Vibrational Spectra and Structure : Vibrational Intensities

VIBRATIONALSPECTRAAND STRUCTURE Volume 22 VIBRATIONAL INTENSITIES EDITORIAL BOARD Dr. Lester Andrews University of ...

Author: B. S. Galabov | T. Dudev

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VIBRATIONALSPECTRAAND STRUCTURE Volume 22

VIBRATIONAL INTENSITIES

EDITORIAL BOARD

Dr. Lester Andrews University of Vir#nia Charlottesville, Virginia USA

Dr. J. A. Koningstein Carleton University Ottawa, Ontario CANADA

Dr. John E. Bertie University of Alberta Edmonton, Alberta CANADA

Dr. George E. Leroi Michigan State University East Lansing, Michigan USA

Dr. A. R. H. Cole University of Western Australia Nedlands WESTERN AUSTRALIA

Dr. S. S. Mitra University of Rhode Island Kingston, Rhode Island USA

Dr. William G. Fateley Kansas State University Manhattan, Kansas USA

Dr. A. Miiller Universitiit Bielefeld Bielefeld WEST GERMANY

Dr. H. Hs. Giinthard Eidg. Technische Hochschule Zurich SWITZERLAND

Dr. Mitsuo Tasumi University of Tokyo Tokyo JAPAN

Dr. P. J. Hendra University of Southampton Southampton ENGLAND

Dr. Herbert L. Strauss University of California Berkeley, California USA

V)lliJBRAT lUOl NAIL SPIEEC ? III1:~A N lid STIIII~UCTUIIII~IIIIE A SERIES

OF ADVANCES

VOLUME

JAMES R. DURIG (Series Editor) College of Arts and Sciences University of Missouri-Kansas City Kansas City, Missouri

22

VIBRATIONAL INTENSITIES

Boris S. Galabov and Todor Dudev Faculty of Chemistry, Sofia University, Sofia 1126, Bulgaria

1996 ELSEVIER A m s t e r d a m - Lausanne - N e w Y o r k - O x f o r d - S h a n n o n - Tokyo

ELSEVIER SCIENCE B.V. Sara Burgerhartstraat 25 P.O. Box 211, 1000 AE Amsterdam The Netherlands

ISBN 0-444-81497-3 91996 Elsevier Science B.V. All rights reserved. No part of this publication may be reproduced, stored in a retrieval system or transmitted in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, without the prior written permission of the publisher, Elsevier Science B.V., Copyright & Permissions Department, P.O. Box 521, 1000 AM Amsterdam, The Netherlands. Special regulations for readers in the U.S.A. - This publication has been registered with the Copyright Clearance Center Inc. (CCC), 222 Rosewood Drive, Danvers, MA 01923. Information can be obtained from the CCC about conditions under which photocopies of parts of this publication may be made in the U.S.A. All other copyright questions, including photocopying outside of the U.S.A,. should be referred to the publisher. No responsibility is assumed by the publisher for any injury and/or damage to persons or property as a matter of products liability, negligence or otherwise, or from any use or operation of any methods, products, instructions or ideas contained in the material herein. This book is printed on acid-free paper. Printed in The Netherlands

P R E F A C E TO THE SERIES

It appears that one of the greatest needs of science today is for competent people to critically review the recent literature in conveniently small areas and to evaluate the real progress that has been made, as well as to suggest fi'uitf~ avenues for future work. It is even more important that such reviewers clearly indicate the areas where little progress is being made and where the chances of a si~ificant contribution are minuscule either because of faulty theory, inadequate experimentation, or just because the area is steeped in unprovable yet irrefutable hypotheses. Thus, it is hoped that these volumes will contain critical summaries of recent work, as well as review the fields of current interest. Vibrational spectroscopy has been used to make significant contributions in many areas of chemistry and physics as well as in other areas of science. However, the main applications can be characterized as the study of intramolecular forces acting between the atoms of a molecule; the intermolecular forces or degree of association in condensed phases; the determination of molecular symmetries; molecular dynamics; the identification of functional groups, or compound identification; the nature of the chemical bond; and the calculation of thermodynamic properties. Current plans are for the reviews to vary, from the application of vibrational spectroscopy to a specific set of compounds, to more general topics, such as force-constant calculations. It is hoped that many of the articles will be sufficiently general to be of interest to other scientists as well as to the vibrational spectroscopist. As the series has progressed, we have provided more volumes on topical issues and, in some cases, single author(s) volumes. This flexibiLity has made it possible for us to diversify the series. Therefore, the course of the series has been dictated by the workers in the field. The editor not only welcomes suggestions from the readers, but eagerly solicits your advice and contributions.

James R. Durig Kansas City, Missouri

P R E F A C E TO V O L U M E 22

The current volume in the series Vibrational Spectra and Structure is a single topic volume on the vibrational intensities in the inflared and Raman spectra. The current monograph describes the various models for interpreting intensities and it presents, in a systematic way, the theoretical approaches that are used in analyzing and predicting vibrational intensities. The book is divided into 10 chapters, with each chapter covering a specific topic. The first part of the book deals with the absorption of infxared radiation whereas chapter eight, nine and ten deal with Raman intensities. It is hoped that the consistent notation used throughout the book will facilitate the understanding of this rather complicated topic. The Editor would like to thank the editorial board for suggesting the topic for this volume and the two authors for their contributions and patience, which was required when producing the monograph. The Editor would also like to thank his Administrative Associate, Ms. Gail Sullivan, and Editorial Assistant, Mrs. Linda Smitka for providing the articles in camera ready copy form and quietly enduring some of the onerous tasks associated with the completion of the volume. He also thanks his wife, Marlene, for copy-editing and preparing the author index.

James R. Durig Kansas City, Missouri

P R E F A C E BY T H E A U T H O R S

The present book appears more than ten years since the publication of "Vibrational Intensities in Infrared and Raman Spectroscopy," a volume edited by W. B. Person and (3. Zerbi. It contains comprehensive reviews describing major developments in the field made during the seventies. Though not at a very fast pace, advances in the field, especially in theoretical approaches, have been made during the past fifteen years. In 1988 the monograph of L. A. Gribov and W. J. Orville-Thomas "Theory and Methods of Calculation of Molecular Spectra" was published. This volume presents, in detail, the progress achieved within the valence optical theory of infrared and Raman intensifies. Vibrational intensities in infrared and Raman spectra are important physical quantifies that are directly related to the distribution and fluctuations of electric charges in the molecule. These spectral parameters can be experimentally determined with good accuracy for many molecules. Additionally, infrared and Raman intensities are presently estimated theoretically by advanced analytical derivative ab imtio molecular orbital methods. These fundamental molecular quantifies are being used in structural, and other studies, on a limited basis. A monograph describing the currently available methods and models for interpreting intensities is needed to stimulate wider application of these important molecular quantities that can be obtained without much ditticulty from experiment and quantum mechanical calculations. It is a principal aim of the present book to present, in a systematic way, the theoretical approaches that are used in analyzing and predicting vibrational intensities. The formalisms developed are illustrated with detailed numerical examples. By experience we have realized that a theory cannot be fully understood and appreciated unless concrete applications are performed. Thus, most of the theoretical models described in the book were obtained and then applied to chosen molecules. We hope that the approach adopted will facilitate the easier understanding of seemingly complicated formulations. We have also used a consistent notation in presenting the different theoretical approaches, thus eliminating another barrier in uaderstanding some methods, especially those developed by the Russian spectroscopic school. The book does not aim at completeness in covering the various aspects of the field. This is, in principle, difficult to achieve in the present times of scientific information explosion. The authors would like to acknowledge many extremely helpful discussions with researchers from different countries that have contributed so much to the development of the field: Bryce L. Crawford, Jr., John C. Decius, W. J. Orville-Thomas, Willis B. Person, Lev A. Gribov, Ian M. Mills, John Overend, Giuseppe Zerbi, Mariangela Gussoni, Peter Pulay, Henry F. Schaefer HI, Derek Steele, Donald C. McKean, William M. A. Smit, Salvador

vii

Montero, Keneth B. Wiberg, and many others. In fact, a substantial part of the content of this book reflects theoretical and experimental developments achieved by these fine scientists. We are particularly grateful to Professor W. J. Orville-Thomas. One of the present authors (B. G.) was first introduced to the field of vibrational intensities while a postdoctoral fellow in the laboratory of Professor W. J. Orville-Thomas at the University of SalforcL The collaboration established lasts for more than twenty years. We are also very indebted to the series Editor, Professor J. IL Durig, for his encouragement and support.

Boris Galabov and Todor Dudev Sofia, Bulgaria

~176176

viii

C O N T E N T S OF O T H E R V O L U M E S

VOLUME 10 VIBRATIONAL SPECTROSCOPY USING TUNABLE LASERS, Robin S. McDoweH INFRARED AND RAMAN VIBRATIONAL OPTICAL ACTIVITY, L. A. Nafie RAMAN MICROPROBE SPECTROSCOPIC ANALYSIS, John J. Blaha THE LOCAL MODE MODEL, Bryan R. Henry VIBRONIC SPECTRA AND STRUCTURE ASSOCIATED WITH JAHN-TELLER INTERACTIONS IN THE SOLID STATE, M.C.M. O'Brien SUM RULES FOR VIBRATION-ROTATION INTERACTION COEFFICIENTS, L. Nemes

VOLUME 11 INELASTIC ELECTRON TUNNELING SPECTROSCOPY OF HOMOGENEOUS CLUSTER COMPOUNDS, W. Henry Weinberg

SUPPORTED

VIBRATIONAL SPECTRA OF GASEOUS HYDROGEN-BONDED COMPOUNDS, J. C. Lassegues and J. Lascombe VIBRATIONAL SPECTRA OF SANDWICH COMPLEXES, V. T. Aleksanyan APPLICATION OF VIBRATIONAL SPECTRA TO ENVIRONMENTAL PROBLEMS, Palricia F. Lynch and Chris W. Brown TIME RESOLVED INFRARED INTERFEROMETRY, Part 1, D. E. Honigs, R. M. Hammaker, W. G. Fateley, and J. L. Koenig VIBRATIONAL SPECTROSCOPY OF MOLECULAR SOLIDS - CURRENT TRENDS AND FIYI'URE DIRECTIONS, Elliot 1L Bemstein

ix

x

CONTENTS OF OTHER VOLUMES VOLUME 12

HIGH RESOLUTION INFRARED STUDIF_~ OF Srl~ STRUCTURE AND DYNAMICS FOR MATRIX ISOLATED MOLECULES, B. I. Swanson and L. H. Jones FORCE FIELDS FOR LARGE MOLECULES, Hiroatsu Matsuura and Mitsuo Tasumi SOME PROBLEMS ONTHE STRUCTURE OF MOLECULES IN THE ELECTRONIC EXCITED STATES AS STUDIED BY RESONANCE RAMAN SPECTROSCOPY, Aldko Y. Hirakawa and Masamichi Tsuboi VIBRATIONAL SPECTRA AND CONFORMATIONAL ANALYSIS OF SUBSTITUTED THREE MEMBERED RING COMPOUNDS, Charles J. Wurrey, Jiu E. DeWitt, and Victor F. Kalasinsky VIBRATIONAL SPECTRA OF SMALL MATRIX ISOLATED MOLECULES, Richard L. Redington RAMAN DIFFERENCE SPEC'I~OSCOPY, J. I.aane

VOLUME 13 VIBRATIONAL SPECTRA OF ELECTRONICALLY EXCITED STATES, Mark B. Mitchell and William A. Guillory OPTICAL CONSTANTS, INTERNAL FIELDS, AND MOLECULAR PARAMETERS IN CRYSTA[~, Roger Frech RF~ENT ADVANCES IN MODEL CALCULATIONS OF VIBRATIONAL OPTICAL ACTIVITY, P. L. Polavarapu VIBRATIONAL EFFECTS IN SPECTROSCOPIC GEOMETRIES, L. Nemes APPLICATIONS OF DAVYDOV SPLITHNG FOR PROPERTIES, G. N. Zhizhin and A. F. Goncharov

STUDIES OF

CRYSTAL

RAMAN SPECTROSCOPY ON MATRIX ISOLATED SPECIES, H. J. Jodl

VOLUME 14 HIGH RESOLUTION LASER SPECTROSCOPY OF SMALL MOLECULES, Eizi Hirota

CONTENTS OF OTHER VOLUMES

xi

ELEL'TRONIC SPECTRA OF POLYATOMIC FREE RADICALS, D. A. Ramsay AB 1NITIO C ~ T I O N

OF FORCE FIELDS AND VIBRATIONAL SPECTRA, Geza

Fogarasi and Peter Pulay FOURIER TRANSFORM INFRARED SPECTROSCOPY, John E. Bertie NEW TRENDS IN THE THEORY OF INTENSITIES IN INFRARED SPECTRA, V. T. Aleksanyan and S. KK Samvelyan VIBRATIONAL SPECTROSCOPY OF LAYERED MATERIALS, S. Nakashima, M. Hangyo, and A. Mitsuishi

VOLUME 15 ELECTRONIC SPECTRA IN A SUPERSONIC JET AS A MEANS OF SOLVING VIBRATIONAL PROBLEMS, Mitsuo Ito BAND SHAPES AND DYNAMICS IN LIQUIDS, Walter G. Rothschild RAMAN SPECTROSCOPY IN ENERGY CHEMISTRY, Ralph P. Cooney DYNAMICS OF LAYER CRYSTALS, Pradip N. Ghosh THIOMETALLATO COMPLEXES: VIBRATIONAL SPECTRA AND STRUCTURAL CHEMISTRY, Achim Miiller ASYMMETRIC TOP INFRARED VAPOR PHASE CONTOURS AND CONFORMATIONAL ANALYSIS, B. J. van der Veken WHAT IS HADAMARD TRANSFORM SPECTROSCOPY?, R. M. Hammaker, J. A. Graham, D. C. Tilotta, and W. G. Fateley

VOLUME 16 SPECTRA AND STRUCTURE OF POLYPEPTIDES, Samuel Krimm STRUCTURES OF ION-PAIR SOLVATES FROM MATRIX-ISOLATION/SOLVA-TION SPECTROSCOPY, J. Paul Devlin LOW FREQUENCY VIBRATIONAL SPECTROSCOPY OF MOLECULAR COMPLEXES, Erich Knozinger and Otto Schrems

xii

CONTENTS OF OTHER VOLUMES

TRANSIENT AND TIME-RESOLVED RAMAN SPECTROSCOPY OF SHORT-LIVED INTERMEDIATE SPECIES, Hiro-o Hamaguchi INFRARED SPECTRA OF CYCLIC DIMERS OF CARBOXYLIC ACIDS: THE MECHANICS OF H-BONDS AND RELATED PROBLEMS, Yves Marechal VIBRATIONAL SPECTROSCOPY UNDER HIGH PRESSURE, P. T. T. Wong

VOLUME 17.4, SOLID STATE APPLICATIONS, R. A. Cowley; M. L. Bansal; Y. S. Jain and P. K. Baipai; M. Couzi; A. L. Venna; A. Jayaraman; V. Chandrasekharan; T. S. Misra; H. D. Bist, B. Darshan and P. K. Khulbe; P. V. Huong, P. Bezdicka and J. C. Grenier SEMICONDUCTOR SUPERLATTICES, M. V. Klein; A. Pinczuk and J. P. Valladares; A. P. Roy; K. P. Jain and R. K. Soni; S. C. Abbi, A. Compaan, H. D. Yao and A. Bhat; A. K. Sood TIME-RESOLVED RAMAN STUDIES, A. Deffontaine; S. S. Jim; R. E. Hester RESONANCE RAMAN AND SURFACE ENHANCED RAMAN SCATTERING, B. Hudson and R. J. Sension; H. Yamada; R. J. H. Clark; K. Machida BIOLOGICAL APPLICATIONS, P. Hildebrandt and M. Stockburger; W. L. Peticolas; A. T. Tu and S. Zheng; P. V. Huong and S. R_ Plouvier; B. D. Bhatmchm3,ya; E. TaiUandier, J. Liquier, J.-P. Ridoux and M. Ghomi

VOLUME 1713 STIMULATED AND COHERENT ANTI-STOKES RAMAN SCATTERING, H. W. Schrrtter and J. P. BoquiUon; G. S. Agarwal; L. A. Rahn and tL L. Farrow; D. Robert; K. A. Nelson; C. M. Bowden and J. C. Englund; J. C. Wright, tL J. Carlson, M. T. Riebe, J. K. Steehler, D. C. Nguyen, S. H. Lee, B. B. Price and G. B. Hurst; M. M. Sushchinsky; V. F. Kalasinsky, E. J. Beiting, W. S. Shepard and tLL. Cook RAMAN SOURCES AND RAMAN LASERS, S. Leach; G. C. Baldwin; N. G. Basov, A. Z. Grasiuk and I. G. Zubarev; A. I. Sokolovskaya, G. L. Brekhovskikh and A. D. Kudtyavt$r

OTHER APPLICATIONS, P. L. Polavarapu; L. D. Barron; M. Kobayashi and T. Ishioka; S. 1L Ahmad; S. Singh and M. 1. S. Sastry; K. Kamogawa and T. Kitagawa; V. S. Gorelik; T. Kushida and S. Kinoshita; S. K. Shanna; J. IL Durig, J. F. Sullivan and T. S. Little

CONTENTS OF OTHER VOLUMES

xiii

VOLUME 18 ENVIRONMENTAL APPLICATIONS OF GAS CHROMATOGRAPHY/FOURIER TRANSFORM INFRARED SPECTROSCOPY (GC/FT-IR), Charles J. Wurrey and Donald F. Gufl~ DATA TREATMENT IN PHOTOACOUSTIC FT-IR SPECTROSCOPY, K. H. Michaelian RECENT DEVELOPMENTS IN DEPTH PROFILING FROM SURFACES USING FT-IR SPECTROSCOPY, Marek W. Urban and Jack L. Koenig FOURIER TRANSFORM INFRARED SPECTROSCOPY OF MATRIX ISOLATED SPECIF~, Lester Andrews VIBRATION AND ROTATION IN SILANE, GERMANE AND STANNANE AND T t ~ I R MONOHALOGEN DERIVATIVES, Hans Biirger and Annette Ralmer FAR INFRARED SPECTRA OF GASES, T. S. Little and J. R. Durig

VOLUME 19 ADVANCES IN INSTRUMENTATION FOR THE OBSERVATION OF VIBRATIONAL OPTICAL ACTIVITY, M. Diem SURFACE ENHANCED RAMAN SPECTROSCOPY, Ricardo Aroca and Gregory J. Kovacs

DETERMINATION OF METAL IONS AS COMPLEXES I MICELLAR MEDIA BY UVVIS SPECTROPHOTOMETRY AND FLUORIMETRY, F. Fernandez Lucena, M. L. Marina Alegre and A. R. Rodriguez Fernandez AB INITIO CALCULATIONS OF VIBRATIONAL BAND ORIGINS, Debra J. Searles and

EUak I. yon Nagy-Felsobuld APPLICATION OF INFRARED AND RAMAN SPECTROSCOPY TO THE STUDY OF SURFACE CHEMISTRY, Tohru Takenaka and Junzo Umemura INFRARED SPECTROSCOPY OF SOLUTIONS IN LIQUIFIED SIMPLE GASES, Ya. M. Kimel'ferd VIBRATIONAL SPECTRA AND STRUCTURE OF CONJUGATED CONDUCTING POLYMERS, Issei Hamda and Yukio Furukawa

AND

xiv

CONTENTS OF OTHER VOLUMES VOLUME 20

APPLICATIONS OF MATRIX INFRARED SPECTROSCOPY TO MAPPING OF BIMOLECULAR REACTION PATHS, Heinz Frei VIBRATIONAL LINE PROFILE AND FREQUENCY SHIFt STUDIES BY RAMAN SPECTROSCOPY, B. P. Asthana and W. Kiefer MICROWAVE FOURIER TRANSFORM SPECTROSCOPY, Alfred Bander AB/N/T/O QUALITY OF SCMEH-MO CALCULATIONS OF COMPLEX INORGANIC SYSTEMS, Edward A. Boudreaux C~TED AND EXPERIMENTAL VIBRATIONAL SPECTRA AND FORCE FIELDS OF ISOLATED PYRIMIDINE BASES, Willis B. Person and K~styna Sz~ze-

VOLUME 21 OPTICAL SPECTRA AND LATTICE DYNAMICS OF MOLECULAR CRYSTALS, G. N. Zhizhin and E. I. Mukhtarov

TABLE OF CONTENTS

P R E F A C E T O THE SERIES ..........................................................................................

v

P R E F A C E B Y THE E D I T O R .......................................................................................

vi

P R E F A C E B Y THE A U T H O R S ..................................................................................

vii

C O N T E N T S OF O T H E R V O L U M E S ..........................................................................

ix

CHAPTER 1 A B S O R P T I O N O F I N F R A R E D R A D I A T I O N B Y M O L E C U L E S .......................... 1 I.

Theoretical Considerations .....................................................................................

2

II.

Selection Rules For Infrared Absorption ...............................................................

12

A.

Harmonic Oscillator Selection Rules ............................................................

12

B.

Symmetry Selection Rules ............................................................................

14

lII.

Experimental Determination o f Infrared Intensities .............................................. 17

CHAPTER 2 C O O R D I N A T E S IN V I B R A T I O N A L A N A L Y S I S .................................................. 25 CHAPTER 3 S E M I - C L A S S I C A L M O D E L S O F I N F R A R E D I N T E N S I T I E S ............................ 35 I.

Introduction ..........................................................................................................

II.

Rotational Corrections To Dipole M o m e n t Derivatives ........................................ 40

II1.

36

A.

The Compensatory Molecular Rotation ........................................................ 40

B.

The Hypothetical Isotope Method ................................................................

43

The Bond Moment Model ....................................................................................

51

A.

Theoretical Considerations ...........................................................................

51

B.

Applications .................................................................................................

63

C.

Atomic Charge -- Charge Flux Model .......................................................... 68

D.

Group Dipole Derivatives as Infrared Intensity Parameters ........................... 72

CHAPTER 4 M O L E C U I ~ R D I P O L E M O M E N T D E R I V A T I V E S AS I N F R A R E D I N T E N S I T Y P A R A M E T E R S ...................................................................................

77

I.

Atomic Polar Tensors (APT) ................................................................................

79

A.

79

General Formulation .................................................................................... xv

B.

lnvariants of Atomic Polar Tensors Under Coordinate Transformation ............................................................................................. 83

C.

Symmetry Properties of Atomic Polar Tensors ............................................. 88

D.

Atomic Polar Temors - Examples of Application ........................................ 93

E.

Interpretation of Atomic Polar Tensors ......................................................... 98

F.

Predictions of Infrared Intensities by Transferring Atomic Polar Tensors ....................................................................................................... 105

H~

Bond Charge Tensors...

HI.

Bond Polar Parameters ........................................................................................ 111

................................................................................ 106

A.

General Considerations ............................................................................... 111

B.

Formulation ................................................................................................. 116

C.

Examples of Application ............................................................................. 120

D.

Physical Significance of Bond Polar Parameters .......................................... 126

E.

Prediction of Vibrational Absorption Intensities by Transferring Bond Polar Parameters ................................................................................ 130

W. Effective Bond Charges from Rotation-Free Atomic Polar Tensors ..................... 131 A.

Rotation-Free Atomic Polar Tensor ............................................................. 131

B.

Effective Bond Charges ............................................................................... 132

C.

Applications ................................................................................................ 134

CHAPTER 5 RELATIONSHIP BETWEEN INFRARED INTENSITY F O R M U L A T I O N S ..............

141

CHAPTER 6 P A R A M E T R I C F O R M U L A T I O N S OF I N F R A R E D A B S O R P T I O N I N T E N S I T I E S O F O V E R T O N E AND C O M B I N A T I O N BANDS ........................ 149 I.

Introduction ......................................................................................................... 150

II.

AnharmonicVibrational Transition Moment ....................................................... 151 A.

Variation Method Formulation .................................................................... 151

B.

Perturbation Theory Formulation ................................................................ 152

Ill.

The Charge Flow Model ...................................................................................... 158

IV.

The Bond Moment Model ................................................................................... 160

CHAPTER 7

AB INITIO M O C A L C U L A T I O N S OF I N F R A R E D I N T E N S I T I E S .................... 163 I.

Introduction ......................................................................................................... 164 xvi

H.

Computational Methods ...................................................................................... 165 A.

m.

Numerical Differentiation ........................................................................... 165

B.

Dipole Moment Derivative from the Energy Gradient ................................. 166

C.

Analytic Dipole Moment Derivatives .......................................................... 167

Calculated Infrared Intensities ............................................................................. 169 A.

Basis Set Considerations ............................................................................. 169

B.

Influence of Electron Correlation on Calculated Infrared Intensities ............ 176

W~ Conclusions ......................................................................................................... 187 CHAPTER 8 INTENSITIES IN RAMAN SPECTROSCOPY ...................................................... 189 I.

Molecular Polarizability ...................................................................................... 190

II.

Intensity of Raman Line ...................................................................................... 199

111. Raman Intensities and Molecular Symmetry ........................................................ 205 IV.

Resonance Raman Effect ..................................................................................... 207

V.

Experimental Determination of Raman Intensifies ............................................... 211 A.

Absolute Differential Raman Scattering Cross Section of Nitrogen ............. 212

B.

Differential Raman Scattering Cross Sections of Gaseous Samples ............. 213

CHAPTER 9 P A R A M E T R I C MODELS FOR I N T E R P R E T I N G R A M A N I N T E N S I T I E S ........................................................................................................... 215 I.

II.

Rotational Corrections to Polarizability Derivatives ............................................ 216 A.

Zero-Mass Method ...................................................................................... 218

B.

Heavy-Isotope Method ................................................................................ 219

C.

Relative Rotational Corrections ................................................................... 223

Valence-Optical Theory of Raman Intensities ..................................................... 223 A.

Theoretical Considerations .......................................................................... 224

B.

Valence Optical Theory of Raman Intensifies: An Example of Application ................................................................................................. 232

C.

Compact Formulation of VOTR .................................................................. 235

D.

Compact Formulation of VOTR: An example of Application ...................... 239

111. Atom Dipole Interaction Model (ADIM) .............................. ............................... 245 IV.

Atomic Polarizability Tensor Formulation (APZT) .............................................. 249 A.

APZT: An Example of Application ............................................................. 253 xvii

V~

Relationship Between Atomic Polarizability Tensors and Valence Optical Formulations of Raman Intensifies ...................................................................... 258

VI.

Effective Induced Bond Charges From Atomic Polarizability Tensors ................ 261 A.

Theoretical Considerations .......................................................................... 261

B.

Applications ................................................................................................ 263

C.

Discussion of Effective Induced Bond Charges ........................................... 266

C H A P T E R 10

A B INITIO C A L C U l a T I O N S O F R A M A N I N T E N S I T I E S ................................. 273 I.

II.

Computational Methods ...................................................................................... 274 A.

Finite Field Calculations of Raman Intensities ............................................. 274

B.

Polarizability Derivatives from the Energy Gradient ................................... 275

C.

Analytic Gradient Methods ......................................................................... 275

Calculated Raman Intensifies .............................................................................. 276 A.

Basis Set Dependence ofAb Initio gaman Intensifies .................................. 276

B.

Influence of Electron Correlation on Quantum Mechanically Predicted Raman Intensifies ........................................................................ 278

REFERENCES ............................................................................................................ 283

A U T H O R INDEX ....................................................................................................... 303

SUBJECT INDEX ....................................................................................................... 317

.~176 XVlll

CHAPTER 1

ABSORPTION

OF INFRARED

RADIATION

BY MOLECULES

I.

Theoretical Considerations .................................................................................... 2

II.

Selection Rules For Infrared Absorption .............................................................. 12

m~

A.

Harmonic Oscillator Selection Rules .......................................................... 12

B.

Symmetry Selection Rules .......................................................................... 14

Experimental Determination of Infrared Intensities .............................................. 17

2

GALABOV AND DUDEV

I. T H E O R E T I C A L

CONSIDERATIONS

The probability of absorption of a photon with energy hvn,n- by a molecule per unit of time leading to a transition between a lower energy state n" to higher state n' is given by [1-3]

8~;3 Wn': =-~ (n'[ X ej(u~ rj) J

I

n#)2

P(Vn,n,,).

(1.1)

In expression (1.1) ej and rj are the electric charge and position vector of atom j ill a molecule, rj refers to an arbitrary molecule-fixed Cartesian system, u x is the position vector of the photon with respect to a space-fixed Cartesian system. The quantity p called radiation density is equal to the number of photons with energy hvn,n- per unit volume. It is understood that the Bohr condition En, - F_.n,= hvn,n- must be satisfied. The polarization vector of the photons ux does not affect the molecular wave functions. A quantity called electric transition dipole may, therefore, be defined

P c : =(n'[ E ejrj In">. J

(1.2)

Since the electric dipole moment is given by

p

=

Xejrj,

(1.3)

J

Eq. (1.2) becomes

Pn'n" = ( n ' l p In").

(1.4)

Pn'n" has components along the x, y and z axes of the molecule-fixed Cartesian system. The directions of ux and Pn'n" need not coincide since molecules are randomly oriented. There is, therefore, an angle 0 between the vectors ux and Pn'n"- Eq. (1.1) may then be rewritten as

8~;3 n' [ p [ n") 2 COS2 0 P(Vn,n-).

Wn,n- = - - ~ (

(1.5)

cos20 should be taken as an average over all possible orientations of the molecule in space

ABSORPTION OF INFRARED RADIATION

3

2rc COS2 0 = --L1 f cos 2 0 Sin 0 dO dO = _1. 4~ 0 0 3

i

(1.6)

The transition probability associated with absorption of a photon with energy hvn,n,, is then given by 8/I;3
(1.7)

The quantity Bn,n,, = (81t3/3h2)(n'lpln") 2 is the well known Einstein coefficient of absorption. Per unit radiation density it is equal to the transition probability. The Einstein coefficient depends on the molecular structure and may, eventually, be used to characterize molecular properties on the basis of experimentally determined intensities of the respective transitions. In the infrared region each vibrational transition is accompanied by a number of quantum transitions between rotational states of the molecules. In lower resolution spectra or if conditions for considerable broadening of absorption lines are present, the accompanying rotational transitions determine usually a non-symmetric PQR structure of the infrared band. In higher resolution spectra the individual rotational lines are separated. The intensity of an infrared absorption band represents, therefore, a sum over the intensities of all fine structure lines associated with the respective vibrational transition. It is, thus, necessary to describe the factors determining intensities of the component rotational lines and, then, see how these sum up into overall intensity of an infrared band. The probability for a reverse quantum transition from higher (n') to lower energy state (n") is also determined by expression (1.7). Therefore, the probability for absorption of electromagnetic radiation will depend on the number of molecules per unit volume of absorbing medium in the lower (Nn,,) and higher (Nn,) states

8~3 (n' I pin") 2 P(Vn,n-)(Nn--Nn,).

(1.8)

3h 2

For every elementary act of interaction of the electromagnetic radiation with a molecule an energy hvn,n, is absorbed provided some conditions are met. The energy absorbed by a differential element with a cross section equal to unity and thickness dl is equal to 81t3 n"

= vn'n" Sh-- <

Ip I

n") 2

- N

(1.9)

4

GALABOV AND DUDEV

Since the radiation beam intensity is equal to I = c p, we obtain 8~3I

n,,)2

(Nn" - Nn') dl"

(1.10)

or

-dlnI

8~3

=

Vn'n"~

(n' I P In") 2 (Nn" -Nn')dl.

(1.11)

For a given electronic state of the molecule the matrix elements (n' [ p[ n") refer to vibrational-rotational wave functions q~VP,- In the approximation in which vibrational and rotational energies are separable we may write [4]

P [n")2:~

WX~R P WVR dxVR (1.12) t,

=

I,

n

n

~Fv ~FR P~Fv ~FR d'rvR .

The same approximation expressed in terms of quantum numbers yields

Vn'n" = VV,W + VR,R--

(1.13)

It is now of interest to follow how the dipole moment operator acts upon the matrix element components [Eq. (1.12)]. The transition dipole matrix element, as given in expressions (1.1) and (1.5) is defined with respect to a molecule-fixed Cartesian reference system. In this system with axes denoted by x, y and z the dipole moment is expressed as p2 = p2x +p2 +pz2 "

(1.14)

In order to describe the rotational motion of the molecule it is necessary to express p in terms of coordinates referring to a space-fixed Cartesian system. The transformation from molecule-fixed (x, y, z) to space-fixed (X, Y, Z) coordinates is defined as [4] PX = OXx Px + OXy Py + OXz Pz PY = ~Yx Px + ~Yy Py + ~Yz Pz PZ = OZx Px + OZy Py + ~Zz Pz 9

(1.15)

ABSORPTION OF INFRARED RADIATION

5

cl)Fg (F = X, Y, Z; g = x, y, z) are the direction cosines between the respective axes. The transition dipole matrix element may then be written in the form

F=X,Y,Z

~ W~r* V~t* (~FxPx + ~ F y p y +~Fzpz)W~W~,dXvR .

(1.16)

To evaluate the dipole matrix element for a transition between two vibrational states v" -~ v' it is necessary to sum expression (1.16) over all rotational quantum numbers R' and R" associated with the vibrational transition. Considering Eqs. (1.12) and (1.13) and the invariance of molecular dipole moment with respect to orientation in space and, therefore, to rotational coordinates, the following expression for the transition matrix element between vibrational-rotational states n' and n" is obtained [4]

(n'lpln")2=

E

F=X,Y,Z

I E

g=x,y,z

(1.17) -

( R' I ~g [R" ) is the rotational wave function matrix element between space - fixed (F) and molecule-fixed (g) Cartesian axes. The matrix elements ( n ' l p l n " ) are primarily associated with vibrational transitions. Since, however, the rotational quanttma numbers R' and R" also change, in a consistent treatment the transition probabilities of all rotational components of a vibrational absorption band must be evaluated. Usually, an approximate approach is adopted [4] and the direction cosine matrix elements replaced by classical averages over the cosines. These are equal to [3-5]

(l)Fg r

= (1/3) ~gg,

(1.18)

where 8gg, is the Kronecker delta symbol (~gg, = 0 if g ~ g' and ~Sgg,= 1 if g = g'). If we now tam back to Eq. (1.8) the population difference (Nn,,- Nn,) must also be approximated analogously to the treatment of the lransitional dipole moment. Using the Boltzmann distribution relations we obtain Nn, = Nn- e

- 0av n, - hvn.) / kT

NV,R, = NV-R,, e

- h(Vv, v. + VR,R. ) / kT

(1.19)

6

GALABOV AND DUDEV

In Eq. (1.19) k is the Boltzmann constant and T the absolute temperature. By omitting the rotational terms and taking into account relation (1.18) expression (1.11) becomes 8/i; 3 - dlnI = Vv,v,, 3oh ( V' I p IV" )2 (Nv, ' _ NV,) dl.

(1.20)

Aside from the frequency factor Vv,v,, the remaining part of the expression correctly accounts for the transition probabilities of all rotational components of an infrared absorption band since they have identical vibrational matrix elements. These, however, differ in frequency in view of relation (1.13). In integral form Eq. (1.20) becomes 8/I;31 ln(Io/I) = Vv,v,, 3oh ( V' I p IV" )2 (Nv, _ NV,) "

(1.21)

Io is the intensity of the incident beam. It is a reasonable approximation to accept that the Einstein coefficient is a constant for a quantum jump between two vibrational states [4]. The integration over the entire band will, therefore, yield the following expression A = _1 ~band In (~) dv = v v ' v " 3sn3 - ~ ( v' I p ]w )2 (Nv. _ NV,) 1

(1.22)

If the integration is to be carried out over the individual rotational components of a band the result would be, evidently, slightly different. This inaccuracy may be treated by placing the firequency factor on the left hand side lr = 1 1

and

In

dirty=-3ch

<

l pl

-Nv').

(1.23)

The integrated intensity of the band F becomes, however, frequency dependent. Thus, the relative intensities of different bands in a molecule, or in different molecules, cannot be directly compared. This inconvenience is, very possibly, the reason that in most studies the integrated absorption coefficient A is preferred. It is now necessary to consider in more detail the matrix element and the population factor appearing in the right-hand side of Eqs. (1.22) and (1.23) in order to arrive at an expression that will directly relate the observed integrated infrared absorption band intensities to quantifies characterizing molecular structure. In the usual notation the (V' I p IV") matrix element reads (V'lp

IV") =j'q"-~* p %

dxv.

(1.24)

ABSORPTION OF INFRARED RADIATION

7

Vibrational wave functions may be presented as a product of linear harmonic oscillator wave functions defined with respect to a set of generalized molecule-fixed coordinates called normal vibrational coordinates [4] tFv = q/l(Ql) tF2(Q2) .-- Vk(Qk) -.. tFsN-6(QsN-6)

(1.25)

or

qJV =

3N-6 rI ~i(Oi).

