever coordinates. Without any classification, representations like the one in Fig. 8 would not be possible. The calculated resonance widths agree in most cases very favorably with those measured by Tobiason et al. in the SEP experiment [42]. Some examples are displayed in Fig. 9. The calculations using the new PES reproduce the general trends much better than the previous calculations employing the BBH potential, especially at high energies [42]. Nevertheless, there are still a few resonances for which substantial disagreement is found; which particular features of the PES are responsible for these failures is not known to us. In accord with common experience, we found that resonance widths are much more difficult to improve by slight modifications of the PES than the resonance energies. The relatively strong dependence of the widths on the excited modes can easily be read off the scales in Fig. 8.

j1

II \ \

1

:jr"" (1 .vz

2)

40

20

0 1

"2

,' 2

3

4

5

6

7

8

"2

Figure 9. Comparison of measured [42] and calculated [33] resonance widths (in cm-') for selected HCO resonances. The terms UI. u2, and u3 are the H-CO stretch, C-0 stretch, and H-C-O bending quantum numbers, respectively. (Redrawn from Ref. 42.)

764

R. SCHINKE et al.

In view of the strong mode specificity of the dissociation of HCO, it is not at all surprising to find that the statistical rate does not reproduce well the quantum mechanical average, as seen in Fig. 7. The statistical prediction is significantly larger than the quantum mechanical average rate (note that the logarithmic scale is quite extended in the figure). However, one must keep in mind that the particular way, in which we determine dissociation rates, namely the extraction of resonance widths from a spectrum, is “blind” to very large rates, because the corresponding resonances are hidden in the background. Including the very large rates would certainly push up the quantum mechanical average and thus improve the agreement with the statistical rate. Tunneling corrections [50] are not incorporated in the statistical rate so that it starts abruptly when the first channel opens at the TS. Next we turn to a brief discussion of the final-state distributions, Z’(”(n,j), of the CO fragment, where n and j are the vibrational and rotational quantum numbers, respectively. Let us emphasize again that each resonance has a distinct final-state distribution and that the dissociation process can be claimed to be completely understood only when all P(i)(n,j)are known. This is a formidable task, not only experimentally. In the case of HCO the final-state distributions for approximately 140 resonance states have been calculated but not fully analyzed at present time. A systematic investigation is currently in progress. Figure 10 depicts the vibrational distributions for the (0, u2, 0) series of resonances for u2 = 8,. ..,11, and a very systematic variation with the CO stretching quantum number is observed: With increasing u2 the distributions shift in a gradual manner to larger and larger n values and always peak near the highest accessible state. This trend can be qualitatively explained by an adiabatic picture in which the CO degree of freedom is adiabatically separated from the other two modes [32]. Originally the energy is mainly stored in the r coordinate and the amount of energy contained in R is insufficient to overcome the barrier, A vibrationally nonadiabatic transition with a Auz value of 4-5 (internu1 vibrational energy redistribution, IVR) is required to pump enough energy into the dissociation mode in order to allow the H-CO bond to break. This scenario, which is quite similar to the An = 1 propensity rule often found in the fragmentation of many weakly bound van der Waals molecules (see Ref. 54 or Chapter 12 of Ref. 4) explains the n = u2 - 5 propensity seen in the final vibrational state distributions. Rather different vibrational distributions will be presented for HO2 in Section VII. Rotational state distributions in fragmentation processes often reflect, in a quite direct manner, the angular dependence of the wave function at the transition state, that is, the y -dependent distribution of dissociating molecules before they enter the exit channel [9, 10, 55, 561. In a semiclassical picture, the modulus square of the TS wave function determines the initial conditions

765

RESONANCES IN UNIMOLECULAR DISSOCIATION

= 11

“co Figure 10. Final vibrational state distributions of CO following the decay of the (0, u2,O) resonances with y = 8,. ..,1 1 . The arrows mark the highest accessible state for the respective

resonance. (Reprinted, with permission of the American Institute of Physics, from Ref. 32.)

of a swarm of classical trajectories that are started at the “point of no return” with positive momentum PR, that is, that immediately lead to fragmentation. HCO nicely exemplifies this “mapping” of the transition-state wave function. Figure 11 shows the rotational distributions following the decay of some of the (0, u2, 0) resonances. They all look rather similar, having a bimodal shape: a main maximum at low values of j, a node near j = 10 or so, and a less intense maximum at higher rotational states. The main maximum is superimposed by a secondary oscillation that can be explained in a Franck-Condon-type picture [20, 321. This bimodality reflects the bimodal shape of the TS wave functions, an example of which is shown in Fig. 12. Since the wave functions have the same qualitative shape for all the (0, u2,O) resonances shown in Fig. 11, it is not surprising that the resulting rotational state distributions are also qualitatively similar. Note that at the TS the wave function has a node in the angular coordinate although inside the well the bending degree of freedom is not excited. This indicates that in this particu-

Figure 11. Final rotational state distributionsof CO following the decay of the (0,u2, 0) resonances with u2 = 4, . . . , 8. The vibrational state of CO is the state with the largest probability. (Reprinted, with permison of the American Institute of Physics, from Ref. 32.)

rotationol state j

i

767

RESONANCES IN UNIMOLECULAR DISSOCIATION

180,

2.0

1

,

I

1

1

3.0

I

1

i

1

1

LO

R Iq,l

1

i

i

I

5.0

i

i

i

i

1 1

6.0

Figure 12. Angular dependence of the (0, 7, 0) resonance wave function of HCO. (Reprinted, with permission of the American Institute of Physics, from Ref. 32.)

lar case the bending degree of freedom is also involved in the IVR process described above. In view of the statistical behavior of the fragmentation of H 0 2 (discussed below), we underline that for HCO, in an adiabatic sense, a single bending level is excited at the TS and the angular dependence of this single level is mapped into the rotational distribution of CO. Other examples are the photodissociations of CINO(Tl) [56, 571 and FNO(S1) [58, 591. Rotational distributions have been measured for only very few vibrational resonances [191 and excited rotational states of HCO; two examples in comparison to our ab initio results are depicted in Fig. 3. They show the same overall behavior as the distributions for J = 0 depicted in Fig. 11, a main peak at low values of j and a broader but less intense maximum at larger rotational states. As discussed in Ref. 32, the first peak originates from an angular region where the rotational-translational coupling in the exit channel is very weak, and therefore it can be explained by a Franck-Condon mapping mechanism (see Chapter 10 in Ref. 4). In such cases it has been demonstrated that the shape of the initial state wave function (the TS wave function in the fragmentation of a resonance state) is more or less directly reflected in the final-product-state distribution. This mapping is mediated by a Fourier transformation between the y a n d j representations [60]. Since the angular parts of the wave functions for the overall rotational states 503 and

768

R. SCHINKE et al.

J13are different, it follows that also the resulting CO rotational distributions differ. This explains the different oscillatory patterns overlaying the first peaks in the distributions shown in Fig. 3. The first system for which this J state dependence has been demonstrated was the dissociation of water in the first absorption band [61]. The second peak in the CO rotational state distributions stems from an angular region where the coupling is not negligible. In such cases details of the angular dependence of the initial-state wavefunction are more or less erased during the final fragmentation [62], with the consequence that t h e j distribution does not depend much on the total angular momentum state. In conclusion, the dissociation of HCO in its ground electronic state is an intriguing example of mode specificity. The dynamics is mainly regular and almost all states can be easily assigned either by a Dunham expansion of the term energies or by considering the nodal structures of the stationary wave functions. Not unexpectedly, the statistical rate is larger than the quantum average by at least a factor of 5. The final-state distributions depend sensitively on the modes that are excited and on the number of quanta in each mode, and they can be explained, at least qualitatively, by simple models. The calculated energies and widths agree quite well with the best resolved measurements available today. It was certainly worth calculating a new ab initio potential-energy surface for this system.

V. DCO: A SPECTROSCOPIC CHALLENGE When hydrogen is substituted by deuterium, the spectroscopy changes dramatically. While almost all bound and resonance states in HCO are readily assignable, only few states in DCO can be unambiguously classified. The reason is a l :l :2 “resonance” (i.e., near degeneracy) of the D-C stretch, C - 0 stretch, and D-C-0 bending frequencies: w1 = 1900.6 cm-’, w2 = 1804.8 cm-’, and 2w3 = 1675 cm-’ (according to the calculations in Ref. 17). Especially the resonance between the two stretching modes destroys the assignment, which is so straightforward in the case of HCO. The nearly 100% mixing between the corresponding wave functions is best seen in an “experiment” that only theorists can perform. Let us consider the system XCO with the mass of the fictitious atom X, mx,changing continously from 1 (X = H) to 3 (X = T). The resulting term energies of the three relevant excited vibrational states are depicted as functions of mx in Fig. 13. For mx = 1 the two stretching states (1, 0, 0) and (0, 1, 0) are well separated in energy and the assignment provides no problem. The variation of mx in principle does not affect the C - 0 stretching frequency and 0 2 would stay constant (in a diabatic sense). On the other and therefore hand, the X-CO frequency scales approximately as 1/&

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decreases with increasing mass (again, in a diabatic sense). However, as a consequence of Wigner's noncrossing rule, the two adiabatic curves are not allowed to cross but avoid each other, and this avoided crossing occurs just in the vicinity of mx = 2 (i.e., DCO). The state that for hydrogen has the clear character of excitation in the CO mode finally turns into excitation in the X-C mode for tritium and vice versa. This situation is completely analogous to avoided crossings of two potential-energy curves as functions of the internuclear distance. The mixing between the two stretching states and the analogy with mixing of electronic states become even more apparent in the inset of Fig. 13 where we show, for one particular state, how the expectation values of the kinetic energies in R and T , (TR)and ( T J , change with mx. These curves are very reminiscent of the diabatic mixing angle near an avoided crossing of two potential curves that belong to electronic states with the same symmetry. As a consequence of the strong mixing of the two stretching states, the corresponding wave functions are rotated in the (R, r ) plane by about 45" away from the axes. This is clearly seen in the top panel of Fig. 14, where we show the wave functions of DCO as functions of R and r for all those states that are unexcited in the angular coordinate. The states that are assigned as (0,1,O) and (1,O. 0) actually are better described in terms of n o m l modes, which account for the mixing, rather than by local-mode quantum numbers referring to the Jacobi coordinates (R, r). However, the degree of mixing changes with the total energy; that is, the avoided crossing seen in Fig. 13 (on page 770) shifts in a complicated manner to other masses mx when the energy is increased. Thus, while some states are reasonably well described in terms of normal modes, an assignment by local modes is more appropriate for other states (see, e.g., the states in the most left column of Fig. 14). Furthermore, to make things worse, the state mixing also changes when the bending degree of freedom is also excited by one or several quanta. In conclusion, a meaningful assignment that is valid over the entire energy regime is not possible. More information about this fundamental question of spectroscopy can be found in the original publications [16, 171. Mixing of states when they are nearly degenerate is, of course, an old theme of spectroscopy (Fermi resonances) and is observed in many polyatomic molecules. Incidentally we note that the same general effect has been found by us in the ground state of HNO. However, there the mixing occurs between the NO stretching and the bending states [38]. In conclusion, although the PESs for HCO and DCO are the same, the spectroscopies of the two isotope variations are very different. While almost all states in HCO can be rigorously assigned, the opposite is true for DCO. Nevertheless, a detailed comparison between the more than one hundred resonance energies measured by SEP [16] and the ab initio calculations [17]

770

R. SCHINKE et al.

HCO

1400

1

DCO

1.5

2

TCO

2.5

3

Figure 13. Excitation energies (with respect to the vibrational ground state) of the second, third, and fourth excited vibrational states of XCO as functions of the mass m x . The dashed lines schematically indicate diabatic energy curves. The inset shows the expectation values of the kinetic energies (measured in terms of the corresponding values in the ground vibrational state) of the fourth excited state [(I, 0, 0) for HCO]. (Reprinted, with permission of the American Institute of Physics, from Ref. 17).

is possible and yields a similarly good agreement as for HCO. The deviations for the widths are slightly larger than for HCO, though. Because of the lack of a clear-cut assignment, the mode-specific behavior of the dissociation rates as illustrated in Fig. 8 for HCO does not exist. Yet, DCO cannot be described as an irregular or “chaotic” system; the wave functions still appear too regular to be considered as chaotic (in contrast to the HO2

RESONANCES IN UNIMOLECULAR DISSOCIATION

77 1

Figure 14. Contour plots of the wave functions for DCO in the ground bending states

v3 = 0. The numbers on the right-hand side denote the polyad number N = U I + ~2 + u3/2. The quantum numbers above each column, (UI, v2. u3), indicate the assignment if the substantial

mixing between the modes were not present. For more details see the text and the original publication (171. For ease of the visualization the relevant potential cut is shown in the upper left comer. (Reprinted, with permission of IOP Publishing, from Ref. 8.)

772

R. SCHINKE et al.

wave functions discussed below). On the other hand, severe mixing between zero-order states is the prerequisite for chaos, and in this qualitative sense DCO might be interpreted as a “precursor” for truly irregular systems like HOz. It will be interesting to investigate in the future whether an assignment based on periodic classical orbits is better suited for DCO than traditional methods [63, 641.

VI. I h O : A MIXED REGULAR-IRREGULAR SYSTEM The potential well for HNO is roughly a factor of 3 deeper than for HCO, which leads to a considerably higher density of states p ( E ) ; while the well for HCO supports only 15 bound states, there exist 215 bound states for HNO. The consequence of this increase of p ( E ) is an increase of accidental (near) degeneracies and thus of the mixing between zero-order states. In other words, the quantum dynamics gradually becomes more complex and irregular with rising energy. Nevertheless, series of quite regular states exist even at high energies where the majority of states are already irregular. Examples of wave functions for a regular and an irregular state, whose energies are both above the H + NO continuum, are depicted in Fig. 15. While the regular wave function has a quite clear nodal structure and can be assigned to 11 quanta of excitation in a mode in the (r, y) plane, the irregular wave function is obviously unassignable. It should be clear that the dissociations of these two states will give quite different rates as well as product-state distributions. A detailed analysis of all bound and resonance states is in progress and will be published at a later date. The coexistence of substantial regular and irregular parts of the classical phase space is also reflected by the statistics of the bound-state energy levels shown in Fig. 16. The nearest-neighbor level statistics on the left-hand panel is between the Poisson distribution, predicted for a regular system, and the Wigner distribution, which according to random-matrix theory describes irregular systems [65,66]. The Brody parameter [67]is 0.55 f 0.1 and clearly bears out the mixed character of HNO. The right-hand panel shows the & statistics and again the numerical results for HNO lie between the predictions for the regular and the fully irregular case. The PES for HNO does not have a barrier between the well region and the exit channel; furthermore, the TS is quite loose so that also the potential curves for the lowest adiabatic vibrational-rotational states are purely attractive at large and intermediate H-NO distances (see Fig. 8 of Ref, 34). Therefore it does not come as a surprise that the dissociation starts right at the threshold with rates that are large compared to HCO (Fig. 7). As for HCO

\

\ %/

H.

1

w

Figure 15. Three-dimensional plots of a regular (a) and an irregular (b)continuum wave function in the dissociation of HNO. The energies are in the same range.

(a)

z

h

G

a'

u)

T

0

.5

1

.5

\

\

5

L \

0

1

d<S>

2 3 4

0

1

2

I

I I

1 I

Figure 16. Left-hand panel: The nearest-neighbor energy spacing distributions P(sl(s));the quantum mechanical results are represented by the histograms (full lines) while the prediction for a regular system (Poisson distribution) and that for an inegular system (Wigner distribution) are represented by the short dashes and the long dashes, respectively. Right-hand panel: The z3(iiL)} distributions; the quantum mechanical results are represented by the full lines while the prediction for a regular system and that for an irregular system are represented by the short dashes and the long dashes, respectively. (For more details see Ref. 36. Reprinted, with permission of the Royal Society of Chemistry, from Ref. 34.)

1

Id0

J

I C

Y

h.

3, I I

1

RESONANCES IN UNIMOLECULAR DISSOCIATION

775

the dissociation rates fluctuate a great deal about some average. However, the fluctuations are much less pronounced than in the former case, encompassing only one order of magnitude rather than three. While for HCO the fluctuations reflect mainly the mode specificity; that is, excitations of the various modes lead to substantially different lifetimes. In the case of HNO they are predominantly the consequence of the irregular behavior of the wave functions, as the one shown on the right-hand side of Fig. 15. Because of the generally larger degree of irregularity (or mode mixing), it is not unexpected that the statistical assumptions are on the average better fulfilled. Thus, although the rates predicted by RRKM seem to be, as for HCO, an upper limit, the deviation from the quantum mechanical average is “only” a factor of 2 or so. In our analysis of absorption-type spectra in order to extract the dissociation rates, the very broad resonances, which make up the “background” of the spectrum, are not taken into account. Therefore, the true quantum average is actually higher than the average of the points shown in Fig. 7, which means that the deviation between the statistical rates and the quantum average is probably smaller than a factor of 2. Although the analysis of HNO is not yet at all complete, we can summarize the presently available data in the following way: HNO is a mixed regular and irregular system, and therefore the requirements for a statistical analysis to be trustworthy are better fulfilled than for HCO. As a consequence the RRKM dissociation rate is on the whole in better agreement with the quantum mechanical average. Nevertheless, regular states exist up to very high energies, even above the fragmentation threshold, and this regularity shows up in the final-state distributions.

VII. HO2: CLASSICAL CHAOS REFLECTED IN DISSOCIATION RATES AND PRODUCT-STATE DISTRIBUTIONS The HO;! PES has roughly the same well depth as the HNO potential. However, since the anharmonicity along the 0 2 coordinate r is significantly larger than for the NO mode in HNO (see Fig. 4), the density of states is also significantly higher. There are altogether 726 bound states (361 with odd and 365 with even parity [36]) compared to “only” 215 for HNO. Since the intramolecular coupling is certainly not weaker than for HNO, it is not difficult to surmise that H02 is more irregular than the former system, and this is clearly born out by the statistics of the bound states shown in Fig. 16. Both the level spacing statistics and the 3 statistics are very close to the predictions for irregular systems; the Brody parameter is 0.92 0.1, compared to 0.55 & 0.1 for HNO. The high degree of irregularity is also illustrated by the wave functions. Figure 17 (on page 777) shows examples of continuum wave functions for neighboring resonances in two energy regimes, the first

*

776

R. SCHINKE et al.

one being just above threshold, which is roughly 0.1 eV. In all cases the wave functions show an irregular nodal pattern that clearly rules out any assignment, at least in the usual spectroscopic sense. Also, there is no systematic trend as one goes from one resonance to the next one; the nodal pattern changes in an unpredictable way. The same kind of fluctuating behavior is, not unexpectedly, also shown by the dissociation rates and the final-state distributions of 0 2 , as we will elucidate below. Extracting the rates from the spectrum shown in the lower part of Fig. 7 is certainly not staightforward. At energies only slightly above threshold the resonances are very narrow and do not overlap, so that in most cases a representation by Lorentzians is unambiguous. However, with increasing excess energy the resonances start to overlap, which makes the extraction of rates cumbersome and finally impossible. For a more detailed discussion see Ref. 37. This inherent problem must be kept in mind when we consider the rates for the dissociation of H02 into H and 0 2 shown, for both parities, in the lower part of Fig. 7. As for HNO, the minimum energy path of the PES does not have a barrier, and therefore the fragmentation starts right at threshold with relatively large rates. Again, the rates fluctuate a great deal about an average; the fluctuations encompass about two and a half orders of magnitude near threshold and are still one order of magnitude at the highest energies considered. Of all the systems discussed in the present contribution, HOz is the most irregular one, and therefore it is not surprising to find that the requirements of the statistical models are fulfilled the best: the RRKM rate goes right through the quantum mechanical points and satisfactorily reproduces the quantum average [37]. In addition, the rates obtained from classical trajectories (the result for one energy is included in Fig. 7) also agree quite well with both the quantum average and the RRKh4 prediction. Without showing results we mention that the distribution of (classical) lifetimes can be well represented by a single exponential over the entire energy range, which also suggests that the fragmentation is random [30]. The unimolecular dissociation of HO2 has recently received much interest; Mandelshtam et al. [68],for example, performed a similar study and obtained similar results for the eigenenergies and the dissociation rates. Let us now turn the discussion to the final-state distributions. If the wave functions show an irregular, unpredictable behavior, it is not difficult to surmise that the final-state distributions behave similarly. Figure 18 shows the ratio of the probabilities with which the n = 0 and the n = 1 vibrational states of 0 2 are populated. While the vibrational state distributions for HCO change in a systematic way as one goes from one resonance to the other, the n = O/n = 1 ratio for H02 fluctuates a great deal as a function of (resonance) energy. There is no apparent systematic behavior, except that with increasing energy the average decreases and approaches 1, as predicted by statis-

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RESONANCES IN UNIMOLECULAR DISSOCIATION

-

-

E 0.15eV

E 0.25 eV

a

t

o

20

LO

60

[degl

80

o

10

20

i

30

o

20

LO

60

T Ideg1

mo

20

10

30

j

Figure 17. Plots of resonant wave functions for H02 as functions of R and y for two energy regimes, E - 0.15 eV and E - 0.25 eV. The 02 coordinate r is fixed. Shown is the logarithm of the modulus square of the total wave function q ~The . right-hand side in each column depicts the corresponding rotational state distributions at these energies. The precise energies (in eV) are from top to bottom: 0.151, 0.152, 0.154, 0.156, 0.158, and 0.252,0.254, 0.257, 0.259, 0.263, respectively. The coordinate R ranges from la0 to 5ao. (Reprinted, with permission of the American Institute of Physics, from Ref. 37.)

778

R. SCHINKE et al. 103

la*

F:--

.

7

100

quanta1

:

.

-

.-. . . .. . : :=-> -.-- - ____ .. . . . . . .

: ’# I **

‘\

Y)z)

3

- classical - -- PST

- ..

10’

I

I

:

la-’

63

*.

-

=

9

-------a

*

0135

6

-

0

--L

.

.

1

0.L5

a5

Figure 18. Vibrational product state distributionof 0 2 following the dissociation of HO2, both quantum and classical, together with the predictions of PST. (Reprinted, with permission of the American Institute of Physics, from Ref. 37.)

tical arguments. The prediction due to phase-space theory (PST [69]) yields a ratio that is on average too small; that is, the vibrational distribution is too “hot” compared to the quantum results. The ratio obtained from classical trajectories is in better agreement with the quantum calculations, especially at the higher energies. At low energies, close to the threshold for the n = 1 channel, violation of zero-point energy overestimates the n = 1 probability and therefore leads to a significant underestimation of the n = O/n = 1 ratio. The final rotational state distributions of the products in the fragmentation of a polyatomic molecule contain additional clues about the intra- and intermolecular dynamics, especially about the coupling in the exit channel. In realistic as well as model studies it has been observed that the rotational state distributions of the photodissociation products “reflect” the angular dependence of the wave function at the transition state and the anisotropy of the PES in the exit channel [4, 9, 101. HO2 is no exception. Examples of final rotational state distributions are shown, together with the corresponding wave functions, in Fig. 17. Several general features are immediately apparent: (i) The number of occupied rotational channels increases steadily with energy, which is simply a result of energy conservation. (ii) While at low energies all states accessible at this energy are populated, at higher energies the levels close to the energetic cutoff remain unfilled; the gap to the highest accessible state increases gradually with E.

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(iii) The distributions show, except for the lowest energies, a very complicated oscillatory behavior, with the number of oscillations generally increasing with energy; the latter observation is in accord with the increase of the number of minima/maxima in the angular dependence of the TS wave function as E increases. (iv) There is no apparent correlation in the oscillatory structures as one goes from one resonance to the next one; the fluctuations seem to be random and unpredictable, just like the nodal structures in the wave functions. The results of the quantum calculations are compared, for four energies, to the results of classical trajectory calculations as well as two types of statistical models in Fig. 19. First of all, except for the pronounced oscillations, which are certainly caused by quantum interferences, the classical rotational state distributions agree on average extremely well with the quantum distributions; especially their extent is accurately reproduced. Phase-space theory, which assumes that for total angular momentum J = 0 all asymptotically accessible states are equally filled, yields poor agreement with the exact results. The PST distributions extend all the way to the highest possible state, while the quantum as well as classical distributions, which incorporate the full dynamics from the inner region to the asymptotic product channel, die off at lower rotational states. The gap increases with the total available energy. On the other hand, the statistical adiabatic channel model (SACM [70, 7 11). which assumes that all energetically accessible adiabatic bending/rotational states are equally populated at the transition state and that there is no further coupling between the adiabatic states from the TS to the products, yields too “cold” rotational state distributions. The vertical bars in Fig. 19 indicate the highest state,jSAcM, that would be populated according to SACM. The true distributions extend to significantly higher rotational states, although in each case ~ S A C Mmore or less marks the beginning of the roughly exponential decay toward the high-j tail. While PST assumes strong coupling among the rotational states all the way to the products, SACM, on the other hand, completely neglects any coupling between the adiabatic states beyond the TS. The truth, at least in the present case, lies inbetween. If Ref. 37 we applied a very simple model that, apart from the oscillations, yields reasonable agreement with the observed rotational distributions. It is based on two assumptions: first, that the wave function at the TS is a linear superposition of all energetically accessible (bending) states with equal weighting and, second, that the final rotational state distribution, on average, is obtained by expanding the TS wave function in terms of the final rotor states (Franck-Condon mapping [4]; see also Refs. 10,72, and 73). Dynamics in the exit channel, which is small but not negligible in the case of HOz, is not taken into account in this model. In conclusion, the dynamics of HOz at high energies, below as well as

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Figure 19. Comparison of the quantum (filled circles, long dashes) and the classical (solid lines) rotational product distributions of 02(n = 0) following the dissociation of HOz for four energies as indicated; the precise energies of the correspondingquantum resonances are 0.1513, 0.2517, 0.3507, and 0.4471 eV, respectively. Also shown are the distributions obtained from PST (short dashes). All distributions are normalized so that the areas under the curves are equal. The arrows on the j axis indicate the highest accessible state at the respective energy and the vertical bars on the classical curves indicate jSACM. the highest populated state according to the SACM. (Reprinted, with permission of the American Institute of Physics, from Ref. 37.)

above the dissociation threshold, is mainly irregular, and therefore the fragmentation rate is on average satisfactorily described by statistical models. The strong fluctuations of the state-specific rate constants are caused by the irregular, randomlike behavior of the corresponding wave functions, which is in accord with the fact that the distribution of the fluctuations seem to

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follow the predictions of random-matrix theory [37]. On the other hand, statistical methods, which make unrealistic assumptions about the dynamics in the exit channel beyond the TS,at least for H02, do not yield satisfactory final vibrational and rotational state distributions for the 0 2 fragments.

VIII. RESUME AND OUTLOOK The series of molecules HCO, DCO, HNO, and HO2 is well suited for a demonstration of the transition from regular to irregular dynamics in unimolecular dissociation processes on ground-state PESs with relatively deep wells. Whether a molecule shows regular or chaotic motion depends intimately on the density of states p(E) and/or the coupling between the various degrees of freedom. In the cases discussed in the present overview both p ( E ) and the degree of internal mixing of zero-order states increase gradually from the mainly regular system HCO to the essentially irregular dynamics of H02, and the dissociation rates as well as the final-state distributions show this. The wave functions for HCO have mostly a regular nodal pattern, and this simple behavior is reflected in the observables. The decay rates and the final-state distributions for H02, too, reflect the corresponding wave functions. But in this case the wave functions have complicated, irregular nodal structures, and the same general behavior is found in the rates and the fragment distributions. They fluctuate a great deal about some average, having no systematic behavior as one proceeds from one resonance to the next one. The wave functions are the central quantities, and therefore we showed examples for all the four systems considered. Inspection of wave functions reveals more or less directly how the observables generally behave. This is, of course, an obvious statement. Knowing the wave functions is the great asset of theory compared to experimental spectroscopy, and it is not difficult to surmise that many spectroscopic misinterpretations could be avoided if the underlying wave functions were known. HCO is a relatively simple system. However, that does not mean that it is uninteresting. On the contrary, the availability of high-resolution experimental data for about 60 resonance states (and about 120 for DCO) calls for high-quality quantum dynamics calculations. The level of detailed comparison between experiment and theory possible for HCO (and likewise for DCO) is unprecedented for a dissociating molecule, which makes it a real challenge for dynamical theory. Because of the large amount of experimental data (energies, widths, rotational constants, and final-state distributions), an accurate PES is required. A less accurate potential might reveal the general dissociation dynamics but certainly not all the fine details. In this respect HCO is actually a rather difficult system; both the PES and the dynamics calculations are required to be highly accurate.

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In the case of H02 the observables show erratic looking fluctuations that are certainly the result of complicated quantum mechanical interferences. Interference structures always depend sensitively on details of the dynamics and ultimately on details of the PES, and therefore, one might be tempted to believe that the potential can be determined by an “inversion” of experimental data. That is very likely not the case, simply because the sensitivity of the observables with respect to the PES is too strong, that is, the variations of rates, for example, with changes of the potential are nonlinear. In other words, no calculated PES will be sufficiently accurate to allow a state-bystate comparison with experimental data (provided they exist). For example, fluctuating dissociation rates have been measured for H2CO [74-761 and CH3O [77,781 and fluctuating rotational state distributions have been observed in the dissociation of, for example, NO2 [79,801. Can one ever hope to reproduce these data on a state-by-state level? Certainly not! One should be content with reproducing the average dissociation rate and the average rotational distribution. However, that can be achieved with less accurate potentials and more approximate dynamics calculations. For example, for the dissociation of H02 we found that the average rate is well described by statistical theories; classical trajectories even reproduce both the average rate and the average final-state distributions of the 0 2 fragments reasonably well. Average values, however, are less sensitive to details of the PES, at least in the case of HO2, and are determined by relatively small regions of the coordinate space. For example, the statistical rate is mainly determined by the potential near the transition state, and the average final-state distribution depends mainly on the forces in the exit channel from the transition state to the asymptote. In this sense HO:! is actually the simpler of the two systems, HCO and H02. In the future we will attempt to extend the calculations to systems with still higher densities of states. In all cases studied up to now, the atomic fragment is hydrogen, leading to a small reduced mass associated with the dissociation coordinate. We are currently calculating PESs for OClO and NO2 in their ground electronic states. For the latter system measured rates as well as rotational and vibrational state distributions are available, and it will be interesting to compare them with the results of classical trajectory calculations. Another interesting question concerns the applicability of statistical models for predicting final-state distributions, depending on the nature of the transition state, that is, whether it is “tight” or “loose.” Furthermore, we will consider the construction of an average quantum mechanical rate directly from time-dependent wavepacket calculations in order to avoid the problems associated with overlapping resonances. This seems to be necessary if one wants to assess the reliability of statistical methods in a more unambigious way.

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Acknowledgments R. S. and H. M. K. gratefully acknowledge financial support by the Deutsche Forschungsgemeinschaft through the Sonderforschungsbereich357 Molekulare Mechanismen Unimolekularer Prozesse. A. J. D. is grateful for a fellowship from the Royal Society/Deutsche Forschungsgemeinschaft under the European Science Exchange Programme. D. H. M. is grateful for a fellowship from the European Union under the Human Capability and Mobility Programme. R. S. and H. M. K. gratefully acknowledge their ongoing collaboration with H.-J. Werner, P. Rosmus, F.Temps, and E. A. Rohlfing on the dissociations of HCO and DCO. Many fruitful discussions with W. L. Hase, J. Troe,C. B. Moore,W. H.Miller, H. Reisler, and many other colleagues have greatly influenced the authors’ general understanding of unimolecular dissociation.

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34. M. Stumpf, A. J. Dobbyn, D. H. Mordaunt, H.-M. Keller, R. Schinke, H.-J. Werner, and K. Yamashita, Faraday Discuss. Chem. Soc. 102, 193 (1995). 35. A. I. Dobbyn, M. Stumpf, H.-M. Keller, W. L. Hase, and R. Schinke, J. Chem. Phys. 102, 5867 (1995). 36. A. J. Dobbyn, M. Stumpf, H.-M. Keller, and R. Schinke, J. Chem. Phys., 103,9947 (1995). 37. A. J. Dobbyn, M. Stumpf, H.-M. Keller, and R. Schinke, J. Chem. Phys. 104,8357 (1996). 38. H. D. Mordaunt, H. Flothmann, M. Stumpf, A. J. Dobbyn, H.-M. Keller, R. Schinke, and

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39. M. R. Pastrana, L. A. M. Quintales, J. Brandb, and A. J. C. Varandas, J. Phys. Chem. 94,8073 (1990). 40. R. N. Dixon, J. Chem. Soc. Faraday Trans. 88,2575 (1992). 41. U. Peskin, H. Reisler, and W. H. Miller, J. Chem. Phys. 101, 9672 (1994). 42. J. D. Tobiason, J. R. Dunlop, and E. A. Rohlfing, J. Chem. Phys. 103, 1448 (1995). 43. D. Wang and J. M. Bowman, J. Chem. Phys. 100, 1021 (1994). 44. D. Wang and J. M. Bowmann, Chem. Phys. Lett. 235,277 (1995). 45. 46. 47. 48. 49. 50.

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53. B. Hartke, A. E. Janza, W. Karrlein, J. Manz, V. Mohan, and H.-J. Schreier, J. Chem. Phys. 96,3569 (1992). 54. J. A. Beswick and J. Jortner, Adv. Chem. Phys. 47,363 (1981).

55. A. Vegiri, A. Untch, and R. Schinke, J. Chem. Phys. 96,3688 (1992). 56. D. Solter, H. J. Werner, M. von Dirke, A. Untch, A. Vegiri, and R. Schinke, J. Chem. Phys. 97, 3357 (1992). 57. C. X.W. Qian, A. Ogai, L. Iwata, and H. Reisler, J. Chem. Phys. 92,4296 (1990).

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A. Untch, K. Weide, and R. Schinke, J. Chem. Phys. 95,6496 (1991). J. M. Gomez Llorente and E. Pollak, Annu. Rev. Phys. Chem. 43, 91 (1992). P. Gaspard, D. Alonso, and I. Burghardt, Adv. Chem. Phys. XC, 105 (1995). T. Zimmermann, L. S. Cederbaum. H.-D. Meyer, and H. Koppel, J. Phys. Chem. 91,4446

(1987). 66. M. C. Gutzwiller, Chaos in Classicafand Quantum Mechanics, Springer, New York, 1990. 67. T. A. Brody, J. Flores, J. B. French, P. A. Mello. A. Pandey, and S. S. M. Wong, Rev. Mod. Phys. 53, 385 (1981). 68. V. A. Mandelshtam, T. P. Grozdanov. and H. S. Taylor, J. Chem. Phys. 103, 10079 (1995). 69. P. Pechukas, J. C. Light, and C. Rankin, J. Chem. Phys. 44,794 (1966). 70. M. Quack and J. Troe, Ber. Bunsenges. Phys. Chem. 78,240 (1974). 71. M. Quack and J. Troe, Ber. Bunsenges. Phys. Chem. 79, 170 (1975). 72. S. A. Reid and H. Reisler, J. Chem. Phys. 101, 5683 (1994). 73. U. Peskin, W. H. Miller, and H. Reisler, J. Chem. Phys. 102, 8874 (1995). 74. W.F. Polik, C. B. Moore, and W. H. Miller, J. Chem. Phys. 89, 3584 (1988). 75. W. F. Polik. D. R. Guyer, W. H. Miller, and C. B. Moore. J. Chem. Phys. 92,3471 (1989). 76. W. H. Miller, R. Hernandez, C. B. Moore, and W. F. Polik, J. Chem. Phys. 93, 5657 (1990). 77. A. Geers, J. Kappert. F. Temps, and J. W. Wiebrecht, J. Chem Phys. 99,2271 (1993). 78. F. Temps, in Molecular Dynamics and Spectroscopy by Stimulated Emission Pumping, H.-L. Dai and R. W. Field, Eds., World Scientific, Singapore, 1994. 79. M. Hunter, S. A. Reid, D. C. Robie, and H. Reisler, J. Chem. Phys. 99, 1093 (1993). 80. S. A. Reid, D. C. Robie, and H. Reisler. J. Chem. Phys. 100, 4256 (1994).

DISCUSSION ON THE REPORT BY R. SCHINKE Chairman: J. Manz

S. A. Rice: Prof. Schinke, can you correlate parts of the distribution of resonances you have calculated with the “scars” of the wave function

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that are associated with unstable classical trajectories? In particular, can the widths of the resonances be grouped so as to show whether or not intramolecular energy transfer competes with the reaction? Using the language of kinetics, I believe that it should be possible to associate the scars with barriers to intramolecular energy exchange. If my assertion is correct, the presence of scars can be taken as a signal, in the quantum mechanical case, of deviation from the statistical limit description of unimolecular reaction rate.

R. Schinke: We did not analyze all resonance wave functions in the case of HO2. But from the limited investigation performed we note that we did not see pronounced scars. This is in accord with your assertion that scars are a signal of deviation from the statistical limit. HO2 is mainly irregular at energies around the dissociation threshold, and this is nicely documented by the classical calculations, which show a single exponential for the lifetime distribution function (Fig. 9 in Ref. 34 of the current chapter). On the other extreme, HCO is mainly regular, and the classical distribution function consists of (at least) three exponentials. The corresponding quantum mechanical wave functions show in most cases clear nodal structures that could be interpreted as scars. Thus, in general, I agree with your assertion that scars are associated with barriers to intramolecular energy transfer whereas the lack of scars indicates very fast internal vibrational energy transfer and therefore statistical decay. R. A. Marcus: Prof. Schinke has certainly described an array of exciting results. In the case where your wave functions showed a complicated pattern, it would be useful (for the case of an isolated internal resonance) to seek out the relevant vibrational periodic trajectories to sort out the series of such states and relate (directly or indirectly) to Kellman’s algebraic analysis of bound states. Another way of calculating the distribution of product states would be to apply an extension of RRKM that Wardlaw, Klippenstein, and I developed. However, judging from your observations, the reaction is highly vibrationally nonadiabatic, considering, for example, the considerable difference in vibrational quantum number uco in HCO and CO and the major change in bending -+ rotational state. In that case a Franck-Condon approach would seem to be much more appropriate than any adiabatic or near-adiabatic or statistically adiabatic model. R. Schinke: Concerning the rotational state distribution of 02, we applied actually a Franck-Condon mapping model that reproduces the rotational state distribution at the transition state. A full description is given in the original publication. However, the coupling is not negli-

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gible in the exit channel, and therefore one must go beyond this model and incorporate the final channel coupling. Classical mechanics is certainly a very good approximation. J. ’ h e : Let me point out that the ab initio calculations of the H02 surface still must be in error in their long-range part. High-pressure H + O2 recombination experiments clearly show that the potential is completely loose and not “semirigid” like the potential you used for illustration. For loose potentials SACM and classical trajectory calculations of product distributions (in the classical range) nicely agree. R. Schinke: The calculations for H02 are certainly model calculations as I underlined in my talk. First, as you pointed out, the potential might not be the most accurate one, especially at large H + 02 separation. Second, there might be other electronic states that are involved within the considered energy range. D. M. Newark: In Lineberger’s photoelectron spectrum of the HCO- ion, resonances above the H + CO dissociation threshold were observed. Were any resonances seen in this experiment that were not seen (at higher resolution) in the stimulated emission pumping work? R. Schinke: The earlier spectroscopicexperiments of the Lineberger group certainly probe the same parts of the potential-energy surface as the experiments of the Rollfing group, with which I compared our theoretical calculations. The Rollfing data are the most complete and the best resolved. M. E. Kellman: Following up on Stuart Rice’s point regarding scars of periodic orbits, are there abrupt changes in reaction rates as nodal patterns of the vibrational wave functions change in the regular regime, for example in DCO, where the local-mode nodal pattern breaks down? R. Schinke: No, we did not clearly see abrupt changes in the dissociation rate as nodal patterns of the underlying wave functions change. The rates fluctuate strongly with energy, and since there is no clear-cut assignment, it is difficult to recognize any schematic trends.

PHOTODISSOCIATING SMALL POLYATOMIC MOLECULES IN THE VUV REGION: RESONANCES IN THE 'E+-fC' BAND OF OCS K. YAMANOUCHI,* K.OHDE, and A. HISHIKAWA Department of Pure and Applied Sciences College of Arts and Sciences The University of Tokyo Tokyo, Japan

CONTENTS 1. Spectra of Dissociating Molecules 11. Absorption Spectrum of O C S in the VUV Region 111. PHOFEX Spectrum of the 'c+-'c' Band of OCS

IV. Fano Profile in the VUV-PHOFEX Spectrum of O C S V. Concluding Remarks References

I. SPECTRA OF DISSOCIATING MOLECULES Recent advances in ultrashort laser technology has enabled us to investigate dynamics of molecules in a time domain, and furthermore, the success of a theoretical interpretation of the results of time-domain experiments by a moving wavepacket on a potential energy surface (PES)impressively demonstrated the importance of time-domain experiments [l]. On the other hand, it is well-known that a spectrum in a frequency domain and an autocorrelation function in the time domain can be transferred with each other via a Fourier transformation [2]. Therefore, it can be said that the spectrum *Communication presented by K. Yamanouchi Advances in Chemical Physics, Volume 101: Chemical Reactions and Their Control on the Femtosecond lime Scale, XXrh Sofvay Conference on Chemistry, Edited by Pierre Gaspard, Irene Burghardt, I. Prigogine, and Stuart A. Rice. ISBN 0-471-18048-3 0 1997 John Wiley & Sons, Inc.

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represented in a frequency domain has essential information regarding the dynamics of a molecular system after it is excited to an electronically excited PES. The interpretation of a spectrum from a dynamical point of view can also be applied to a spectrum containing a broad feature associated with direct and/or indirect dissociation reactions. From such spectra dynamics of a dissociating molecule can also be extracted via the Fourier transform of a spectrum. An application of the Fourier transform to the Hartley band of ozone by Johnson and Kinsey [3] demonstrated that a small oscillatory modulation built on a broad absorption feature contains information of the classical trajectories of the vibrational motion on PES, so-called unstable periodic orbits, at the transition state of a unimolecular dissociation. In this decade dissociation dynamics of a variety of molecules has been investigated both experimentally and theoretically 141. On the experimental side, rich information regarding photodissociation dynamics has been derived by measuring a product-state distribution of photofragments, a photofragment excitation (PHOFEX) spectrum, and a Doppler profile of photofragments. From these photodissociation experiments as well as from the measurements of absorption spectra exhibiting broadened features, it has been known that most polyatomic molecules dissociate very rapidly after absorbing VUV light. However, as far as photodissociation studies in the vacuum ultraviolet (VUV) wavelength region are concerned, laser spectroscopy has afforded only fragmental information at fixed wavelength such as 193 and 157 nm. Until very recently, almost no elaborate effort has been made in order to extract the dissociation dynamics directly from an observed absorption spectrum in the VUV region. In absorption spectra measured under the bulb conditions, it is in general difficult to identify homogeneous linewidths associated with a lifetime shortening separately from inhomogeneous contribution, such as overlapping vibrational hot bands and rotational broadening. Even when free jet expansion is employed to cool down molecules, it has been difficult to sample molecules only in the coldest central region of the free jet. In the present study, we demonstrate that PHOFEX spectrum of jet-cooled molecules is most suitable to derive an absorption profile in the VUV region representing only the homogeneous broadening. 11. ABSORPTION SPECTRUM OF OCS IN THE VUV REGION Among the absorption spectrum of relatively simple polyatomic molecules, a spectral structure of the strong 'C+-'C' band of OCS in the 160-140-nm region [5] is noteworthy. Though there is a broad background-like structure in the entire absorption band, there are seven distinct features with a sep-

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aration of -800 cm-' . The wavelength positions for these seven peaks are 156.1,154.5,152.6,150.6,149.0,147.3,and 145.7nm. The absorption cross cm2. From the Fourier sections of these absorption peaks are as high as transformation of the absorption spectrum recorded by McCarthy and Vaida 161 under jet-cooled conditions, we identified recently a period of 41 fs, corresponding with intervals of -800 cm-', as a clear recurrence time in the autocorrelation function [7].This period should represent some vibrational dynamics on the PES of the excited 'C+ state. Previously, based on an absorption spectrum, two vibrational progressions were identified and were ascribed to two stretching modes; that is, a progression of the C-S stretching mode carrying strong intensity and a weaker progression of the C-S stretching mode built on one quantum of the C-O stretching mode. It is known that the excitation to the upper 'E+ electronic state leads to only one dominant dissociation channel of CO(X C+)+ S( S), and a quantum yield of S('S) was derived to be >80% in the entire 160-140 nm of the absorption band [S]. If the PES for the excited 'E+state is mostly repulsive along the dissociation coordinate and is correlated exclusively to CO(X IC+) + S('S), an intense progression should be assigned to the C-O stretch rather than the C-S stretch, since a spectral structure of a dissociating state reflects vibrational modes orthogonal to a reaction coordinate.

'

111.

'

PHOFEX SPECTRUM OF THE 'E+-lC+ BAND OF OCS

In order to clarify the vibrational dynamics on the PES of the C+ electronic state of OCS, we attempted to record a high-resolution ' F - ' E + absorption profile of OCS free from the inhomogeneous broadening by introducing a technique of PHOFEX spectroscopy with a tunable high-resolution V W laser light source [7,9, 101. Advances in nonlinear laser spectroscopy have made it possible to generate an intense, coherent, and tunable light in the VUV wavelength region by four-wave sum and difference-frequency mixing of tunable visible and UV laser beams in a nonlinear medium like rare-gas and metal vapor [ll]. It has been shown that the VUV light in the wavelength region between 200 and 115 nm can be continuously covered by using Sr, Mg,Hg, Kr, and Xe as nonlinear media. One advantage of this VUV light is its high resolving power as high as (x/6X) - lo6 originating from the high-resolution nature of visible and UV laser light adopted in the nonlinear mixing. In the measurements of the PHOFEX spectrum, we scanned the VUV wavelength while the photofragment of S( IS)was monitored by exciting the S(3DI)-S('S) transition by UV laser light. Since only the fluorescence emitted from the S(3D1)fragments in the central region of the free-jet expansion was collected, the photoabsorption of ultracold (-5 K) OCS was selectively

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detected. Since the quantum yield of S('S) is S O % , the PHOFEX spectrum thus measured is almost identical with an absorption spectrum measured under the ultracold conditions. Therefore, the spectrum should be free from rotational broadening and overlapping of vibrational hot bands, and the spectral feature in the PHOFEX spectrum represents only a homogeneous contribution, reflecting directly the photodissociation dynamics. As described later, the recent ab initio molecular orbital (MO) calculation of the excited-state PES of O C S by Hishikawa et al. [lo] shows that the 'C+ state is mostly repulsive along the dissociation coordinate r(0C- - -S). When we measured the PHOFEX spectrum of the lowest energy peak located at 154.5 nm, we found that the spectral peak exhibits significantly narrower width (-42 cm-I) [9] than that observed previously by McCarthy and Vaida under the jet-cooled conditions. By subtracting (i) the instrumental resolution and (ii) the rotational contribution estimated by using the rotational constant of the electronic ground 'E+state from the observed peak width, the homogeneous width was derived, and then the lifetime of the broadened feature was estimated to be 133 fs. The rotational temperature was estimated from the rotational structure of the A-X transition of CO contained in the sample gas by a trace amount. If the vibrational period of 41 fs derived from the Fourier transform of the absorption spectrum of the 'C+-'C+band [6] represents that of the vibrational motion near the transition-state region of the unimolecular dissociation, the lifetime associated with the 154.5-nm peak indicates that OCS vibrates more than three times along the direction perpendicular to the dissociation coordinate prior to the complete dissociation. There are two vibrational modes, the in-phase stretch and the O X - S bend, which are orthogonal to the dissociation coordinate, corresponding with the C-S stretch. Since both of the two C+ electronic states involved in the transition are known to be linear at their equilibrium position, a long progression may not be expected for the bending excited states. Therefore, the distinct progression with an interval of -800 cm-' observed in the 'C+-'C+ band is ascribed to the vibrational progression for the in-phase stretch. The interesting findings are that the PHOFEX peak at 152.6 nm is significantly broader than the peak at 154.5 nm and the peak at 150.6 nm is much broader. The lifetimes for the peaks at 152.6 and 150.6 nm were derived from the broadened peak widths to be 44 and 27 fs, respectively. It can be said that OCS vibrates only once along the in-phase stretching mode when it is excited to the 152.6-tun peak and less than once when excited to the 150.6-nm peak. In Fig. 1, the PHOFEX spectrum of the three most intense vibronic bands of OCS are compared with the corresponding absorption spectrum measured under the jet-cooled conditions by McCarthy and Vaida [63. It is clearly seen in this figure that the PHOFEX peaks were significantly narrower than the absorption peaks. This difference in the peak width demonstrated that (i)

PHOTODISSOCIATING SMALL POLYATOMIC MOLECULES

MOO0

65Ooo

660v

WAVENUMBER / cm-

793

67000

Figure 1. Comparison of the VUV-PHOFEX spectrum of a part of the I C+-' I? band of the jet-cooled OCS and the corresponding absorption spectrum measured by McCarthy and Vaida [6].

the broadening due to the rotational structure and vibrational hot bands is largely decreased in the PHOFEX measurements, since only the coldest spatial region of the free-jet expansion is sampled in the PHOFEX experiments, and (ii) the spectral resolution in the PHOFEX measurements, determined by the resolution of a tunable VUV laser, is very high.