(1.26)

i

The coordinates Qi are determined in the process of a semiclassical treatmem of molecular vibrations [3-6]. The principal aim of these calculations is to define the specific coordinates Qi, in the basis of which the Schr6dinger wave functions for the vibrational motion of a molecule are reduced to 3N-6 simple linear harmonic oscillator wave functions. One of the mathematical expressions of this result is Eq. (1.25). 3N-6 is the number of vibrations in an N-atomic molecule (3N-5 in the case of linear molecules). Described in terms of Qi the complex vibrational motion of a molecule is expressed as a superposition of 3N-6 linear harmonic oscillator vibrations, each having specific form as described by Qi and own frequency of oscillation. More comments about normal coordinates will be given in the following section. For a complete description of the theory of normal vibrations the reader is referred to a number of monographs [3-6]. For small vibrations the molecular dipole moment may be expressed as a Taylor series along the displacement coordinates Qi

p=po+E 1 +'6

k

[

0

/)3p

,

/

)

0

Q, (1.27)

E (~Qk~--~lOQm Ok Ol Om + --k,l,m 0

For small displacements from the equilibrium configuration, under conditions of mechanical harmonicity (the potential energy is a quadratic function) and electrical harmonicity (the dipole moment is a linear function of vibrational coordinates), the higher terms in Eq. (1.27) are neglected. In the second part of this section we shall discuss in some detail the selection rules that govern vibrational transitions associated with absorption of a photon in the infrared region. We need first to derive, however, an expression relating the measured integrated absorption intensities with quantifies reflecting the electric charge fluctuations taking place in molecules with vibrational distortions. As we shall see, the harmonic oscillator selection rule restricts the allowed transitions only to those involving a change of a single vibrational quantum number by

8

GALABOV AND DUDEV

• For a fundamental transition from ground to one of the first excited vibrational states associated with a normal vibration described by the coordinate Qk the transition dipole matrix element is reduced simply to

J Vk

(Qk) P ~FI~(Qk)dQk =

0

J Vk

(Qk)Qk ~F~'(Qk)dQk-

(1.28)

Substitution with the analytical expression for the wave function qJk(QQ into the integral part of the right-hand side of Eqn (1.28) results in [4] ))1/2 _,v+l h (v k + 1 . J" ~Fk (Qk)Qk V~" (Qk)dQk = 8X2t,Ok

(1.29)

This expression is valid for all excitations involving a change in a vibrational quantum number by 1. Therefore, in the harmonic approximation the integration is carried out also over transitions from higher vibrational states associated with the so-called hot bands. Since these energy levels become populated at higher temperatures, it follows that within the approximations introduced so far, the intensities of absorption bands (Avk = 1) will be temperature independent. In expression (1.29) cok is the harmonic frequency of the vibration. Taking into account relations (1.28) and (1.29) the following expression for the integrated intensity of an infrared absorption band is obtained

A k = ~t'Vk (v k + 1) (Nv,, - N V,) 3cto k

(1.30)

In harmonic approximation, by using the Boltzmann distribution relation, the number of molecules in a given state with vibrational quantum number vk can be expressed as Nvk = Nvo e

(- hVk / kT) Vk

(1.31)

where Nvo is the number of molecules in the ground vibrational state (vibrational quantum number equal to 0). Nv0 is related to the total number of molecules per unit volume N through the relations

ABSORPTION OF INFRARED RADIATION

9

N = Nv0 + Nvl + Nv2 + ... + Nvk + ... = Nvo + Nvo e (-hvk/kT) + Nvo e(-hvk/kT)2 +...+ Nv0 e (-hvk/kT)vk +...

(1.32)

oo

= Nv0

E e(-hvk/kT)vk = Nvo ( 1 - e - h v k / k T ) -1 = Nvo Ok, Vk=0

where Ok = (1 - e

- hv k / kT

)-1

(1.33)

is the vibrational partition function associated with the k-th mode. Thus, from Eqs. (1.31) and (1.32) we have

N

Nvo = - - Ok

(1.34)

Nv k = N e(-hvk/kT)vk.

(1.35)

and

Ok

The number of molecules in a higher energy vibrational state with vibrational quantum number vk+l is

N e(_hvk/kT) (Vk+1)

Nv k +1 = Ok

(1.36)

Combining (1.35) and (1.36) we obtain N (1-e-hVk/kT) e(-hvk/kT)vk " Nvk - N v k + l =~-k

(1.37)

Taking into account the result of summation in Eq. (1.32) and that oo

Vk e(-hvk/kT)vk = (1-e-hVk/kT) -2 e-hVk/kT Vk =0 we finally obtain

(1.38)

10

GALABOV AND DUDEV

oo

E (Nvz -Nv k+1)(Vk + 1)

v k =0

-__~ Ok

(1.39)

(,-e-~k'kT)

oo

X

e(-hvk/kT)vk (v k + 1)= N.

Vk=0

The integrated absorption coefficient [Eq. (1.30)] is then expressed as Ak =~1; band in ( ~ ) d v = Nn (V~k_k)(3Q~k)2

(1.40)

The normalized with respect to molar concentration (N = m No) absorption coefficient becomes 1 A~=~I~. '~

Nox v k d~---~ ~ ~

(1.41)

(t3px/tgQk) 2 + (o3p/t3Qk) 2 + (t3pz/CqQk) 2 .

(1.42)

with (t3p/t3Qk) 2 =

In expression (1.41) m is the molar concentration and N Othe Avogadro number. Usually, in practice, wavenumbers are used to determine band positions instead of classical frequencies. Thus, Eq. (1.41)expressed in terms ofwavenumbers (~ = v/c) reads:

A~=~f~ 1.

d~=~ ~

(1.43)

In further discussion vibrational frequencies will be expressed in cm-1. Therefore, for the sake of simplicity hereafter symbols v and o will be used to denote the respective wavenumbers. Since for most polyatomic molecules the harmonic frequencies (Ok) are difficult to determine experimentally, expression (1.43) is approximated by omitting the frequency factor to

(/2

Ak = N0___..~x.,Op 3c 2

it

(1.44)

ABSORPTION OF INFRARED RADIATION

11

The analogous expression for the quantity F k is

(1.45)

3C2Vk

For the reasons mentioned above cok is replaced by the frequency of the observed band center Vk. Ak and F k are related by the approximate relation Ak = F k v k .

(1.46)

The approximation comes from the different approach in integrating the observed band areas in evaluating Ak and F k [see expressions (1.22) and (1.23)]. IfEqs. (1.44) and (1.45) are expressed in terms of SI units (with exception of cm-1 for wavenumbers), as suggested first by Steele [7], for A k and F k we obtain:

(1.47)

3c2(4mz0)

(1.48)

Fk = 3c 2 (4~O)V k

where so is the permitfivity of vacuum. Thus, Ak is measured in km mo1-1 and F k in cm 2 tool-1. The contribution of rotational quantization to the integrated absorption coefficient has been treated for symmetric rotor molecules [8]. Summation over rotational quantum numbers for parallel and perpendicular bands introduces a correction factor in the expression for the absorption

Ak = f 1

tI11+exp( 0

<

0,

(149,

In relation (1.49) B is the rotational constant, c the light velocity and VkOthe frequency of the pure vibrational transition. Substitution of typical values for B and Vk~ shows that the error in 8p/SQk dipole moment derivatives due to neglecting rotational quantization may not exceed 5%. For heavier molecules the error will be less than 1 percent. Therefore,

12

GALABOV AND DUDEV

the use of expressions (1.47) and (1.48) in deriving the dipole moment derivatives with respect to normal coordinates is justified in most cases. For molecules with higher symmetry some vibrational transitions may be degenerate. In such cases the following expression relating the observed integrated absorption intensities and the dipole moment derivatives is used

Nor:

f(0Pgx)2

(~pg)2

/~)pg/2 t

(1.5o)

The summation is over all degenerate transitions with the same energy. Vibrations in such symmetric molecules are usually polarized along a single axis, provided that the x, y and z directions are chosen in accordance with the symmetry properties of molecules. For such degenerate vibrations the derivatives 01ag /0Qk, 0p~/0Q k and 0pgz/0Qk are equal. The absolute values of these quantifies may, therefore, be determined without difficulty. It is seen from relations (1.47) and (1.48) that absolute values of dipole moment derivatives can only be evaluated fxom experimental integrated intensifies. Therefore, the directions of charge shifts accompanying particular vibrational distortions remain undetermined. This is a major difficulty in any further reduction of vibrational intensity data. For many years the sign ambiguity problem for the dipole moment derivatives has been a cause for the limited application of vibrational intensities in structural analysis. Another formidable problem arises from difficulties in deriving individual Cartesian components of o~/~:~k derivatives from experimental measurements. For molecules in the gas-phase the components of the t~/c3Qk vector may only be determined for a molecule with sufficient symmetry, such that one of c~p~//~k components only is different from zero. In the general case, it is necessary to establish the direction of polarization of vibrations in order to further rationalize the structural information implicit in the measured absorption intensities. It is, therefore, not surprising that most vibrational intensity studies have been restricted to relatively small and symmetric molecules.

II, S E L E C T I O N R U L E S F O R I N F R A R E D A B S O R P T I O N

A. Harmonic Oscillator Selection Rules In deriving the relations between infrared absorption intensities and dipole moment derivatives we have restricted the treatment to transitions involving a change of a single

ABSORPTION OF INFRARED RADIATION

13

vibrational quantum number by +1. The respective selection rule arises from the properties of the linear harmonic oscillator wave function. In general, the selection rules indicate: (a) allowed transitions with absorption bands observed in the spectrum; (b) transitions that are forbidden with absorption bands of zero or small intensity. As mentioned, some selection rules are determined by properties of the harmonic oscillator wave functions. A second set of selection rules are associated with the symmetry properties of vibrations. The harmonic oscillator selection rules for vibrational transition can be evaluated using the expression of the dipole moment as a power series with respect to normal coordinates. On the basis of expressions (1.25) - (1.27) the dipole moment matrix element for a transition between vibrational states V' and V" may be written as

(v'l pIv")=IVv" p % d~v = P0 I"I I ~Fk ~Fk dQk k

.+

Z r

f C Q~ ~ dQ~FIf v;',ei' dQ1

k

1,k

l Z 2p /c)Qk2 )

+2 1

o

(1.51)

I ~Ij'k*Q2 ~k dQk H I ~PI*~I'11' dQl l~k

[~2P/(c~QkaQl)~I~FkQk ~Fk dQk f~FI*QI~FI'dQI I-[ I ~ W " m dQm m#l#k

+ higher terms. The first term in expression (1.51) is equal to zero except for the case with Vk'= Vk". This is determined by the orthogonality of vibrational eigenfunctions. In fact, it is zero for any transition between different vibrational states. The linear term is not zero for transitions involving a change of a single quantum number by + 1 only. Because of orthogonality of the function ~Pl the integral PPl'* ~Pl" dQl will be zero if any other vibrational quantum number except Vk is changed (Vl, l~k). The harmonic approximation restricts the dipole moment expansion to the constant and linear terms. Thus, the selection rule associated with the approximation of electrical harmonicity, states that transitions involving a change by + 1 of just one of the vibrational quantum numbers of the linear harmonic oscillator functions defining the vibrational states of molecules are only allowed. Aside from this, for a transition to take place at least one of the Cartesian components of the (0p/0Qk) 0 derivatives should differ from

14

GALABOV AND DUDEV

zero. We shall later see how the symmetry of vibrations determines the selection rules associated with these derivatives. The third term in expression (1.51) considers transitions associated with a change in a vibrational quantum number by 2. It governs the intensities of the usually weak overtone bands. The possibility of observing such transitions is determined by the fact that vibrations in real molecules are not strictly harmonic, both with respect to potential energy and dipole moment functions. When one quantum number is changed by 2, the linear term has also a f ~ t e , usually small, contribution to the matrix element. The fourth term is associated with the intensities of the weak combination bands (vibrational sum or difference bands). These bands are due to transitions involving changes by • 1 of two vibrational quantum numbers. Absorption bands associated with higher terms in the dipole moment function expansion are also observed. As expected, their intensities are orders of magnitude lower than the intensities of fundamental transitions. Nevertheless, studies on higher overtone transitions flourished during the past fifteen years [9,10]. The main reason for these developments is, very possibly, the unexpected "local mode" rather than "normal mode" properties for some of these transitions. Interesting opportunities for studying properties of individual chemical bonds in complex molecules have been revealed. Applications have mostly concentrated on higher overtones of C-H stretching modes [9,10].

B. Symmetry Selection Rules As stated before, a transition between different vibrational states may take place ff the respective coefficients in the dipole moment expansion as a power series to normal coordinates are not equal to zero. For a fundamental transition at least one of the components/~px//R~, i ~ p y / ~ and/)pz//~Qk must not be zero. For overtone transitions a non-zero value is required for at least one of the derivatives ~px/C3~ 2, ~ p y / ~ 2 and oa2pz/~2. A group theoretical analysis may determine the infrared active transitions by considering the symmetry properties of the vibrational wave functions of the interacting states and of molecular dipole moment. In most general terms, a transition between vibrational states V' and V" will take place if at least one of the component matrix elements differs from zero ( V ' [ Px IV")

(V' [ py IV")

( V ' [ Pz I V " ) .

(1.52)

If all three dipole matrix elements are zero the transition will be symmetry forbidden or infrared inactive.

ABSORPTION OF INFRARED RADIATION

15

The symmetry selection rules are derived by studying the effect of symmetry operations on the matrix elements. The rules stem from the property of dipole moment matrix elements to be invariant with respect to a symmetry operation (R) R ( ( V ' I p g [ V " > ) = (V' I pg IV")

(g = x , y , z ) .

(1.53)

Since the transition dipole is a physical observable, it is evident that its value should be independent under symmetry operations. In other words, the intramolecular charge distribution and fluctuations are invariant with respect to symmetry operations. The representation of transition dipole moment element is given by the direct product of the representations of the respective vibrational wave functions and dipole moment component rk = rv, • rpg • r w

(g = x, y, z)

(1.54)

where k is an index of the k-th normal mode. Since the dipole moment is a vector with components directed along the axes of the reference Cartesian system, it is clear that under a symmetry operation its components will transform as the respective x, y and z Cartesian axes. The same arguments hold for the igpx/0Qk,/gpy/~ and C3pz/~ dipole derivatives. Therefore, the symmetry representations of the dipole moment components, as present in expression (1.52) coincide with the representations of the respective Cartesian axes. The symmetry properties of vibrational wave functions are treated in detail elsewhere [3] and will not be discussed here. The representation of the ground vibrational state wave function belongs to the point group of the molecule at equilibrium configuration. The direct product F k coincides with one of the irreducible representations of the molecular point group. The component matrix element (V'[pg[V") will be different from zero only if the resulting F k coincides with the totally symmetric representation of the point group of the molecule F 0. In the case of degenerate vibrations F k is a reducible representation. The infrared active (allowed) transitions must have the totally symmetric irreducible representation in the structure of F k associated with the respective degenerate mode. The synunetry selection rule may also be expressed in alternative ways. An inflated transition is not forbidden only in the case where the direct product of the presentations of the two interacting states Fv, XFv,, coincides with the representation of at least one of the dipole moment Cartesian components. For a fundamental transition (v' k = 1, v" k = 0) the above requirement concerns the irreducible representation of the excited level (v'). The selection rule for such transitions is simply r v, • rpg=

to.

(1.55)

16

GALABOV AND DUDEV

Similar symmetry restrictions also apply for overtone and combination bands. As already discussed, these transitions are not aUowed under the harmonic oscillator selection rules. R should be pointed out that even ff a given transition is not forbidden under both symmetry and harmonic oscillator selection rules, it may have a very low intensity. This will be determined by the particular form of the vibration and the electronic structure of the molecule. The assignment of a given band to infrared active or forbidden transition is, therefore, not always a straightforward task. The derivation of expressions relating observed integrated infrared absorption coefficients with dipole moment matrix elements for the respective transitions shows that the experimental quantifies contain important structural information. It is related with the distribution and dynamics of electric charges in molecules. We should bear in mind, however, that there are a number of restrictions associated with the possibility to determine dipole moment derivatives with respect to normal coordinates. In many regions of the observed infrared spectrum for any molecule of a medium size, a strong overlap of closely situated bands is usuaUy present. Thus, individual intensifies for all fundamental transitions may not be accurately determined. Lately, with the development of advanced software for band deconvolution and curve fitting this difficulty has been, to some extent, overcome. As already mentioned, another formidable problem arises from the necessity to know the exact direction of polarization for each vibrational mode, so that individual Cartesian components of the Op/0Qk derivatives are evaluated. So far, no general approach to solve this problem experimentally for molecules in the gas-phase has been developed. Thus, in most cases, an elaborate molecular analysis of observed vibrational absorption intensities is only possible for higher symmetry molecules. Existing perturbations in the spectra associated with Fermi resonances, Coriolis interactions and strong anhannonicity effects may often hamper the interpretation of experimental intensity data. The sign ambiguity problem for dipole moment derivatives, as already discussed, is also present. Finally, it should be emphasized that the quantifies 0p/0Qk contain in a rather obscure form the structural information sought. This is due to the very complex nature of normal coordinates. It is, therefore, essential to further reduce the experimental 0p//~)k derivatives into quantifies characterizing electrical properties of molecular sub-units -atomic groupings, chemical bonds or individual atoms. Various theoretical formulations for analysis of vibrational intensities have been put forward. The approaches developed a r e quite analogous to the analysis of vibrational frequencies in terms of force constants. As known, force constants may be associated with properties of molecular sub-units. If such a rationalization of intensity data is successfully performed, another important aim of spectroscopy studies may become possible: quantitative prediction of vibrational intensities by transferring intensity parameters between molecules containing the same

ABSORPTION OF INFRARED 1LM)IATION

17

structural units in a similar environment. The analogy with transferable force constants should again be underlined.

III.

EXPERIMENTAL DETERMINATION OF INFRARED INTENSITIES

The theoretical models for interpretation of infrared intensities presented in the subsequent chapters have been largely applied in analyzing gas-phase experimental data. Gas-phase intensities provide an unique opportunity to study in a uniform approach the mterrelatiom between molecular structure and intensity parameters. This is due to the fact that, in contrast to vibrational frequencies, the absorption coefficients depend strongly on the phase state and on solvent effects. Intensities of different modes of the same molecule are not influenced in a systematic way by the solvent. The variations of absorption coefficients may reach tens and hundreds percent. Accurately determined gasphase intensities are, therefore, of fundamental importance as a source of experimental information on intramolecularproperties. Past difficulties in experimental measurements of integrated infrared intensities have been associated mostly with the low resolving power of spectrometers, poor accuracy on the ordinate and absence of computer facilities for band integration, deconvolution and curve fitting in overlap parts of the spectra. It is clear that presently we have far better experimental means for accurate determination of the integrated intensities of individual absorption bands. Still, however, careful considerations of a number of possible sources of errors are needed in order to obtain sufficiently accurate intensity data. Some of these problems will be discussed later on. It is interesting that the methods developed for experimental determination of vibrational intensities in the gas-phase were aimed at resolving problems arising mostly from the low resolution power of the available spectrometers at the time. It may appear that nowdays, when the researchers have access to instruments with resolution of the order of a few hundredths or even few thousandths of a wavenumber, these techniques may be of lesser importance. Although this is partly true, the current experimental approaches for experimental determination of vibrational intensities fully rely on the original developments. This is determined by the fact that these methods not only compensate for the effect of low resolution on intensities but also provide criteria for the accuracy of measurements and the influence of such phenomena as adsorption of sample gas or slow diffusion process. Thus, the extrapolation method of Wilson and Wells [ 11], further developed by Penner and Weber [12], is the standard approach for experimental intensity studies.

18

GALABOV AND DUDEV

Early attempts for experimental measurements of infrared intensities [13,14] resulted in greatly divergent values for the same molecule. It was soon realized that most of the difficulties were associated with the low resolving power of the spectrometers used [15]. The problems arise from the fact that the incident infrared beam emerging from the monochromator is not strictly monochromatic, but contains a band of frequencies around the frequency v' determined by the slit function g(v,v'). There is, thus, a perfectly good chance for the intensity of the transmitted radiation I by a cell I = Io exp (- ~ p 1)

(1.56)

to differ from the value corresponding to a monochromatic beam with frequency v'. This is particularly possible if the measurements are carried over rotational f'me structure of an infrared band where sharp variations of the absorption coefficient ct are expected. In expression (1.56) p is the pressure of the sample gas, 1is the optical path length of the cell and I0 is the intensity of the incident radiation. The quantity of principal interest in intensity measurements is the integrated absorption coefficient A as defined by Eqs. (1.43), (1.44) and (1.47). For gas samples the respective expression is (1.57)

Due to the finite width of the slit function g(v,v') the measured intensity at a setting v' will not be equal to the true absorption intensity I. An apparent intensity T(v') will be in fact determined. It is given by the integral oo

r ( v ' ) = I I(v)g(v,v')dv.

(1.58)

g(v,v') is the portion of the light of frequency v that reaches the detector when the spectrometer is set at frequency v'. The integration can be carried out from --oo to +oo since g(v,v') is different from zero only in the immediate vicinity of v'. The width of the frequency band is of the order of the spectrometer resolution. Analogous expression can be written for T o oo

T0(v') = I Io (v)g(v,v')dv.

The apparent absorption coefficient J3(v') can, therefore, be defined as

(1.59)

ABSORPTION OF INFRARED RADIATION

]3(v') = (1/pl) In [To(v')/T(v')].

19

(1.60)

The integration over the entire interval of an vibration-rotation band will produce the apparent integrated absorption coefficient B

B : Ibaad~ dr'

:lSb=d pl

(1.61)

The Wilson and Wells [ 11] extrapolation theorem proves that Lira B = A. pl-~

(1.62)

Since both p and 1 can be varied in experimental conditions, the theorem (1.62) provides a convenient way of determining the true absorption coefficient A. The entire derivation of Wilson and Wells [ 11] will be presented since important considerations associated with the accuracy of intensity measurements emerge at different stages of proving the theorem. The difference (A-B) is examined

1 Sbandhl(TIO/dv'= 1 ~bandin( f(v') ~ f(v)g(v,v')_:_v/ It T0I ) p-I ~ g(v, v')dv) d r ' .

A - B = p--~

(1.63)

In expression (1.63) f(v) is f(v) = e x p ( - t x p l )

- I/I o

(1.64)

and I0 is assumed to be constant over the frequency range of the slit width. In presence of foreign gases, such as H20 and CO 2, this assumption may not hold. It is of interest to consider here the function f. Let us suppose that f is constant over the frequency range of the slit width. It becomes a constant multiplier of the inner integral and the entire expression (1.63) vanishes. B will then be equal to A. If, however, the width of individual rotational lines is of the order of the slit width, the exponential function f can vary considerably and B may be very different from A. The limit of A-B with p l y 0 needs now be considered. Expression (1.63) is differentiated with respect to pl to yield [ctfgdv l ~ g d__v1j dr'. Lira(A-B)= Lim I {c~-'.~Tg?~ J dv'= I ~/ct - SIgdv

(1.65)

20

GALABOV AND DUDEV

The function f = exp (- a p 1) approaches unity at the limit. Expression (1.65) will vanish and the theorem proven under certain conditions. First, it will vanish if the absorption coefficient {x is constant over the integration range which is the slit width. Note that expression (1.63) vanishes under the condition that f as an exponent of a, is a constant. In Eq. (1.65) the analogous requirement is for c~. Small variations in ~ can result in much larger changes in f. On the second place, expression (1.65) may vanish if the resolving power of the monochromator is constant over the width of the band regardless of {z. In mathematical terms this condition is expressed as g(v,v') = g(v - v')

(1.66)

g ( v - v') = g(v'- v).

(1.67)

and

If conditions (1.66) and (1.67) are satisfied, we have g(v - v') dv = ~ g(v - v') d(v-v') = G.

(1.68)

G is independent of v'. Another simplification follows f~{xgdvdv' = ~c~fgdvdv' : Gfr

(1.69)

Substituting (1.68) and (1.69) into (1.65) we obtain ~ct dv' - ~ctdv = 0 .

(1.70)

Consequently, Lim B = A pl-,0

(1.71)

if a number of conditions are met. The first is that I 0 is a constant in a resolved range. This can be achieved by removing external absorption fxom atmospheric H20 and CO 2. It is also clear that working under conditions of higher resolution betters the constancy of I0 in the narrower resolved range. In addition, the theorem (1.71) will hold if {z does not vary over the range of the resolution of the spectrometer, or if the resolution is constant over the entire vibrational-rotational band. If appropriate care is taken so that the above

ABSORPTION OF INFRARED RADIATION

21

o:0r V

J

F Bcl

I

50~1

0

I

I

I

1

1

I

1

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 c/,equivolent poth length in cm at S.T.P.

Fig. 1.1. The dependence Bpl/pl for the infrared active modes of methane (Reproduced from Ref. [ 16] with permission).

mentioned conditions are satisfied, Eq. (1.71) provides a convenient approach of determining the ~ue absorption coefficient A by extrapolating B to zero value of pl. It should be noted that the requirement for constant resolution over the integration range of a band is a specific property of the insmxment. It may not always be fully satisfied. For small molecules in the gas-phase at low pressure with clearly expressed fine rotational structure, the variations of r and f = exp(-~pl) at each rotational line is so extreme that the conditions for accurate extrapolation measurement may not be present. Following an early approach of Bartholome [14], Wilson and Wefts [11] recommended that for such molecules a foreign non-absorbing gas under higher pressure is introduced in the sample cell. It will induce a collision broadening of rotational lines and at sufficiently high pressure will lead to a collapse of the fine structure. Obviously, under such conditions the extrapolation method may be used with confidence. In the original method of Wilson and Wells the true integrated absorption coefficient A is determined as the slope of tangent at the origin of the dependence Bpl versus pl. Typical plots of this dependence are shown in Fig. 1.1. There is, however, certain arbiuminess in choosing the exact direction of the tangent at the origin. The method also requires that the measurements be done at very low partial pressures of the absorbing gas where the accuracy is low. Considerable errors can, therefore, result. The problem is resolved by working at sufficiently high pressures of the inert transparent

22

GALABOV AND DUDEV

160 iO.lcm CH4 ~i 120

,

.

....

U

T uE 8 0

4.95 cm CH4

O I

o ,e-

d~ 4o

0

.

.

.

.

.

2".50 cm

CH 4

100 200 300 400 Pressure of nitrogen inotmospheres

Fig. 1.2. Dependence between the apparent integrated absorption of the v 3 band of CH4 and the pressure of added nitrogen (Reproduced from Ref. [ 16] with permission). 1.2

O.6 ------,0.6

In(1~

---~1,0.4 I

! ---4o.2

~lOAtrn/

i

~OAt~~ 42Atm~

OAtm 2800

L

___---- ~ ~ ' N ~ 3000

1 3200

1~ ~C m -I

Fig. 1.3. v 3 band of methane under different pressures of nitrogen used as an external gas. The partial pressure of methane is unchanged (Reproduced from Ref. [ 16]

with permission).

ABSORPTION OF INFRARED RADIATION

23

foreign gas (Ar, N2, etc.) so that the Beer's low plot becomes a straight line passing through the origin. This procedure has been suggested by Penner and Weber [ 12]. Depending on the size of the molecules, the type of rotational fme structure or band shape, the adequate pressure may vary significantly. For larger molecules a pressure of one atmosphere may be sufficient. For small molecules a pressure of up to I00 atm may be needed to reach the linear region of the dependence Bpl/pl. An example is shown in Fig. 1.2. The high pressures used may cause, however, some additional complications. For larger molecules under high pressure a certain amount of the gas sample may deposit on the cell walls, though in absence of foreign gas when the partial pressure is measured, the sample is in a gas-phase. This is a particular property of the molecule under study and each case needs careful consideration so that appropriate conditions for measurements are chosen. Overend [16] has pointed out that pressure-induced absorption can affect the apparent absorption coefficient value. The effect is attributed to intermolecular interaction. It is manifested in the slow rise of the apparent absorption coefficient B as the pressure is increased. The phenomenon is clearly shown in Fig. 1.3. If such effects are present, the pressure-induced absorption has to be eliminated. This is achieved by extrapolating the linear part of the curve to zero pressure of the external gas for each B value determined.

This Page Intentionally Left Blank

CHAPTER 2

COORDINATES IN VIBRATIONAL ANALYSIS

25

26

GALABOV AND DUDEV

Accurate potential force field of the molecule is an essential prerequisite for the molecular interpretation of experimental dipole moment derivatives. The transformation of the 0P//~k dipole moment derivatives into quantifies characterizing the electronic structure of the molecule is only possible if the forms of the vibrations are known with satisfactory accuracy. Vibrational forms are determined in the process of normal coordinate analysis on the basis of data for the atomic masses, molecular geometry and potential force field [3-6]. In solving the mathematical problem for the vibrations of a molecule, a special set of 3N-6 or more coordinates describing the variations of molecular configuration with vibrational motion irrespective with the position or orientation of the molecule in space is needed. Most suitable are the internal or, as also called, natural vibrational coordinates [4,6]. These represent changes of bond lengths, interbond angles, out-of-plane angles and torsional angles [3-6]. For small displacement, in the harmonic approximation, the potential energy is represented in the space of internal coordinates Ri by the expression 11

2V = ~ Fij R i Rj, i,j=l

(2.1)

where n is the number of internal coordinates. The coefficients Fij are the harmonic force constants and, as known, represent the second derivatives of V with respect to the vibrational coordinates. The respective expression for the kinetic energy in terms of R i is I1

2 T = ~ G]] 11~i l~j. i,j=l

(2.2)

/tj are time derivatives of the internal coordinates. The coefficients Gij-1 are determined from data for the atomic masses and molecular geometry. The most rational approach in solving the classical vibrational motion problem includes simplification of the expressions for kinetic and potential energy by determining normal vibrational coordinates, Qk. These represent a special type of displacement coordinate in the basis of which the expressions for the kinetic and potential energy acquire diagonal form with coefficients of the diagonal terms equal to unity. Qk are related with the internal coordinates through the expression [4,6] Ri = ~ Lik Qk, k

i = 1, 2,3,..., n

(2.3)

Q = L-1R.

(2.4)

or in a matrix form R = L Q

COORDINATES IN VIBRATIONAL ANALYSIS

27

Here we do not aim at presenting the standard methods of solving the Newton vibrational equations. It should be emphasized that essential results of these calculations are the transformation coefficients Lik that define the relative contribution of each internal coordinate to the respective normal vibrations in the molecule. As underlined, the availability of accurate vibrational form coefficients are needed in intensity analysis. This is determined simply by the fact that vibrational intensities in the infrared spectra of molecules in the gas-phase (at low pressure so that no considerable intermolecular interaction is present) are governed by two principal factors: (1) the intramolecular charge rearrangements accompanying vibrational distortions and (2) the form of the normal vibrations as expressed in the coefficients of the normal coordinate transformation matrix L. The elements of L are determined by solving systems of linear equations of the type [4,6] X. {(GF)ij - X fiij} Lj = 0, J

i = L2,3 ..... n.

(2.5)

2. is the frequency parameter (Xk = 47r2 v 2 ) and 80 the Kronecker delta symbol. The L matrix elements are evaluated with accuracy depending on the reliability of both kinematic coefficients and force constants. Molecular geometry is usually derived from alternative experimental sources such as microwave spectra, X-ray and electron diffraction. Accuracy in vibrational analysis is limited to difficulties in evaluating force constants. As is known, in the inverse vibrational frequency problem, there is a multiplicity of solutions that satisfactorily reproduce the observed vibrational wavenumbers. On the other hand, the number of force constants defining the harmonic force field of a molecule is usually much higher than the number of observed vibrational frequencies in the infrared and Raman spectra. Experience has shown that a satisfactory solution of the inverse vibrational problem for the frequencies can be obtained by using an extensive set of experimental data that depend on the potential field [3-6,17-20]: observed wavenumbers in the infrared and Raman spectra, wavenumbers of isotopically substituted molecules, isotopic wavenumber shifts, constants of vibrational-rotational interaction (Coriolis constants), centrifugal distortion constants, mean amplitudes of vibrations and others. It is obvious that such detailed experimental data can only be obtained for relatively small and symmetric molecules. Since the coefficients of the potential field may, however, be transferable between molecules having the same structural elements, real possibilities exist for the analysis of vibrational spectra of complex molecular systems. Theoretical predictions of force constants from ab imtw molecular orbital calculations have proved extremely useful in deriving reliable force fields for many molecules [21,22]. Considerable simplifications in describing molecular vibrations are attained by introducing symmetry coordinates. These are related to the ordinary internal coordinates by the transformation, in matrix notation [4]

28

GALABOV AND DUDEV

S = UR.

(2.6)

U is an orthogonal matrix. The vibrational problem is solved separately for vibrations belonging to different symmetry species of the molecular point group. In this way the number of independent force constants defining the potential field is considerably reduced. The relation between symmetry and normal coordinates is given by the expression S = LS Q.

(2.7)

Aside from purely mathematical simplifications, the solution of the vibrational frequency problem in the space of internal coordinates offers considerable advantages from a physical point of view. As derivatives of the potential energy with respect to changes in bond lengths and valence angles, the evaluated force constants characterize in a clear manner the forces binding the atoms in a molecule. The relationships between force constants and electronic structural parameters of molecular sub-units are well established [23-26]. The transferability properties of potential force constants between structurally related molecules are also best expressed if defined in terms of internal coordinates. There are, however, certain shortcomings in using internal coordinates that need to be mentioned. These are mostly associated with the redundancies that exist between angular internal coordinates for some structural arrangements of atoms in a molecule, e.g. in the case of a common atom of three or more bonds. It is obvious that individual force constant values defined as derivatives of the potential energy with respect to redundant internal coordinates do not carry a definite physical information about molecular structure. This difficulty can be overcome by expressing the force constants in internal symmetry coordinate basis. Particularly advantageous is the use of local group symmetry coordinates describing the vibrations of atomic groupings such as CH 3, CH 2, CO, ring vibrations, etc. The redundancy problem is treated quite efficiently while, at the same time, preserving the direct association of force constants with basic structural units in a molecule. Earlier [27] and more recent studies [21,28-30] have shown the efficacy in applying local group symmetry coordinates in normal coordinate analysis. We discuss this problem in some length since very similar arguments are valid in vibrational intensity analysis as well. The principal aim in intensity analysis is, again, to transform the observed integrated absorption band intensities into parameters that will, hopefully, characterize molecular structure in a clear and physically sound way. As it happened, this goal became particularly difficult to achieve in analyzing vibrational intensities. Outside the treatment in the space of purely vibrational coordinates, such as internal, symmetry and normal coordinates, vibrational motion can be described on the

COORDINATES IN VIBRATIONAL ANALYSIS

29

basis of Cartesian coordinates as well [3-6]. Atomic Cartesian displacement coordinates find an extensive application in theoretical analysis of molecular vibrations, especially in theoretical developments associated with intensities in infrared and Raman molecular spectra. This is determined, very possibly, by the fact that, in contrast to the energies of vibrational transitions, the experimental dipole moment derivatives possess spacedirectional properties. It is, therefore, only natural that all theoretical formulations, aimed at reducing experimental integrated intensities into molecular parameters, employ at certain stages atomic Cartesian displacement coordinates in describing vibrational motion. As mentioned, these coordinates are also used in the vibrational frequency calculations. Defining the potential force field in the space of Cartesian coordinates results, however, in the loss of the simple physical sense of force constants as quantifies characterizing molecular structure. The transferability properties of the potential field coefficients are also obscured. Further in the present section, the transformation relations between various types of coordinates used in vibrational analysis are given. The motion of an N-atomic molecule possessing 3N-6 vibrational degrees of freedom (3N-5 in the case of linear molecules) can be defined by 3N atomic Cartesian displacement coordinates Axa, AYa and Az~ where a is an atomic index. In describing vibrational motion it is necessary to express the vibrational kinetic and potential energies into a basis set of coordinates that are independent of the translation and rotation of the molecule in space. This is achieved by imposing on the 3N Cartesian displacement coordinates conditions that eliminate the coupling between the three types of molecular motion. In explicit form these conditions as defined by Eckart [31] and Sayvetz [32] are Z m a Ax a = 0 Ct

~_~ ma Aya =O Ot

maAz a =0 Ot

ma (YOa Aza -zOa Aya)= 0

(2.8)

Ot

ma(Z~ Axa-x~ AZa)=O Ot

o o 2 ma(xaAya-yaAxa) =0 Ct

In expression (2.8)m a are atomic masses, x O, yO and z O are the equilibrium coordinates of the a-th nucleus. As defined, the Eckart-Sayvetz conditions neglect the vibrational-rotational interactions accounting for a part of the kinetic energy of the

30

GALABOV AND DUDEV

molecule, called Coriolis energy. It has been shown that, compared to vibrational kinetic energy, the Coriolis interaction terms are small and may be neglected. The Eckart-Sayvetz conditions imply that, if during the vibration a small translation of the center of masses is invoked, the origin of the Cartesian reference system is displaced so that no linear momentum is produced. The second Sayvetz condition, expressed in the last three equations of (2.8), imposes the constraint that, during vibrational displacements, no angular momentum is produced. Eq. (2.8) implies that the reference Cartesian system translates and rotates with the molecule in such a way that the displacement coordinates Ax~ Aya and Aza reflect pure vibrational distortions. It is evident that through Eq. (2.8) certain mass-dependency is imposed on the atomic Cartesian displacement coordinates. It is sometimes convenient to treat the vibrational motion in terms of massweighted Cartesian coordinates qax= m ~ a Axa qay = m ~ a Aya

(2.9)

qaz= m~~a Aza. The Eckart-Sayvetz conditions can easily be expressed in terms of the coordinates qag (g = x, y, z). Summarizing, the vibrational motion of an N-atomic molecule with 3N-6 vibrational degree of freedom can be described by 3N nuclear Cartesian displacement coordinates forming a column matrix X. Six degrees of freedom are related with translational and rotational motions of the molecule. These motions can be described by the external coordinates p (three translations and three rotations). In a transposed form the different types of vibrational coordinates may be presented as follows = (Q1--- Qk-.. Q3N-6) =

fi = (Arl ...Am ...ArN_I AVl ...ATm-.. )

= (S1 ...Sj ...$3N_6)

x,~

(2.10)

COORDINATES IN VIBRATIONAL ANALYSIS

31

Arn and AYm are stretching and angular internal coordinates. X(=) and q(a) are combinations of the three ordinary and mass-weighted Cartesian displacement coordinates of atom Qt. The analytical expressions for the external coordinates are [3,4]

~x = ~

ma Axa

'UY=Z Ot

maAya

Zz = ~

ma Aza

t~

Rx-Z m. (y".

Ay.)

Ry = Z mot (z O Axa - x O Aza) (t

Rz=X ma( x~ AYa-Y ~ Axa).

(2.11)

0~

The relations between the above sets of coordinates are given by the following matrix transformations R

B - (aR/ax)

A - (ax/aR)

s = tmx

tm = (as/ax)

AU = (ax/as)

p = I~x

~ = (ap/ax)

,~ - ( a x / a p )

R

LQ

X

X

Dq

D = (aR/aq)

R

=

=

=

BX

=

AR

=

(2.12)

ALQ

c = (aq/aR)

a = (a~aq).

All transformation matrices between the various coordinates except B and U are mass dependent. The combination between vibrational and rototranslational parts of expressions (2.12) leads to the following relations [33-37]

x

x : (A:o,

In terms of mass-weighted Cartesian coordinates Eq. (2.13) translates into

(2.13)

32

GALABOV AND DUDEV

(2.14)

Additional relations between the matrices defined in Eqs. (2.12) are AB + r 13 = E3N

Bet = 0

13A = 0

BA = E3N.6

13r = E 6

BM -1 B = G

13M-1 ~ = A

~.M cx = N -1

BM -1 ~ = 0

(2.~5) DC = E3N.6

~Sy= E 6

X = M - 89q

q=M 89

DT=0

8C = 0

DM 89 B

8M 8 9 13 .

E3N is a unit 3Nx3N matrix, 0 is a null matrix, G is the matrix of kinematic coefficients, M is a diagonal matrix containing triplets of the atomic masses. A is a diagonal matrix with element triplets of the reciprocal molecular mass and the reciprocal values of the principal moments of inertia. The matrices B, G and U are determined in the process of normal coordinate analysis [3-6]. The matrix A def'med for the ftrst time by Crawford and Fletcher [38] is obtained according to the expression A = M-1 B G -1

(2.16)

A - M-1 BUG~1 U ,

(2.17)

or

where G s is the symmetrized matrix of the kinematic coefficients. The matrix A can also be given by the expression [4] A = M- - ~ C .

(2.~8)

COORDINATES IN VIBRATIONAL ANALYSIS

33

The matrix a with dimensions 3Nx6 defines the relations between Cartesian displacement coordinates and the three normal translations and three normal rotations of a molecule. Vectors of the type X ~ ) (= 3X(a)/0xg) and X ~ ) (= 8X(a)/SRg) with g = x, y, z are formed. By arranging the vectors X(xag) and X (a) Rg in rows the matrices 8X//~g and OX/ORg are formed. Their elements are calculated according to the expressions 0X(a)/C~g = M-1 E3

(2.19)

ax(%aRg=

(2.20)

(( r (a))) [ - 8 9

M is the molecular mass matrix and 1 - 8 9= diag ( Ix- 89 Iy- 89 Iz- 89 Ix, Iy and I z are the principal moments of inertia of the molecule, r(a) is the position vector of the a th atom. By the symbol ((n)) an antisymmetric Cartesian vector is designated of the type

0 -n z

nz 0

ny

-n x

-ny / nx .

(2.21)

0

For each atom a 3x6 array is formed with elements of the derivatives 0X(a)/&g and

axo~)/aRg.

The elements of 13 [Eqs. (2.12)] are calculated according to [33,34] pl(Zx):l~l,3a. 2 = trha/M

131,3a.1 = 0

131,3a = 0

p2(xT):132,3a.2 = 0

132,3a.1 = ma/M

132,3a = 0

p3(Zz):~3,3a.2 = 0

133,3a.1 = 0

~3,3a = rna/M (2.22)

P4(Rx):~4,3a.2 = 0

~4,3ct-1 = -rnctz~/Ix 89

~4,3ct = mixY~/Ix 89

P5(ILy):~5,3a.2 = maz ~/Iy 89

135,3a_1 = 0

135,3a = -rnax~ t/Iy 89

P6(Rz)'136,3a.2 = - r n ~ / I z 8 9

136,3a-1 = max ~/Iz 89

136,3a = 0.

In expression (2.21) a is an atomic index (a = 1, 2, 3, ..., N), x~ t, y~ and z~ are the distances of atom a to the center of mass of the molecule and Ix, Iy, and I z are the inertial moments with respect to the equilibrium principal axes. The Eckart-Sayvetz conditions were explicitly presented with regard to the set of Cartesian displacement coordinates [Eqs. (2.8)]. From the relations R = BX and S = UR it is clear that the conditions of zero linear and angular momenta are also imposed on the coordinates Ri and Sj. Thus, certain mass-dependency is implicit in the definition of

34

GALABOV AND DUDEV

internal and synunetry coordinates. This has important consequences for molecular quantifies possessing space-directional properties such as dipole moment or polarizability derivatives when expressed in the space of these coordinates. In order to satisfy the Eckart-Sayvetz conditions in the cases of vibrational modes belonging to synunetry species that contain rotation, a certain compensatory rotation of the molecule is implicit in the respective internal or symmetry coordinates. These compensatory rotations will not be equal for different isotopic species of the molecule. Thus, it becomes necessary to calculate correction terms for the respective dipole moment or polarizability derivatives in order to account for these rotational contributions. Crawford first showed the relations between dipole moment derivatives in different isotopes [35] and later devised a method for calculating the respective correction terms [36].

CHAPTER 3

SEMI-CLASSICAL

MODELS

OF INFRARED

INTENSITIES

I.

Introduction ......................................................................................................... 36

II.

Rotational Corrections To Dipole Moment Derivatives ....................................... 40

gig.

A.

The Compensatory Molecular Rotation ...................................................... 40

B.

The Hypothetical Isotope Method .............................................................. 43

The Bond Moment Model .................................................................................... 51 A.

Theoretical Considerations ......................................................................... 51

B.

Applications ............................................................................................... 63

C.

Atomic Charge -- Charge Flux Model ........................................................ 68

D.

Group Dipole Derivatives as Infi'ared Intensity Parameters ......................... 72

35

36

GALABOV AND DUDEV

I. I N T R O D U C T I O N The theoretical analysis of the observed wavenumbers of vibrational transitions is aimed primarily at determining the potential force field of molecules and the form of vibrations as reflected in the elements of the normal coordinate transformation matrix L [Eq. (2.4)]. Two important goals are simultaneously achieved: (1) The intramolecular binding forces are quantitatively represented through the force constants matrix F. It is now recognized that the physically most plausible way of expressing inu'amolecular forces is by defining the F matrix coefficients as second derivatives of the potential energy with respect to internal vibrational coordinates; and (2) the normal vibrational coordinates are evaluated by solving the vibrational secular equation. As known, in terms of normal coordinates the complex vibrational motion of a molecule with 3N-6 degrees of freedom is reduced to a superposition of 3N-6 simple linear harmonic oscillator motions along the respective coordinates Qk. Thus, an essential and vital bridge between quantum mechanical and classical treatment of vibraffonal motion is established, and the theory of molecular vibrations acquires an integral and complete form. This unique combination of quantum and classical mechanics integrated treatment provides a solid basis in interpreting, calculating and predicting the frequencies of vibrational modes. The theory developed is also a basis for analyzing many other characteristics of the observed spectra, including vibrational intensities. The theory of the interaction of electromagnetic radiation with vibrating molecules defines a simple relation between absorption intensities in the infrared spectra and derivatives of molecular dipole moment with respect to normal coordinates as given by Eqs. (1.47) and (1.48). The success of the theory of vibrational frequencies is determined by the discovery of physically plausible approaches in describing vibrational motion and in devising efficient mathematical methods for evaluation of the respective molecular parameters. Valuable information about molecular structure has been accumulated over the years [36,17-26]. Numerous successful predictions of spectral properties of molecules confirm the validity and solid physical foundation of the models developed. The theory of vibrational transitional intensities is aimed at finding analogous approaches that would enable one to interpret, in a physically sensible and mathematically straightforward way, the observed intensifies of the absorption bands. Though the theoretical developments on vibrational intensities have now more than fifty years of history [39] the progress has been relatively slow. There are a number of factors that have negatively influenced the field. For many years the nature of the intramolecular charge fluctuation effects determining the intensities of vibrational absorption bands have remained more or less unknown. Only after the development during the 1960's and 1970's of semiempirical and, especially, ab mitio quantum theoretical methods combined with the tremendous progress in computational technology, it became possible for

SEMI-CLASSICAL MODELS OF IR INTENSITIES

37

researchers to study in detail the intramolecular charge reorganizations induced by vibrational distortions that determine the intensities of infrared bands [40-51]. As it turned out, these charge fluctuations are extremely difficult to describe in terms of a much desired simple physical model. In the second place, accurate experimental determination of vibrational intensities was, until recently, a formidable task due primarily to instrumental deficiencies and the presence of strong band overlap in many parts of the spectra. Thus, there were very few reliable experimental data for molecules in the gas-phase that could serve as a basis to test the theoretical formulations put forward. The recent progress of insmunentation and specific computer software developed to deal with overlapping bands have provided reliable approaches in the experimental measurements. Unfortunately, even under these much more favorable conditions, the available gas-phase intensity data are still scarce. In this chapter we shall present theoretical approaches for interpretation and calculation of vibrational intensities in the infrared spectra. The formulations developed are aimed at reducing the experimental dipole moment derivatives into molecular parameters characterizing properties of simple structural sub-units. In spite of many difficulties, a steady progress in analyzing vibrational intensities has been achieved, especially during the past two decades. A volume on the subject comprising papers from different laboratories appeared a decade ago [52]. We were tempted, however, to present in a unified treatment the various theoretical models developed, to discuss the relations between these approaches and also to present the progress achieved during the past decade. Examples of applications are also given so that the sometimes seemingly rather obscure formulations become clearer. The theories developed are best applied to experimental data for isolated molecules. Translated into realistic experimental conditions this refers to molecules in the gas-phase at low partial pressure so that no substantial intermolecular interactions are present. As mentioned, the main reason for these restrictions is that in contrast to vibrational transition frequencies, the intensities of absorption bands are extremely sensitive to environmental variations such as change of phase, intermolecular interactions and solvent effects. A good illustration in this respect is offered by the measured experimental intensities of CH2C12 in the gas-phase, in solution and as a pure liquid as summarized in a review by Person and Steele [53] and given in Table 3.1. The experimental data reveal a strong dependence of the observed intensities on phase state and medium. It is clear that fully reliable information on the relationship between intensities and molecular structure can be derived from gas-phase experimental data. As a consequence, the range of molecules that can be investigated is much limited. Theoretical models for quantitative assessment of the influence of solvents on vibrational absorption intensities have been developed [54-58]. Using these approaches, experimental data determined in solution are transformed to expectation values in the gas-phase. These transformations

38

GALABOV AND DUDEV TABLE 3.1. Integrated intensities (in km mo1-1) of infrared absorption bands in CH2CI2 in gas-phase (Av), solution in CCI4 (As) and as a pure liquid (AL)a

Symmetry class

Vibration

A1

Vl v2 v3 v4

B1

v5 v6

Av

AS

AL

3137 1430 714 283

6.90 0.60 8.00 0.60

4.60 2.10 9.00 0.60

1.90 3.80 19.90

3195 896

1.20

2.60 3.00

4.50 3.80

v7 1268 v8 aReprinted from Ref. [53] with permission.

26.60

27.00

28.80

B2

Wavenumber (cm-1)

are, however, always based on approximations and the reliability of estimated intensities is under question. An alternative possibility is to carry out studies in solvents accepted as standard, e.g. chloroform or carbon tetrachloride. The range of molecules that can be investigated is greatly extended and some of the experimental difficulties encountered in gas-phase measurements are avoided. These opportunities have been extensively explored [59-62]. Many problems, however, remain. From the single example provided in Table 3.1 it is seen that the intensities of vibrational modes do not change in any regular manner in the transitions between gas, liquid and solid phases. Thus, the observed quantifies cannot, in our opinion, be considered as reflecting the intramolecular electronic structure and dynamics of electric charges. In this respect, the treatment of solid-state intensity data appears to offer better opportunity for theoretical considerations because of the structural regularities and constancy of interactions within crystals and other solid-state materials. Provided aU problems associated with band overlaps, sign indeterminacy for ~ / / ~ k derivatives or other factors are overcome, the first step in transforming the experimental data is the evaluation of dipole moment derivatives with respect to symmetry vibrational coordinates. Hereafter in our discussion we shall assume that the molecules treated possess some reasonable symmetry. As already mentioned, for such molecules individual Cartesian components of t h e / ~ / ~ derivatives can be determined. The equations presented are valid for an arbitrary molecule. Realistically, however, the reduction of experimental intensity data into molecular parameters, or solution of the socalled inverse intensity problem, can be performed without introducing considerable

SEMI-CLASSICAL MODELS OF IR INTENSITIES

39

additional uncertainties for molecules with higher symmetry only. The /gp//gQk derivatives form a rectangular array PQ with dimensions 3x(3N-6) with the following structure

(0px / 0Q1 PQ =/0Py/OQI / 0QI

0px / t)Q2 ~Py/r ~pz / OQ2

..... .--

0px / t)Q3N-6 r J 9 ~pz / OQ3N-6

(3.1)

The transformation to c3p/tgSjdipole derivatives is given by the relation

ap/asj =2 LZJ(ap/aQ )

(3.2)

k

or in a matrix form (3.3)

PS = PQ L s - I

The matrix LS-I is the inverse of L s defined by Eqn (2.7). The PS matrix has the structure

PS =

/~i/igS1 /~)SI / ~)SI

~x/OS2 ~)Py/~)$2 ~Pz / t)S2

...."'"

igPx/~)$3N-6 / 0py/t)S3N. 6 . 0Pz / ~$3N-6

(3.4)

The elements of LS-1 are determined in the process of normal coordinate analysis. As already underlined, due to relation (3.3) the molecular interpretation of vibrational intensities may be carried out with some confidence only if the force field is accurately determined. At least for now, this, again, is only achievable for relatively small and symmetric molecules. By the transformation (3.3) a substantial step is made in the transition from experimental intensities into quantifies characterizing molecular structure. At the first place, a natural separation between dipole derivatives associated with bond stretchings and angle deformations is achieved. In some cases the ~/3Sj derivatives can be associated with vibrations localized within certain atomic groupings. Such distortions may be described by local group symmetry coordinates. Snyder [27] first applied dipole moment derivatives with respect to group symmetry coordinates as basic parameters in infrared intensity analysis on a series of crystalline n-alkanes. The procedure described in his work will be discussed later in this section. The intensities in infrared spectra are determined by the intramolecular charge shifts accompanying the normal vibrations of molecules. Thus, the measured band envelopes contain quite essential information about the distribution and dynamics of

40

GALABOV AND DUDEV

electric charges in molecules. It is the extraction of this information that is a principal goal of the theoretical formulations of infrared intensities. Appropriately defined molecular parameters are sought that would, hopefully, represent in a plausible way the complex picture of charge reorganizations with vibrational distortions. Since experimental data associated with intramolecular electric charges are of considerable importance for both chemistry and physics, the development of theoretical approaches for deriving this information is, no doubt, a task of prime interest. Experience has shown, however, that in spite of early hopeful developments [39,63] this goal became especially difficult to achieve. This is hardly surprising in view of the complexity of intramolecular charge rearrangements determining intensities in the infrared spectra. A second important aim of theoretical formulations is to provide possibilities for quantitative predictions of intensities by transferring intensity parameters associated with polar properties of basic structural sub-units between molecules having the same fragments. In recent years there has been certain competition between purely theoretical predictions provided by ab imtw MO calculations and the semi-classical approaches based on transferable parameters determined from experimental data. The arguments in favor of one approach or the other are hardly of any relevance and have the same validity as the analogous discussions concerning experimental and quantum theoretical force fields. The answer here is that both purely theoretical and experiment based approaches are complementary to each other, thus significantly enhancing our understanding of molecular spectra and sm~cture.

II. R O T A T I O N A L C O R R E C T I O N S T O DIPOLE MOMENT DERIVATIVES

A. The Compensatory Molecular Rotation The Eckart-Sayvetz equations [Eqs. (2.8)] imposed on a vibrating molecule require that the condition of zero linear and angular momenta is fulfilled. The molecular motion is considered as ff it is purely vibrational. Rotations and translations of the molecule as described by the six external coordinates Xx, Zy, Zz, Rx, Ry and Rz, are ignored. 3N-6 vibrational coordinates are defined with respect to a Cartesian system stuck to the molecule and moving with it. The condition for zero linear and angular momenta stems from the more fundamental theorem for momentum conservation. The momentum of a system with respect to a given axis preserves its value and direction ff the momentum of external forces acting upon the system is zero [64]. It is interesting to recall in this respect the

SEMI-CLASSICAL MODELS OF IR INTENSITIES

41

well known example with the falling cat which rotates its tail in direction opposite to that of the body, thus compensating for its angular momentum. In this way the cat preserves zero angular momentum and, as is known, it always touches the ground by its paws. Let us turn now to the vibrating molecule. When it undergoes an antisymmetrical distortion with frequency v i the direction of the molecular principal axes changes and, as a result, the molecule acquires some angular momentum. Then, a compensatory rotation with the same frequency v i and opposite direction is invoked in obeying the angular momentum conservation condition. This rotation has no effect on the frequency of the vibrational line but it may contribute to the overall intensity of the i-th band. This is the case with molecules possessing a permanent dipole moment or non-spherical equilibrium polarizability ellipsoid. The contributions could be quite significant in some cases, especially for small-sized highly polar or anisotropic molecules. Evidently, the contributions arising from the compensatory molecular rotation have to be eliminated in the course of vibrational intensity analysis. This procedure is an essential step in the process of decomposing the experimental band intensities into parameters characterizing molecular properties. Intensity parameters free from contributions from compensatory rotation can only be considered as pure intramolecular quantifies. As already discussed, an important step towards reducing the experimental absorption intensities into molecular intensity parameters is the evaluation of the matrix PS [Eq. (3.3)]. Dipole moment derivatives with respect to symmetry coordinates may, however, contain contributions from molecular rotation for certain vibrations. These conlributions can be eliminated by using the following relation p~orr = PS - R S '

(3.5)

where p~orr matrix contains /gp/aSj derivatives corrected for contributions from the compensatory molecular rotation and R s is the 3 x(3N-6) matrix of rotational contribution terms. Its structure is analogous to that of PS and will be illustrated with some examples in the succeeding parts of this section. The elements of R S can be evaluated by the equation [35] Rj = wj • PO,

(3.6)

where wj is the compensatory rotation accompanying the j-th vibration and Po is the equilibrium dipole moment of a molecule, wj is a vector which could be presented in a pseudo-tensor form as 0

w~

- w .y J

0

(3.7) 0

42

GALABOV AND DUDEV

In Eq. (3.7) w jX, w~ and w jZ are the components of wj, where x, y and z denote the respective axes of rotation. Combining Eqs. (3.6) and (3.7) we obtain Rj = - ((wj)). P0 9

(3.8)

The value of the absolute compensatory rotation arising for a particular non-fully symmetric vibration could be determined if a hypothetical non-rotating isotope of the actual molecule is created (hypothetical isotope method). This hypothetical reference species is characterized with negligibly small compensatory rotation. Historically, nonrotating isotopes of molecules were introduced by Crawford, et al. [35,36] who derived the basic mathematical expressions for treating the problem. It was suggested that the masses of some appropriately chosen atoms of the molecule were set equal to zero. Usually these are atoms non-lying along the main symmetry axis of the molecule. Since difficulties were encountered in applying this approach to some classes of molecules [34], another reference isotope was proposed by Van Straten and Smit in the mid-seventies [34]. According to this method, the non-rotating isotope is built up by multiplying the masses of some atoms by a factor of 1000 or more. The choice of atoms to be weighted depends on the symmetry of the molecule and on the form of the particular vibration which is treated. The application of these two approaches in evaluating rotational correction terms to dipole moment derivatives with respect to symmetrical vibrational coordinates will be illustrated with several examples in the following part. The compensatory rotation wj can be calculated with the aid of the following expression [36] w j = ~ p r A ja.

(3.9)

In this equation superscripts r and a stand for the reference and the actual molecule, respectively, and Aj is the j-th column of the Crawford's A matrix [38] [Eqs. (2.16) and (2.17)]. The rotational part of the 13matrix (lip) used in the calculations is in the form 0

-maz~/Ix

~,-may~/Iz

o

may~/Ix / -max~ t / Iy

maxS:/Iz

0

where a = 1,2,3, ..., N is an atom index. Thus, combining Eqs. (3.8) and (3.9) we arrive at

Rj =-(([3prAja)). po .

(3.11)

SEMI-CLASSICAL MODELS OF IR INTENSITIES

43

B. The Hypothetical Isotope Method 1.

The Zero-Mass Approach

The formaldehyde molecule will be used as an example to illustrate the zero-mass approach [35,36] for evaluating absolute rotational correction terms to dipole moment derivatives. H2CO has C2v symmetry and infrared active vibrations belong to A 1, B 1 and B 2 symmetry classes. The non-fully symmetric vibrational distortions of B 1 and B 2 species contain contributions from compensatory molecular rotation. Structural parameters for formaldehyde and symmetry coordinates used in the analysis are presented in Table 3.2. Cartesian reference system and internal coordinates are shown in Fig. 3.1. Following the prescriptions given in Ref. [36] the masses of the two hydrogen atoms are set equal to zero in order to create a non-rotating formaldehyde isotope. The application of Eq. (3.9) yields (in units A -1 for stretching and rad -1 for bending coordinates)

$4 w=~;A~=

0.

$5

$6

o

1 -0.147 0

(3.12)

y, z

or in a pseudo-tensor form

S4 ( ( 1= ~3A ) )_~

0 0 -0.051 0 0 0 0.051 0 0

S5 0 0 0.147 0 0 0 -0.147 0 0

S6 0 0 0 0 0 -0.098. 0 0.098 0

J

(3.13)

The equilibrium molecular dipole moment is directed along z-axis and in vector form can be written as

(0)D

Po = -2.33

Hereafter, the following convention for the dipole moment sign will be used: a negative charge shift towards the positive direction of the Cartesian axis results in a negative value of p or of the respective dipole moment derivative.

44

G A L A B O V AND DUDEV

Table 3.2 Structural parameters, symmetry coordinates and rotational correction terms to dipole moment derivatives for H2CO Geometry a,b rCH = 1.09 A ,

Z H C H = 120 ~

r c o = 1.213

po = -2.33 D S.ynunetry coordinates c

Rotational correction terms (D/A or D/rad)

B1

B2

$4 = (At 1 - At2)/x/2

X

-0.119

S 5 = ( A l l - A~2 ) /

X

0.342

$6 = A0d

Y

--0.228

aFrom Res [65]. blA = 10-1~ m; 1D = 3.33564x 10-30 C m. eInternal coordinates are defined in Fig. 3.1. dOut-of-plane mode.

X

FIG. 3.1. Cartesian coordinate reference system, numbering of atoms and defimtion of internal coordinates for H2CO.

SEMI-CLASSICAL MODELS OF IR INTENSITIES

45

Eventually, applying Eq. (3.11) separately for each symmetry vibrational coordinate we obtain the rotational correction term matrix R s (in units of D/A or D/rad) 54

$5

$6

0 0

-0 28

X

RS=

y. Z

These quantifies are given in Table 3.2 as well. The same procedure can be followed in the case of other X2CY molecules with X = D, F, C1, Br and Y = O, S. The rotation-free isotope is created by setting the X-masses equal to zero. Application of the zero-mass approach in evaluating rotational correction terms to polarizability derivatives will be illustrated with an example in the second part of the book.

2.

The Heavy Isotope Approach

The method was introduced by Van Straten and Smit [34] in order to overcome problems arising with the zero-mass approach for certain types of molecules. For bent X 2Y and pyramidal X 3Y molecules the creation of an isotope with zero X-masses results 1in indefinite 13p elements, thus hampering the evaluation of rotational correction terms [34]. No such problems are encountered following the procedure of Van Straten and Smit. It will be illustrated with several examples. a.

Ammonia

Ammonia belongs to the group of X3Y pyramidal molecules possessing C3v symmetry. The vibrations belonging to the doubly degenerate E symmetry class contain contributions from molecular rotation. Application of the heavy isotope method requires that the mass of the nitrogen atom be multiplied by a factor of at least 1000 [34]. Molecular geometry data and symmetry coordinates used are given in Table 3.3. The orientation of the molecule in the Cartesian reference frame and definitions of internal coordinates are shown in Fig. 3.2. r sa for NH 3 is as follows (in units A-1 or rad -1) The compensatory rotation matrix 13pA

S3x

S4x

I~PrA a ( 0 s = 0.046 0

0 0.026 0

S3y

S4y

--0.046--0!26) X y. 0 0 Z

(3.16)

46

GALABOV AND DUDEV

TABLE 3.3 Geometry data, symmetry coordinates and rotational correction terms for NH 3 Geometrya rNH = 1.0116 A ,

ZHNH = 106.67 ~

po=-1.47 D Rotational correction terms

Symmetry coordinates b

(D/A or D/tad) E'

Et!

S3x = (2Ar 1 - Ar2 - Ar3)/,~/'6

X

-0.068

S4x = (2Aot1 - Act2 - Atx3)/,~

X

-0.038

S3y ~ (Ar2- - At3)/~f2

Y

-0.068

S3y = (Atx2 - Atx3)/'~

Y

-0.038

aGeometry data are taken from Ref. [66] and dipole moment value from Ref. [67]. bDefinifions of internal coordinates are given in Fig. 3.2.

z

x

FIG. 3.2. Cartesian reference system and definition of internal coordinates for NH 3.

After presenting this matrix in a pseudo-tensor form and multiplication with the equilibrium dipole moment vector we obtain the rotational correction terms matrix R s (in units of D/A or D/tad)

SEMI-CLASSICAL MODELS OF IR INTENSITIES

47

TABLE 3.4 Molecular geometry, symmetry coordinates and rotational correction terms for 1,1-dichloroethylene Geometrya rcc = 1.324 A , rcc I = 1.710 A , rCH = 1.070 A ZHCH = 120", ZC1CC1 = 114.5 ~ Po = 1.34 D Syrmnetry coordinates b gl

B2

Rotational correction terms (D/A or D/rad)

57 = (AR l - AR2) /

X

0.124

S8 = (A~I - A~2 ) / ~ f 2

X

0.775

S9 = (At 1 - Ar2)/~f2

X

0.014

SI0 = (Abl - Ab2)/~J2

X

--0.019

S 11 = A0CCI

yC

0.546

S12 = AOcH

yc

0.050

astmetural data are taken from Ref. [68] and dipole moment value from Ref. [69]. bIntemal vibrational coordinates are defined in Fig. 3.3(A). cOut-of-plane vibrations.

S3x

RS-

i:oo,

S4x 0 0

S3y

o

-0.068 0

0ix

S4y

-0.038 0

y. z

(3.17)

These quantifies are tabulated in Table 3.3. b.

1,1-Dichloroethylene

The 1,1-dichloroethylene molecule belongs to C2v point group and has vibrations distributed among the following symmetry species F V = 5A l + A 2+4B1 + 2 B 2 .

(3.18)

Vibrations belonging to B 1 and B 2 have to be treated for contributions from compensatory molecular rotation. The geometric parameters and the definition of symmetry coordinates are given in Table 3.4. Cartesian reference system and definition of internal coordinates are shown in Fig. 3.3(A).

48

GALABOV AND DUDEV

A

R1

~1

bI rl~~ )

) x

z

R1 B

~1

r3 r2

~

a3

FIG. 3.3. Cartesian reference system and definition of internal coordinates for (A) 1,1-dichloroethylene and (B) 1,1,1-trifluoroethane.

The most appropriate non-rotating isotope of the molecule is that containing two heavy carbon atoms. Thus, the C-C bond coinciding with the C2 symmetry axis, will maintain fixed direction during vibrational distortions and the rotational correction terms for both B 1 and B 2 vibrations can be evaluated. A question for the weighting factor magnitude, however, arises. A simple survey of values for weighting factors given in Ref. [34] reveals that very significant weighting is needed for molecules containing relatively heavy atoms (CI, Br, I) which do not lye along the main synunetry axis. In order to determine a proper weighting factor in the case of 1,1-dichloroethylene, the compensatory rotations were calculated by employing different isotopes of the molecule.

SEMI-CLASSICAL MODELS OF IR INTENSITIES

49

The carbon atoms masses were multiplied by factors of 103, 104, 105 and 106. Compensatory rotation values wj obtained via Eq. (3.9) are presented in Table 3.5.

TABLE 3.5 Compensatory rotations for 1,1-dichloroethylene obtained by using different weighting factors for the two carbon atoms Symmetry coordinate x 103 B1

B2

Compensatory rotation (in A-1 Or tad-l) x 104 • 105

• 106

$7

Y

-0.095

-0.093

-0.093

-0.093

S8

Y

-0.562

-0.577

-0.579

-0.579

$9

Y

-0.010

-0.010

-0.010

-0.010

Sl0

Y

0.014

0.014

0.014

0.014

S 11

X

0.400

0.408

0.409

0.409

S12

X

0.038

0.038

0.038

0.038

The analysis of the results shown implies that factors of 103 and 104 are insufficient to effectively fix the directions of inertial axes of the molecule when it undergoes deformational distortions of the CCI2 group (symmetry coordinates 8 and 11). It is seen that compensatory rotations approach constant values when factors of 105 or higher are used, indicating that proper weighting of the molecule has been reached. Therefore, a multiplication with factor of 105 or more is recommended in this case. The values given in the third column of Table 3.5 (weighting factor of 105) were used to calculate the rotational contribution terms to dipole moment derivatives. These are presented in Table 3.4. c.

1,1,l-Trifluoroethane The 1,1,1-trifluoroethane molecule possesses C3v synunetry with infrared active

vibrations belonging to A 1 and E symmetry classes. Vibrations belonging to E symmetry class are accompanied by a compensatory rotation. Geometric parameters used in evaluating the respective rotational correction terms and the synunetry coordinates are given in Table 3.6. The orientation of the molecule in the Cartesian space is shown in Fig. 3.303). A heavy isotope of the molecule created by multiplying the masses of the two carbon atoms by a factor of 104 was employed in the calculations. The following compensatory rotation matrix was obtained (in units of A -1 and rad-1).

50

GALABOV AND DUDEV

T A B L E 3.6 Structural parameters, definition of symmetry coordinates and rotational correction terms for 1,1,1-trifluoroethane Geometrya r c c = 1.530 A ,

rCF = 1.335 A ,

Z C C F = 111.03",

rCH = 1.085 A ,

Z C C H = 108-32~ Po = 2.32 D

Symmetry coordinates b

Rotational correction terms (D/A or D/rad)

E

E

!

M

S7x = ( 2 A R 1 - A R 2 - A R 3 ) / ' f 6

X

0.020

SSx = (2Aa I - Aa 2 - A a 3 ) / i f 6

X

-0.221

S9x = (2A131 - A[32 - A133) /

X

1.092

Sl0 x = (2Ar 1 - Air2 - A t 3 ) / ' f 6

X

-0.044

S l l x = (2Aa I _ Aa 2 _ Aa3)/,r

X

-0.040

S12x = (2Ab I - Ab 2 - Ab3)/.f6"

X

0.069

S7y = (AR 2 - AR 3 ) / ' 4 2

Y

0.020

SSy = (Aa 2 - Aa3) /

Y

-0.221

S9y = (AI32 - A~3 ) / , J 2

Y

1.092

Sl0y = (Ar2 - At3) [,r

Y

-0.044

S 1 ly = (Aa2 - Aa3) /

Y

-0.040

S 12y = (Ab2 - Ab3) /

Y

0.069

aCnmmetric parameters are taken from Ref. [70] and the dipole moment value from Ref. [69]. bIntemal coordinates are defined in Fig. 3.3(B).

I31~A ~ =

S7x

Ssx

0 -0.009 0

0 0.095 0

S9x

S lOx

S 1ix

S 12x

0 0 --0.471 0.019 0 0

0 0.017 0

0 --0.030 0

S7x

Sgx

S9x

Slox

Sllx

S12x

0 0.009 0

0 -0.095 0

0 0.471 0

0 -0.019 0

0 -0.017 0

0 0.030 0

(3.19)

J

SEMI-CLASSICAL MODELS OF IR INTENSITIES

51

Values for rotational correction terms to dipole moment derivatives with respect to symmetry coordinates evaluated via Eq. (3.11) are tabulated in Table 3.6. The hypothetical isotope method is aimed at calculating absolute rotational correction terms. An alternative method for evaluating relative rotational corrections in a series of isotopically related molecules was proposed by Escribano, del Rio and Orza [71]. The formalism of this approach will be briefly presented in the second part of the book.

HI. THE BOND M O M E N T M O D E L

A. Theoretical Considerations The first complete formulation of vibrational intensities was put forward some fifty years ago by Volkenshtein et al. [39,63], and timber developed by other authors [72,73] later. The formulation, known as the valence-optical theory (VOT), is based on the assumption that the molecular dipole moment may be represented as a vector-additive sum of bond moments P = E Ok k

(3.20)

where k is a bond index. The changes of molecular dipole moment determining the intensities of infrared bands are attributed to charge fluctuations associated with the dipole moments of individual bonds. At the start, an important feature from physical point of view is introduced: the intensities in the observed absorption spectra are associated with polar properties of valence bonds. Later, the validity of relation (3.20) has been subject to criticism [47,48,53,74]. It does appear, however, that the hypothesis has some physical grounds. Bond moments derived l~om equilibrium molecular dipole moment data have been shown to predict satisfactorily the permanent dipole moments in certain types of molecules [75]. We need also remember that, at the time, very tittle was known about electron density distribution in molecules and its behavior under vibrational distortions. In the original variant of the valence-optical theory a second significant assumption is made. It is assumed that during small vibrations the bond moments remain constant in magnitude and directed along the bonds. In other words, the changes in dipole moment induced by vibrational distortions have purely geometrical origin. The problem is reduced to an appropriate coordinate description of separate bond distortions. A number of approaches have been put forward aimed at representing the forms of

52

GALABOV AND DUDEV

vibrations in terms of bond coordinates. In describing stretching vibrations the ordinary internal coordinates representing changes in bond lengths are used. In describing the changes in spatial orientations of bonds in the case of deformational modes different displacement coordinates have been employed: changes in bond direction cosines, changes in bond direction angles and polar angles [6,63,72,73,76,77]. Considerable progress, in comparison with the earlier formulations of the valence-optical theory, was achieved by introducing in the equations of the zero angular momentum condition [72,77,78]. Thus, the molecular quantifies determined, called electro-optical parameters (eop), are associated with changes in the dipole moment induced by purely vibrational distortions. Gribov has provided an early detailed description of the valence optical theory in a well known book [72]. The derivation is based on expressing the dipole moment as (3.21)

p = ~ ektt k k

ttk are scalar quantifies - the magnitudes of bond moments, and ek the unit vectors directed along the bonds. Differentiating this expression with respect to internal vibrational coordinates leads to equations of the type: ~p/~)Qi = ]~ ]~ ek ( ~ k / t)Rj) Lji + 2 ~ ~tk (t)ek/t)Rj) Lji. k j k j

(3.22)

Qi are normal coordinates, Rj internal coordinates, and Lji elements of the normal coordinate transformation matrix. From expression (3.22) we can see that electro-optical parameters are the following quantifies: ~tk, magnitudes of the bond moments, and 0gk/0Rj, derivatives of bond moments with respect to all internal coordinates. There is a particular advantage of the local intensity parameters defined above. In contrast to other intensity formulations, the eop's do not have space-directional properties. Provided that the approximations introduced are acceptable and the solution of linear equations (3.22) possible, such type of parameters may serve quite well the principal purposes of intensity calculations. Essential elements in solving the problem are the determination L matrix coefficients [Eq. (2.5)] and the derivatives 0ek/0Rj forming a rectangular matrix (0e/0R). In a matrix form equations (3.22) are expressed as PQ = { '~(r / ~R)+ tt(3e / r

L.

(3.23)

As initial quantifies, instead of 0p/0Q i, the dipole moment derivatives with respect to symmetric vibrational coordinates may be used. This offers certain advantages. Quite often important experimental information on the relative signs of dipole moment

SEMI-CLASSICAL MODELS OF IR INTENSITIES

53

derivatives expressed as 0p/0Sj may be obtained when a'eating intensity data for different isotopic species. Averaged values over a number of isotopes are expected to provide more reliable initial sets of dipole derivatives. Besides, symmetric properties of molecular vibrations are introduced in a natural way with all consequent simplifications. In particular, reduction in the number of independent intensity parameters is achieved. Equation (3.21) is transformed into expressions of the form 0 P / 0 S j = E E Z e k ( ~ t k / 0 R i ) U i j + Z Z Z ~tk(0ek/0Ri)Uij 9 (3.24) k i j k i j Uij are elements of the orthogonal matrix U [Eq. (2.6)]. The total number of eop's in a completely defined problem is equal to (N-1) • (n+ l ), where N-1 = m is the number of bonds in an N-atomic non-cyclic molecule and n the number of internal coordinates. In absence of redundancies, the number of eop's is given by (N-1)+(N-1)• It is evident that the total number of eop's exceeds by far the number of experimental observables from which these parameters can be determined. These are integrated intensities of the infrared absorption bands and the equilibrium value of molecular dipole moment. The matrix (0e/aR) defining bond reorientations in space with vibrational distortions has the following structure

(a~/aR)=

Oelx / aR 1

Oeix / ~)R2

...