IV. FAN0 PROFILE IN THE WV-PHOFEX SPECTRUM OF OCS It can also be noticed in Fig. 1 that spectral features for these three peaks are not symmetrical; that is, their spectral shape deviates considerably from a simple Lorentzian line shape. Since the rotational contribution in the peak width in the PHOFEX spectrum is -I cm-', which is significantly smaller than the observed peak width, these asymmetrical spectral features are regarded as Fano-type profiles, which can appear in a spectrum for quasibound states. A Fano profile was originally derived to interpret an asymmetrical spectral feature of autoionizing atoms [ 121, but it can also be identified in the electric spectrum of some simple molecules, which indirectiy or directly dissociate. It has been known that a transition from an electronic ground state to a resonance state in the excited-state PES,formed through a mixing between zero-

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K. YAMANOUCHI, K. OHDE, AND A. HISHIKAWA

order bound and continuum states, can exhibit an asymmetrical line shape, when there are nonzero transition moments to both of the bound and continuum zero-order states. It has been reported that H2 [13], Cs2 [14], and 0 2 [15] predissociate via a coupling with a dissociation continuum and Fano profiles were identified in their spectra. In the case of Cs2, Kim and Yoshihara identified a clear q-reversal in the progression of vibronic bands, each of which exhibits an asymmetrical line shape. In the optical-optical double resonance (OODR) spectrum of the 7 3 S ~Rydberg state of HgNe, Okunishi et al. [ 161 observed a characteristic asymmetrical line shape caused by a potential barrier. The resonances resulting in the asymmetrical spectral profiles for HgNe can be classified as the shape resonance, since the dissociation occurs on a single electronic potential energy curve, while those found in H2, Csz, and 0 2 can be classified as the Feshbach resonance [ 181. In the present measurements of the PHOFEX spectrum of OCS, the observed Fano profile is categorized into the Freshbach resonance, in which a zero-order bound state associated with the vibrational motion perpendicular to the dissociation coordinate couples with the zero-order continuum state associated with the motion along the dissociation coordinate. This type of resonances was also identified of the photodissociation of FNO by Reisler and co-workers [18]. By using a Fano line shape formula, the observed PHOFEX peaks were fitted, and an asymmetry parameter q and a width are determined for these three peaks. It is clearly seen in Fig. 1 that the peak width increases substantially as energy increases and that these three peaks exhibit asymmetric line profiles, though a degree of asymmetry decreases as energy increases. We performed the fitting of the observed peak profile to a Fano line formula, expressed as

where e = (E--Er)/(I'/2), with E, and r representing the energy and the full width at half maximum (FWHM) of the resonance state, respectively, and an asymmetric parameter q reflecing the interference between the discrete and dissociative states. From the least-squares fitting, r = 105(10) cm-' and q = -8.1(7) were derived for the 152.6-nm peak and I' = 186(20) and q = -20(3) for the 150.6-nm peak. Previously, we determined the corresponding values for the 154.5-nm peak as r = 41.6(2) cm-' and q = - 3 3 9 ) [7]. The observation indicates that the dissociation rate increases sensitively as energy increases in the energy range covering these three peaks. It is noted that the q value becomes negatively large as energy increases. In other words, the

PHOTODISSOCIATING SMALL POLYATOMIC MOLECULES

795

observed asymmetrical line shape approaches a symmetrical Lorentzian line shape. Recently, Hishikawa et al. [lo] calculated ab initio PESs for the electronically excited states of O C S and showed that the excited state in the 160-140-nm region is rather isolated from the other electronic states and that the slope of the PES along the dissociation coordinate is extremely flat in the Franck-Condon region from the ground vibrational state of the electronic ground state. The PES for the '72 state has a shallow valley along the dissociation coordinate r(OC-- -S) from the Franck-Condon point, and a shallower slope extends along r(0-- -CS). From the wavepacket calculation on the ab initio PES,it was demonstrated that a wavepacket moves along the direction perpendicular to the dissociation coordinate while the dissociation proceeds rapidly. The autocorrelation function derived from the wavepacket dynamics exhibits a period of 48 fs as a recurrence time for the wavepacket associated with the transition-state vibrational motion, which is comparable with the experimental period of 41 fs [6]. This reasonable agreement of a vibrational period near the transition state confirmed that the overall feature the 'I7state PES is described well by the theoretical PES. An asymmetrical line shape is known to characterize a resonance state in a bimolecular reaction. Sadeghi and Skodje [ 191derived a line shape formula representing a spectral profile for a barrier resonance in D + H2. They showed that their formula can represent well a spectral profile for resonances associated with a reactive periodic orbit as well as that for barrier resonances associated with a periodic orbit dividing surface. We have attempted to apply their formula to the PHOFEX spectral peaks of OCS. The least-squares fit to the observed peaks using their formula for an even wavepacket results in a reasonable fitting as a simple Fano formula. It should be noted that a physical significance of determined parameters in Sadeghi and Skodje's formula may not be straightforwardwhen the formula is applied to resonances other than barrier resonances. However, their analysis of resonance wave functions near the transition state of the bimolecular reaction is informative in the sense that a spectral feature such as a width and an extent of asymmetry is dependent on a type of resonance. For example, the width of the resonance peaks need not increase (or decrease) monotonically as energy increases. Indeed, in the case of the reaction D + H2 barrier resonances in the lower energy region exhibit much broader profile than resonances for reactive periodic orbits located in the higher energy region. In the photodissociation of OCS, the resonance width can also be influenced by a shape of the excited-state PES,which governs a type of resonance at a particular energy. It is expected that the PES near the transition state could be further characterized from the observed characteristicvariation of the peak profiles by simulating the absorption spectrum on the basis of a theoretically derived PES.

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V. CONCLUDING REMARKS

Through measurements of the PHOFEX spectrum of jet-cooled OCS, it has been demonstrated that detailed information of the dissociation dynamics on the mostly repulsive PES can be derived from the PHOFBX spectrum of jet-cooled molecules. The vibronic transitions at 154.5, 152.6, and 150.6 nm in the E+- C+ transition of OCS exhibit significantly narrower peak widths in the PHOFEX spectrum recorded by monitoring the S( S) fragments than those in the previously recorded absorption spectra, because (i) only the coldest region of a free jet is sampled in the PHOFEX measurements, resulting in a spectrum free from rotational and vibrational congestion, and (ii) the spectral resolution of the employed light source in the VUV region (i.e., a so-called VUV laser) is significantly high. The PHOFBX spectrum of OCS clarified for the first time that the peak shape of the three vibronic transitions are all asymmetric. The asymmetry of the peak profile was interpreted as a Fano line shape, which appears through the mixing between bound and continuum zero-order wave functions resulting in a dissociative eigen-wave function. The extent of the mixing depends sensitively on the shape of the PES in the transition-state region. It is certain that rich information is contained not only in the width and intensity of a vibronic transition peak but also in its profile. Such an asymmetric line profile can be regarded as a general feature for structured absorption profiles observed in the VUV region.

'

Acknowledgments This work is supported in part by a Grant-in-Aid for the Priority Area (No. 07240106) from the Ministry of Education, Science, Sports and Culture. The authors thank K. Yamashita (University of Tokyo) for his helpful discussion on the potential-energy surface. They also thank C. D. Pibel, S. Liu, and R. Itakura for their involvement in the present study.

References 1. A. H.Zewail, J. Phys. Chem. 97, 12427 (1993). 2. E. J. Heller, in Chaos and Quantum Physics, NATO Les Houches Lecture Notes, A. Voros, M. Gianonni, and 0. Bohigas, Eds., North-Holland, Amsterdam, 1990. 3. B. R. Johnson and J. L. Kinsey, Phys. Rev. Lett. 62,1607 (1989); J. Chem. Phys. 91,7638 ( 1989). 4. R. Schinke, Photodissociation Dynamics, Cambridge University Press, Cambridge, 1993. 5. J. W. Rabalais, J. M. McDonald, V. Scherr, and S. P. McGlynn, Chem. Rev. 71,73 (1971). 6. M. I. McCarthy and V. Vaida, J. Phys. Chem. 92,5875 (1988).

7. K. Yamanouchi, K. Ohde, A. Hishikawa, and C. D. Pibel, Bull. Chem. Soc. Jpn. 68,2459 (1995). 8. G. Black, R. L. Sharpless, T. G. Slanger, and D. C. Lorents, J. Chem. Phys. 62, 4274 (1975).

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9. C. D. Pibel, K. Ohde, and K. Yamanouchi, J. Chem. Phys. 101, 836 (1994). 10. A. Hishikawa, K. Ohde, R. Itakura. S. Liu, K. Yamanouchi, and K. Yamashita, J. Phys. Chem., 101,694 (1997). 11. K. Yamanouchi and S. Tsuchiya, J. Phys. B: Opt. Ar. Mol. Phys. 28, 133 (1995). 12. (a) U. Fano and J. W.Cooper. Rev. Mod. Phys. 40,441 (1968); (b) U. Fano, Phys. Rev. 124, 1866 (1961). 13. M. Glass-Maujean, J. Breton, and P. M.Guyon, Chem. Phys. Lert. 63,591 (1979). 14. (a) B. Kim and K. Yoshihara, J. Chem. Phys. 99, 1433 (1993); (b) B. Kim, K. Yoshihara, and S. Lee, Phys. Rev. Lett. 73,424 (1994). 15. S. T. Gibson and B. R. Lewis, J. Elec. Spec. Rel. Phenom., 80,9 (1996). 16. M. Okunishi, K. Yamanouchi, K. Onda, and S . Tsuchiya, J. Chem. Phys. 98,2675 (1993). 17. M . S. Child, Molecular Collision Theory, Academic, London, 1974. 18. (a) A. Ogai, J. Brandon, H. Reisler, H. U. Suter, J. R. Huber, M. von Dirke, and R. Schinke. J. Chem. Phys. 96,6643 (1992); J. T. Brandon, S. A. Reid, D. C. Robie, and H. Reisler, J. Chem. Phys. 97, 5246 (1992). 19. R. Sadeghi and R. T. Skoje, J. Chem. Phys. 102, 193 (1995).

DISCUSSION ON THE COMMUNICATION BY K. YAMANOUCHI Chainnun: J. Manz D. M. Neumark: Prof. Yamanouchi, which photofragment are you looking at?

K. Yamanouchi: In the VUV-PHOFEiX measurements, the photofragment of S('S) was monitored by exciting it to the S(3D,)state by the UV laser light and by detecting the laser-induced fluorescence emitted from S(3Dl).Since only the fluorescence from the S fragments produced in the central region of the free-jet expansion was collected, the photoabsorption of ultracold (-5 K) OCS was selectively detected.

PHASE AND AMPLITUDE IMAGING OF EVOLVING WAVEPACKETS BY SPECTROSCOPIC MEANS MOSHE SHAPIRO Department of Chemical Physics The Weizmann Institute Rehovot 76100, Israel CONTENTS 1. Introduction 11. Theory of Wavefunction Imaging 111. Imaging of a Highly Rotating Na2 Molecule

Acknowledgments References

We show how one can image the amplitude and phase of bound, quasibound and continuum wavefunctions, using time-resolved and frequencyresolved fluorescence. The case of unpolarized rotating molecules is considered. Explicit formulae for the extraction of the angular and radial dependence of the excited-state wavepackets are developed. The procedure is demonstrated in Na2 for excited-state wavepackets created by ultra-short pulse excitations.

I. INTRODUCTION Since the development of ultra-fast pump-probe methods [l], in which one detects the temporal evolution of molecular absorption [l, 21, or emission

Advances in Chemical Physics, Volume 101: Chemical Reactions and Their Control on the Femfosecond Time Scale, XXth Solvay Conference on Chemistry, Edited by Pierre Gaspard, Irene Burghardt, I. Prigogine, and Stuart A. Rice. ISBN 0-471-18048-3 0 1997 John Wiley 8t Sons, Inc.

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MOSHE SHAPIRO

[3]-[5], the general feeling [6]has been that such “real time” measurements contain enough information to yield the phase as well as the amplitude of the time-evolving wavefunction. In recent publications [7,81 we have presented a practical procedure for using spectroscopic experiments to derive complex wavefunctions. Our solution, which involves using both time-resolved and frequency-resolved data, makes no prior assumptions about the fluorescence frequency-internuclear distance correspondence [ 1, 51, the degree of localization of the wavepacket, or the degree of harmonicity. So far, our wavefunction imaging method was limited to non-rotating molecules. In the present paper we deal with rotating molecules. In particular, we show how to image unpolurized time-evolving mixed excited states. We explore the accuracy of our method by demonstrating the imaging of the density-matrix of a highly rotating Na2 molecule.

11. THEORY OF WAVEFUNCTION IMAGING

Consider the fluorescence from a molecular wavepacket excited from the ground electronic state by a short pulse of light. We assume that the initial energy of the molecule is E,,g,,s, where u, j denote, respectively, vibrational and rotational quantum numbers, with well defined magnetic quantum m,

where xuS,j,(R) is a ro-vibrational wavefunction which is a function of R(= (RI,R2, ...,RN)- the collection of internuclear distances specifying the shape of the molecule. The molecule is subjected to the action of a linearly-polarized pulse of the form, <(t) = eR,t(f) exp(iw,t)

(2)

where Re signifies the real part, 2 is the polarization direction, chosen for convenience to lie along the z axis, w, is a “carrier” frequency and t ( t ) is a (complex) pulse envelope. Choosing the time axis such that at t = 0 the pulse is practically over, the excited wavepacket is given as,

PHASE AND AMPLITUDE IMAGING OF WAVEPACKETS

801

$7 are electronically-excited vib-rotational wavefunctions, @ = Y ~ X with ~ , ~energies , E,, (s = u, j ) . y, are (radiative or non-radiative) decay

where

rates. Because the pulse is over at t = 0, u?, the “preparation coefficients”, are given, in first order perturbation theory, as [9, 101,

where ?(a)is the Fourier transform of the pulse, = (Eu,j - Eug,jg)/h7 and d r g= ~ “ , j I ~ ~ , j g I m ) l x u , , jand , ) , jdj,jgIrn)is an angle-averaged transitiondipole operator, given, for parallel transitions, as,

where [XI denotes (2x + l)l/z. The excited-state wavepacket spontaneously emits photons while undergoing transitions to any of the electronically-ground vibrational wavefunctions @, (where we have lumped the final state quantum numbers uf,j f in a single index f). The rate of emission from a given @ component of the is given in terms of the Einexcited wavepacket to a given ground state stein A-coefficient [9],

The (“non-dispersed”) rate of fluorescence, F m ( t ) , from the entire wavepacket is given [ 10, 111 as a coherent (double) sum of amplitudes from the excited states comprising the wavepacket, summed over all the possible final states. It can be expressed, in matrix notation, by treating s, s’ as a single index k, as

where

= F”(t), Xr I u:u?*,

is the vector of unknown coefficients, and

e“‘ is a “molecular” matrix, given as,

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By strobing the time intervals such that their number equals the number of k values, we can try to invert the e"' matrix of E!q. (7) to obtain the unknown X m vector. However, it follows from Eq. (8) that the em matrix cannot be inverted, as it contains a number of columns, explicitly all the s = s' columns, composed of a single number. This is due to the fact that for s = s' the E, - Est terms vanish, leaving the ys decay rates as the only source of time-dependence. Since for spontaneous radiative decay (and many other processes), the decay times, l/ys, are orders of magnitude longer than the duration of the sub-picosecond measurement, the e;s,s matrix elements are essentially time-independent and hence identical to one another at different times. As a result, the em matrix, which becomes nearly singular, cannot be inverted. We can solve this problem by recognizing that we can split F m ( t ) into two, by defining its F r f f ( t ) component as, (9) where

We see that F r f f ( t )contains only the s $ s' terms in Eq.(7). Defining the em matrix of Eq. (8) to include k indices that only correspond to s # s', allows us to solve Eiq. (7) as,

What remains therefore is to find a way of deriving FT'Jf) from the observed signal Fm(t).This can be done by recording, in addition to the timedependent non-dispersed fluorescence, the dispersed signal. The dispersed

PHASE AND AMPLITUDE IMAGING OF WAVEPACKETS

803

signal is composed of a series of lines at different q Sfrequencies, each line-strength yielding a single ATslaFI2exp(-y,t) term of Eq. (10). Since the Einstein A-coefficients are known, or can be measured separately, the strength of each line yields directly &I2, while the sum of all the linestrengths is exactly F&,(t = 0). The procedure can be repeated to deal with unpolarized initial states [12]. It is possible to sum analytically over the m quantum numbers of Q.(3) [13] to obtain the unpolarized excited-state density matrix p e x ( t )= C , I\km)(\km) from an equation similar to Eq. (1 1).

III. IMAGING OF A HIGHLY ROTATING Na2 MOLECULE In order to check our imaging procedure we have to first stimulate the fluorescence emitted by excited polarized (and unpolarized) Na2 wavepackets. In these simulation we assume that the molecule, which exists initially in a (xu,,j g ) Na;! (X vib-rotational state, is excited by a pulse to a superpositionof (xs) vib-rotational states belonging to the Na2(B nu)electronic-states. In order to make this a realistic simulation, we have used the ub-inirio Na2 curves of Meyer et al. [I41 to calculate the df,, dipole matrix-elements. Given these matrix elements, the simulated preparation coefficients, u:, are obtained from Eq. (4) and the time-dependent fluorescence signal, Eq. (7). F(t)-from The resulting unpolarized fluorescence, shown in Fig. 1, is a result of

' xi)

'

-

c

4

m-averaged total fluorescence

Na, B-X I

t 0

I

I

2

I

I

4

,

I

I

I

I

I

6 8 10 time (in psec)

I

I

12

I

t

I

14

I

16

Figure 1. The time-dependent Na2 fluorescence signal, F(t), following excitation of the 1200 cm-' wide pulse. - 90 points strobed for the inversion procedure. The data is for the X t B case with pure radiative decay. The pulse, whose center frequency is 22,400 cm-', simultaneously excites the &jto 615Na2(B 'nu) vibrational states.

xg ground state by a

+++

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MOSHE SHAPIRO

excitation with a pulse whose bandwidth is 1200 cm-’, i.e., of -8 fsec in duration. This pulse simultaneously excites -10 xs vib-rotational states in the excited electronic state. We have computed the fluorescence due to transitions to the first 35 vib-rotational states in the ground state. Also shown in Fig. 1 are the time points actually sampled in the solution of Eq. (11). As clearly seen in Fig. 1, the time-dependent fluorescence F(t) is made up of a complicated beat pattern, with almost complete recurrence at the Na2(B111,) vibrational period. Of interest are the split peaks, due to the appearance of a transient on its way in and out of a turning point [l, 151. As explained above, in order to affect the density-matrix imaging we need, in addition to the time-dependent fluorescence, to compute Fdiag(t),derived from the dispersed fluorescence. We have dispersed the fluorescence of Fig. 1 to obtain a series of lines, from which Fdiag(t) is obtained using Eq. (10). Given Foff ( t ) from Eq. (9) we can now use the line-strengths of the dispersed spectrum to compute the e matrix, and, using Eq. (1 1). to derive the imaged coefficients, G. Snapshots taken at two different times comparing an “Image” wavepacket, derived by the above procedure from data analogous to that of Fig. 1, to its “Source”, are given in Figs. 2a,b. We see that the imaged wavefunctions are practically indistinguishable from the true wavefunctions for all the times considered.

+ -6

i

0.04

Exact and imaged B state wavepacket I

I

I

0.02 0

0.04 0.02

I

1

t=O

I

1

1

+ = .0.00 -0.02

I 4

1

5

I

I

6 7 R(Na-Na) (in a.u.)

a I

(4

I

9

Figure 2. The real and imaginary parts of the source 9(t)and its image at different times. Source real part -, source imaginary part - - - - - - - - -. Images 6 6 . a) B state wavepacket at f = 0. The left hand ordinate scale is of the real part, the right hand ordinate scale is of the imaginary part. b) B state wavepacket at t = 1.7 psec.

+

PHASE AND AMPLITUDE IMAGING OF WAVEPACKETS

t=1.7 psec

0.02 3

2

-

0-

-0.02

805

I

I

I

I

I

The procedure was repeated for the density matrix matrix of unpolarized excited states. In Fig. 3 we display I@', 8 = OlpeX(t)lR,8 = 0)l2 for j = 134, derived from the data of Fig. 1, at different times. As in the pure case of Figs. 2, our procedure is able to prefectly reconstruct the true density-matrix at all times considered.

I

6

8

R

6

I

8

Figure 3. The square of the density matrix, I(R.0 = O(pex(r)lR',8 = 0)l2,at different times. (a) The Image at t = 0. (b) The Source at r = 0. (c) The Image at r = 19 psec. (a) The Source at f = 19 psec.

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Acknowledgments This work was supported by the Israel Academy Fund for Basic Research.

References 1. A. H. &wail, in Femtosecond Chemistry J. Manz and L.Woste, editors, (VCH Weinheim Germany, 1995) p. 15; ibid Faraday Disc. Chem. SOC. 91 207 (1991). 2. M. Gruebele, G. Roberts, M. Dantus, R. M. Bownman, and A. H. Zewail, Chem. Phys. Lett. 166 459 (1990); R. B. Bernstein and A. H. Zewail, Chem. Phys. Lett. 170 321 (1990); M. H. M. Janssen, R. M. Bowman, and A. H. Zewail, Chem. Phys. Lett. 172 99 (1990); M. Gruebele, and A. H. Zewail, J. Chem. Phys. 98 883 (1993).

3. T. J. Dunn, J. Sweetser, I. A. Walmsley, and C. Radzewicz, Phys. Rev. Lett. 70 3383 (1993). 4. T. J. Dunn, I. A. Walmsley, and S. Mukamel, Phys. Rev. Lett. 74 884 (1995). 5. P. Kowalczyk, C. Radzewicz, 1. Mostowski and I. A. Walmsley, Phys. Rev. A 42 5622

(1990). 6. J. L. Krause, M. Shapiro, and R. Bersohn, J. Chem. Phys. 94 5499 (1991); M. Shapiro, Farad. Dis. 91 352 (1991). 7. M. Shapiro, J. Chem. Phys. 103 1748 (1995); ibid, “Spectroscopic Wavefunction Imaging and Potential Inversion”, J. Phys. Chem. 100 7859 (1996). 8. M. Shapiro, Chem. Phys. Lett. 242 548 (1995).

9. R. Loudon, The Quantum Theory of Light, 2nd ed. (Clarendon Press, Oxford, 1983). 10. M. Shapiro, J. Phys. Chem. 97 7396 (1993).

11. S. Haroche, in High Resolution Laser Spectroscopy, K. Shimoda, Ed. (Springer-Verlag, Berlin, 1976). p. 253. In Eq.(7.8) of that paper an approximation is made in that the different w& photon phase-space factors are replaced by their average. This approximation is not made in Ref. [lo]. 12. M. Shapiro, Chem. Phys. 207 317 (1996). 13. R. N. Zare, Angular Momentum, (Wiley Interscience, New York, 1988). 14. I. Schmidt, Ph.D. Thesis, Kaiserslautern University, 1987.

15. J. R. Waldeck, M. Shapiro, and R. Bersohn, J. Chem. Phys. 99 5924 (1993); M. Shapiro and J. R. Waldeck, in Femtosecond Reaction Dynamics, D. A. Wiersma, Editor, p. 91

(North Holland, Amsterdam, 1994).

DISCUSSION ON THE COMMUNICATION BY M. SHAPIRO Chairman: J. Manz S. A. Rice: Prof. Shapiro, is your method extendable to spectra that include resonances, either those that generate L2 states after complex rotation, or possibly other resonances? If the answer is yes, I infer that we can put to rest the issue of the existence of quantum chaos.

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M. Shapiro: 1. Yes, our analysis applies to a sum of Fano type resonances as

well as to the sum of Lorentzians (giving rise to the exponential decays which I showed) studied here. 2. The existence of a wave function does not tell us anything about quantum chaos. Whether chaos exists or not, there is always a wavefunction which can either be computed or extracted from experiment.

G. R. Fleming: I want to make the comment to Prof. Shapiro that another way to solve the phase problem is by controlling the phase with light, as described in Scherer et al., J. Chem. Phys. 95, 1487 (1991). M. Shapiro: Could this possibly involve experimental difficulties, even though the approach is conceptually simple? G. R. Fleming: Our experiments are not difficult to perform. M. Shapiro: The spectra I have analyzed are obtained from emission, as in Walmsley’s experiments, or absorption, as in Zewail’s experiments, and are in a sense simpler to measure than experiments requiring the definition of the relative phase between two pulses. R. A. Marcus: Perhaps the following remark relates to Stuart Rice’s question regarding invertibility and “quantum chaos”. One might regard “quantum chaos’’ as corresponding to the overlapping of a number of internal resonances. In that case, if your method doesn’t apply when several resonances overlap, it wouldn’t, of course apply and probably wasn’t intended to apply, to “quantum chaos”. M. Shapiro: If the resonances overlap to such an extent that we can no longer break the frequency resolved spectrum to a sum of Fano lines in a unique way, then my analysis would not be unique. However this is an extreme situation and even in this case one can try to fit the spectrum (admittedly in a non-unique way) to a sum of Fano lines (or complex energy poles).

Advances In Chemical Physics, Volume101 Edited by I. Prigogine, Stuart A. Rice Copyright © 1997 by John Wiley & Sons, Inc.

GENERAL DISCUSSION ON TRANSITION STATE SPECTROSCOPY AND PHOTODISSOCIATION Chairman: J. Manz

M. Herman: There will be no experiment reported here, but this should rather be taken as an extended comment on R. Schinke’s talk. The question I wish to address is: at what stage should one shift from detailed and accurate spectroscopic parameters to a more statistical approach? It is for instance well known that one can statistically predict the number of vibrational energy levels. However, detailed spectroscopic observations, as those performed by Jost on NO2 and by Troe and Abel on HOCI, lead to different puzzling results. I wish to address that problem with regard to the specific example of acetylene, following Bob Field’s talk. We know that a11 vibrational levels can be gathered into polyads or clusters. The generic structure of each cluster is presented in the first figure, according to Ref. 1. The dynamical interpretation is that these clusters represent the first tier of the energy flow. Time-resolved predictions can be performed by Fourier transforming the energy picture, thus simulating a coherent excitation of the whole cluster. The stronger and more numerous the anharmonic resonances within the cluster, the faster the energy flow. We have started considering the rotational structure in those vibrational levels, thus getting closer to real life. One can connect clusters with identical n,, n, pseudo quantum numbers, but with nk quantum numbers varying by 2, through the wellknown rotational Z-doubling off-diagonal matrix elements. This is presented in the second figure. Each diagonal element is now a cluster, as just discussed, in which one adds terms in Bo, DO,and Ho,and their vibrational dependencies. If one constrains those extra parameters, as well as those of the off-diagonal elements to their values reported in the infrared literature, thus without any fitting procedure, one can calculate the rotational energy levels. The resulting B,, D,,and H, parameters can then be retrieved and compared to their experimental value. This 809

810

GENERAL DISCUSSION

TABLE 1 Calculated Rotational and Centrifugal Distortion Constants BO = 1.17664632, Do = 1.627 x "0

Bv (Obs)*

B, (Calc)

Ho = 0 . 0 0 1 6 ~

All the values are in cin-'.

D, x I O - ~ D, x I O - ~ H , (Obs)*

(Calc)

x 10-9

(Obs)*

H, x 10-~ (Calc)

(2, lO,J, e, u} rovibrational cluster 6413.90 6449.57 6556.46 6623. I 4 6654.24 6690.57

1.170010 1.168130 1.163778 1.167978 1.171783 1.169312

1.170043 1.167774 1.163913 1.167567 1.171674 1.169316

-9.24 -20.35 1.61 4.02 8.03 2.49

-11.43 -18.98 1.63 3.81 8.4 2.01

-8.70 -29.9 0.0028 0.39 1.66 1.53

-11.12 -28.16 0.0025 0.38 2.33 3.97

is performed in Table 1, for one of the clusters. The agreement is striking. Similar agreement is obtained for higher energy vibrational levels, but not detailed here (see Ref. 2). The next step is to take care of the avoided crossings, whose presence was pointed out in DCO by R. Schinke. We believe that this can be achieved with the help of a very small number of extra parameters, of Coriolis type, also included in the second figure. Thus a well-defined and limited set of parameters is probably able to reproduce all rovibrational energy levels and spectral features. This should make possible refined predictions of dynamical nature. It is important to point out that the picture of pseudo quantum numbers, designed by M. Kellman, holds all the way through. We start with vibrational clusters characterized by n,, nr and nk. We then extend the cluster size to allow clusters with different nk values to interact, still keeping however ns and n,. This picture allows us to reproduce the rotational parameters. Making clusters with different n, values to interact, keeping thus n, as the last valid quantum number, ought to image all interactions, including accidental Coriolis induced crossings. We are on our way testing if similar approaches can be applied to other, larger molecules such as pyrrole, thus with the hope to find regular patterns, supported by pseudo quantum numbers, allowing to frame the whole rovibrational structure, and therefore the IVR. 1. M. Abbouti Temsamani and M. Herman, J. Chem. Phys. 102,6371 (1995).

2. M. Abbouti Temsamani and M. Herman, J. Chem. Phys. (19%) in press.

TRANSITION STATE AND PHOTODISSOCIATION

where nS=V1+V2+V3 nr = 5V1+ 3V2 + 5V3 + V4 + V5 nk(= k) = E4 + fs W i j : anharmonic resonance

For nk > 0, +/- are omitted. ~~~~~

Figure 1. The vibrational cluster.

For each rotational quantum number J

-~ ~

~

Figure 2. The ro-vibrational cluster.

81 1

GENERAL DISCUSSION

812

W. H. Miller: I would like to ask Prof. Schinke the following question. Regarding the state-specific unimolecular decay rates for H02 -+ H + 0 2 , you observe that the average rate (as a function of energy) is well-described by standard statistical theory (as one expects). My question has to do with the distribution of the individual rates about the average: since there is no tunneling involved in this reaction, the TST/Random Matrix Model used by Polik, Moore and me predicts this distribution to be x-square, with the “number of decay channels” being the cumulative reaction probability [the numerator of the TST expression for k(E)];how well does this model fit the results of your calculations?

R. Schinke: Although the information on the rate for H02 is rather limited, we performed a statistical analysis and found reasonable agreement with the prediction of random matrix theory. A picture is given in the original publication [A. J. Dobbyn et al., J. Chem. Phys. (15 May 1996)]. J. Manz: Let me add a comment on Professor W. H. Miller’s remark that he would never make himself, but I can express this as the chairman of this session. In fact, Professor Miller’s extension of the standard RRKM-theory allows to predict not only the statistical mean values of the rate coefficients, but also their fluctuations. This is an important achievement in the theory of chemical reaction theory over the past couple of years and it should be adequate to call it the [ 11. RRKMM theory (Ramsperger-Rice-Kassel-Marcus-Miller) In addition, I would like to point to the complementary theory of R. D. Levine on the fluctuations in reaction rate coefficients [2], with applications to mode selective decay rates of near-degenerate resonances in model systems [3], similar to Schinke’s results [4]. I.

W.H. Miller, R. Hernandez, C. B. Moore, and W. F. Polik, J. Chem. Phys. 93,5657

(1990); W. F. Polik, D. R. Guyer, W. H. Miller, and C. B. Moore, J. Chern. Phys. 92, 3471 (1990); R. Hernandez, W. H. Miller, C. B. Moore, and W. F. Polik, J. Chem. Phys. 99,950 (1993). 2. R. D. Levine, Be,: Bunsenges. Phys. Chem. 92,222 (1988). 3. R. H. Bisseling, R. Kosloff, J. Manz, F. Mrugala, J. Romelt, and G . Weichselbauer, J. Chem. Phys. 86,2626 (1987). 4. R. Schinke, H.-M. Keller, M. Stumpf, C. Beck, D. H. Mordaunt, and A. J. Dobbyn, Adv. Chem. Phys., Vol. 101, Resonances in Unimolecular Dissociation: From ModeSpecific to Statistical Behavior.

F. Remade: In the case of HO2 described by Prof. Schinke, the resonances overlap and I would expect a prompt and a delayed branch

TRANSITION STATE AND PHOTODISSOCIATION

813

in the distribution of the decay rates [I], as is typical of highly congested structures embedded in the products’ channels, like for example in the dynamics of high Rydberg states. The dissociation rates that you report for HO2 fluctuate around the statistical value and do not segregate into a prompt and a delayed branch. Since you extract the rates from an absorption-type cross section where in the case of overlapping resonances, the prompt lifetimes appear as a broad background and cannot be easily identified, could it be possible that you are missing the prompt rates in your analysis? In this connection, what is the number of resonances that you detect in the spectrum compared to the dimension of the basis set used for the computations? And how sensitive is the distribution of the rates to the initial state used to compute the spectrum? 1.

F. Remade and R. D. Levine, J. Phys. Chem. 100, 7962 (1996).

R. Schinke: We extracted the resonance widths from the “spectrum’’. It is clear that resonances are missed, especially the broader ones. Moreover, the widths have some uncertainty, especially at higher energies. Therefore, the statistics of rates is not unambiguously defined. The only point which I want to make is that our results are in qualitative accord with the predictions of random matrix theory. R. Jost: The presented nearest neighbor distribution P(S) which demonstrates the chaotic behavior of the vibrational levels of HO2 is an averaged property, from the ground state to the limit of dissociation. This chaotic behavior can (and should) be studied as a function of energy because the degree of chaos is expected to increase with energy. In addition, besides the short range correlation properties (i.e. the nearest neighbor distribution which “saturates” to a Wigner distribution even for incomplete chaos), the long range correlation properties (C2 or A3 or the Fourier transform) should be analyzed because these statistics are more stringent and give a deeper insight on the chaotic behavior. In addition the study of the isotopes, like DO2, in the same potential energy surface may allow interesting comparisons on the appearance of chaos which is governed by the lowest overlapping resonances (which in turn depend on the vibrational frequency ratios, i.e. on the masses).

R. Schinke: You are certainly right, the nearest neighbor distribution including all bound states is an averaged quantity. One should analyze the distribution in different energy regimes in order to see how the degree of regularity and irregularity varies with excitation energy.

814

GENERAL DISCUSSION

J. Troe: Prof. Schinke, in your HCO results, when you compare the fluctuating quantum results with RRKM results, you observe that the RRKM curve is above the average quantum data. Can the reason be that you use an inadequate p ( E ) for the continuum energy range?

R. Schinke: The definition of the density states p ( E ) in the continuum is certainly a problem, especially for a molecule like HCO whose potential is quite shallow. We took all bound states (15) plus a number of low-lying resonance states to determine an analytical function N(E), and from this the density of states. R. A. Marcus: Dr. Troe has made an interesting point. Have you calculated the density of states p(E) for your systems, to avoid using some harmonic or other approximation for that density?

R. Schinke: In the case of HNO and H02, we calculated the number of states and simply extrapolated this number into the continuum. We believe that this is the best what can be done, provided a global potential energy surface and full dimensionality dynamics calculations for this potential are available. Because of the much smaller number of states, for HCO this procedure is less well defined. In our final analysis (Ref. 33 of our paper in this volume) we tested the extrapolation from the bound to the continuum region and an estimation of the density of states based on a Dunham expansion of the term energies and found that both recipes give essentially the same result. W. H. Miller: One refinement that might lower the TST k(E) for HCO + H + CO - and bring it into better agreement with the average of your quantum results-would be to use a better transition state. This is because any TST k(E) is an upper bound to the true (average) rate. R. A. Marcus: It certainly is a good point that transition state theRRKM,provides an upper bound to the reactive flux (apart from nuclear tunneling) as Wigner has noted. Steve Klippenstein [I] in recent papers has explored the question of the best reaction coordinate, e.g., in the case of a unimolecular reaction ABC -+ AB + C, where A, B, C can be any combination of atoms and groups, whether the BC distance is the best choice for defining the transition state, or the distance between C and the center of mass of AB, or some other combination. The best combination is the one which yields the minimum Aux. In recent articles Steve IUippenstein has provided a method of determining the best (in coordinate space) transition state [I]. ory, and hence

1. S. J. Klippenstein, Chem. Phys. Lett. 214,418 (1993); and references cited therein.

TRANSITION STATE AND PHOTODISSOCIATION

815

R. Schinke: Actually, Dr. Klippenstein is currently applying his methods of defining the best transition state to HNO and it will be interesting to see how his results agree or disagree with our estimations of the statistical rate using the usual scattering (Jacobi) coordinates.

R. A. Marcus: It is certainly necessary to include all of the reactive trajectories, those that lead to immediate dissociation and those that do not. As Wigner pointed out, in effect, one needs to include all of the phase space occupied by the assumed transition state. Exclusion of any of these trajectories would include part of that phase space. Transition state theory is (classically) an upper bound to the rate since the trajectory may include parts of the TS phase space twice (multiple crossings of the transition state). J. Maw: Let me close this session with three suggestions which are stimulated by the superb contributions, by D. Neumark and R. Schinke, and by the participants in the discussion: 1. The four examples presented by R. Schinke, from regular resonance decay of HCO to the irregular behavior of HO;?, are cases for the textbooks [l]. They should serve also as touchstones for approximate theories such as RRKM or SAC [2], and, if I may add, I think it is not primarily the task of R. Schinke, but of the proponents of RRKh4, SAC, etc., to verify their statistical theories. 2. I would also like to suggest that the exact quantum results of R. Schinke [ 11 should serve as touchstones for the complementary theories developed by P. Gaspard [3]. His methods [3] are based on classical trajectories, and it would be nice to compare, e.g., regular resonances in HCO versus irregular ones for HOz in the complementary quantum and classical-based theories. 3. Finally, I would like to suggest to Professor D. Neumark that he may consider to extend his new two-photon-(w I , w+photoelectron detachment technique for radicals [4] from continuous-wave (cw) to ultra-short (fs) laser pulses. This should allow to investigate the wealth of competing dynamical processes in the radicals prior to reaction or decay.

1. R. Schinke, H.-M. Keller, M. Stumpf, C. Beck, D. H. Mordaunt, and A. J. Dobbyn,

Adu. Chem. Phys., Vol. 101, Resonances in Unimolecular Dissociation: From ModeSpecific to Statistical Behavior. 2. J. Troe, Adu. Chem. Phys., Vol. 101, Recent Advances in Statistical Adiabatic Channel Calculations of State-Specific Dissociation Dynamics.

816

GENERAL DISCUSSION

3. P. Gaspard and I. Burghardi, Adv. Chem. Phys., this volume. 4. D. L. Osborn, H. Choi, and D. M. Neumark, Adv. Chem. Phys., this volume.

D.M. Neumark We are currently carrying out somewhat different femtosecond experiments in which time-resolved photoelectron spectroscopy is used to probe the photodissociation dynamics of negative ions. In these experiments, an anion is photodissociated with a femtosecond laser pulse. After a time delay, the dissociating anion is photodetached with a second femtosecond pulse and the resulting photoelectron spectrum is measured. The photoelectron spectrum as a function of delay time provides a detailed probe of the anion photodissociation dynamics. First results have recently been obtained for the photodissociation of I;.

REACTION RATE THEORIES

RECENT ADVANCES IN STATISTICAL ADIABATIC CHANNEL CALCULATIONS OF STATE-SPECIFIC DISSOCIATION DYNAMICS

Institut fir Physikalische Chemie Universitat Gottingen Gottingen, Germany

CONTENTS I. Introduction

11. Adiabatic Channel Potential Curves

111. Thermal Capture Rate Constants IV. Specific Rate Constants for Dissociation V. Comparison of Statistical Adiabatic Channel and Variational Transition-State Treatments A. Comparison of SACM and VTST for Isotropic Charge-Locked Permanent Dipole Systems B. Comparison of SACM and VTST for Anisotropic Charge-Permanent Dipole Systems C. Comparison of SACM and VTST for General Potentials VI. Recent SACM Applications to More Complex Reaction Systems References

I. INTRODUCTION Statistical theories present particularly useful approaches to the quantitative characterization of dynamical phenomena in chemical kinetics. On the one hand, they provide a shortcut to the overall rate of the reaction, avoiding the explicit treatment of the dynamics before and after reaching the reaction botAdvances in Chemical Physics, Volume 101: Chemical Reactions and Their Control on rhe Femtosecond Time Scale, Xyrh Solvay Conference on Chemistry, Edited by Pierre Gaspard, Irene Burghardt, I. Prigogine, and Stuart A. Rice. ISBN 0-471-18048-3 0 1997 John Wiley & Sons, Inc.

819

820

J. TROE

tleneck; on the other hand, they account for the relevant molecular properties in the vicinity of this region in a most transparent way. The price to pay is ambiguities in the formulation of “statistical” theories, which can be eliminated only by comparison with true dynamical calculations and, possibly, the appearance of “nonstatistical” effects. Statistical theories can be applied in many fields of reaction kinetics, for example, in the characterization of the dissociation of molecules in bonding electronic states. In order to judge the merits and limitations of this approach, more basic studies about the validity of statistical theories are required. Much progress in this direction has been made in recent years. A system in which statistical approaches can be tested particularly well against dynamical calculations is the bimolecular charge+.de capture process or the reverse unimolecular dissociation process of molecular ions into charged and dipolar neutral fragments. This system can serve as a prototype for dissociation/association processes in general, and the results can qualitatively be transferred to other reaction systems. The situation here is particularly clear, because the interaction potentials between two reactants are simple and well characterized. Classical trajectory calculations have been performed extensively, and the results have been represented in simple parametric manner [1]-[3]. Quantum scattering calculations are available as well [4]. Statistical calculations [5-161 have been made in various ways, many of them shown to be variants of the statistical adiabatic channel model (SACM), differing essentially only in the extent of numerical simplifications [13]. Likewise, a variational transition-state theory (VTST) was applied. Where the conditions for an application of classical trajectory calculations are fulfilled, SACM and classical trajectory calculations were shown to agree perfectly [15] (i.e., within an accuracy of better than 1% of the rate constant). The chargedipole interaction potential shows the simplest form of an anisotropy. It is well known that this anisotropy causes deviations from phase-space theory (PST), the latter providing an upper limit of statistical reaction rates. It should be emphasized that PST, in our work, corresponds to a treatment using the real minimum-energy path but neglecting anisotropies of the potential perpendicular to this path. It is convenient to express the deviation from PST by a r i g i d d y f k m r f r i g i d [17]. For instance, thermal capture rate constants k,ap(T) include a thermal rigidity factorfigid(T) that depends on the temperature T.:

One may go one step further and determine thermal rigidity factors for capture of dipoles in individual rovibrational states i, that is, derive frigid, i(T).

STATISTICAL ADIABATIC CHANNEL CALCULATIONS

82 1

By taking into account microscopic reversibility, one may consider ion fragmentations, which involve the same potentials of the charge-dipole type, and express the difference between the true fragmentation rates and the predictions from PST by an energy (E) and angular-momentum- (J)- specific rigidity factor ffigid(E,J). In this case, the specific rate constants for fragmentation k(E, J ) are expressed by [171

Provided that microscopic reversibility is properly accounted for, thermal averaging of specific rate constants leads to thermal capture rate constants such that a relation between f+&) and frigid@, J ) can be established. As the charge-dipole potential is particularly simple, the behavior is transparent such that a reference for comparison with other types of interaction potentials is available. Of particular interest is the relation between the various forms of VTST and SACM. The charge-dipole system is particularly well suited to investigate this aspect in a quantitative way. In the following we show that, for potentials without pronounced energy barriers, VTST and SACM in general are not equivalent, the numerical differences depending on the chosen variant of VTST. Because SACM agrees well with classical trajectory calculations, the comparison of VTST with SACM may help to identify artifacts of the VTST treatment. 11. ADIABATIC CHANNEL POTENTIAL CURVES We consider a situation where the interaction potential between two associating/dissociating species is described sufficiently well by the simple charge-dipole potential

is the dipole moment of the neutral where q is the charge of the ion, dipole, r is the distance between the centers of mass, and y is the angle between the dipole axis and the line connecting the two centers of mass. In some considerations, an isotropic polarizability term is also included

822

J. TROE

where (II denotes the isotropic polarizability of the dipole. However, this is done for illustration only; other contributions to the potential will generally become relevant as well whenever the polarizability term develops its influence. Formulating the Hamiltonian of the system with the potential of Eq. (4) and solving the eigenvalue problem of the system for fixed r lead to a set of eigenvalues Vi(r)that define adiabatic channel potential curves. It was shown by treating the neglected kinetic-energy part of the Hamiltonian as a perturbation [ 8 ] or by comparing with classical trajectory calculations [ 3 ] , [15] that, up to moderately high translational energies, the reacting system stays on individual adiabatic channel potential curves (nonadiabatic effects, nevertheless, may become observable under certain circumstances; see, e.g., Refs. 15 and 16). The hypothesis of the statistical adiabatic channel model is that the channels i are populated with equal statistical weight and the maxima of the channel potentials V i ( r )along the reaction path r determine the channel-specific threshold energies Eoi for reaction. The eigenvalue problem for the simple cos y potential of Eq. (4)can be solved easily by matrix diagonalization using a basis of free-rotor wave functions. For practical purposes, however, it is also useful to have approximate analytical expressions for the channel potentials Vi(r).The latter can be constructed by suitable interpolation between perturbed free-rotor and perturbed harmonic oscillator eigenvalues in the anisotropic potential for large and small distances r, respectively. Analogous to the weak-field limit of the Stark effect, for linear closed-shell dipoles at large r, one has [7]

with 3m2 - j ( j + 1)

F(h m, = j ( j + 1)(2j - 1)(2j + 3 ) and F(0,O) =

5

(7)

The rotational quantum numbers of the free linear dipole are j and m (with Iml Ij ) , and the rotational constant is denoted by B. Analogous to the strong-field limit of the Stark effect, for linear dipoles at small r, one has

STATISTICAL ADIABATIC CHANNEL CALCULATIONS

823

The corresponding expressions for symmetric and asymmetric top dipoles and for open-shell dipoles are also available [15, 181. Higher terms of the two perturbation expansions of Q s . (7)and (8) have been elaborated. However, it is more important to have an adequate interpolation between the two limiting expansions of the channel potentials of Eqs. (6H8).This can be obtained by continuing the series of Eqs. ( 5 ) and (8) by one further term and by fixing its coefficient by a fit to numerically calculated Vi(r) in intermediate ranges of r values [15, 181, Figure 1 gives examples for adiabatic channel potential curves for the H<-HCl system [2, 151 that show a particular channel-specific pattern of the threshold energies when the channels open up. The complete adiabatic channel potentials are finally obtained by adding As long as channel maxima are located at a centrifugal energy term Ecent(r). comparably large values of r, the quasi-diatomic expression

with the quantum number 1 for orbital motion and the reduced mass p of the two species is adequate (for small quantum numbers J of rotation of the charge-dipole complex, a suitable interpolation between J and 1 is advisable; if small values of r become relevant, Eq. (9) should be modified as well). Figure 2 demonstrates the I dependence of the adiabatic channel potential curves for the Hi-HCl system. Determining the maxima of the complete adiabatic channel potential curves Vi(r) + Ecent(r)along the reaction path r defines quantum-number-specific channel threshold energies Eoi = Eoi( j , m,( J , l } ) , where the curly bracket indicates the relevant choice in the overall characterization of angular momenta. 111. THERMAL CAPTURE RATE CONSTANTS

Assuming that vibrational excitation, apart from an r-independent energy shift, does not influence the pattern of the threshold energies Eoi, we now proceed to the calculation of the thermal capture rate constants kcaP(T).Neglecting tunneling, channels i are considered to be closed at energies below the threshold energies Eo; and to be open at energies above Eoi. The thermal rate

824

J. "ROE

Figure 1. Adiabatic channel potential curves Vi(r) for the interaction between H; and HCI (calculations for ( j , m,I ) = (0,0, I ) with 1 = 0, 40, 60 from Ref. 15; full curves: real potential; dashed curves: locked-dipole potential used in PST).

constant then is equal to k C a p ( T ) =kT h

h2 (-)2upkT

with the activated complex partition function

3/2

Qd

Qrot

STATISTICAL ADIABATIC CHANNEL CALCULATIONS

j

1OC \

I

I

\

= 2

i I

\

SC

: m

825

a t

c

E a

--

\

L

5

- 5(

-10

-1 5

10

20

30

40

r/ W Figure 2. As Fig. 1, for ( j , m , l ) = (2, Irnl,l) with Iml = 0, 1, 2 from Ref. 15 (full curves:

1 = 0 dashed curves: 1 = 40).

and the partition function Qmt of the linear dipole,

where Ei( j, m) = Bj( j + 1) denotes the energy levels of the linear dipole rotor and g( j, m) or g( j, m, I ) are the respective degeneracies (vibrational excitations are neglected throughout this chapter). Before evaluating @ and k, for the charge-dipole system with its real anisotropy, we consider the hypothetical isotropic case with cos y in Eq. (4) replaced by unity. We understand this situation as that characterized by

826

1. TROE

phase-space theory and we call this the “locked-dipole case.” As the adiabatic channel potentials now are all parallel, being displaced by B j ( j + 1) relative to the potential V(r)+ Ecent(r),the channel threshold energies are given by

where Eo(Z) denotes the centrifugal barriers of the potential V ( r ) . If V ( r ) is of the locked-dipole form [cos y= 1 in Fq. (4)], for Z(Z+ I) I 2pqpD/ti2, one obtains Eo(Z) = 0 and

for Z(2 + 1) 1 2pq/.kD/tt2. Introducing Eqs. (13) and (14) into Eq. (11) and passing from the sum to the integral give

Combining Eqs. (lo), (12), and (15) finally leads to

Equation (16) represents the locked-dipole capture rate constant. The first term, which at the same time is the high-temperature limit, denotes the wellknown Langevin rate constant

-

In the low-temperature limit (T 0), the first term of the weak-field expansion of the channel eigenvalues of Eqs. (5)-(7) suffices for the determination of the channel potentials Vi(r). Combining Eqs. (5), (7), and (9) then leads

STATISTICAL ADIABATIC CHANNEL CALCULATIONS

827

to channel threshold energies f o r j = m = 0 of

where

Introducing this into the activated complex partition function @ of Eq. (11) gives [7]

k,,,(T

-

0) = 2wq

1+-

3aB

Comparison of Eqs. (20) and (16) leads to a low-temperature limiting thermal rigidity factor of

which approaches zero at T .--t 0 K. For higher temperatures, one may consider a pure harmonic oscillator model. Here one obtains particularly transparent results when the polarizability term is neglected, that is, when a = 0. In this case one has [7] Eoi

B(2j - Iml+ 1)2 2(1 - G )

and, after evaluating Eqs. (10-(12),

828

J. TROE

In this approximation, hence, the thermal rigidity factor is given by

Below we shall compare this result with that from the corresponding classical trajectory calculations as well as from variational transition-state treatments. In the following we consider the complete statistical adiabatic channel calculations, going beyond the limiting low-temperature results from the quantum perturbed-rotor expansion and beyond the limiting high-temperature result from the classical harmonic oscillator model. The results from the detailed SACM treatment were found [3], [15] to agree perfectly with the analytical representation of the trajectory results as long as k T / B >> 1. The agreement was within the numerical uncertainties of the trajectory results and their analytical representation. Only for very large dipole moments and moderately low temperatures were realized small differences, in comparison with the results from Ref 1; however, these could uniquely be attributed to artifacts of the first trajectory calculations, that is, to insufficient ranges of capture distances. These deviations were corrected for by comparison with the SACM results [3], [15]. At lower temperatures, k T / B I1, SACM and classical trajectory results differ to a major extent [6], [7] which obviously is due to the inadequacy of a classical treatment. It was discovered in the trajectory work of Ref. 1 that the ratio k,,,/kL can be expressed in reduced form in terms of the parameter X =

PD

(2c~kT)'/~

Using this parameter, the following relationship was proposed [ 11:

"={ kL

0.4767~+ 0.6200 (x

0.5090)2 10.526

+

+ 0.9754

for x 2 2

(26)

for x I2

(27)

The SACM results agree within about 0.3% with Eqs. (25x27) in the range

x 5 3 [15]. The above-mentioned deviations were found for larger x in

ranges where quantum effects did not yet matter. The classical result of Eqs. (25)-(27) could be corrected on the basis of SACM results by replacing Eqs. (26) and (27) by [15]

STATISTICAL ADIABATIC CHANNEL CALCULATIONS

0.5642(x - 11) + 6.2765

-k:& -

kL

-

for x 2 11

829 (28)

0.4963(x - 3)'.03 + 2.0501 for 3 I x I 11

(29)

0.4767~+ 0.6200

(30)

(x + 0.5090)2 + 0.9754 10.526

for 2 Ix I 3 for x I 2

[A partial extension of Eqs. (26) and (27) was also given in Ref. 1.1 For low temperatures, where kT is below a few B, the classical result for kfap (denoted by k:ip) of Eqs. (28x31) also had to be replaced by a transition to the It was limiting quantum result of Eq. (20) [denoted by k!zp = kCap(T-O)]. shown in Ref. 15 that this transition can also be expressed in parametrized universal manner by the equations

kcap - k~ = 1 - 0 . 0 6 5 ~ '. ~0.069y3.* for y I 1 q" - kL kcap kcap - k~ k",gp - kL

= 1- 0 . 1 3 4 ~ - ~ . ~

for y 2 1

(33)

where

with kz:p = kcap(T-+O) and k:ip given by Eqs. (20) and (28)-(31), respectively, whereas kL is given by Eq. (17). (An alternative representation of kELp was proposed in [3]). The quality of the given parametrized expressions for capture rate constants is illustrated in Figs. 3-5, where classical trajectory and SACM results are compared for a series of chargedipole capture systems. The good agreement confirms that a satisfactory dynamical and statistical solution of the problem is at hand. The final result for kcap also allows for a rigorous determination of the thermal rigidity factors figid( T). This is particularly illuminating when approximate models such as the pure oscillator model of Eqs. (22x24) are compared with the complete result. The charge-permanent dipole capture (i.e., a 0), for x >> 2, using Eqs. (25) and (26), would be characterized

-

830

J. TROE

T*= B / k

TIK

Figure 3. Thermal rate constants for capture of HCI by Hi (PST locked-dipole capture corresponding to phase-space theory, Eq. (1 6); SACM: statistical adiabatic channel model, Eqs. (26)-(34) [ 151; SACMcl: classical SACM, Eqs. ( 2 8 x 3 1 ) [ 151; C T classical trajectories, Eqs. (26) and (27) ill).

T*

T/K

Figure 4. As Fig. 3, capture of HCN by Hi.

83 1

STATISTICAL ADIABATIC CHANNEL CALCULATIONS

\ P

u O

A

o* 0.20

2

1

Figure 5. Low-temperature rate constants for capture by ions of HCl CS (a) (SACM calculations from Ref. 15; dashed curve: Eqs. (32)-(34)].

HCN (x) and

(0).

by a capture rate constant

that is, by a thermal rigidity factor

One might suspect that the discrepancy between the thermal rigidity factors of Eqs. (24) and (36) is due to inadequacies of the SACM treatment in general or of the pure harmonic oscillator model. However, after having corrected the analytical representation of the trajectory results through Eqs. (28)-(31), for x >> 11, Eq. (28) leads to

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J. TROE

that is, to the thermal rigidity factor frigid = 0.5

(38)

which agrees with that of the corresponding pure harmonic oscillator model of SACM from Eq. (24). We conclude this section by summarizing that classical trajectory calculations of thermal capture rate constants agree within calculational accuracies of about 0.3% with SACM calculations 131. In addition, the results can be well represented in general analytical and parametrized form by Eqs. (28)-(3 1). At lower temperatures (kT / B < lo), where the classical treatment fails, the SACM treatment can be expressed in general analytical form as well by Eqs. (32) and (33). Because of the very good agreement of the statistical SACM and the dynamical trajectory calculations on the same potential, we shall use this as a reference for inspecting other statistical treatments later on. One may note in passing that earlier treatments of charge-dipole capture rates are superseded by the present convenient and accurate approach.

IV. SPECIFIC RATE CONSTANTS FOR DISSOCIATION Through equilibrium constants K,,the thermal capture rate constants k,,, which are identical with thermal association rate constants in the high-presare directly related to thermal dissociation rate constants sure limit k,, kdiss,- of the adducts in their high-pressure limit. In the following we consider the corresponding specific rate constants k(E, J) for molecular ions dissociating into ions and linear neutral dipole fragments. In statistical theory, k ( E , J ) is given by

with the density of states p(E, J) and the number of open channels W(E,J). We do not discuss here the calculation of densities of states. We just recall that their calculation is less trivial than generally assumed, because the anharmonicity contributions so far are not well understood (see, e.g., Ref. 19). Instead we only inspect W(E,J) for the dissociation/association processes considered here. The complete W(E,J) is obtained by convolution of the contributions of conserved modes and transitional modes. For the charge-dipole potential, the transitional modes are the free-rotor modes of the ion and two perturbed rotor modes of the linear neutral fragment, only the latter being governed by

STATISTICAL ADIABATIC CHANNEL CALCULATIONS

833

the anisotropy of the considered potential. At first we only analyze the latter modes, calculating their number of states W(E,J) as the number of channels whose threshold energies Eoi are smaller or equal to E. The numerical calculation is straightforward with the explicit channel potential curves Vi(r) used in the present approach [15]. An upper limit of W ( E , J ) is given by the free-rotor expression

[ ( ?(J)) +

WPST(E,J)= (U + 1) Int

-

I]

which characterizes PST and where Int(x) denotes the largest integer below PST corresponds to the real interfragment potential but neglects its anisotropy). In reality, because of the anisotropy of the potential, W(E,J) falls below WPST(E, J). For the considered charge-permanent induced dipole system, this decrease becomes less pronounced with increasing E - Eo(J) because with increasing energy the channel maxima are found at increasingly shorter interfragment distances where the isotropic polarizability part of the potential becomes relatively more important than the anisotropic permanent-dipole part. This behavior is illustrated in Fig. 6, where W(E,J) and WPST(E, J) are shown for the H3-HCl system with its permanent- and induced-dipole contributions to the potential. We do not discuss here analytical representations of W(E,J) and the corresponding specific rigidity factors frigid(& J ) for the general case. These again can be obtained in parametrized form. Instead we only consider the pure permanent-dipole case (i.e., a = 0) with its “uniform anisotropy” over the full range of r values. For this case, one observes that, apart from its step character, W(E,J), similar to WPST(E, J), essentially becomes a linear function of the energy with a J-dependent but E-independent specific rigidity factor frigid(& J). A satisfactory representation is given by

x (it is emphasized again that our use of

[

W ( E , J )= (U + 1) Int

- EO(J)I

(frigid(E,J)[E

B

+ ll

(41)

where

(minor details here are omitted, J and 1 being approximately identical for

834

J. TROE

E /cm-’

Figure 6. Number of open channels for the interaction between H; and HCI (SACM calculations from Ref. IS; PST phase-space theory; full curves: permanent + induced dipole; dashed smoothed curves: permanent dipole; J : total angular momentum of H;-HCI complex).

large values of J or Z; E ~ ( J is) nearly zero as long as J(J + 1)h2 s 2pqpD and infinite otherwise). The corresponding smoothed curves are included in Fig. 6. It is easily shown that, in the classical limit, Eqs. (41) and (42) are consistent with the thermal capture rate constants for the oscillator model of charge-permanent dipole capture. The relevant part of the activated complex partition function, instead of Eq. (I l), can be written as

With the simplifications mentioned for Eq. (42) and with W(E,J ) approxi-

STATISTICAL ADIABATIC CHANNEL CALCULATIONS

835

mated by

Then

ef

is evaluated, giving

in agreement with Eqs. (15) and (38). It should be emphasized that finer details of W(E,{J,1}) may be required, such as given by SACM, when the fine structure of W is of interest and can be resolved. The complete W(E,J) is obtained by convolution of Eq. (41) with the contributions from the free-rotor transitional modes of the ionic fragment and from the conserved modes. The first part follows the conventional PST treatment [see Eqs. (C27) and (C28) of Ref. 20 and the low-J-high4 interpolation from Ref. 21); the second part, for example, employs the Beyer-S winehart algorithm, convoluting harmonic oscillators with transitional modes (see the appendix in Ref. 22).

V. COMPARISON OF STATISTICAL ADIABATIC CHANNEL AND VARIATIONAL TRANSITION-STATE TREATMENTS

Statistical rate theories often are also formulated using variational principles. Like the adiabatic principle, variational principles are intuitive and have to be proven (or disproven) by comparison with true dynamical treatments. As SACM in the previous chapters has been shown to give identical results with trajectory calculations at high temperature for the considered simple reaction system, differences between SACM and VTST would speak against the latter. The charge-dipole system, because of its simplicity, can be used particularly well for a quantitative comparison between SACM and VTST and, hence, for a quantitative test of VTST. Variational transition-state theory has been formulated on various levels [5, 23-27]. At first, there is the group of canonical VTST (CVTST) treatments, which correspond to the search for a maximum of the free energy AG(r) along the reaction path r [23, 241. It was noticed early that for barrierless potentials this approach leads to an overestimate of the rate constant because, in the language of SACM, channels are included that are closed. Therefore, an improved version (ICVTST) was proposed [25] that truncates ef at the position rs of the minimum of @ ( r ) by including only states

836

J. TROE

with energies above the lowest dissociation threshold. In the present case, ICVTST in only includes channels with positive energies. In the following, we test the performance of these two canonical versions. Besides the group of CVTST, there are microcanonical VTST treatments (pVTST). In this case, the minimum of the number of channels W(E,r ) with fixed total energy E is searched along r. One may improve this pVTST by a J-conserving version (HJVTST)in which W(E,J, r ) is minimized. Both treatments can be modified by a truncation analogous to ICVTST, omitting channels with energies below the minimum threshold energy (IpVTST and IpJVTST). In the following we also test the performance of pVTST and pJVTST and their improved versions. It should be emphasized that further specification of variables, such as specifying individual channel quantum numbers, would bring VTST even closer to SACM such that, for complete specification of channel quantum numbers, SACM may be understood as a channel-specific VTST where the position of the adiabatic channel threshold Eoi identifies a minimum of W(E,J , i, r). It should further be mentioned that SACM and the various variational treatments approach each other for potential-energy surfaces with high-energy barriers. In the following we elaborate VTST expressions for various charge-dipole potentials. For demonstrative purposes, we further consider the isotropic locked permanent-dipole case where SACM and PST are identical. We also consider the real anisotropic permanent-dipole case in the quantum low-temperature and classical high-temperature oscillator limits. Finally we show comparisons for real permanent and induced-dipole cases. We always employ explicit adiabatic channel eigenvalues for calculating partition functions or numbers of states.

A. Comparison of SACM and VTST for Isotropic Charge-Locked Permanent-Dipole Systems The locked permanent-dipole capture system may serve as our first example for an isotropic potential where SACM and PST coincide. In this case, the adiabatic channel potential curves have the form

with a degeneracy g ( j , rn, 1) = (21 + 1)(2j + 1). In CVTST, one minimizes

STATISTICAL ADIABATIC CHANNEL CALCULATIONS

837

Considering temperatures T sufficiently above B / k , the sum may be replaced by an integral leading to

The minimum of Q(r) is found at

which decreases with increasing T. Identifying @ with Q(r') gives

In other words, the thermal capture rate constant of CVTST exceeds that of SACM/PST by a factor e = 2.718. The result may be somewhat improved by ICVTST, that is, by replacing @ = Q(@) by Q(#, V ( j ,m,I, rs) > 0). Again the calculation is straightforward, giving

In pVTST, W(E,r ) is calculated as

where C* includes all ( j , m , l ) for which V ( j , m , l ,r) is smaller than E. Replacing the summation by an integration leads to

W(E, r) = fir2 (E + 5y)2 A2B which has a minimum at

838

J. "ROE

Here, r* decreases with increasing energy, Identifying W(E) with W(E,r'), one obtains (54)

which exceeds the corresponding SACM/PST result by a factor of 2. The corresponding expression for IpVTST is equal to

thus falling below the SACM/PST result by a factor of 2. Conserving J -- 1 in pJVTST, W(E,J, r ) is calculated as

W E ,J , r ) = E*g(j, m,J )

(56)

where C* includes all (j,m) for which V ( j ,m,J , r ) is smaller than E at fixed J. Integrating Eq. (56) now gives W ( E , J ,r) = (ZI+ 1)

E

(Bf

q p o / r 2 - J(J + l)A2/2pr2 B

with a minimum at r'(E,J) =

-

(58)

such that

in agreement with Eq. (55) from SACM/PST, which is valid for J(J + 1) < 2t(qpD/h2. The truncations in IpJVTST now do not change the result such that

Thermal averaging of pVTST or IpVTST results, via Eq. (43), leads to thermal capture rate constants, with

STATISTICAL ADIABATIC CHANNEL CALCULATIONS

Q

*

p

=

~QS,ACM ~ ~

~

~

839 (61)

and

Although the results are better than the CVTST and ICVTST results from Eqs. (49) and (50), the differences from SACM/PST are still inacceptably large. Only with the pJVTST and IpJVTST results is agreement with SACM/PST obtained for the present locked-dipole potential.

B. Comparison of SACM and VTST for Anisotropic Charge-Permanent Dipole Systems As a second example we analyze the anisotropic charge-permanent dipole potential where SACM and PST differ from each other. Here, for demonstration, we only consider the low-energy perturbation and the high-energy harmonic oscillator limits. In the former limit, the adiabatic channel potential curve for the lowest channelj = m = 0 has the form

whereas, in the latter limit, one has

The corresponding channel degeneracies, for Eq. (63), are g ( l ) = (2l+ 1) and, for Eq. (64), g(u, I ) = (u + 1)(2l+ 1). Calculating Q(r) for the low-temperature limit with Eq. (47a) leads to

)

Q W = ( 2pr2kT h2 enp(s+-) The minimum of Q(r) is found at

q2Pi 6BJ'kT

J. TROE

840

such that, in the limit, T

--t

0,

The SACM with Eqs. (10) and (20) would have given

such that, at kT/B < 1, CVTST markedly differs from SACM. While the discrepancy between Eqs. (67) and (68) for T --+ 0 is not surprising, it is of larger interest to investigate the high-temperature range. With Eq. (22), ef in SACM is easily determined, giving

On the other hand, with Eq. (64)one derives

The minimum of Q(r) is found at r' = 0.792

(w) qPD

(71)

The anisotropy of the potential shifts the position of r' toward smaller values than for the isotropic potential [see Eq. (48)]. Now Q(r') follows as

where

efAcMis given by Eq. (69). Again, the CVTST result exceeds the

STATISTICAL ADIABATIC CHANNEL CALCULATIONS

84 I

SACM result, although the difference is smaller than in the isotropic case of Eq. (49). Evaluating ICVTST gives worse results than for the isotropic case described by Eq. (50); one obtains

As pJVTST appears much more useful then pVTST, we calculate W(E, J, r ) for pJVTST. The minimum of W(E,J, r ) now is found at

leading to

In this case, pJVTST and SACM agree such that W ( E , J )and thermally averaged capture rates are identical in both approaches. However, truncation of W in IpJVTST leads to

that is, the result by the truncation is not improved but deteriorated.

C. Comparison of SACM and VTST for General Potentials

A numerical comparison of SACM and quantum corrected ICVTST has been performed in Ref. 26 for the capture of HCl, HCN, or CS by Hi. Permanentand induced-dipole terms were included. For a system with aBlj.4; = the following results were obtained: for kT/B values of 125, 31.25, 13.89, 3.472, 0.8681, and 0.4883, ratios klCVTST/kSACM Of 1.33, 1.39, 1.41, 1.40, 1.22, and 1.18, respectively, were obtained. These results are in line with the conclusions from the previous sections, that is, that the truncation in ICVTST does not sufficiently improve the CVTST treatment. Comparisons of classical trajectory calculations with various versions of VTST have also been performed for quite a series of other reaction systems such as neutral radical association/dissociation and radical-surface association processes [27-301. In these studies, the various treatments employed

842

J. TROE

consistent potentials. Quite generally it was observed that microcanonical VTST in the J-conserving version (pJVTST) gave good agreement with trajectory results. This observation is in accord with the results of our present analysis. However, it should be emphasized that SACM and VTST become truly idential only when completely quantum-number-selected single channels are considered. In concluding this section we summarize our observations: (i) The VTST gives satisfactory results only when the E- and J-specific microcanonical versions are employed. In this case, pJVTST, SACM, and classical trajectory calculations provide closely similar results. However, truncation procedures (IpJVTST) should not be applied. Also, the range of application of the classical trajectory calculations is limited to sufficiently large values of the ratio RT/B (larger than about 10). (ii) Canonical and truncation-improved canonical VTSTs (i.e., CVTST and ICVTST) in all cases lead to unacceptable errors. These approaches, therefore, should be used with caution, keeping in mind that errors up to factors of about three can arise. Also, CVTST and ICVTST do not embrace the correct result, CVTST mostly providing upper bounds whereas ICVTST may be lower or higher than the true results. (iii) One may ask why one should use a variational treatment on the pJVTST level at all when the major task is the determination of r-dependent energy levels (i.e., of adiabatic channel potential curves). It may be that variational techniques provide some numerical advantages when complicated potential-energy surfaces are analyzed by Monte Car10 sampling of phase space [31]. However, in this case the quantization of the phase-space volume also presents problems. When quantum effects become relevant, SACM appears to provide the only acceptable statistical approach. (iv) The situation becomes more serious for the case of dipole-dipole capture, i.e., for the interaction between two non-monatomic species. There are four transitional oscillator modes of the adduct, which arise from four rotational modes of two separated linear dipoles. In addition, the adiabatic channel potential curves are characterized by numerous multiple avoided crossings, except for the lowest channels [ 161. For this reason, one cannot expect that VTST, even in the pJVTST form, correctly reproduces the dynamics, which on the other hand is correctly described by the SACM/CT treatment.

VI. RECENT SACM APPLICATIONS TO MORE COMPLEX REACTION SYSTEMS We shall conclude this chapter by briefly mentioning a series of rigorous SACM treatments of more complicated reaction systems. In addition to the

STATISTICAL ADIABATIC CHANNEL CALCULATIONS

843

capture of linear dipoles by ions, the capture of symmetric and asymmetric top dipoles as well as of open-shell dipoles has been treated in Ref. 15. Figure 7 shows examples of adiabatic channel potential curves for the C+-OH system [9]. Besides fine structure, one may include hyperfine-structure effects [32]. Fine-structure effects, for example, were shown to generate subtle differences in the adiabatic channel potential curves and the low-temperature thermal capture rate constants by ions of closed-shell quadrupole N2 and open-shell quadrupole 0 2 [33]. Figures 8-11 illustrate these fine details for the lowest channel potential curves and low-temperature capture rate constants. The capture of polarizable quadrupoles was treated by SACM and compared with classical trajectory calculations in Ref. [34]. The capture of dipolequadrupole species has been treated in Ref. 35. More complicated anisotropies of the potential are, for example, encountered in the association of two linear dipoles. Adiabatic channel potential curves for this case have been calculated and expressed analytically in Ref. 16. More systematic studies, also comparing SACM and trajectory results, were reported in Ref. [36]. One may as well consider open-shell effects; for example, the association of two open-shell HO radicals in their lowest rotational state was treated in Ref. 37. Figure 12 shows the lowest rovibronic adiabatic channel potential curves for this system. The ultimate goal

10

20

30 r

(h

40

50

Figure 7. Lowest adiabatic channel potential curves for the interaction of C+ with (curves labeled by rotational quantum numbers m for given j , n; SACM calculations from Ref. 9).

844

J. TROE

qQ/Br3 Figure 8. Lowest adiabatic channel potential curves [33] for the interaction of electronic ground state N2 with ions (q = ionic charge, Q = N2 quadrupole moment; N, M = free-rotor quantum numbers; k, u = harmonic oscillator quantum numbers; for more details, see Ref. 33).

-20

-10

0

10

20

qQ/BP Figure 9. Lowest adiabatic channel potential curves [33] for the interaction of electronic ground state 0 2 with ions (q = ionic charge, Q = 0 2 quadruple moment; N,J , M = free-rotor quantum numbers; M ,k, u = harmonic oscillator quantum numbers; N = electronic angular momentum quantum number; see Ref. 33).

845

STATISTICAL ADIABATIC CHANNEL CALCULATIONS

1.3

-

4

.p:

n

21

1.2

1.1

Figure 10. Thermal rate constants for capture of N2 by an ion (SACM calculation [33] with channels generating from rotational states N = 0, 1, 2, accounting for nuclear statistical weights; left figure: positive ion;right figure: negative ion).

I

1.1

X-1

\

n 0 V

Y

I

I

-

-

;L f 0.24

I L:

-

1.0 -

-

I

I

1

0.2.1

-

1

-

Figure 11. Thermal rate constants for capture of electronic ground state 0 2 by an ion (SACM calculation [33] with channels generating from rotational states J = 0, 1, 2, accounting for the open-shell character of 0 2 ; left figure: positive ion; right figure: negative ion).

846

J. TROE

1

I

I

1

7

I

I

r

3

k

5

2

0

$ -2 -4 -6 0

&r3B

Figure 12. Lowest rovibrational adiabatic channel potential curves of two interacting

OH(2113/2) radicals [classified in W ( s , k )nomenclature, s = +1 states shown; see Ref. 371.

of SACM developments of this type will be the treatment of an association of two asymmetric top species with simple model anisotropies. A parametrized analytical representation of the results then should also be envisaged. More fundamental extensions of adiabatic channel treatments are also being made and should be continued; for example, a “postadiabatic” channel model was designed in Ref. 38. Tunneling contributions could be included for the individual adiabatic channel potential barriers. Nonadiabatic effects have been characterized elsewhere [3, 8, 15, 16, 34, 391. Nonadiabatic couplings will not be the same for various groups of channels, influencing product-state distributions in fragmentations that may be different from SACM predictions [39]. In summary, one may emphasize that a series of reliable dynamical and statistical treatments of rate processes on bonding potential-energy surfaces are at hands that allow for sufficiently accurate treatment of the reaction. Attention, therefore, should focus predominantly on the determination of adequate potential-energy surfaces that, in the range of adiabatic channel maxima, often are not characterized sufficiently well.

Acknowledgment Financial support of this work by the Deutsche Forschungsgemeinschaft (SFB 357 Molekulare Mechanismen unimolekularer Prozesse) is gratefully acknowledged.

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References 1. T. Su and W. J. Chesnavich, J. Chem. Phys. 76, 5183 (1982); T. Su, J. Chem. Phys. 88, 4102 (1988); ibid., 89, 5355 (1988). 2. N. Markovic and S. Nordholm, Chem. Phys. 135, 109 (1989). 3. A. I. Maergoiz, E. E. Nikitin, J. Troe, and V. G. Ushakov, J. Chem. Phys. 105, 6263 (1996). 4. D. C. Clary, Mol. Phys. 53,3 (1984); ibid., 54,605 (1985);Ann. Rev. Phys. Chem. 41,61 (1990); D. C. Clary, T. S. Stoecklin, and A. G. Wickham, J. Chem. Soc. Faraday Trans. 89,2185 (1993). 5. M. Quack and J. Troe, Ber: Bunsenges. Phys. Chem. 78,240 (1974); in Theoretical Chemistry, Vol. 6B, D. Henderson, Ed., Academic, New York, 1981, p. 199. 6. J. Troe, Chem. Phys. Lett. 122,425 (1985). 7. J. Troe, J. Chem. Phys. 87, 2773 (1987); Adv. Chem. Phys. Se,: 82,485 (1992). 8. E. E. Nikitin and J. Troe, J. Chem. Phys. 92,6594 (1990). 9. M. L. Dubernet and R. McCarroll, Z Phys. D13, 255 (1989); Z Phys. D15, 333 (1990); M. L. Dubemet, M. Gargaud, and R. McCarroll. Astron. Astrophys. 259, 373 (1992). 10. K. Takayanagi, in Physics of Electronic and Atomic Collisions, S . Datz, Ed., North-Holland, Amsterdam, 1982, p. 343; J. Phys. Soc. Jpn. 45,976 (1978); ibid., 51, 3337 (1982); K. Sakimoto, Chem. Phys. 85,273 (1984); Chem. Phys. Lett. 116,86 (1985). 11. D. R. Bates and I. Mendas, Proc. Roy. Soc. London A402, 245 (1 985). 12. B. Pezler, J. Turulski, and J. Niedzielski, J. Chem. Soc. Faraday Trans. 89, 655 (1993). 13. M. Ramillon and R. McCarroll, J. Chem. Phys. 101, 8697 (1994). 14. J. Turulski and J. Niedzielski, Inr. J. Muss. Spec. Ion Phys. 139, 155 (1994). 15. J. Troe, J. Chem. Phys. 105, 6249 (1996). 16. A. I. Maergoiz, E. E. Nikitin, and J. Troe, J. Chem. Phys. 95,5117 (1991); Z Phys. Chem. 172, 129 (1991). 17. J. Troe, Z.Phys. Chem. NF 161,209 (1989);J. Chem. Soc. Faraday Trans. 93,885 (1997); Be,: Bunsenges. Phys. Chem. 101,438 (1997). 18. A. 1. Maergoiz and J. Troe,J . Chem. Phys. 99, 3218 (1993); A. 1. Maergoiz, J. Troe,and Ch. Weiss, J. Chem. Phys. 101, 1885 (1994). 19. J. Troe, Chem. Phys. 190, 381 (1995). 20. J. Troe, J. Chem Phys. 79,6017 (1983). 21. M. Olzmann and J. Troe, Ber. Bunsenges. Phys. Chem. 96, 1327 (1992); ibid., 98, 1563 ( 1994). 22. D. C. Astholz, J. Troe, and W. Wieters, J. Chem. Phys. 70, 5107 (1979). 23. S. Glasstone. K. J. Laidler, and H. Eyring, The Theory ofRute Processes, McGraw-Hill, New York, 1941; M. A. Eliason and J. 0. Hirschfelder, J. Ckem. Phys. 30, 1426 (1959); W. A. Wong and R. A. Marcus, J. Chem. Phys. 55,5625 (1971). 24. M. Quack and J. Troe, Ber Bunsenges. Phys. Chem. 81,329 (1975). 25. S. N. Rai and D. G. Truhlar, J. Chem. Phys. 79,6046 (1983). 26. N. Markovic and S. Nordholm, J. Chem. Phys. 91, 6813 (1989). 27. X.Hu and W. L. Hase, J. Chem. Phys. 95, 8073 (1991). 28. E. E. Aubanel, D. M. Wardlaw, L. Zhu, and W. L. Hase, Int. Rev. Phys. Chem. 10, 249 ( I99 I ).

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29. G . H. Peslherbe and W. L. Hase, J. Chem. Phys. 101,8535 (1994). 30. H. Wang, L. Zhu, and W. L. Hase, J. Phys. Chem. 98, 1608 (1994). 31. D. M. Wardlaw and R. A. Marcus, Chem. Phys. Lett. 110, 230 (1984); J. Chem. Phys. 83, 3462 (1985). 32. J. Troe, Be,: Bunsenges. Phys. Chem. 99, 341 (1995).

33. A. 1. Maergoiz, E. E. Nikitin, and J. Troe, Z. Phys. D 36, 339 (1996). 34. A. I. Maergoiz, E. E. Nikitin, J. Troe, and V. G. Ushakov, J. Chem. Phys. 105, 6270 (1996). 35. S. C. Smith and J. Troe,J. Chem. Phys. 97,5451 (1992).

36. A. 1. Maergoiz, E. E. Nikitin, J. Troe, and V. G. Ushakov, J. Chem. Phys. 105, 6277 ( 1996). 37. A. I. Maergoiz, E. E. Nikitin, and J. Troe, J. Chem. Phys. 103, 2083 (1995). 38. E. E. Nikitin, J. Troe, and V. G. Ushakov, J. Chem. Phys. 102,4101 (1995).

39. E. I. Dashevskaya, E. E. Nikitin, and J. Troe,J. Chem. Phys. 93,7803 (1990).

DISCUSSION ON THE REPORT BY J. TROE Chainnun: R. Jost

J.-C.Lorquet: The proper definition of a bottleneck when the potential-energy function does not exhibit a bottleneck involves a rotational barrier. But the potential is known only at large internuclear distances that correspond to low energies and therefore is no longer adequate at high energies. The SACM corrects for that by interpolating between the asymptotic range and the equilibrium geometry where the quantization is assumed to be known. Would you describe SACM as involving a gradual (rather than abrupt) transition-state switching and can you recognize systematic trends in this evolution? Can one apply SACM at high energies although the argument starts from an assumed and necessarily simplified long-range potential? J. IIkoe: I have chosen the simple charge-dipole potential because it gives very transparent results. In reality, one can and one has to use the real potential with its short-range valence and its long-range electrostatic contributions. It is important to consider the attraction potential along the reaction coordinate together with its anisotropy and to provide proper switching expressions. As long as the anisotropy has the simple cos 8 or cos28 form, the results still can be expressed analytically. Such calculations have, for example, been made for H202 HO + OH where the maximum attraction and maximum repulsion energies have been calculated along the reaction coordinate.

-.

STATISTICAL ADIABATIC CHANNEL CALCULATIONS

849

J. Manz 1. Prof. Troe has presented to us the “capture cross sections” for two colliding particles, for example, an induced dipole with a permanent dipole interacting via the potential ~ ( re), = (rq/2r4- qpDcos 8 / r 2 (see “Recent Advances in Statistical Adiabatic Channel Calculations of State-Specific Dissociation Dynamics,” this volume). The results have been evaluated using classical trajectories or SAC theory. But quantum mechanically, a colliding pair of an induced dipole and a permanent dipole could never be “captured” because ultimately they have to dissociate after forming some sort of a collision complex. I would therefore like to ask for the definition of the “capture cross section.” 2. I would like to ask Prof. J. Troe whether he could discuss some typical situations where the SAC approximation may fail. For example, consider the F + HBr --c FHBr* -HF(u) + Br reaction with energy E just above the potential barrier V s. In this situation, the adiabatic channels in the transition state ($) should be populated only in the vibrational ground state, and they should, therefore, yield products HF(u = 0 ) + Br, according to the assumption of adiabatic channels. This is in contrast with population inversion in the experimental results; that is, the preferred product channels are HF(J) + Br, where u’ = 3, 4 [l]; see also the quantum scattering model simulations in Ref. [2]. The fact that dynamics cannot be rigorously adiabatic (as in the most literal interpretation of SAC) has been discussed by Green et al. [3], and the most recent results (for the case of ketene) are in Ref. 4. 1. J. P. Sung and D. W. Setser, J. Chem. Phys. 69, 3868 (1978); N. B. H. Jonathan, P. V. Sellers, and A. J. Stace, Molec. Phys. 43, 215 (1981); L. S. Dzelzkans and F. Kaufman, J. Chem. Phys. 79, 3836 (1983); P. M. Aker, D. J. Donaldson, and J. J. Sloan, J. Phys. Chem. 90, 3110 (1986); L. L. Feezel and D. C. Tardy, J. Phys. Chem. 93, 3124 (1989). 2. J. Manz and H. H. R. Schor, Chem. Phys. Leu. 107,549 (1984); P. L. Gertitschke, J. Manz, J. Romelt, and H. H. R. Schor, J. Chem. Phys. 83, 208 (1985); J. Manz and J. Romelt. J. Chem. SOC. Faraday Trans. I1 86, 1689 (1990). 3. W. H. Green Jr., C. B. Moore, and W. F. Polik, Ann. Rev. Phys. Chem. 43, 591 (1992). 4. 1. Garcia-Moreno, E. R. Lovejoy, and C. B. Moore, J. Chem. Phys. 100,8890,8902 ( 1994).

1. I have treated capture processes because they correspond to a series of real systems: For example, exothermic bimolecular reactions

J. TROE

850

without redissociation of the adduct fall into this class; equally association processes in high-pressure environments are described by capture theory, because collisions stabilize the adducts. 2. Nonadiabatic effects of course occur. When they are not too important, they can be treated by perturbation theory. When they are strong, SACM breaks down. Barrier crossing rates may be less sensitive to nonadiabatic effects than detailed product-state distributions. The classical trajectory treatments from refs. [3], [34], and [36] of my article quantitatively describe the transition from adiabatic to non-adiabatic dynamics, where the former corresponds to small and the latter to large relative kinetic energies between the reactants. R. A. Marcus: My interests in variational microcanonical transition state theory with J conservation goes back to a J. Chem. Phys. 1965 paper [l], and perhaps I could make a few comments. First, using a variational treatment we showed with Steve Klippenstein a few years ago that the transition-state switching mentioned by Prof. Lorquet poses no major problem: The calculations sometimes reveal two, instead of one, bottlenecks (transition states, position of minimum entropy along the reaction coordinate) [2], and then one can use a method described by Miller and partly anticipated by Wigner and Hirschfelder to calculate the net flux. Regarding the reactions themselves, it is clear that some of them, for example those discussed by Schinke, are highly vibrationally nonadiabatic and so it is best, I believe, not to try to cast the reaction products’ state distribution in an adiabatic or statistically adiabatic mode: There are innumerable curve crossings in this case, which, unless one is prepared to (1) trace them along the reaction coordinate, (2) calculate curve crossing probabilities, and (3) ignore any overlapping of such curve crossing regions, make any detailed tracing of the reaction channels along the reaction path not very fruitful: One might as well try to solve the entire dynamical problem in the exit channel numerically. For the reaction rates themselves, as we discussed in the 1960s, the best hope for any relatively simple reaction rate theory is to assume a locally vibrationally adiabatic process or, as I termed it there, a little more generally, a statistically adiabatic process. The effect of nonoverlapping avoided crossings, discussed in J. Phys. Chem. 83,204 (1979), was inferred to be relatively small.

R. A. Marcus, J. Chern. Phys. 43,2658 (1965). 2. S. J. Klippenstein and R. A. Marcus, J. Phys. Chem. 92, 5412 (1988); J. Chern. 1.

Phys. 91, 2280 (1989); ibid., 93, 2418 (1990).

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J. Troe

1. Transition-state switching may also occur in SACM, either for single-channel potential curves or for groups of channels such as in microcanonical variational versions with adiabatic channel states. 2. Microcanonical transition-state theory and SACM are identical only for single channels, not for groups of channels. However, in some cases the results come close to each other; in other cases they differ. 3. SACM may very well serve as the adiabatic reference against which nonadiabatic effects are definable. It is a question of the strength of the coupling forces that determines how adiabatically a system behaves. 4. Obviously, strong nonadiabatic coupling of the states of a longlived dissociating species forms an essential ingredient of statistical unimolecular rate theory and governs the “one-channel lifetimes” h p ( E , J ) of the species. However, the dynamics of the capture motion considered here, which determines the “number of open channels” WfE, J ) , under typical conditions is globally adiabatic, such as shown in [3], [34], [36]. Nevertheless, multiple avoided crossings with local nonadiabatic behavior arise if the two associating species are not atomic as, for example, in dipole-dipole capture [ 161, [36]. In this case an SACM/CT treatment apparently is the method of choice. We expect that pJVTST in this case differs from SACM/CT, and we are presently investigating the extent of discrepancies.

M. Herman: Prof. Troe explicitly referred to the Stark effect in his talk. In the strong-field limit, the Stark effect will mix thej levels. Does his model explicitly take that mixing into account? J. The: My answer to Prof. Herman is that the high-Stark-field description of the close approach of a dipole to an ion can very well be represented in terms of the relevant quantum numbers. The linear dipole-free rotor quantum numbers j and m are converted to the oscillating dipole quantum number u with the identity u = 2j - lml.

QUANTUM AND SEMICLASSICAL THEORIES OF CHEMICAL REACTION RATES W.H. MILLER Department of Chemistry, University of Cali$ornia, and Chemical Sciences Division, Lawrence Berkeley National Laboratory Berkeley, California

CONTENTS I. Introduction 11. Quantum Theory 111. Semiclassical Approximation for the CRP IV. Concluding Remarks References

I. INTRODUCTION The rate constant for a chemical reaction is conveniently expressed in terms of the cumulative reaction probability [ 11 (CRP) N(E),

where the S matrix elements must in general be determined by solving the state-to-state reactive scattering Schrodinger equation (n, and np denote the asymptotic quantum states of reactants and products). The microcanonical rate constant for total energy E, typically the quantity of interest for unimolecular reactions, is given in terms of the CRF’ by

Advances in Chemical Physics, Volume 101: Chemical Reactions and Their Control on the Femtosecond Time Scale, XXth Solvay Conference on Chemistry, Edited by Pierre Gaspard, Irene Burghardt. I. Rigogine, and Stuart A. Rice.

ISBN 0-471-18048-3

0 1997 John Wiley & Sons, Inc.

853

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W. H.MILLER

k ( E ) = [271.)lp,(E)]-’N(E)

(1.2)

where p,. is the density of reactant states per unit energy, and the canonical (or thermal) rate constant is the Boltzmann average of the CRP, m

k ( T ) = [2rhQr(T)]-’

(1.3)

where Qr is the reactant partition function per unit volume. Considerable progress has been made in recent years [2-4] in learning how to calculate the CRP more directly than via Eq. (1 . l ) , that is, without having to solve explicitly for the S matrix, yet still correctly, without any inherent approximations. It is still necessary to determine the quantum dynamics of the system, but typically only for short times and only in the interaction region where the reactive flux is determined, not in the longer range regions that determine the distributions of reactant and product quantum states. The overall approach has very much the “feel” of transition state theory [5, 61, though it is a fully rigorous (“exact”) quantum treatment. Section I1 briefly reviews this rigorous quantum approach for obtaining the CRP and its various applications to date. Even with the progress that has been made in rigorous quantum approaches, it is nevertheless possible to carry out such calculations only for relatively simple chemical systems. For example, the largest molecular system for which such calculations have been carried out is for the reaction Hz + OH -+ H 2 0 + H. There is clearly interest, therefore, in the development of approximate versions of the approach that can be applied to more complex systems. Section 111 describes a semiclassical approximation for doing this, and Section IV concludes. 11. QUANTUM THEORY

Miller et al. [ 2 ] showed more than 10 years ago that the CRP can be expressed exactly in terms of the microcanonical density operator,

where I% is the Hamiltonian operator of the molecular system and 6 is a flux operator defined with respect to a dividing surface that separates reactants and products. [The value of N(E) is independent of the location of the

THEORIES OF REACTION RATES

855

dividing surface.] The task is then to fin$ an efficient way to compute matrix elements of the density operator S(E - H). Thirumalai et al. [7]noted that one can use a Gaussian prelimit representation of the delta function to do the job:

6(E -

If Aa = a / M is sufficiently small, then a “short time”-like approximation can be used to obtain the matrix representation of e x p i - h ( H - E)2], and then the full result in Eq. (2.2) is obtained by matrix multiplication. This approach was seen to work well in some simple examples. A more powerful approach was presented by Seideman et al. [3],

with

&E) = ( E + i;

-

h)-’

(2.3b)

where is an absorbing potential energy operator [8]. If were chosen to be a constant, as in the conventional definition of the Green’s function, then Eq. (2.3) is a Lorentzian prelimit representation of the delta function,

but the power of Eq. (2.3) is that E^ need not be a constant. By allowing it to be a potential energy operator, one can choose it to be essentially zero in the interaction region, where the relevant reaction dynamics takes place, and “turn it on” gradually in the reactant and product exit valleys; this is effect imposes outgoing-wave boundary conditions for an L2 representation of the Green’s function in Eq. (2.3b). Figure 1 shows a contour plot of the H2 + H reaction and the absorbpotential energy surface for the H + H2 ing potential. (The points in Fig. 1 are the grid points for a discrete variable representation of the Hamiltonian operator.)

-.

958 Figure 1. Solid lines are contours of the potential energy surface for the H

+ H2

-

H2

+ H reaction. Broken lines are contours of the absorbing potential (which is zero in the central

part of the interaction region and turned on at the edge) for three possible choices of it. The points are the grid points that constitute the basis set for the evaluation of the quantum trace, Eq. (2.5).

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From a time-dependent point of view, the absorbing potential damps the time-dependent wavepacket as it exits the interaction region and thus prevents reflection from the edge of the grid, thereby enforcing the outgoingwave boundary condition. In practice, one wishes to turn on the absorbing potential as rapidly as possible, so as to keep the grid (i.e,, basis set) as small as possible, but not so rapidly as to cause unphysical reflections. With the microcanonical density operator given by Eq. (2.3), Seideman et al. [3] then showed that Eq. (2.1) for the CRP takes the simpler form N(E) = 4 td&(E)*gP&(E)zr]

(2.5)

with the Green’s function given by Eq.(2.3b) and where ;&,) is the part of the absoring potential in the reactant (product) exit valley. Note that in Eq. (2.5) there is no longer any reference to a “dividing surface,” now only a “dividing region,” the space between the two absorbing potentials. Equation (2.5) corresponds effectively to solving the Schrodinger equation (equivalent to constructing the Green’s function) through this interaction region, revealing again the qualitative “transition-state” nature of the formulation. Finally, Manthe et al. [4] have shown that for large systems Eq. (2.5) can be evaluated most efficiently by writing it as N(E) = tr[&E)]

(2.6a)

where

&E) = 4;

&(E)*gp&(E)g:I2

(2.6b)

The reaction probability operator h of Eq. (2.6b) is Hermitian and clearly positive, and one can furthermore show [4] that it is bounded by the identity operator, that is, 0 < F(E)< 1

(2.7)

The eigenvalues of k, ( p k ( E ) } , thus lie between 0 and 1 and can therefore be interpreted as probabilities, the eigen reaction probabilities, the sum of which gives the CRP,

The reascp that Eqs. (2.6H2.8) are so efficient computationally is that the rank of P(E), that is, the number of its eigenvalues that are significantly

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W. H. MILLER

different from zero, is in general quite low compared to the dimension of the matrix itself (which is the dimension of the Hamiltonian matrix). The Lanczos algorithm for determining the eigenvalues {pk(E)} is thus very efficient, the number of iterations required being only the rank of the matrix (i.e., the number of nonzero eigenvalues). For the H2 + OH .+ H 2 0 + H reaction, for example, the size of the Hamiltonian matrix is the order of I d x lo5, but there are only -10-20 nonzeto eigenreaction-probabilities{pk }, so only -1&20 Lanczos iterations with are required to obtain them. Every Lanczos iteration with P, however, requires two operations of the Green’s function onto a vector,

<

-

G(E)(or G(E)*) v = x

(2.9a)

meaning that one must solve the linear system

(Efi~-H).x=v

(2.9b)

each time, and this is the major computational task.The most powerful way discovered so far for doing this is the quasi-minimum residual (QMR) algorithm [9, 101, an iterative method that has proved to be extremely efficient (when used with preconditioning) and very economical of computer memory, requiring only a few vectors and no matrices to be stored. By way of example, Fig. 2 shows the eigen-reaction-probabilitiesfor the

E lev1

Figure 2. Dotted lines are the eigen-reaction-probabilities ( p k ( E ) } for the collinear H + HZ reaction. The solid line is their sum,the cumulative reaction probability N ( E ) .