~)elx/ ~)Rn

~)ely / ~)R1

~)ely / OR2

...

~)ely/ ~)Rn

0elz / ~)R1

~elz / ~R 2

9

~elz / ~)Rn (3 9

~emx / t)R 1

Oemx/ OR2

...

~)emx/ t)R n

0emy / OR1

/gemy/ ~)R2

...

~)emy/ ~)Rn

~)emz / t)R 1

~)emz/ ~)R2

9

Oemz/ ORn

(0e/OR) contains, in fact, derivatives of the bond unit vector components ekg (g =x, y, z) with respect to internal coordinates. The elements of (0e/0R) are calculated according to Ref. [72] (ae/aR)

---

r 1 (AA

-

I E:OI)

(3.26)

where r - 1 is (N-1)x(N-1) diagonal matrix with elements the reciprocal values of bond lengths. The kth element of the 3(N-1)• matrix A is equal to: 0 if atom a is not included in bond k; -1 if ct is the initial atom of bond k; + 1 if tz is the end atom of bond k. A is the matrix defining the relation between atomic Cartesian displacement coordinates and internal coordinates [Eqs. (2 916) and (2.17)]. E is a matrix of the bond

54

GALABOV AND DUDEV

unit vector components with different anangement of the elements ekx, eky and ekz as compared to matrix e. 0 is a null matrix with appropriate dimensions. Expressed in terms of derivatives of the dipole moment with respect to symmetry coordinates Eq. (3.23) transforms into PS = { ~(i~t / aR)+ ~t(Be / aR)} U .

(3.27)

In a more detailed form Eq. (3.27) becomes

PS = { e(911 / c3R)+ I1r-I(AA -I E:0 [) } U.

(3.28)

It should be emphasized that the condition of zero angular momentum is implicit in Eqs. (3.23), (3.27) and (3.28). In particular it is contained in the second term of Eq. (3.28). Thus, the usually troublesome problem associated with the compensatory molecular rotations as required by the Eckart-Sayvetz conditions is treated in an elegant way. Solutions of equations of the type (3.27) are not straightforward. First, all equilibrium bond moment values must be known. In the general case, a correct quantum mechanical definition of a "bond moment" is difficult to produce, if possible at all. Attempts to define bond moments quantum-mechanically are always based on severe approximations [79]. Another formidable problem in the above parametric expression of vibrational intensities arises fi'om the very large number of parameters appearing in Eqs. (3.23) and (3.27). These exceed by far the number of experimental observables. To arrive at a defined inverse electro-optic problem, i.e. to evaluate the set of electro-optical parameters from the observed intensities, a considerable number of elements of the matrix (0~aR) have to be necessarily constrained to zero. While this is physically acceptable for some distant interactions, the practice shows that it is necessary to neglect many close interactions as well. The examples presented later in the text illustrate the problem quite clearly. To overcome these difficulties a least squares approach is usually adopted and a set of initial eop's is optimized to fit the experimental intensities of a molecule or, preferably, of a series of structurally related molecules. While experience and intuition can be of considerable help in the choice of structure and values for the elements of the initial eop matrices, the arbitrariness involved is still quite considerable. The valence-optical scheme can be applied, as often done in the past, in a zeroorder approximation. All derivatives of the type ath~/aRj are constxained to zero except 0gk/0rk, where rk is a change of a bond length. The number of eop is reduced to such a degree that the problem becomes completely defined, provided, of course, that static bond moment values are known. The zero-order approximation, however, has long ago been shown to lead to inconsistent results and has been abandoned [72,74].

SEMI-CLASSICAL MODELS OF IR INTENSITIES

55

The bond moment model in the variant formulated by Gribov [72] has been applied in interpreting and predicfin~ vibrational intensities in numerous studies from different laboratories [52,53,60-62,73,80,81]. A comprehensive survey of publications on the subject until 1970 is contained in the monograph of Sverdlov, Kovner and Krainov [73]. The contradictory opinions regarding the reliability of the mathematical approach and the physical sense of intensity parameters obtained in view of the numerous approximations implicit in the formulation and, in particular, in its practical applications, require a more detailed discussion on the subject. The basic molecular parameters employed are bond dipole moments. The remaining parameters are derivatives of bond moments with respect to internal coordinates. It is, therefore, of prime importance to define as fully as possible the physical significance of bond dipoles and their contribution to the dipole moment fluctuations determining the intensities of vibrational transitions. An early and quite satisfactory definition of bond dipole moments is given by Coulson [82,83]. The electric moment of a bond is presented as a sum of three terms (3.29) The first term is determined by the non-equal sharing of bonding electrons, and is defined by the so-called net atomic charges. The second term ~ arises from the asymmetric distribution of electron density around the nuclei in molecules. This term is usually referred to as the atomic dipole. It includes the electric moments due to lone pair electrons. The third term is the homopolar or overlap moment. The term ~ is physically associated with the so-called point charge approximation. Viewed as vectors, gq are bond directed and comply, therefore, with the additive approximation for the molecular dipole moment. The second term ttn is formed by dipole contributions that are not, in the general case, bond directed. For example, the electric moment associated with the lone pair of electrons in ammonia is, obviously, not directed along any of the N-H bonds. Therefore, the bond moments, as determined from the permanent dipole moment of the molecule include projections from the lone pair atomic dipole along the bond axes. It is clear, however, that there are atomic dipole components in perpendicular directions to the bond axes as well. The third term, the overlap moment, for the chemical bond in a diatomic molecule is directed along the bond. In polyatomic molecules, however, overlap moments arise from overlap of orbitals centered on non-bonded atoms as well. The above arguments are substantiated by a detailed LCAO MO analysis on dipole moment contributions to infi'ared intensities carried out by Orville-Thomas and coworkers [41-44,48]. In the framework of the LCAO MO approach a Cartesian

56

GALABOV AND DUDEV

component of the dipole moment for closed shell molecules described by a single determinant wave function may be written as

Px = - e Z A

PvvFvv-XAZA -c Z Z Z P~vF~tv-CZ Z PttvFttv A~B ~t v A ~t#v

(3.30)

with (3.31) and

Pttv= 2 ~ C~Cvs, s(occ)

(3.32)

where O~t are atomic orbitals and Pity elements of the electron density matrix expressed in the usual notation. In expression (3.32) the summation is over occupied orbitals. X A and Z A are the Cartesian coordinate and charge of the Ath nucleus. Expression (3.30) provides a convenient interpretation of the molecular dipole moment in terms of contributions associated with the constituent atoms and atomic orbitals. The first term in Eq. (3.30) defines the contribution to the molecular moment from the net charges assigned to nuclei. It can be regarded as a point charge contribution provided that the basis set consists of orbitals symmetrical with respect to the nuclei (pure s, p, d orbitals). This term may be denoted by pq. The second term gives a measure of the contribution of the dipole moment arising from the overlap density of orbitals on different atoms, usually referred to as the homopolar or overlap moment Po- The third term is associated with dipole integrals from orbitals centered on the same nucleus. It is designated as hybridization or atomic dipole term. For the molecular vibration described by a normal coordinate Qk all three terms may change. Thus, ~x

+...--_

(3.33)

"lf~

~

+ ~Qk

{

A~B v

~t~:v

v

SEMI-CLASSICAL MODELS OF IR INTENSITIES

57

Contributions from these three terms to the dipole moment changes determining infrared intensifies will be considered in more detail. The first term in Eq. (3.33) may rigorously be related to bond moments. Therefore, the corresponding contributions to the 0p/0Qk derivatives are correctly described by the bond moment model. Early semiempirical [41,42] and later ab jniao MO calculations [41-44,46-49] have shown that a substantial amount of electronic charge may be transferred between the atoms as the molecule vibrates. The contribution to the overall dipole moment variation due to the atomic charge terms may, therefore, be derived into two parts. First, the change in p during vibration due to the motion of the equilibrium charges, Pqe, and, secondly, changes in p due to the flow of electronic charge induced by vibrational motion, p ~ . These two terms may be expressed as follows: Pqe = - e 2 2 Ptttt (Fl~g- Fttg) A ix

(3.34)

PAq

(3.35)

=

- e 2 ]~ (Pl~.t-P~o) F~j.. A l.t

The superscript ( ' ) denotes that the associated term refers to the molecule in a distorted geometry, while the absence of prime indicates that the term refers to the molecule in its equilibrium geometry. The appearance of a charge flux term does not contradict the bond moment description of the molecular dipole moment, but the corresponding bond moment fluctuations have to be taken into account. The second term in Eq. (3.33) arises from the overlap density Of orbitals on different atoms. As in the former case, the corresponding dipole changes induced by vibrational distortions can be expressed as: AB

Poe = - e

Z E 2 P~tv(F~tv- Fttv) A~eB tt v

(3.36)

AB

PAo = - e

~

~ E (Pgv-Pity) F~tv-

A~B ~

(3.37)

v

These dipole terms cannot immediately be associated with bond moments since Pttv and Fttv are taken over all pairs of atoms in a molecule. It may, however, be argued that the dipole integrals between bonded atoms make the greater contribution to the overlap

58

GALABOV AND DUDEV

dipole term. Thus, in a kind of a "tight-binding approximation" the bond moment representation still retains certain merits as far as this dipole term is concerned. The last term in Eq. (3.33) can also be sprit into a part representing the change in p arising from distortions of equilibrium charges and a part determined by charge density fluctuations. A PTIe = -- e ]~ ~ Pttv (F~tv - Fttv) A

pA,i = - e ~

A Y. (P~v-Pttv) F/iv . A ttc-v

(3.38)

(3.39)

Even equilibrium PTI is not formed by bond directed partial moments. Neither can one predict the change of magnitude and direction of these atomic dipoles during vibrations. Therefore, the expression of PTI in terms of bond moments seems little justified. It has been shown that the role of the atomic dipoles in determining infrared intensities, can be quite substantial [44,84]. The above treatment of the various contributions to the total dipole moment change determining infrared intensities is bound to the LCAO representation of molecular wave functions, and, therefore, not to observable physical quantifies. The various terms defined by Eqs. (3.34) through (3.39) reveal, however, how the dipole moment and its derivatives, as calculated by ab initio MO methods, are affected by vibrational distortions. Let us emphasize here that a lot of arguments in recent literature referring to model description of vibrational intensities are based on ab imtio calculated wave functions and charge densities [44-46,81,85,86]. An alternative, and possibly physically more acceptable, interpretation of the intramolecular factors determining the dipole moment changes associated with vibrational motion may be based on consideration involving the electron density function, the spatial part of which is an observable physical quantity. The function p(R) defining the electron density at position R in LCAO expansion reads ~'R) = ~ Pttv Ott(R) Or(R) 9 ~v

(3.40)

Pity are elements of the charge density matrix. Integration of p(R) over the entire molecular space gives the number of electrons in a molecule. It is possible, in principle, to partition the molecular space into sub-spaces related to the constituent atoms and to integrate ~ ) over these sub-spaces. As a result, estimates of the charges associated with individual atomic sites are obtained. If the atomic sub-spaces are considered as points in

SEMI-CLASSICAL MODELS OF IR INTENSITIES

59

the molecular space, we arrive at a point charge model of the electric charge distribution in molecules. There is a continuing argument as to the best way of partitioning the electron charge density and the definition of formal atomic charges [47,49-51,87-90]. We shall later discuss in some detail the point-charge approximation as applied to theoretical formulations of infrared intensities. Here we shall mention only results from electron density function analysis referring to bond moment rather than point-charge model. The two representations of molecular charge distribution are, of course, very close. Wiberg and Wendoloski [47,49] using results from large basis set ab mitio MO calculations for a number of hydrocarbons concluded that infrared intensities are determined by: (1) Motion of static bond moments; (2) Creation of a bent bond moment arising l~om incomplete orbital following; and (3) Changes in bond moments due to charge flux effects. In their studies no example of molecules with lone-pairs of electrons is considered. It may be expected that in such cases other factors would also interfere. Generalizing two different approaches - the first based on analysis of contributions of the LCAO composite dipole terms, and the second based on studies of properties of the electron charge-density f u n c t i o n - reveal a complex picture of intramolecular electronic effects determining band intensities in infi'ared spectra. The application of a bond-moment model in describing this particular molecular property is far from straightforward. Carefully considering the existing problems in the physical definitions of the bond dipole moment, Gribov [72] accepts that the electro-optical parameters are effective quantifies characterizing an atomic grouping, such as CH3, CH2, etc., as a whole. The entire combination of eop's associated with a functional grouping is of significance. Individual parameter values, including those denoted by ttk, do not possess a well-defined physical sense. In order to compensate for some of the above difficulties of the bond moment model, Sverdlov et al. [73,91] have proposed a modification of the original formulation. In their treatment certain adjustments of the physical prerequisites are made by allowing the presence of bond moment components perpendicular to bond directions. The additive representation for the molecular dipole moment is retained: 3

P = ~ Pk = ~ ~ ~ki ei. k

k

(3.41)

i=l

In expression (3.41) the bond moments ~ are expressed by three components (gkl' ~k2' gk3) referring to a bond axis Cartesian system, in which the axis 1 coincides with the bond direction. The vectors ei refer to molecular Cartesian frame. Differentiating (3.41) with respect to internal coordinates and using relation (2.4) one arrives at a set of linear equations defining the relation between the experimental dipole moment derivatives 0p/0Qk and the eleca'o-optical parameters associated with

60

GALABOV AND DUDEV

expression (3.41). With the modifications introduced by these authors the physical foundations of the valence-optical theory appear to be improved since it is clear that the charge distribution around the bonded atoms are determined by local dipoles that deviate from the bond directions. On the other hand, however, the number of intensity parameters is tripled. It becomes necessary to go even fmther on the way of approximations, neglect of parameters, and other assumptions in the mathematical sohtiom. These are, very possibly, the reasons for the limited application of this variant of the theory [73]. As early as 1972 it was recognized that an explicit inclusion of terms associated with the charge-flux effects accompanying molecular vibratiom may offer an opportunity for better understanding of the physical significance of parameters in infrared intensity models based on the bond moment concept [42]. The bond charge parameter formulation developed by Van Straten and Smit [92] describes infrared intensities in terms of static bond charges and bond charge fluxes induced by vibrational distortions. In the mathematical procedure many common features with the apparatus of the valence-optical theory are present. The molecular dipole moment is treated in the additive approximation. Each bond moment is presented as lak = qk rk ek"

(3.42)

The molecular dipole moment is, then, defined as P = ~ qk rk ekk

(3.43)

rk is a bond length, qk an effective bond charge, and ek a unit vector directed from the negative towards the positive end of a polar bond. Differentiating Eq. (3.42) with respect to an internal coordinate leads to the expression

aWo

=

+ (akmR

)qk'k + (

daRi)

rk"

(3.44)

If dipole moment derivatives with respect to symmetry coordinates are used as initial parameters, which in most cases is appropriate, the following matrix equation is obtained

Ps = re(~7"~) U+ qe(br/;~R)U+~ (3e/bR)U.

(3.45)

The structure of the PS array was already given [Eq. (3.4)]. (Oq/OR) is a matrix containing derivatives of the effective bond charges with respect to internal vibrational coordinates. These quantifies reflect the charge reorganizations with vibrational motion.

SEMI-CLASSICAL MODELS OF IR INTENSITIES

61

r is a diagonal matrix containing the equilibrium bond lengths, while the symbol ( A ) over r denotes a matrix having triplets of the bond lengths on the diagonal. The matrix e is identical with the respective array in the VOT formulation [Eq. (3.23)]. (0r/0R) contains derivatives of bond lengths with respect to internal coordinates, q is a diagonal matrix with elements the effective bond charges while q with A over it is an extended bond charge matrix having on the principal diagonal triplets of the bond charges. The array (0e/0R) is as defined by expression (3.25). A principal part of the calculations is the evaluation of the matrices (0r/0R) and (0e/0R). Though equation (3.45) appears overcrowded with matrices and is too complex, it can in practice be calculated without much difficulty with standard programs for vibrational analysis. The necessity to determine simultaneously for each bond in molecular static and flux bond charges imposes again the sensitive problem associated with the number of parameters involved. Applications have been limited to small symmetric molecules [93]. These studies have underlined once again the significant role of charge-flux effects in determining infrared band intensities, and the necessity of their proper consideration in intensity models. Other approaches aimed at explicit presentation of charge flow effects associated with xa'brational displacements of individual valence bonds have also been put forward [46,94]. These investigations have confirmed the importance of orbital rehybridization effects that accompany vibrational distortions and are a cause for considerable charge rearrangements. It should be emphasized that the charge-flux effects are implicitly included in electro-optical parameters of the type agk/aR i that appear in the first-order bond moment model [72]. Thus, in standard applications to various molecules it does not seem necessary to extend the original formulation since this would result in further increase in intensity parameters. As mentioned, to deal with the considerable gap in number between experimental dipole derivatives and electro-optical parameters, a least squares approach is usually adopted [72]. From the available literature it does not become entirely clear how individual parameter values, especially equilibrium bond moments, are derived from the sets of linear equations as defined by Eqs. (3.23) and (3.24). It can be assumed that extended applications to a large number of molecules that have the same type of structural elements may enable, by multiple testing, to arrive at physically significant elecr parameters. In fact, a library of such intensity parameters has been created and used together with standard transferable force fields to predict the infrared spectra of a number of aliphatic and aromatic hydrocarbons [60-62]. Comparisons between calculated and predicted spectra do not, however, appear to be satisfactory for many molecules. Thus, the accuracy of both force constants and eop's is under question. On the other side, the main purpose of these libraries of empirical vibrational parameters

62

GALABOV AND DUDEV

q~

a.

-~ (e)t

-0

1.0

2.0

I

(=•

I xs

6r

b.

Xcl

,

F

4.0 !

!/

.I

~r

(•

I

c. 1.5

6.0t _

(lOSdyne cfn-1)

=

2.0 t R~

I

cw,,....~~

Cl

F

FIG. 3.4. Plots of the dependencies between experimental atomic equilibrium charges of halogens in CH3a and (a) difference of electronegativity between halogen and carbon, (b) stretching force constant of the bond r and (c) interatomic distance c~C (Reproduced from Ref. [81] with permission. Copyright [1984] American Chemical Society).

+

q:

~ '~c'"~c,,,+ 3H4

0.10

~cc.

C~H,,~ CzHi

1.o

1:2

1:4

1:6 R:o,,,

FIG. 3.5. Plot of dependence between the experimental atomic equilibrium charges of hydrogen in hydrocarbons and the interatomic distance of the adjacent CC bond (Reproduced ~om Ref. [81] with permission. Copyright [1984] American Chemical Society).

SEMI-CLASSICAL MODELS OF IR INTENSITIES

63

appears to be their-application in an artificial intelligence software for spectral-structural correlations where only principal features of the calculated spectra need to match the observed spectra. In such a perspective the project seems to fulfill its principal objectives. More detailed work on applications of the valence-optical theory is produced by Gussoni, Zerbi et al. [80,81,95]. These authors have obtained eop values that appear to correlate very well with independent structural parameters and quantum mechanical data. These results appear to support the validity of some of the basic assumptions of the bond moment model. On the other hand, we have the more reserved statement of Gribov about the physical si~ificance of bond moments and derivatives as effective quantifies [72]. With these words of caution we reproduce in Figs. 3.4 and 3.5 some interesting results reported by Gussoni et al. [81] on the dependencies between bond moments as derived from expe"rtmental infrared intensities and other structural parameters. The atomic equilibrium charges are obtained from the respective bond moment values. Some of the problems and ditticulties associated with applying the bond moment model will be seen fxom the examples of application described next.

B. Applications 1.

H20

The analysis of gas-phase infrared band intensities of water using the valenceoptical theory is a case with the inverse intensity problem completely defined in terms of available experimental data. These are the measured integrated intensities of the three fundamental vibrations and the permanent dipole moment value. Experimental intensity data employed are as determined by Clough et al. [96]. The force field of Mills [23] is used to derive the normal coordinate transformation matrix L S. The respective L S is then applied in calculating 0p/0Sj from the 0p/0Q i dipole moment derivatives. In accordance with 1UPAC recommendations [97] the matrix L S is given together with dipole moment derivatives with respect to normal coordinates so that no ambiguity associated with the phase of normal coordinates is present. The signs of 0p/0Q i are as determined by Zilles and Person [98]. The standard sign convention was defined in the beginning of the present chapter. The L S transformation matrices for H20 obtained from the force field of Mills [23] are (in units of ainu- 89or rad ainu- 89A-1 )

A1

S1 S2

Q1 1.0171 0.0111

Q2 -0.0660 1.5284

B2

S3

Q3 1.0345

(3.46)

64

GALABOV AND DUDEV

7

)

x

FIG. 3.6. Definitions of internal coordinates, bond directions, Cartesian reference system and numbering of atoms for H20.

TABLE 3.7 Observed integrated gas-phase infrared intensifies of H20 and calculated dipole moment derivatives with respect to symmetry coordinates Aia

0p/0Sj b

~mol)

(D/A or D/tad)

A 1 = 2.24

Pl = -0.234

A 2 = 53.6

P2 -

0.726

A 3 = 44.6 P3 = -0.992 a From Ref. [96]. b For the sign choice and force field used see text.

The geometric parameters of H20 are: rOH=0.9572 A, LHOH = 104.5 ~ [99]. The reference Cartesian system and internal coordinates are defined in Fig. 3.6. The symmetry coordinates have their usual form. The non-zero elements of PS obtained are given in Table 3.7. From these the application of Eq. (3.24) produces the following equations relating 0p//~Sj derivatives with eop's for the two symmetry classes [ 100] AI:

-0.866 (aia/0r) - 0.866 (ag/&') = - 0.234 -1.224 (ala/a0) + 0.791 ~ -

B2:

0.726

-1.118 (a~Or) + 1.118 (a~/~') - 0.051 ~ = - 0.992.

(3.47)

SEMI-CLASSICAL MODELS OF IR INTENSITIES

65

Bond moments are in Debye units, 0p/at are in units D/~-1 and a~O0 are in units D tad -1. Conversion factors to SI units are given in Table 3.2. An additional equation relating the permanent dipole moment value Po =-1.85 D [34] to the OH bond moment has the form: - 1.224 tx = - 1.85.

(3.48)

Eqs. (3.47) and (3.48) contain four observables and four unknowns. Therefore, all eop's can be uniquely determined. The values obtained are laOH = 1.511D c91a/c30 = 0.383 D rad -l (3.49) a~igr = 0.544 D/~-! ait/&' = - 0 . 2 7 4 D A -1 In the third equation of the set (3.47) the term -0.05 lit is the rotational correction term. It should be noted that for H20 the inverse electro-optic problem is entirely defined, simply because of the very small size of the molecule and its symmetry allowing the OH bond moment to be evaluated from the permanent dipole moment. Obviously, in molecules where at least two non-equivalent bonds are present, experimental determination of the bond moments is not possible and certain assumptions have to be made. 2.

CH3Cl

In spite of the high symmetry and small size of methyl chloride the application of VOT in analyzing infrared intensities of this molecule reveals quite clearly the computational difficulties that are usually encountered. Because of the closeness of all bonds no intramolecular interactions can be neglected on acceptable physical grounds. The experimental gas-phase infrared intensities used are those determined by Kondo et al. [ 101]. The L S matrix is obtained with the force field of Duncan et al. [24]. The definitions of internal coordinates and of the Cartesian reference system are given in Fig. 3.7. Geometric parameters and symmetry coordinates are shown in Table 3.8. The L s matrix for CH3CI is as follows: A1

1.0081 --0.0939 -0.0433

0.0011 1.3897 0.0840

-0.0039 0.1668 0.3202

E

1.0510 0.1131 --0.1012

0.0209 1.5134 0.3207

-0.0013 -0.2284 0.9136

(3.50)

66

GALABOV AND DUDEV

X

~Z o2

R

FIG. 3.7. Definitions of internal coordinates, bond directions, Cartesian reference system and numbering of atoms for CH3C1.

TABLE 3.8 Geometry and symmetry coordinates for CH3C1 Geometrya rCH = 1.095 A, rcc I = 1.778 A, Z H C H = 110.83 ~ S3nnmetry coordinates b A1

S 1 = (At I + Ar 2 + A t 3 ) / ~ f 3 S 2 = a (Act 1 + A ~ 2 + Act3) - b (AI31 + A~32 + A133)

a =0.397249,

b=0.418959

S3 = AiR E'

S4a = (2Ar 1 - Ar2 - Ar3)/,~/6

E"

S4b = (Ar 2 - Ar3)/,r

S5a = (2Ac~1 - Act 2 - Act3) /

S5b = (Act2 - A~3) /

S6a = (2AI31 - AI32 - AI33)/~t'6

S6b = ( A ~ 2 - A~3 ) / ' f 2

a From Ref. [24]. b As defined in Ref. [ 102].

The PS matrix obtained is presented in Table 3.9. The signs of dipole moment derivatives are as determined by ab imtio MO calculations [ 103 ]. Application of Eq. (3.24)yields the following relations between 0p/0Sj and electro-optical parameters [100]

SEMI-CLASSICAL MODELS OF IR INTENSITIES

67

T A B L E 3.9 Dipole moment derivatives with respect to symmetry coordinates for methyl chloride A1

AI:

Pl = -0.625

E

P4 -- -0.386

P2 = 0.182

P5 -

0.285

P3 = 2.237

P6 - -0-166

-1.732 (cgM/&) + 0.538 (c01a/0r) + 1.075 (egg/or') = -0.625 1.192 (c9M/c9~) + 1.257 (tgM/013) + 0.370 (c91.t/c9o0 + 0.740 (cgl.t/0a') -0.390 (c9~c913) - 0.780 (0~c913') -

1.196 ~t = 0.182 (3.51)

- (0M/0R) + 0.931 (al.t/0R)= 2.237. E:

1.164 (0~0r) - 1.164 (01x/0r') + 0.038 M - 0.035 kt = -0.386 1.164 (Og/o~) - 1.164 (c3~0o~') + 0.037 M + 0.769 ~t = 0.285 1.164 (al.ffOJ3) - 1.164

(a /af3')

(3.52)

- 0.070 M - 0.695 ~t : - 0 . 1 6 6 .

To these an expression relating Po and ~tk can be added 0.931 la - M = 1.870.

(3.53)

In Eqs. (3.51) through (3.53) M is the C-CI bond moment and ~t the C - H bond moment. Bond moments are in Debye units, eop's of the type a~tk/orj are in units D A -1, while a ~ / ~ i with Yi an angular internal coordinate are in units D rad -1. The total number of electro-optical parameters is 13 while the number of observables that can be used in their evaluation is just 7. It is evident that to solve the sets of linear equations some eop's have to be put equal to zero. Another difficult problem is how to decompose Po into particular bond moment values. Even if g and M are assigned particular values [73] a number of different solutions are obtained depending on which eleca'o-optical parameters are ignored. It is, therefore, of particular importance to discuss in detail the approximations adopted in calculations employing the bond moment model.

68

GALABOV AND DUDEV

C. Atomic Charge- Charge Flux Model The bond moment model of infrared intensities is associated with the assumption that chemical bonds are the most basic molecular units of interest. Indeed, the qualitative distinction between isolated atoms and molecules is, perhaps, best reflected in the chemical bond. The characterization of bond polar properties by an appropriate reduction of intensity data is, therefore, quite welcome. On the other hand, the concept of atoms in molecules also has support since theoretical calculations show that some basic atomic properties, including electronegativity, are partially retained in bonded systems. The NMR phenomenon also quite strongly supports, from the experimental side, the concept of atoms in molecules. Though there is little resemblance in the physical factors determining nuclear shielding in NMR and absorption of infrared radiation, it may be argued that both reflect in one way or another the electronic structure of the molecule. Thus, the possibility of describing the electronic structure of a molecule in terms of electric charges assigned to individual atoms is indeed appealing. In 1926 Dennison [104] introduced the concept of fixed atomic charges in describing the dipole moment variations with vibrational motion. The approximation was, of course, too far reaching to be practically applicable in explaining the observed spectral profiles for even very simple molecules. The atomic charge representation of the dipole moment function applied to theoretical formulation of infrared intensities was reviewed some fifty years later by several authors. King et al. [ 105] defined a quantity called effective atomic charge that is derived from the sum of integrated infrared absorption intensities of a molecule in a formulation based on expressing the dipole function in terms of atomic Cartesian displacement coordinates. The exact physical si~ificance of the King's effective atomic charges was not quite clear at the time, though it was shown to be constant for a number of hydrocarbons. The effective atomic charges of King found considerable application later in the atomic polar tensors representation of infrared intensifies [45,106]. In 1973 Samvelyan, Aleksanyan and Lokshin [107] presented in brief an effective atomic charge model of infrared intensities in terms of mathematical approach strongly correlated with the atomic polar tensors formulation [33,108]. It will therefore, be more appropriate to discuss these developments in conjunction with the atomic polar tensors method. In this part we shall concentrate on the effective atomic charge - charge flux model of infrared intensities (ECCF) developed by Decius [109]. In the basic definitions the Dennison idea of assigning point charges to individual atoms in molecules is retained. By introducing the charge-flux terms, however, a si~ificant step is made on the way to describe more correctly the intramolecular charge variations determining band intensities in infrared spectra. The molecular dipole moment is approximately described as

SEMI-CLASSICAL MODELS OF IR INTENSITIES

P = E ~t rot.

69

(3.54)

131,

~a is the effective charge associated with atom a, and rot is the position vector of ct. The summation is over all atoms. To account for the charge, reorganization accompanying vibrational distortion terms related with these effects are introduced. A Cartesian component of the molecular dipole is defined, within the harmonic approximation, as px = px (~ + E ( r ~ ( o ) a x ~ + ~ (x

x~(O)) =

(3.55)

= Px0 + AlPx + A2Pxpx(o) is the equilibrium value of the dipole moment component. The term A 1 Px is associated with the equilibrium atomic charges and A2 Px is the charge flux term. The entire formulation is expressed in terms of internal symmetry coordinates. As stressed before, this is natural since great simplifications are achieved. On the other side, individual Cartesian components of the experimental dipole moment derivatives can only be known if the polarization of the vibrational transition moment is fixed by symmetry. This is not the case for molecules with symmetry point groups C 1, Ci, Cs, C2 and C2h. For such systems the inverse intensity problem cannot be treated explicitly in any intensity formulation. For symmetric distortions the terms A 1 Px and A2 Px are represented as functions of the respective symmetry coordinate A1 Px = Z ~-xz(0) E Axa(J) Sj ct j

(3.56)

A2 Px = X X ~x (j) x~ (0) Sj ot j

(3.57)

ax~O) = aax~/asj

(3.58)

~..~x(J) = O~/OSj

(3.59)

where

Ax~ are atomic Cartesian displacements; ~(0) are the equilibrium atomic charges and ~(J) are the atomic charge fluxes for a symmetric distortion defined by Sj. The solution is reduced to determining the quantifies ~x~(0)and ~a(J) for each atom. It is clear that the number of parameters is much higher than the number of experimental quantifies and explicit solutions are only possible in cases where, due to high symmetry, the equilibrium atomic charges can be immediately determined from the equilibrium dipole moment

70

GALABOV AND DUDEV

TABLE 3.10 ECCF intensity parameters for diatomic molecules a AB

PO dp/dr (D) (D/A) -1.82 -1.68 I-IF -1.085 -0.88 HC1 0.112 -3.10 CO aReprinted from Ref. [ 109] with permission.

r(O) (A) 0.917 1.274 1.128

~A(0) (D/A) 1.98 2.13 -0.099

~A(1) (D/A2) -0.33 -1.2 2.84

value. From the condition of conservation of the total charge an additional relation between the parameters C~x(0)follows. The general equation relating the effective atomic charges and charge fluxes to the experimental dipole derivatives is as follows

apx/aSj = E Ix

+ E xjO)(

;josj).

(3.60)

Ix

From the definitions it can be seen that there is a considerable similarity between the VOT formulation and the ECCF model. The evaluation of intensity parameters meets also comparable difficulties. Explicit solutions are possible only for small symmetric molecules such as diatomics, bent AB 2, pyramidal AB3, etc. Least-squares refinement of parameter values can be easily introduced, thus enabling bigger and less symmetric molecules to be treated. The author of the ECCF model did not, however, propose such an approach. Most clearly, the basic ideas of the Decius model are illustrated in the case of a polar diatomic molecule AB. In such simple systems no assumptions need to be made. The equilibrium atomic charges for the diatomics can be unambiguously derived from the permanent dipole moment. There are two intensity parameters to be determined: ~A(0) (~B(0) =--~A(0)) and 0~A/0S 1 where S 1 = ARAB. Let us assume that the atom B is the center of negative charge and is situated along the positive direction of the X axis of a reference Cartesian system. The permanent dipole will then have a negative value. Application of Eq. (3.60) yields 0px/0S 1 = - (~A(0) + CA(1) r(0)) = (Po/r(0)) - ~A(l) r(0).

(3.61)

r(0) is the equilibrium bond length. Decius [ 109] provides very interesting results for a number of diatomic molecules, some of which are shown in Table 3.10. Similar formulation, based on representing a molecular dipole moment in terms of effective atomic charges, has been put forward by Aleksanyan et aL [ 107,110]. Basically the treatment refers to analogous intensity parameters. These are effective atomic charges and atomic charge fluxes expressed in the space of atomic Cartesian displacement

SEMI-CLASSICAL MODELS OF IR INTENSITIES

71

coordinates. We shall use here the notation already adopted. Starting from the expression for the dipole moment as given by Eq. (3.54), the change in p induced by a vibrational distortion is given by N N N Ap = Z ~xz(0)Ara+ X ra(0) Z (O~/Orl3)0Arj3. a=l a=l 13=1

(3.62)

~x(0) is the equilibrium (static) effective charge of atom a and ra(o) is the equilibrium radius vector of ct. The charge flux term is expressed in a Cartesian coordinate space. In a matrix form Eq. (3.62) is given by the expression ap =

E'

[C + xo (a~tax)]

x

(3.63)

.

Ap is a column matrix with components APx, Apy and Apz. The matrix E' has the sm~ct~e

E ' = II E3 . . . E3 II

(3.64)

with appropriate dimension where E 3 is a 3x3 unit matrix. ~ is 3x(3N) matrix containing triplets of the static effective charges (~1

0

0

~2

0

0

...

~N

0

0~

=/00

~1 0

0 ~1

0 0

~2 0

0 ~2

... -.-

0 0

~N 0

0 J . ~N

(3.65)

The matrix X 0 is a 3 x(3N) array with elements the equilibrium atomic coordinates in an arbitrmy Cartesian reference system Xl 0) x2 (0)

Xo=

... XN(0) (3.66)

YI(~ y21~l . . yN . (~ . Zl (0) z 2 ... ZN(0)

The matrix (OUOX)has a block structure

(a4sax)

= II aedOXl I aedaX21 . . .

I oxN II.

(3.67)

Each block has the followingform

a;/ax<< = It)~l/o-Jxa

aN)az<=j

(3.68)

72

GALABOV AND DUDEV

X is a row matrix containing atomic Cartesian displacement coordinates (AXl A Y l Axa AYctAza . . . AXNAYN AZN ). Finally, an important relation is provided

AZl

+ x0 (ac/ax)= Px.

9 9-

(3.69)

PX is 3x(3N) array of atomic polar tensors as originaUy developed by Biarge, Herranz and Morcillo [ 108]. It contains derivatives of the Cartesian components of the molecular dipole moment with respect to atomic Cartesian displacement coordinates. The derivation of PX from experimental dipole moment derivatives will be discussed in the following part. It is clear that Px can be devided into static effective charges and charge flux terms only if the elements of ~ are known in advance. This is possible, as in the treatment of Decius [ 109] only for small symmetric molecules. The charge terms expressed in Cartesian coordinate space are very ditiicult to interpret. Besides, as we shah later see, its elements will carry some contributions from the compensatory rotation of polar molecules stipulated by the Eckart-Sayvetz conditions. In the Decius approach, on the other hand, the consideration of these rotational effects is explicit and, as result, both the effective charges and charge fluxes determined are purely intramolecular quantifies reflecting the charge reorganization effects accompanying vibrational distortions.

D. Group Dipole Derivatives as Infrared Intensity Parameters In 1965 Snyder [27] offered a formulation of intensifies which provides good opportunities for applications to larger molecules, especially polymer systems. Dipole moment derivatives with respect to local group symmetry coordinates are employed as intensity parameters. In formulating the intensity problem it is assumed that the direction of 0p/0Sj is determined by the symmetry of the respective group. It is also inferred by this assumption that the charge fluctuation accompanying distortions as defined by the local symmetry coordinate are largely localized within the group. To what extent these approximations are correct can be judged only after specially designed quantummechanical studies. Molecular orbital calculations for a homologous series of dialkyl ethers [111] show that distortions within CH2, CH 3 and C-O groups result in certain charge shifts outside the respective group. In a non-synunetric environment, e.g. O-CH2-C, the direction of 0p/0Sj may not coincide with the site symmetry axis of the group. The problem with the compensatory molecular rotation is not considered. Since the procedure is applied to a series of n-alkanes [27] that have zero or smaUer dipole moment the respective rotational terms are negligible. In the general case of more polar

SEMI-CLASSICAL MODELS OF IR INTENSITIES

73

molecules, rotational contributions to dipole moment derivatives need also be accounted for. These can be derived separately provided that following known procedures [34-36] all Cartesian components of the molecular dipole moment are known. If the molecule is not symmetric or more than one conformer is present the evaluation of rotational correction terms becomes a very difficult task. In spite of these shortcomings the group moment derivatives approach has appropriate spheres of applications mostly in the case of large molecules where rotational contributions are negligible, and especially polymer systems. For such system calculations of intensities following the method prescribed by Snyder appear to be one of very few choices. With these preliminary comments we describe below the mathematical formulation as given by Snyder [27]. Expression (1.50) defines the relation between observed absorption intensity of the kth molecular vibration and the Cartesian components of the dipole moment derivatives with respect to the kth normal coordinate. A transformation fxom / g p / ~ to /gp/aSj derivatives is then performed using the relation igPG/aQk = E Lkj (/gPG/c3Sj). J

(3.70)

The index G refers to a molecular-fixed Cartesian system in which the dipole moment derivatives with respect to normal coordinates are obtained. The summation is over the group symmetry coordinates. Eq. (3.70) can be solved if the Cartesian components of ~i9~ are known. Introducing Eq. (3.70) into Eq. (1.50) the following relation between the dipole derivatives/gpG//gS j and the intensity A k is obtained A k = 3c2(4~,e0) G~

o]2

Lkj - ~ j j

(3.71)

Such initialization of the mathematical approach provides the opportunity to determine the quantities 0pG/aSj in an iterative refinement process. A second Cartesian reference system, still molecule-fixed, oriented as dictated by the symmetry elements of the group of atoms associated with Sj is introduced. It is designated by gji (i = 1, 2, 3). The dipole derivatives/gpG/OSj are expressed in the group local Cartesian system as igPG//gSj = Z a(G, gji) (0Pgji//gSj). gji

(3.72)

a(G, gji) are direction cosines between the axes of the two reference Cartesian frames. A relation between Ak and the dipole moment derivatives with respect to the group symmetry coordinates can then be obtained

74

GALABOV AND DUDEV

Ak = 3c2(4r~0) ~G

Lkj gjiE a(G,gji ) (~Pgji '~Sj

(3.73) "

The notation is then simplified by the substitution M m = (/gp/igSj)eji.

(3.74)

m is all index for the three axes gji associated with all group local Cartesian systems. Therefore, m equals 30-1) + i. eji are unit vectors directed along the gji axes. Eq. (3.73) becomes k Ak = E Cmm'MmMm'.

(3.75)

mm ~

k The coefficients Cram' are equal to C ~ , = {N0rd[3c2(4~c0)]} I ~ L~, E a~ a~, , G

(3.76)

where G

am

=

a(G, gn0

=

cos(G, gn0.

(3.77)

Snyder [27] then allows particular group moment derivatives M m to have the same values in different molecules Akp = Sp E

Cm~'M mMm,.

(3.78)

mm*

The index p refers to a molecule of the series, while Sp is an appropriate scale factor Sp = Akp (relative) / AM (absolute) .

(3.79)

This is a particularly useful proviso since relative intensities can be used instead of absolute values. One of the molecules in the series is taken as a standard. To obtain satisfactory accuracy in an iterative process the number of observed intensities should considerably exceed the number of intensity parameters to be determined. Snyder [27] provides a detailed description of how the least-squares refinement process should be performed. The following parameters are defined Ykp - Akp(0) - Akp(~ m i = M i 0 - Mi(~

(3.80)

SEMI-CLASSICAL MODELS OF IR INTENSITIES

Sp =

S.(') P

Sp(0)

~

75

.

Akp(~ is the observed intensity of the kth vibration in molecule p and Mi (0) and Sp(0) are initial values for the group moment derivatives and the scaling factor. These give an initial estimated value for the intensity A ~ (~ Using Eq. (3.78) the following expression for Ykp is obtained Ykp = SO X uP mt + V~ Sp

(3.81)

t

with u~t = 2S0 X Ctt' MO

(3.82)

t'

and

VP = X C~kp' Mt Mt'

(3.83)

tt'

Equation (3.81) can be simply expressed as y = Ax.

(3.84)

x is a vector with components the correction factors appearing in Eq. (3.80) x = [m's].

(3.85)

Solution of Eq. (3.84) is obtained from the least-squares equation A'py = A'pAx.

(3.86)

Different approaches can be adopted in assigning weighting factors. In the particular case of crystalline n-alkanes studied by Snyder [27] the weighting factors to the observed intensities are inversely proportional to the respective observed values. Aside from the refinement procedure the significance of group moment derivatives depends greatly on the validity of the basic assumptions implicit in the formulation. Nevertheless, the method appears to be quite convenient for calculations in systems where to apply non-approximate theories is not possible for symmetry reasons or the size of the molecule. It is, therefore, not surprising that the method of Snyder has been applied in calculations of intensities of various molecules [112-114] including proteins

[115].

This Page Intentionally Left Blank

CHAPTER 4

MOLECULAR

DIPOLE MOMENT

AS INFRARED

INTENSITY

DERIVATIVES

PARAMETERS

Atomic Polar Tensors (APT) ...............................................................................

79

A.

General Formulation ..................................................................................

79

B.

Invariams of Atomic Polar Tensors Under Coordinate Transformation ....... 83

C.

Symmetry Properties of Atomic Polar Tensors ........................................... 88

D.

Atomic Polar Tensors -- Examples of Application ...................................... 93

E.

Interpretation of Atomic Polar Tensors ....................................................... 98

F.

Predictions of Infrared Intensities by Transferring Atomic Polar Tensors ...............................................................................

105

II.

Bond Charge Tensors ........................................................................................

106

III.

Bond Polar Parameters ......................................................................................

111

111 116 Examples of Application .......................................................................... 120 Physical Significance of Bond Polar Parameters ....................................... 126

A.

General Considerations ............................................................................

B.

Formulation ..............................................................................................

C. D. E.

Prediction of Vibrational Absorption Intensities by Transferring Bond Polar Parameters ......................................................... 130

IV'.

Effective Bond Charges from Rotation-Free Atomic Polar Tensors ................... 131 A.

Rotation-Free Atomic Polar Tensor .......................................................... 131

B.

Effective Bond Charges ............................................................................

132

C.

Applications .............................................................................................

134

77

78

GALABOV AND DUDEV

It is interesting that a formulation of vibrational absorption intensities that is not based on the additive representation for the molecular dipole moment was developed much later than the approximate theories. In 1961 Biarge, Herranz and Morcillo [108] put forward a complete theoretical model of infrared intensities incorporating as basic parameters molecular dipole moment derivatives with respect to atomic Cartesian displacement coordinates. These quantifies were termed atomic polar tensors (APT). In 1965 Mayants and Averbukh [116] proposed an alternative formulation employing as parameters dipole moment derivatives with respect to bond Cartesian displacement vectors. As will be seen, the two approaches are very closely interrelated. Not until the mid seventies, however, following the works of Person and Newton [33] and of King and Mast [45], the atomic polar tensor model became well established. By this time quantum mechanical calculations have shown that the nature of intramolecular factors determining vibrational absorption intensities is so complex that the semi-classical approaches may not be expected to provide a satisfactory interpretation of these spectral quantifies. In most general terms, a representation of infrared intensities in terms of dipole moment derivatives with respect to some type of local coordinates describing vibrational distortions of basic structural units is quite natural. The analogous quantifies associated with vibrational frequencies - the force constants - represent derivatives of the molecular potential energy with respect to coordinates describing motions of simple structural elements. Many physical ambiguities implicit in the approximate intensity formulations are avoided. Besides, for molecules where the directions of the vibrational transition dipoles are determined by symmetry, the inverse intensity problem becomes completely defined. Thus, the di~cult problem with the large number of intensity parameters implicit in all semi-classical models is simply not present. What is lost is the simple physical si~ificance of parameters. In the approximate formulations the respective quantifies are, supposedly, directly related to the electric charge distribution in molecules. The physical interpretation of molecular dipole moment derivatives is by no means simple. Additional difficulties in this respect are present if dipole derivatives with respect to Cartesian displacement coordinates are used. As already discussed, force constants expressed in terms of Cartesian displacements lose their simple physical interpretation. The search for appropriate representations of the dipole moment variations upon vibrational distortions is, therefore, still continuing. In this chapter we shall describe formulations of vibrational absorption intensities employing, as basic parameters, derivatives of the molecular dipole moment.

MOLECULAR DIPOLE MOMENT DER/VATIVES

79

I. A T O M I C P O L A R T E N S O R S ( A P T )

A. General Formulation In our presentation of the atomic polar tensor formulation we shall follow the notation introduced by Person and Newton [33] since it is now generally accepted. The dipole moment changes induced by vibrational distortions are represented as functions of individual atom displacements N Ap = ~ PX(a) ra.

(4.1)

~=1

PX (a) are the atomic polar tensors and ra are displacement vectors that can be defined as r= = Ax~i + Ayaj + Az~k.

(4.2)

i, j and k are unit vectors directed along the Cartesian axes and ct is an atom index. PX (=) are matrices with structure

PX(a)

/pyz/ P

=/~o~;/~ 3Px//)Ya t)Px/t)za)=(P~ P~Ya /~gxot ~Py / 0yot t)Py/~z a Pyy 9 ~/gPz//~xot /)Pz/OYa 3Pz//)z a [,pazx P~ Pazz)

ip x

(4.3)

Arranged in a row all atomic polar tensors of a molecule form the matrix PX PX = (Px (1) Px(2) . .. px(ct) . . .

Px(N)).

(4.4)

The elements of PX are determined from the dipole moment derivatives with respect to normal coordinates using relations of the type

0 ~t=l ~)X(t~) 0 ~Qt The 3N atomic Cartesian displacement coordinates describe not only vibrational motion but the translation and rotation of the molecule in space as well. Therefore, Qt in expression (4.5) include also the six rototranslational normal coordinates. Thus, (o~h~Qt)o can be divided into two parts. The derivatives of p with respect to normal vibrational coordinates form a polar tensor PQ with dimensions 3x(3N-6) as defined by

80

GALABOV AND DUDEV

Eq. (3.1). The dipole moment derivatives with respect to normal vibrational coordinates are transformed into derivatives with respect to Cartesian coordinates by the transformation (4.6)

V X = PR B = PQL - l B .

PR is a matrix of dipole moment derivatives with respect to internal coordinates. Vx is called vibrational polar tensor matrix. B and L are defined by relations (2.4) and (2.12). Since molecular wanslations do not change the dipole moment, all derivatives of the type/~/&, where ~ are the translational normal coordinates, are zero. The dipole moment derivatives with respect to rotational coordinates are obtained according to the relation cqp/0Rp = ((p)) 1- 89.

(4.7)

The structure of the pseudotensor ((p)) is defmed by expression (2.20). Combining the translational and rotational parts, the matrix P p is obtained [33] 0 0 0

0

Pp= 0 0 0 - p z / I ~ 2 0 0

py/I~ 2

pz/I( 2

- p y / I z1/2

0

Px/Iz1/2

-Px/I( 2

0

.

(4.8)

The relation between rototranslational and Cartesian coordinates is given by the matrix 13 [Eq. (2.12)]. The product Po13 is termed a rotational polar tensor RX = P0 13"

/

(4.9)

The RX matrix acquires the following structure mo~Z~z / Iy + m~yOo.py/ Iz -muY~x/Iz - n ~ d 3 x / Iy

- n ~ X ~ z / Iy -mo~x~y / Iz -m~y~tPz / Ix nkxZ~z/Ix + n ~ x ~ x / Iz -motZ~y / Ix nkLy~tPy/Ix + n ~ x ~ / I y

. (4.10)

The entire representation of (~//~Qt)o in Eq. (4.5) in terms of Cartesian displacement coordinates is simply the sum of vibrational and rotational polar tensors PX = VX + RX

(4.11)

MOLECULAR DIPOLE MOMENT DERIVATIVES

81

or

PX = PQ L-1 B + Pp [3.

(4.12)

Since PQ L -1 = PR, we may also write PX = PRB + Pp13.

(4.13)

Expressed in terms of symmetry coordinates Eq. (4.13) becomes PX = PS U B + Ppl~.

(4.14)

With the addition of the Ppfl term the atomic polar tensor matrix becomes mass independent since it refers to a space-fixed coordinate system. Thus, the transformation of measured infrared intensities of different isotopic species of a molecule with identical symmetry will result in approximately the same PX matrix, within the experimental uncertainties. This is an important feature of the atomic polar tensor representation of infrared band intensities. The troublesome problem of rotational correction terms is treated in a straightforward and general way. The treatment, however, introduces some difficulties in the physical interpretation of the elements of atomic polar tensors. These will be discussed later in conjunction with some examples of calculations. The inverse transformation from Px to PR is given by PR = Px A .

(4.15)

The dipole moment derivatives with respect to normal coordinates are then obtained from the relation PQ = PX A L .

(4.16)

Equation (4.16) is a basis for applications of the APT formulation in predicting vibrational intensities by transfemng atomic polar tensors between molecules having the same atoms in a similar environment. The invariance of the dipole moment with respect to molecular translation leads to the expression igp/~ = 03 where 0 3 is a 3•

r

(4.17)

zero matrix. Since aX(a)/~ are equal for all atoms, we obtain

N E Vx (~) = 03.

(4.18)

82

GALABOV AND DUDEV

The change of dipole moment with infinitesimal rotation of molecules yields another relation between the elements of the atomic polar tensors [ 110,117,118] N E PX(a) ((r(a)))= ((Po)).

(4.19)

0~=1

r(a) is the position vector of the a th atom with respect to the center of mass. ((P0)) is a second rank pseudotensor containing Cartesian components of the equilibrium dipole moment. In a complete form Eq. (4.19) becomes

N X

P~

a Pxy

P a~

0

zO

_yO

/ P~

o Pyy

a Pyz

_z 0

0

x0

yO

_x O

0

(X

a=l ~pazx

Pzy

(X

Pzz

=

0

Pz

-Py

-Pz

0

Px

Py

-PI

0

.

(4.20)

In expression 0.20) x ~ y O and z ~ are components of the radius vector of atom a as defined in relation (4.19). The analytical expressions relating the APT elements that follow fIom equation (4.20) are as follows [118] X [-~)z

0 + (p~)yO]

= 0

Ot

Ot Ot X [(Pyx) zO - (Pyz)xO] = 0 Ot

a x0 ] = 0 X ['-(P~) yO _ (Pzy)

0.21)

Ot

Xt(Pxx)a z 0 + ( p ~ ) x O ]

=_X[_(pyy)a

X

xx)yO + (p~y)X 0] = _ ~:

I1

ft.

z 0 + (pyz)yO]a

"

z~

+

(p~)yO]

a yO + (pyy) a x o ] - - X [(Pzx) a zO - (Pzz~) xO] 9 X [-(Pyx) Ot

Ot

The values of the elements of atomic polar tensors as evaluated from Eq. (4.12) depend on the choice of a reference Cartesian coordinate system. Applications in describing structural properties of molecules require to tabulate atomic polar tensor elements for different atoms in various molecular environments. With values depending

MOLECULAR DIPOLE MOMENT DER/VATIVES

83

on the particular choice of a reference Cartesian frame this is quite inappropriate. The difficulty is dealt with by transforming the APT elements into values referring to a specific Cartesian system with diagonal elements independent on the particular arrangement of neighboring atoms. Biarge, et al. [ 108] proposed that a local Cartesian system is employed with one of the axes directed along the bond containing the atom a, while the directions of the other two axes are appropriately defined in accord with the bond site symmetry. The following relation may be used (4.22)

(PX(a))0 = T x PX (a) T x .

(PX(a))o refers to the bond axis system. T x is a standard transformation matrix between Cartesian systems [119]. Relation (4.22) may appropriately be used to test the transferability of APT elements. In view of relations (4.18) and (4.19) such attempts should, however, be carried out with a great deal of caution.

B. I n v a r i a n t s o f A t o m i c P o l a r T e n s o r s Under Coordinate Transformation

A quantity called effective atomic charge can be derived from the atomic polar tensors. Originally it has been defined by King, et al. [105] from considerations of intensity sum rules using Cartesian coordinate representation of the dipole moment derivatives. The derivation stems from the Crawford's G-sum rule for infrared intensifies. The experimental dipole moment derivatives are expressed as a column vector (0p/0Q) whose transpose (Op/O0)' has a structure (Op/OQ)'

=

(Opx/OQl , Opy]OQl,... , Opx/OQk , Opy/OQk .... ).

(4.23)

The sum over all intensities is given by

k

Ak =

Non 3c2(4rr~0) (Op/~Q)' (~)p/~gQ).

(4.24)

The respective G-sum rule is then expressed as

k

Non 3r (4he 0) g

where g = x, y, z.

()R)' G

0R)

(4.25)

84

GALABOV AND DUDEV

The dipole derivatives with respect to Cartesian coordinates are given by relation (4.13). In Eq. (4.25) the respective dipole derivative arrays are expressed as vectors

(o~g/aX)' = (o~g/aR)'B + (Opg/ap)' 13.

(4.26)

Using the relation B M -1 B = G [4] and B M -1 ~= 0 [Eq.(2.15)] and substituting (4.26) into (4.25) the following relation for the stun of intensities is obtained

[

~Ak =

ax)' M-1 (4.27)

The matrix product I] M-l [l= A is defined in expression (2.15). A is a diagonal 6x6 matrix with elements a triplet of the reciprocal of molecular mass and the three principal moments of inertia. The second term in Eq. (4.27) denoted by F2, which represents the rotational part of Eq. (4.27), is equal to fl = (p2y +p2z)/ixx + (pc2 + p2x)/iyy + (p2x + p2y)/izz . Finally the stun of intensities is expressed as

k

Ak=

3C2(4~:e0)

-

(4.28)

o]

.

n ~ is the mass of the ,th atom. The quantity ~2 is equal to the sum of squares of the elements of the respective atomic polar tensor (4.30) g

g2 may also be expressed as ~2 = Tr (V~ P). (Vet P)' ,

(4.31)

where V is the gradient operator. The elements of atomic polar tensors are in units of electric charge. The quantity (=~/~'~) is, therefore, also in units of electric charge. Tensor properties require that the sum of squares of all elements are independent on the particular choice of a reference

MOLECULAR DIPOLE MOMENT DERIVATIVES

85

Cartesian system. It is hoped that the ~x reflect electric properties of individual atoms in molecules. King et al. [105] in their original study have found out that ~ for hydrogen atoms are almost constant for many types of hydrocarbons with the exception of acetylene. An accurate physical interpretation of effective atomic charges as derived from Eq. (4.31) is, however, difficult to produce. The problem will be discussed in the following part. An alternative derivation of intensity sum rule and effective charges is offered by Person and Newton [33]. A matrix product PQPQ' with dimensions 3x3 is defined

k

(/)Px / CK~k)2 ~ (()Px/ OQk)(~gPy/ t)Qk) ~ (t)Px/ OQk)(~Pz/ OQk) k

PQPQ, =

k

E(C)Py ' (~Qk)2 k

E (OPy ' t)Qk)(C)Pz' ~Qk) k E( pz/ Qk) 2

.

(4.32)

k

The trace of PQPQ' is equal to Tr (PQPQ')= ~ [(/~ax/~k) 2 + (c~py/c~Q~2 + (/~pz/C~k)2] . k

(4.33)

The sum of vibrational absorption intensities is obtained by multiplying Eq. (4.33) by [N0g/3cE(4ge0)].

(4.34)

NOn Tr(PQPQ,)= ~ A k . 3c2 (4r~0) k

The trace of the product PQ' PQ multiplied by [N07t/3c2(4rce0)] is also equal to the sum of intensities since

pQpQ_

r

(0p/0QI) 2

.

(igp / aQ1)(~ / aQ2)

~

.

(~/~Q2

)2

sym. ~ 1 7 6 1 7 6

~

(4.35)

~ 1 7 6

(3p / 3Q3N-6) 2

L(~ / igQl)(~ / igQ3N-6) So, we have Non 3r

(4.36) k

GALABOV AND D U D E V

86

The product of augmented matrices of dipole moment derivatives with respect to normal coordinates that include also the derivatives with respect to external coordinates is (4.37)

(PQ" Pp) 0PQ" Pp)' = PQ PQ' + Pp Pp' . PpPp' is equal to PPPP' = l p 2 / I nsym. + p2y/Izz

-PyPx /Izz pz2 / Ixx + p2x/ Izz

-PxPz /Iyy / PyPz / Ixx 9 p2/Ixx +p2x/Iyy

(4.38)

Therefore, Tr [0PQ" rp) (PQ" Pp)'] = Tr (PQ PQ') + (pz2 + p2)/Iyy + (p2 + p2)/Ixx + (p2 + p2)flzz.

(4.39)

Since the sum of the last three terms of Eq. (4.39) is equal to the rotational term f2 [Eq. 0.28)] we may write Tr [0PQ" Pp) (PQ" Pp)'] = Tr (PQ PQ') + f2.

(4.40)

Atomic polar tensors may also be represented in mass-weighted Cartesian coordinates q as defined by relations (2.9). The mass-weighted Cartesian displacements qa referring to a space-fixed Cartesian frame are related to the normal coordinates Qk by an orthogonal transformation [4]

Mass-weighted atomic polar tensors are derived by an expression analogous to (4.13) [33] Pq = PR D + P p S .

(4.42)

The matrices D and ~ are defmed in expressions (2.12). Pq has a structure analogous to the PX matrix. Using Eq. (4.41) Pq may be expressed as Pq = (PQ.Pp) r

and (PQ" Pp) = Pql.

(4.43)

MOLECULAR DIPOLE MOMENT DERIVATIVES

87

Substituting Pqi for (PQ" Pp) in Eq. (4.40) we obtain Tr 0Pq I r Pq') = Tr (Pq Pq') = ~ (Op/Oqk)2 k (4.44)

= Tr(PQPQ') + ~ . If K = [N0rd3c2(4xe0)] we also obtain Tr 0Pq Pq') = (I/K) ~ A k + f 2 . k

(4.45)

Another expression of (4.45) is (4.46)

Tr [pq(OO (pq(a)),] = (I/K) ~'~ A k + f2.

a

k

Transformed to ordinary Cartesian coordinate representation Eq. (4.46) becomes a

(I/m~ Tr [Px (a) (Px(a)) '1 = (l/K) ~ A k + f~.

(4.47)

k

The sum rule given by expression (4.47) is first derived by Biarge, et al. [108]. It is identical to the sum rule derivation of King, et al. [ 105]. Thus, the square of an effective atomic charge ~t is simply equal to the trace of the product Px(a)(Px(a)) ' 2 = Tr [Px(a)(Px(a))'].

(4.48)

later King [ 120] introduced a different quantity for an effective atomic charge denoted

byx X 2 = (1/3) { 2 .

(4.49)

Two alternative invariants with respect to coordinate transformation can be derived from the atomic polar tensors - the mean dipole derivative, pa, and the anisotropy, 13a2 [33,108]. pa reflects the invariancy of the trace of a tensor m

m

I

~a = (1/3) Tr (Px(a)).

(4.50)

The anisotropy is defined according to the expression 2 = 89[(Pxx - pyy)2 + (pyy_ pzz)2 + (Pzz - Pxx)2 + 3 ( p ~ + p 2 +Pyx2 + p2 z + p 2 + p2zy)] where Pn~ =

~n/~

(11, e = x, y, z).

(4.51)

88

GALABOV AND DUDEV

The invariants of atomic polar tensors are related by the expression

~2= 3( p-~)2 + (2/3) ~ 2 .

(4.52)

C. Symmetry Properties of Atomic Polar Tensors Atomic polar tensors, defined with respect to an arbitrary Cartesian system, may be transformed into quantifies referring to a bond axis system using Eq. (4.22). If a molecule has sets of equivalent atoms the transformation (4.22) will produce identical atomic polar tensors, provided the local reference systems are chosen in a consistent way. With such representations it is clear that the number of independent atomic polar tensor elements is smaller for molecules with higher symmetry. Decius and Mast [117] analyzed in detail the site symmetry properties of atomic polar tensors expressed in terms of bond axis system. The treatment covers molecules with sufficient symmetry so that the directions of ~ransitional dipole moments are uniquely determined. Molecules with symmetry point group G = C2h, C 2, C s, C i and C 1 are excluded from the analysis. As already discussed, all elements of the PQ matrix for such molecules cannot be determined from experiment. The site symmetry of a bond containing a representative atom of an equivalent set o is a subgroup Ho of the molecular point group G. If the representative atom is designated by ot the respective polar tensor will have nine elements of the type 0p~/0Xq(a) where ~, TI =x, y, z. The following rule determines the structure of the 3x3 tensor: if the pg component and Xq coordinate belong to the same symmetry species of the site subgroup, the respective derivatives will differ from zero. In all other cases the elements will vanish. Decius and Mast [117] proposed a convenient notation to express the non-vanishing elements of atomic polar temors. An AB4 (To) molecule will have two types of atomic polar tensors for the two representative types of atoms A and B. The coordinate axes orientation is as shown in Fig. 4.1. Atom A is invariant under the full point group T0. The non-vanishing elements of the atomic polar tensors will be associated with the triply degenerate symmetry class F2 and, also, 0px/0XA = 0py/0yA = 0pz/0zA. All other elements of Px(A) will be zero. The structure of the tensor will, therefore, be

(,ooXo / Ax 0

00 .

(4.53)

Ax

The atom B situated along an AB bond axis with site bond symmetry C3v will, in the present treatment, be the atom along the z direction (Fig. 4.1). The character table for T d

MOLECULAR DIPOLE MOMENT DERIVATIVES

89

)

Y FIG. 4.1. Coordinate axes for AB 4 (Td) molecule.

TABLE 4.1 Character table and symmetry species for Cartesian coordinates for T d molecules

Td

E

8C 3

3C 2

6S 4

66 d

A1 A2 E F1 F2

1 1 2 3 3

1 1 -1 0 0

1 1 2 -1 -1

1 -1 0 1 -1

1 -1 0 -1 1

~Xxx+r (Ctxx+Ctyy-2CXzz,Ctxx-Ctyy)

Rx, Ry, R~ Tx, Ty, Tz

(etxy, tXxz, Otyz)

molecules is given in Table 4.1. Symmetry species A l (z-polarization) and the doubly degenerate E class (x,y) will be associated with the non-vanishing dipole moment derivatives. Since Opx/OXB equals Opy/OyB the atomic polar tensor will have a structure

o ~/

Bx 0

0. Bz

(4.54)

In the case of the AB 2 (C2v) molecule A is invariant under the full point group and the symmetry species associated with non-zero dipole derivatives will be A 1 (z), B 1 (x) and B 2 (y). The character table for C2v point group is shown in Table 4.2.

90

GALABOV AND DUDEV

TABLE 4.2 Character table and symmetry species for Cartesian coordinates for C2v molecules

C2v

E

C2(z)

A1

1 1 1 1

1 1 -1 -1

A2 BI B2

Gv(Zy) (;v(ZX) 1 -1 1 -1

1 -1 -1 1

Tz Rz Ty, Rx Tx, Ry

axx, an,, azz axy ayz axz

The structure of the polar tensor for atom A will simply be Ax

0

0

Ay 0

0 (4.55)

0 . Az

The bond site synmletry for the B atom is C s. The non-vanishing elements will be associated with symmetry species A' (x and z) and A" (y). The polar tensor of atom B has the form :x

~Bzx

0

Bxz

By

0

0

Bz

.

(4.56)

The coordinate axes orientations for the AB 2 molecule are as defined in Fig. 3.6. Following such an approach Decius and Mast [ 117] have tabulated the structure of atomic polar tensors for a number of simple molecules representative of several point groups. Translational and rotational dependencies arising from Eqs. (4.18) and (4.20) are also explicitly defined. The structures obtained are summarized in Table 4.3. Decius and Mast [117] proved also that the total number of independent APT elements is exactly equal to the number of infrared active modes in molecules where symmetry determines the polarization axes of all vibrations. The number of atomic polar tensor elements associated with an equivalent set of atoms o is given by 1o = ~ [no(n)]2 9 11

(4.57)

MOLECULAR DIPOLE MOMENT DERIVATIVES

91

TABLE 4.3 Structures of atomic polar tensors of symmetric molecules (Reprinted from Ref. [ 117] with permission) A AB2 (Dooh) z IIC|

z 11C2 moleculein xz

Translational conditionsa

B

lo o/(ooXxoo Oo oOX / Ayo o// ox 0

Az

By

~,Bzx 0

0

/

Bz

Rotational conditionsb

Ax+2Bx=~c A,+2B,%

Ax+2Bx=~ Ay+2By=~ Az+2Bz=~

Axcosl3-2Bxzsinl3---p/r 0 Aycosl3=p/r 0

AB3 fD3h) zil C3

AB3 (D3v) zllC3

AB4 (Td)

A2B2 (DootO z IICoo

A2B4 (D2h) moleculein xy

/mr 00/ A A0z 0x

/B00x 0/BY0 0 nz

o)/ x0 Byo Byoz/

x 0 Az

(A00z 0Az0 Az,,0/

0 Bzy Bz

~mx+(3/ mz+3Bz= x + B 2y)(B) = ~

+By)=~ Az+3Bz=~

AxCOST-(3/2)Bz .siny=p/r0

( A0xz +Bx00 (4/B 3z~ ) (/ 2 B x+Bz)=~ B

0 :/fooX B0x 00 / 22Az+2Bz= x+2x~

Ax 0 Az

oOX / Ay~0 Az00

0 Bz

By 0 2Ay+4By=~ 0 Bz 2Az+2Bz=~ aThetranslationalconditionsare expressedfor a neutralmolecule. If ionsare considered,the sumswillbe equalto thetotalioncharge. bp is the equilibriumdipolemomentand r0 - the ABbondlength;bondanglesare definedin Ref. [117]. c~is the net chargeof the molecule.

92

GALABOV AND DUDEV

no(n) is the number of times the irreducible representation Fo(tl) is found in the reduction of a Cartesian representation of the subgroup Ho. Since molecules with higher symmetry are only considered, i.e. other than these with molecular point group C2h, C2, Cs, Ci and C1, symmetry coordinates may be constructed either in internal or Cartesian coordinate space. The number of Cartesian symmetry coordinates associated with the equivalent set of atoms o is of interest to the treatment here. It is given by the expression

= E E

(4.58)

y* TI

Summation is carried out over the infrared active species y,. The coefficients c ~ are obtained from c.rq =(1/h ) ~ Z(Y)(R) Z(rl)(R) RCHo

(4.59)

where h is the order of Ho, while Z('/)(R) and Z(rl)(R) are the characters of the respective symmetry species. The summation is over the symmetry operations of the subgroup. For the higher symmetry molecule considered, the following relation holds y.

Z(Y)(R) = Xp(R)

(4.60)

where Zp(R) is the character of the molecular dipole moment. The character of the dipole moment in the subgroup Ho is determined by the relation gp(R) = Z no0q') z(TI')(R) 9

(4.61)

Vl'*

Introducing Eqs. (4.60) and (4.61) into (4.59) and summing over the infrared active modes result in

y*

cyq = (l/h) ~ RCH~

Z no(n') Z(n')(R) ~(rl)(R) =no(n). tl'*

(4.62)

Substitution in (4.58) gives m o = ~ [no(n)12 . vl*

(4.63)

MOLECULAR DIPOLE MOMENT DERIVATIVES

93

Comparing (4.63) and (4.57) the following result is obtained Io =

(4.64)

E n~

or

l = E 1o = ~

~

[n~(rl)]2 .

(4.65)

The total number of elements 1 of the atomic polar tensor of a molecule is, therefore, equal to the number of Cartesian symmetry coordinates in the infrared active species. The set of Cartesian symmetry coordinates describes, in the general case, vibrational distortions as well as translations and rotations belonging to the same symmetry species as the infrared active modes. The translational and rotational conditions can be explicitly written as shown in Table 4.3. The important conclusion is that the net number of independent atomic polar tensor elements is exactly equal to the number of infrared active modes. In the case of AB 2 (C2v) molecule 1 = 3+5 = 8. For such molecules, however, there are three translational and two rotational conditions relating the APT elements as shown in Table 4.3. Subtracting these from 1 yields exactly the number of infrared active vibrations of the molecule.

D. Atomic Polar Tensors - Examples of Application In this section we will present examples of how atomic polar tensors are derived from experimental infrared intensity data for three molecules - the non-polar molecule of ethane and for CH3CI and H20, two molecules that possess a permanent dipole moment. Data for the last two molecules were also used to illustrate the valence-optical scheme. 1.

Ethane

The experimental PQ matrix for ethane is taken from the work of Kondo and Satki [ 121]. The reference Cartesian system, numbering of atoms and internal coordinates are shown in Fig. 4.2. Symmetry coordinates are given in Table 4.4 together with molecular geometry data [122]. Symmetrized normal coordinate transformation matrix L s is calculated using the force field of Nakagawa and Shimanouchi [ 123]. The L s matrix is given in Table 4.5. Dipole moment derivatives with respect to symmetry vibrational coordinates evaluated from the experimental c3p/0Qi quantifies are presented in Table 4.6. Atomic polar tensors are calculated from the relation PX = PQL-1Bs- The rotational

94

GALABOV AND DUDEV

x

R

FIG. 4.2. Definitions o f internal coordinates, bond directions, Cartesian reference system and n u m b e r i n g o f atoms for ethane.

TABLE

4.4

Structural parameters and s y m m e t r y coordinates o f ethane Geometrya rCH = 1.095 A ,

rCC = 1.531 A ,

Z C C H = 111.5 ~

S y m m e t r y coordinates b A2u

$5 = (At 1 + Ar 2 + Ar 3 - Ar 4 - Ar 5 - Ar6) / . f 6 S 6 = [a ( A a l + Aa2 + Act3 - Aa4 - Aa5 - Aa6) -

b (A[31 + A~2 + A[33 - A[34 - A~5 - A l 3 6 ) ] / . ~ a = 0.414957,

E u'

b = 0.401428

S10a = (2At 1- Ar 2 - Ar 3 - EAr 4 + Ar 5 + A t 6 ) / . ~ i 2 S l l a = ( 2 A a l - Aa2 - Aa3 - 2Aa4 + Aa5 + Aa6) / S12a = (2A]31- A~ 2 - A[~3 - 2A~4 + A~5 + A~6 ) / ~ / ~ "

Eu"

S10 b = (Ar 2 - Ar 3 - Ar 5 + At6) / 2 S l l b = (Act2 - A a 3 - A a 5 + Aa6) / 2

S 12b = (A[32 - A[33 - A[35 + A[~6) / 2 a From Ref. [122]. b Internal coordinates are defined in Fig. 4.2.

MOLECULAR DIPOLE MOMENT DERIVATIVES

95

TABLE 4.5 L S matrix for ethane (in units of ainu- 8 9or rad amu- 8 9A-1) A2u

1.0125 -0.1437

0.0216 1.4640

Eu

1.0485 0.1353 -0.0091

-0.0112 1.4403 0.4311

-0.0303 -0.2779 0.8094

polar tensor Ppl3 [Eq. (4.9)] has zero elements. The entire PX matrix for ethane is given below (in units D A -1) [124]

Px =

2.

--0.626 0 0

CI 0 0.626 0

-0.611 0 -0.508

H3 0 0.194 0

-0.333 0 0.014

-0.008 0.349 0.254

H4 0.348 -0.409 -0.439

-0.008 -0.349 0.254

H5 -0.348 -0.409 0.439

0.167 0.288 0.014

-0.611 0 -0.508

H6 0 0.194 0

-0.008 0.349 0.254

H7 0.348 -0.409 -0.439

0.167 -0.288 0.014

-0.008 -0.349 0.254

H8 -0.348 -0.409 0.439

0 0 -0.042

0.626 0 0

C2 0 0.626 0

0 0 -0.042

0.167 -0.288 0.014

-0.333 0 0.014

0.167 0.288 J . 0.014

(4.66)

Methyl Chloride

Experimental data, definitions of coordinates, molecular geometry data, and L S and PS matrices for methyl chloride are given in section 3.3. The calculated atomic polar tensor matrix is given below (in units ofD A -1) [124].

96

GALABOV AND DUDEV

TABLE 4.6 PS matrix for ethane (in units of D A -1 or D rad-1) A2u

P5 = -1.144 P6 = -0.365

Px/~

C1 0 0.565 0

C12 -0.847 0 0 0 -0~47 0 0 0 -2237

0 0 2950

H4 0248 0267 0.086 0267 -0.060 -0.149 0.151 -0262 -0238

H3 0 -0172 0.402 0 0 -0238

-0214 0 -0302

0248 -0267 0151

Pl0 = -1.196 Pll = 0.251 P12 = -0.297

(4.67)

H5 -0267 0.086 / -0.060 0149 . 0262 -0238

PX is a sum of two terms - vibrational polar tensor (PsBs) and rotational polar tensor (Ppl3). The rotational polar tensor is evaluated using an equilibrium dipole moment value of -1.87 D [34]. The PsBs and Ppl3 matrices are given below. C1 PsBs =

- 0 i62

0 -0.162 0

C12 0 0 2.950

0.114 0 0

H3 -0.292 0 -0.120 0 0.324 0 -0.302 0 -0.238

0

0 0 -2.237

H4 0.170 0.267 0.060 0.267 -0.138 -0.104 0.151 -0.262 -0.238

H

5 0.170 -0.267 0.060 -0.267 -0.138 0.104 0.151 0.262 -0.238

0 0.114

J

(4.68)

MOLECULAR DIPOLE MOMENT DERIVATIVES

0.!27 Pal3= I

[

0.078 0 0

CI 0 0.727 0

C12 -0.961 0 -0.961 0 0 0

0 0 0

H3 0 --0.052 0.078 0 0 0

H5 0.078 0 0 0.078 0 0

97

0.026 0.045 0

0 0 0

H4 0 O.026 0.078 --0.045 0 0.