THEORIES OF REACTION RATES

859

collinear H + H2 -+ H2 + H reaction as a function of energy. One clearly sees the qualitative correspondence of these eigenvalues with the transmission probabilities of transition state theory (TST),

where Pld(E1) is a one-dimensional transmission probability as a function of the energy El = E - e! available in the reaction coordinate. (Here, e! is the vibrational energy level for motion orthogonal to the reaction coordinate, that is, for states of the “activated complex.”) In TST each transmission probability rises from zero at the various thresholds of the activated complex energy levels and remains at unity. The eigen-reaction probabilities are seen to behave qualitatively like this but do not follow this behavior in detail because of TST-violating dynamics, that is, trajectories (in a classical picture) that recross the dividing surface due to the short-lived collision complex of the H + H2 system. Figure 3 shows the CRP N ( E ) for the collinear and three-dimensional versions of the H + H2 reaction. The nonmonotonic energy dependence seen in N ( E ) for the collinear case (Fig. 3a) is a result of transition state-violating dynamics. In the 3d case (Fig. 3b), however, N(E) is monotonically increasing; that is, the additional averagings involved in 3d causes TST to be a better approximation (a well-known phenomenon). Finally, Fig. 4 shows N ( E ) for the H2 + OH + H 2 0 + H reaction [ 111, and here one sees not only monotonic energy dependence but also the disappearance in the step structure seen in Fig. 3. This is because of the higher density of states (because of the additional degrees of freedom), causing the “steps” to overlap and thus not be apparant to the eye. 111. SEMICLASSICAL APPROXIMATION FOR THE CRP

Even with the advances described in Section 11, however, rigorous quantum calculations are still feasible only for relatively simple chemical reactions. The largest molecular system studied to date using these approaches is H:! + OH H 2 0 + H for total angular momentum J = 0, a six degree-offreedom system [ 111. The fundamental limitation in these rigorous quantum approaches is that the number of basis functions (or grid points in a discrete variable representation [ 121) grows exponentially with the number of degrees of freedom for the system, and the linear algebra calculation necessary to obtain the Green’s function grows in difficulty more than linearly with the number of basis functions.

-

W. H. MILLER

860

I

w

I

I

I

I

I

I

I

I

I

I

16.0 -

-

-

-

12.0 -

-

-

-

-

8.0 -

4.0 -

-

-

-

-

0.0

I

I

I

I

I

I

I

I

I

I

-

H2 + H reaction: (a) Figure 3. Cumulative reaction probability for the H + H2 collinear geometry (Ref. 3a); (b) three-dimensional space for total angular momentum J = 0 (Ref. 3b).

One is thus clearly interested in approximate approaches that have the potential for treating far more complex systems but that are also of useful reliability. To this end, we have pursued a semiclassical, classical trajectorybased approach for evaluating the CRP. The semiclassical approximation for the CRP is in principle quite straightforward: One uses Eq. (2.5) with a semi-

86 1

THEORIES OF REACTION RATES

0.0

0.1

0.2

E (eV

0.3

0.4

0.5

( a1

E (eV) (b)

-

H 2 0 + H reaction as a Figure 4. Cumulative reaction probability for the Hz + OH function of total energy for total angular momentum J = 0 (Ref. 11): ( a )logarithmic scale; (b) linear scale.

classical approximation for the Green’s function. (A similar kind of semiclassical approximation for the Green’s function has also been used to obtain the S matrix itself [13].) For a generic Cartesian Hamiltonian with f degrees of freedom, the primitive, or Van Vleck [ 14, 151, approximation for the Green’s function is

W. H. MILLER

862

where xt= x(x1, pl;t ) is the classical trajectory with the indicated initial conditions, S is the classical action along the trajectory, Y (the Maslov index) is the number of zeros experienced by the Jacobian determinant along the trajectory, and el is the exponential damping factor arising from the absorbing potentials,

The sum in Eq. (3.1) is over all values of the initial momentum pI that satisfy the boundary condition

The absorbing potential factor e-er/tt is the only nonstandard feature in

Eq. (3.1). Fortunately, it is much simpler to deal with the absorbing poten-

tial semiclassically than it is quantum mechanically: it cannot cause any unwanted reflections in the above semiclassical expression because we have implicitly made an infinitesimal approximation for it. Thus, it does not affect the dynamics and only causes absorption; see also the disscussion by Seideman et al. [3b]. The key feature to making a semiclassical approach practical is to avoid having to deal explicitly with the double-ended boundary conditions in Eq. (3.3) [16-201. (The initial condition x(xl,p,;O) = xI is obviously easy to deal with.) To do this, one uses the standard coordinate space representation of (2.5),

a.

(3.4) and the following initial-value representation (IVR) for the Green’s function [19b]:

863

THEORIES OF REACTION RATES

This comes about by making the primitive semiclassical approximation in a momentum representation and then Fourier transforming this to obtain (x~~G(E)IxI). We found, however, that the direct use of Eq. (3.5) gives quite unstable and erratic results (vide infra). Excellent results are obtained, though, if Eq. (3.5) is subjected to a modified Filinov transformation (MFT) of the type used by Makri and Miller [21]; this replaces Eq. (3.5) by

In the limit A -+0, the IVR of Q. (3.5) is regained, and as A that

-

00,

one sees

and it is not hard to show that Eq.(3.6) then reverts to the original primitive semiclassical result for G, Eq.(3.1) [with the boundary condition, Q. (3.3)]. Equation (3.6) is also closely related to the “cellular dynamics” approach of Heller and co-workers [18] and to some of the IVRs used by Herman et al. [17] and by Kay [20]. The final expression used to generate the results discussed in the next section is thus

864

W. H.MILLER

Regarding practical aspects of the calculation, we note that for given values of the initial conditions ( X I pl), , the time integral in Eq. (3.8) is evaluated along the trajectory while one is computing it; that is, one does not need to store all the time-dependent quantities and then do the integral over t . Doing the t integral thus entails essentially no extra effort in the calculation. (One continues the integral over t until the absorption factor e-Qlh damps the integrand effectively to zero.) Also due to the structure of Eq. (3.8), one can do the calculation for many different values of the integration variable x2, and also for many different values of energy E, all from the same set of initial conditions ( X I , pl), that is, from the same set of trajectories. The most crucial numerical aspect of the calculation is the integral of initial momenta pI,because the integrand has positive and negative contributions (while the integrals over X I and xz involve positive-definite quantities). The Filinov damping factor, exp[- kA(x2 - x,)~],greatly improves this aspect of the calculation and is the principal reason why the modified Filinov representation of the Green’s function is preferable to the IVR of Eq. (3.5). Finally, note that in Eq. (2.12) we have included no Maslov phase factor related to the square root of the Jacobian determinant D,

This is because in this “hybrid representation” D(t) will in general never vanish for any value of t (because ap,/apI and axr/ap, are each real). It is easy to see that the initial value (t -+ 0 ) of D(t) is unity (since Xr - x1 +plr/m, pr= pl), and to obtain the correct phase of D(t)’12for all later t, one simply follows the complex value of D(t) along the trajectory, adding the necessary

865

THEORIES OF REACTION RATES

phase 8"anytime the branch line, D(t) = a negative real number, is crossed so that ~ ( t ) ' is/ ~a continuous function of t. As a first test [22] of the semiclassical approach described above we have computed the transmission probability through the Eckart potential barrier,

(3.10)

V(x) = Vosech2(x)

with a barrier height V, = 0.425 eV and the mass m = 1060 a.u. chosen, as before [3a],to correspond approximately to the H + H2 H2 + H reaction. The first important feature to show is the extent to which the Filinov "smoothing" transformation simplifies the integral over P I . For the values XI = -4.75 a.u. and x2 = 4.25 a.u., Fig. 5 shows the real part of the integrand of Eq.(3.6) for several values of the parameter A (0, 1.0, and 10.0) and two values of the energy E, one below the barrier (E = 0.3 eV) and one above the --t

10oo.o

E = 0.300eV

E = 0.600eV

500.0

0.0 -500.0

1 a

10oo.o 500.0

0.0 -500.0 1m.0

500.0 0.0

-500.0

-1Ooo.O' 0.0

"

1.0

2.0

' 3.0

.

I

4.0

.

"

1.0

I

2.0

'

' 3.0

'

4.0

61 Figure 5. Real part of integrand of &. (3.6) as a function of p i , for x i = -4.75 a.u. and x2 = 4.25 a.u., for the two indicated values of total energy and three different values of the Filinov parameter A = 0.0, 1.0, 10.0 (top, middle, and bottom, respectively).

866

W. H.MILLER

Energy (eV) Figure 6. The N ( E ) vs. energy E for several values of the Filinov parameter [(O) A = 0.0,(0)A = 1.0, ( 0 ) A = 5.0, ( 0 ) A = 20.01 compared to exact quantum result (-).

barrier (E = 0.6 eV). One clearly sees the effect of the Filinov smoothing (A > 0) on damping the oscillatory nature of the integral. One also sees the stationary phase region of the integrand for the case above the barrier, E = 0.6 eV, near the value of pi = 1.0. There is no stationary phase region for the energy below the barrier, E = 0.3 eV, and this is of course why the transmission probability is small in this case (i.e., in the tunneling regime). Figure 6 shows the results obtained for N ( E ) for several values of A. We do not obtain satisfactory results for A = 0, but for a wide range of A > 0 we obtain quite stable results that are relatively insensitive to the particular value of this smoothing parameter. This is precisely the behavior one wishes to see. It is also significant that the results in Fig. 6 are accurate for some ways into the “classically forbidden” tunneling regime, in this case for energies as much as 0.1 eV or so below the barrier, down to a transmission probability o f = 10-3.

IV. CONCLUDING REMARKS One thus has a theoretical methodology that allows one to compute the rate constant for a chemical reaction “directly,” without having to solve the com-

THEORIES OF REACTION RATES

867

plete state-to-state reactive scattering problem, yet also “correctly,” without inherent approximation. One can of course carry out the correct quantum calculation only for relatively small molecular systems, but is is possible to utilize a semiclassical approximation (which includes the effects of interference and tunneling) that should be applicable to larger systems.

Acknowledgment This work was supported by the Director, Office of Energy Research, Office of Basic Energy Sciences, Chemical Sciences Division of the U.S. Department of Energy under Contract NO. DE-AC03-76SF00098.

References 1. W. H. Miller, J. Chem. Phys. 62, 1899 (1975). 2. W. H. Miller, S. D. Schwartz, and J. W. Tromp, J. Chem. Phys. 79,4889 (1983). 3. (a) T. Seideman and W. H. Miller, J. Chem. Phys. 96,4412 (1992); (b) T. Seideman and W. H. Miller, J. Chem. Phys. 97, 2499 (1992). 4. U. Manthe and W. H. Miller, J. Chem. Phys. 99, 3411 (1993). 5. (a) An interesting set of papers by many of the founders of the theory-Wigner, M. Polanyi, Evans, Eyring-is in Trans. Faraduy Soc. 34, Reaction Kinetics-A General Discussion, 1938, pp. 1-127; (b) F? Pechukas, in Modern Theoretical Chemistry, Vol. 2: Dynamics of Molecular Collisions, Part B, W. H. Miller, Ed., Plenum, New York, 1976, Chapter 6. (c) D. G. Truhlar, W. L. Hase, and J. T. Hynes, J. Chem. Phys. 87, 2664 (1983). 6. W. H. Miller, Accrs. Chem. Res. 26, 174 (1993). 7. D. Thirumalai, B. C. Garrett, and B. J. Berne, J. Chem. Phys. 83, 2972 (1985). 8. (a) A. Goldberg and B. W. Shore, J. Phys. B 11, 3339 (1978); (b) C. Leforestier and R. E. Wyatt. J. Chem. Phys. 78, 2334 (1983); (c) C. Cerjan, D. Kosloff. and T. Teshef, Geophysics 50, 705 (1985); (d) R. Kosloff and D. Kosloff, J. Comput. Phys. 63, 363 (1986); (e) D. Neuhauser and M. Baer, J. Chem. Phys. 90,4351 (1989); (f)D. Neuhauser, M. Baer, and D. J. Kouri, J. Chem. Phys. 93, 2499 (1990); (g) D. Neuhauser, J. Chem. Phys. 93, 7836 (1990). 9. R. W.Freund, SIAM J. Sci. Star. Comput. 13, 425 (1992). 10. H. 0. Karlsson, J. Chem. Phys. 103,4914 (1995). 11. (a) U. Manthe, T. Seideman, and W.H.Miller, J. Chem. Phys. 99, 10078 (1993); (b) U. Manthe, T. Seideman, and W. H. Miller, J. Chem. Phys. 101,4759 (194). 12. J. C. Light, I. P. Hamilton, and J. V. Lill, J. Chem. Phys. 82, 1400 (1985). 13. S. Keshavamurthy and W. H. Miller, Chem. Phys. Leu. 218, (1994) 189. 14. J. H. Van Vleck, Proc. Natl. Acad. Sci. 14, 178 (1928). 15. (a) W. H. Miller, Adv. Chem. Phys. 25,69 (1974); ibid., 30,77 (1975); (b) V.P. Maslov and M.V. Fedoriuk. Semiclassical Approach in Quantum Mechanics, Reidel, Boston, 1981; (c) M. C. Gutzwiller. Chaos in Classical and Quantum Mechanics. Springer, New York, 1990. 16. (a) W. H. Miller, J. Chem. Phys. 53, 3578 (1970); (b) W. H. Miller and T. F. George, J. Chem. Phys. 56, 5668 (1972); (c) W. H. Miller, J. Chem. Phys. 95, 9428 (1991). 17. (a) M. F. Herman and E. Kluk, Chem. Phys. 91, 27 (1984); (b) E. Kluk, M. F. Herman,

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and H. L.Davis, J. Chem. Phys. 84,326 (1986); (c) M.F. Herman, J. Chem. Phys. 85, 2069 (1986).

18. (a) E. J. Heller, J. Chem. Phys. 94,2723 (1991); (b) M. A. Sepulveda and E. J. Heller, J. Chem. Phys. 101, 8004 (1994); (c) E. J. Heller, J. Chem. Phys. 95,9431 (1991). 19. (a) G. Camplieti and P. Brumer, J. Chem. Phys. 96,5969 (1992); (b) G. Camplieti and P. Brumer, Phys. Rev. A 50,997 (1994).

20. (a) K.G. Kay, J. Chem. Phys. 100,4377 (1994); (b) K. G. Kay, J. Chem. Phys. 100,4432 (1994); (c) K. G. Kay, J. Chem. Phys. 101, 2250 (1994). 21. (a) N. Makri and W. H. Miller, Chem. Phys. Lett. 139, 10 (1987); (b) N. Makri and W. H. Miller, J. Chem. Phys. 89, 2170 (1988). 22. B. W. Spath and W. H. Miller, J. Chem. Phys. 104,95 (1996).

DISCUSSION ON THE REPORT BY W.H. MILLER Chairman: R. Jost J. Maw: Prof. Miller has presented to us a quantum theory for rate coefficients, based on the expression [ 1-51

+

A

where Q is the partition function, and the symmetrized flux operator, = (P+ P ~ / Z In practice this integral has to be evaluated only over a short time t, and the trace (tr) involves only wavepackets propagated in small regions close to the transition state. Since this is a conference that centers much attention on femtosecond chemistry, I would like to ask Prof. Miller whether he could specify the decisive time scale that he needs to evaluate k ( T ) for some specific system. If this time scale is short enough (<<1 ps or so), then one may use, indeed, modem computational techniques [6-101 that allow to propagate wavepackets in many (hundreds!) of dimensions; that is, one can evaluate k ( T ) for large plyatomic molecules using femtosecond quantum propagation techniques. 1. 2. 3. 4. 5. 6.

W. H. Miller, J. Chem. Phys. 61, 1823 (1974). W. H. Miller, S. D. Schwartz, and T. W. Tromp, J. Chem. Phys. 79, 4889 (1983). T. Seideman and W. H.Miller, J. Chem. Phys. 97, 2499 (1992). U. Manthe and W. H. Miller, J. Chem. Phys. 99,3411 (1993). U. Manthe, T. Seideman, and W. H. Miller, J. Chem. Phys. 101,4759 (1994). R. B. Gerber, V.Buch, and M. A. Ratner, J. Chem. Phys. 77, 3022 (1982); R. B.

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Gerber and M. A. Ratner, Adv. Chem. Phys. 70.97 (1988); R. B. Gerber, A. B. McCoy, and A. Garcia-Vela, in Femtosecond Chemistry, J. Manz and L. Waste. Eds., Verlag Chemie, Weinheim, 1995, Chapter 16. 7. H.-D. Meyer, U. Manthe, and L. S. Cederbaum, Chem. Phys. Lett. 165,73 (1990); U. Manthe, H.-D. Meyer, and L. S. Cederbaum, 1. Chem. Phys. 97,3199 (1992); G . A. Worth, H.-D. Meyer, and L. S. Cederbaum, J. Chem. Phys. 105,44 12 (1996). 8. P. Jungwirth and R. B. Gerber, J. Chem. Phys. 102,6046,8855 (1995); P. Jungwirth and R. B. Gerber, J. Chem. Phys. 104,5803 (1996). 9. G. Stock, J. Chem. Phys. 103, 2888, 10015 (1995). 10. S. Krempl, M. Winterstetter, H. Plohn, and W. Domcke, J. Chem. Phys. 100,926 (1994); S. Krempl, M. Winterstetter, and W. Domcke, J. Chem. Phys. 102, 6499 ( 1995).

W. H. Miller: Yes, the calculation is especially efficient because only short-time dynamics is required to determine the net reactive flux (while much longer time dynamics would be required to determine state-to-state reaction probabilities). H2 + H reaction, for example, the reactive flux For the H + H2 requires time evolution of only -25 fs. The requirement of only shorttime dynamics does indeed suggest the utility of a variety of approximations, such as the time-dependent self-consistent field approximation. J. Troe. I have two questions for Prof. Miller:

-

-

1. Do you think that the H + 0 2 + HO;, H0; .+ HO + 0 system, which describes the H + 0 2 HO + 0 reaction, can be separated into three decoupled steps? 2. For the rigid entrance/rigid exit complex-forming bimolecular reaction HO + CO --t H + COz, which passes through HOCO’, a separated-step conventional Rice-Ramsperger-Kassel-Marcus (RRKM) treatment extremely well reproduces the experimental temperature and pressure dependences of this four-atom system. W.H. Miller: For a reaction such as H + 0 2 -+OH + 0, or ketene isomerization H2CC’O OCC’H2, which involve a metastable intermediate, one can certainly do better than simple TST by using a model with two transition states and the “unified statistical model” to approximate the net reactive flux. Such an approach, though often useful, is nevertheless an approximate model that can never be made into a rigorous description. Such a model, for example, cannot describe the resonance tunneling structure in the ketene isomerization that I described. P. Gaspard: I would like to know the opinion of Prof. Miller on the following point. The formula that gives the reaction rate in terms of the flux-flux correlation is of the same kind as the other Green-Kubo

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formulas. However, the flux is an observable that is singular. In that regard, is not there a difference with respect to the other Green-Kubo formulas?

W. H. Miller: The expression for the reaction rate (in terms of a flux-flux autocorrelation function) obtained by myself, Schwartz, and Tromp in 1983 is very similar (though not identical) to the one given earlier by Yamamoto. It is also an example of Green-Kubo relations. R. A. Marcus: Prof. Miller, has your insightful quantum mechanical flux-flux correlation expression for the rate been used to test some of the simplified quantum mechanical tunneling calculations for reactions? I recall that Coltrin and I found a simple path that agreed to a factor of 2, over six or so orders of magnitude of tunneling, with the quantum mechanical results for the collinear symmetric reaction H + H2 H2 + H [l]. Truhlar has proposed an extension for asymmetric reactions.

-.

1. R. A. Marcus and M. E. Coltrin, J. Chem. Phys. 67,2609 (1977).

W. H. Miller: For many years one has had a variety of approximate models for describing chemical reaction rates, mostly based on transition-state theory with tunneling corrections that come from semiclassical theory (e.g., the “instanton” and related models). One of the most useful of these, I believe, is recent work [W. H. Miller, R. Hernandez, N. C. Handy, D. Jayatilaka, and A. Willetts, Chem. Phys. Lett. 172,62 (1990)l based on the conserved action variables associated with the saddle-point region of a potential-energy surface. The emphasis in the present reactive flux formulation, however, has been to obtain a rigorous, ab initio, if you like, approach that would permit calculations independent of any modelistic assumptions. Rather than testing various simple models on reactions that are transition-state-like (that is, that are essentially “direct” dynamics, transmission through a simple saddle point) we have concentrated on applications to a variety of more general situations for which the simple models simply do not apply.

R. A. Marcus: In the case of the reaction Klippenstein and I studied, which showed two transition states [11, the motion was that largely of heavy atoms rather than hydrogen atoms. We assumed incoherent motion between the two, though in some systems, such as the one you treated, coherence can certainly be important. In your ketene system was tunneling involved in passage through the two barriers? 1. S. J. Klippenstein and R. A. Marcus, J. Phys. Chem. 92, 5412 (1988); J. Chem. Phys. 91, 2280 (1989); ibid., 93, 2418 (1990).

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W. H. Miller: Tunneling is surely involved in the present case because the isomerization involves a large amount of hydrogen atom motion. Also, with no tunneling the resonances would be infinitely narrow (and thus unobservable). Our calculations, though, are a fully quantum mechanical treatment and thus do not explicitly identify what is tunneling and what is not. This would only be meaningful in an approximate (e.g., semiclassical) treatment.

FEMTOSPECTROCHEMISTRY: NOVEL POSSIBILITIES WITH THREE-DIMENSIONAL (SPACE-TIME) RESOLUTION V. S. LETOKHOV Institute of Spectroscopy Russian Academy of Sciences, Troitzk, Moscow Region 142092, Russia

CONTENTS I. Introduction 11. principal Idea 111. Femtosecond MPI of Chromophores IV. Laser Resonance Photoelectron Spectromicroscopy V. Toward Femtosecond Laser Photoion Microscopy VI. Conclusion References

I. INTRODUCTION The laser, one of the most modem and universal twentieth-century tools at our disposal, is of importance to chemistry in two respects. 1. Laser light is at the root of a great variety of laser spectroscopic techniques, allowing many chemical problems to be solved. Let us list these briefly. First, laser light helps one to determine the atomic-molecular composition

Advances in Chemical Physics, Volume 101: Chemical Reactions and Their Control on the Femtosecond Time Scale. XXth Solvay Conference on Chemistry, Edited by Pierre Gaspard, Irene Burghardt, I. F’rigogine, and Stuart A. Rice. ISBN 0-471-18048-3 0 1997 John Wiley & Sons, Inc.

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of substances with an extraordinary high sensitivity, the sensitivity being so high as to reach the natural sensitivity limit (single atoms and molecules). This is of special value in laser analytical spectrochemistry [l]. Second, it makes it possible to deternine not only the species but also the quantum state of particles entering into a chemical reaction or produced by a chemical reaction. This opens the possibility of gaining a much deeper insight into the kinetics of chemical reactions, including the elementary processes of fairly complex reactions such as the combustion of various mixtures. Third, it makes it possible to observe on a real-time basis the evolution of the most short-lived transient of reacting particles, that is, to study the molecular dynamics of chemical reactions in a femtosecond time scale [2]. Many papers presented at the XXth Solvay Conference were devoted to this possibility of implementing one-dimensional resolution along the reaction coordinate. The time resolution Ar corresponds to the spatial resolution Az = At(u), where (u) is the average velocity along the reaction coordinate, for example, the velocity of the reaction products. At (u) = 3 X 10'' cm/s, the quantity At = 100.fs corresponds to a one-dimensional (longitudinal) resolution of Az = 0.3 A. b

2. Laser light enables one to deposit energy in the internal degrees of freedom of molecules in a resonant manner, greatly disturbing the equilibrium internal energy distribution among the particles and their degrees of freedom, which allows a number of chemical problems to be solved. First, high-intensity laser radiation is capable of depositing energy at an exceptionally high rate, the rate of stimulated transitions between different quantum levels depending directly on the radiation intensity or, for many multiple-photon stimulated transitions, even more strongly [3]. For example, the energy of electronic or vibrational motion can be raised at a rate of 109-10'4 eV/s (1-10 eV in 10-9-10-'3s), which substantially exceeds the rate of energy relaxation to the equilibrium state. Second, a tunable laser radiation can excite atoms or moiecules of a certain species (or even of a certain isotopic composition) in a mixture, that is, can ensure the inremolecular selectivity of the subsequent photochemical processes 131. Third, ultrashort laser pulses can provide for selective electronic or vibrational excitation of molecules, which will probably allow one to effect a high selective excitation of certain vibrational degrees of freedom, that is, to effect mode-selective photochemical reactions. Fourth, coherent excitation by means of femtosecond laser pulses of a controllable pulse interval makes it possible to control to some extent the coherent evolution of molecules in an intermediate excited state and

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thus coherent control to the pathways (or channels) of chemical reactions ~41. The following laser light characteristics play a key part in both these approaches. (a) The monochromaticity and availability of laser light of any wavelength [from Vacuum UV (VUV) to infrared (IR)] ensure selective excitation of the desired quantum states or modes of a certain molecular species. This forms the basis for photochemistry with intermolecular selectivity and quantumstate-selective photochemistry. (b) The short and controllable laser pulse duration (from micro- to femtoseconds) makes it possible to study with a high time resolution not only the kinetics of chemical reactions but also their dynamics. Ultrashort coherent laser pulses will probably help one to observe mode-selective and pathwayselective chemical processes. (c) The spatial coherence of laser light and its capability to focusing provide for the study and stimulation of chemical processes with a spatial resolution down to the light wavelength. What is more, there are possibilities of bringing the spatial resolution down to a molecular size. It is exactly this challenging possibility that is the objective of investigations conducted at my laboratory in the last few years. For the time being we are far from the solution of this problem but have obtained the first encouraging results.

II. PRINCIPALIDEA The laser femtosecond ionization and fragmentation of molecules with formation of charged particle (photoelectrons or photoions) gives the new possibility to realize to lateral two-dimensional spatial resolution (femtosecond laser visualization of molecules). This possibility is based on two key features: (1) the point of photoionization fragmentation [5] (or point of photoion or photoelectron detachment) is localized with accuracy much better than the photoionizating laser wavelength and (2) the electron or ion microscopy technique allows to observe the points of ejection of charged particles with great magnification ( 105-106).This is basis for an interesting possibility of detection of molecules and chromophores with spatial resolution of several nanometers or even angstroms, that is, the development of femtosecond photoionization microspectroscopy. The idea of photoionization spatial localization of the molecular bonds or chromophore of a large molecule based on a combination of spectrally selective photoionization of a chosen bond or chromophore of the molecule with electron or ion microscopy is shown on Fig. 1 for a particular case of projective microscopy. A hemispherical tip of a needle with a radius of curvature

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/ / A -

;1

\ f2

:reen

laser pulses w, ,w2.. W

n-I

\ Ebnd AB

fl

Figure 1. (a) Schematic diagram of a laser photoion projection microscope and (6) spectrally selective multistep photoionization scheme for absorbing centers (color centers, molecular chromophores, etc.) by ultrashort laser pulses.

r is used as a cathode, and a confocal screen with a radius of curvature R is used as an anode. A molecule is adsorbed on the surface. Under sequence of ultrashort (femtosecond) light pulses the resonant multiphoton ionization (MPI) of the chosen molecular bond or chromophore takes place, that is, the extraction of molecular ion M ' . In presence of a strong electric field near the cathode the photoions move along radial lines toward to screen. The electric field only serves to transfer the photoions to the anode rather than to ionize the adsorbed molecule. For this purpose the strength of the electric field can be reduced to values at which field-induced nonselective and destructive molecular evaporation and decomposition are eliminated. Here it is the difference between the field-ionization ion Muller microscopy and this laser-induced ion microscopy [6]. The resolution of photoion laser microscopy is limited by two fundamental factors [7]:the Heisenberg principle of uncertainty and the presence of the nonzero tangential component of the velocity of the ejected photoion (photoelectron). The same factors restrict the spatial resolution of the field-ion microscopy. It must be emphasized again that the key difference lies in the fact that for photoion microscopy there is no need for a strong (ionizing) electric field that distorts and desorbs the molecules. And also, the femtosecond laser radiation allows the photoion to be photoselectively extracted from certain parts of a molecule.

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The uncertainty principle causes a spread of the transversal velocity of the photoion, Av = h/2M Ax; its coordinate is accurately determined to be Ax, where M is the mass of the photoion. This spread, Avx, results in a circle of ion scattering on the screen with a diameter d = 27&, where 7 is the flight time of the particle from the tip to the screen and R is the distance between the tip and screen. On the other hand, the circle of scattering on the screen, d, is also related to the indeterminacy of the coordinate at the point of detachment ~ ( A =Xd / K ) , where K = R/r is the projector magnification coefficient. The simple estimations show that the uncertainty principle limits the spatial resolution of photoelectron microscopy on the level 20-30 A; for photoion microscopy this is below 1 A. An ejected photoion can have nonzero transversal velocity, Aut,, if a part of the excitation energy is converted to kinetic energy during the photodetachment from the surface. It is assumed that the average kinetic energy of the transverse motion of a photoion equals &in. Then, in a way similar to the above procedure, it is possible to estimate the spatial resolution due to this effect. To achieve the spatial resolution Ax of 3-5 A at F,,, = 0.2 eV/A it is necessary to realize a very precise MPI of the chroand r = lo3 mophore on the cooled tip with excess of transversal kinetic energy of only ~t~~ = 1 0 - ~eV. The simple estimates show that with photoion microscopy it will be possible to reach resolutions of several angstroms, which may be sufficient for the visualization of large molecules.

A,

111. FEMTOSECOND MPI OF CHROMOPHORES To develop a photoion microscope, it is necessary to learn how to detach with a laser pulse a chromophore from a large molecule adsorbed on the surface. Apparently femtosecond laser excitation is in principle capable of depositing through multiple-photon processes a substantial amount of energy in the chromophore, which gives reason to expect that it will be ionized and detached from the molecule. But to observe the formation of ions directly on the projector tip surface, with desorption on the neutral molecule being excluded, use should be made of subpicosecond laser pulses, for in that case one can guarantee that no neutral molecule will have enough time to move more than 0.3 nm away from the surface during the laser pulse. Our experiments with tryptophan (Trp) and a tryptophan-containing tripeptide revealed some specific features that are especially evident from comparison between the mass spectra resulting from irradiation with femtosecond and nanosecond laser pulses in the UV (308 nm) and visible (620 nm) regions of the spectrum. The degree of fragmentation depends on both

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the laser pulse duration and wavelength. The UV pulse effects the multiplephoton ionization of Trp through the resonant SIstate, while the visible pulse effects the multiple-photon ionization and two-photon excitation of the SI state [8]. Of particular interest is the pulse duration dependence of the threshold energy density Ethr for the appearance of mass spectra (Fig. 2) in the case of the multiple-photon UV ionization of the Trp GZy Ala molecule. It is clearly seen that the threshold energy .density is reduced by a factor of 10 in the case of irradiation with femtosecond pulses. The same effect is observed with a pure tryptophan powder. The threshold energy density for the appearance of J/cm2 (3 x lo9 W/crn2), is an the 300-fs pulse mass spectrum, Efg, = order of magnitude lower than that of the 15-11s mass spectrum, Er{r = J/cm2 (6 x lo5 W/cm2). ' yield for The intensity dependence of molecular chromophore ion R the femtosecond MPI of the tryptophan is shown in Fig. 3. Two-photon resonance-enhanced MPI requires an absorption of the five visible (620-nm) photons.

b

I Eth

I

I I

I I

5 10-2

h

-

'

266

266 266

Y

5

desorption

in vacuum

10-3 -

Photoionization I on surface I I I

I

I

I

Pulse duration .+

Figure 2. Threshold energy fluence for the appearance of the tripeptide mass spectrum with distinct chromophore R+ photoion as a function of the UV laser pulse duration (from Ref. 8).

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w (cm*)

lo-’

105 -

-3

104 -

v 0)

a

2

E

nm

103 -

fs

102 -

’

10’ 10-4

I

I

10-3

I

I

10-2

I

i

J km2)

Figure 3. Intensity dependence of chromophore molecular ion R+ yield for two-photon resonance-enhanced MPI of tryptophan.

The experiments demonstrate that femtosecond laser pulses offer new opportunities for multiple-photon ionization of bioorganic molecules on surface. The fast femtosecond excitation makes it possible to produce molecular and fragmentation ions directly on the surface being irradiated. The two-photon excitation with an intense femtosecond pulse allows the selectivity of ionization of the chromophore (tryptophan in our case) in large molecules

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to be improved. This is of interest in the future development of a laser photoion microscope with an ultrahigh resolution for mapping chain bioorganic molecules. Femtosecond photoionization mass spectrometry might be useful in the study of the three-dimensional structure of large biomolecules. When a selectively excitable and ionizable chromophore is located on the outer (surface) part of large molecule, one can be detached in the picosecond time scale. However, when the excitable chromophore is located in the inner part of the big molecule, its detachment will require a much longer time, which is needed for spatial rearrangement of the molecule. So, even the simple mass spectrometry of bioorganic molecules with femtosecond laser ionization can reveal some details of their spatial structure. However, the main important potentialities of femtosecond laser photoionization in future lie in combination with photoion microscopy. When this was first realized, we decided to demonstrate the photoelectron version of resonance MPI spectromicroscopy, for in that case the problem of the reproducibility of the results is simplified because there is no thermal desorption of the molecules and their photofragmentation is absent.

IV. LASER RESONANCE PHOTOELECTRON SPECTROMICROSCOPY Figure 4 presents a schematic diagram of a laser photoelectron projection microscope. The needles were fabricated from LiF-F2 crystal fragments by etching. They were a few millimeters long, and their pointed end had an almost conical shape with the radius of curvature of the tip, r, as small as 1 pm and less. The experiments were conducted by Konopsky and Sekatsky [9]. The needle was fastened to an electrode with a voltage of U ranging between 0 and 2.5 kV applied to it. The vacuum in the microscope chamTom. The needle tip was irradiated with the 488ber amounted to 3 x and/or 514-nm lines of a continuous-wave (CW) Ar+ laser. The photoelectrons emitted from the needle tip were directed by an electric field to the input of a microchannel plate (MCP) combined with a fluorescent screen at a distance R = 10 cm from the tip. The optical image formed on the screen was registered with a charge-coupled device (CCD) camera combined with an Argus-50 model computer image processor (Hamamatsu Photonics K.K., Japan). The photoelectron collection solid angle was determined by the size of the working area of the MCP (diameter 32 nm) and amounted to some 20". Therefore, it was only a small part of the needle tip that was displayed on the screen. The cleaning of the needle surface from adsorbed molecules was effected on account of photostimulated ion desorption by applying a stepped-up (up to 20 kV)voltage of opposite polarity to the needle. Immedi-

Ar' ion laser

MCP and phosphor screen

Figure 4. Schematic diagram of the laser photoelectron projection microscopy experiment.

LE9

Pumping

Liquid nitrogen cooling system

Vacuum chamber

Sample holding electrode

d

ir

Gas inlet system

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ately following this cleaning the polarity of the voltage applied to the needle was reversed and stable, repeatable photoelectron images of the needle tip were registered. A typical photoelectron image (PEI) of a LiF-F2 needle tip is presented in Fig. 5. The laser intensity was varied in the range 4 x lo3-8 x 103W/cm2, determined by the dynamic range of the registration system used. Since the photoemitted electrons possess some nonzero transverse motion kinetic energy E&, each photoelectron must be imaged as a spot with a diameter defined by the expression

= I eV and U = 2.5 kV, we get a = 30 nm, which agrees with Putting the size of individual bright spots in Fig. 5.

Figure 5. Photoelectron image of the surface of a LiF-FZ needle tip with a radius of a curvature r = 1 pm at a lo5 magnification. Individual F2 centers are visible as spots with white ring on this black and white photograph.

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The photoemission of electrons from LiF crystals under the effect of visible light is due to the photoionization of defects in the surface region of the crystals, and the main defect in the specimens under study are the F2 centers (adjacent anionic vacancies in the LiF crystal lattice that have captured two electrons). It is therefore only natural to associate the bright spots observed in photoelectron images with the visualization of individual F2 centers. With the electron escape length l,, in the LiF crystal equal to some 3-10 nrn and the Fz center concentration n = 10’6cm-3,the average distance between the photoelectron images of these centers must be 1 = (hescn)-’/*= 100-170 nm. This agrees with the experimentally observed distances between the bright spots in Fig. 5. The magnification attended in the experiment with the photoelectron microscope was A4 = lo5, and the spatial resolution was around 30 nm, which proved sufficient for the visualization of individual color centers in a LiF crystal with the concentration of such centers less than l O ” ~ m - ~The . results obtained in Ref. 9 may be considered the first successful implementation of laser resonance photoelectron microscopy possessing both subwavelength spatial resolution and chemical selectivity (spectral resolution). It will be necessary to increase the spatial resolution of the technique by approximately an order of magnitude and substantially improve its spectral resolution by effecting resonance multistep photoionization by means of tunable ultrashort laser pulses. In this experiment, no use is made of femtosecond laser pulses, although the photoionization process itself is a femtosecond event. The next natural steps are experiments using femtosecond pulses and the traditional (not projection) electron optics. This materially extends the class of possible experiments.

V. TOWARD FEMTOSECOND LASER PHOTOION MICROSCOPY Especially interesting are the prospects for the development of laser photoion spectromicroscopy that potentially possesses a spatial resolution of a few angstroms, that is, potentially allows visualizing absorbing centers in large molecules, biomolecules in particular. To develop a photoion microscope, it is necessary to learn how to detach a chromophore from a large molecule adsorbed on the projector tip surface with a laser pulse. The formation of photoions upon irradiation of a surface with a powerful nanosecond UV laser pulse resulted from the MPI of neutral molecules undergoing laser-induced thermal desorption [lo]. To avoid this effect, which leads to a loss of spatial resolution, the laser pulse duration must satisfy the following obvious condition:

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where Tdel is the time it takes for a photoion with a size of amolto move at an average velocity of uo = lO%m/s for a distance of the order of umo1from the tip surface and ~ t r a n s f is the time of excitation transfer from the excited chromophore to the other parts of the molecule. It is obvious that to realize the MPI of a molecular chromophore directly on the tip surface necessitates the use of femtosecond laser pulses. The above-mentioned experiments on the femtosecond MPI of molecules on a surface have shown that under femtosecond pulse irradiation conditions the formation threshold of molecular ions is reduced by a factor of around 10 compared to that in the case of irradiation with femtosecond pulses, This is evidence that the MPI of the chromophores under the effect of femtosecond pulses occurs directly on the tip surface and prior to their desorption (Fig. 2). To avoid too fast an energy transfer and enhance the selectivity of the laser excitation of the chosen sites in large biomolecules, it seems very promising to use their chemical labeling. This is especially important in the mapping of the sequences of DNA nucleotide bases having very close spectral properties. The first experiments on the MPI of dye-labeled nucleic acid bases were quite a success [ll]. The next natural steps are the study of the mechanisms of photodetachment of dye chromophore ions from molecules adsorbed on surfaces and the search for conditions necessary to effect preferably the MPI of the labeled chromophores while causing no laser-induced desorption of intact molecules. This is necessary in order to make it possible to irradiate a large molecule on a surface repeatedly and thus accumulate information on the location of the chromophores in its various parts.

VI. CONCLUSION In conclusion I would like to emphasize that the suggested approach (femtosecond laser spectromicroscopy)is not a simple modification of the Muller microscope [6], for the electric field here is not the decisive factor but serves solely to form the image. Table I lists the comparative characteristics of the Muller projection field-ion microscope (FIM)and proposed laser resonance photoelectron (photoion) spectromicroscope (LRFSM). The above-described experiments are actually the first successful steps on the new trend of laser spectroscopy for which the very high temporal resolution (At) allows to achieve the very high spatial resolution (Ax). This new possibility based on the simple relation. The Ax = uAt, where u is

885

FEMTOSPECTROCHEMISTRY TABLE I Comparison of Characteristics of Field-Ion Microscopy (FIM) and Laser Resonance Photoion (Electron) Spectromicroscopy (LRFSM) Characteristic Ionization Imaging Sample Spectral resolution

FIM Nonselective by strong electric field By projection Conductive, refractory metals, sharp needles No

LRFSM Photoselective by laser pulses By projection or electron (ion) optics Any Yes

the velocity fragments of the dissociation or ionization molecule. For the u = 104-105 cm/s and 100-fs laser pulse the potential spatial resolution Ax = 10-9-10-8 cm (0.1-1 A), that is, comparable with the atomic structure of the molecule. In this way we can anticipate new possibilities in the study of molecular dynamics and molecular structure, particularly for molecules on surface and bioorganic molecules. Femtosecond photoion (photoelectron) microscopy combines the merits of two types of microscopy: the high spatial resolution of ion projective microscopy and the high spectral (energy) resolution of optical spectroscopy. From this point of view, photoion microscopy is an interesting example of wave-corpuscular microscopy. Indeed, there are two well-known types of microscopy: wave (optical) and corpuscular (electron, ion). In optical microscopy, good spectral resolution can be obtained since light reflection (transmission) is responsive to small changes in the photon energy (AE = 0.001-0.01 eV). This is quite understandable since the photon energy is usually commensurate with energy of the molecular bonds. However, the spatial resolution of optical microscopy is low (Ax = 103-104 A). It depends on the wavelength A of the light and the aperture of the beam 2a. In corpuscular microscopy, high spatial resolution can be realized. For example, with an electron energy of 1 MeV, a de Broglie wavelength X = A, and the angular aperture 2a = rad, $e spatial realization of an electron microscope may be as high as Ax = 1 A. However, the spectral resolution of an electron microscope is not high since a 1-MeV electron is almost insensitive to any type of chemical bond. The fact that electron interaction with a substance depends on the electron density of the atom makes it possible to form an image with a low contrast. Therefore, to attain angstrom resolution with a high contrast, it is necessary that heavy atoms be incorporated into the molecules under study. Thus, laser femtosecond pulses allow us to combine wave (optical) and corpuscular (ion electron) microscopy. This kind of microscopy is based on

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the detachment of molecular ions (or electrons) from certain parts of macromolecules using femtosecond multiphoton ionization. With it, it is possible, in principle, to realize high spatial (several angstroms), spectral (0.001-0.01 eV), and time (pico- and femtosecond) resolution, all at the same time.

Acknowledgment

The author expresses his gratitude to his colleagues at the Institute of Spectroscopy for joint work at various stages of this project and Kamamatsu Photonics K.K., Japan for kind lending of part of equipment.

References 1. V. S. Letokhov, Ed., Laser Analytical Spectrochemistry, Adam Hilger, Bristol and Boston, 1986. 2. J. Manz and L. Woste, Eds., Femtosecond Chemistry, Vols. 1, 2, Verlag Chemie, Weinheim, 1995. 3. V. S. Letokhov, Laser Nonlinear Chemistry wirh Multiple Photon Excitation, SpringerVerlag, Berlin, 1983. 4. S . A. Rice, Report on the XXth Solvay Congress on Chemistry, Brussels, Nov. 28-Dec. 2, 1995. 5. V. S. Letokhov, Laser Photoioniwtion Spectroscopy, Academic, Orlando, 1987, Chapter 12. 6. E. W. Muller and T. T. Song, Prog. Surf. Sci. 4 (1973). 7. v. S. Letokhov. Comm. Atom. Molec. Phys. 11, 1 (1981). 8. S. V. Chekalin, V. V. Golovlev, A. A. Kozlov, Yu. A. Matveyets, A. P. Yartsev, and V. S. Letokhov, J. Phys. Chem. 32, 6855 (1988); in Ultrashort Phenomena, Vol. 6, T. Yajima,

H.Yoshihara, C. B. Hams, and S. Shionoya, Eds., Springer-Verlag, Berlin, Heidelberg, 1988, p. 414.

9. V. Konopsky, S. Sekatsky, and V. S. Letokhov, in Proceedings of XI1 International Laser Spectroscopy Conference, June 95, Capri, Italy, World Sci., Singapore, p. 433 and Optics Comm. 132,251 (1996). 10. S. E. Egorov, V. S. Letokhov, and A. N. Shibanov, Sov. J. Quant. Elecr,: 14,940 (1984); Chem. Phys. 85,349 (1984). 11. S. V. Chekalin, V. V. Golovlev, Yu. A. Matveyets, A. P. Yartsev, V. S. Letokhov, K.-0. Grenlich, and J. Wolfrum. IEEE Quant. Elecr,: 26, 2158 (1990).

DISCUSSION ON THE COMMUNICATION BY V. S. LETOKHOV Chairman: R. Jost

G. R. Fleming: The suggestion of Letokhov has been realized by the group of CheTla at Lawrence Berkeley Laboratory, who have achieved 1-ps (10-A) resolution of electron flow in a transmission line with a sensitivity of one electron. V. S. Letokhov: I heard recently about this work but had no chance

FEMMSPECTROCHEMISTRY

887

to read it. There were a number of attempts to combine the Scanning Tunneling Microscopy (STM) with laser irradiation. I think it is really a very interesting experiment.

ACADEMIC SESSION AT THE CASTLE OF LAEKJSN: PRESENTATION TO THE KING ALBERT I1

Modem Photochemistry S. A. RICE University of Chicago Chicago, Illinois

SUMMARY OF REMARKS MADE BY S. A. RICE This Twentieth Solvay Congress is concerned with the characteristic features of chemical reactions induced by absorption of light and the control of those reactions at the level of the molecular dynamics. In one sense the subject matter of this Congress can be thought of as a natural extension of what was discussed at the Thirteenth Solvay Congress, in 1965, under the overall title “Reactivity of the Photoexcited Organic Molecule.” However, because of advances since 1965, this Congress focuses attention on issues very different from those examined earlier. In particular, this Congress addresses questions related to several of the most fundamental issues associated with the dynamical evolution of photoexcited molecules. The placement of these questions and issues in the overall framework of the quantum theory of matter suggests that this Solvay Congress can also be thought of as a continuation of previous Solvay Congresses in Physics that focused attention on the character and description of elementary physical processes. Photochemical reactions are widespread in nature. We are able to see by virtue of a reaction induced in molecules in the retina of the human eye when light is absorbed. We are able to eat by virtue of the absorption of light by Advances in Chemical Physics, Volume 101: Chemical Reactions and Their Contml on the Femtosecond Time Scale, XXth Solvay Conference on Chemistry, Edited by Pierre Gaspard, Irene Burghardt, I. Prigogine, and Stuart A. Rice. ISBN 0-471-18048-3 0 1997 John Wiley & Sons, Inc.

889

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ACADEMIC SESSION AT THE CASTLE OF LAEKEN

plants, which starts the chain of reactions that generate the conversion of carbon dioxide and water into carbohydrates. We are affected by the amount of short-wavelength ultraviolet light that reaches the surface of earth, which is influenced by the concentration of photochemically generated ozone in the upper atmosphere. Many other examples of the influence of photochemical processes on everyday life can be cited. The study of photochemistry involves a spectrum of activities, ranging from determination of the sequence of chemical reactions that follows the initial formation of an excited molecule to the experimental and theoretical characterization of the elementary events that occur when light is coupled to a molecule. The modem focus, and the subject of this Solvay Congress, is on the study of the elementary processes in photochemistry. The study of these elementary processes has been made possible by a revolution in technology that, in a span of about 40 years, has provided light sources that can have exceptional spectral purity, can have very large intensity, and can be pulsed in extremely short bursts. In those 40 years the typical duration of a light pulse used to study the time evolution of a photochemically induced process has s) to femtoseconds s). The ability shortened from milliseconds to make measurements on the femtosecond time scale means that the detailed motions of the atoms in a molecule can be studied. The most recent stage in the technological revolution is the development of techniques that permit the generation of light pulses with specific temporal shape and spectral content. The availability of such light pulses permits the development of methods to control the outcome of a reaction by altering the molecular dynamics during the reaction. In the past three decades there have been numerous important accomplishments in the study of elementary photochemical processes. A few of these are: The elucidation of the mechanism of so-called radiationless transitions, that is, the decrease and eventual loss of expected emission of light from an isolated photoexcited polyatomic molecule when the energy of the molecule is increased. The seminal work of Robinson, Jortner, Berry, Freed, and others provided the first quantitative understanding of the influence on intramolecular energy transfer processes of the trade-offs between the number of molecular states per unit energy and the strength of the interactions between those states and to enhanced understanding of the mechanism of intramolecular energy transfer in a chemical reaction. This work also led to recognition of the possibility of coherent energy transfer processes even when the number of molecular states per unit energy is very large. The direct determination of the rates of elementary photofragmentation reactions. These studies have provided information concerning the

PRESENTATION TO THE KING ALBERT I1

891

molecular features that determine, for example, how energy is distributed among the products of a reaction, and they provide hints concerning when it is possible to achieve mode-selective dynamics in large molecules. The determination of the properties of transition states of reacting molecules. Much of the theory of the rate of chemical reactions is based on the notion that the transformation from reactant to products passes through a particular geometry of the excited reactant molecule, called the transition state. Until recently there were no experimental tools with which the transition state could be studied, but it is now possible to determine some of the properties of transition states and thereby test theoretical models of the photochemical reaction. The development of a new form of spectroscopy based on the exploitation of the time evolution of the coherence associated with the rotational motion of an excited molecule. Conventional spectroscopies depend on the measurement of differences between the energy levels of a molecule, which become more and more difficult to measure and to interpret as the size of the molecule increases. In contrast, the intervals between recurrences in the coherent rotational motions of large molecules are directly related to the moments of inertia of the molecules and can be used to determine their structures. 0 The elucidation of the early steps in the photosynthetic process, in the solvation of excited solute molecules, and other important reactions. Two lines of inquiry will be important in future work in photochemistry. First, both the traditional and the new ,methods for studying photochemical processes will continue to be used to obtain information about the subtle ways in which the character of the excited state and the molecular dynamics defines the course of a reaction. Second, there will be extension and elaboration of recent work that has provided a first stage in the development of methods to control, at the level of the molecular dynamics, the ratio of products formed in a branching chemical reaction. These control methods are based on exploitation of quantum interference effects. One scheme achieves control over the ratio of products by manipulating the phase difference between two excitation pathways between the same initial and final states. Another scheme achieves control over the ratio of products by manipulating the time interval between two pulses that connect various states of the molecule. These schemes are special cases of a general methodology that determines the pulse duration and spectral content that maximizes the yield of a desired product. Experimental verifications of the first two schemes mentioned have been reported. Consequently, it is appropriate to state that control of quantum many-body dynamics is both in principle possible and is

892

ACADEMIC SESSION AT THE CASTLE OF LAEKEN

experimentallyfeasible. The immediate perspective for the use of the control methodology is as a tool for learning more about molecular dynamics and for testing concepts advanced to describe aspects of molecular dynamics. The cost of photons is sufficiently great that it seems unlikely that commercial chemical syntheses can be based on the enhanced selectivity of product yield generated by control fields. However, there are likely other practical applications, for example, in telecommunications, where both the interference method and the pulse separation method have been shown to generate fast semiconductor optical switches. Since the underlying principles of control theory as applied to quantum physics are very broadly applicable, it is likely that many other applications will be developed.