0.078 0 0

(4.69)

J

It is important to underline here that the components of the rotational polar tensor Ppl3 for the carbon and chlorine atoms have large values. These are, in fact, determining both the sign and magnitude of the final PX matrix elements for these two atoms [Eq. (4.67)]. Since the elements of Ppl3 depend on the equilibrium dipole moment value, the numerical results obtained are quite significant. The mixing of vibrational and rotational terms in the PX matrix has important consequences with regard to the interpretation of the atomic polar tensors. 3.

Water

Coordinate definitions, L S and PS matrices for H20 used in evaluating the atomic polar tensor elements are as given in section 3.3. With the aid of relation (4.14) the PS matrix is lransformed into vibrational polar tensor, while the rotational polar tensor is calculated using a permanent dipole moment value of-1.85 D [34]. The two submatrices obtained are as follows (in units of D A-l):

-1.109 PsBs =

O1 0

0

0554

H2 0

0.429

0

0

0

0

0

0

0

0

- 1.402

-0.333

0

0.701 (4.70)

0354 0 0.333

H3 0 0 0

- 0.429 0 0.701

J

98

GALABOV AND DUDEV

O1 0 - 3.157 0

H2 0 0 0

0348 0 0

0 1.578 0

-0.798 0 0 (4.71)

0348 0 0

H3 0 0.798 "~ 1.578 0 J. 0 0

Finally, the PX matrix evaluated as a sum of vibrational and rotational polar tensors is

-2!06 PX =

O1 0 0 -3J57 0 0 -1.402

1.102 0 -0333

H2 0 -0.369 1.578 0 0 0.701 (4.72)

H3 LI02 0 0333

0 1378 0

0369 "~ 0 /" 0.701

It is easily seen that some elements of PX, i.e. all second row elements (pyy), come directly from the rotational polar tensor. Since the non-zero elements of Ppl3 are derived from the Cartesian components of the equilibrium dipole moment, it is evident that the final Px matrix also contains such non-vibrational contributions. Therefore, the term PoI3 does not simply correct for rotational contributions to dipole moment derivatives in the sense discussed by Crawford [35,36]. It directly introduces terms arising from the permanent dipole moment into the PX matrix.

E. Interpretation of Atomic Polar Tensors So far we have seen that the APT representation of infxared absorption intensities offers a mathematically efficient way of reducing the experimental data to quantifies associated with motions of individual atoms in molecules. In general terms such a rationalization of intensity data appears acceptable since intramolecular charge distribution may be approximately expressed in terms of partial atomic charges provided, of course, that an effective way is found to relate the charge distribution with atomic polar tensors.

MOLECULAR DIPOLE MOMENT DERIVATIVES

99

For simple systems such as a diatomic AB molecule it has been shown [ 125] that the polar tensor of atom A has the structure (~thB/r o 0 0 / Px(A)=~ ~ ~tAB/r o 0 0 3~thB / ~r o

J

(4.73)

where ~tABis the AB bond moment and r 0 is the eqtfilibrium bond distance, lXABis equal to the permanent AB dipole moment P0. Biarge, et al. [108] have also discussed similar interpretation of atomic polar tensors for the atoms of a bond if the respective PX (a) elements are expressed in bond axis system. Provided that the bond moment hypothesis holds perfectly, including the condition that the dipole moment changes are directed along the bond axis, the atomic polar tensor will have strictly diagonal structure. The appearance of off-diagonal terms is considered as a measure for the deviations from the bond moment hypothesis. Results for many molecules show that the off-diagonal terms are usually quite significant. The simple diatomic case discussed earlier cannot provide a clue for a general interpretation of the atomic polar tensors derived from experimental infrared intensities. It was emphasized earlier that the usual Cartesian representation of the potential force field does not provide satisfactory physical basis for analyzing vibrational fi'equencies outside of purely computational convenience in some cases. The intramolecular forces are not described in a clear and coherent way. The transferability properties of force constants are entirely hidden. Similar arguments are also valid with regard to the dipole moment derivatives representation. There is, however, an important proviso. The molecular dipole moment is a vector and expressing its derivatives in a Cartesian space is more acceptable.

1.

Physical Significance of Effective Atomic Charges

Simple tests for assessing the physical significance of the effective atomic charges may be obtained in the cases of diatomic AB molecules on the basis of experimental data alone. The atomic polar tensor for diatomic molecules have the simple form as given by expression (4.73). Using experimental data for I.tAB (=P0) and dipole derivative values as given in Table 3.10 the following results are obtained. The reference Cartesian system is the same as already defined in conjunction with the description of Table 3.10. The units are proton charges (0.2082 p.u. = 1 D A-l). a.

HF

'3

~

---0.413 0

~/

0 . -0.350

(4.74)

100

GALABOV AND DUDEV

The effective atomic charge Za for the fluorine atom calculated by the trace of the product 3 Z 2 = Tr (V F P). (V F P)'

(4.75)

is equal to -0.393 p.u. The experimental value of the net charge on the fluorine as evaluated from the bond moment value is ~F = -0.413 p.u. The agreement between ~F and ~F appears quite satisfactory. b.

HC1 The atomic polar tensor for the chlorine atom in HCI is as follows -0"i 17 PCI

=

0 -0.117 0

0 / 0 . -0.183

(4.76)

The calculated value of XCI is -0.179 p.u. The experimental value of ~Cl is -0.177 p.u. The agreement in this case is perfect. The next example for the molecule of carbon monoxide will, however, show that simple interpretation of the effective charge Za is not at hand. c.

CO The atomic polar tensor in proton units for the oxygen atom in CO is as follows

PO =

19 o

-0.119 0

Oo/ .

(4.77)

-0.645

The calculated value of ~O is -0.373 p.u. while the experimental value of ~O is only 0.019 p.u. The examples presented here show that in the general ease the effective charge obtained from the trace of the product (Va P).(V a P)' does not find an immediate physical interpretation.

2.

The Charge-Charge l~ux Overlap Model for Interpretation of Atomic Polar Tensors

The examples discussed so far show that the elements of atomic polar tensors are determined by intramolecular charge fluctuations that are rather complex. The invariant

MOLECULAR DIPOLE MOMENT DERIVATIVES

101

under coordinate transformation- the effective atomic charge -- does not have a simple physical sense. Nevertheless, it was felt that a direct association of atomic polar tensors with the intramolecular electric charges is quite desirable. King and Mast [45], following quantum mechanical considerations regarding the definition of atomic charges in molecules, proposed a rather appealing, because of its simplicity, method for analyzing atomic polar tensors. The approach is known as the charge-charge flux overlap method (CCFO) [45]. The quantum mechanical treatment of King and Mast is based on the LCAO representation for the molecular dipole moment. As basic quantifies to describe the electrical properties of atoms in molecules, the authors accept the net atomic charges as derived from Mulliken population analysis [87]. The experimental effective charges or Xa are associated with the trace of the product (Va P).(V a P)'

~2= 3 ;2 = (V a p).(Va p), p2+p2+p2 =

0

0

p2yx + p 2 +p2yz

0

0

0 0

(4.78)

p2+p2+p2

where p ~ n = ~ d ~ with ~5,11= x, y, z. In their approach King and Mast aim, basically, to divide the various possible dipole contributions to atomic polar tensors into "perfect following" and "non-perfect following" parts. The "perfect following part" is associated with the static net charge densities on atoms, as provided by the Mulliken population analysis. The remaining contributions form the "non-perfect following part". If the elements of atomic polar tensors are determined by the "perfect following part" alone, these quantifies will have strictly diagonal form with identical elements. All differences in the diagonal terms arise, therefore, from the "non-perfect following" contributions. The rationalization of these basic concepts is done, as mentioned, by analyzing the LCAO composite dipole terms. We shall not follow the entire derivation of King and Mast [45]. Their analysis leads to the following representation of atomic polar tensors

F~P-

~ I + I g O r a(13 - [V~(I)1313.

(4.79)

f-,a and ~13are net atomic charges, I is a unit diagonal vector and RI3 is the position vector of atom 13. The term (1)1313has a rather complex nature. In general terms it comprises all dipole contributions that do not arise from the net charges. By definition it is E (~Pi(rl3)I r13 I~Pk(r~)) N~k~(R). @13130R) = i,k

(4.80)

102

GALABOV AND DUDEV

R is the set of individual vectors R~, R = {RI3}. The integrals are the usual dipole moment integrals between atomic orbitals ~Pi and ~Fk and 9 " jk N~8 = ~ 2 C~v Svf)" v j

(4.81)

Cpv are the LCAO coefficients for the basis set atomic orbitals 13 and v. The i and j are indices of molecular orbitals, sJv~ are overlap integrals. By expressions (4.80) and (4.81) a partitioning of the non=classical dipole term arising from orbitals centered on different atoms is accomplished so that all dipole contributions are associated with atomic sites. The last term in Eq. (4.79) comprises the dipole contributions ~om the second and third terms as defined in Eq. (3.30). On the other hand, the first and second terms in Eq. (4.79) are associated with the same dipole contributions as defined in Eqs. (3.34) and (3.35), respectively. The first two terms of Eq. (4.79) represent directly the point charge model. The first reflects the static charges of atoms and the second the charge fluxes. The third term has no classical analog or easily visualized explanation. King and Mast denote it as the overlap term. It is, perhaps, more satisfactory to describe it as an effective atomic dipole tenn. Indeed, it certainly includes the atomic dipoles contribution as defined by Eqs. (3.38) and (3.39). There is no experimental way of obtaining the overlap (atomic dipole) term in the CCFO interpretation of atomic polar tensors. Thus, applications are limited to quantum theoretically calculated polar tensors. The overlap term is not included in any standard ab mitio wave function calculation packages. It is estimated indirectly by subtracting from the calculated atomic polar tensors the charge and charge=flux terms. These calculations may be used to analyze the factors determining vibrational intensities in infrared spectra. A modification of the CCFO model was proposed by Gussoni et al. [126,127]. Basically, it resorts to expressing atomic polar tensors in terms of two sub-matrices rather than three as in the original formulation. A polar tensor of an atom a is represented as

(a~l3 / ~)xp

(a~l~/ az,x)Xl~

(4.82)

~z is the static charge associated with atom r The second matrix comprises the charge flux terms. ~ are presumed to reproduce the molecular dipole moment via the relation

MOLECULAR DIPOLE MOMENT DERIVATIVES

p = ~

~ rtt,

103

(4.83)

{z

where r~ is the position vector of atom ix. The values of ~ may be estimated from the experimental equilibrium dipole moment in the cases of simple systems such as AB, AB 2 (CEv), AB 3 (Cav), etc. More options appear to be available for evaluation of Qx fi'om quantum mechanical calculations [51,127]. 3.

Vibrational Atomic Polar Tensors

Person and Kubulat [86,106] have emphasized the importance of analyzing the vibrational part of atomic polar tensors. Both experimental and theoretical calculations have shown that rotational contributions to atomic polar tensors can be quite significant and, sometimes, overwhelming. For the water molecule, for instance, it accounts for twothirds of the total intensity Y_.Ak+f2 [86]. The rotational terms, therefore, dominate the atomic polar tensors of this molecule. Since the interpretation of vibrational intensities is aimed at deriving quantifies associated with molecular vibrations, it is desirable to separate the rotational terms explicitly. As can be seen from expressions (2.12) and (4.8) the rotational polar tensors are uniquely determined by the permanent dipole moment, atomic masses and equilibrium molecular geometry. We must conclude that the interpretation of atomic polar tensors in terms of intramolecular effective charges, etc. need always consider the complex nature of the individual elements of atomic polar tensors. This has been clearly shown by Person and Kubulat [86]. In their derivation the atomic polar tensor of an individual atom is given by the expression (4.84)

P~ = (P~)vib + (IP~)rot 9

The squared mass-weighted effective charges 3 z~/m~ may also be separated into vibrational and rotational part 3 Z 2/m~ = (3 Z 2/m~vib +

(3 Z 2/m~z)rot .

(4.85)

The sum over all atoms of the rotational part gives the total rotational intensity f2 f2 = k ~ (3 Z 2/m~rot .

(4.86)

ot

The sum of vibrational absorption intensities may also be represented in terms of contributions from individual atoms

k

A k = k ~ (3 Z 2/m~vib . ct

(4.87)

104

GALABOV AND DUDEV

The vibrational squared mass-weighted effective charges may in turn be defined as (3 X2/ma.)vib = 3 X2/m~ - (3 X2 / n ~ r o t .

(4.88)

A particular Cartesian component of PXa is Oven by r

ot

1~

o~

l,P~)xx = ( Px,vib )xx + ( PX,rot )xx"

(4.89)

The square of (P~)Xx that will contribute to (3 X2/too) is then [(P~)xx ]2 : [(Px,viba )xx]2 + 2 [(P~(,vib)XX] [(PX,rota )xx] + [(PX,rottX)xx]2 .

(4.90)

The last term in Eq. (4.90) contributes to the rotational part of the square of mass-weighted effective charges. Thus, cross-terms appear in the expression for the (3 Z~2/mo.)vib (3 g 2/ma)vib = (I/n~ Tr[(Px,vib a ) (P~,vib )']

3

+ (I/ma) ~

ij=l

2( Px,vib a a )ij (Px,rot )ij -

(4.91)

In general matrix formulation the vibrational polar tensor of a molecule is given by rx(v) : rx-

l'p13

: Vx.

(4.92)

The elements of V X are derived by the simple transformation VX = PS U B

(4.93)

provided the intensities are first transformed into dipole moment derivatives with respect to symmetry coordinates. As already discussed, the Op/0Sj derivatives may contain some contributions from the compensatory molecular rotation. To obtain 0p/0Sj quantifies that reflect purely vibrational distortions rotational correction terms must be subtracted [3436]. It is evident that Vx for different isotopes of such molecules will be different. The

MOLECULAR DIPOLE MOMENT DERIVATIVES

105

interpretation of such mass-dependent quantifies in terms of intramolecular charge dynamics is, therefore, not quite satisfactory.

F. Predictions of Infrared Intensities by Transferring Atomic Polar Tensors Atomic polar tensors derived from measured integrated infrared intensities may be used to predict intensifies in other molecules containing the same atoms in a similar environment. A comprehensive review on the subject has been published by Person [128]. By using Eq. (4.16) and Px matrix constructed from transferred atomic polar tensors, the matrix of dipole moment derivatives with respect to normal coordinates in the new molecule is obtained PQ=PxAL

=0PRB+Pp13) A L - P R B A L

= PRL

(4.94)

since 13 A = 0 and B A = E [Eq. (2.15)]. The procedure is, however, not so simple. Eq. (4.22) has to be twice used in order to transform first the APT elements into values referring to bond axis system and then once more to the molecular Cartesian reference system in the new molecule. In transferring APT elements the assumptions that need to be made with regard to the rotational part of atomic polar tensors are of great significance. By definition, there are contributions in atomic polar tensors associated with the permanent dipole moment value and other properties of the particular molecule that do not necessarily refer to individual atomic sites. Therefore, an approximation is implicit that the rotational part is also transferable. This problem is, evidently, not present in molecules that do not possess a permanent dipole moment. In spite of the approximation implicit in transferring atomic polar tensors, actual calculations have shown reasonable success [ 128]. Atomic polar tensors for the fluorine atom determined from experimental data for CH3F and for the hydrogen atom from methane are used to predict with considerable success infrared intensities for other fluoromethanes. The predicted intensities for CF4 are in very good agreement with the experimental values. For CF2H 2, however, some predicted band intensities differ significantly from the observed values. Some of the difficulties encountered may be avoided by transferring vibrational polar tensors only, while calculating the rotational part for the particular molecule considered if the dipole moment is known. These options appear not to have been pursued.

106

GALABOV AND DUDEV

II. B O N D C H A R G E T E N S O R S An alternative approach for analysis of vibrational intensities has been put forward by Mayants and Averbukh [116,129,130]. An extensive review on the method has been published by Rupprecht [37]. Hereafter, we shall follow with few exceptions the notation used by Rupprecht which is closer to the notation used so far. Instead, on the basis of atomic Cartesian displacement coordinates, the 0p/0Q i quantifies are transformed into the coordinate space of bond displacement vectors. The change of dipole moment is defined as

N-1 Ap = ~ D(k) xs(k) . k=-I

(4.95)

The summation is over the number of bonds N - l , with k a bond index. X6(k) may be expressed as Xs(k) = (Axa(k)- Axb(k)) i + (AYa(k) - AYb(k)) j + (Aza(k)- Azb(k)) k or

Xr

= Ax6(k) i + Ays(k) j + Az6(k) k,

(4.96)

where a and b are indices of initial (a) and final (b) atoms of the kth bond and i, j and k are unit vectors along the reference Cartesian frame. Also Axe(k) = Axa(k)- Axb(k) AYs(k) = AYa(k)- Ayb(k)

(4.97)

with the terms on the fight hand side of Eq. (4.97) being the atomic Cartesian displacement coordinates for the respective atoms. The total number of bond coordinates is 3N-3. By expressing X6(k) as differences between the displacements of the two atoms forming the bond the ~anslational motion is eliminated and, therefore, no translational dependencies are implicit in the resulting equations. It may also be argued that expressing intensifies in terms of bond displacements rather than atomic displacements in a molecule is more satisfactory from a physical point of view. D(k) in Eq. (4.95) are matrices with structure

MOLECULAR DIPOLE MOMENT DERIVATIVES

107

~ x / 0 y ~ (k) 0px/~zs (k) y / Ox8 (k) ~3y/~hj'~i(k) 0py/()z8 (k)

0px / 0xs(k) D(k) =

~

z / 3x~i (k)

(4.98) =

(dkxx dkxy d k / /dkyx dkyy dky-z ~dkzx d k

.

dkzz

The elements are in units of electric charge. D(k) are termed bond charge tensors. All D(k) matrices form the molecular bond charge tensor matrix D D = (D(1) D(2) . . .

D(k)...

D(N-1)) .

(4.99)

The dipole moment derivatives with respect to normal coordinates are related to the elements of the D matrix by the general equation

0

'OX? k)

= ~k

Xs't

"

(4.100)

In the Mayants-Averbukh formulation the rotational terms are not treated separately as in the APT method. An augmented PQ matrix is used that comprises both PQ and the three dipole moment derivatives with respect to the rotational coordinates. The resulting matrix is similar to the (PQ:Po) matrix [Eqs. (3.1), (4.8), (4.37)]. The three zero elements in Po are deleted since translational motion is eliminated through the defimtion of x6(k). For consistency of notation this matrix is denoted as P~i- It can be expressed as (4.101)

P8 = [PQ-(c3p/0Rp)] = [PQ'((Po)) I-%]

where ((P0)) has already been defined in section 4.1. The relation between P8 and D can be expressed as follows P8 = O L 8 .

(4.102)

L 8 is a normal coordinate transformation matrix of rank 3N-3 which includes the three rotations. It defines the relation between the bond displacement coordinates X~ and an

(o/

augmented normal coordinate column matrix with structure R'p

Essential stage of the procedure is the determination of the transformation matrix L 8. It is necessary to introduce a matrix A6 defining the relation between the column matrices X

108

GALABOV AND DUDEV

of atomic Cartesian displacement coordinates [Eq. (2.10)] and the matrix X8 of bond displacement vectors X 8 = A6 X .

(4.103)

A6 is of dimensions (3N-3)x3N. The row associated with a given bond has a triplet of +1 for the end atom of the bond and three -1 for the initial atom. The remaining elements of the row are zeros. The relations between X and X~5 and the matrix of normal vibrational coordinates Q are conveniently represented by the expressions [37] X - ALQ

(4.104)

= LxQ

X~ = A s A L Q

(4.105)

= Lx~Q.

The transformation matrices L x and LX~ define the contribution of each atomic or bond Cartesian displacement coordinate to the normal vibrations of the molecule. The matrix L~i needed for evaluating the elements of D has the following structure 9 ((sl)) I-1/2

Lti = Lx~i " ((s 2 ))1-1/2 .

.

.

.

.

9

(4.106)

.

9 ((sN-1)) I-1/2

The terms ((sk)) I - 8 9define the relation between X~5 and the normal rotations of the molecule

0xs(k)/0Rp = ((sk))I- 89

(4.107)

where 1-% = diag (IxV2,IyT~,Iz y2) in the principal axes system. ((sk)) replaces ((ra)) in the analogous expression (2.19). The matrix L 8 is most easily evaluated in an indirect way. Using the expression Lp = [L X " a ] ,

(4.108)

where L X is obtained from Eq. (4.104) and the elements of ct are given by Eqs. (2.18) and (2.19), L8 may be obtained from the matrix product A.Lp by simply deleting the three zero columns resulting from the action of A on the three terms in a associated with the Xg (g = x, y, z) coordinates [Eq. (2.18)]. The L~ matrix is non-singular. The elements of D are, therefore, conveniently calculated from the relation D ffi P8 L8 -1.

(4.109)

MOLECULAR DIPOLE MOMENT DERIVATIVES

109

As mentioned, in contrast to the atomic polar tensors formulation, in the MayantsAverbukh approach no explicit distinction is made between the contributions to D from vibrational and rotational terms. As we have seen in the case of the APT method such a separation is helpful in understanding the physical significance of individual tensor elements. The bond charge tensors, similar to the atomic polar tensors, are mass-independent quantifies. The physical interpretation of individual elements is rather difficult simply for the fact that vibrational and rotational terms contribute to the components of D. The origin of these conlributions is quite different. The vibrational terms arise from the intramolecular charge reorganizations induced by vibrational distortions. The rotational terms are associated with the equilibrium dipole moment value. The mixing of such principally differing terms hampers, in our understanding, the physical interpretation of these terms. Also, the transferability of DO') between molecules having the same type of bonds is also under question since the permanent dipole moment components are included in the last three columns of the P8 matrix. Permanent dipole moments are not transferable quantities. In general terms, there is a considerable similarity between the APT and BCT (bond charge tensor) formulations of infrared intensities. Formulas connecting the elements of Px and D matrices have been derived [ 131]. The authors have shown that if the coordinate system and numbering of atoms are conveniently chosen the following relation holds [ 131] D - PX C-1

(4.110)

0D(1)D(2) ... D(k) ... D(N-1)) = 0Px(l)Px(2)... PX (~) ... Px(N)) C-1

(4.111)

or

The matrix C is a non-singular matrix of dimensions (3N-3)• with elements +1, -1 and 0 appropriately placed to define the connection between the coordinate X and Xa. This has the effect that the elements of D are numerically equivalent to the elements of Px less the polar tensor of the first atom referring to the column array Xa. Mayants and Averbukh provided general expressions for the elements of D(k) referring to the bond axis system [ 129,130]. The bond axes are defined as follows: the z0(k) axis coincides with the bond direction, the x0(k) axis lies in any existing plane of symmetry with respect to the bond site symmetry and y000 axis is oriented in a way to form a right-handed coordinate system. Since analysis of vibrational absorption intensities is restricted to molecules with higher symmetry, the directions of the bond axis system are usually uniquely fixed. This transformation allows comparisons of bond charge tensors for different types of bonds and in different molecules. The transformarion is an essential step in any attempt of intensity predictions by transferring

110

GALABOV AND DUDEV

parameters. The bond displacement vectors referring to bond axis system are denoted by ~(k). The coordinate 8z0(k) coincides by definition to the usual bond stretching internal coordinates [37]. A standard orthogonal transformation between different Cartesian systems can be used [ 119,132] Xs(k) = Tx (k) 8(k),

8(k) = TX'(k) xs(k),

(4.112)

where TX'(k) is the transpose of Tx(k). The transformation between Do (k) and D(k), where DO(k) is the respective tensor in a bond axis system, is defined as D00 = Tx (k) Do00 Tx'(k).

(4.113)

D0C~) = Tx'(k) DO,) Tx0,).

(4.114)

Also

In expressions (4.113) and (4.114) D(k) refers to the molecular coordinate system common for all bond displacement vectors. In terms of Do(k) vectors the dipole moment derivatives with respect to normal coordinates are expressed as (0p/0Qt)o = ~ Tx (k) DO(k) St(k) . k

(4.115)

St(k) defines the contribution of 6(k) to the normal coordinate QtTranslational motion does not affect the elements of bond charge tensors. The values are, however, dependent on rotational contributions. The change of dipole moment with infinitesimal rotation of the molecule has been shown to produce six relations among the elements of the D matrix as shown by Averbukh [ 118]. The general expression is as foUows k

D(k) ((X60(k))) = ((P0))

(4.116)

where Xs0(k) is the kth bond vector at equilibrium and ((P0)) is an antisymmetric tensor [Eq. (2.20)] containing the Cartesian components of the equilibrium dipole moment. In a complete form Eq. (4.116) reads

NI ~., k---1

/dkyx dkyy dkyz [d k d k dkzz

i0

-z~ [~y~

0 -x~

x~ 0

= k

/0p. Py/ -Pz Py

0 -PI

Px 0

9

(4.117)

MOLECULAR DIPOLE MOMENT DERIVATIVES

111

The relations between the BCT elements resulting fxom Eq. (4.117) are [ 118] X [-(dkxy)Z 2 ) + (dkxz)Y2)] = 0 k X [(dkyx)Z2 ) - ( d k ) x 2

)] = 0

k

X [ - ( d k ) y 2 ) - ( d k z y ) X 2 )] = 0 k

(4.118)

X {[(dkxx)-(dk)] z~O(k) +(dkxz)X80(k) + (dkyz)Y~O(k)} = 0 k X {[(dkxx)- (dk)] Y2 ) + (dkzy)z2 ) - (dkxy) x 2 ) } = 0 k {[(dkyy)- (dk)] x~O(k) + ( d k ) z 2

) -(dk)y2

)} = 0.

k

III. B O N D P O L A R P A R A M E T E R S

A. General Considerations

The theoretical formulations described so far in this chapter provide alternative ways of interpreting the observed vibrational intensities in infrared spectra. Quite often some of the models developed are used in analyzing theoretically estimated intensities or dipole moment derivatives from ab initio MO calculations. The latter applications are certainly important as convenient approaches to gain more information about molecular properties from the calculated molecular electronic functions. We believe, however, that the principal application of the theoretical models lies in analyzing experimental intensity data. Vibrational intensities are directly related to the chemically most important property of a molecule -- the charge distribution. With the availability of more and more accurately determined infrared intensities, it is of considerable importance that these data are txansformed into molecular parameters characterizing intramolecular electric charge properties. Purely theoretical information about charge distribution in molecules can be obtained directly from the calculated wave functions. Properties of density functions are

112

GALABOV AND DUDEV

A

xh;;.;-9 .;. ] .~;_..-

~ I ~ //,""i""" I FIG. 4.3. Charge densities and their associated gradient vector fields in the plane of the carbon nuclei of cyclopropane at (A) equilibrium geometry, RCC = 1.50 A and for two distorted geometries; (B) RCC =1.80 A; and (C) RCC = 1.88 A (Reproduced with permission from Ref. [133]. Copyright [1983] American Chemical Society).

most conveniently expressed in the fascinating charge density maps [133]. An example is given in Fig. 4.3. The transition from ab mitio calculated vibrational absorption intensities to charge distribution is certainly less direct and introduces unnecessary approximations. With these comments we would like to underline the importance of theoretical formulations of intensities as an indispensable mean for analyzing experimental data. Accurate quantum mechanical calculations are, on the other hand, quite essential in studying trends of changes of intensity parameters, the intramolecular factors determining vibrational intensities as well as the validity of approximations inherent in the models developed. The second principal aim of intensity formulations is to provide convenient and reliable approaches for quantitative predictions of intensities. The success in this respect has been so far limited. The main reason is the very high sensitivity of intensities, respectively dipole moment derivatives, to structural variations. Transferability of intensity parameters is, therefore, much less pronounced if compared with force constants. The moderate success of the theories developed in predicting intensities is due, therefore, to the specific nature of this molecular property. With the increasing availability of sophisticated and efficient programs for ab initio MO calculations of intensities sufficient accuracy in predicted intensities may be reached. As we shall see later, however, such a stage is not yet in grasp, at least for infrared intensities.

MOLECULAR DIPOLE MOMENT DERIVATIVES

113

1.110 HCOS~(tr~

..... Nero o ~ 0 C

t

oCHF~

,'.mc.'~"~ o Mcoo~

1.100

C:,~"C,~'~_.~CN

, . , , ~ ' C o~,,,c - C~F~ " ~ l

- < 1.090

CH3|~C3H~ c-C3~

o 1.080

H~'

-

c~2~ 1.070 -

. _ ..~

2700

O.CN

- HCCCN

~c~~cc~ ~.J:~-L'-_ I.ICNO

1.060 I 2800

l 2900

l,,, 3000

i 3100

t 3200

i 3300

t 3z,00

"~ CH / cm-1

FIG. 4.4. Plot of the dependence between r 0 C-H (A) and isolated VCH (cm-1) for different molecules (Reproduced from Ref. [ 135] with permission).

Some of the intensity theories presented so far transform the experimental intensifies into quantifies characterizing bond properties, while others are aimed at reducing the experimental data to parameters associated with individual atoms. From a general physical point of view, the concepts of atoms in a molecule and molecules constructed of chemical bonds have their own place in describing molecular properties. The reduction of vibrational spectroscopic data to quantities characterizing bond properties is more satisfactory for a number of reasons. First, the interpretation of frequencies in terms of valence force constants has proved much superior to any other potential force field models [3-6]. Valence force constants are easily interpreted in terms of bond or group properties. As already mentioned, a number of spectral parameters are naturally interpreted in terms of chemical bond parameters. The local mode interpretation of high overtones of stretching modes in many molecules offers a nice example in this respect [9,10]. Illuminative and convincing arguments in this respect are provided by the perfect correlations between isolated bond stretching frequencies and bond lengths discovered by McKean [134, 135]. An example is shown in Fig. 4.4. Of even greater relevance to the present discussion are the experimental results reported recently fIom the same laboratory on the variation of infrared intensities of C-H stretching modes of isolated C-H bonds [136]. The precision measurements of high purity isotopes with only a single non-deuterated C-H bond reveal remarkable characteristics. Some of the results

114

GALABOV AND DUDEV

TABLE 4.7 Isolated C-H stretching frequencies and intensifies for different molecules a Speciesb

vi

CH4

2993 2950 2950

C2H6 C3H8 CH3

(CH3)3CH

c-C6H12

Ai

(cm-l) (lan/mol)

,CHH 2936 2918

29.2

2942

27.0

.CHH

2922 2893 2923 2891 2972

24.6 37.6 27.2 11.2d

,CHCI 2945 2983 CHH 2971

18.7d 12.7 11.3

CH C C 'CHH

CH3CH2Cl CH3

CH2 CH3CH2Br CH3

28.5

'CHC

CH2

CH3

10.8 27.3

t f

Species b

CH3F CH3Cl CH3Br CH3I CH2F2 CH2CI2 CHF3 CHCI3 CH3NH2

(CH3)3N CH3OH (CH3)20

CH'"

CHH CH'" cHMe CH'"

CHH

CH'" CHMe

vi (Ore-1) 2976 3012 3027 3030 2984 3025 3018 3034 2879 2955 2799 2952 2921 2979 2883 2985

Ai 0an/tool) (28.3)c 10.1 7.8 4.1 (32.9)c 3.1 24.8, 23.9 0.32 55.0 33.0 80 29.5 40.2 35.9 47.5 25.8

CHBr 2936 19.7 2996 7.4 aRel~.rintedwith permissionfrom Ref. [136]. bcI~X denotes the C-H bond in Cans positionto group X. CValues estimated fromundeuterated species. dUpdated values; D.C. McKean, personal communication.

CH~

are shown in Table 4.7. The data show that individual C-H stretching band intensities are very sensitive to structural variations. As can be seen, the isolated C - H intensities vary from 0.32 km tool-1 in CHC13 to 55.0 km tool -1 for the C-H bond in CH3NH2 situated off-plane with respect to C-N-lone-pair plane. The variation of frequencies and stretching force constants are, as known, on a much smaller scale, though fully characteristic. The chances of correctly predicting intensities by transfer of the C-H bond intensity parameters do not appear good, except between very close stn~ctural analogs. Another important result that could be deduced by inspecting Table 4.7 is the pronounced difference between isolated C-H stretching infrared intensities for bonds within the same grouping depending on its position with respect to polar substituents. A C-H bond positioned trans to the lone-pair of electrons in N(CH3) 3 has only about 37% of the intensity of the alternatively placed C-H bonds. Similar observations were made earlier by Wiberg et al. [137] about isolated C-H stretching band intensities for equatorial and axial C-H bonds in cyclohexane. The equatorial C-H bonds have nearly

MOLECULAR DIPOLE MOMENT DERIVATIVES

115

0/+9 042

C--N

a

OaorHa

035" L~

z

028-

o

021

<~ rn rr wq

OlZ~ 007000 3090

2990

2890

2790

WAVENUMBERS

FIG. 4.5. C-H stretching region of the infrared spectrum of CHD2NH2:57 tort partial pressure with 10 bar H 2 at 0.25 cm-1 resolution (Reproduced from Ref. [136] with permission).

40% higher intensity. These results are illustrated in Fig. 4.5 for the case of methyl amine [ 136]. The C-H bond situated trans to the nitrogen lone-pair has about 40% lower intensity. The important conclusion resulting from these data is that it is not entirely satisfactory to analyze observed infrared intensities in terms of group dipole moment derivatives. Significant differences between contributions to overall group intensifies due to different bonds are neglected and averaged. Thus, essential structural information can be lost. Similar experimental data were reported in the case of Raman intensities by Gough and Murphy [ 138]. These results strongly support the representation of vibrational intensities in terms of bond rather than individual atom or group parameters. It should also be noted that vibrational speclra are one of a few experimental sources of information about properties of individual chemical bonds. In fact, valence bonds as distinct physical entities are, perhaps, most clearly manifested in the infrared and Raman spectra of molecules. A theoretical formulation of intensities, proposed originally by Galabov [ 139] and later reformulated by Galabov et al. [140], that offers a possibility to convert the observed intensities into molecular parameters associated with bond vibrational distortions is described in this section.

GALABOV AND DUDEV

116

B. Formulation Several physical argmnents form the basis for the theoretical model presented in this section. First, as already noted, vibrational intensities in infrared spectra are expressed in terms of parameters related with vibrational distortions of separate bonds. Such an approach is supported by the experimental results discussed above. On the second place, the additive representation for the molecular dipole moment is not employed. Thus, the severe approximations that are implicit in it are avoided. The parameters represent derivatives of the Cartesian components of the total dipole moment with respect to bond displacement coordinates. By an appropriate choice of coordinates distinction is made between parameters associated with stretching and deformation mode intensities. This leads to a simplified physical sense of the parameters obtained. Such a separation is not present in atomic polar tensors as well as in bond charge tensors. The local intensity parameters in these formulations are determined, in the general case, by mixed con~ibutions from stretching and bending mode intensities. The problem with rotational contributions to intensities is dealt with by eliminating the rotational terms from both sides of the resulting linear equations. As a consequence, the parameters obtained are determined from purely vibrational distortions in the molecules. As noted in an early review by Overend [16], subtraction of contributions to dipole moment derivatives arising from the compensatory molecular rotation present in particular modes of polar molecules is required to consider the quantifies obtained as purely intramolecular parameters that depend solely on the electronic structure of molecules. A satisfactory treatment of rotational contributions is implicit in the valence optical scheme. In contrast, in atomic polar tensors and bond charge tensors, due to the requirement that intensities are expressed on the basis of parameters referring to space-fixed Cartesian systems, a considerable amount of rotational intensity is introduced into the respective tensor elements, as shown by Person and Kubulat [86]. The parameters obtained have space-directional properties. In order to compare parameter values for different molecules, a transition to a bond axis system is required. This is done following a similar approach as in the APT and BCT formulations. The vibrational motion is described in terms of the following coordinates: (i) changes of bond lengths, Ark, describing stretching modes, and (ii) changes of bond polar angles, A~, and bond azimuthal angles A~k associated with an arbitrary molecule-fixed Cartesian reference system. The angular coordinates A ~ and A~k describe the changes in spatial on'entation of bonds caused by non-stretching vibrations. A0k defines the change of the angle which the k 9 bond forms with the z axis of the reference Cartesian system, while A~k reflects changes in the (x,y) plane of the bond projection rotation angle around the z axis.

MOLECULAR DIPOLE MOMENT DERIVATIVES

117

The vibrational motion of an N-atomic non-cyclic molecule having N - I bonds is, thus, described by 3N-3 coordinates which equals the vibrational degrees of freedom plus the three rotations. The intensity parameters employed, termed bond polar parameters (BPP), represent derivatives of the Cartesian components of the molecular dipole moment with respect to the variables Ark, A0k and A~. These form two types of parametric matrices for each bond:

/

Sk = ~~~~p:/~)rk/~rk

dk = ~~~/~)0k/~)0k

/

/~)Ok/~)t~k .

(4.119)

The elements of sk are determined from dipole moment derivatives associated with stretching modes, while dk are determined from dipole derivatives associated with angle deformational modes. Arranged in a row, all Sk and dk form the matrix of bond polar parameters Pb

Pb = (Sl dl s2 d2 "-. Sk dk-.- SN-ldN-1)-

(4.120)

The solution of the inverse intensity problem, which in this formulation implies the evaluation of bond polar parameters from experimental dipole moment derivatives with respect to normal coordinates, can only be performed for a molecule possessing higher symmetry. The situation in this respect is the same as in the alternative theoretical formulations. The requirement is that the direction of the vibrational transition dipole is fixed by symmetry. In other words, there should be only one non-zero element in each column of the PQ matrix [Eq. (3.1)]. Again, all calculations are considerably simplified if the/gp/SQ i derivatives are first transformed into dipole moment derivatives with respect to internal symmetry coordinates. The determination of the elements of Pb can then be realized using the following general expression, in matrix notation ~__pp= ~)p BI 3X ~R ~S

~91 ~X ~R ~S

9

(4.121)

I is a linear array containing the set of bond displacement coordinates Ark, A0k and Ark. Using the standard notation we can also write PS = Pb V A U

or

(4.122)

GALABOV AND

118

PS = Pb V A S .

DUDEV

(4.123)

The matrix A s is derived through the expression AS

=

M -1 B S GS-1

4.124)

equivalent to Eq. (2.16). The matrix V defines the bond coordinates Ark, A0k and A~k in terms of atomic Cartesian displacement coordinates. The elements of V are determined according to the formulas

IOrk/Oxil =l sinOk cosOkl [Ork/~Yi] =[ sinOk sin*kl l On,/azil =1 cosOkl cos Ok cos ~k rk cOS0k sin~k

=

rk [00k/Ozi] =

sin0k rk sin r rk sin0k

l~k/ayil =

cos~,k

(4.125)

rk sinOk

[a~k/azil = 0 with aIk/~i = -aIk/a~j (~ = x, y, z). i and j are indices of the initial and end atoms of a bond k, and rk, Ok and ~k are the equilibrium values of the respective geometrical parameters. The signs of the elements of V depend on the particular orientation of the bond and can be determined by numerical differentiation [ 141]. The condition of zero angular momentum is implicit in the A s matrix in the fight-hand side of Eq. (4.123). Thus, the elements of Pb, similar to the electro-optical parameters [Eq. (3.28)], are free of any possible rotational contributions. As it stands, however, Eq. (4.123) can only be applied in solving the inverse intensity problem for non-polar molecules. In the cases of polar molecules, in symmetry classes containing both infi'ared active vibrations and molecular rotation, additional terms arising fi~om the compensatory molecular rotation

MOLECULAR DIPOLE MOMENT DERIVATIVES

119

will appear in the right-hand side of the equation. Two complications arise immediately. First, the number of intensity parameters is increased and the set of linear equations [Eq. (4.123)] cannot be solved. Secondly, terms of the type 0pg/00k and 0pg/04~ (g = x, y, z) arising flom molecular rotation appear that have completely different physical origin than the respective derivatives associated with dipole moment changes due to purely vibrational distortions. To overcome this difficulty the following approach is adopted. The PS elements are corrected for rotational contributions following known methods [3436]. In the BPP formulation the heavy isotope method of Van Straten and Smit is used [34]. The right-hand side of Eq. (4.123) is also corrected for possible contributions fTom the compensatory molecular rotation using A s matrix for a heavy, non-rotating isotope, denoted by Ags [34]. The corrected equation acquires the form PS - R s

(4.126)

= PbVAg S .

R s is 3x(3N-6) array containing rotational correction terms [Eq. (3.5)]. Both sides of Eq. (4.126) are free of rotational contributions and form complete sets for each symmetry class. The elements of Pb are then easily calculated. A requirement exists that the axes of the reference Cartesian system are oriented in accordance with molecular symmetry. The values of bond polar parameters obtained using Eqs. (4.123) and (4.126) depend on the specific orientation of the reference Cartesian system with respect to which the analysis is performed. In order to compare parameter values in different molecules it is necessary to convert the initially obtained sets into values corresponding to a local bond axis system. The transformation from a molecular (x, y, z) to a bond system (xo, Yo, zo) is achieved by applying the following relations: Sk0 = T x sk

(4.127)

dk0 = Txd k Ta.

(4.128)

T x is a standard transformation matrix between two Cartesian coordinate systems [ 119,132]. TQt defines the variables A0k and A~k referring to a molecular Cartesian frame in terms of the corresponding angular coordinates which refer to a bond axis system. It has the following structure O0/000 "

~0/~0

/

-

The specific expressions for the T~ elements are given elsewhere [ 142].

(4.129)

120

GALABOV AND DUDEV

In choosing the directions of the bond axis system the following rules apply: (1) one of the local axes (x0) passes through the bond direction (from the initial to the final atom); (2) the second axis (z0) passes through any present plane of symmetry with respect to the bond site symmetry; and (3) the direction of the third axis (Y0) is chosen to form a right-handed coordinate system. Equations (4.123) and (4.126) can be used in predicting intensities, i.e. solving the direct intensity problem. This has been done with success for a series of n-alkanes [ 142,143]. In view of the discussion in the be#nning of this section, intensity predictions by tramfetmg parameters can only be recommended between molecules with very similar structure. The high sensitivity of intensity parameters to structural variations makes such endeavors particularly difficult.

C. Examples of Application I.

H20

Observed intensities, coordinate definitions and force fields used in evaluating the elements of Pb are as already given in section ]II.C. In constructing the linear equations (4.126) corrected for rotational contribution PS matrix is used. The heavy isotope created to eliminate rotational terms in the right-hand side of Eq. (4.126) refers to a molecule with the mass of the oxygen multiplied by 1000. The same isotope is, of course, used to evaluate the rotational correction to the P3 derivative. Application of Eq. (4.126) produces the following equations relating 0p/0Sj to bond polar parameters: AI"

B2:

0.707 0pz/0rl + 0.707 0pz/0r 2 =

--0.234

--0.500/3pz/t301 -- 0.500/3pz,/C~2 =

0.726

(4.130)

0.707 C~x/OrI - 0.707 C3px/t3r2 = -0.914 .

Considerations of molecular symmetry show that there are only three independent values for the bond polar parameters, since /3pz/t3r1 = O~z,/O~2 c3Pz/C301 = c3Pz/C~2 o~xl&

I =

_ OPxl& 2 .

(4.131)

MOLECULAR DIPOLE MOMENT DERIVATIVES

121

b01

x

9

R

I

FIG. 4.6. Definition of bond displacement coordinates for methyl chloride. The following Pb matrix is determined (sk in units of D/lt -1 and dk in units of D rad-l) s1 Pb =

(_0~46/ [ 1~-0.165)

Z

dl 00 -0.726

s2

d2

i)f0"~46/

(

~

i)

"

(4.132)

~,--0.165) -0.726

CH3C,!

Experimental intensity data, reference Cartesian system, internal coordinates, force fields and L matrices for methyl chloride are the same as given in section III.C. Bond displacement coordinates are defined in Fig. 4.6. Rotational corrections to the dipole moment derivatives with respect to symmetry coordinates are evaluated using the heavy isotope method [34]. The rotational correction terms are given in Table 4.8. To illustrate the calculations in more detail the entire V matrix of methyl chloride is presented in Table 4.9. To remove the rotational terms from the sets of linear equations for symanetry TABLE 4.8 Rotational correction terms to dipole moment derivatives with respect to symmetry coordinates for methyl chloride (in units D/~-1 and D rad-1) A1

R 1 =0 R2=0 R3=0

E

ILl =-0.071 R5 = -0.068 R6= 0.131

TABLE 4.9 V matrix for methyl chloride c1

Ax1

A63

Cl2

Az~

Ayl

Ax2

A22

Ax3

Hg

H4

H1

Ay2

Ay3

AX^

623

Ay4

A24

Ax5

A23

Ay5

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0.562

0

0

0

0

0

0

0

0

0

0

0

0

0

0.951

0

0.310

0

0

0

0

0

0

0.868

0

0

0

0.283

0

-0.868

0

0

0

0

0

0

-0.961

0

0

0

0

0

0.961

0

0

0

0

0

0

0.475

-0.823

-0.310

0

0

0

0

0

0 -0.475

0.823

0.310

0

0

0.142

-0.245

0.868

0

0

0

0

0

0 -0.142

0.245 -0.868

0

0

0.832

0.480

0

0

0

0

0

0

0 -0.832

-0.480

0

0

0

0

0.415

0.823

-0.310

0

0

0

0

0

0

0

0

0

-0.415

-0.823

0.310

0.142

0.245

0.868

0

0

0

0

0

0

0

0

0 -0.142

-0.245

-0.868

-0.832

0.480

0

0

0

0

0

0

0

0

0

0

0.832 -0.480

0

0

0

1

0

0

-0.562

-0.562

0

0.562

0.562

0

-0.562

0

0

-0.95 1

0

-0.310

-0.283

0

0

-1

8c

M

MOLECULAR DIPOLE MOMENT DERIVATIVES

123

class E u' and Eu" the masses of the heavy atoms are multiplied by a factor of 1000 in evaluating the Ag s matrix. The equations obtained are as follows: AI:

0.577/gpz./O~rl+ 0.577 3pz/3r 2 + 0.577/gpz/O~r3

-0.625

=

0.419 i3pz/i301 + 0.419/)pz/i302 + 0.419/3pz/ig0 3

(4.133)

=

0.182

=

-0.315

3pz/3R = 2.237

Eu':

0.816/gpx/tgr I

--

0.408/3Px/3r 2 - 0.408/gPx/O~r3

(4.134)

-0.488/~px/~2 + 0.488 ~x/C3~)3 = 0.353 -0.816/~px/301 + 0.408/~px/302 + 0.231 apx/a~2 + 0.408 C3px/803 - 0.231 O~Px/O~3 = -0.297

Eu":

0.707 3py/~ 2 - 0.707 3py/3r3 = -0.315 (4.135)

0.535 c3py/c~l - 0 . 2 6 8 C3py/0~2 - 0 . 2 6 8 C3py/a~b3 = 0.353 -0.253 3py/C~l - 0 . 7 0 7 O~y/C~2 + 0.127/3py/0~) 2 + 0.707 O~y/303 + 0.127 0py/a~3 = -0.297

Considering molecular symmetry, e.g. 3pz/3r 1 = 3pz/& 2 =/3pz/tgr3, etc. we arrive at completely determined sets of linear equations. The Pb matrix for CH3CI obtained is as follows (in units D A -1 and D rad -1)

( /(0 0/I,7/ro.06 o) SR

Pb =

dR

Sl

~

0

0

2237

0

0

d1

0.440

~ -0361)

~0.145

0 (4.136)

s2 |-0222| ~-0361]

d2 0.092 0.145

s3

d3

|0.222| |-0.092-0220| ~-0361] ~, 0.145

The numbering of s and d matrices is in accordance with the definition of stretching internal coordinates, as shown in Fig. 3.7.

124

GALABOV AND DUDEV

x

bq~ 5 7

bO 4 FIG. 4.7. Definition of bond displacement coordinates for ethane.

3.

Ethane

Experimental infIared intensities, Cartesian reference system, definitions of internal and symmetry coordinates and normal coordinate transformation matrices used in evaluating bond polar parameters are as given in section W.A. Bond displacement coordinates are defined in Fig. 4.7. Since ethane does not possess an equilibrium dipole moment, bond polar parameters are evaluated using the simpler equation (4.123). The following sets of linear equations are obtained for the different symmetry species:

A2u:

0.408 0pz/~ 1 + 0.408 3pz/~ 2 + 0.408 Opz/&3 - 0.408/~z/&4 - 0.408 3pz/~ 5 - 0.408 3pz/~ 6 = -1.144

(4.137)

0.280 o3pz/t~1 + 0.280 Opz/t~ 2 + 0.280 C3pz/c~3 + 0.280 O~z./O~4 + 0.280 Opz/c~5 + 0.280 3pz/O06 = -0.365

0.577 6 ~ / & 1 - 0 . 2 8 9 3px/& 2 -0.289/~px/Or3 - 0 . 5 7 7 3px/& 4 + 0.289/~px/~5 + 0.289 3Px/& 6 = -1.196 - 0.366 c3Px/C3~2+ 0.366 3Px/C~3 + 0.366 C3Px/C34~5 -0.366 t~/~6

(4.138)

= 0.251

- 0.577 c3~/c301 + 0.289 Opx/C302+ O. 193 c313x/C3~2+ 0.289 O~x/C303 -0.193 31~/c3~3- 0.577 c3Px/C304+0.289 C3Px/C305-0.193 C3Px/C~5 + 0.289 O~x/306 + O. 193

3px/3~6 =

-0.297

MOLECULAR DIPOLE MOMENT DERIVATIVES F,U It[

125

0.500 o p / * 2 - 0.500 0 p p r 3 - 0.500 ~ / 0 r 5

+ 0.5000Px/&6 = - 1.196 0.4220py//~l - 0.2110py//~2 - 0.2110py//~3 - 0.422 ~ 0 r + 0.211 ~ / ~ 5 + 0.2110py//~6 : 0.251

(4.139)

- 0.2230py/0r - 0.5000py/a02 + 0.1120py/Or + 0.5000py/a03 + 0.1120py//~ 3 + 0.223 ~)y//~4 - 0.5000py//~05 - 0.1120py//~5 + 0.5000py//~06 - 0.112 c~//~6 = -0.297 In these equations the numbering of bonds corresponds to the defufition of internal vibrational coordinates Oven in Fig. 4.2. Symmetry considerations of the equivalent set of C-H bonds simplify Eqs. (4.137) through (4.139) and explicit solutions for all elements of Pb are obtained. The Pb matrix of ethane is given below. The elements of sk and dk are in units D A-1 and D tad -1, respectively.

SR Pb =

(

0 0) 0/

dR 00

Sl

dl

000 I [ - 0 " 06 9 51 _) [0 . 4-0.2018 6 7 0 0.19800 )

[ ~176176

~1760.171/

s2

d2

s3

-0598 -0.467

0.082 -0.099 -0.218 0

0.598 -0.467

-0.082 -0.099 -0.218 0

s5

d5

(069,/(009, s4

d4

0 0.467

0 -0.198 -0.218 0

s6

d6

-0598 0.467

-0.082 -0.218

0.099 0

0.598 0.467

d3

0.171/

0.082 0.099 -0.218 0

(4.140)

126

GALABOV AND DUDEV

D. Physical Significance of Bond Polar Parameters In contrast to atomic polar tensors, bond polar parameters need not be flaker transformed into effective charges, bond moments, charge fluxes, etc. These should be considered as quantifies influenced by all kinds of short and long distance intramolecular charge reanangement effects. The physical sense of these intensity parameters will emerge as parameter values for different types of bonds in varying molecular environments are compared. For these purposes it is first necessary to eansform parameters obtained from the general equations (4.123) and (4.126) into values referring to bond axis system via Eqs. (4.127) and (4.128). In analyz~ intensities in different molecules using the BPP formulation, a point of principal interest is to establish dependencies between parameter values and bond properties. Bond polar parameters for C-H, C-C, C=O and C=S bonds derived from experimental intensities of several series of organic molecules are summarized in Table 4.10. Certain general conclusions can be drawn f~om the survey of the data presented. A stretching of a C-C bond is accompanied by very small charge shifts. This is in full accord with expectations because of the low polarity of the C-C bonds. Interesting trends are found in the variation of the 0Px0/0rc_H parameters. It changes from relatively high negative values in ethane and propane to positive values in acetylene and propyne. A negative value of this parameter implies that a flow of a negative charge in the direction of the hydrogen atom is induced upon extending the bond. It has been argued that this is primarily due to charge flux effect, while the equilibrium charge distribution along the bond is C-'H+. Careful ab mitio quantum mechanical analysis [49] reveals, however, that the equilibrium charge distribution for the C-H bonds in most hydrocarbons is C+H-. It seems, therefore, that the 0Pxo/0rc_H parameters reflect to a greater extent the equilibrium charge distribution along a C-H bond. The positive value for these quantifies in HC-=CH and HC-=C-CH3 is in accord with the acidic character of the acetylenic C-H bonds. The C=O and C=S bonds have much greater parameter values reflecting the higher polarity of these bonds. Very good linear correlations between C-H bond polar parameters and halogen electronegativities have been found for the series of methyl halides [48]. An intriguing point to consider in relation to the data presented in Table 4.10 is the ~ransferability of parameters between molecules having the same type of bonds in a similar environment. In the methyl halide series the variation of 0Px0/0rC_H and /~xo/~C_H is substantial and the quantifies are, evidently, not transferable between these molecules. This finding is in contrast to the overall similarity of the force fields of methyl halides [24]. The variations of bond polar parameters in these molecules illustrate clearly the high sensitivity of infrared band intensities to structural variations. The ~xo/&C_H parameter values for Csp2-H bonds in ethylene, aUene and benzene are

MOLECULAR DIPOLE MOMENT DERIVATIVES

127

TABLE 4.10 Bond polar parameters for C-H, C--C, C=O and C=S bonds in some organic molecules a Bond

C-H

Molecule

cgpxo/&C_x (D/A)

CH4 -0.650 CH3CH3 -0.814 CH3CH2C.H3-0.923 CH3CH2CH 3 -0.570 CH2-CH 2 -0.461 CH2-C=CH2 -0.217 C6H6 -0.496 HC=-CH 0.938 HC~--C-CH3 0.871 HC=-C-CH3 -0.417 CH3-C-C-CH 3 -0.511 CH3F -0.675 CH2F2 -0.759 CHF 3 -0.625 CH3C1 -0.339 CH3Br -0.268 CH3I --0.197 C-C CH3CH2CH3 0.044 HCe--C--CH3 -0.052 CH3--C--C-CH3 -0.106 C=O F2CO -3.970 C12CO -4.700 C=S F2CS -3.070 CI2CS -3.710 aReprmted from Ref. [141] with permission fromthe publishers.

C3Pxo/C~C_X (D/rad) -0.511 -0.007 -0.034 0.102 0.145 0.112

0.005 0.007 0.174 0.293 -0.153 -0.228 -0.283 0.130

somewhat more divergent than expected. The absolute differences are, however, not so big and an averaged value of (-0.35 + 0.15) D A-1 for this parameter may well represent the charge fluctuations accompanying the stretchings of Csp2-H bonds. The C-H bond polar parameters of ethane and propane are quite close and seem, therefore, transferable to higher hydrocarbons. The respective quantifies in methane are rather different. The local intensity parameters for C-H bonds in acetylenic hydrocarbons appear also to be transferable.

128

GALABOV AND DUDEV

TABLE 4.11 Bond polar parameters for carbon-halogen bonds in some organic moleculesa 0Pxo/Orc_X (D/A) C-F CH3F -4.540 CH2F2 -4.126 CHF 3 2.626 CF3CI -6.250 CF3Br -6.053 CF3I -5.245 F2CO -4.923 F2CS -4.754 C-C1 CH3CI -2.150 CF3C1 -2.712 C12CO -3.987 C12CS -4.168 C-Br CH3Br -1.546 CF3Br - 1.414 C-I CH31 --0.533 CF3I -0.804 aR~rinted from Ref. [141] withpermissionfi-omthe publishers. Bond

Molecule

OPxo/OOc_x (D/rad) 0.824 1.961 1.496 1.022 0.963 0.281 0.982

-0.417 0.257

Table 4.11 contains bond polar parameter values for carbon-halogen bonds determined from experimental infrared intensities in halomethanes and carbonyl halides. As emphasized before, bond polar parameters are not directly related to the equilibrium charge distribution and, therefore, any relationship between the magnitude or sign of these quantifies and the actual polarity of valence bonds may only be qualitative. It should also be pointed out that parameter values are dependent on the accuracy of experimental measurements and the force fields employed. In analyzing the data presented in Table 4.11 some general trends are found and worth discussing. It can be seen that the magnitude of bond polar parameters for C-F, C-C1, C-Br and C-I bonds is much higher than for the respective quantities of C-H and C-C bonds (Table 4.10). It may safely be concluded that carbon-halogen bonds are much more polar than the C-H and C-C bonds, in accord with expectations. If we survey the relative magnitudes of OPxo/&C_x and OPxo/00c_X parameters for the different types of carbon-halogen bonds we can easily arrive at the logical conclusion that the polarity of these bonds is changing in the order C-F > C-CI > C-Br > C-I. We went into some detail in discussing bond polar parameter values in different molecules since only such an analysis can reveal the physical significance of these molecular quantifies. Let us emphasize here, that no approximations in transforming the

MOLECULAR DIPOLE MOMENT DERIVATIVES

S

100

TORR

129

TORR

T

T

o.g 0.5 0.1

'

" "

l"

'

"

I

|

l

TG 0.9 0.5 o.1 I

|

-

9

I

i'

,

i""

9

I

9

GG

Ud (J : . 0.9 <: m 0 0.5 0.1 i

-

i

9

I

r

0.9

l

i

9

i

!

'

TT+TG+GG

0.S 0.1

EXP. 0.g 0.5 0.1

3600

~soo WAVENUMBER

~bo

~oo

(~a'n-1)

FIG. 4.8. Simulated infrared spectra for trans-trans (TT), tram-gauche (TG), gauche-gauche (GG), an equilibrium conformer mixture and experimental survey gasphase spectrum of n-pentane.

elements of the PS matrix are introduced in evaluating bond polar parameters. Since all possible rotational terms are eliminated, the quantifies obtained are determined by purely

130

GALABOV AND DUDEV

vibrational distortions of the molecules. Trends of changes of parameter values in different molecules that are in accord with physical expectations have been established.

E. Prediction of Vibrational Absorption Intensities by Transferring Bond Polar Parameters The transfer of intensity parameters between molecules for quantitative intensity predictions encounters various problems that need careful consideration. As already emphasized in this section, due to the very high sensitivity of intensities to structural changes transferability properties of intensity parameters are expected to be much less pronounced compared to other molecular quantifies. Secondly, certain parameters will be dependent on the particular site symmetry of the chemical bonds or atoms considered. Additional complications can arise if rotational correction terms are to be calculated. Predictions by transfer of parameters should, therefore, only be attempted for closely related molecules, such as homologous series. Bond polar parameters have been used in predicting intensities in irffrared spectra in fluorinated methanes [144], alkylacetylenes [145] and medium-size n-alkanes [143]. In Fig. 4.8 the predicted infrared spectra of different conformers of n-pentane using bond polar parameters from n-butane are presented [143]. In more quantitative terms the predicted intensities are compared with the experimental values in Table 4.12. As can be seen from Table 4.12, the agreement between calculated and observed intensities is quite satisfactory.

TABLE 4.12 Observed and calculated infrared band intensities for n-pentane a AiObservedb tool -1)

Interval of Integration (era-1)

AiCalculated (lan tool-l)

3000- 2800

336.09

355.42

1550- 1420

17.02

19.45

1400 - 1300

6.89

6.64

1320-1240

2.51

1.17

1200-1120

1.45

1.30

1100-1050

0.33

0.42

1050-1000

1.94

1.34

950- 820

2.06

2.53

750- 700

2.51

2.73

420- 380

1.62

aFrom Rff. [143].

bin the gas-phase.

MOLECULAR DIPOLE MOMENT DERIVATIVES

131

IV. E F F E C T I V E B O N D C H A R G E S FROM ROTATION-FREE ATOMIC POLAR TENSORS Among the different models for interpretation of vibrational absorption intensities the atomic polar tensor formulation is by far the simplest to apply in transforming the experimental 0p/0Qi dipole derivatives into quantifies associated with molecular subunits, atoms in molecules in the particular case. Besides, the transformation does not involve unnecessary approximations and assumptions. The APT formulation provides also the possibility to directly compare experimental data and theoretical ab initio results. The physical interpretation of atomic polar tensors is, however, hampered by the redundancies between the elements of atomic polar tensors as expressed by Eqs. (4.18) and (4.19). Rotational atomic polar tensors associated with the permanent dipole moment value can make, in the general case, substantial contributions to APT elements. In the present section a theoretical framework for analysis of vibrational intensities recently developed by Galabov et al. [ 146] is presented. Fully corrected for rotational contributions atomic polar tensors are transformed into quantifies termed effective bond charges. The effective bond charges are expected to reflect in a generalized manner, polar properties of the valence bonds in molecules. Aside from the usual harmonic approximation no other constraints are imposed on the dipole moment function.

A. Rotation-Free Atomic Polar Tensor The starting molecular quantifies in evaluating effective bond charges are atomic polar tensors as expressed in the PX matrix. From PX one easily obtains the vibrational polar tensor matrix VX from the relation VX = Px - Pp[3-

(4.141)

If experimental data are treated, Vx is calculated from the relation (4.93). In Eq. (4.93) PS is the array of dipole moment derivatives with respect to symmetry coordinates [Eq. (3.4)]. As underlined earlier, VX refers to a molecule-fixed reference coordinate system as also do the experimental Op~/0Qi derivatives (~ = x, y, z). The array VX may contain implicitly contributions originating from the compensatory molecular rotation in the case of polar molecules. These contributions are also present in the PS matrix. Rotationally corrected PS may, however, be used to derive a fully rotation-l~ee atomic polar tensor matrix. This is achieved through the equation PX(v) = P s U B - R s U B

(4.142)

132

GALABOV AND DUDEV

R s is the matrix of rotational correction expressed in terms of symmetry coordinates [Eqs. (3.5) and 0.11)]. The elements of Px(v) are determined by purely vibrational distortions. From the rotation-free atomic polar tensor an invariant with respect to Cartesian axes reorientation can be deduced from the trace of the product P

(4.143)

= Tr

The quantity ~-~z(v),an effective atomic charge, is determined by purely vibrational contributions to the dipole moment changes. The elements of Px(v) are not interconnected by the equations resulting from relation (4.19). Still, nine redundancy conditions as defined by Eq. (4.18) are present. The removal of these remaining redundancies can be accomplished by defining the problem in bond Cartesian displacement coordinate space [Eqs. (4.96) and (4.97)].

B. Effective Bond Charges Bond displacement coordinates are defined by relations (4.96) and (4.97). By expressing the bond coordinates Xs(k) as differences between the respective Cartesian displacements of the two atoms forming a bond the translational motion is eliminated. Thus, there are no redundancies associated with translational motion between the elements of bond charge tensors [129]. The elements of D matrix [Eq. (4.98)] are expressed in terms of a space-fixed Cartesian reference system. The elements of D matrix may contain considerable contributions associated with the equilibrium dipole moment value. The resulting implicit redundancies are expressed by six relations [Eq. (4.118)]. A polar tensor, associated with bond distortions, which is flee from both translational and rotational redundancies can be easily evaluated from the rotation-free atomic polar tensor [Eq. (4.142)]. It is denoted by Dk(v) and has the following structure

dkxx(V)

Dk(v) = dkyx(v) dkzx(V)

d k (v)

dkxz(V)

dkyy( v ) d k y z (v) d~(v)

d~(v)

0.144)

MOLECULAR DIPOLE MOMENT DERIVATIVES

133

Dk(v) refers to a molecule-fixed Cartesian system with value determined by the changes of dipole moment resulting from purely vibrational distortions. Arranged in a row Dk(v) form the following matrix

]D(V)- 0D(1)(v) ]D(2)(v)...

D(k)(v)... ]D(N-1)(v)) .

(4.145)

The elements of D(v) are obtained from Px(v) using the relation D(v) = Px(v) C-1.

(4.146)

The structure of the C -1 matrix will be shown later in this section. D(v) matrices are numerically equivalent with the respective Px(v) arrays less the atomic polar tensor for the first atom. The elements of D(v) have dimension of electric charge. The charge fluctuations reflected in D(v) are determined, as already emphasized, from vibrational distortions of the respective bonds. The elements of D(v) are expected to be connected with the electronic structure of the bonds undergoing vibrational distortions. More polar bonds should give rise to higher values for the elements of D(v) and, also, to higher integrated intensities of the respective irLfrared absorption bands. Of particular interest are the invariants with respect to reorientation of the Cartesian reference system of the tensors Dk(v). An effective bond charge has been defined [ 146] --

[Ok(v). Bk(V)].

(4.147)

The quantifies 61, are expected to reflect, in a generalized way, electrical properties of the respective bonds. As is seen, no approximations outside the harmonic approximation are introduced in evaluating effective bond charges. The ~ values can be evaluated without any difficulty fTom experimental vibrational absorption intensities providing the molecule possesses sufficient symmetry and that individual 0p~/~i values are determined. The physical significance of effective bond charges is of particular interest. It should be emphasized that no direct numerical link is expected with static bond dipole moments. First, as already discussed, static bond dipoles cannot be unambiguously defined in principle, except in the case of diatomic molecules. Secondly, by defimtion are complex quantifies depending on the charge fluctuations induced by the different vibrational modes. Simple interpretations of effective bond charges, though tempting, are, therefore, not feasible. The dipole moment derivatives contained in l)k(v) are associated with the dynamic dipoles resulting from the vibrational distortions. These charge fluxes do not necessarily reflect the static charge distribution. Each element of

134

GALABOV AND DUDEV

Dk(v) is determined by contributions arising from all possible vibrational displacemems of the bond k. Distortions in the immediate neighboring bonds may give rise to non-zero 8k values even for non-polar bonds. An example in this respect is offered by the ab initio results of Wiberg and Wendoloski [49]. The authors have calculated the charge fluxes associated with individual atomic sites for distorted configurations of acetylene corresponding to its vibrations. In the case of the antisymmetrical stretching mode the following charge distribution is obtained [6-31G(d,p)] basis set, molecule deformed along the stretching symmetry coordinate by AS -- 0.1, charges in units of electron] -->

H~ +0.099

---}

C--------- C ~ -0.108 -0.123

H +0.133

It is seen that a dynamic dipole is created for the non-polar at equilibrium C~-C bond. These dipole terms will eventually be reflected in final values for 8C_c. The physical significance of effective bond charges will eventually emerge from applications of the formulation outlined above in analyzing infrared intensities of various types of molecules.

C. Applications The effective bond charge (EBC) formulation is aimed uniquely at interpreting vibrational absorption intensities. It is quite tempting to illustrate the theory with ab initio estimated dipole moment derivatives since these data are free of the usual experimental uncertainties and inaccuracies. As will be shown in the subsequent chapter, however, different levels of ab mitio molecular orbital calculations produce very divergent sets of predicted infrared intensities. Though the overall shape of the spectrmn is reproduced qualitatively, the ratios computed and experimental intensities for a given molecule do not follow any definite trend at a given basis set. Thus, there is no certainty that ab initio intensity results can provide a sufficiently reliable source of data for assessing the physical significance of local intensity parameters in different molecules. Thus, in this section we shall present results for gas-phase experimental intensity data. The transformation of the infrared intensities for water and ammonia into effective bond charges will be followed in detail so that the computational procedure becomes clear. Results from the application of EBC formulation in interpreting intensity data for a number of medium size molecules will then be presented. Comparisons of effective bond charges for different bonds in varying molecular environments will provide a basis for assessing the physical significance of these molecular quantifies.

MOLECULAR DIPOLE MOMENT DERIVATIVES

135

TABLE 4.13 Experimental dipole moment derivatives with respect to symmetry coordinates for ammonia (in units D A -I or D rad-l)a,b AI

Pl = 0.290 P2 = 1.620

E

P3x = -0.195 P4x = -0.366

aFrom Ref. [147l. definition of symmetrycoordinates is given in Table 3.3.

Cartesian reference systems, geometric parameters and symmetry coordinates for H20 and NH 3 are given in Chapter 3. Dipole moment derivatives with respect to symmetry coordinates for H20, evaluated in analyzing experimental absolute infrared intensities, are also presented there, igp/aSj dipole moment derivatives for ammonia used in the present calculations were taken from Ref. [147] and are presented in Table 4.13. The signs of these quantifies have been fixed with the aid of ab initio MO calculations [147]. Elements of the respective R s matrices for both molecules were evaluated by employing the heavy isotope method [34], weighting the respective heavy atoms by a factor of 1000. The rotational correction terms for ammonia are tabulated in Table 3.3. The R s matrix for H20 has the following form (in D A -1 or D rad -1)

RS =

Sl

S2

S3

0

0

0

y

0

0

0

z

(4.148)

Rotation-free atomic polar tensors were calculated using Eq. (4.142). Finally, D(v) matrices were obtained with the aid of the respective C -1 matrices [Eq. (4.146)]. As an example, the structures of C and C -1 arrays for the water molecule are shown below.

C

-1 0 0 -1 0 0

0 -1 0 0 -1 0

0 0 -1 0 0 -1

1 0 0 0 0 0

0 1 0 0 0 0

0 0 1 0 0 0

0 0 0 1 0 0

0 0 0 0 1 0

0 0 0 0 0 1

(4.149)

136

GALABOV AND DUDEV

C-1

0 0 0

0 0 0

0 0 0

0 0 0

0 0 0

0 0 0

1

0

0

0

0

0

0 0 0 0 0

1 0 0 0 0

0 1 0 0 0

0 0 1 0 0

0 0 0 1 0

0 0 0 0 1

(4.15o)

PS U B, R s U B, Px(v) and D(v) matrices for H20 and NH 3 are given in Tables 4.14 and 4.15, respectively. Calculated effective bond charges [Eq. (4.147)] for O-H and N-H bonds are represented in Table 4.16 together with the 5k values evaluated liom analysis of experimental gas-phase infrared intensities of three other series of molecules. These include hydrocarbons (methane, ethene, ethyne and propyne), methyl halides (CH3F, CH3C1, CH3Br and CH3I) and X2CY molecules (H2CO, F2CO, F2CS, C12CO and C12CS). More details concerning the computations for these molecules are presented elsewhere [ 148]. By comparing ak values for O-H and N-H bonds, it can be seen that these are quite close in values. This is in accordance with expectations because of the similarity of the two bonds. It is of particular interest to examine the changes of C-H bond charges in the series of hydrocarbons as a function of the varying hybridization of the carbon atom and changes in environment. Analysis of the data collected in Table 4.16 reveals a tendency towards higher values with increased s-character of the carbon valence orbitals, beginning with 0.162 e for methane (Csp3-H), 0.179 e for the Csp2-H ~k value in ethene, and ending with much higher effective charges for the acidic Csp-H bonds in ethyne and propyne. The Csp3-H effective bond charge in various molecules do not appear to be very sensitive to environmental changes in the series of hydrocarbons and methyl halides with variations in the range 0.129 e (CH3Br) to 0.163 e (CH3F). Much more pronounced variations in ~k values are found for the polar carbon-halogen, carbon-oxygen and carbon-sulphur bonds. The C-X effective bond charge in methyl halides is changing from 0.946 e in CH3F, 0.470 e in CH3CI , 0.329 e in CH3Br to 0.135 e in CH3I. This is in full accord with expectations and the values evidently may be related to the electronegativity of the halogen atoms. This dependence is shown in Fig. 4.9. The C=O effective bond charges determined are higher than the respective C=S values in the sulphur analogs. This finding is also in accord with the lower polarity of the C=S bonds due to lower electronegativity of the sulphur atom.

z

9m

TABLE 4.14

Experimental PsUB, RsUB, Px(v) and D(v) matrices for H 2 0 (in units of eIectrons)ahc

2

H2

-0.23 1 PsUB =

0 0

-0.292 -0.0 18

(-0.213

0.115

0 -0.069

0 0 0

0.089

0 0 0

-0.089

0.146

0.1 15 0 0.069

0

0.146

0 0 0

0.009

0 0 0

0.007 0 0

0.009 0 0

0 0

-0.007

0 0

0

0

0 0 -0.292

0.106 0 -0.069

0 0 0

0.082

0.107 0 0.069

0 0 0

-0.082

0 0.146

0.106 =

[-0:69

0 0 0

0.146

r?

rl

D(v)

%

H3

0.082 0 0.146

0.107 0 0.069

0

-0.082

0 0

0.146

aReprinted from Ref. [ 1481 with the kind permission from Elsevier Sci. Ltd, The Boulevard, Langford Lane, Kidlington OX5 IGB, UK, Copyright (1995) blD A-1 = 0.2082 e. values given correspond to Cartesian reference frame shown in Fig. 3.6.