Femtochemistry A. H. ZEWAE

California Institute of Technology Pasadena, California

SUMMARY OF REMARKS MADE BY A. H. ZEWAIL Mankind, ever since the invention of photography, has been searching for better time resolution to directly observe the motion of objects in our universe. In 1872, the invention of fast photography gave the ability of recording animals and humans in motion with time resolution of milliseconds, onethousandth of a second. A century later new developments with lasers made it possible to see atoms and molecules in motion in femtoseconds, a millionth of a billionth of a second, reaching the limit of time for all microscopic molecular, nuclear motions. The principles of this new field of femtochemistry and its scope of applications were reviewed in this lecture with examples covering different elementary motions, from the simple table salt to the complex systems of biological vision and plant photosynthesis. (See Refs. 1-3.) 1. A. H. &wail, Sci. Am. 262 (12). 76 (1990). 2. G. Taubes, “Welcome to Femtoland,” Discover, 83 (1 994). 3. A. H. Zewail, Femtochemistry: Ultrafast Dynamics of the Chemical Bond, Vols. I and 11, World Scientific, Singapore, 1994.

CONCLUDING REMARKS S . A. RICE

This XXth Solvay Congress has been concerned with questions related to several of the most fundamental issues associated with the dynamical evolution of photoexcited molecules and with the control of that dynamical evolution at the molecular level. We have discussed chemical reaction dynamics on the femtosecond time scale, the active control of branching and product formation in chemical reactions, the use of nonlinear spectroscopy to study chemical and physical relaxation processes relevant to photochemistry, the new information that can be and is obtained from very high resolution spectroscopy of ordinary molecules and van der Waals molecules, the character of photoinduced relaxation phenomena in clusters, the information available from new spectroscopies such as ZEKE (zero-kinetic-energy electrons), the dynamical properties of Rydberg states of molecules, and much more. The subjects we have focused attention on differ from those discussed at the XIIIth Solvay Congress, in 1965, under the overall title “Reactivity of the Photoexcited Organic Molecule.” The technology of that time, which included flash photolysis, permitted chemical reaction dynamics to be followed on the microsecond to submicrosecond time scale but did not permit the study of the faster processes that dominate the first stages of photochemistry. Consequently, the major interest in photochemistry concerned the sequence of chemical reactions that followed photoexcitation. The striking difference in the character of the topics discussed at the XIIIth Solvay Congress and this Solvay Congress is one consequence of the revolution in technology associated with the development of lasers. Indeed, in a span of about 30 years, laser technology has provided light sources that can have exceptional spectral purity, can be tuned over a wide-wavelength region, can have very large intensity, and can be pulsed in extremely short bursts. All of these characteristics of laser light sources have been put to use in photochemistry. Since 1965 the improvement in spectral resolution has Advances in Chemical Physics, Volume 101: Chemical Reactions and Their Control on the Femtosecond lime Scale, XXth Solvay Conference on Chemistry. Edited by Pierre Gaspard. Irene Burghardt, I. Prigogine, and Stuart A. Rice. ISBN 0-471-18048-3 0 1997 John Wiley & Sons, Inc.

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S. A. RICE

permitted the determination of molecular structures and force fields with very high precision, the combination of spectral purity and high intensity has permitted the introduction of new spectroscopic techniques that reveal features of the molecular structure and dynamics previously unavailable from experiment, and the reduction of the duration of a light pulse to femtoseconds has permitted measurements of the detailed motions of the atoms in a molecule. The most recent stage in the technological revolution is the development of techniques that permit the generation of light pulses with specified temporal shape and spectral content. The availability of such light pulses permits the development of methods to control the outcome of a reaction by altering the molecular dynamics during the reaction. And, although there were also other motivations, each stage of the improvement in experimental capability stimulated improvement in the theory of elementary photochemical processes. It is worthwhile recalling some of the numerous important accomplishments in the study of elementary photochemical processes that occurred in the past 30 years. My biased selection of these accomplishments includes: 1. Elucidation of Mechanism of Molecule-Preserving Radiationless Transitions. The theoretical analysis of radiationless processes provided the first

quantitative understanding of the influence on intramolecular energy transfer processes of the trade-offs between the number of molecular states per unit energy and the strength of the interactions between thoses states and thus enhanced understanding of the mechanism of intramolecular energy transfer in a chemical reaction. The theory of molecule-preserving radiationless processes also offered an opening for the development of the theory of photofragmentation and photoisomerization reactions. Experimental studies of radiationless processes led to recognition of the preservation of level-specific dynamical processes even under circumstances where the density of states of the molecules is large and to the recognition of the possibility of coherent energy transfer processes even when the density of states is very large. 2 . Direct Determination of Rates of Elementary Photofragmentation Reactions. These studies have provided information concerning the molecular features that determine, for example, how energy is distributed among the products of a reaction, and they provide hints concerning when it is possible to achieve mode-selective dynamics in large molecules. 3. Determination of Properties of Transition States of Reacting Molecules. Much of the theory of the rate of chemical reactions is based on the notion that the transformation from reactant to products passes through a particular geometry of the excited reactant molecule, called the transition state. Until recently there were no experimental tools with which the transition state could be studied, but it is now possible to determine some of the prop-

CONCLUDING REMARKS

895

erties of transition states and thereby test theoretical models of the photochemical reaction: The development of new forms of spectroscopy based on nonlinear optical processes and on the exploitation of the time evolution of the coherence associated with the rotational motion of an excited molecule. The elucidation of the early steps in the photosynthetic process, in the solvation of excited solute molecules, and other important reactions.

I believe that the content of our discussions clearly implies that a new stage has been reached in our studies of photochemistry and that two lines of inquiry will be important in future work. First, both the traditional and the new methods for studying photochemical processes will continue to be used to obtain information about the subtle ways in which the character of the excited state and the molecular dynamics defines the course of a reaction. Second, there will be extension and elaboration of recent work that has provided a first stage in the development of methods to control, at the level of the molecular dynamics, the ratio of products formed in a branching chemical reaction. These control methods are based on exploitation of quantum interference effects. Two of the simplest schemes, one based on the manipulation of the phase difference between two excitation pathways between the same initial and final states and the other based on the manipulation of the time interval between two pulses that connect various states of the molecule, have been experimentally verified. These schemes are special cases of a general methodology that determines the pulse shape, duration, and spectral content that maximizes the yield of a desired product. Consequently, it is appropriate to state that control of quantum many-body dynamics is both in principle possible and experimentally feasible. The immediate perspective for the use of the control methodology is a toll for learning more about molecular dynamics and for testing concepts advanced to describe aspects of molecular dynamics. It has been pointed out at this Congress that because producing photons of the type required for controlling a chemical reaction is costly, commercial chemical syntheses based on the enhanced selectivity of product yield generated by control fields are not economic. It has also been pointed out that production of a small amount (says tens of kilograms) of a very high value product such as a pharmaceutical could be based on a manufacturing scheme that uses biotechnology techniques to greatly amplify the small amount of material prepared by the use of controlled laser fields. Perhaps, at the next Solvay Congress on Photochemistry we will live in a world where this method of manufacture of new chemicals has become commonplace. For the present, however, there is already evidence of the use of control methods for other practical

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S . A. RICE

applications, for example, in telecommunications, where both the interference method and the pulse separation method have been shown to generate fast semiconductor optical switches. Since the underlying principles of control theory as applied to quantum dynamics are very broadly applicable, it is likely that many other applications will be developed.

V. S. LETOKHOV In my concluding remarks let me first express my deep satisfaction to the Solvay Conference Chemistry Committee for selecting the topic of laser photochemistry for active discussion on this XXth Solvay Chemistry Conference. This certainly represents the leading edge of modem physical chemistry or chemical physics, with many future surprising results in fundamental science and in chemical technology. Particularly promising areas of laser femtochemistry are, first, biological molecules such as DNA (as mentioned briefly by A. H. Zewail even in the presence of the King Albert 11) and, second, coherent control of chemical reactions as considered by S. A. Rice. Let me discuss the practical prospects of the second problem. First, to realize the coherent control of a chemical reaction with high yield, we should photoexcite the molecule to the intermediate quantum state with high probability. For a typical cross section of excitation u,,, = 10-17-10-18cm2we will need a femtosecond laser pulse with energy fluence = hu/uexc= 0.03-0.3 J/cm2. For the coherent interaction the duration of the laser pulse 7p must be shorter than the phase relaxation time T2 or intramolecular vibrational relaxation time T ~ V R(Fig. 1). This means that the intensity of the laser pulse Z = will be in the range 10”-10’2 W/cm2 at least. This intensity is acceptable for molecular beam experiments but can be less suitable for condensed-media experiments because of other nonlinear (damage) effects: self-focusing, optical breakdown, and so on (Fig. 1). We should take into account these effects when we are discussing, for example, potential applications of femtosecond coherent control for photochemical synthesis of pharmaceutical molecules in solutions. In this respect we have an “experimental window” for coherent control of photochemical reactions. Perhaps to make this experimental window larger, we should consider the possibility of coherent control of molecules on the interface of solution/gas. From the laser technology point of view the prospects are very optimistic. The costs of femtosecond visible light are becoming quite acceptable: -0.001-0.01 $/J of femtosecond photons

-100- 1OOO $/mol of femtosecond photons

CONCLUDING REMARKS

1 0-l2

10-15

I 0-9 ns

PS

fs Coherent range

--1

4

897

S

Noncoherent range

T2, W R

Figure 1. Diagram of the intensity I (W/cm2) vs. duration of laser pulse 7 p ( s ) with various regimes of interaction of the laser pulse with a condensed medium being indicated very qualitatively. At high-intensity and high-energy fluence @ = zpl optical damage of the medium occurs. Coherent interaction takes place for subpicosecond pulses with rp 5 T2. r i m . For low-energy fluence (@ < 0.001 J/cm2) the efficiency of laser excitation of molecules is very low (linear interaction range). As a result the experimental window for coherent control occupies the restricted area of this approximate diagram with flexible border lines.

In the case of high yield of the chemical reactions we can expect the “photon part” of the cost of products to be

- 1 $/g of products which is quite acceptable for pharmaceutical materials.

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S.A. RICE

If the “Nature” will be agreeable with us, I hope that the participants of the Twenty-Fifth or Thirtieth Solvay Conference on Chemistry will read the future high-tech magazine with the title “Industrial Femtochemistry.”

Advances In Chemical Physics, Volume101 Edited by I. Prigogine, Stuart A. Rice Copyright © 1997 by John Wiley & Sons, Inc.

Numbers in parentheses are reference numbers and indicate that the author’s work is referred to although his name is not mentioned in the text. Numbers in italic show the pages on which the complete references are listed. Abbouti Temsamani, M.,465(7), 488(7), 490, 521(112, 114). 529-530(112), 53 I ( 112, 123), 532(123), 534-536(114), 579, 580, 809(1). 810(2), 810 Abramson, E.,493(3), 575 Adams, C. S., 189-190(9), 191 Adams, J. E., 259(64), 262-263(64), 272 A g m , O., 503(38), 518(88), 577-578 Agmon. N., 393-394(19, 23), 402 Aharonov, Y.,725 Aicher, P., 626(22), 645 Aker, P. M., 849(1), 849 Akesson, E.,394(27), 399(27), 402 Akimoto, H., 731(7), 733(7), 741 Akiyama, H., 405(5), 406 Akulin, V. M., 659(1), 659 Alagia, M., 86(2), 87 Alber, G., 565( 155). 570(155). 581 Albeverio, S., 517(79), 578 Albrecht, A. C., 146154). 180, 433(72), 437(72), 441 Alcaraz, C., 669(22), 697 Alexandrov, I. V., 393(12), 402 Alicki, R., 238(38), 271 Aliev, M. R., 496(17), 498(17), 576 Allen, J., 216(16), 271, 328(4), 339(4), 341 Allen, L., 304(12), 312 Alonso, D.. 495(14). 497(14), 500(14, 31). 501(14), 504(14), 510(14). 514(14), 521(14), 524(14). 526-527(14), 529(14), 534(14), 54 l-542( 14). 554(114). 558(114), 57q114). 576,785(64), 785 Alonso Ramirez, D., 512-514(61), 578 Alt, C., 682(38), 697 Alt, C. E.,626(9, 15). 629(9, 15). 645 Al’tshuler, B. L., 518(88), 519(92), 5785 79 Amat, G., 486(17), 490

Ambartsumian, R. V., 327(3), 339(3), 341, 661(2), 662 Ambegeokar, V., 393-394(14), 402 Amirav. A., 419(45), 440 Amrein, A. H.,626(25), 645 Amstmp, B., 48(10), 59(10), 75, 218(19). 250(59), 252(59, 61). 271-272, 317(6), 322. 328-329(20), 332(20), 335(20), 339(20), 342 Anandan, J., 725 Anderson, E. M., 4 I2( 13), 440 Anderson, S . L., 669(21), 697 Andr, J. C., 279(2), 280 Andreev, A. V., 518(88), 578 Andreev, S . V., 661(5), 662 Andresen, P., 327(3), 339(3), 341, 768(61), 785 Antonov, V. S . . 661(5-7). 662 Apanasevich, P.A., 333(33), 343 Apkarian, V. A., l46(21), 179, 373(7), 374 Arcuni, P. W.,705(19-20). 708 Arimondo, E., 302(10), 305(10), 307(10), 312 Arndt, M.,542( 142). 580 Arnett, D. C., 348(20,22), 371 Arnold, C. C . , 668(5), 697 Arnold, V. I., 496(19), 501(19), 509(19), 542-543(19), 546-547(19), 551(19), 576 Asano, T., 395(40). 399(40), 403 Asano. Y.,194 Ashfold, M . N.R.. 668(9), 697, 726(1), 726 Ashijian, P., 373(7), 374 Aspect, A.. 302(10), 305(10), 307(10), 312 Assion, A., 55(19), 60(37), 64(43), 76, 79(5), 79 Astholz, D. C., 835(22), 847 Astrom. K.J., 319(9), 322

899

900

AUTHOR INDEX

Atabek. 0.. 48(12-13), 76, 703-704(10), 708 Athanassenas, K., 626(27), 646 Atkinson, J. B., 87, 89(5) Aubanel, E., 328( I I), 339( 1l), 342 Aubanel, E. E., 841(28), 847 Augst, S., 374(1), 376(1), 377 Aumayr, F., 626(28), 646 Aurell, E., 501(34), 577 Aurich, R., 518(87), 578 Avouris, P., 411(8), 439 Backhaus, P., 87, 89(4) Badar, J. S., 394(33), 403 Baer, M.. 855(8), 867 Baer, T., 668(3), 697 Bagchi, B., 142(5), 145(5), 172(5), 179, 394(25-26, 35). 402403 Baggott, J. E., 327(3), 339(3), 341, 373(6), 374 Bagratashvili, 451( 1). 451 Bahatt, D., 434(83), 437(83), 441,626(18), 628(18), 645-646, 702(3), 707 Bahns, J., 308(16), 312 Bain, A. J., 400(49), 403 Baker, A. D., 609(2), 616(2), 623 Baker, C., 609(2), 616(2), 623 Baklshiev, N. G., 394(39), 403 Balakrishnan, N., 201(18), 202, 332(29) Balian, R., 236(34), 271 Balint-Kurti, G. G., 768(61), 785 Balling, P.,65(48), 77 Balucani, N., 86(2), 87 Balykin, V. I., 185(4), 189(4, 8, 10). 190(10), 191 Banared, L., 399(47), 403 Band, Y. B., 423(57), 425(64), 441,444(1), 444 Bandrauk, A., 302(4), 312 Bandrauk, A. D., 48(11), 65(11), 76, 286(3), 292, 375-376(2), 377 Banin, U., 196(8-9), 198 Baranger, M.,521(109). 546(144), 579, 581 Baranov, L. Y.,434(85), 437(85), 442, 626(3), 629(41), 634(3), 643(3a), 645646 Baranova, B. A., 286(8), 292 Baras, F., 514(64), 578 Barbara, P. F., 394(27, 36), 399(27), 402403

Bardeen, C. J., 60(36), 62(36), 76 Barends, E. J., 573(161), 581 Bartana, A., 196(7-9), 196, 198, 239(40), 272,308(17), 312 Bartmess, J. E., 731(10), 741 Basilevsky, M. V., 394(30), 399(30), 402 Baskin, J. S., 40, 85, 391(1), 400401(1), 401 Bhsman, C., 620(25), 623 Bates, D. R., 820(11). 847 Bauder, A., 416(36), 440 Bauer, C., 748-749(17), 751(17,32-33). 752(32-33). 753-754(32), 756(32), 761(32-33). 763(33), 764-767(32), 768-771(17), 783-784 Baumert, T., 49(14), 52(17-18). 55(19), 60(37), 63(41), 64(43), 65(18, 55, 49). 67(49), 72(54), 76-77, 78(1, 3). 79(5), 79, 90(2), 90, 103(6), 117(14), 131, 135(8), 137, 196(6), 196, 217(17). 229-230(17). 271, 328(10), 339-340(10), 341 Bauschlicher, Jr., C. W.,731(11), 741 Bayfield, J., 584(5), 585 Bear, T., 612(8-9). 614(10), 623 Beck, C., 812(4), 822, 815(1), 815 Beck, M., 346(15), 371 Beddard, G., 4(8), 43 Beece, D., 405(1-2). 405406 Beenakker, C. W. J., 519(92) Beil, A., 377(3), 379 Bekov, G. I., 661-662(8), 662,663(2), 663 Ben-Nun, M., 95-96, 153-154(29), 156(29), 157(31), 179-180, 195, 626(19), 645 Bennemann, K. H., 116(11), 131 Benvenuto, F., 584(6), 585 Berendzen, J., 405(4), 406 Berezhkovskii, A. M., 393(18), 394(21), 402 Berghout, H.L., 327(3), 339(3), 341 Bergmann, K.,328(5, 9), 339(5, 9), 341, 423(56), 424(60), 425(60,62-63), 441 Berkovitz, J.. 610(3), 623 Berkowitz, M., 393(17), 402 Berne, B. J., 855(7), 867 Bernstein, R. B., 4(11), 39(11), 43, 86(1), 87, 698, 799(2), 806 Berry, M.V.,493(13), 503(38), 505(41), 506-507(13), 511(58), 5 17(81-82), 518(84), 519(91), 576579

AUTHOR INDEX

Berry, R. S., 114-115(9), 131, 634(47), 646, 657( 1). 657 Berry, S., 412(11), 439 Bersohn, R., 203(9), 204, 434(78), 441, 800(6), 804(15), 806 Beswick, J. A., 637(55), 646, 76404). 785 Bisht, P. B., 394(29), 399(29), 402 Bisseling, R. H., 200(3), 201(10), 201-202, 458(3), 458, 761(51), 784, 812(3), 812 Bittman, 1. S., 760(48), 784 Bixon, M., 411(9), 434(88), 437(88), 439, 442, 537(128), 580, 629(39), 642(62), 646,668(13), 681-682(13), 691-692(13), 697 Black, G., 791(8), 796 Blake, N. P.,201(13), 202 Blanc, J., 102-103(1), 131 Blanchet, V., 57(27), 76 Blank, D. A., 732(12), 737(12), 741 Bleher, P. M.,5 16(75), 5 18(75), 578 Blodgett-Ford, S. J., 510(51), 577 Bloembergen, N., 40 Blomberg, C., 393(9), 401 Bludsky, 0..416(34), 440 Blumel, R., 511(57), 528(57), 541(141), 577,580 Boers, B., 65(47), 77 Boesl, U.,620(25), 623 Bogomolny. E., 503(38), 577 Bohigas. 0.. 516(76), 517(77), 518(76), 578 Bohm, A., 510(52), 577 Bohmer, W., 712(6), 715 Boller, K.-J., 302(7), 312 Bolte, J., 517(78), 518(87), 578 BonaZiL-Koutecki, V., 79(9), 80, 103(4), 114-l15(10), 117(10, 13, 16-17), 118(13), 122(13). 129-130(25), 131-132, 133(4, 6-7). 134(4), 135(4, 6). 135, 136(4), 137, 200(6), 201, 203(4-5). 203 Bordas, C., 434(81), 441 Bordas, M. C., 538(129), 580 Borkovec, M.,392(3), 401 Born, M., 189(10), 191, 630(43), 634(43), 646 Borne, T.B., 612(7), 623 Botter, R., 612(9), 623 Bouchene, M. A., 57(27), 76 Bowman, J. M., 760(43-44, 48). 784 Bowman, R. M., 4 1 2 2 , 393(10), 400(10),

901

401,561(151), 566(151). 581,799(2), 806 Bowne, S. F., 405(2), 406 Bownman, R. M., 799(2), 806 Boyer, M.,538(129), 580 Bradforth, S. E., 146114, 18), 152(18), 154(18), 157(18), 158(18, 34), 159(18), 179-1 80 Bramley, M. J., 201(11), 202 Brandgo, J., 752(39), 784 Brandon, J., 794( 18a. 1Bb), 797 Brandon, J. T., 767(58), 785 Brandrauk, A. D., 328(1I), 339(11). 342 Brauman, J. I., 732(14), 741 Braun, A. M., 279(2), 280 Brechignac, C., 103(3), 129(3), 131 Breen, J. J., 41 Breton, J., 146(15), 160(15), 179, 794(13), 797 Breuer, H. P., 328-329(19), 335-336(19), 342 Brevet, P. F., 434(81), 441 Brickmann, J., 519(94), 579 Brinkman, H. C.. 393(9), 401 Brody, T.A., 505(44), 577, 772(67), 785 Broers, B., 59(31), 76 Brown, F. B., 259(71), 272 Brown, S. S., 327(3), 339(3), 341 Broyer, M., 68(50), 72(50, 53). 73(53), 77, 102(1-2), 103(1), 121(19), 122(2). 131, 133(2), 135, 135(2), 203(7), 203-204, 434(81), 441 Brucker, G. A., 86(2). 87 Brumer, P.,48( 1-2). 57(1). 75, 2 15-21 6( 1-3). 219( I, 26). 221-223(27), 224(26-27). 249(53), 270(79), 270-272, 274(1), 274,286(1,7, 14). 287(16), 288( 14). 291-292, 295( 1-2). 296(3-7). 297(2, 5). 300(5-6), 300, 302(2), 312, 315(3), 319(3), 322, 327(2), 339(2), 341, 373(4, 8). 374, 381(2), 382, 388, 419(4748), 440, 505(44), 518(86), 577-578, 862(19). 868 Brundle, C. R.,609(2), 616(2), 623 Brupbacher, Th., 416(36), 440 Buch, V., 868(5), 868 Buchanan, M., 270(80), 273, 286(10), 292 Buchleitner, A., 542(142), 580 Buehler, B., 63(41), 76 Buhler, B., 52(18), 65(18), 76

902

AUTHOR INDEX

Bullough, P. A., 157(32), 180 Burak, I., 749(19), 765(19), 767(19), 783 Burger, H.,453(5), 453 Burghardt, I., 493-494(10), 495(14), 497(14), 500-501(14), 504(14), 510(14), 514(14), 521(14), 524(14), 526-527(14), 529(14), 534(14), 541(14), 542(10, 14), 544-546( 10). 548( 146), 553-556( lo), 554(10, 14, 146). 555(10), 556(10, 14), 558(lO, 14),558(10), 566(146), 570(14, 146). 576, 581, 772(64), 785, 816(3), 816 Burke, P.G., 538-539(133), 580 Bums,M.J., 39 Butler, L. J., 730(3), 741 Bylicki, F.,521(111), 528(111), 579 Cabaiia, A., 485(15), 490 Cabrol, O.,57(27), 76 Cahuzac, Ph., 103(3), 129(3), 131 Calef, D. F., 393(13), 402 Callegari, C., 626(30), 646 Callomon, J. H., 412(18), 440 Campargue, A.,521(111),528(111),579 Campbell, E. E. B., 626(23), 645 Campolieti, G.,862(19), 868 Carlier, F.,103(3), 129(3), 131 Carlson, R. J., 48(10), 57(21), 59(10, 211, 75-76, 217(18), 218(19), 242(18), 253(61), 271-272, 303(11),312 Carmeli, B., 393(9), 401 Carrington, A., 723(I), 723 Carrington, T.,201(11), 202 Carrington, Jr., T., 536(124),580 Carruthers, P., 512(59), 578 Carter, E. A., 142(9), 1439). 163(9), 173(9), 179 Cartier, S. F., 626(32), 646 Casati, G.,518(89), 519(89, 96). 578-579, 583(l-3), 584(4-6), 584-585 Casavecchia, P., 86(2), 87 Castleman, Jr., A. W., 4(3), 43 Castner, Jr., E. W., 173(47),180, 394(38), 403 Cederbaum, L. S., 200(9), 201(9), 201-202, 493(5), 510(53), 518(5),528(5),537(5), 540(5), 575, 577, 772(65), 785, 868(7), 868-869 Cejan, C., 565(154),569(154),581, 855(8), 867

Cespiva, L., 129-130(25), 132 Chachisvilis, M.,146(19), 152(19), 160(19), 179 Chambaud, G.,528(119), 580 Chan, C. K., 221-224(27), 271 Chandler, D., 163(36), 180, 394(33), 403 Chandler, D. C., 394(33), 403 Chandrasekharan, V., 711-712(3) Chang, Y.J., 173(47), 180, 394(38),403 Chapman, S., 656(1), 657 Charalambidis, D.,286(15), 292 Charutz, D. M., 698 Charvat, A., 521(111), 528(111), 579 Chatfield, D. C., 493(9), 538(9), 575 Che, J., 59(35), 76, 273(1), 274(8), 274-275, 346(7), 370 Chebotayev, V. P., 420(50), 423424(50), 441 Chekalin, S. V., 878(8), 884(1 I), 886 Chelkowski, S . , 48(1 I), 65(1 I), 76, 328(1 I), 339(11), 342 Chen, C., 57(22-24). 76, 286(4,9). 292 Chen, G., 91(1), 91, 466(10), 489(10),490 Chen, P. Y.,41-42 Chen, Y., 218(22), 268-269(77), 271-272, 5 18(85),578, 642-643(58), 646 Chen, Y.-T., 4650). 484(5), 485(12), 486-487(5), 488(5, 12), 489(5), 490 Chen, Z.,286(14), 287(16), 288(14), 292, 419(48), 440 Chenevier, M.,521(111), 528(111), 579 Cheng, Zh.,516(75), 518(75), 578 Chergui, M.,4(1),9(1), 25(1), 34(1), 43, 664(1),664, 711(3), 712(3,6-9), 713(1&11), 714(14), 714-715, 717-718 Chernyak, V., 91(2), 91-92, 200(1), 200, 386(2), 387, 5 14(68), 578 Cheshnovsky, 0.. 646 Chesnavich, W. J., 820(1), 828-830(1), 847 Chevaleyre, J., 102-103(1), 131, 434(81), 441 Chewter, L. A., 617-618(17),623 Child, M. S., 208(1), 208, 498(25), 544(25), 545( l43), 548( l43), 550( 143), 554(l43), 560(143), 571(143), 576,581, 596(1), 598, 634(50), 646,656(1),657, 794(17), 797 Chin, S. L., 374(1), 376(1), 377 Chirikov, B.,518-519(89), 578 Chirikov, B. V., 583(1-3), 584-585

AUTHOR INDEX

903

Cho, M.,145(11, 13). 169(38-39). 171(39), Corkum, P. B., 48(11), 65(11), 76, 270(80), 172(11, 13,4445). 173(13), 176(11. 13). 273, 286(3, lo), 292, 374(1), 376(1), 179-180, 394(30), 403 377 Cho, S.-W., 750(22), 784 Coulston, G. W., 328(5), 3390). 341, Choi, H., 816. 816(4) 424(60), 425(60, 63), 441 Chong, S.H., 41, 85, 391(1), 400-401(1), Courtney. S.H., 400(50), 403 401 Coy, S. L.,465(2), 468(11), 476(2), Chudinov, A. N., 286(8), 292 484(11). 489( 11). 490 Chudinov, G. E., 394(30), 399(30), 403 Cozzens, R. F., 279(1), 280 Chupka. W. A,, 434(82. 84), 437(82, 84). Crim, F. F., 79(7), 79-80, 274(11), 275, 441, 629(37), 639137). 646, 668(11), 327(3), 339(3), 341, 373(6), 374, 679(37), 681-682(11), 697,699, 702(2), 747(11-12). 768(60). 783 707 Cukier, R. J., 394(24), 402 Cina, J., 204(1), 204 Culot, F.,541(140), 580 Cina, J. A., 57(21), 59(21), 76, 150(27), Cvitanovii, P., 503(37), 512-513(60), 179, 217(18), 241(18), 271, 303(11), 514(64), 577-578 312 Cyr, D. R., 730(4), 733(16), 739(16), 741 Ciordas-Ciurdariu, C., 360(26), 371 Czege, J., 405(2), 406 Clark, J. W., 247(46-50). 248(45, 47). 268(48), 271-272 Dabrowski, 721 Clary, D. C., 86(2), 87, 668(20), 697, Dabrowski, I., 705( 14-16), 708 820(4). 847 Dahleh, M., 48(6, 9). 75, 248-249(51), Cogdell, R. J., 158(33), I80 272, 274(3), 275, 302(3), 312, 315(1), Cohen, L., 346(13), 371 316(1, 4). 319(1), 322, 328(8), 339(8), Cohen, M. J., 497(20), 576 341, 346(5), 370, 373(2), 373 Cohen, S.J., 465(1), 467(1), 490 Dahleh, M. A., 249-250(54), 272 Cohen-Tannoudji. C., 745(2), 783 Dahleth, M. A., 215(8-9). 227(8), 236(8-9). Cohen-Tannoudji, C. N., 302(9-lo), 2 70 305(9-lo), 307(9-lo), 312 Dai, H.-L., 747(15), 783 Colin, R., 529(122), 580 Dalton, B. J., 286(15), 292 Colin de Verdiire, Y.,505(45), 577 Dam. N., 423(58), 441 Collings, B. A., 296(8), 300, 626(25), 645 Daniel, C., 79(6-7). 79-80, 328( 17). Colson, S. D., 619-620(22-23). 623, 335(17), 339(17), 341(17), 342 668(6), 697 Dantus, M., 41-42. 52(16), 76, 86(3), 87, Coltrin, M. E., 870(1), 870 89(3-4), 90(6), 90, 185(2), 191(2), 191, Combariza, J. E., 79(7), 79-80, 274(5), 320(12), 322, 561(151), 566(151), 581, 275, 328(14, 17). 329(14, 27). 330(14), 799(2), 806 335(14, 17). 336(14), 339(14, 17). Darling, B. T., 431(66), 441 341(14, 17, 27). 342, 373(3), 374, 375(3), Dashevskaya, E. I., 846(39), 848 377 Date, G., 517(80), 578 Cong. P., 41, 44 Davies, G. J., 173(48), 180 Cononica, S., 400(50), 403 Davies, K. T. R., 546(144), 581 Continetti, R. E.. 730(4), 733(16), 737(19), Davis, E. B., 238(39), 271 739(16), 741-742 Davis, H. L., 862-863(17), 867-868 Cook, R. J.. 189(7), 191 Davis, M. J., 557(150), 581, 642-643(61), Cooper, J., 87, 89(7) 646 Cooper, J . W., 793(12a), 797 De Aguiar, M.A. M.,546(144), 581 Cordes, E., 619(24), 623 de Boeij, W. P., 348(21), 371 Corey, G. C., 201(11), 202 de Frutos, M., 103(3), 129(3), 131 Corkum, P., 456(6), 456 De Silvestri, S.,61(39), 76

904

AUTHOR INDEX

de Vivie-Riedle, R., 79(6, 8-9). 79-80, 111(8), 117(13, 15), 118(13), 120-121(15), 122(13, 15). 124(8), 131, 132(1), 133(4, 6), 134(1,4), 135(1, 4, 6). 135, 136(1,4), 137, 196(1-2), 196, 197(1), 200(6), 201, 202(1, 3). 203f4-5). 203 DeBruijn, D. P., 732(15), 741 Dehmer, J. L., 459( I), 459, 706(27-29), 708 Dehmer, P. M., 459(1), 459, 706(27-29), 708 Delacritaz, G., 68(50), 72(50-51, 53), 73(53), 77, 102(2), 117(12), 122(2. 20). 131, 133(3), 135(3), 135, 203(8), 204 Delande, D., 5 10(50), 5 19(98), 542( 142). 557(50), 577, 579-580 Delon, A., 493(6), 518(6), 521(111), 528( 11I), 537(6), 540(6), 575, 579 Delone, N. B., 419(46), 423(46), 440 DeLong, K. W., 59(34), 76 Delos, J. B., 493(11), 510(11, 51). 537(6), 576577 Demiralp, M., 249-250(56), 272, 317(5), 318(8), 320(8), 322 Demtroder, W., 493(5), 518(5), 528(5), 537(5), 540(5), 575 Dennison, D. M., 431(66), 441 Deshmukh, S . , 732(13), 737(13), 741 Desouter-Lecomte, M., 541( 140). 580, 637(56), 646 Dexheimer, S. L., 146(16), 179 Dhar, L., 346(3), 362(3). 370 Dietrich, H.-J., 629(35), 646 Dietrich, P., 286(3), 292, 374(1), 376(1), 377, 456(6), 456 Dietz, B., 511(57), 528(57), 577 Dietz, K., 328-329(19), 335-336( 19), 339(19), 342 Dietz, T., 339(35), 343 Dikshit, S. N., 158(34), 180 Dill, D., 706(22), 708 DiMauro. L. F., 286(11), 292, 731(8), 733-734(8), 741 Diu, B., 745(2), 783 DiVincenzo, D. P., 302(6), 312 Dixon, R. N., 726(1), 726, 754(40), 784 Doanyy, F. E., 400(48). 403 Dobbyn, A. J., 80(1), 80, 746(8), 747(34), 751(34-37), 752(38), 753-754(34), 755(37), 758(34, 36-37), 760(34), 762(8),

767(59), 769(38), 771(8), 772(34), 774(34, 36). 775(36), 776-781(37), 783-785, 786(34), 812(4), 812, 815(1), 815 Dohle, M., 329(26), 331(26), 340-341(26), 342,377(1), 379 Domcke, W., 201(12), 202, 510(53), 577, 868(10), 869 Donaldson, D. J., 341(25), 342, 849(1), 849 Dong, Y.,394(27), 399(27), 402 Donhal, A., 85(1), 85 Doolen, R., 394(30), 399(30), 402 Doom, S. K., 394(27), 399(27), 402 Dopfer, O., 619(22, 24), 620(22), 623, 668(6), 697 Dorfman, J. R., 502(35), 543-544(35), 577 Doron, E., 511(57), 528(57), 577 Doublet, M. L., 573(161), 581 Doucet, Y., 485(15), 490 Douglas, A. E., 412(14), 440 Douhal, A., 42 Doyla, Z. E., 328(13), 332(13), 334(13), 339( 13), 342 Dreschsler, G., 620(25), 623 Dressler, K., 704(12). 708 Drullinger, R. E., 87, 89(7) Du, M., 57(21), 59(21), 76, 142(3), 145(3), 172(4445), 173(3), 179-180, 217(18), 242(18), 271, 303(11), 312, 394(31), 403 Du, N. Y., 706(24), 708 Dubal, H. R., 454-455(1), 455, 587(1), 588 Dubernet, M. L., 820(9), 843(9), 847 Dudko, S. A., 394(21), 402 Dugourd, Ph., 102-103(1), 121(19), 131, 132(2), 135(2), 135, 203(7), 203-204 Dunlop, J. R., 759(42), 761(42,49), 763(42), 784 Dunn, H. K., 346(8), 370 Dunn, T. J., 8W3-4). 806 Dunning, F. B., 434(73), 441 Dupont, E., 270(80), 273, 286(10), 292 Duppen, K.,178(52), 180, 348(19), 371 Dupuis, M., 731(6), 741 Dyson, F. J., 516(75), 518(75), 578 Dzelzkans, L. S., 849(1), 849

Ebata, T., 422(53), 441, 674(32), 697 Eberly, J. H., 304(12), 312 Eckhardt, B., 501(34), 503(37), 504(40),

AUTHOR INDEX

512-513(60), 570(40), 573(160),

577-578.581 Eckmann, J.-P., 5 14(64), 578 Egorov, S. E., 883(10), 886 Ehlich, R., 626(31), 646 Eisenbud, L.,538-539(133), 580 Eisenstein, L.,405(1-2). 405406 Eisenthal, K.B.. 393(lo), 400(10), 401 El-Sayed, M.A., 4(9), 43, 411(8),439 Eliason. M. A., 835(23), 847 Ellert, Ch.,129(24), 132 Elliott, D.S.. 57(22-24). 76, 286(4, 9, 12). 292 Engel, V.,60-61(38), 64(42), 65(49), 67(49),76-77, 81(1), 81, 196(6), 1%. 274(10), 275,327(1), 339(1), 341, 37316). 374,565(153). 566(159), 570(153), 573(159-160). 581, 768(61), 785 Ernst, W. E., 72(52), 77, 691(47),698 Etchepare, J., 173(47), 180 Evans, M.,173(48), 180 Even, U.,416(35). 419(45),434(78. 83, 85), 437(83,85),440-442, 626(1,4,

16, 18-19), 628(4, 18). 629(1), 634(1),

644(1), 645,652,668(14), 681-682(14), 697,702(3),707, 724(1), 724 Ewing, G.E., 418(39). 440 Eyler, E. E., 634(49),646 Eyring, H.,835(23), 847 Ezra, G.S., 497(21), 521(110), 526(110), 576, 579, 590

Fabre, C., 382(10), 385 Fan, H.-Y., 382(13,15), 385 Fang, J.-Y., 514(66), 578 Fano, U., 686(40). 698, 703(7), 708, 793(12a, 12b), 797 Fantucci, P.,103(4). 114-115(10), 117(10, 16-17), 129-1 30(25), 131-1 32 Farantos, S.C.,484(13), 490 Faucher, 0.. 286(15), 292 Fayet, P.,122(20). 131 Fedoriuk, M.V., 861(15), 867 Feezel, L. L., 849(1), 849 Feit. M.D., 200(2), 201 Feldstein, M.J., 348(20), 371 Felker, P. M., 12(13), 39-41, 43, 412(12), 439 Ferreira, L. F. A., 612(9), 623 Feshbach, H.,636(52),642(52). 646

905

Field, J. E.. 302(7). 312 Field, R. W.,412(10), 421(52),439, 441.

465(1-5), 467(1, 3), 476(2), 484(5), 485(12), 486(5), 487(5, 18). 488(3-5, 12). 488(3), 489(5), 490, 493(3). 518(85), 529(122), 536(125), 575, 578, 580, 707(34), 708, 747(13, 15). 783 Fielding, H. H., 670(28),687(4345), 688(43, 45). 689(43), 697698, 707(35-36), 708 Fink, R., 767(59), 785 Fischer, I., 626(10), 629(10),645, 668(7), 682(7), 697, 726(4), 726 Fischer, M., 279(3), 280 Fishman, S., 503(38), 504(40), 570(40), 577 Fitzcharles, M.S., 86(2), 87 Flannery, B. P., 332(30),342 Fleck, Jr., J. A., 200(2), 201 Fleischhauer, M.,302(8), 312 Fleming, G.R., 57(21), 59(21), 76, 94, 96, W3.4, 7). 144(4), 145(3,4, 7, 11). 146(14, 18. 20). 147-148(22. 24), 149(24), 150(22, 24), 152(18, 24,28). 153(29), 154(18, 28-30), 155(28), 156(29), 157(18, 31), 158(18, 34), 159(18), 160(24), 162-163(7), 164(37), 166(37), 168(37), 169(3840), 171(39-40), 172(11,37. 44). 173(34,7, 37,49). 174-175(37). 176(11), 177(22), 179-180, 195, 217(18), 241(18), 271, 282, 303( 1 l), 312, 346(4), 370, 393(1 I), 394(25-26, 31, 34). 400(50), 401403 Floethmann, H., 751(33),752(33, 38). 761(33),763(33), 769(38), 784 Flores, J., 505(44), 577, 772(67), 785 Flugge, S.,332(31). 343 Foltin. M.,626(26). 645 Fonseca, T.,393(16), 394(32), 402403 Ford, J.. 583(3), 585 Forst, W.,515(72), 539(72),578, 750(27), 784 Fotakis, C., 286(15), 292 Fourkas, J. T., 346(3), 362(3), 370 Frauenfelder, H., 405(I-2), 405-406 Fredin, S., 647(1), 648, 707(32), 708 Freed, K.F., 411(7), 439 Freer, A. A., 158(33), 180 Fremacle, F., 813(1), 813 French, J. B., 505(44), 577, 772(67), 785 Frenkel, A., 511(57), 528(57),577

906

AUTHOR INDEX

Frensley, W. R., 513(62), 578 Freund, R. W., 858(9), 867 Fried, L. E., 394(37), 403, 497(21), 576, 590 Friedman, H. L., 142(6), 145(6), 173(6). 179 Friedman, R. S., 493(9), 538(9), 575 Friesner, R. A., 147-148(22), 150(22), 177(22), 179 Frillon, G., 173(47), 180 Frishman, A. M., 393(18), 402 Frosch, R. P., 419(43), 440 Fugol, I. Ya., 711(2), 714 Fujii, A., 674(32), 697 Furukawa, M., 422(53), 441 Furuta, H., 395(40), 399(40), 403 Fyodorov, Y. V., 519(101), 579 Gabriel, W., 328-329(20), 332(20), 335(20), 339(20), 342 Gabrielse, G., 379(9), 381 Gallagher, R. F., 668( 19), 670(27), 697 Gandhi, S. R., 65(45-46), 77 Garcia, M. E., 116(11), 131 Garcia-Moreno, I., 849(4), 849 Garcia-Sucre, M., 746(7), 783 Garcis-Vela, A., 201(14), 202, 868(5), 868 Gardecki, J., 142(2), 145(2), 173-174(2), 178, 394(34), 403 Gm,A., 393-394(14), 402 Gargaud, M., 820(9), 843(9), 847 Garland, C. W., 143(53). 180 Gameau, J.-M., 485(15), 490 Garraway, B., 4(12), 43 Garrett, B. C., 259(68-70), 264(75), 272, 493(9), 538(9), 575, 855(7), 867 Gaspard, P., 59(28), 76, 209, 215(6), 218(6), 227(6), 231(6), 233(6), 236(6), 249(52), 270-272, 286(2), 291, 302(18), 313, 3 17(6), 322, 458( l), 458, 493-494(10). 495(14), 497(14,23), 500(14, 31), 501(14, 33). 502(35), 504(14), 509(48), 510(14, 56), 511(56), 512-513(61), 514(14, 33, 61, 64,67). 515(70), 517(33), 519(95, 97, 102-103). 521(14, 108, 112, 114), 522(108), 524(14, 108). 525-526(14), 528(56), 529(14, 112), 530(108, 112). 531(112), 534(14, 114). 535(114), 53q108, 114). 541(14), 542(14, 23, 33), 543(33, 35, 48), 544(10, 3 3 ,

545-546( lo), 548( 146). 553( lo), 554( 10, 14, 146), 555(10, 149). 556(10, 14). 558(10, 14), 559-560(33), 565(33), 566(146), 570(14, 146). 576-579, 581, 772(64), 785, 816(3), 816 Gaubatz, U., 328(9), 339(9), 341, 425(62), 441 Gaus, J., 79(9), 80, ll7-118(13), 122(13), 131, 133(4,6-7), 134(4), 135(4,6), 135, 136(4), 137, 200(6), 201, 203(4-5). 203 Gauthier, J.-M., 286(3), 292 Gauyacq, D.,647(1), 648, 707(32), 708 Gavrila, M., 338(34), 343 Gavrilov, N. K., 551(147), 581 Gea-Banacloche, J., 382(7), 385 Geers, A., 782(77), 785 Gehrke, Ch., 407 Gejo, T., 731(9), 741 Gelbart, W. M., 411(8), 439 George, T. F., 862(16). 867 Georges, R., 493(6), 518(6), 521(111), 528(6, 111). 537(6), 540(6), 575, 579 Gerber, G., 49(14). 52(17-18), 55(19), 60(37), 63(41), 65(18,49), 67(49), 72(54), 76-77, 78(1, 3). 79(5), 79, 90(2), 90, 103(6), 117(14), 131, 135(8), 137, 196(6), 196, 217(17), 229-230(17), 271, 328(10), 339-340(10), 341 Gerber, R. B., 201(14-15), 202, 868(5, 8). 868-869 Gerdy, J. J., 42 Gertitschke, P.L.. 458(3), 458, 849(2), 849 Gertner, B. K., 393(8), 399(8), 401 Geva, E., 239(42), 271 Ghosh, R., 157(32), 180 Giacobino, E., 382(10), 385 Giannoni, M. J., 517(77), 578 Gibson, S. T., 794(15), 797 Gilbert, R. D., 634(50), 646 Gilbert, R. G., 750(28), 784 Gillilan, R., 317(6), 322 Gillilan, R. E., 235(28), 245(28), 265(28), 286(2), 292 Giniger, R., 393(11), 401 Girard, B., 57(27), 76 Girardeau, M. D., 236(33), 271 Giusti-Suzor, A., 48(12-13), 76 Glasbeek, M., 39 Glass-Maujean, M.. 794(13), 797 Glasstone, S., 835(23), 847

AUTHOR INDEX Godar, D. E., 39 Goldberg, A., 82(2), 82, 711-714(5), 715, 855(8), 867 Goldberg, J., 519(91), 579 Goldfield, E. M., 86(2), 87 Golovlev, V. V.,878(8). 884(11), 886 Gomez-Llorente, I. M., 557(150), 581 Gomez Llorente, 1. M., 772(63), 785 Good, D., 405(1-2). 405-406 Goodman, M. F.. 85( I), 85 Gorden, R. J., 216(15), 223(15), 225(15), 271 Gordon, J., 216(12), 270 Gordon, R. G., 286(6), 292 Gordon, R. J., 57(25-26), 76, 216(13-14, 16), 270-271, 286(5, 13), 292, 328(4), 339(4), 341 Gorini, V., 238(37), 271 Gortler, S., 329(27), 341(27), 342 Goswami, D., 65(45), 77 Gould, P., 308(16), 312 Govers, T.R., 612(9), 623 Grabert, H., 393(16), 402 Graham, R., 320(11), 322, 382(12), 385, 393(16), 402 Grant, E., 117(12), 131, 286(12), 292, 668( I), 696 Grant, E. R., 72(51), 77, 133(3), 135(3), 135, 203(8), 204, 433(72), 437(72), 441, 616(16), 623 Gray, S. K.,86(2), 87, 760(47), 784 Green, Jr., W. H., 750(29), 784, 849(3), 849 Greene, B. I., 433(71), 441 Greene, C. H., 706(24-25). 708,726(3), 726 Grenlich, K.-0.. 884( I I), 886 Groebner, I., 379(9), 381 Gross, P.,215(11), 218(21-22). 268-269(76-77). 270-272, 315(2), 317(6), 318(7), 322 Grosser, M., 52(17), 63(41), 76, 103(6), 131 Grote, R. F.,393(8, 16). 399(8), 401402 Grotemeyer, J., 626-627(21), 634(21), 645, 656 Grozdanov, T. P., 760(46), 776(68), 784-785 Grubb, S. G., 433(72), 437(72), 441 Gruebele, M.,40,492(1), 561(151), 566(151), 575, 581, 799(2), 806

907

Guameri, I., 519(96), 579, 583(1-2). 584(4-5). 584-585 Guckenheimer, J., 496(19), 501(19),

509( 19), 542-543( 19), 546-547(19), 551(19), 576 Guelachvili, G., 465(7), 488(7), 490 Gutzwiller, M., 517(83), 578 Gutzwiller, M. C., 493(12), 499(12), 510(12), 576, 772(66), 785. 861(15), 867 Guyer, D. R., 493(4), 541(4, 138), 575, 580, 782(75), 785, 812(1), 812 Guyon, P.M., 612(9), 623,669(22), 697, 794( 13). 797

Haake, F., 510(50), 519(97, 99). 557(50), 577,579

Habenicht, W., 434(74), 441,615(11), 623, 701(1), 707

Haberland, H., 129(24), 132 Haggerty, M. R., 521(109), 579 Hahn, 0.. 557(150), 581 Halberstadt, N., 86(2), 87 Hall, J. L., 382(5), 385 Haller, E., 493(5), 518(5), 528(5), 537(5), 540(5), 575

Hake, E. J., 675(33), 697 Halstead, D., 200(7), 201 Hamilton, C. D., 529(122), 580 Hamilton, C. E., 421(52), 441, 747(13), 783 Hamilton, I. P.,859(12), 867 Hammerich, A. D., 196(5), 196, 304(19), 313 Hamoniaux, G., 173(47), 180 Han, C. H., 400(49), 403 Hanazaki, I., 731(9), 741 Handy, N. C., 259(64), 262-263(64), 272, 496(18), 497(20), 576, 870

Hgjlggi, P., 392(3), 393(8, 16). 399(8), 401402

Hannay, J. H., 509(49), 546(49), 557(49), 577

Hansen, K., 626(29), 646 Harding, L. B.. 760(48), 784 Hariharan, A., 65(45-46), 77 Harmin, D. A., 686(41), 698 Harcche, S., 801(11), 806 Harris, N. A., 707(34), 708 Harris, S. E., 302(7), 312 Hartke, B., 79(4), 79, 196(3), 196, 281(2), 281, 328(12), 339(12), 342, 750(21), 761(21, 52-53), 784-785

908

AUTHOR INDEX

Hase, W. L., 750(22), 751(30-31, 3 3 , 776(30), 784, 835(27), 841(27-30), 847-848, 854(5), 867 Hasegawa, H., 519(97), 579 Haseltine, J. N., 400(49), 403 Hashimoto, N., 194 Hatano, Y., 405(5), 406 Hattori, T., 382(20), 385 Hausler, D., 327(3), 339(3), 341, 768(61), 785

Hawthornthwaite-Lawless,A. M., 158(33), 180

Hayes, J. M., 171(41), 180 He, G. Z., 425(63), 441 He, X., 48(12-13), 76 Heaven, M., 731(8), 733-734(8), 741 Heidenreich, A., 82(2), 82 Heikal, A. A., 41, 85, 391(1), 4CKM01(1), 401

Heilweil, E. J., 400(48), 403 Heitz. M.-C., 79, 79(6) Helbing, J., 60(37), 64(43), 76, 79, 79(5) Heller, E. J., 504(39), 505(47), 521-522(108), 524(108), 530(108), 536( 108). 538-539( 130). 577,579-580, 602(1), 602,790(2), 796, 862-863(18), 868

Helman, A., 414(26-26). 440 Helman, A. B., 393(13), 402 Henri, G., 619-620(22), 623 Henriksen, N. E., 328-329(20), 332(20), 335(20), 339(20), 342 Hepbum, J. W., 669(26), 697 Herek, J. L., 41-42, 44, 56(20), 65(55), 76-77.90(1,3), 90,328(10), 339-340(10), 341 Herman, J., 529(122-123), 531-532(123), 580

Herman, M., 465(7), 488(7). 490, 521(112, 1141, 529-531(112), 534-536(114), 579, 809(1), 810(2), 810 Herman, M. F., 862-863(17), 867-868 Hermann, R. H., 610-611(5), 623 Hernandez, R., 496(18), 497(20), 541(139), 576, 580, 782(76), 785, 812(1), 812, 870 Henmann, F. W., 72(52), 77 Herzberg, G., 634(46), 646,705(14-16). 706(21), 708, 721,726(2), 726 Hess, W. P., 732(13), 737(13), 741 Hessel, M. M., 87, 89(7)

Hessels, E. A., 705(18-20). 708 Hill, R. K., 189(7), 191 Hintsa, E. J., 737-738(18), 737(19), 741-742

Hioe, F. T.,425(61), 441 Hirai, K., 521-522(108), 524(108), 530(108), 536(108), 579 Hirata, F., 142(6), 145(6), 173(6), 179 Hirschfelder, J. O., 835(23), 847 Hishikawa, A., 791(7), 794(7), 796 Hoburg, E. A., 395(41), 400(41), 403 Hobza, P., 416(34), 440 Hochstrasser, R. M.. 393(11), 395(42, 46). 399(42), 400(48-49), 401, 403. 412(17), 440

Hoffman, B. M., 393(16), 402 Hogervorst, W., 493(11), 510(11). 576 Hohmann, H., 626(30-31). 646 Holbrook, K. A., 515(71), 539(71), 578, 750(26), 784 Holle, A., 493(11), 510(11), 576 Hollenstein, H., 453(54), 453 Holloway, S., 200(7), 201 Holmes, D., 296(6), 300, 300(6) Holmes, J. L., 731(10), 741 Holmes, P.,496(19). 501(19), 509(19), 542-543(19), 546547(19), 551(19), 576 Holthaus, M., 328-329( 19). 335-336( 19). 339(19), 342 Hong, C. K., 382(9), 385 Hopfield, J. J., 393-394(19), 402 HoriEek, J., 746(6), 783 Horani, M., 647(1), 648, 707(32), 708 Honn, N. B., 416(35), 440 Horn, B. A., 42 Homg, M. L., 142(2), 145(2), 173-174(2), 178, 394(34), 403 Hougen, J. T., 423(55), 441, 485(14), 490 Houston, P. L., 749(18-19). 759(18), 765(19), 767(19), 783 Hsu, Y.P., 419(44), 440 Hu, X., 835(27), 841(27), 847 Hu, Y., 178(51), 180 Huang, G., 247(46), 272 Huang, G . M., 218(21), 247(47-50), 248(47), 268(48, 76). 269(76), 271-272, 318(7), 322 Hubbard, L. M., 536(124), 580 Huber, J. R., 746(9), 765(9), 767(9), 778(9), 783, 785,794(18a), 797

AUTHOR INDEX Huber, K. P., 693(48), 698, 705(13), 707(13), 708 Hudson, A. J., 69600). 698 Huet, T. R., 529(122), 580 Hunter, C. N., 146(19), 152(19), 160(19), 179 Hunter, M., 782(79), 785 Hunziker, H. E.. 731(5), 733(5), 741 Hupp, J. T., 394(27), 399(27), 402 Hupper, B., 572(157). 573(160), 581 Huppert, D., 393(1I), 401 Hutson, J. M., 80(1), 80 Hynes, J. T., 142(9). 145(9), 163(9), 173(9), 179,393(8, 14, 16). 394(14, 31). 399(8), 401403, 854(5), 867 Ikeda, N., 642-643(60), 646 Imamoglu, A,, 302(7), 312 Imamoto, Y.,405(5), 406 Impey, R. W., 394(33), 403 Imre, D., 493(3), 575 Imre, D. G., 274(10), 275, 327(1), 339(1), 341, 373(6), 374 Innes, K. K., 493(3), 575 Inoue, G., 731(7), 733(7), 741 Ionov, S. I., 86(2), 87 Isaacs, N. W., 158(33), 180 Isaacson, A. D., 259(68-69), 272 Isaacson, X.,26(71), 272 Ishikawa, H., 465(5), 484-489(5), 490 Itakura, R., 791-792(10), 795(10), 797 Ito, K., 705(13), 707(13), 708 Ito, M., 422(53), 441, 674(32), 697 Ivanov, L. N., 662(8), 662,663(2), 663 Ivanov, Yu., 286(3), 292 Iwata, L.,767(57), 785 Izrailev, F. M.. 541(140), 580, 583(3), 585 Jackson, J. D., 363(27), 371 Jacobson, M., 468(11), 484(11), 489(11), 490 Jaffe, C., 388 Jain, S. R., 517(80), 578 Jakubetz, W.. 274(7), 275, 281(3-4), 281, 328(8, 23). 329(23, 27). 334(23), 339(8, 23), 341(23,27), 341-342, 373(2), 373 Janik, G. R., 670(27), 697 Janssen, M. H. M.,52(16), 76, 651(1), 652, 730(2), 741, 799(2), 806 Janza, A. E., 200(8), 201, 761(53). 785

909

Janzky, J., 382(8, 11, 14, 17, 19). 385 Jaques, C., 86(2), 87 Jayatilaka, D., 496(18), 576, 870 Jean, J. M., 147-148(22-25). 149(24-25), 150(22-24), 151(25), 152(24-25). 153(25), 160{24-25), 177(22, 25), 179, 195 Jeannin, C.,714(14), 715 Jeschke, H., 116(1 I), 131 Jessen, B., 520(105), 579 Jhe, W.,379(9), 381 Jia. Y., 146(20), 164(37), 166(37), 168(37), 169(40), 171(40), 172-175(37), 179-180 Jimenez, R., 146(18), 152(18), 154(18), 157(18), 158(18, 34). 159(18), 179-180 Jiminez, R., 394(31, 34), 403 Jin, Y.,233-234 Joens, J. A., 566(158), 572(158), 581 Johnson, A. E., 394(27, 29, 36), 399(27, 29). 402403 Johnson, B. R., 521(107), 572(107), 579, 651,790(3), 796 Johnson, P., 668-669(4), 697 Johnson, P. M., 419(44), 434(76), 440441 Joly. A. G., 173(47), 180 Jonas, D. M., 94, 96,146(14,20), 152(28), 153(29), 154(20,28-30). 155(28), 156(29), 179, 195,465(1), 467(1), 490 Jonathan, N. B. H., 849(1), 849 Jones, K. E.,39 Jones, M. R., 146(19), 152(19), 160(19), 179 Joo, T., 146(20), 164(37), 166(37), 168(37), 169(3940), I7 l(39-40). 172(37), 173(37, 49). 174-175(37), 179-180 Joosen, W., 59(31). 76 Jortner, E. J., 711-712(1, 4-5). 713-714(4-5, 13). 714-715 Jortner, J., 81(1), 82(1-2). 82, 393(15), 402,411(9), 412(11, 17), 416(35), 419(45), 434(88), 437(88), 439440, 442, 454455(1), 455, 537(128), 580, 588(5), 589, 629(39), 637(55), 642(62). 646,668(13), 681-682(13), 691-692(13), 697, 764(54), 785 Joseph, T.,200(4), 201, 274(6), 275, 328(22), 332(22), 339(22), 341(22) Jost. R., 493(6), 515(68), 518(6), 521(111, 114). 528(6, 11I), 534-536( 114). 537(6), 540(6), 575, 578, 579

910

AUTHOR INDEX

Joyeux, M., 521(113), 526(116-117). 527( 117). 528( 113, 118-121), 579-580 Judson, R. S., 251(60), 272, 319(10a), 322 Julienne, P. S . , 423(57), 441 Jumenez, R., 142(34), 144(4), 145(34), 173(3-4), 179 Jung, C., 511(57), 528(57), 577 Jungen, C . , 634(46), 646, 647(1), 648, 693(48), 698.703-704(10), 705(13-16), 706(21-23, 26, 30), 707(13, 32, 34). 708, 721 Jungwirth, P., 201(15), 202, 868(8), 869 Just, B., 79(7), 7 W 0 , 274(5), 275, 328(14, 17, 19, 23-24), 329(14, 19, 23-24, 27), 330(14), 334(23), 335(14, 17, 19,24), 336(14, 19,24), 339(14, 17, 19,23-24). 341(14, 17, 23-24, 27). 342, 373(3), 374, 375(3), 377 Kades, E., 79(7), 79-80, 274(7), 275, 281(3), 281, 328(8, 17), 335(17), 339(8, 17), 341(17), 341-342, 373(2), 373 Kahn, K. H., 302(7), 312 Kaiser, R., 302(10), 305(10), 307(10), 312 Kakitani, T.,405(5), 406 Kalinowsky, H., 379(9), 381 Kaluza, M., 258(63), 261(63), 272 Kalyanaraman, C., 201(18), 202, 332(27), 342 Kane, D., 59(33), 76 Kane, D. J., 346(9-lo), 370 Kanfer, S., 388 Kapelje, K. A., 510(51), 577 Kappert, J., 782(77), 785 Kappes, M., 103(4), 131 Karasch. S., 157(32), 180 Karlsson, H. 0.. 858(lO), 867 Karrlein, W., 200(8), 201, 761(53), 785 Kash, P.W., 730(3), 741 Kasha, M., 418(42), 440 Kassakowski, A., 238(37), 271 Kastberg, A., 305(15), 312 Kaufman, F., 849(1), 849 Kause, J. L., 800(6), 806 Kawashiria, H.,176(50), 180 Kay, K. G., 862-863(20), 868 Kazmina, N. P., 327(3), 339(3), 341 Keating, J. P., 503(38), 517(81), 577-578 Keldish, L. V., 374(1), 376(1), 377

Keller, H.-M., 484(13), 490, 746(8, 10). 747(16--17, 34). 748(1&17), 749(17, 20), 750(20), 751(16-17.32-37). 752(32-33, 38), 753(32, 34). 754(32-34), 756(32), 758(34, 36-37), 760(34), 761(16, 32-33), 762(8, 32), 763(33), 764(32), 765(10, 32). 766-767(32). 768(17), 769(16-17, 38). 770(16-17), 771(8, 17), 772(34), 774(34, 36). 775(36-37), 776-777(37), 778-779(10, 37), 780-781(37), 783-784, 786(34), 812(4), 812, 815(1), 815 Kellman, M. E., 466(9-10). 489(9-lo), 490, 590, 591(1), 594 Kempl, S., 201(12), 202 Kendall, D. J. W., 705(14), 708 Kennedy, R. A., 723(1), 723 Keshavamurthy, S . , 861(13), 867 Khidekel, V., 182(2), 182, 360(26), 371, 386(2), 387, 514(68), 578 Khundkar, L. R., 41, 86(1), 87, 492(1), 575 Kim, B., 794(14a, 14b), 797 Kim, H. J., 393(16), 402 Kim, K. G., 236(33), 271 Kim, S., 320(12), 322 Kim, S . B., 42, 185(2), 191(2), 191 Kim, S. K., 4 1 4 2 , 85(1), 85, 400(50), 403 Kim, W.-H., 171(41), 180 Kimble, H. J,, 382(5), 385 Kinsey, J. L., 421(52), 441,493(3), 518(85), 521(107), 529(122), 572(107), 575, 578-580, 747(13), 783, 790(3), 796 Kirmse, B., 521(111), 528(111). 579 Kistiakowski, G. B., 412(13), 440 Kitsopoulos, T.N., 668(5), 697 Kleiman, V., 216(14-16), 223(15), 225(15), 270-271, 286(13), 292, 328(4), 339(4), 34 I Kleiman, V. D., 286(6), 292 Klein, M. L., 394(33), 403 Klein, S., 172(43), 180 Kleinman, V. D., 57(26), 76 Kliner, D. A. V., 394(36), 403 Klippenstein, S . J., 814(1), 814, 850(2), 850, 870(1), 870 Klosek-Dygas, M. M.,393(16), 402 Kluk, E., 862-863(17), 867-868 Knee, J. L., 668(8. 17). 669(17), 681(17), 692(17), 697 Kneppe, H.,731(5), 733(5), 741 Knight, P.L., 286(15), 292, 382(3), 385

AUTHOR INDEX Knittel, 73..413(22), 428(22), 435(22), 440 Knospe, 0.. 657( I), 657 Knyazev, I. N., 327(3), 339(3), 341, 661(5-7). 662 Kobayashi, T.,382(17, 20). 385 Kobe, K., 121(19), 12&125(22), 131, 132(2), 133(7), 135(2), 135, 137, 203(5, 7). 203-204 Kwh, E. E., 711-712(1), 714 Koenig, W.. 346(8), 370 Kohen, D., 204(1), 206 Kohler, A., 713(12), 715 Kohler, B., 18(15). 43, 59(34-35). 76, 173(47), 180, 235(30), 265(30), 271, 273( I), 274(8), 274, 275( I), 275-276. 302(5), 312. 346(6-7), 370 Kolba. E., 79(7), 79-80, 328(12, 17), 335(17), 339(12, 17). 341(17), 342 Kollman, M., 519(99), 579 Kompa, K. L., 457 Kong, W., 669(26), 697 Konig, A., 510(51), 577 Kono, T., 731(9), 741 Konopsky, V.,880(9), 883(9), 886 Konz, E., 423(56), 441 Koperski, J., 87, 89(5) Koppel, H., 203(6), 203,493(5), 510(53), 5 I8(5), 528(5), 537(5), 540(5), 575, 577, 772(65), 785 Korolkov, M. V., 274(5), 275, 328(15-17). 329(15-16), 332(16), 333(15), 335(17), 337(16), 339(15-17). 341(15-17). 342 Kosloff, D., 200(1), 201, 855(8), 867 Kosloff, R., 48(3). 59(28), 75-76, 78(2), 79(2. 4), 79, 90(4), 90, 196(3, 5-10), 196, 198, 200(1, 3, 5), 201(18), 201-202, 215(5-6), 216(5), 218(6, 20). 227(6), 228(5), 231(6), 233(6), 236(6), 239(4@42), 246(20), 253(20), 255(20), 257-258(20), 270-271, 274( I), 274, 286(2), 291, 302(18). 304(19), 308(17), 312-313,317(6), 322,328(6), 332(29), 339-340(6), 341-342, 373(1), 373, 393-394(23), 402, 458( I , 3). 458, 761(51), 784, 812(3), 812, 855(8), 867 Kosman, W. M., 693(49), 698 Kosower, E. M., 393(1 I), 401 Kosygin, D. V., 516(75), 518(75), 578 Koszykowski, M. L., 410(2), 439, 519(94), 579

91 1

Kouri, D. J., 855(8), 867 Koutecky, J., 103(4), ll4-ll5(lO), 117(10, 16-17), 129-130(25). 131-132 Kowalcyzk, P., 800(5), 806 Kozlov, A. A., 878(8), 886 Krainov, V. P., 419(46), 423(46), 440 Kramers, H. A., 392(6), 401 Krasnopolsky, K. M., 746(6), 783 Krause. H.. 416(32-33). 434(77), 440-441, 630(43), 646,669(25), 697 Krause, J., 302(5), 312 Krause, J. L., 18(15), 43, 59(35), 76, 218(25), 235(29, 30-32). 236(32), 265(25, 29, 30-32), 267(25), 271, 273(1), 274(8), 274, 275(1), 275, 276(2), 276,296(3), 300, 328(7), 339(7), 341, 346(6-7). 370 Krause, L., 87, 89(5-6) Krauss, M., 87 Kreisle, D., 626(33), 646 Krempl, S., 868(10), 869 Krim, L.,86(2), 87 Kris, Y.,629(41), 646 Kroes, G. J., 573(161), 581 Kudriavtsec, Yu. A.. 327(3), 339(3), 341 Kuharski, R. A,, 394(33), 403 Kuhling, H., 102-103(1), 117-1l8(13), 121(19), 122(13), 131, 132(2), 133(7), 135(2), 135. 137, 203(5, 7). 203-204 Kuhn, A., 328(9), 339(9), 341,425(63), 441 Kuhn, 0..332(32), 333(32-33). 343 Kukulin, V. I., 746(6), 783 Kulander, K. C., 565(152, 154). 566(152), 569(154). 581 Kumar, P. V.,142(3-4, lo), 144(4). 145(34, 10). 163(10), 173(34, lo), 179, 394(31, 34), 403 Kuo, Q., 465(7), 488(7), 490 Kurizki. G., 270(79), 272, 286(7), 292 Kurokawa, K., 382(20), 385 KGs, M., 510(50), 519(99), 557(50), 577, 579 Kutzelnigg, W., 451(1), 453, 587(2), 588 Kuzmin, M. V., 451(1), 451 Labastie, P., 68(50), 72(50), 77, 102(1-2). 103(1), 122(2), 131, 434(81), 441, 538(129),580 Lacy, L. Y.,346(8), 370

912

AUTHOR INDEX

Ladanyi, B. M., 145(12), 172(12,43), 176(12), 179-180, 181(1), 181 Lagendijk, A.. 59(31), 76 Laidler, K. J., 835(23), 847 Lakshmanan, M.,519(90), 520(103), 579 LaI&, F., 745(2), 783 Lamb, W.E., 243(44), 271 Lambert,I. R., 668(9), 697

Lambert,W.R., 40

Lambropoulos, P., 286(15), 292 Lambry, J.-C., 146(15), 160(15), 179 Lan, B. L., 281(4), 281, 329(27). 341(27), 342

Landau, L., 457(1), 457 Landau, L. D.,162(35), 180 Landauer, R., 393(16), 402 Landi, K., 238(38), 271 Landman, U., 81-82(1), 82, 711-714(4), 715

Lane, A. M., 538-539(133), 580 Lane, N. F., 703(9), 708 Lang, M. J., 164(37), 166(37), 168(37), 172-175(37), 180 Langer, J. S., 392-393(7), 401 Lankhuijzen, G. M.,537(127), 580 Laporta, P., 61(39), 76 Lauder, M. A., 286( 15). 292 Lauritzen, B., 521(109), 579 Leahy, D.J., 730(4), 741 Leaird, D.E., 59(29-30). 76 Lebowitz, J. L., 516(75), 518(75). 578 Lee, D., 146(54), 180 Lee, M.,400(49), 403 Lee, S.,794(14b), 797 Lee, S.-Y., 196(4), 196, 382(18), 385, 538-539(130), 580 Lee, Y. T., 434(86-87), 437(86-87), 442, 626(17), 629(17), 634(17), 645, 668(15), 681(15), 697, 732(12), 737(12, 18-19), 738(18), 741-742 Lefebvre-Brion, H., 487(18), 490,702(4), 707

Lefebvre-Brion, M.,412(10), 439 Lefivre, G., 647(1), 648, 707(32), 708 Leforestier, C., 855(8), 867 Legnier, J., 103(3), 129(3), 131 Lehmann, K.,454(2), 455 Lehmann, K. K., 485(12), 488(12), 490 Leisher, T., 114-115(9), 131 Leisner, T., 626(27), 646, 657(1), 657

Lemaitre. D.,465(7), 488(7), 490 Lester, Jr., W. A., 731(6), 741 Letokhov, V. S., 4(4), 43, 79(7), 79-80, 185(3-5), 189(4, 8, lo), 190(10), 327(1, 3). 328(11), 339(1, 3, ll), 341, 372(5), 420(50), 423424(50), 441,451(1), 451, 661(1-9), 662(1, 8). 662, 663(1-2). 663, 874(I , 3), 875(5), 876(7), 878(8), 880(9), 883(9-lo), 884(1 I), 886 Levin, R. D.,731(10), 741 Levine, R. D.,95-96, 153-154(29), 156(29), 157(31). 179-180, 195, 434(78, 83, 85), 437(83, 85), 441442, 454-455(1), 455, 457, 519(94), 540(137), 579-580, 588(5). 589, 615(13), 623, 626(34, 20). 627(3c), 628(3c, 4-5). 629(2), 631(45), 634(2-3, 5,20. 45a), 636(3c, 5.45a. 45b. 45f. 53). 637(53), 639(3b, 3c, 3d, 4%). 640(45), 642(45b, 53). 643(45b, 45d). 645646, 649-652, 656, 659, 668(14, I@, 681-682(14, 18), 691(18), 697698, 702(3), 707, 724(1), 724, 812(2), 812, 813(1), 813 Levinger, N. E., 394(27, 36). 399(27), 402-403

Lewerenz, M., 587(3), 588 Lewis, B. R., 794(15), 797 Lezius. M., 626(26), 645 Li, X.,57(26), 76, 216(14-15), 223(15), 225(15), 270-271, 286(6, 13), 292, 328(4), 339(4), 341, 747-748( 16). 751(16), 761(16), 769-770(16), 783 Li, Z., 146(21), 179, 373(7), 374, 514(66), 578

Lias, S. G., 731(10), 741 Lichtenberg, A. J., 496(19), 501(19), 509( 19). 542-543( 19), 546-547( 19), 551(19), 576 Lieberman, A. J., 496(19), 501(19), 509(19), 542-543(19), 546-547(19), 551(19), 576 Liebman, J. F., 731(10), 741 Lienau, C., 41-42 Liivin, J., 521(112), 529-531(112), 579, 637(56), 646 Lifshiz, E. M.,162(35), 180 Light, J. C., 565-566(152), 581, 778(69), 785, 859(12), 867 Likar, M. D.,327(3), 339(3), 341, 373(6), 374

AUTHOR INDEX Lill, J. V., 859(12), 867 Lin, Q.,195 Lin, S. H., 610-611(5), 623 Lindblad, G., 238(36), 271 Lindner, R., 617(20), 619-620(22), 623, 626(10), 629(10), 629(35), 645-646, 726(4), 726 Lineau, C., 195 Lineberger, W.C., 732(14), 741 Linskens, A. F., 423(58), 441 Lipert, R. I., 619-620(23), 623 Little, D. D., 327(3), 339(3), 341 Littlejohn, R. G., 510(54), 529(54), 577, 601 Littman, M., 315(2), 322 Liu, H. C., 270(80), 273, 286(10), 292 Liu, Q.,4 1 4 2 , 56(20), 76, 90(1), 90, 328(10), 339-340(10), 341 Liu, S., 791-792(10), 795(10), 797 Liu, W. K., 39 Logan, D. E., 642(59), 646 Logvin, Yu. A., 2740). 275, 328-329(16), 332(16), 337(16), 339(16), 341(16), 342 Lohr, L. L., 490 Lombardi, M., 493(6), 518(6), 528(6), 537(6), 538(129), 540(6), 575, 580 Lopez-Delgado, R.. 4 12(18). 440 Lorents, D. C., 791(8), 796 Liirincz, A., 250(59), 252(59), 272, 317(6), 319(10b), 322 Lorquet, J. C., 541(140). 580 Los, J., 732(15), 741 Loudon, R., 382(3), 385, 801(9), 806 Lovejoy, E. R., 849(4), 849 Lu, D., 262(72-73), 272 Lu, D.-H., 750(22), 784 Lu, S.-P., 57(25), 76, 216(12-13, 15). 223(15), 225(15), 270-271, 286(5, 13). 292, 328(4), 339(4), 341 Lu, Z., 591(1), 594 Lu. 2-M., 218(23), 268-269(78), 271-272, 320-321(13), 322 Luckhaus, D., 377(3). 379 Lundeen, S. R., 705(17-20). 708 Luo, X., 749(19), 7 6 3 19). 767( 19). 783 Lykke, K. R., 626(24), 645, 732(14), 741 Lyon, S. A., 315(2), 322 Ma, Si, 382(16), 385 Maas, D. J., 65(48), 77

913

MacInnis, J., 394(34), 403 Mackenzie, S. R., 629(40), 646, 669(24), 670-671(29), 672(29-30). 675(33), 684-685(24), 697 Mackey, I., 538-539(133), 580 Maergoiz, A. I., 820(3, 16). 822(3, 16). 823(18), 828-829(3), 832(3), 842(16), 843-845(33-34,36-37), 846(3, 16, 34, 37). 847-848 Magnes, O., 425(64), 441,444(1), 444 Magnus, W., 597(3), 598 Mahon,C. R., 670(27), 697 Main, J., 493(11), 510(11), 576 Makarewicz, J., 416(36), 440 Makri, N.,263-264(74), 272, 863(21), 868 Malisch, J., 328(17), 335(17), 339(17), 341(17), 342 Malisch, W., 79(7), 79-80 Mallard, W.G., 731(10), 741 Malta, C. P., 546( 144). 581 Malzahn, D., 332(32), 333(32-33), 343 Mandel, L., 346(14), 371, 382(9), 385 Mandelshtarn, V. A., 760(46), 776(68), 784-785 Manners, 3.. 526(116-117). 527(117), 580 Mantegna, R. N., 542(142), 580 Manthe, U., 200(9), 201, 854(4), 857(4), 859(11). 861(11). 867, 868(4-5, 7), 868-869 Manz, J., 4(2-3). 18(2),43, 79(6-9), 79-80, 86(2), 87, 89(4), 103(5), 117-118(13), 122(13), 131, 133(4, 6-7), 134(4), 135(4, 6). 135, 136(4), 137. 185(1), 191(1), 191, 196(1), 1%. 197(1), 200(34, 6, 8). 201, 202(1), 202, 03(4-5), 203, 274(5-7). 275, 281(1-3). 281, 328(8, 12, 14, 17, 21-23), 329(14, 21, 23, 26-27), 330(14), 331(26-27), 332(21-22). 334(23), 335(14, 17, 21), 336(14), 339(8, 12, 14, 16, 21-23), 340(26), 341(14, 17, 22-23, 26). 341-342, 373(2-3). 373-374, 3733). 377(1), 377, 379, 458(3), 458, 545(143), 548(143), 550(143), 554(143), 560(143), 571(143), 581,750(21), 761(21.51-53). 784-785, 812(3), 812, 849(2), 849, 874(2), 886 Mao, J.-M., 493(11), 510(11, 51), 576577 Marcus, R. A., 96(1), 96, 391(2), 393(20), 394(20, 22). 395(2, 44), 396(44), 398-399(44), 401403,405,406(1),

914

AUTHOR INDEX

Marcus, R. A. (Conrinued) 406407,410(1-2), 414(25-26). 439-440, 454(1), 454, 519(94), 539(135), 579-580, 750-751(23), 784, 835(23), 842(31), 847-848, 850(1-2), 850, 870(1), 870 Marcus, R. M., 493(2), 575 Marden, M. C., 405(2), 406 Markovic, N., 820(2), 823(2), 835(26), 847 Marks, J., 732(14), 741 Maroncelli, M., 142(2-4, 7-8, 10). 144(4), 145(2-4, 7-8, 10). 162(7), 163(7-8, lo), 173(2-4, 7-8, lo), 174(2), 178-179, 394(31, 34), 403 Marquardt, R., 93(3-5), 93, 377(3), 379(7), 379, 381(1-2). 381, 588(7), 589, 590(1), 591 Marque, J., 405(2), 406 Martens, C. C., 373(7), 374, 514(66), 578 Martin, J.-L., 146(15), 160(15), 179 Marvet, U., 86(3), 87, 89(3) Maslov, V. P., 861(15), 867 Masnou-Seeuws, F., 647( 1). 648, 707(32), 708 Mastenbroek, J. W. G., 730(2), 741 Masters, C. C., 146(21), 179 Matemy, A,, 42, 90(3), 90 Mathews, C. W., 412(14), 440 Mathies, R. A., 146(16-17). 151-152(17), 178(17), 179, 196(4), 196, 382(18), 385 Mathis, J., 750(21), 761(21), 784 Matkowsky, B. J., 393(16), 402 Matro, A., 48(10), 57(21), 59(10, 21), 75-76, 150(27), 179, 204(1), 204, 217(18), 218(19), 241(18), 253(61), 271-272, 303( 1I), 312 Matsumoto, A., 332(31), 343 Matsunaga, F. M., 570(156), 581 Matthies, C., 517(81), 578 Matveyets, Yu.A., 878(8), 884(11), 886 Matveyev, 0.I., 661-662(8), 662 Matyuk, V. M., 661(&7), 662 Maurette, M.-T., 279(2), 280 Maushart, Th., 416(38), 440 Mauta, K., 730(2), 741 May, B. D., 626(32), 646 May, V., 332(32), 333(32-33), 343 Mazurenko, Yu. T.,394(39), 403 McCammon, J. A., 393(17), 402 McCann, J. F., 286(3), 292

McCarroll, R., 820(9), 843(9), 847 McCarthy, M. I., 791-792(6), 795, 796 McCarthy, P. J., 400(49), 403 McCormack, E. F., 459(1), 459 McCoy, A. B., 590, 868(5), 868 McDermott, D., 158(33), 180 McDonald, J. M., 791(5), 796 McGlynn, S. P., 791(5), 796 McKoy, A. B., 201(14), 202 McKoy, V., 617-618(18), 668(1), 696 McKoy, V. B., 616(16), 623 Mead, R. D., 732(14), 741 Mease, K., 218(21), 268-269(76), 271-272, 318(7), 322 Mecke, R., 331(28), 342 Meerts, W. L., 423(55), 441 Mehta, A., 454(1), 454 Mehta, M. L., 516(74), 518(74), 578 Meier, C., 60-61(38), 64(42), 65(49), 67(49), 76-77, 196(6), 196 Melinger, J. S., 65(45-46), 77 Mello, P. A., 505(44), 577, 772(67), 785 Mendas, I., 820( 1 I), 847 Meredith, D. C., 521(109), 579 Merkt, F., 434(75, 79), 441, 610(4), 623, 626(11-12), 629(11-12, 40), 634(11-12), 639(57), 644(12), 645-646, 668(2, 10, 12), 669(24), 670(28), 672(30), 675(2, 33), 681(12), 684-685(24), 686(12), 689490(46), 697-698 Messiah, A., 745( I), 783 Messina, M., 59(35), 76, 218(25), 235-236(32), 265(25,32), 267(25), 271, 273(1), 274(8), 274-275, 276(2), 276, 328(7), 339(7). 341, 346(7), 370 Metiu, H., 201(13), 202 Metz, R. B., 733(16), 739(16), 741 Meyer, H.-D., 200(9), 201(9), 201-202, 772(65), 785, 868(7), 868-869 Meyer, W., 79(8), 80, 196(1), 196, 197(1), 202( I), 203 Meyerhofer, D. D., 374(1), 376(1), 377 Michaille, L., 521(113), 528(113), 579 Mies, F. H., 48(12), 76.87, 89(6) Mikeska, H. J., 519(95), 579 Miladi et al., 718 Miller, T. A., 731(8), 733-734(8), 741 Miller, W., 259(66-67), 263(67), 272 Miller, W. H., 259(64-65), 262(64), 263(64, 74), 264(74), 272, 493(4), 496(18),

AUTHOR INDEX 497(20,24), 499(29), 510-51 1(55), 512(29), 528(55), 536(124), 538(24), 540(136), 541(4, 139). 555(24), 575-577, 580, 759(41), 761(50), 764(50), 779(73), 782(74-76), 784-785, 812(1), 812, 853(1), 854(24,6). 855(3), 857(34), 859(11), 860(3), 861(11, 13, 15a. 16). 862(3b), 863(21), 865(3a, 22), 867, 868(1-5). 868, 870 Mills, I. M., 496(16), 576 Mishin, V. I., 661462(8), 662, 663(2), 663

Mlynek, J., 189-190(9), 191 Mohan, V., 201(18), 202, 761(53), 785 Mohrschladt. R., 393(8), 399(8), 401 Moiseyev, N., 760(45), 784 Mojtabai, F., 393(8), 399(8), 401 Mokhtari, A., 41. 44 Moldauer, P. A., 505(44), 577 Molin, Y., 4(9), 43 Molinari, L., 519(96), 579 Momose, T., 734(17), 741 MOOR,C. B., 493(4), 541(4, 138-139). 575, 580,750(29), 782(74-76). 784-785, 812(1). 812, 849(34), 849 Moore, E. J., 400(48), 403 Mordaunt, D. H., 668(9), 697, 747(34), 752(34, 38). 753-754(34), 758(34), 760(34), 769(38), 772(34), 774(34), 784, 786(34), 812(4), 812, 815(1), 815 Morgan, J. D., 393(17), 402 Morgan, M. C., 405( I), 405 Morikawa et al., 718 Morillo, M., 394(24), 402 Morley, G. P.,668(9), 697 Morokuma, K., 484( 13). 490 Mostowski, J., 800(5), 806 Movshev, V. G., 661(6-7). 662 Mrugala, F., 812(3), 812 Muckerman, J. T., 258(63), 261(63), 272 Muhlpfordt, A., 626(1, 16), 629(1), 634(1), 644(1), 645, 668(14), 681-682(14), 697 Mukamel, S., 91(1-2). 91-92, 142(1), 146(1), 160-162(1), 178-180, 178(51), 182(1-3). 182, 200(1), 200, 210, 235(28-30). 245(28), 265(28-30). 271, 274(8), 275(1), 275-276, 317(6), 322, 346(6), 347(17), 349(17), 350(23), 351(17), 356(17), 357(24), 360(26), 363(17), 369(17), 370-371, 386(1-2),

915

387, 394(35, 37), 403, 514(65, 68), 578. 800(4), 806 Muller, E. W., 876(6), 884(6), 886 Muller, H. G., 59(31), 76 Muller. K., 504(40), 570(40), 577 Muller-Dethlefs, K., 434(74), 441, 615(1l),

616(14-16), 617(17, 20). 618(17), 619(21-22, 24), 620(22), 623,626(6a), 629(6), 645, 668(6), 676(34-35). 678(35), 668(1), 696497, 701( 1). 707, 726(4), 726

Mulliken, R. S., 63(40), 76, 649, 702(5), 708

Munster, M., 636(51), 641(51), 646 Muradaz, M. A., 279(1), 280 Murthy, M. V. N., 517(80), 578 Myers, J. D., 732(13), 737(13), 741 Nadler, W., 394(22), 402 Nagasawa, Y.,169(39), 171(39). 173(49), 180,394(29), 399(29), 402 Nakamura, K., 519(90,95, 102), 520(103), 579

Nakata, R. S., 570(156), 581 Nazarova, N. B., 328(13), 332(13), 334(13), 339(13), 342 Negus, D. K., 393(11). 401 Nelson, K., 304(14), 312 Nelson, K. A., 59(32), 76, 173(47), 176(50), 180, 346(3), 362(3). 370 Nemeth, G. I., 626(3b, 13), 629(3. 13). 645 Nenner, I., 612(9), 623 Neugebauer, F., 333(33), 343 Neuhauser, D., 215(11), 270, 855(8), 867 Neuhauser, R., 410(5-6), 416(30), 42q5-6, 30), 420(30), 423(59), 424-425(6), 427(6), 428(6, 30). 43 1(30), 433(69), 435-436(89), 438(89), 439-442 Neumark, D. M., 458(2), 458, 631(44), 646, 668(5), 697, 730(4), 733(16), 739(16), 741, 816(4), 816 Neusser. H. J., 410(3-6), 413(3-4, 19-22), 414(3-4, 23-24, 27), 415(3, 28-29), 416(30-33, 37). 417(29), 418(37), 420(30), 423(59), 427(6), 428(6, 22, 30). 431(30, 65), 433(69-70). 434(77), 435(22, 89). 436(89). 438(89), 439442, 630(43), 646, 669(25), 697 Neyer, D. W., 749(18-19). 759(18), 765(19), 767(19), 783

916

AUTHOR INDEX

Ng, C. Y.,668(3), 697 Ni, G. Q., 72-73(53), 77 Nibbering, E. T. J., 178(52), 180, 348(19), 371 Nibler, J. W., 143(53), I80 Nicolis, G., 514(64), 578 Niedzielski. J., 820(12, 14). 847 Nielsen, H. H., 486(17), 490 Nielsen, U., 200(7), 201 Nikitin, E. E., 820(3, 8, 16), 822(3, 8. 16), 828-829(3), 832(3), 842(16), 843(33-34, 36-37), 844-845(33), 846(3, 8, 16, 34, 37-39), 847-848 Nishiyama, K., 194 Nitzan, A., 171(42), 180, 393(9, 16). 401-402 Noid, D. W., 410(2), 439 Noordam, L. D., 59(31), 65(47-58). 76-77, 537(127), 580 Nordholm, S., 636(54), 642(54), 646, 820(2), 823(2), 835(26), 847 North, S., 732(12), 737(12), 741 Northrup, F. J., 747(14), 783 Northup, H., 393(17), 402 Neskov, J. K., 200(7), 201 Nussenzweig, H. M., 347-348(18), 371 Nygibd, J., 520(104), 579 OBrien, J. P., 465(3-4), 467(3), 488(34), 490, 536(125), 580 Ogai, A., 767(57-58), 785, 794(18a), 797 OHailoran, M. A., 706(27-29), 708 Ohde, K., 791(7, 9-10), 792(9-lo), 794(7), 795(10), 796-797 Ohrnine, I., 145(11), 172(11), 176(11), 179 Ohshirna, Y.,465(5), 484-489(5), 490 Okada, T., 194 Okunishi, M.,794( l6), 797 Olender, R., 171(42), 180 Oliveros, E., 279(2), 280 Olzrnann, M., 835(21), 847 Onda, K., 794(16), 797 Ong, C. K., 247(48-49), 268(48), 272 Onuchic, J. N., 393-394( 14), 402 Orel, A. E., 565(154), 569(154), 581 Orlik, W., 129(24), 132 Orlowski, T. E., 39 Ormos, P., 405(2), 406 Orszag, A., 173(47), 180

Osborn, D. L., 730(4), 733(16), 741, 816(4), 816 Otis, C. E., 433(72), 437(72), 442 Ott, E., 548(145), 581 Ovchinnikov, Yu B., 189(8), 192 Oxtoby, D. W., 394(25), 402 Ozier, I., 423(55), 442 Ozorio de Almeida. A. M., 500(32), 509(49), 546(32, 49), 557(49), 577 Pack, R. T., 538(131), 580 Palit, D. K., 395(42), 399(42), 403 Pandey, A., 505(44), 577, 772(67), 785 Papazyan, A., 142(2), 145(2), 173-174(2), 178, 394(34), 403 Papiz, M. Z., 158(33), 180 Papousek, D., 496(17), 498(17), 576 Paramonov, G. K., 79(7), 79-80, 274(4-5). 275, 328(13-18, 21, 23), 329(14-16, 18, 21, 23, 26-27), 330(14), 331(26), 332(15-16, 18, 21), 334(13, 18, 23), 335(14, 17-18, 21). 336(14, 18). 337(16), 339(13-18, 21, 23), 340(26), 341(14-17, 23, 26-27), 342, 373(3), 374, 375(3), 377(1), 377, 379 Parasuk, V., 281(4), 282, 329(27), 341(27), 342 Park, H. K., 669(23), 697 Park, N. S., 395(41), 400(41), 403 Park, S. M., 57(25), 76, 216(12-13), 270, 286(5), 292 Parkin, J. E.. 412(18), 440 Pannenter, C. S., 412(15), 440 Passino, S., 173(49), 180 Passino, S. A., 146(14), 169(39), 171(39), 179-180 Pastrana, M. R., 752(39), 784 Pate, B. H., 454(2), 455 Patel, J. S., 59(29), 76 Patterson, M. R., 634(48), 646 Paye, J., 346(11-12), 370-371 Peatrnan, W. B., 612(6-8), 623, 626(7), 629(7), 645 Pechukas, P., 519(90), 545(143), 548(143), 550(143), 554(143), 560(143), 571(143), 579, 581, 776(69), 785, 854(5b), 867 Pedersen, S . , 42, 56(20), 76, 90(1), 90, 328(10), 339-340(10), 341,399(47), 403 Peirce, A., 248-249(51), 272, 316(4), 322 Peirce, A. P., 215(8-9), 227(8), 236(8-9),

AUTHOR INDEX

249-250(54), 270, 272, 328(8). 339(8). 341 Peng, L. W., 41 Pipin, C., 485(15),490 Percival, I. C., 519(93), 579 Pemg, B.-C., 142(6), 145(6). 173(6), 179 Persch, G.,493(5), 518(5), 528(5), 537(5), 540(5), 575 Person, M. D., 730(3), 741 Peskin, U.,759(41), 779(73), 784-785 Peslherbe, G.H.. 841(29), 848 Peteanu, L. A., 146(16-17), 151-152(17), 178(17), 179 Peters, E.-M., 339(35), 343 Peters, K.,339(35), 343 Petrov, V., 339(36), 343 Petsko, G.,405(3), 406 Petto, J . P., 434(81). 441 Pezler, B., 820(12), 847 Heifer, P., 93(6), 93 Phelps, D.K.,394(28), 399(28), 402 Phillips, D.,379(9), 381 Phillips, Jr., G. N., 405(4),406 Phillips, W. D., 302(9), 305(9), 307(9), 312 Physics Today, 382(1). 385 Pibel, C. D., 791(7,9), 792(9), 794(7), 796-797 Pilling, M. J., 515(71),539(71),578 Pique, J . 2 , 493(7), 518(85), 526(116-117), 527( 117). 528(7), 528(119). 575, 578, 580,597-598(3), 598, 642-643(58). 646 Pittner, J., 114-115(IO), 117(10). 131 Pliva, J., 465(6),486(6), 490 Plohn, H.,201(12), 202, 868(10),869 Polanyi, J . C., 4(10), 43, 296(8), 300 PolQek. M., 329(27), 341(27). 342 Polik, W.F.,465(34), 467(3), 488(34), 490, 493(4),536(125), 541(4, 138-139). 575, 580, 750(29), 782(74-76). 784-785, 812(1), 812, 849(3), 849 Pollak. E., 393(16, 18). 402, 545(143), 548( 143). S O ( 1431,554( 143), 560( 143), 571( 143). 581, 772(63), 785 Pollard, W. T., 146(16), 179, 196(4), 196, 382(18),385 Pollicott, M., 514(63), 578 Pomphrey, N., 519(93),579 Portella-Oberli. M.-T., 714(14). 715 Porter, C.E., 505(44), 516(73), 518-5 19(73),577-578

917

Potapov, V. K., 66I(6-8), 662 Potter, E.D., 42, 56(20), 76 Potter, E. P, 90(1), 90,328(10), 339-340(10), 341 Powis, I., 668(3), 697 Pratt, S. T., 434(80), 441, 459(1), 459, 629(38), 646, 668(16), 681( 16), 697, 706(27-30), 708 Press, W. H.,332(30).342 Primas, H.,93(6), 93 Prince, S. M., 158133). 180 Pritchard, H.0.. 493(8). 575 Provost, D., 521(109), 579 Pshenichnikov, M. S., 348(21), 371 Puget, P., 485(15), 490 Pugliano, N., 395(42), 399(42), 403 Pullerits, T.,146(19), 152(19), 160(19), 179 Pullman, B., 454-455(1),455, 588(5), 589 Qian, C. X. W., 767(57), 785 Qiu, P., 86(2), 87 Quack, M., 84(1-2). 84, 93(2-5, 7). 93, 377(24), 378(4), 379(7-8). 379, 380(6), 443(1),443, 451(1-4),453(4-8). 453, 454(1,3). 454, 45X1-6). 455-456, 537(126), 539(134), 580, 587(14), 588(5-7). 588-589, 590(1), 59J, 595( l-2), 595, 750(24), 779(7&71), 784-785, 820(5), 835(5, 24). 847 Quaid et al., 716 Quast, H., 329126). 331(26), 339(35), 340-341(26), 343 Quiniales, L. A. M., 752(39), 784 Quint, W.,379(9), 381 Rabalais, J. W., 791(5), 796 Rabani, E., 434(85), 437(85),442, 626(3), 627(3c, 5), 628(3c), 629(2, 3a), 634(2, 3, 5). 636(3c, 3,639(3b), 643(3a. 3d). 645,650,659, 668(14),681-682(14), 697698, 724(1), 724 Rabinovich, S., 393-394(23), 402 Rabitz, H.,215(7-10, 11). 218(7,21-23), 226(7-8), 236(8-9). 248(51),249(5I, 54, 56), 250(56-59), 251(57-58, 60). 252(59), 268-269(76-78), 270-272, 274(3), 275, 281(3), 281,286(2), 292, 302(3), 312, 315(1-2), 316(1, 4). 317(5-6), 318V-8). 319(1, IOa, lob),

918

AUTHOR INDEX

Rabitz, H. (Continued) 32&321(13), 322, 328(8), 339(8), 341, 346(5), 370, 373(2), 373 Rabitz, H. J., 48(5-9), 75 Radzewicz, C., 800(3, 5 ) , 806 Raftery, D., 395(46). 403 Rai, S. N., 835(25), 847 Raineri, F. 0..142(6), 145(6), 173(6), 179 Rajaram, B., 465(1, 5), 467(1), 484-489(5), 490

Rakowsky, S., 72(52), 77 Raksi, F., 18(15), 43, 274(8), 275(1), 275-276, 302(5), 312, 346(6), 370 Ramakrishna, V.,218(22), 248-249(51), 268-269(77), 271-272, 315(2), 322 Ramillon, M., 820(13), 847 Rankin, C., 778(69), 785 Raoult, M., 678(36), 697, 706(23, 26). 707(33), 708 Rappaport, F., 146(15), 160(15), 179 Rasaiah, J . C., 394(24), 402 Raseev, G., 746(7), 783 Raski, F., 235(30), 265(30), 271 Ratner, M. A., 393(16), 402, 868(5), 868 Raymer, M. G., 346(15), 371 Redfield, A. G., 147(26), 179 Refaey, K., 679(37), 697, 699 Reid, S. A., 779(72), 782(79-80), 785, 794( 18b). 797 Reilly, J. P., 617(19), 623 Reimers, Th., 129(24), 132 Reinginger, R., 7 13(12). 715 Reinisch, L., 405(1-2). 405406 Reinot, T., 171(41), 180 Reischl, B., 79(8-9). 80,1l7( 13, 15). 118(13), 120-121(15), 122(13, 15). 131, 132(1), 133(4-5, 7), 134(1,4-5). 135(4-5), 135, 136(4-5), 137, 196(1-2). 196, 197(1-2), 200(6), 201, 202(1, 3). 203(4-5). 203 Reischl-Lenz, B., 133(6), 135(6), 137, 203(5), 203 Reiser, G., 434(74), 441, 615(11), 619(21-22). 620(22), 623, 659(1), 659, 668(6), 676(34), 697, 701(1), 707 Reisler, H., 746(10), 759(41), 765(10), 767(57-58), 778(10), 779( 10, 72-73), 782(79-80), 783-785, 794( 18a, 18b), 797 Reitze, D. H., 59(30), 76

Remade, F., 434(85), 437(85), 442, 541(140), 580,631(45), 634(45a), 636(45a, 45b. 45f. 51). 639(45c), 640(45), 641(51), 642(45b), 643(45b, 45d). 646, 649, 652, 668(18), 681-682(18), 691(18), 697 Repinec, S. T.,395(42), 399(42), 403 Resat, H., 142(6), 145(6), 173(6), 179 Retterling, W. T., 332(30), 342 Reuss, J., 423(58), 440 Reynaud, S.,382( lo), 385 Reynolds, A. H., 405( I), 405 Rhodes, W.,382(16), 385 Rice, 0. K., 410(1), 439 Rice, S., 57(21), 59(21), 76, 317(6), 322 Rice, S. A., 48(34, lo), 59(4, 10, 28). 75-76, 78-79(2), 79, W(4-5). 90. 196(10), 198, 200(5), 201, 215(4-6), 216(4-5), 217(18), 218(6, 19-20, 24). 226(4, 6), 228(5), 231(6), 233(6), 241(18), 246(20), 249(52, 55), 250(55), 253(20 61), 255(20), 257-258(20), 262(24), 270-272, 273-274(1), 274, 282(1), 286(2), 291, 302(18), 303(11), 312-313, 328(6), 339-340(6), 341, 346(1), 370, 373(1). 373, 412(17), 418(40), 419(49), 440, 458(1), 458, 501(33), 514(33), 517(33), 519(95, 102). 520(103), 542-543(33), 555( 149), 559-560(33), 565(33), 579, 581, 636(54), 642(54), 646, 875(4), 886 Richard, E., 453(6), 453 Riedle, E., 410(3-4), 413(3-4, 19-22), 414(34, 24, 27). 415(3, 28-29), 416(31, 37). 417(29), 418(37), 428(22), 431(65), 432(68), 435(22), 439-441 Rieger, D., 619(21), 623 Rigrod, W. W., 187(6), 191 Ring, H., 528( 119). 580 Rinneberg, H., 510(51), 577 Rips, I., 393(15), 402 Roberts, G., 561(151), 566(151), 581, 799(2), 806 Roberts, R., 279(1), 280 Robertson, S. H., 515(71), 539(71), 578 Robie, D. C., 782(79-80). 785, 794(18b), 797 Robinson, G. W., 412(16), 419(43), 440 Robinson, P.J., 750(26), 784 Roche, A. L., 707(31), 708

AUTHOR INDEX Rodgers, D., 669(26), 697 Rohlfing, E. A., 759(42),761(42, 49). 763(42), 784 Romelt, J., 545(143), 548(143). 550(143), 554(143), 560(143),571(143),581, 812(3), 812, 849(2), 849 Romero-Rochin, V., 57(21), 59(21), 76, 217(18), 242(18), 271, 303(11), 312 Ronkin, J., 505(44),577 Rose, T. S., 41, 522(115),525-526(115), 579 Rose-Petruck, C.. 235(30), 265(30), 271, 274(8), 275(1). 275-276, 346(6), 370 Rosenblit, M.,714(13), 715 Rosenthal, S.J., 142(3), 145(3), 173(3), 179, 394(31, 34).403 Rosker, M.J., 41, 90(6), 90, 522(115), 525-526(115),579 Rosmus, P., 528(119),580, 748-749(17), 751(17, 32-33), 752(32-33), 756(32), 761(32-33), 763(33),764-767(32), 768-771(17). 783-784 Ross, G.C., 746(7),783 Ross, S . C., 490. 704(11), 706(30), 708 Rosums, P., 328-329(20), 332(20),335(20), 339(20), 342 Rothenberger, G.,393( I I), 401 Rottke, H., 493(11), 510(11),576 Rouben, D.C., 521(110), 526(110).579 Rouleau, G..379(9), 381 Roy, S.,142(5), 145(4), 172(5), 179. 394(34),403 Rubahn, H. G.,423(56),441 Rudecki, P., 328(9), 339(9), 341, 425(62), 441 Rudolph, H., 617-618(18), 623 Ruelle, D., 514(63),578 Ruff, A., 124-125(22), 126(23), 128-129(23), 131 Ruggiero, A. J., 57(21), 59(21), 76, 217(18), 242(18),271, 303(11), 312 Ruhman, S., 79(4), 79, 173(47),180, 196(3, 8-9). 196, 198 Ruscic, B., 610(3),623 Russek, A., 634(48),646 Russell, M.E., 679(37),697, 699 Rutz, S., 79(8-9), 80, 104-106(7), 1 I I@), ll4-ll5(9), I17(13. 15), 118(13), 120(15), 121(15, 19). 122(13, 15), 123(21), 124(21-22), 124(8), 125(22),

919

126(23), 128-129(23), 131, 132(1-2). 133(4,7). 134(1. 4). 135(1,4,7),135, 136(1, 4). 137, 196(1-2), 196, 197(1-2), 200(6), 201, 202( l-3), 20314-5,7), 203-204, 657(1), 657 Ryaboy, V., 760(45), 784 Saalfrank, P., 79(6), 79, 203(5), 203, 333(33),343 Sadeghi, R., 538(132),571-572(132), 580, 795(19), 797 Saher, D.,519(97), 541(140), 579-580 Saito, S., 145(11), 172(11), 176(11),179 Sakimoto, K.,686-687(42),698, 820(lo), 847 Salamon, P., 239(43), 271 Salapaka, M.V., 248-249(51),272 Saleh, B. E., 382(4), 385 Saltiel, J., 395(41),400(41) Samson, A. M.,274(4), 275, 328(13), 332(13),334(13),339(13), 342 Sander, M., 616(14).617-618(17), 623, 676(35), 678(35), 697 Sargent, 111, M., 243(44), 271 Sarkisian, A. A., 327(3), 339(3), 341 Sartakov, B.,423(58), 441 Sassara, A., 712(9),715 Satchler, G.R., 745(3), 783 Sathyamurthy, N.,201(18),202, 332(27), 342 Savva, V. A,, 274(4). 275, 328(13-14,23). 329(14,23). 330(14),332(13), 334(13, 23). 335-336(14), 339(14, 13, 23). 341(14, 23). 342 Schaeffer, 111, H. F.,536(124), 580 Schafer, F. P., 328(11), 339(11),341 Schalg, E. W., 663 Scharf, D., 81-82(1), 82, 711-714(4), 715 Schatz, G.C., 86(2),87 Scherer, N. F., 41, 57(21), 59(21),76, 86(I), 87, 145-145(28), 152(28), 172(44), 179-180, 217(18), 242(18),271, 303(11), 312, 346(4), 348(20,22), 370-371, 394(31),403, 807 Scherr, V., 791(5),796 Scherzer, W.G., 615(12),623,626(9, 14-15), 629(9,15, 36). 630(36),645446, 682(38),697 Scheurer.. C... 333(33). . ,. 343 Schichida, Y.,405(5),406

920

AUTHOR INDEX

Schiemann, S., 328(9), 339(9), 341. 423(56), 425(6243), 441 Schilcher, R. R., 382(7), 385 Schinke, R., 274(10), 275, 327(1, 3), 339(1, 3). 341, 373(6), 374, 484(13), 486(16), 490, 565(153), 566(159), 570(153), 573(159), 581, 730(1), 741, 745(4), 746(4, 8-10), 747-748(16-17, 34), 749(17, 20), 750(20), 751(16-17, 32-37), 752(4, 32-33, 38). 753(32, 34). 754(4, 32-34), 755(4), 756(32), 758(34, 36-37), 760(34), 761(16, 32-33), 762(8), 763(33), 764(4, 32). 765(9-10, 32, 55-56), 766(32), 767(4, 32, 56, 58-59), 768(17, 60-62). 769(16-17, 38), 770(16--17). 771(17), 772(8, 34), 774(34, 36). 775(36-37). 776-777(37), 778(4, 9-10, 37). 779(4, 10, 47). 780-781(37), 783-785,786(34), 790(4), 794(18a), 796797, 812(4), 812, 815(1), 815 Schlag, E. W., 413(19), 414(23-24, 27). 416(34, 37). 418(37), 434(74), 440-441, 610-61 1(5), 612(7-8). 615(11-13), 616( 14-1 6). 6 17-618(17), 619(21-22), 620(22, 25). 623, 626(6a, 7, 20-21). 627(21), 629(6-7), 634(20-21). 645-646, 656, 659( 1). 659, 668( I , 6), 676(35), 678(35), 682(38), 696-697,701(1), 707 Schlautmann, M., 320( 1I), 322 Schleich, W., 382(6), 385 Schlichting, I., 405(4), 406 Schmidt, B., 87, 89(4), 274(5), 275, 328-329(16), 332(16), 337(16), 339( 16), 341(16), 342 Schmidt, C.,517(77), 578 Schmidt, I., 288(18), 292, 803(14), 806 Schmidt, M., 129(24), 132 Schmidt, R., 657(1), 657 Schoenlein, R. W., 146(17), 151-152(17), 178(17), 179 Schon, J., 203(6), 203 Schor, H. H. R., 200(3), 201, 281(1), 281, 328(12), 339(12), 342, 849(2), 849 Schreiber, E.. 79(8-9), 80, 104-106(7), 111(8), 114-115(9), 117(13, 15), 118(13), 120(15), 121(15, 19). 122(13, 15). 123(21), 124(8, 21-22), 125(22), 126(23), 128-129(23), 131, 132(1-2). 133(4, 7). 134(1, 4), 135, 135(1-2, 5), 136(1,4), 137, 196(1-2), 196, 197(1-2), 200(6),

201, 202(1-3). 203(4-5, 7). 203-204, 657(1), 657 Schreiber, M., 333(33), 343 Schreier, H.-J., 274(7), 275, 281(3), 281, 328(8, 23). 329(23)), 334(23), 339(8, 23), 341(23), 341, 373(2), 373, 761(53), 785 Schriider, T., 748-749(17), 751(17), 768-771(17), 783 Schriidinger, E., 52(15), 76,92(1), 93 Schroeder, J., 392(5), 393(8), 395(5, 45). 399(8), 401, 403,407 Schubert, U., 410(4), 413(4, 19, 21). 414(4, 27). 439440 Schuss, A., 393(16), 402 Schuss, Z., 393(16), 402 Schutte. C., 329(27), 341(27), 342 Schwartz, S. D., 854(2), 867, 868(2), 868 Schwartzer, D., 393(8), 399(8), 401 Schwarzer, D., 407 Schweizer. W., 519(91), 579 Schwenke, D. W., 493(9). 538(9), 575 Schwentner, N., 59(35), 76, 273(1), 274(8), 274-27.5, 711(1, 3), 712(1, 3, 6-7). 713(10-11), 714-715 Scoles, G., 454(2), 455 Scott, J. L., 768(60), 785 Scully, M. O., 243(44), 271, 302(8), 312 Sears, T.J., 747(14), 783 Seaton, M. J., 686(39), 698, 703(8), 708 Seba, P., 517(79), 578 Segev, E., 274(10), 275, 327(1), 339(1), 341, 373(6), 374 Seideman, T.,296(4), 300, 854-855(3), 857(3), 859(11), 860(3), 861(11), 862(3b) 865(3a), 867,868(3, 3, 868 Seifert, F., 339(36), 343 Sekatsky, S., 880(9), 883(9), 886 Sekreta, E., 617(19), 623 Sekyia, H., 617(20), 623 Seligman, T. H., 538(129), 580 Sellers, I? V., 849(1), 849 Selzle, H. L., 416(34), 440, 61M11(5), 615(12), 623. 626(6b, 14). 629(6, 36), 630(36), 634(3), 645-646, 682(38), 697 Sension, R. J., 395(42, 46). 399(42), 403 Seplilveda, M. A., 504(39), 514(65), 577-578, 862-863( 18). 868 Setser, D. W., 849(1), 849 Seyfried, V., 55(19), 60(37), 64143). 76, 79(5), 79

AUTHOR INDEX Shank, C. V., 60(36), 62(36), 76, 146(16-1 7). 151-1 52(17). 178(17), I79 Shapiro, M., 48(1-2). 57(1), 75, 2 15-216( 1-3). 219( 1, 26). 221-223(27), 224(26-27), 249(53), 270(79), 270-272. 274(1, lo), 274,286(1, 7, 14). 287(16), 288( 14), 291-292, 295( 1-2). 296(3-7). 297(2,5), 300(5-6), 300,302(2), 312, 315(3), 319(3), 322,327(1-2). 339(1-2). 341,373(4, 6, 8), 374,381(2), 382, 419(47-48), 440,505(44), 577,651, 800(6-8), 801(10), 803(12), 804(15), 806 Sharpless, R. L., 791(8), 796 Shaw, J., 493(11). 510(11), 576 Shayegan, M., 315(2), 322 Sheeny, B., 286( 1I), 292 Shehadeh, R., 286(12), 292 Shen, Y. R., 338(34), 343 Shepelyansky, D. L., 583( 1-2), 584(4), 584-585, 584(6), 585 Shi, S., 48(5, 7-43). 75,215(7, lo), 218(7), 227(7), 270,274(3), 275,281(3), 281, 286(2), 291,328(8), 339(8), 34J.373(2). 3 73 Shibanov, A. N.,883(10), 886 Shida, T., 734(17), 741 Shil'nikov, L. P., 551(147), 581 Shimamura, I., 538-539(133), 580 Shnirelman, A. I., 505(45), 577 Shoemaker, D. P., 143(53), 180 Shore, B. W., 255(62), 272,423(54), 425(54), 441,855(8), 867 Sibert, 111, E. L., 590 Sidorov, A. I., 189(8), 191 Sieniutycz, S., 239(43), 271 Sigel, M., 189-190(9), 191 Siglow, K., 435436(89), 438(89), 442 Silky, R. J., 465( I), 467( I), 490 Simon, J. D., 394(30), 399(30), 402403 Simons, B. D., 518(88), 519(92), 578-579 Sinai, Y.G.,516(75). 518(75), 578 Singh, H., 218(21), 268-269(76), 271-272, 318(7), 322 Sinha, A., 327(3), 339(3), 341,373(6), 374 Sipes, C., 86(1), 87 Sitja. G., 493(7), 526(116-117), 527(117), 528( 119). 528(7), 575,580,597-598(3), 598 Skodje, R. T., 259(68), 272,538(132), 557(150), 571-572(132), 580-581

921

Skoje, R. T., 795(19), 797 Slanger, T. G., 791(8), 7% Slator, T., 321(14). 322 Sleva, E. T., 39-40,304(13), 312 Sloan. I. I., 849(1), 849 Smale, S., 552(148), 581 Small, G. J., 171(41), 180 Smilansky, U., 511(57), 519(91), 528(57), 577,579 Smith, 111, A. B., 400(49), 403 Smith, A. M.,415(29), 417(29), 440 Smith, A. V.,57(24). 76,286(9), 292 Smith, B. C.,465(8), 467(8), 488(8), 490 Smith, E. W.,87,89(7) Smith, J. M., 626(8), 629(8), 634(8), 645, 668(8, 17), 669(17). 681(17), 692(17), 697 Smith, M. A,, 296(8), 300 Smith, S. C.,750(28), 784,843(35), 848 Smithey, D. T., 346(15), 371 Sobol, P. E., 705(17), 708 Soep, 8.. 86(2), 87 Softley, T.P., 434(75), 441,610(4), 623,639(56), 646,668(2), 669(24), 670(28-29), 67 1(29), 672(29-30). 675(2, 33). 684-685(24). 687(4345), 688(43, 45). 689(43), 691(47), 696(50), 697698, 707(35), 708 Sokol, D. W., 584(5), 585 Sokolov, V. V..541(140), 580 Solina, S. A. B., 465( 1-4). 467( I , 3). 476(2), 488(3-4), 490,536(125), 580 Solter, D., 76356). 767(56), 785 Sommerer, G., 114-1 15(9), 124-125(22), 131,657(1), 657 Sommers, H.-J., 519(101), 579 Song, T.T.,876(6). 884(6), 886 Sorensen, L. B., 405( I), 405 Sparpaglione, M., 394(35), 403 Spath, B. W., 865(22), 868 Sprik, M., 394(33), 403 Squiet, J., 59(34), 76 Srednicki, M., 505(46), 577 Stace, A. J., 849(1), 849 Staemmler, V., 566(159), 573(159), 581 Stark, G., 705(13). 707(13), 708 Steckler, R., 259(71). 272 Steiger, A., 200(2), 201 Steil, G., 517(78), 578 Stein, J., 519(99), 579

922

AUTHOR INDEX

Steiner, F., 517(78, 81). 518(87), 578 Stephens, J. A,, 706(25), 708, 726(3), 726 Stephenson, T. A., 418(40), 440 Stepp, H., 413(20), 440 Stevens, W. J., 87, 89(6) Stickland, R. J., 726(1), 726 Stiick, C., 747(16), 748-749(16-17), 749( l7), 751( 16-17), 761( l6), 768( 171, 769(16-17), 770(16-17). 771(17), 783 Stock, G., 201(16), 202, 868(9), 869 Stkkmann, H.-M., 519(99), 579 Stoeckel, F., 521(111), 528(111), 579 Stoecklin, T. S.. 820(4), 847 Stoffregen, U., 519(99), 579 Stohner, J., 454(3), 455 Stolow, A., 296(8), 300, 668(7), 682(7), 697 Stolte, S., 730(2), 741 Stolz, H., 346(16), 361(16), 371 Stranges, D., 86(2), 87 Stratt, R., 145(11), 172(11), 179 Stratt, R. M., 145(12-13), 172(12-13), 173(13), 176(12-13), 179, 181(1), 181 Strickland, D., 374(1), 376(1), 377 Stuchebrukhov, A. A,, 451(1), 451, 454(1), 454

Stumpf, M., 486(16), 490, 746(8), 747(34), 748-749(17), 751(17, 32-37), 752(32-33, 38). 753(32, 34). 754(32, 34). 75602). 758(34, 36-37), 760(34), 761(32-33), 762(8), 763(33), 764-767(32), 768( 17). 769(17, 38). 770(17). 771(8, 17). 772(34), 774134, 36). 775(36), 776-781(37), 783-784, 786(34), 812(4), 812, 815(1), 815 StUmS, W. G., 705(17-20), 708 Su, S. G., 394(30), 399(30), 403 SU. T., 820(1), 828-830(1), 847 Subbotin, M. V., 185(4), 189(4), 191 Sudarshan, E. C. G., 238(37), 271 Suhm, M. A., 201(17), 202 Sumi, H., 393-394(20), 395(40), 399(40), 402403,406( I), 406 Sun, Y.-P., 395(41), 400(41), 403 Sundbexg, R. L., 493(3), 575 Sundstrom, V., 146(19), 152(19), 160(19), 179 Sung, I. P., 849(1), 849 Suominen, K-A., 4(12), 43 Sussmann, R., 410(5-6), 415(29), 416(3@31), 417(29), 420(5-6, 30).

424-425(6), 427(6), 428(6, 30). 43 l(30). 433(69-70). 439-441 Sutcliffe, E., 595(1-2), 595 Suter, H. U., 746(8), 765(8), 767(58), 778(8), 783, 785, 794(18a), 797 Suzuki, T., 422(53), 441 Svelto, O., 61(39), 76 Swamy, K. N., 750(22), 784 Swanson, J. A., 393(16), 402 Sweetser, J., 800(3), 806 Swimrn, R. T., 497(22), 576 Swwet, R. M., 405(4), 406 Syage, J. A., 41 Szabo, G., 250(59), 252(59), 272 Szakics, T., 317(6), 322 Szarka, A. Z., 395(42), 399(42), 403

Taatjes, C. A., 730(2), 741 Tabche-Fouhaile, A,, 612(9), 623 Tabor, M., 493(13), 506-507(13), 519(94), 576, 579 Takahashi, A., 91(1), 91 Takami, T., 519(97), 579 Takayanagi, K., 820( lo), 847 Takayanagi, M., 731(9), 741 Talkner, P., 392(3), 401 Tanaka, I., 4(9), 43 Tang, H., 218(20), 246(20), 253(20), 255(20), 257(20), 258(20), 271 Tanimura, Y., 182(1), 182 Tannor, D., 233-234, 239(40), 271, 304(19), 313 Tannor, D. J., 48(34), 59(4, 28). 75-76, 78-79(2), 79, 90(4), 90, 138, 196(5, 7, 10). 196, 198, 200(5), 201, 204(1), 206, 215-216(4-6), 218(6), 226(4, 6), 228(5), 23116). 235-236(6), 270, 273(2), 274, 282, 286(2), 291, 302(4, 17, 18), 312-313, 317(6), 322, 328(6), 339-340(6), 341, 346(1), 370, 373(1), 373,419(49), 440, 458(1), 458 Tardy, D. C., 849(1), 849 Tarn, T. J., 247(46-50), 248(47), 268(48), 2 72 Tarr, A. W., 296(8), 300 Tarrago, G., 486(17), 490 Taubes, G., 892 Taylor, H. S., 557(150), 581, 760(46), 776(68), 784-785 Teich, M. C., 382(4), 385

AUTHOR INDEX Temps, F., 747(16), 748(16-17), 749(17), 75l( 16-17), 76l( 16), 768( 17). 769-770(1&17), 771(17), 782(77-78), 783, 785 ten Wilde, A., 59(31), 76 Terasaki, A., 382(20), 385 Tersigni, S . , 59(28), 76, 215(6), 218(6), 227(6). 23 1(6), 235-236(6), 249(52), 270, 272,286(2), 291, 302(18), 313, 317(6), 322,458( I), 458 Teshef, T., 855(8), 867 Teukolsky, S . A., 332(30), 342 Thalweiser, R., 52(17-18). 63(41), 65(18), 72(54), 76-77, 78(1), 79, 90(2), 90, 103(6), 117(14), 131, 135(8), 137, 217(17), 229-230117). 271, 328(10), 339-340(10), 341 Thiele, E., 596(2), 598(2), 598 Thimng, W., 495(15), 576 Thirumalai, D., 855(7), 867 Thomas, R. G., 505(44), 538-539(133), 577,580 Ticich, T. M., 327(3), 339(3), 341, 373(6), 3 74 Tobiason, I. D., 759(42), 761(49), 784 Toglhofer, K., 626(28), 646 Tominaga, K., 394(29), 399(29), 402 Tominga, K., 394(36), 403 Tomkins, F. S., 706(27-29), 708 Tomsovic, S., 504(39), 577 Tornehave, H., 520(105), 579 T6th. G. I., 250(59), 252(59), 272, 319(1Ob), 322 Trebino, R., 59(33-34), 76, 346(3, 9-10), 362(3), 370 Trentelman, K.. 216(15), 223(15), 225(15), 271, 286(13), 292, 328(4), 339(4), 341 Troe, I., 392(5), 393(8), 395(5, 45). 399(8), 401-403, 407, 539(134), 580, 750(24-25), 779(70-71), 784, 815(2), 81.5, 820(3, 5-8, 15-17), 821(17), 822(3, 7-8, 15-16), 823(15, 18), 824(15), 827(7), 828(3, 6-7, 15), 829(3, IS), 830-831(15), 832(3, 15, 19). 833(15), 835(5,20-22,24), 842(16), 843(32-37). 844-845(33), 846(3, 8, 15-16, 34, 37-39), 847-848 Tromp, I. W., 201(11), 202, 854(2), 868(2), 867-868 Truhlar, D.G., 259(68-7 I), 262(72-73),

923

264(75), 272, 493(9), 538(9), 575, 7435). 783, 835(25), 847, 854(5), 867 Tsuchiya, S., 465(1), 467(1), 490, 791(11), 794(16), 797 Tsuji et al., 716 Tucker, S., 393(18), 402 Turner, D. W., 609(2). 616(2), 623 Turulski, I., 820(12, 14). 847 Udaltsov, V. S., 394(39), 403 Ulbricht, M.,102-103(1), 131 Ulmer, G., 626(23), 645 Ungar, H., 610-611(5), 623, 626(13), 629( 13). 645 Untch, A., 746(8), 765(8, 55-56), 767(56), 768(62), 778(8), 783, 785 Ushakov. V. G.. 820(3), 822(3), 828-829(3), 832(3), 843(34, 36), 846(3, 34, 38). 847-848

Vaida, V., 791-792(6), 795-796 Valachovic, L., 86(2), 87 Van Craen, J. C., 529(122), 580 van der Avoird, Ad, 432(67-68), 441 van der Velt, T., 493(11), 510(11), 537(6), 5 76 van der Zwan, C., 393(14), 394(14. 31). 402403 van Ede van der Pals, F?, 520(103), 521(114), 534-536(114), 579 van Grondelle, R., 146(18), 152(18). I54(18), 157-159(18), 179 van Linden van den Heuvell, H. B., 59(31), 65(47), 76-77 van Mourik, F., 146(18), 152(18), 154(18), 157-159(18), 179 Van Vleck, J. H., 703(6), 708, 861(14), 867 vander Auwera, J., 465(7), 488(7), 490, 529( 122). 580 Vander Wal, R. L.,327(3), 339(3). 341, 373(6), 374, 768(60), 785 Vansteenkiste, N., 302(10), 305(10), 307(10), 312 Varandas, A. J. C., 752(39), 784 Varkking, M. I. J., 668(7), 682(7), 697 Vassen, W., 493(11), 510(11), 576 Vegiri, A., 765(55-56), 767(56), 785 Velsko, S . P., 393(11), 401 Vereoni, M., 236(34), 271 Vervloet, M.,705(15-16). 708, 721

924

AUTHOR INDEX

Vidolova-Angelova, E. P., 662(8), 662, 663(2), 663 Vigliotti, F., 664(1), 664, 712(8-9), 715 Vilallonga, E., 315(2), 322 Villeneuve. D. M., 668(7), 682(7), 697 Vinogradov, An. V., 382( 17, 19), 385 Viriot, M. L., 279(2), 280 Visticot, J. P., 86(2), 87 Viswanathan, K. S., 617(19), 623 Vohringer, P., 348(20, 22), 371, 393(8), 399(8), 401 Volpi, G. G., 86(2). 87 von Bargen, A., 415(28), 440 von Dirke, M., 566(159), 573(159), 581, 765(56), 767(56, 58-59), 785, 794(18a), 797 von Neumann, J., 238(35), 271 von Oppen, F., 519(100), 579 von Puttkamer, K., 454-455(1), 455 von Schnering, H. G., 339(35), 343 Voros, A., 505(42), 520(36), 577 Vos, M. H., 146(15), 160(15), 179 Voss, F., 407 Vrakking, M. J. J., 434(86-87), 437(86-87). 442, 626(17), 629(17), 634(17), 645, 668(15), 681(15), 697 Waldeck, D. H., 393(11), 395(41, 43), 399(43), 400(41). 401403 Waldeck, J. R., 804(15), 806 Walker, B., 286( 1 I), 292 Walker, G . C., 394(27), 399(27), 402 Wallace, S., 693(49), 698 Walls, D. F., 382(2), 385 Walmsley, I. A., 360(25), 371, 800(3-5), 806 Walther, H., 542(142), 580 Wang, D., 760(43-44), 784 Wang, H., 841(30), 848 Wang, H. X., 91,91(1) Wang, J., 465(5), 484-489(5), 490 Wang, J.-K., 4 1 4 2 , 195 Wang, K.. 616(16). 623, 668(1), 696 Wang, Q., 60(36), 62(36), 76, l46(16-17), 151-152(17), 178(17), 179 Wang, X.-J., 515(70), 578 Wardlaw, D. M., 539(135), 580, 750-751(23), 784, 841(28), 842(31), 847-848

Warmuth, B., 79(7), 79-80, 328(17), 335(17), 339(17), 341(17), 342 Warren, W. S . , 40,48(9), 65(4546), 75, 77, 274(3), 275, 302(3), 312, 315-316(1), 319(1), 322, 346(5), 370, 373(2), 373 Wasilewski, Z. R., 270(80), 273, 286(10), 292 Watanabe, K., 570(156), 581, 607(1), 623 Watson, J. K. G., 485(14), 490, 529(122), 580 Watson, R., 696(50), 698 Watt, D. M., 485(12), 488(12), 490 Weaver, M. J., 394(28), 399(28), 402 Weber, Th., 413(22), 415(28-29). 416(31, 37), 417(29), 418(37,41), 428(22), 431(65), 435(22), 44W41 Wefers, M. M., 59(32), 76, 176(50), 180 Weichselbauer, G., 812(3), 812 Weide, K., 768(62), 785 Weidele, H., 626(33), 646 Weidenmann, E.. 55(19), 76 Weidenmuller, H. A., 393(16), 402 Weigert, S., 510(54), 529(54), 577, 601 Weiner, A. M., 59(29-30). 76, 346(2), 362(2), 370 Weinfurter, H., 321(14), 322 Weinkauf, R., 626(22), 645 Weiss, Ch., 823(18), 847 Weiss, U., 328(10), 339-340(10), 341 Weiss, V., 55(19), 63(41), 76, 78(1), 79, 90(2), 90, 135(8), 137, 217(17), 229(17), 230(17), 271 Welge, K. H., 493(11), 510(11), 576 Wendoloski, J. J., 731(6), 741 Wendt, H. R., 731(5), 733(5), 741 Werner, H.-J., 747(34), 748( 17), 749(17,20), 750(20), 751(17, 32-34), 752(32-33). 753-754(32, 34). 756(32), 758(34), 760(34), 761(32-33), 763(33), 764(32), 765(32, 56), 766(32), 767(32, 56), 768-771(17), 772(34), 774(34), 784 Western, C. M., 668(9), 697 Westervelt, R. A., 348(20), 371 Weston, T., 596(1). 598 Wheeler, J. A., 382(6), 385 Whetten, R., 117(12), 131 Whetten, R. L., 68(50), 72(50-51, 53). 73(53), 77, 133(3), 135(3), 135, 203(8), 204, 433(72), 437(72), 441

AUTHOR INDEX

White, A. M., 412(15), 440 Whitnell, R., 317(6), 322 Whitnell, R. M., 18(15), 41.43, 59(35), 76, 235(28-3 1). 245(28), 265(28-3 I), 271, 273(1), 274(8), 274, 275(1), 275-276, 286(2), 292, 302(5), 312, 328(7), 339(7), 341, 346(6), 370 Wickham, A. G., 820(4), 847 Wiebrecht, J. W., 782(77), 785 Wiebusch, G., 493(1 I), 510( 1 I), 576 Wiedemann, E., 63(41), 76 Wiersma, D. A., 4(6), 43, 178(52), 180, 348(19, 21). 371 Wiesenfeld, J. M., 433(71), 441 Wieters. W., 835(22), 847 Wiggins, S., 548(145). 581 Wigner, E. P., 538-539(133), 580 Wilkie, J., 388, 518(86), 578 Willberg, D. M., 41 Willets, 870 Willetts, A., 496( 18). 576 Williamson, J., 320(12), 322 Williamson, J. C., 41-42, 86(1), 86, 185(2), 191(2), 191 Wilson, D. J., 596(2), 598(2), 598 Wilson, K., 317(6), 322 Wilson, K. R., 18(15),41, 43, 59(34-35). 76, 218(25), 235(28-32). 236(32). 245(28), 265(25, 28-32), 267(25). 271, 273(1), 274(8), 274, 275(1), 275. 276(2), 276, 286(2), 292, 302(5), 312. 328(7), 341, 346(6-7). 370, 393(8), 399(8), 401 Winkler, S., 597(3), 598 Winn, J. S., 465(8), 467(8), 488(8), 490 Winterstetter, M., 201(12), 202, 868(10), 869 Wintgen, D., 504(40). 521(106), 570(40), 577,579 Wittenmark, B., 319(9). 322 Wittig, C., 86(2). 87 Witzel, A., 339(35), 343 Wodtke, A. M.. 737-738(18), 741 Woermer, M., 339(36), 343 Wolf, E.,189(10), 192, 346(14), 371 Wolf, J. P., 68(50), 72(50, 53). 73(53), 77, 102(1-2), 103(1), 121(19), 122(2, 20). 131, 132(2), 135(2), 135, 203(7), 203-204 Wolf, S.,114-115(9), 131, 657(1), 657 Wolfrum, J., 884( 1I), 886

925

Wolynes, P. G., 173(46), 180, 393(13), 402, 642(59), 646 Wong, S. M., 505(44), 577 Wong, S. S. M., 772(67). 785 Wong, V., 360(25), 371 Wong, W. A., 835(23), 847 Woody, A., 215(7), 218(7), 227(7), 270, 274(3), 275, 281(3), 281, 328(8), 339(8), 341, 373(2), 373 Worth, G. A., 201(9), 201-202, 868(7), 868-869 Woste, L., 4(2), 18(2). 43, 68(50), 72(50-51, 53). 73(53), 77, 79(8-9). 80, 86(2), 87, 102(1-2), 103(1, 5 ) , 104-106(7), 114-115(9), 117(12-13), 118(13), 121(19). 122(2, 20). 124-125422). I26(23), 128-1 29(23), 131, 132(2), 133(3-4,7), 134(4), 135(2-4), 135, 136(4), 137, 185(1), 191(1), 191, 196(1), 196, 197(1), 200(6), 201, 202(1-2), 203(4-5, 7-8), 203-204, 657(1), 657, 874(2), 886 Wright, J. S., 341(25), 342 Wright, T. G., 619(24), 623 Wu, H., 382(5), 385 WU,L.-A., 382(5), 385 Wullert, J. R., 59(29), 76 Wunner, G., 519(91), 579 Wunsch, L., 414(23). 440 Wurz, P.,626(24), 645 Wyatt, R. E., 855(8), 867 Xantheas, S. S., 732(13), 737(13), 741 Xavier, Jr., I . M., 304(13), 312 Xie, J., 674(31), 697 Xie, X., 394(3 I), 403 Xie, X.L., 142(3), 145(3), 173(3), 179 Xie, Y., 216(13). 270, 286(5), 292 Xu, H., 668( 10). 697 Ya Zel’dovitch, B., 286(8), 292 Yakolev, V. V., Yakovlev, V. V., 18(15), 43, 59(34-35), 76, 235(30), 265(30), 271, 273(1), 274(8), 274, 275(1), 275-276, 302(5), 312, 346(6-7), 370 Yamaguchi, M., 734(17), 739(20), 741-742 Yamanouchi, K., 465(l), 467(1), 490, 64243(60), 646, 715, 791(7, 9-11), 792(9-10), 794(7), 794(16), 795(10), 796-797

926

AUTHOR INDEX

Yamashita, K., 484(13), 490, 747(34), 751(34), 752(38), 753-754(34), 758(34), 760(34), 769(38), 772(34), 784,786(34), 791-792(10). 795(10). 797 Yan, Y., 18(15), 41, 43, 59(35), 76. 273(1), 274, 286(2), 292, 302(5), 312, 317(6), 322, 328(7). 339(7), 341, 346(6), 370 Yan, Y. J., 235(28-30, 32), 236(32), 245(28), 265(28-30, 32), 271,274(8), 275(1), 275, 276(2), 276, 328(7), 339(7), 341, 357(24), 371, 394(35, 37), 403 Yan, Y. Y., 235(31), 265(31), 271 Yang, T.-S., 348(22), 371 Yartsev, A. P., 394(29), 399(29), 402, 878(8), 884( 11). 886 Yeretzian, C., 610-611(5), 623, 626(29), 646 Yin, Y-Y., 57(22, 24), 76, 286(4, 9, 12), 292 Yokoyama, A., 737(19), 742 Yoshihara, K., 394(29), 399(29), 402, 794(14a, 14b), 797 Yoshino, K., 705( 13), 707( 13). 708 Yu, J.-Y., 146(20), 164(37), 166(37), 168(37), 169(39), 171(39), 172-175(37), 179-1 80 Yu, S., 704(12). 708 Yue, K. T., 405( 1-2), 405-406 Yukawa, T., 5 19(90), 579 Yushin, Y., 382(8, 11, 14), 385 Zachariasen, F., 512(59), 578 Zadoyan, R., 146(21), 179, 373(7), 374 Zaidi, H. R., 382(13, 15). 385 Zakrzewski, J., 519(96, 98). 542(142), 579-580 Zare, R. N., 626(12), 629(12), 634(12), 644(12), 645,668(10, 12), 669(23), 674(31), 681(12), 686(12), 691(47), 697, 803(13), 806 Zeglinski, D. M., 395(43), 399(43), 403 Zeigler, L. D., 346(4), 370 Zelditch, S., 505(45), 577 Zeller, G., 519(91), 579 Zerza et al., 716 Zerza, G., 664(1), 664, 712(8-9). 715

Zewail, A. H., 4(5, 7, 10-ll), 12(13-14), 14(5,.7), 15(14), 22(5), 25-26(5), 34(5), 37(5, 7). 39(7, 11, 14). 3940, 52(16), 56(20), 65(44), 76-77. 85(1), 85, 86(1), 86-87, 90(1, 3, 6). 90, 185(2), 191(2), 191, 195, 203(9), 204, 302(1), 304(13), 312, 320(12), 322, 328(10), 339-340(10), 341, 373(7), 374, 391(1), 399(47), 4-01(1), 401-404,412(12), 4-39, 492(1), 522(115), 525-526(115), 561(151), 566(151), 575,579, 581, 789(l), 796, 799(1-2). 800(1), 806, 892 Zhang, D. H.. 86(2), 87 Zhang, H., 250-251(57), 272 Zhang, J., 274(10), 275, 286(15), 292, 327(1), 339(1), 341, 373(6), 374 Zhang, J.-S., 393(16), 402 Zhang, J. Z. H., 86(2), 87 Zhang, X., 626(8), 629(8), 634(8), 645. 668(8, 17), 669(17), 681(17), 692(17), 697 Zhao, M., 218(24), 249-250(55), 262(24, 72), 271-272 Zhao, X., 737(19), 742 Zhao, Y., 668(5), 697 Zhong, D., 41-42 Zhu, J. J., 394(24), 402 Zhu, L., 57(26), 76, 216(14-16), 223(15), 225(15), 270-271, 286(6, 13). 292, 328(4), 339(4), 341, 434(76), 441, 668-669(4), 697, 841(30), 841(28), 847, 848 Ziegler, L. D., 172(44), 180, 394(31), 403 Zimmennann, T., 493(5), 5 18(5). 528(5), 537(5), 540(5), 575, 772(65), 785 Zitserman, V. Yu., 394(21), 402 Zittel, P. F., 327(3), 339(3), 341 Zobay, O., 565(155), 570(155), 581 Zoller, P., 320(1I), 322 Zubairy, M. S., 382(7), 385 Zusman, L. D., 393(12), 402 Zwanziger, J., 133(3), 135(3), 135, 203(8), 204 Zwanzinger, J., 72(51), 77, 117(12),131 Zyczkowski, K.. 517(79), 519(96), 578579

Advances In Chemical Physics, Volume101 Edited by I. Prigogine, Stuart A. Rice Copyright © 1997 by John Wiley & Sons, Inc.

SUBJECT INDEX Ab initio calculated spectra, compared with ZEKE spectroscopy, 617-6 18 Ab initio simulations, 202-203 Absorption spectrum, OCS, VUV region, 790-79 1

Acetylene: Darling-Dennison resonance, 600 dispersed fluorescence spectra, 465-468, 602-603

electronic transitions, 602 Hamiltonian, 533 Lyapunov exponents of periodic orbits, 534, 536

order in chaotic region, 591 Poincak mappings, 533-534 polyad model, 595 spectral reorganization, 591, 593 stretching Darling-Dennison interaction, 533

vibrational motion, 530-536 vibrogram, 532 Adaptive learning algorithm, 252 Adiabatic channel: statistical calculations, 8 19-847 compared with VTST, 835-842 comparison of SACM and VTST anisotropic charge-locked permanentdipole systems, 839-841 general potentials, 841-842 isotropic charge-locked permanentdipole systems, 836-839 dissociation, specific rate channels, 832-835

number of open channels, 832-835 potential curves, 821-824 SACM applications to more complex reaction systems, 842-846 thermal capture rate constants, 823-832 threshold energies, 827 Ag2, ZEKE spectroscopy compared with photoionization efficiency, 6 1 M 1 1 Amplitude imaging. See Phase and amplitude imaging

Anharmonic resonances, 466-467, 473, 476, 488

Argon: autoionizing Rydberg states matrix diagonalization approach, 69 1-692

MQDT calculations, 689-692 total ion compared with threshold spectrum, 614, 616 Asymmetric double well, quantum dynamics, 150-1 5 1 Autocorrelation function, 51 1-512 classical behavior, 521 Fourier transform, 6 0 1 4 2 polarization operators, 365 Autocorrelation signal: gated, 348-349, 359-362 ideal, 349 Autoionimtion states, 662-663 7-Amindole, DNA base-pair model, 35-36 Barrier reactions, 22-25 Base pairs, photoinduced tautomerization, 85

Basis functions, energy-dependent, 755 Benzene: coupling between Rydberg series, 446 intramolecular coupling, 430-43 1 intramolecular dynamics. 412-415 intermediate vibrational excess energy, 414-415

low excess energy, 413-414 photoelectron compared with ZEKE spectrum, 617-619 Rydberg spectrum, 435-437 Benzene-12 complex, electron transfer reaction, 83 Benzeneliodine bimolecular reaction, 3 1-34 Berry-Tabor periodic-orbit amplitudes, 516

Berry-Tabor trace formula, 506-509, 573 CZHD, 530 CS;?, 527-528

927

928

SUBJECT INDEX

Schrginger equation, eigenstate Bifurcation: solutions, 219 associated with transition to chaos, variants, 224 545-552 Bubble formation, on Rydberg state area-preserving mappings, 545-546 excitation, 7 1 1-7 I8 periodic-orbit dividing surfaces, cage radius, 715 545-547 subcritical antipitchfork, 548-550 supercritical antipitchfork, 546-548 CaF, quantum defect/frame transformation, tangent, 550-552 720 decay modes, 631-632 Carbon clusters, delayed detachment rate, Bifurcation theory, 591 excess energy dependence, 656 Bimolecular charge-dipole capture process, Carbon dioxide: 820 Smale horseshoe, 565, 568-569 Bimolecular reactions, ground-state subcritical antipitchfork bifurcation, dynamics, 25-27 548-550 Bimolecular scattering, 295-300 ultrashort-lived resonances, 565-57 1 general superposition states, 299 Carbon ions: BohrSommerfeld orbit, effect of frequency electron emission time profiles, 654 of perturbation of core, 625-627 experimental and calculated emission rate, Boltzmann average, cumulative reaction 655 probability, 854 resonant multiphoton detachment spectra, Bom-Oppenheimer approximation, 701-702 653, 655 multichannel quantum defect theory, 7 19 time-of-flight photoelectron spectra, parameters, 65 I 654 Rydberg series, 722 Carbon monoxide, state-selected ions, Bom-Oppenheimer channels, 704 672-675 Bom-Oppenheimer Hamiltonians, 219-220 CD31, photodissociation, 730 Bom-Oppenheimer potential-energy surface, Centrifugal energy term, 823 721 CF,, symmetric stretching, 451. 453 ground/excited, 303 CF2HCI, strongly coupled CH stretching Bom-Oppenheimer regime, 623-624, and bending vibrations, tirne626-627 dependent entropy, 379-380 versus inverse Bom-Oppenheimer CH, acetylenic stretching, 454 regime, 724-725 Chaos, signatures of, 388 Bom-Oppenheimer theory, coordinateChargedipole potential, 821, 832-833, dependent electronic energies, 706 848 Bowman-Bitman-Harding potential-energy Charge transfer reactions, femtochemistry, surface, 760-761, 763 30-34 Br + I, exchange reaction, femtosecond CH3 + CO channel, 737-739 dynamics, 26-27 C2HD, vibrational motion, 529-53 1 Brillouin light-scattering spectroscopy, 193 Chemical bonding, 91-92 Brody parameter, 775 CH31, photodissociation. 742 Brumer-Shapiro method, 219-226, 276-277 Chiral molecules, symmetries, 377-379 branching ratio for formation of products, Chirped pulses, phonon squeezing, 382-384 22 1 Chromophore: key element, 283 femtosecond multiphoton ionization, molecule irradiated with two 877-880 electromagnetic fields, 220 optical transition frequency, 160 probability of forming a product Chronocyclic representation. See Wigner molecule, 220-221 wavepackets

SUBJECT INDEX Clusters: boiling off solvent molecules, 404 coherence in, 14, 16 Coherence, 7-9 control by phase-coherent multiple pulses, 9-10 in electron diffraction, 18-20 in ionization, exploiting, 450-45 1 in orientation, 11-13 in reactions, 14 in solvation, 14-16 in states of isolated molecules, 9, 1I wavepacket control, 14-1 8 Coherence transfer, 148- 149, 195 Coherent control, 47-97 above-threshold ionization process, 77 bimolecular scattering, 295-300 “bubble-type” mechanism, 82 intense laser pulses, 65-74 diabetic and adiabatic dressed states, 68, 71 direct ionization of Na2, 68-70 multiphoton ionization of Naz, 66 Rabi-type cycles, 65 signal drop for zero pump-probe delay, 66, 68, 72 strongly attenuated pulses, 69, 7 1, 73 laboratory experiment, 49-5 1 laser pulse duration effect, 63-65 linear superposition of degenerate continuum eigenstates, 296 localized vibrational motion, 95 macroscopic motions, 92 multiphoton ionization pathways, 79 phase-modulated femtosecond laser pulses, 59-63 phase-sensitive pumpprobe experiments, 57-59 practical applications, 278 primary aim, 286 pumpprobe schemes, 52-57 strategies, 78-79 two-photon ionization spectroscopy, 81 Coherent ion dip spectroscopy, 419, 425428,438439,445 experimental procedure, 428,430 experimental setup, 428-429 spectra, 430433 Coherent state, 198-200 Coherent transients, 7-8

929

Coherent vibrational motion, negative ions, 313 C+-OH system, adiabatic channel potential curves, 843 Colliding pair, kinetic energy, optical control, 296-300 Colliding pulse modelocked ring dye laser, 49, 51 C o h e a r configuration, 542-543 Collinear model, COz dissociation, 568-570 Collision energy, resolution, 679-680 Collisions, production of desired superposition states, 297 Complex liquids, relaxation time, 194 Condensed phases, 141-178 constant-temperature, 39 1-392 fluorescence Stokes shift function, 143-145 system-bath interactions, 160-175 echo spectroscopies, 165-175 line shape function, 160-164 vibrational dynamics, 148-160 experimental studies, 152-160 multilevel Redfield theory, 148-152 Continuous-wave interference control, one photon-three photon, 223-224 Continuous-wave spectroscopy, 122 Control methodology, 895-896 Cooling: nonevaporative, 305, 307-3 12 using shaped pulses, 308 Coordinate probability distributions, 152-153 Coriolis interaction, 48f3-488 Coriolis coupling, 720-721 Correlation function: expression, spontaneous light emission, 347-353 four-point, fluorescence spectra, 365-368 Coulomb explosion, 77-78 Coulomb interaction, between two charges, 634-635 Coulomb systems, Heisenberg time, 52 1 Coupling, separation of intrastate and interstate, 442 CPT symmetry violation, 377, 379

cs2:

dispersed fluorescence spectrum, 597-598 spectral decomposition of symmetricstretch wavepackets, 598-599 vibrational motion, 526-528

930

SUBJECT INDEX

Cumulative reaction probability, 853-854, 857 coordinate space representation, 862 H + H,, 859-860 H, + OH, 859, 861 hybrid representation, 864-865 semiclassical approximation, 859-866 Curvature distribution, 519 Curve-crossing processes, 177 Cycle expansion, 558 Cyclobutone-ethylene system, addition/cleavage reaction, 26-29 DABCO, experimental lifetimes, 629-630 Darling-Dennison resonance, 473-475, 591 between symmetric and antisymmetric stretch, 598, 600 D + CD$O channel, 739-740 DCO: dissociation dynamics, 768-772 excitation energies, 769-770 potential-energy surfaces, 769 stimulated emission pumping, 747-749 wave functions, contour plots, 769, 77 1 Delta function, 855 Dense fluids, coherence and solvation, 14-15 Density matrix, 805 equation, coherent population dynamics, 423-424 Density operator, microcanonical, 854, 857 Dephasing, 7-9, 171, 240 fast, 374 pure, 206 separating homogeneous from inhomogeneous, 8 Diatomic molecules: NeNePo spectroscopy, 1 1 1-1 14 vibrational motion, 524-525 Dichloroanthracene, decay lifetime, 628 para-Difiuorobenzene, photoelectron compared with ZEKE spectrum, 619 Dilution: external perturbations as source of, 644 versus trapping, high Rydberg states, 639-644 Dimer-hopping model, 158-159 Dirners, rotational temperature, 137 Dipole force, 185-1 86 Dipole moment, instantaneous, 303-304

Dipole operator, multitime correlation function, 209-210 Diradicals, role in cleavage, closure, and rotation, 26-21, 29 Direct overtone pumping, 747 Dispersed fluorescence spectrum: acetylene, 465468, 602-603 CS2,597-598 early-time dynamics of ZOBS, 472-473 extracting information, 469-470 unzipped polyads, 470472 Dissipation: in competition with vibrational transitions, 336-337 role in quantum mechanics, 204 source, 343 Dissociation: above-threshold, vibrational transitions, 336-338 HCO, rotational state distributions, 749-750 molecular, spectra, 789-790 polyatomic molecules, 632 on potentials with saddle bifurcation, associated with transition to chaos, 545-552 classical properties, 541-545 semiclassical quantization, 555-561 Smale horseshoes, 552-555 specific rate constants, adiabatic channels, 832-835 ultrafast, 574-575 see also HCO, dissociation dynamics; Unimolecular dissociation Dissociation rate: as function of energy, 751 RRKM result, 632 unimolecular, 539-54 1 Dissociative molecules, resonance constants,

566

Distortion constants, rotational and centrifugal, 809-8 10 DNA models, tautomerization reactions, 34-37 photoinduced tautomerization of base pairs, 85 Doorway wavepacket. 353-355, 357 phase-space fluorescence, 368-369 pump-probe signals, 369-370

SUBJECT INDEX Doppler broadening, eliminating, 410 Double-well, banierless, population decay, 151-152 Doubly-many-body-expansionsurface, 572 Dunham expansion, 496-497 diagonal and nondiagonal parts, 531 domain of validity, 497, 555 without anharmonic resonances, 529 Dynamics-inverse scattering duality, 267-269 Schriidinger equation, 268-269 Echo spectroscopies, 165-175 dephasing, 171 experimental setup, 165 instantaneous normal modes, 172, 181-1 82 three-pulse echo, 166-169 peak shift data, 170-1 7 1 Eckart potential barrier, 865 Effective Hamiltonian: diagonalizing, 640,642 time evolution of high Rydberg states, 636-639 Eigenfunctions: averages, periodic-orbit expression, 504-505 Berry’s conjecture, 505 Wigner transforms, 508 Eigen reaction probabilities, 857-858 Eigenstates: high Rydberg states, 639 spectrum, extracting information from, 469 Einstein A-coefficient, 801 Einstein B coefficient, generalized, 303-304 Electron: free, polarizability in optical field, 186 reflection by evanescent laser wave, 189-190 Electron beam: coherence length, 192 focusing, 187-189, 191 monochromaticity, 191-1 92 Electron diffraction, coherence, 18-20 Electronic coherence, 91 Electronic states: ground, nuclear wavepacket, “hole burning”, 195-198 pair, field-matter interactions, 145-146

931

Electronic transition frequency correlation function, normalized, 174-175 Electron-phonon coupling, 656 Electron transfer reactions: adiabatic elimination of fast variables, 393-394 barrierless, 394 dielectric dispersion of solvent, 394395 femtochemistry, 30-34 linear response approximation, 406407 in solution, 400 Sumi-Marcus treatment, 394 Electron velocity, normal component, variation, 190 Elimination, two-center, 29-30 Enantiomers, tunneling between, 38 1 Energy: Gaussian distribution, initial wavepacket, 522 levels, curvature statistics, 5 19-520 Energy difference operator, 162 Energy relaxation, 193-194 Energy shell, 507 Energy spectrum: bounded systems, 5 14-5 19 average level density. 515-516 beyond Heisenberg time, 520 periodic-orbit structures, 5 16 scale below mean spacing, 516-519 irregular, mechanisms, 537-538 open systems, 538-539 Entropy barrier, 5 17 Equilibrium points, quantization, 556-557 isolated, 496-498 Escape rate, 5 14 Escape-time function, 544 Evolution operator, 494 Excitation pathways, phase control, 274 Fabri-Perot &talon,361 Fano profile, OCS, VUV-PHOFEX spectrum, 793-795,797 Fast Fourier transform propagation techniques, 200 Feedback: in molecular control design. 316-318 quantum dynamics control, 3 15-325 effective Schriidinger equation, 3 17-3 I8

SUBJECT INDEX

932

Feedback (Conrinued) Heisenberg’s equation of motion, 320 inversion of molecular dynamics, 320-321. 323-324 laboratory control, 3 18-320 nonlinear Schriidinger equation, 3 18, 320, 322 tracking control, 3 18 Femtochemistry, 3-45, 892 barrier reactions, 22-25 complex organic reactions, 26-30 electron transfer reactions, 30-34 ground-state dynamics, 25-27 key concepts, 6 reaction dynamics, 4-5 reactions studied, 37-38 resonances in unimolecular reactions, 20-22 tautomerization reactions of DNA models, 34-37 two-center elimination, 29-30 see also Coherence, 14 Femtosecond laser pulses, phase-modulated, 59-63 Femtosecond NeNePo, 658 Femtosecond transition-state spectroscopy, 20

Femtospectrochemistry, 873-887 femtosecond multiphoton ionization of chromophores, 877-880 laser photoion microscopy, 883-884 laser resonance photoelectron spectromicroscopy, 880-883 principal idea, 875-877 Fermi bifurcation, CS2, 527 Fermi resonance, 591 Feynman diagrams: double-sided, 161-162 r > 0, 167-168 Feynman-Vernon-Hellwarth diagram, 304-307

FHS

Smale horseshoes, 554 tangent bifurcation, 550-552 Field-ion microscopy, 884-885 Filed-matter interactions, pair of electronic states, 145-146 Fluctuation-dissipation theorem, 386 Fluorescence: anisotropy function, 157-158

four-point correlation function, 365-368 phase-space doorway-window wavepackets, 368-369 rate, 801-802 Stokes shift function, 143-145, 162-163 correlation with line-broadening function, 164 time-dependent Na2 signal, 803-804 Fourier-Laplace transform, response function, 163 Fourier transform, windowed, 52 1-522, 601 Four-wave mixing, heterodyne-detected, wigner wavepackets extension to, 358-359 Fragmentation: deconvolution, 127-128 rate constants, 821 rates, 128-129 Fragmentation model, 127 Fragmentation reaction, 220-22 I Franck-Condon active normal modes, configuration space, 469, 589-590 Franck-Condon active vibrational modes, 464-465 Franck-Condon bright state, 464-465 fractionation, 470 Franck-Condon bright ZOBS, 468-470 survival probability, 477 Franck-Condon mapping, 767-768 model, 786-787 Franck-Condon principle, 63 Franck-Condon region, 325 initial excitation in, 457 Franck-Condon transition, 227, 230 squeezing and, 382 Franck-Condon window, 121, 135-136 Free radicals: reactive, 730 see also Vinoxy radical Free-rotor expression. 833 Frequency-resolved optical gating technique, 346 Gaspard-Rice lengthening of lifetimes, 560, 565 Gated autocorrelation signal, 348-349 Gaussian component, line broadening, 181 Gaussian fluctuations, classically chaotic systems, 5 18

933

SUBJECT INDEX

Green-Kub*Yamamoto-Zwanzig formulas, 505 Green's function: imaginary part, 637 initial-value representation, 862-863 onto a vector, 858 semiclassical approximation, 860-862 Wigner wavepackets, 363 Ground-state dynamics, bimolecular reactions, 25-27 Gutzwiller periodic-order amplitudes, 5 16 Gutzwiller trace formula, 573 extending to scattering, 5 10 isolated periodic orbits, 498-502

H3: periodic orbits, 6OO-601 subcritical antipitchfork bifurcation, 548-550

ultrashort-lived resonances, 57 1-572 H + H,: cumulative reaction probability, 859-860 potential energy surfaces, 855-856 H, + OH, cumulative reaction probability, 859, 861

Hamiltonian: anharmonic resonance, 590 diagonal and nondiagonal parts, 498 full rotational-vibrational, scattering resonances, 567 spectral determinant, 495 spectroscopic raising and lowering operators, 590 water, 591 stability borders, 665 Hamiltonian matrix, diagonalizing, 725 Harmonic osciIlators, 589 Hartley band, ozone, 572 Hartree approximation, time-dependent, 265-267

HCI, thermal capture rate constants, 830, 83 1

HCI,: Smale horseshoes, 554 tangent bifurcation, 55&552 HCN, thermal capture rate constants, 830-83 1

HCO: comparison of fluctuating quantum and RRKM results, 814

dissociation dynamics, 759-768, 781 bending mode, 762 CO stretching mode, 762 final-state distributions of CO fragment, 764-766

Franck-Condon mapping, 767-768 IVR process, 764, 767 resonance series as function of CO stretching quantum number, 762-763

vibrational dynamics, 761 dissociation rates, 759-760 excitation energies, 769-770 lifetime function, 756, 758-759 photoelectron spectrum, 787 potential-energy surfaces, 752-754, 769 potential well, 772 resonance wave functions, 756-757 angular dependence, 765, 767 rotational state distributions following dissociation. 749-750 HCP, stimulated emission pumping spectra, 484488

Healing process, 648-649 Heisenberg operator, 161 Heisenberg's equation of motion, 320 Heisenberg time, beyond, 520 Hermitian rate operator, 636-638 Hexane molecules, small clusters, viscosity and solvent friction, 405 HgI, coherence, 94.96 Hgl,: antipitchfork bifurcation, 561-563 higher order perturbation theory, 596 periodic orbits, 586 Smale horseshoe, 563-565 supercritical antipitchfork bifurcation, 546

ultrashort-lived resonances, 56 1-565 HgNe van der Waals dimer, Rydberg states, 7 15-7 16

HI, potential-energy diagram, 225 High-resolution rotationally resolved UV spectroscopy, van der Waals complexes, 414416 HNO: best transition state, 815 dissociation dynamics, 772-775 dissociation rates, 759-760 lifetime function, 756, 758-759

934

SUBJECT INDEX

HNO (Continued) nearest-neighbor energy spacing distributions, 772, 774 potential-energy surfaces, 752-754 potential well, 772 Huang-Tam-Clark theorem, 248 Hydrogen: Fourier transform infrared emission spectra, 721 ion-molecule reaction, 678481, 698-699 collision energy resolution, 679480 Rydberg state perturbation, 680-68 1 transmission effects, 680 pulsed-field ionization, 723-724 Rydberg states, 703-704 singlet-triplet interaction, 721 state-selected ions, 672 vibrational and rotational autoionization, 706 vibrational states, 723 IBr: photodissociation, 222-224 potential energy curve, 222 IHgI system, barrier reactions, 22-24 Incoherent interference control, 286-287, 290-29 1,293 Incoherent population dynamics, timedependent, 421 Infrared fields, very intense, 455-456 Infrared multiphoton dissociation, vinoxy radical, 738 Infrared multiple-photon excitation and dissociation, 45 1-452, 454 Integral operator, Hilbert-Schmidt type, 250 Intensity parameter, 186 Interference, between isolated periodic orbits, 502-504 Internal molecular degrees of freedom, 301-3 14 instantaneous dipole moment, 303-304 nonevaporative cooling, 305, 307-3 12 vibrational heating, using nondestructive optical cycling, 304-307 Intramolecular coupling, 642 Intramolecular dynamics, 409-439, 442459,463-489 bottlenecks, 633 coherent population dynamics, 422-424 special pulse sequences, 424-425

electronically excited S, state of benzene, 411419 effect of van der Waals bonded noblegas atoms, 414419 intermediate vibrational excess energy, 414-415 mechanism, 411-412 states at low excess energy, 413414 high Rydberg states in polyatomic molecules, 433438 experimental results. 435-438 experimental setup, 435 incoherent population dynamics, 420422 IVR control, 449-45 1 polyatomic systems, 4 10 resonance structure change, 484488 spectroscopic effective Hamiltonian model, 464-466 see also Coherent ion dip spectroscopy Intramolecular energy redistribution, 42 1 Intramolecular vibrational density redistribution, Na3(B), 134 Intramolecular vibrational-energy redistribution, 9, 11, 85-86, 103, 586-587 unzipped polyads, 473-474 Invariant set, 543-545 Inverse Born-Oppenheimer approximation, 630-63 1,647,649-65 I Inverse Born-Oppenheimer regime, versus Born-Oppenheimer regime, 724-725 Inversion algorithm, 321, 323-324 Iodjne: B state, 152-155 evolution of vibrational wavepacket, 273 in hexane curve-crossing problem, 209 solution spectra, 142-143, 145 interaction with solvent, 195 Morse-type model, vibrogram, 524-525 polarization-detected pumpprobe signal, 154 solvent-induced dissociation, 154 stretched target state, 267-268 Ionic model clusters, classical trajectory studies, 657 Ionization: exploiting coherence, 450-45 1

935

SUBJECT INDEX probability and microwave, field strength, 584-585 Ionization potential, 628 Ion-molecule reaction: hydrogen, 678-68 1, 698-699 coilision energy resolution, 679-680 Rydberg state perturbation, 680-681 transmission effects, 680 state-selected, 669672 IR144, in ethanol, three-pulse echo peak shift, 170-173 Isolated molecular dynamics, 9, 11 Isomerization, vibrational transitions, 338-340 Jahn-Teller effect, 726 Jahn-Teller interaction, 725 Jahn-Teller splitting, 624 Kaluza-Muckerman reduction scheme, 258, 26 1 Karplus-Porter surface, 571 Keldish limit, 374-376 Kicked-rotator model, 583 39*39K2 isotopomer, 105-106, I37 39*41K, isotopomer, 105-106, 137 Kramer's equation, 392-393 microviscosity, 400 Laboratory feedback control, 3 18-320 Lagrange multipliers, constraints, 245 Lambda-type double-resonance experiment, 420 Landau-von Neumann superoperator, 512-514 Langevin rate constant, 826 Laser, 893-894 importance to chemistry, 873-875 light characteristics, 875 ultrashort pulses, 48 Laser control, 185-191, 373-388 domains, 327-328 electron beam focusing, 187-1 89 enantiomers with different parities, 381 intramolecular vibrational distribution rate, 449-45 1 optimal conditions, 375, 377 parity and chirality, 377-379 product branching ratios, 458 response functions, 386-387

stability matrix, 387-388 symmeuies, time dependence, 377-378 Laser-induced continuum structure, 286 Laser-induced transparency, 302 Laser lens, spherical aberration, 189 Laser photoion microscopy, femtosecond, 883-884 Laser photoion projection microscope, 875-876 Laser pulse: duration effect, 63-65 intense, coherent control, 65-74 intensity versus duration, 896-897 ultrashort, 373 up- and down-chirped. 60-63 Laser resonance ionization spectroscopy with mass spectroscopy, 663 Laser resonance photoelectron spectromicroxopy, 880-883 Laser resonance photoion spectromicroscopy, 884-885 Laser wave, evanescent, electron reflection, 189-190 Learning algorithm, laboratory feedback control, 319 Lie algebra, 248 Lifetime function, 756, 758-759 LiF-F2 needle tip, 880, 882-883 Light emission, spontaneous: correlation function expression, 347-353 molecular density matrix expansion, 350-351 Line shape function, 160-164 Liouville operator, 388 Liouville space diagram, 350-351, 366 Liouville-von Neumann equation, 238, 309

Liouvillian eigenvalues, 5 13 Liouvillian operator, quasiclassical regime, 5 12-5 13 Liquids: dynamics, very short time, 193 hard-sphere model, 407-408 Locked-dipole capture rate constant, 826 London-Eyring-Polanyi-Sato surface, 565-567 Lyapunov exponents, 496-497 dynamical instability, 5 17-5 18 periodic orbit, 500 acetylene, 534. 536

936

SUBJECT INDEX

Lyapunov exponents (Continued) bifurcated, 500-501 resonance lifetime and. 556 Markov approximation, 199-200 Maslov index, 556, 559 Mass-analyzed threshold ionization spectroscopy, 668, 670, 723 Mass reflectron, 661 Maxwell wave equation, 362 Metal clusters: characteristics, 102 multiphoton ionization, 102 transient two-photon ionization spectroscopy, 103-104 see also Ultrafast relaxation Microviscosity, 400 Mixed time-frequency representation, 346 Molecular control: design, feedback role, 316-318 issues in, 280 Molecular design, potential advantage of physical methods, 277 Molecular dynamics: control theory applications, 235 simplified picture, 594-595 Molecular eigenstates, 442 Molecular motion, 588 Molecular spectra, 588 Molecular structure, 11-13 coherence in orientation, 11-13 complex, coherence, 18-20 Molecules, heating or cooling in electronic ground state, 313-314 Morse-like vibrational spacings, 485 Morse oscillator tailored to OH bond, 329-330 Multichannel quantum defect theory, 686, 701-707 autoionizing Rydberg states, 686-696 argon calculations, 689-692 calculation method, 687-689 nitrogen calculations, 692696 reactance matrix, 688489 Schrodinger equation, 686-688 Bom-Oppenheimer approximation, 719 coordinate-dependentquantum defect matrices, 707 frame transformations and bound states, 702-704

nonadiabatic effects, 720 rotation-electron coupling, 703 states in electronic continuum, 706 Multiphoton ionization, metal clusters, 102 Multiple-pulse echo experiments, 209 Multiple pulses, phase-coherent, coherence control, 9-10 Na+ fragment, TOF spectra, laser pulse duration effect, 64-65 Na2+: frequency-filtered pumpprobe signal, 57-58 transient signals, 52-54 Na3: B state, 132 fragmentation rates of C state vibrational bands, 123-1 25 pseudorotating, 121-122 B state, 139 wavepacket propagation during picosecond pumpprobe excitation, 120-121 Na3(B). intramolecular vibrational density redistribution, 134 NaI: double-peak structure, 44 femtosecond dynamics of dissociation reaction, 20-21 unimolecular dissociation, 16, 18 vibrogram, 525-526 Na*+/Na+, signal ratio as function of pumpprobe delay, 53, 55-57 Na*+/Na+ ratio, as function of pulse delay, 230 Nd-Yag laser, frequency-doubled, 286-287 Negative ions, coherent vibrational motion, 313 NeNePo spectroscopy: diatomics, 11 1-1 14 large clusters, 129-130 triatomics, 114-1 17 Nitric oxide: in argon matrices, 7 12 monomers, fluorescence, 717 state-selected ions, 676-678 vibronic states, 607-609 ZEKE spectroscopy compared with ab initio calculated spectra, 617418 photoelectron spectra, 6 16-6 I7

937

SUBJECT INDEX Nitrogen: autoionizing Rydberg states, MQDT calculations, 692696 state-selected ions, 675-676 thermal capture rate constants, 845 Nitrogen ion: TDF profile following Rydberg state excitation, 68 1-685 ZEKE spectroscopy compared with photoionization efficiency, 608410

Nitrogen-ion system, adiabatic channel potential curves, 844 N02.528-529 vibrational motion, 528-529 Noble-gas atoms, van der Waals bonded, effect on intramolecular dynamics, 414419 Nonevaporative cooling, 305, 307-3 I2 atomic cooling scheme. 305, 307 density operator, 308-309 Hamiltonians, 308-309 molecular, 307-308 phase-angle diagram for zero mass transport, 309-3 12 population change equation, 309 Normalization constant, 163 Nuclear motions, 14 Nuclear spin symmetry, conservation, violation, 381 Number variance, 518 ocs: absorption spectrum, VUV region, 7 9 6 7 9I Fano profile, VUV-PHOFFiX spectrum, 793-795.797 photofragment excitation spectrum, 791-793 OH: neutral ground-state spectrum, 61 1, 613 rovibrational adiabatic channel potential curves, 846 Optical Bloch equations, 147-148 Optical cycling, nondestructive, vibrational heating using, 304-307 Optical polarization, Wigner wavepackets, 362-364 Optimal control theory, 217-218, 231

Organic reactions, complex, femtochemistry, 26-30 Orientation, coherence in, 11-1 3 Orthogonal coordinates, role, 209 Oxygen: Franck-Condon mapping model, 786-787 thermal capture rate constants, 845 Oxygen-ion system, adiabatic channel potential curves, 844 Ozone: photodissociation, I38 ultrashort-lived resonances, 572 Partition function: activated complex, 824, 827, 834 linear dipole, 825 Periodic-orbit dividing surfaces, 545-547 Periodic-orbit quantization, 556-557 Periodic orbits, 491-575 bifurcating, 509-5 10 bounded systems, 5 14-538 average level density, 5 15-5 16 beyond Heisenberg time, 520 CzH2.530-536 C2HD. 519-531 CS2,526-528 diatomic molecules, 524-526 emergent classical orbits and vibrograms, 520-524 energy scale below mean spacing, 516-519 energy spectrum, 514-5 19 I,, Morse-type model, 524-525 level curvature statistics, 5 19-520 NaI, vibrogram, 525-526 periodic-orbit structures, 5 16 synthesis, 536-538 tetra-atomic molecules, 529-536 time domain, 520-524 triatomic molecules, 525-529 bulk periodic, 506, 508-509 expression, eigenfunction averages, 504-505 nonisolated, Berry-Tabor trace formula, 506-509

off-diagonal, 559-560, 567 open systems, 538-573 bifurcation associated with transition to chaos, 545-552 classical dynamics, 542-545

938

SUBJECT INDEX

Periodic orbits (Continued) C02, 565-571 dissociation on potentials with saddle, 541-561 energy and time domains, 538-539 fully chaotic regime, 552-555 H,, 571-572 Hg12, 561-565 0 3 , 572-573 periodic-orbit quantization in fully chaotic regime, 557-561 periodic regime, quantization, 555557 transition regime, quantization, 557 ultrashort-lived resonances in triatomic molecules, 561-573 unimolecular dissociation rates, 539-541 period-energy diagram, 568,574 role in scarring phenomena, 505 shape, 595-596 topological characterization, 595-596 triatomic molecules, 585-586 see also Quantization, semiclassical Perturbation model, 207-208 Perturbation theory, 587, 590 higher order, 596 Phase and amplitude imaging, 799-805 highly rotating Na2 molecule, 803-805 theory, 8W803 Phase coherence, 302 Phase-modulated femtosecond laser pulses, 59-43 Phase space: spiraling frequency, 205 Wigner wavepackets, 353-355 Phase-space theory, 778-779 deviations caused by anisotropy, 820 fragmentation rates, 821 free-rotor expression, 833 Phase-space wavepackets, 346-347 Phenol-methanol, ZEKE spectrum, 621 Phenol-water complex: intermolecular normal modes, 619,621 total ion signal compared with ZEKE spectrum, 619-620 Phonon squeezing, 382-384 Phonon states, Q-functions, 382-383 Photoabsorption, cross section, 570 Photoassociation, 87-88

Photochemical nonradical chain production processes, 279-280 Photochemical reactions, 889-890 Photochemistry: future work, 891-893, 895 important accomplishments, 890-891 study of, 890 vinoxy radical, 731 Photodetachment, 116 Photodissociation: CD& 730 CHiI, 742 C02, 565 HgI,, 561 Na2, 285-293 ozone, 138 produces linear combination of internal states, 297-298 small polyatomic molecules, VUV region, 789-796 total, temporal evolution, 127 trends in experiments, 730 vibrationally mediated, 747 vinoxy radical anisotropy parameter, 735-736, 742-743 CH3 + CO channel, 737-739 D + CD2C0 channel, 739-740 energetics, 737-738 experiment, 732-733 photofragment TOF spectrum, 735-736 results, 733-736 translational energy distributions, 734-735.742-744 see also Vinoxy radical Photoelectron spectroscopy: compared with threshold photoelectron spectroscopy, 614-615 ZEKE spectroscopy, 616-6 19 excited states, with resonance ionization, 661 Photofragmentation reactions, elementary, rates, 894 Photofragment excitation spectrum, O C S , 791-793 Photoionization efficiency, compared with ZEKE spectroscopy, 608-6I 1 Photoionization mass spectrometry, 66I Photoionization spectroscopy, versions, 660

SUBJECT INDEX Photoisomerization, stilbene, 84-85, 456 Photon echo, time-integrated, 348 Poincar6 mappings, acetylene, 533-534 Poincar6 surface of section, 596 Poisson distribution, spacing, 5 16 Polarizability: free electron in optical field, 186 isotropic, 821-822 Polarization operators, autocorrelation function, 365 Pollicott-Ruele resonances, 5 L4 Polyad, 466, 468 fractionation patterns within, 470-473 intrapolyad matrix elements, 477478 unzipped, 470-472 Polyatomic molecules: dissociation, 632 spectra, 789-790 fluorescence, 743 highly excited states, 443 intramolecular dynamics, 4 10 high Rydberg states, 433438 mechanism, 411412 laser control of IVR, 454 photodissociation in VUV region, OCS, 789-796 absorption spectrum, 790-79 1 Fano profile, VUV-PHOFEX spectrum, 793-795, 797 photofragment excitation spectrum, 791-793 potential-energy surface, 746747 unimolecular reactions, 648-649 Population dynamics: coherent, 422424 special pulse sequences, 424425 incoherent, 420-422 Porter-Thomas distribution, 540-54 1 Potassium clusters: pumpprobe experiments, 126 transient two-photon ionization, 123-124 Potential-energy surface, 782 ab initio, OCS, 795 Bowman-Bitman-Harding, 760-761,763 electronic states, 106-107 H + Hz,855-856

HCO,752-754

HNO, 752-754

polyatomic molecules, 746-747 reduced-dimension,262

939

shaping, 374 unimolecular dissociation, 752-754 water, 752-754 XCO, 769 Potentials, distinguishing between types, 538-539 Predissociation process, 41 8 Preparation coefficients, 801 Probability densities, as function of time, 588-589 Product branching ratio, photodissociation of Na2, 285-286 Protein dynamics, solvent viscosity effects, 405 Pseudorotations, 133-134 Pulsed-field ionization: H,, 681 hydrogen, 123-724 Rydberg states, 668670 Pumpprobe experiments: bound excited trimer states, 117-122 phase-sensitive, 57-59 potassium clusters, 126 Pump-probe schemes, 52-57 Pumpprobe signal: doorway-window phase-space wavepackets, 369-370 NO in argon matrices, 713-714 Purple photosynthetic bacteria, lightharvesting protein, 157 @functions, phonon states, 382-383 Quantization: periodic-orbit, fully chaotic regime, 557-561 periodic regime, 555-557 semiclassical, 494-5 14 around isolated equilibrium points, 496498 Beny-Tabor trace formula, 506-509 bifurcating periodic orbits, 509-510 Gutzwiller trace formula, 498-502 rate and relaxation behavior emergence, 511-514 semiclassical scattering, 510-5 11 time evolution, 494-496 zeta function and interference between isolated periodic orbits, 502-504 transition regime. 557 Quantum computing, 302

940

SUBJECT INDEX

Quantum defect functions, 659 Quantum defects: body-frame, 705 determination from experiment, 706-707 internuclear distance dependence, 721 matrix, diagonal and off-diagonal, 721 Quantum dynamical localization, 584-585 Quantum dynamics, control. See Feedback, quantum dynamics control Quantum excitation, classical behavior, 583 Quantum many-body dynamics, 213-283 active control, 214-215 attainability of control, 247-253 adaptive learning algorithm, 252 feedback control, 251 finite dimensional bilinear control representation, 248 Huang-Tarn-Clark theorem, 248 integral operator, Hilbert-Schmidt type, 250 Lie algebra, 248 pump-dump experiment, 252 state-to-state transformation, 250-25 1 Brumer-Shapiro method, 219-226, 276277 control within linear response approximation, 275-276 density matrix, 276 dynamics-inverse scattering duality, 267-269 excitation pathways, phase control, 274 extension of control theory, 218 generic conditions for control, 237-247 combined density operator, 237 conservation of population, 239 density operator, 243, 245 dephasing. 240 .equation of motion, 238-239 globally optimum control field, 244-246 Hamiltonian. 237 Lagrange multipliers, 245-246 loading term, spatial derivative, 240 phase angle, 241-243 radiative coupling term, 237-238 resonant electric field components, 240 transfer equations, 240-24 1 issues, 2 18-2 19 optimal control theory, 217-2 I 8.23 1 IR laser puke variant, 274

phase-space formulation, 276 product selectivity control, 215-216 shaped pulse control, experimental status, 273 Tannor-Rice control scheme, 216-217 see afso Reduced space analyses; Tannor-Rice-Kosloff-Rabi tz method Quantum mechanics, time evolution, 494-496 Quantum numbers, 809 pseudo, 809-8 10 Quantum operators, Weyl-Wigner transforms, 499 Quantum packet, localized, propagation, 43-44 Quantum systems, transition to chaotic motion, 584 Quantum theory, 854-859 Rabi frequencies. 427 Rabi oscillations, 8 1 Radiationless transitions, moleculepreserving, 894 Raman scattering: coherent, shift, 444-445 stimulated, for probing wavepacket dynamics, 109-1 10 Rampsberger-Rice-Kassel-Marcus theory. See RRKM theory Random matrices, 518 Rare gases, condensed, bubble formation on Rydberg state excitation, 7 11-7 18 Reaction path Hamiltonian analysis, 259 Reaction probability operator, 857 Reactions: coherence in, 14 control, imposing our will on molecules, 594-595 femtosecond dynamics, 14, 16 unimolecular. See Unimolecular reactions Redfield equation, Wigner representation, 205 Redfield superoperator, frequencydependent, 200 Redfield theory, 147, 195 multilevel, 148-152 Reduced space analyses, 253-267, 281 in coordinate space, 259-260. 262-265 density matrix element, time dependence, 257

94 1

SUBJECT INDEX equation of motion, 254 factorization, 265-267 frequency-independent effective operator, 256

kinetic energy, 262 optimal control theory, reaction coordinate representation, 264 polyatomic reactant, 262-263 population dynamics, 256 three-level system, 257-258 potential-energy surface, reduceddimension, 262 projection operator, 255 reaction path Hamiltonian, 259, 263-264 reduced density matrix, 265 in state space, 253-259 time evolution matrix, 255 time evolution operator, 255-256 Zhao-Rice reduction scheme, 264-265 Refractive index, 186 Relaxation, emergence, quasiclassical regime, 511-514 REMPI technique, 661, 669 Renner-Teller effect, 720 Repeller. 5 10-5 11,543-545 C02, 567-571 fully chaotic regime, 551-552 HgI2, 561-565 Smale horseshoes, 552-554, 557-559 Resolvent, classical Liouvillian, 5 12-5 13 Resonance: distribution chi-square probability, 540 unimolecular dissociation rates, 539-541

dominant, lifetimes, 497 mode-specific behavior, 750 statistical behavior, 750 see nlso Unimolecular dissociation, resonances Resonance-enhanced multiphoton ionization, 661,669

Resonance photoionization spectroscopy, 660

Resonance wave functions: angular dependence, HCO, 765,767 HCO, 756-757 water, 775-777, 786 Resonant continuous electric field, 254 Resonant torus, 507-508

Response function: linear, 386 nonlinear, 386-387 chaos and, 388 nth-order, 388 Reverse unimolecular dissociation. 820 Rose oxide, manufacture, 280 Rotational coherence spectroscopy, 11-13 Rotation4ectron coupling, 703 Rotation-vibration eigenstate, 468-469 Ro-vibrational cluster, 811 Rovibronic reaction matrix, 703 Rovibronic transitions, eliminating Doppler broadening, 410 RRKMM theory, 812 rate constant, calculation, 648 RRKM theory, 456457,750-75 1 extension, 786, 812 unimolecular dissociation rates, 53954 I see also Solvent dynamics, RRKM theory of Ruelle topological pressure, 501-502 Rydberg electron: bulb, nitric oxide, 716 coupled to vibration, 635, 659 effects of environment, 664 frequencies, 702 high orbital angular momentum states, 705-706

orbiting rotating ionic core, 65 1 perturbation, 644 Rydberg formula, 435 Rydberg molecules, in electric fields, 657-658

Rydberg-Rydberg transitions, 7 12 Rydberg series: Born-Oppenheimer approximation, 722 coupling, benzene, 4 4 W 7 Rydberg states: adjacent, energy spacings, 445-446 autoionizing, 685 multichannel quantum defect theory, 686-696

complex molecules, 669 congestion, 630 crossings, 443-444 dynamics, 668 excitation, bubble formation in condensed rare gases, 711-718

942

SUBJECT INDEX

Rydberg states (Continued) HgNe van der Waals dimer, 715-716 high, 625-645 average decay width, 631 bifurcation, decay modes, 63 1-632 bottlenecks, 631-633 bound phase space, 631, 633 condensed-phase environment, 664 decay, physical nature, 630 lifetime, 615-616, 626, 629, 682-683 external perturbers and, 632433 motivations for study, 625-626 quantum defect functions, 659 sensitivity to external perturbations, 629 time evolution, 634-644 Coulomb interaction between two charges, 634-635 coupling constants, 634 diagonalizing effective Hamiltonian, 440,642 effective Hamiltonian, 636-639 intramolecular coupling, 642 K eigenstates, 639 separation of time scales, 636 trapping versus dilution, 639-644 time-resolved ZEKE spectroscopy, 628-629 time scales, 625-626 ZEKE spectroscopy, 626, 632 lifetime distribution, 656-657 lifetimes, 68 1-683 enhancement by electric fields, 682 experimental measurements, 683-686 low, 634 decay behavior, 68 1 dipole moment, 719 metastability, 671 molecular, 701-702 optical excitation, 644 overtones and, 659 perturbation by collision, 6 8 M 8 I p l y atomic molecules, intramolecular dynamics, 433438 pulsed-field ionization, 668-670 relevance to photoselective excitation, 458 triplet, 44-7 as zero-order basis, 650

SACM, 75&751,779,848 applications to more complex reaction systems, 842-846 compared with VTST anisotropic charge-permanent dipole systems, 839-841 general potentials, 841-842 situations where it breaks down, 849-850 thermal capture rate constants, 828-829 transition-state switching, 85 1 Saddle-point avoidance, 407 Saddle-point transition state, 22-25 Scattering, semiclassical theories, 510-51 1 Scherer-Fleming wavepacket interferometry experiment, 282 Schrainger cat state, 382 Schriidinger equation, 231-232, 68-87 controlling dynamical evolution of system, 248 dynamics-inverse scattering duality, 268-269 eigenstate solutions, 2 19 nonlinear, 318, 320, 322 time-dependent, 227, 265, 329 time evolution in quantum systems, 494 time-reversed, 386 Schriidinger representation, dynamics of n-state system, 253-254 Schriidinger-type equations, coupled cubically nonlinear, 3 17 Semiclassical theory, 522-523 Silver dimers, NeNePo spectrum, 112-1 14 trimer, NeNePo spectra, 114-1 16 Sisyphus effect, 307 Smale horseshoes, 552-555 CO,, 565, 568-569 HgI,, 563-565 repeller, 557-559 Sodium: highly rotating molecule, phase and amplitude imaging, 803-805 photodissociation, 285-293 comparison of experimental and theoretical yields, 288-289 experimental Na(3d) and Na(3p) emission, 281-288 incoherent interference control, 290-29 I , 293

SUBJECT INDEX Na(3d) fluorescence for different w l frequencies, 290-29 1 product branching ratio, 285-286 potential energy curves, 55-56 for excitations, 52-53 Solvation: coherence in, 14-16 dynamics, ultrafast component, 173 Solvay Conferences, 4

Solvent:

role in ?ram-stilbene photoisomerization. 456 components, structure and function, 405 viscosity, effects on protein dynamics, 405 Solvent dynamics: RRKM theory of clusters and, 391408 microcanonical, 395-399 one-coordinate type treatment, 395 rate constant, 398-399 saddle-point avoidance, 407 slow and fast coordinates, 406 vibrational assistance treatment, 395-399 survey of developments, 391-392 Specific rate constants, dissociation, adiabatic channels, 832-835 Spectral rigidity, 5 18 Spectroscopic effective Hamiltonian model, 464-466 anharmonic resonances, 46-67. 473, 476 Darling-Dennison resonance, 473475 eigenstates, 469 IVR.473474 matrices, 476 survival and transfer probabilities, 479484 ZOBS survival probability, 476 Stability matrix, 387-388 Stark effect, 851 Stark level-crossing spectroscopy, 541 Statistical adiabatic channel model, See SACM Steradiancy techniques, 61 1-612 Stilbene, photoisomerization, 84-85, 456 in solution, 404

943

S tiibene-hexane: energy-specific excited stilbene lifetimes, 404 structure, 408 Stimulated emission pumping: K O , 747-749 HCP, 484488 STIRAP, 444 Stretching perturbations, asymmetric, 555-556,559 Stuckefberg-Landau-Zener theory, 208 Sumi-Marcus treatment, 394 Symmetric-stretch classical stability diagram, 597-598 System-bath interactions, 160-1 75 line shape function, 160-164 see also Echo spectroscopies System-bath model, 263 Tannor-Kosloff-Rice scheme, 52, 59 Tannor-Rice control scheme, 2 16-2 17 more sophisticated, 217 Tannor-Rice-Kosloff-Rabitz method, 226-236 density matrix representation, 235-236 excited-state-potential-energy surface, 230, 234-235 fragmentation, 227 ground-state potential-energy surface, 228 Hamiltonian, 227 modified objective functional, 23 1 optimal control field. 236 optimal pulse shape calculation, 232-234 product yield as function of pulse separation, 229 pump-dump scheme, 226 pulse separation, 228-229 Tannor-Rice perturbation theory, 227 Tautomerization reactions, DNA models, 34-37 Tetra-atomic molecules. vibrational motion, 529-536 Thermal capture rate constants, 820. 823-832 HCI, 830, 831 HCN, 830-831 locked-dipole, 826 nitrogen, 845 oxygen, 845 SACM. 828-829

944

SUBJECT INDEX

Thermal rigidity factor, 828, 831-832 low-temperature limiting, 827 Thermionic emission, 656 Threshold photoelectron photoion coincidence, 669 Threshold photoelectron spectroscopy, 611-612, 614 compared with photoelectron spectroscopy, 6 14-6 15 see also ZEKE spectroscopy Time-dependent self-consistent-field methods, 201 Time evolution, 92 Time evolution operator, 255-256 Time-of-flight spectrometer, 5 1 Time-resolved spectroscopy: bound-free trimer transitions, 122-1 26 electronically excited states, 103-109 ground electronic states, 109-1 1 I Ti-sapphire laser system, 49-50 Trace formulas, 494-496 Tracking control, 318 Transient two-photon ionization spectrum, Na,, 117-118 Transition-dipole operator, angle-averaged, 80 1 Transition states, reacting molecules, properties, 894-895 Transition state spectroscopy, 809-81 6 Transition-state switching, SACM, 85 I Transition state theory, transmission probabilities, 859 Transparency, laser-induced, 302 Trapping, versus dilution, high Rydberg states, 639-644 Triatomic molecules: NeNePo spectroscopy, 114-1 17 periodic orbits, 585-586 ultrashort-lived resonances, 561-573 C02, 565-571 H3.571-572 HgI2, 561-565 H20, 572-573 lifetime comparison, 573 0 3 , 572 vibrational motion, 525-529 Triatomic monohydride, linear, B value, 485486 Tryptophan, multiphoton ionization, 878-880

Tunnelling, between enantiomers, 38 1 Two-pendulum model, 206-207 Ultrafast nonlinear spectroscopy, 142 Ultrafast relaxation, 101-130 dimers NeNePo spectroscopy of diatomics, 111-1 I4 time-resolved spectroscopy electronically excited states, 103109

ground electronic states, 109-11 1 two-color experiments, 106-108 larger clusters, 126-130 bound-free transitions into excited states, 126-129 NeNePo experiments, 129-130 triatomics, 114-126 ab initio full CI energy surfaces, 117, I19 NeNePo spectroscopy, 114-1 17 pumpprobe experiments, 117-122 time-resolved spectroscopy, 122-126 Unimolecular dissociation: resonances, 745-782 DCO, 768-772.781 HCO, 759-768.781 HNO, 772-775 lifetime function, 756, 758-759 potential-energy surfaces, 752-754 quantum mechanical calculations, 755-759 water, 775-78 1 reverse, 820 Unimolecular reactions: polyatomic molecules, 648-649 resonances. 20-22 Vacuum state, 199 van der Waals complexes: coupling, 446 intermolecular vibrations, 43 1 4 3 3 Van Vleck contact transformations, 497 Variational transition-state theory, 407 compared with SACM anisotropic charge-permanent dipole systems, 839-841 general potentials, 841-842 isotropic charge-locked permanentdipole systems, 836-839

SUBJECT INDEX compared with statistical adiabatic channel treatment, 835-842 microcanonical, 850-85 1 Vibrational cluster, 811 Vibrational coherence, retention, 176. 178 Vibrational dynamics, 148-1 60 asymmetric double well, 150-151 banierless double well, 151-1 52 multilevel Redfield theory, 148-152 Vibrational entropy: change, 396 as function of slow and fast coordinates, 396397 Vibrational heating, using nondestructive optical cycling, 304-307 Vibrational levels, Franck-Condon factor, 77 Vibrationally mediated photodissociation, 747 Vibrational motion, delocalization, 95-96 Vibrational relaxation, energy dependence, 194-195 Vibrational spacing, Morse-like, 485 Vibrational spectra, with Wignerian level spacing, 493 Vibrational-state-to-vibrational-state transitions, individual, 333-335 Vibrational transitions, 327-340 above-threshold dissociation. 336-338 in competition with dissipative processes, 336337 corresponding time-dependent Schriidinger equation, 329 double-well potential, 329, 331 equation of motion, reduced-density operator, 332-333 individual vibrational-state-to-vibrationalstate transitions, 333-335 IR femtosecond/picosecond laser pulses, 328 isomerization, 338-340 laser parameters in model systems, 335 models and techniques, 328-333 Morse oscillator tailored to OH bond, 329-330 OH and SBV potential-energy surfaces, 329-331 semibullvalenes, 329, 331 series, 335-337

945

Vibrogram, 574 emergent classical orbits and, 520-524 recurrences, 536-537 Vib-rotational wavefunctions, 800-801 Vinoxy radical, 729-741 B state, 731 intermediate in hydrocarbon combustion, 73 1 photochemistry, 731 photodissociation anisotropy parameter, 735-736, 742-743 CH3 + CO channel, 737-739 D + CD2C0 channel. 739-740 energetics, 737-738 experiment, 732-733 photofragment spectrum, 735-736 results, 733-736 translational energy distributions, 734-735,742-744 transition bands, 73 1 VUV-PHOFEX spectrum, OCS, Fano profile, 793-795, 797 Water: dissociation dynamics, 775-781 ab initio calculations, 787 erratic fluctuations,782 final-state distributions, 776, 778 nearest neighbor distribution, 8 13 rotational product distributions, 779-780 RRKM rate, 776 state-specific unimolecular decay rates, 812-81 3 dissociation rates, 759-760 lifetime function, 756, 758-759 motion, 751 potential-energy surfaces, 752-754 potential well, 775 resonance wave functions, 775-777, 786 ultrashort-lived resonances, 572-573 Wavefunctions: DCO, 769,771 imaging, theory, 800-803 partial, 755-756 resonance, HCO, 756757 Rydberg-coupled and -uncoupled, 703 total, 755-756

946

SUBJECT INDEX

Zakrzewski-Delande curvature distributions, Wavepacket. 14 autocorrelation function, 324-325 519 coherence control, 14-18 ZEKE spectroscopy, 80, 82, 607-622, 667-696 control, 382 dispersive spreading, 209 aims of, 663 excited-state potential-energy surface, compared with 216-217 photoelectron spectroscopy, 616-619 fast Fourier transform propagation photoionization efficiency, 608-61 1 techniques, 200-201 total ion signal, 619-620 initial excited ionic core, 630-631 Gaussian distribution in energy, 522 extraction pulse, 67 1 Wigner function, 521 field ionization behavior, 670 motion, 146-147 ground-state neutral systems, 610-61 1, in dissociation and barrier reactions, 613 25 high Rydberg electron, 647 multiple-pulse preparation, 16-17 high Rydberg state, 626, 632 phase and amplitude imaging. See Phase lifetime distribution, 656-657 and amplitude imaging historical development leading to, on potential-energy surface domains, 607-608.659-662 457458 inverse Bom-Oppenheimer zero-order spreading, 93-94 basis, 650 symmetric-stretch, spectral decomposition, ion-molecule reactions, 678-681 598-599 collision-energy resolution, 679-680 transfer, 207-208 H,+ + Hz.678-679 pumpprobe spectroscopy, 355-358 Rydberg state perturbation, 680-68 1 see also Wigner wavepacket transmission effects, 680 Wentzel-Kramers-Brillouincondition, laser system, 671-672 503 lifetimes, 659 Weyl-Wigner transforms, quantum long decay times, origin, 643 operators, 499 mass-selected anion spectrum, 619, Wigner distribution, 352, 355 622 spacing, 517 molecular systems studied, 610, 612 Wigner function, 360-361, 382, 385, 504 MQDT approach, 647, 650 initial wavepacket, 521 new sequential technique, 657 Wigner transforms, eigenfunctions, 508 peaks, 647 Wigner wavepacket, 345-370 Rydberg level structure, 723 correlation function expression, Rydberg state lifetimes, 681-683 spontaneous light emission, spectral resolution, 607 347-353 state-selected ion-molecule reactions, extension to heterodyne-detected four669-672 wave mixing, 358-359 state-selected ion preparation Green function, 363 carbon monoxide, 672-615 optical polarization, 362-364 hydrogen, 672 in phase space, 353-355 nitric oxide, 676-678 pumpprobe spectroscopy, 355-358 nitrogen, 675-676 see also Doorway wavepackets; Window stationary, 658 wavepackets time-resolved, 628-629 Window wavepackets, phase-space: time-resolved, 652 fluorescence, 368-369 types of systems studied, 614 pumpprobe signals, 369-370 ZEKE states, 701-702

SUBJECT INDEX Zero electron kinetic energy spectroscopy. See ZEKE spectroscopy Zero ion kinetic energy spectroscopy, 662 Zeta function, 502-504 classical. 5 13-5 14

inverse Ruelle, 560 periodic regime, 555 product over periodic, 558 semiclassical quantization, 57 1 Zhao-Rice reduction scheme, 264-265

947