L

W

4

TABLE 4.15 Experimental PsUB, RsUB, Px(v) and D(v)matrices for NH3 (in units of electrons)a,b

0 0

0.044 0 0.182

0.071 0.031 -0.022 0.031 0.036 0.038 0.056 -0.097 0.182

0.071 -0.031 -0.02 -0.031 0.036 -0.038 0.056 0.097 0.182

0

0.008 0 0

0.009 0 0

-0.547

0

0.009

0 0.009

0

0 0

RsUB =

-0.132

H4

0.018 0 -0.112

0 0

H3

H2

N1

0 -0.547

0

0.089 0

0.009 0

0

0.036

0.080

0

0.062 0.031

-0.112

0

0.182

0.056

D(v) =

[&

0

0.080 0

-0.004 0.007

0

0

0.031 -0,018 0.027 0.031 -0.097 0.182 r2

‘1

0.009

0 0.009

0,036 0 0.182

0.062 0.031 0.056

0.031 -0.018 0.027 0.031 4.097 0.182

0 0

0.009

-0.ocn

0

0

0.062 -0.031 0.027 -0.031 0.056 0.097 0.182

-0.031

‘3

0.062 -0.031 -0.031 0.027 -0.031 0.056 0.097 0.182

aReprinted from Ref. [ 1481 with kind permission from Elsevier Sci. Ltd, The Boulevard, Langford Lane, Kidlington OX5 IGB, UK,Copyright (1995). bThe values given correspond to Cartesian reference fiame shown in Fig. 3.2

MOLECULAR DIPOLE MOMENT DERIVATIVES

139

TABLE 4.16 Effective bond charges from experimental infrared intensities (in units of electrons)a Bond

Molecule

5k

O-H N-H C-H

H20 0.210 NH 3 0.231 CH 4 0.162 CH2--CH2 0.179 CH-CH 0.352 0.139 CH#-=C-CH3 ~ ~0.349 CH3F 0.163 CH3C1 0.130 CH3Br 0.129 CH3I 0.139 H2CO 0.267 C-F CH3F 0.946 F2CO 1.096 F2CS 1.064 C-C1 CH3C1 0.470 C12CO 0.866 C12CS 0.879 C-Br CH3Br 0.329 C-I CH3I 0.135 C=O H2CO 0.789 F2CO 0.931 C12CO 1.070 C=S F2CS 0.650 C12CS 0.781 aReprinted from Ref. [ 148] with kind permissionfrom ElsevierSci. Ltd, The Boulevard, LangfordLane, Kidlington OX5 1GB, UK. Copyright(1995).

By comparing 5k data for lower polarity bonds (C-H) to the much polar C=O, C=S, C-F, C-C1, C-Br, O-H and N-H bonds it is seen that higher polarity is reflected in higher effective bond charges, with values varying in most cases with expectations. The analysis of data for effective bond charges as obtained from experimental gasphase intensity data for a number of molecules reveals some definite trends of changes. The variations found may, in most cases, be related to polar properties of the bonds considered. As already stressed, the effective bond charges are solely determined by

140

GALABOV AND DUDEV

F r

0.8

cO ILl r re" ,<: "r" (J

CH3CI/~

El

Z 0 m

i,!

0.4

i--I

i,I i, LI.. i,i

0.0

/~H3I i

2.0

i

ii

i'1

ii

i ii

2.5

i ii

i

i i

i

i i'l

3.0

ii

i1'1

ii'1I

ii

3.5

i i i i I i i

i ii

4.0

! I i

I !

i

ELECTRONEGATIVITY [Pauling units)

FIG. 4.9. Plot of the dependence between 8C_x in CH3X (X = F, CI, Br, I) and the electronegativity [149] of halogens (Reproduced from Ref. [148] with kind permission fxom Elsevier Sci. Ltd, The Boulevard, Langford Lane, Kidlington OX5 1GB, UK, Copyright (1995).)

purely vibrational distortions of the molecules. The data obtained show that these quantifies may be regarded as an experimental parameter that characterizes the intrinsic electric properties of valence bonds.

CHAPTER 5

RELATIONSHIP BETWEEN I N F R A R E D INTENSITY F O R M U L A T I O N S

141

142

GALABOV AND DUDEV

A number of different theoretical models for vibrational absorption intensity analysis have been developed. As described in Chapters 3 and 4, they are aimed at reducing the observed or theoretically estimated intensities into molecular quantifies characterizing the electronic structure and properties of basic molecular sub-units. The theories put forward provide also a possibility for calculating vibrational spectra using transferable force constants and intensity parameters. The accuracy of intensity predictions depends on the approximations inherent in the particular method and, to a great extent, on the properties of the molecule treated. While reasonably accurate intensities can be obtained from high level ab mitJo MO calculations, transferred empirical intensity parameters are indispensable in calculations aimed at simulating vibrational spectra of complex molecular systems such as proteins [115,150]. Because of the diverse approaches applied in different laboratories in interpreting intensity data, it is essential to analyze and define the relationships between the theories developed. Some of these relations were already given. Eqs. (4.110) and (4.111) in section 4.2 show the numerical equivalency between atomic polar tensors and bond charge tensors. Eq. (3.69) defines the relation between the atomic polar tensor theory and the atomic effective charge model developed by Aleksanyan et al. [107,110]. Equation (4.73) gives an exact expression of atomic polar tensors of diatomic molecules in terms of electro-optical parameters. General relations between intensity formulations have been derived [37, 100,124,131,151,152]. The existence of equations relating different intensity parameters does not imply that the various formulations are equivalent from a physical point of view. Neither is it correct to claim that using one type of parameters or another is simply a question of choice and preference. We have seen that there are basic physical differences in the definitions of intensity parameters. The relations between parameters in the semiclassical formulations and parameters representing derivatives of the total molecular dipole moment have physical relevance only to the extent that the assumptions in the approximate theories are valid. We shall limit our presentation only to defining relations between different intensity parameters and atomic polar tensors [33,108]. There are two main reasons for this. Atomic polar tensors are evaluated fTom experimental infi'ared intensities of symmetric molecules (the directions of transition dipoles are fLxed by symmetry) without any further assumptions or approximations to be made. Secondly, quantum mechanical estimates of infrared intensities are presently obtained from calculations by analytical differentiation atomic polar tensors. These quantifies appear in the standard output of current ab mitio program packages [153,154]. Thus, most intensity parameters can be interconnected by defining their relations to atomic polar tensors. Gussoni and Abbate [ 151] have derived a general formula relating atomic polar tensors and electro-optical parameters. Eq. (3.27) is an expression of the dipole moment derivatives with respect to symmetry coordinates in terms of electro-optical parameters.

RELATIONSHIP BETWEEN INFRARED INTENSITY FORMULATIONS

143

In general form atomic polar tensors are represented in terms of electro-optical parameters by the matrix equation:

~p

01a

ax = e ~ + t t a x

~e

"

(5.1)

Using the relation aR/3X = B [Eq. (2.12)] the following relation is obtained

~p

~t

~e

(5.2)

As an illustration, in Table 5.1 explicit relations between APT and electro-optical parameters for CH3X molecules derived by Gussoni and Abbate [ 151] are shown. The expressions refer to the local Cartesian systems for X and H atoms defined in Fig. 5.1. Decius and Mast [117] have defined the relations between atomic polar tensors and effective charges and charge-fluxes [ 109]. Explicit expressions for AB 2 (DooQ, AB2 (C2v), AB3 (D3h), AB3 (C3v), AB4 (Td), A2B2 (DooQ and A2B4 (D2Q have been given. In Table 5.2 the relations for AB 2 (DooQ and AB 2 (C2v) are shown. The structure of atomic polar tensors and the notation are as defined in section 4.1.3. The Cartesian axes orientation for the molecules of the two symmetry groups is given in Fig. 5.2. The symmetry coordinates for the infrared active modes used in defining the respective expressions are: AB2 (Dooh)

S2 = r Ar S3 = 2- 89(Ar2 - Arl)

AB2 (C2v)

S 1 = 2- 89(Arl + Ar2) $2 = r A~ S3 = 2- 89(At2 - Arl)

(5.3)

where r is the bond angle and r is the equilibrium bond length. In Table 5.2 the righthand sides of the equations relating atomic polar tensors with either electro-optical parameters or effective atomic charges and charge fluxes are overcrowded with parameters employed in the respective semiempirical models. The equations become fully determined only for very small symmetric molecules. Nevertheless, the formulas derived define the links between semiclassical and non-approximate formulations of infrared intensities. The relationship between atomic polar tensors and bond polar parameters has also been analyzed by Galabov et al. [100,124]. Multiplying both sides of the equation PS - RS = Pb V AgS [Eq. (4.126)] by a synunetrized B matrix we obtain

PsBs - RsBs =PbVAg SB S

(5.4)

TABLE 5.1. Atomic polar tensors for X and H1 atoms in methyl halides (CH3X) expressed in terms of electro-optical parameters. MO and RO are the equilibrium dipole moment and bond length of C-X bond and po and fl are the respective quantities for the C-HI bond. Local coordinate systems are shown in Fig. 5.1 (Reproduced from Ref. [ 15 11 with permission).

1

j

{";[$-$I+

XX

Mo} I Ro

0

Yx

=X

0

0

{f i [$-$1 + }

Mo / Ro

E

0

0

j'

0

0 0

0

c

RELATIONSHIP BETWEEN INFRARED INTENSITY FORMULATIONS

i

145

R

FIG. 5.1. Internal coordinates and local reference coordinate systems for CH3X (X = F, CI, Br, I) molecules.

TABLE 5.2

Relations between effective atomic charges, charge flux parameters and atomic polar tensors for AB 2 (Dooh) and AB 2 (C2v) molecules (Reprinted from Ref. [ 117] with permission) AB2 (Dooh)

AB2 (C2v)

/~px/O~2 = - 8 9~A(0) Cgpz/aS3 = 2- 89(~A(0) + 2br(O))

C3pz/~ 1 = 2- 89 + ar(0)cosl3 C3pz/aS2 = - 8 9sinl3~A(0) + br(0)cosl3 0Px/aS3 =2- 89 1~A(0) + 2 89

C] =

M sin~;

2mBsin2~+ m A

Ax = ~A(~ A z = ~A(0) + 2br(0) B x = ~B(0) B z = ~B(0)- br(0) Ax=~A(0)+2(sin213)cr(0) Az=~A(0)+(2 89 cosl3 -2b sinl3) r(~ Bx=~B(0)--(sin213) cr(0) Bz=~B(0)-(2- 89 cosl3 -b sin~) r(O)cosl3

Bxz=Cr(0)sinl3cosl3 Bzx--(2- 89 sinl3 +b cosl3) r(O)eosl3

146

GALABOV AND DUDEV

Z

Z

X~ FIG. 5.2. Definition of Cartesian reference frames for AB 2 (Dooh) and AB 2 (C2v) molecules.

Adding to both sides of Eq. (5.4) the matrix product Ppl3 yields

PsBs-

RsBs + Pp13=PbVAg SB S + Pp13.

(5.5)

Making use of equation (4.14) the following relation expressing atomic polar tensors in terms of bond polar parameters is obtained PX = PbVAg s B s + R s B s + Pp13.

(5.6)

It is interesting to note the presence of two rotational terms in the right-hand side of Eq. (5.6). This comes as no surprise because of the different approaches in the treatment of rotational contributions to dipole moment derivatives adopted in the two intensity formulations. It is of interest to analyze the matrix product Pb V AgS B s. Comparing Eqs. (4.14) and (5.6) we see that

PbVAg SB S + RsBs = PsBs

(5.7)

RELATIONSHIP BETWEEN INFRARED INTENSITY FORMULATIONS

147

where PS BS is the vibrational polar tensor in the APT formulation. It follows that Pb V AgS B S = PS BS - RS BS

9

(5.8)

The expression PS BS - RS BS appearing in the fight-hand side of Eq. (5.8) represents a vibrational polar tensor corrected for contributions arising from the compensatory molecular rotation accompanying some vibrational modes. This relation was used in section 4.4 to obtain rotation-free atomic polar tensor Px(v) [Eq. (4.143)]. As already mentioned, in contrast to the usual atomic polar tensors PX, the rotation-free tensor Px(v) refers to a molecule-fixed Cartesian system. Because of the presence of the term R s BS, the elements of PX(V) will be the same for all isotopes of the molecule with identical symmetry.

This Page Intentionally Left Blank

CHAPTER 6

PARAMETRIC FORMULATIONS OF INFRARED ABSORPTION INTENSITIES OF OVERTONE AND COMBINATION BANDS

I.

Introduction

II.

Anhannonic Vibrational Transition Moment ..................................................... 151

....................................................................................................... 150

A. Variation Method Formulation ..................................................................... 151 B. Perturbation Theory Formulation ................................................................. 152 Ill.

The Charge Flow Model ....................................................................................

IV.

The Bond Moment Model .................................................................................. 160

149

158

150

GALABOV AND DUDEV

I. I N T R O D U C T I O N The theories presented so far are limited to treating intensities of fundamental vibrational transitions. It has been of desirable interest that analogous theoretical developments are extended to overtone and combination bands. In general terms such formulations are aimed at deriving the structural information contained in these observable quantifies. These formulations will provide opportunities of gaining information on the intramolecular factors determining the intensities of overtone and combination bands and, thus, of expanding our understanding of vibrational spectra in general. Overtone and combination bands appear in the spectra as a result of mechanical and electrical anharmonicity. Higher order terms are needed to present more accurately the potential energy and the dipole moment function. Experimental evidence, as manifested in the observed overtone and combination bands, clearly shows that molecular vibrational wave functiom are not perfect harmonic oscillator wave functions. The higher order terms of the dipole moment expansion, in terms of normal coordinates, play a major role in determining the intensities of overtone and combination bands. The theoretical analysis of overtone and combination band intensities encounters considerable difficulties. There are complications associated with the greatly increased number of coefficients in the higher order terms of the potential energy expansion. The necessity to employ curvilinear coordinates in describing nuclear displacements hampers the simple interpretations with the aid of mathematical and computational approaches used in the treatment of fundamental transitions [3-6,155]. On the other hand, complete experimental data for frequencies and intensities of overtone and combination transitions can only be obtained for very small molecules. Experimental studies have been restricted mostly to diatomic and triatomic molecules [3,155,156]. The determination of the signs of the higher terms in the dipole moment expansion is even more difficult than for the linear terms. Only recently calculations of such terms by ab mitw quantum methods have been reported [ 157]. Studies on overtone and combination transition intensities are aimed at determining first and higher order terms in the dipole moment expansion along the normal coordinates using appropriate anharmonic wave functions [158-160]. Two theoretical representations of overtone intensities in terms of empirical parameters have also been formulated [ 155,161]. These developments will be presented in this section.

IR INTENSITIES OF OVERTONE AND COMBINATION BANDS

II. A N H A R M O N I C

VIBRATIONAL TRANSITION

151

MOMENT

A. Variation Method Formulation The transition dipole between arbitrary vibrational states is given by (nlplm)=

f wn* P ~am d% .

(6.1)

In the general case wn and ~Pam are artharmonic vibrational wave functions, p is the dipole moment operator. The anharmonic wave functions can be represented as a linear combination of harmonic basic functions [155]

~Fa = E Cv ~Fv(0) 9

(6.2)

V

The coefficients Cv are found by a variational technique in combination with a potential function expressed in a forth order Taylor series. The basis set functions ~Pv~u) are defined in the usual normal coordinate space. The transitional dipole for higher order transition can then be expressed in terms of dipole matrix elements referring to normal coordinates and in a second order expansion is

Mnm

k

0

1

( ~)2p / ~n[(QkQ

2 k,1

)]Cm

+

(6.3)

0

[(Qk)] and [(QkQI)] are square synunetric matrices containing matrix elements of the type f wn*(Q~ Qk q,m(Q~ dQ k

(6.4)

and ~Pn*(Qk) Qk qJm(Qk) dQk 9~ ~n*(Ql) QI qJm(Ql) dQl 9

(6.5)

All combinations of basis set functions are present in [(Qk)] and [(QkQI)]. C--nand C m are row and column arrays of the variational coefficients cv of the interacting vibrational states. Once the variational coefficients are determined, the probabilities for overtone and combination transitions are expressed in terms of first and second dipole moment

152

GALABOV AND DUDEV

derivatives with respect to normal vibrational coordinates. Further transformations are then possible which are aimed at representing the transitional dipole moments, respectively, overtone and combination band intensities, in terms of local intensity parameters such as electro-optical parameters, effective charges and polar tensors. The variational approach is quite approximate in the sense that it allows for large amplitude vibrational motion to be described by coordinates that refer to a lineafized coordinate system. Such coordinates can be used in describing motions with infinitesimal amplitudes. A more accurate representation of the transition dipole moment should, therefore, involve wave functions defined in the space of ctavilinear coordinates. Appropriate transformations between different coordinate spaces can then be applied.

B. Perturbation Theory Formulation A perturbation theory treatment known as the method of contact transformations [162] provides a convenient approach in defining the anharmonic potential energy and transitional dipole moment [158-163]. Both mechanical and electrical anharlnonicities influence the intensities of overtone and combination bands. In general terms, since an exact solution of the vibrational equation in terms of anharmonic wave function is not possible, use is made of the fact that for finite displacements at each step of the potential energy or transitional dipole moment expansion, the higher terms are much smaller than the respective lower terms. A perturbation theory treatment becomes, therefore, feasible. The potential energy may be expressed in the form [3] V = V 0 + ~ V 1 + 2L2 V 2 + ' " .

(6.6)

and the energies E = ~ h c c o i ( v i + 8 9) + 2. E 1 + K2E2 + " ' ' .

(6.7)

i

Z. is an ordering parameter defining the degree of smallness of the terms appearing in the expressions. The method of contact transformations simplifies the perturbation theory treatment. The perturbation Hamiltonian can be expressed as H = H 0 + ~ , H 1 + X 2 H 2 + ~,3H3 + . . . .

(6.8)

IR INTENSITIES OF OVERTONE AND COMBINATION BANDS

153

In applying contact transformations to the Hamiltonian, terms referring to a particular order can be removed from the expression with H 0 remaining unchanged. Two successive contact transformations provide a new form of the Hamiltonian H ' = T l H T1-1 H + = T 2H'T2-1 = T 2 T I H T l - i T 2 -1.

(6.9)

(6.10)

Expressions (6.9) and (6.10) define similarity transformations with unitary functions T 1 and T 2. The form of the function T i is such that, as mentioned, terms of a particular order are removed from the expression for the Hanfiltonian. Thus, after two transformations H § is expressed by zero order terms and term of order ~3 H+ = H 0 + ~.3H3+ .

(6.11)

The functions T have the form T = eiZS

(6.12)

and, since T -1 = e-ikS

(6.13)

the wansformation is indeed unitary. The specific form of the S terms appearing in Eqs. (6.12) and (6.13) is given in Ref. [162]. The general expression for the transformed wave functions is q~v' = T qJv"

(6.14)

Two successive contact transformations remove from the expression of the Hamiltonian the first and second degree terms. Thus, the wave functions of the zero order term which, in our case, are the standard linear harmonic oscillator wave functions, are also eigenfunctions of the Hamiltonian H + up to the second order. If a standard perturbation theory is applied, there will be an extensive number of off-diagonal matrix elements of the first-order perturbation Hamiltonian appearing in the expressions for any molecular quantity estimated from second order matrix elements. By the contact transformations the matrix elements will be diagonal through second order which greatly simplifies the calculations. If the linear harmonic oscillator wave function is denoted by ( n [, the matrix element ( n [ H + i m ) may be expressed as [Eq. (6.10)]

154

GALABOV AND DUDEV

(n Ill + [m> = ( n [ T 2 T 1H T/-1

T/-2Im)

.

(6.15)

Through second order the matrix element on the right-hand side of Eq. (6.15) is diagonal. Therefore, the functions (Ti'lTi "2 [ m ) ) can be considered as second order eigenfunctions of H. Similar transformation can be applied to any molecular quantum mechanical operator. The dipole moment operator can be conveniently expressed as p = p(0) + gp(1) + ~2 p(2) + ;L3 p(3) + - - ' .

(6.16)

The first contact transformation function has the form z2

TI = e+ikS1 = l+ikS 1 - - 7 S 2 - i T S3 + . . . .

(6.17)

The expression for the transformed dipole will, therefore, be p'=T1 p T1-1= l + i ) ~ 1 ---~x (p(0)+ kp(1)+ ~,2p(2)+ ~3 p 0 ) + . .

.)

(6.18)

X 1-i~S 1 - - 7 - S 1 2 + i - 6 s3 + ' ' " p' can also be expressed as p ' = p'(0) + ~,p'(1) + ~2p,(2) + ),3 p'(3) + ' "

.

(6.19)

The different order terms appearing in Eq. (6.19) are equal to p'(0) = p(0) (6.20)

p'(1) = p(0)+ i ( S l P ( 0 ) - p(0)Sl)

p'(2) = p(2)+ i (S lP(1 ) - p(1)S 1) - ~

+ s p(o))

or

(6.21)

IR INTENSITIES OF OVERTONE AND COMBINATION BANDS

155

The second contact transformation of p can be written as p + = T2P' T~ 1 = T 2 T 1 p TI"1 T~ 1 .

(6.22)

The transition dipole matrix element is then expressed as

M+mu = (nl T 2TIp

Ti-1

T~Itm).

(6.23)

The important conclusion that follows from Eq. (6.23) is that the double contact transformed transitional dipole moment can be evaluated using zero-order wave functions. The second transformed electric dipole moment may also be expressed in an order of magnitude series p + = p+(0) + ~.p+(1) + ~2 p+(2) + ~3 p+(3) + - - ' .

(6.24)

Following the approach outlined by Eqs. (6.20) and (6.21), explicit expressions for the various order terms appearing in Eq. (6.24) can be obtained. Since rotational motion has only a small effect on vibrational absorption intensifies (Chapter 1), the vibrational part of the Hamiltonian may only be considered to a good approximation, ff the vibrational Hamiltonian defined by Eq. (6.8) is expressed as a function of dimentionless normal coordinates ~ and their momenta conjugate m s, the respective ordering terms will have the form [ 159] hc

It0 = - ~ - ~

(ms + q2) t~s

s

H l=hc ~

kss's"qs qs'qs"

(6.27)

S,St,S t'

H 2 = he

]~ kss's"s" qs qg qs" qs" 9 s,s',s",s"

kss,s, and kss,s,,s,, are cubic and quartic anharmonic normal coordinate force constants [ 164]. qs are related to the usual normal coordinates Qs by

qs =

h~ s

Qs.

(6.28)

In terms of qs the different order terms of the dipole moment operator appearing in Eq. (6.16) are [158,161]

156

GALABOV AND DUDEV

(6.29)

pg(O) = pg + ~ pg qj J 1

(6.30)

Pg(1) = ~ ' Z Pfk qjqk j,k 1

1 2) = g

j,k,l

(6.31)

qjqkq

where Pgo (g = x, y, z) is a Cartesian component of the equilibrium dipole moment and pg =Bpg ~qj

(6.32)

~2pg Pfk = ~qj~qk

(6.33)

c33pg Pgkl = 3qj3qk~ql "

(6.34)

Overend and Hylden [161] have derived the ordering terms of the second contact transformed dipole moment operator as defined by Eq. (6.24) 1

pg+(0) = pg(0) = p g + ~ .

p gqj

(6.35)

J

1

~ Pgm pg+(1) = ~'~"E Pgk qJ qk + j~k m j,k x [Sjn~ qj qk + Sjkm (l+~Sjm +~Skm)mj mk]

(6.36)

IR INTENSITIES OF OVERTONE AND COMBINATION BANDS

157

I pg+(2) = -~ X Pgkmqj qk qm + j,k,m

,

+ --2.~ Pgk

[1 a~
4"--

+--1 k ] 2 a~b (SJabqaqbqk + Sabqaqbqj)

1

{

S abj (1 + ~aj + ~BJ)

a.~
•

d,f

[Sfb(mamfqd +qdmfma)(l+~Sbd) +Sf (mbmfqd + qdmfmb) (1+ ~ad)]

-Z

SJab[d~_<e(2'S~dqbqdqe+ 2Sbeqaqdqe ) a.~
+1~. pg[ ~ s~bj(qdmamb+mambqd)(l+~Saj+~ibj) j kd,a_
9

1

"

(6.37)

Evaluation of the transition moments using linear harmonic oscillator wave functions and second contact-transform dipole operator shows that intensities of the fundamental absorption bands are determined predominantly by first-order contributions from the matrix dement (v[pg+(0)tv+l) and negligibly small contributions from (vlpg+(2)lv+l). The intensities of binary overtone and combination bands are

158

GALABOV AND DUDEV

determined by dominant contributions arising from the second order term pg+(1). The last contains comparable contributions from the second and first dipole moment derivatives.

IlL THE CHARGE FLOW MODEL The various dipole moment derivatives appearing in the expressions for the contact-transformed transition dipole moment matrix elements may be expressed in terms of electric dipole moment parameters such as effective charges and charge fluxes, polar tensors, electro-optical parameters or other local quantifies. Thus, intensities of overtone and combination bands can in principle be interpreted in terms of parameters closely related to those used in analyzing absorption intensities of fundamental transitions. Overend and Hylden [161] proposed a formulation that employs as basic parameters effective atomic charges and charge flows (charge fluxes) as defmed by Decius [109]. First and second order derivatives appear in the expression for the dipole moment matrix element in the case of binary overtones and combination transitions [Eq. (6.36)]. Therefore, as basic parameters the following quantifies are defmed

~(a~ ~

=

~)~a ~t,b bRa'

=

O2~a

aR~aRb

"

(6.38)

~(0) is the effective charge of atom a and Ra,b internal vibrational coordinates. Derivatives with respect to non-symmetrized internal coordinates are preferred. Symmetry considerations are introduces at a later stage following the approach suggested by Decius[109]. The parameters defined in expression (6.38) appear in the Taylor expansion of the effective atomic charges in terms of internal coordinates

~a =

~)

+ ~a

(C}~a/ 1 <~b( t)2~a ' ) R a R b + " " . ~,~Ra)Ra +'~ ~,~RR~Rb

(6.39)

Most of the calculations are performed on the basis of Cartesian displacement coordinates. A curvilinear transformation from internal to Cartesian coordinate space is carried out to yield ~ia = ~ ba ~ t a

(6.40)

IR INTENSITIES OF OVERTONE AND COMBINATION BANDS

a

a

(6.41)

~ija = X bij ~x + X b i b b ~ a

159

a,b

a =~

bijk~a + ~ bij a,b + X babjb b~c~a abc"

bj +bjk

a

(6.42)

a,b,c

Formulas for evaluating the partial derivatives of internal coordinates with respect to Cartesian displacement coordinates (bia= 8Ra/8~i, ~ = x, y, z) are available [ 162,164]. The same coefficients are used in analogous transformation of the anharmonic potential function. Since in the effective charge model the molecular dipole moment is def'med by p = Z ~t rot [Eq. (3.54)], a Cartesian component of p may be represented as follows Px = X g Ot +

x. = X

xa+X

Ot

a

1Z ij(~ i 2 a i<j

X ~i(~i-~e)xaa i

e

(6.43)

e

In expression (6.43) ~ = x, y, z and ~e are equilibrium position coordinates. Eq. (6.43) is then used to evaluate as intermediary quantifies dipole moment derivatives with respect to Cartesian coordinates in terms of charge and charge flow parameters defined in a Cartesian space. The following expressions are obtained

(Spx/~ = ~(a0) + Z (a(.xx/~ x13

(6.44)

(~y/C3x~ = ~ (O~l}/~ y~

(6.45)

(~px/o~ 2) = 2 (a~.~/Sx~ + Z (o~13/o~ 2) x13 13

(6.46)

(02px/~/gx~) = (a~13//~) + (0~I3/~D) + ~ (~v//~/gx13) x v 7

(6.47)

(a2px/aX~ay~)= (ar.Aay~)+

Z Y

(a2~ax~ay~)x~

(6.48)

160

GALABOV AND DUDEV

(/)2px/0ya/~yl3) = ~ (c32~o~a/~y13) xy, etc.

(6.49)

Finally, from the set of first and second dipole moment derivatives with respect to Cartesian displacement coordinates the dipole moment derivatives with respect to dimcntionless normal coordinates are obtained s

(C3px/t~s) = ~ Lj (C3px/C3~j) J S

St

(~px//kls/~qs,) = ~ L j L k (~px/0~j0~k). j,k

(6.50)

(6.51)

Equations (6.40) through (6.51) define the transformation from the basic parameters of the charge flow model ~(a~ , 0~/0R a and ~2Qz/0Ra0R,o to dipole derivatives with respect to normal coordinates entering the respective expressions for the transition dipole moment, as defined by relations (6.32) through (6.34). It is clear that the calculations are quite elaborate. As formulated, the charge flow model provides a possibility for simultaneous calculation of infrared band intensities associated with fundamental and binary overtone and combination transitions. A least squares optimization procedure for these calculations has been applied [161]. The parameters ~(o) and c3~JORa are determined from the ,intensities of fundamental modes. Initial guess values of the higher overtone terms are iteratively optimized to fit the observed intensities of the binary overtone and combination bands. Reliable experimental data for overtone and combination band absorption intensities can be determined for very small molecules. Purely mathematical restrictions arise from the appearance of a large number of parameters in the equations, as already discussed in Section 3.4. Hylden and Overend [ 165] have applied the charge flow model in analyzing the high derivatives of the dipole moment for a number of oxygen containing triatomics.

IV. T H E B O N D M O M E N T M O D E L The transformation of vibrational absorption intensities of binary overtone and combination bands in terms of bond moments and derivatives has been formulated by Gribov [72,155]. The transition dipole moment is expressed in terms of first and second dipole moment derivatives with respect to normal coordinates as shown in Eq. (6.3). Further expansion of (/~P//g~o and (/)2p//~Qk/~l)0 terms appearing in the expansion in terms of bond moments and derivatives with respect to internal coordinates is

IR INTENSITIES OF OVERTONE AND COMBINATION BANDS

161

straightforward. The formulation for the first derivatives was already described in Section 3.3. A particular dipole moment derivative with respect to normal coordinates is given by Eq. (3.22). In matrix notation Eq. (3.22) becomes

()

~oi i g P = ~

+~t~-~

(6.52)

Li

or

e+ ~

-i

i.tj.

(6.53)

L i is the ith row of the L matrix. The analogous expression for the second dipole moment derivatives is much more elaborate [ 155]:

~2p = Li [~-~-~-~-jLje+ ~ ~)Qi~j 0

+~--ff

~R LIj + "'" + ~R Lnj

~R L0+'"+ ~R L~j +5-fft,5-ff) (6.54)

-- "LiL~R ~~-~(~'~Lje}+ Li{(~~--~-~/(~~--~)+(~~-~--~)(~R)}Lj ~,~R) + Li [~-~ ~,~-~'jL j

9

The matrix product ~-Rk~-RjLj

(6.55)

is then expressed as ( b2~k

The arrays entering expression (6.56) have the structure

(6.56)

162

GALABOV AND DUDEV

~RI 2

/ O2~tk

.

.~176.

.

. ~, o~R12 . /0

~o~l~'-~n /0 (6.57)

Ra b)o -

/ t)R~2~1 nOR1)0 "'" ~~,~Rn ()21'2It )0" ~,'()R~ (ct21"tm1')0 "'" ~,~t)2~ t~Rnm2 )0

and el

0

0

I

0 0

el "'" 0 0 ..- el

I

.

.

.

.

.

.

em

"--

.

.

.

0 "--

0

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

0

0 "'" e m

}n" )

~m.

(6.58)

I

J

m is the number of bonds while n is the number of internal coordinates. Using the notation used in Section 3.3 and the expression for the matrix (0e/0R) [Eq. (3.26)] the following result for the second dipole derivatives is obtained

0Qi(~Qj 0

Lk0Ra0Rb

(6.59)

kOR~b M Lj. The matrix M consists of bond moment values arranged as the matrix E [Eq. (6.58)] while the array 02ek/ORaORb has a form analogous to matrix/~k/ORaOR b [Eq. (6.57)]. The procedure for evaluating elements of the matrix 02ek/0Ra0Rb is quite elaborate and will not be presented. It is described in the book of Gribov and OrviUe-Thomas [ 155]. Three types of intensity parameters enter Eq. (6.59). These are the bond dipole moments ~ , the first derivatives of bond moments with respect to internal coordinates, 0~k/0Ra, and second derivatives 02~k/0Ra0Rb . The latter quantifies are termed electrooptical anharmonic parameters. These terms reflect the non-linear dependence of ~k on the vibrational coordinates Rj and are determined by the electrical anharmon/city of molecular vibrations. As in the charge flow formulation, harmonic terms enter the expressions for intensities of binary overtone and combination bands.

CHAPTER 7

AB/N/NO

MO

OF INFRARED

CALCULATIONS INTENSITIES

I.

Introduction .......................................................................................................

164

N.

Computational Methods .....................................................................................

165

A.

Numerical Differentiation ........................................................................

165

B.

Dipole M o m e n t Derivative from the Energy Gradient .............................. 166

C.

Analytic Dipole M o m e n t Derivatives ....................................................... 167

gig~

IV.

Calculated Infrared Intensities ...........................................................................

169

A.

Basis Set Considerations ..........................................................................

169

B.

Influence o f Electron Correlation on Calculated Infrared Intensities ...................................................................................

176

Conclusions .......................................................................................................

187

163

164

GALABOV AND DUDEV

I. I N T R O D U C T I O N The semiclassical theories described so far are aimed mostly at interpreting the experimentally determined vibrational absorption intensities of molecules in terms of quantifies associated with the charge distribution and dynamics. Fewer attempts have been made for quantitative predictions of intensities based on transferable intensity parameters. Successful predictions are difficult to achieve because transferability properties are not so well expressed as in the case of force constants. This is determined by a number of factors: (1) the high sensitivity of vibrational intensities associated with particular modes to changes in molecular environment; (2) the physical limitations of the approximate point-charge models; and (3) mathematical difficulties in applying non-approximate models such as polar tensors or bond polar parameters for larger molecules. All these problems in intensity predictions are, a prwri, not present in quantum-mechanical calculations of intensities based on ab mitio methods. The development of effective theoretical approaches for analytic evaluation of first, second and third derivatives of molecular properties, energy and dipole moment in particular, have made considerable impact on the computations of vibrational spectra by ab mitw methods [166-169]. As we will see, however, in spite of the tremendous progress in theoretical developments and computational effectiveness, the accuracy of predictions based on even the most sophisticated wave functions is not fully satisfactory. At present, it is not possible to quote any particular percentage of deviations from experimental intensities for different levels of calculations because of considerable variations from molecule to molecule. Thus, the predictive power of semiclassical models [33,72,108,129,140] may be higher than that of ab miao quantum mechanical calculations. The mathematical difficulties in applying these methods, however, as well as the approximate nature of the physical concepts involved - transferability and additivity properties for molecular subunits - determine limited use of these approaches to special classes of molecules. On the other hand, calculations of vibrational spectra parameters are currently incorporated in the most advanced packages for ab initio MO calculations [153,154,170-172] and are by far easier to perform by simply running an ab mitw program. The predictions of vibrational intensities using high level ab initio calculations is, therefore, preferable in view of universality of application, effectiveness and more solid physical grounds. There is a second important side of the application of ab mitw MO calculations to the study of vibrational intensities. As for many other physical and chemical phenomena, the quantum mechanical studies provide essential information about the free mechanisms and factors determining the observed physical quantifies, in our case the integrated absorption intensities of bands in the infi'ared spectra of molecules. As we have seen,

AB INITIO IR INTENSITIES

165

such information, accumulated over the years, has been widely used in the development of different theoretical models for interpretation of infrared intensifies. This chapter is not aimed at describing the history of the application of semi-empirical followed by ab initio molecular orbital methods in evaluating vibrational absorption intensities. There are a number of comprehensive review articles covering the early and later stages of these studies [21,52,53,168,169,173-176]. We also do not intent to discuss details of the theoretical background of analytic derivative methods for calculations of intensities, also presented in excellent reviews [168,169,175]. Our principal aim is to describe the current state-of-the-art in these calculations that are changing so frequently and so dramatically as a result of developments in computer technology, improved theoretical approaches or further perfectness in ab mitw software.

II. C O M P U T A T I O N A L M E T H O D S The evaluation of the first and second derivatives of molecular energy with respect to an appropriate set of coordinates defining the positions of nuclei is required for calculations of frequencies and intensities in vibrational spectra. The second derivatives of energy, the force constants, are used in determining the frequencies and the normal coordinates. The normal coordinate transformation matrix is applied together with theoretical estimates of the dipole moment derivatives in evaluating vibrational absorption intensities.

A. Numerical Differentiation Initially, with older versions of ab mitio programs, energy and dipole moment derivatives were calculated by numerical differentiations using fmite-difference techniques. In general terms, the purely numerical differentiation of energy and dipole moment with respect to the appropriate set of molecular coordinates can result in sufficiently accurate values for the respective molecular parameters. Such an approach requires, however, a great amount of calculations at vibrationally distorted geometries. As stressed by Pulay et al. [177], it is not always obvious how to model small displacements along exact curvilinear coordinates. These authors have designed an algorithm for the generation of distorted molecular geometries represented as Cartesian displacements resulting from small internal coordinate values [ 177]. Since in the approximation of electrical harmonicity the dipole moment is a linear function of vibrational coordinates [Eq. (1.27)], the dipole derivatives can be determined by numerical differentiation f~om the relation

166

GALABOV AND DUDEV

c3p/Oqi = [ p (qi= + A i ) - p ( q i = - Ai) ] / 2 Ai .

(7.1)

p is the molecular dipole moment, qi vibrational coordinates, and Ai small values of the vibrational displacements.

B. Dipole Moment Derivative from the Energy Gradient As underlined above, the ab imtio calculations of dipole moment derivatives are strongly interrelated with the respective force field calculations. In fact, as shown by Komomicki and McIver [178], the dipole derivatives can be derived efficiently from the energy gradients 0E/0qi, where qi are coordinates describing nuclear motions. A particular Cartesian component of the dipole moment p~ of a system of fixed point charges is given by [177] p~ =

aE

/ 0f~

(7.2)

.

E is the energy of the system in a constant weak electric field f, so that the polarization of the intramolecular charges is negligibly small. Finite field perturbation theory [ 179] can then be applied to express the energy as a function of the electric field strength f (7.3)

E ( f ) = ( V I H - p f IV

If the energy is obtained at several values of the electric field f, the magnitude of p can be determined l~om expression (7.2). The dipole moment derivative with respect to a vibrational coordinate qi may be derived from the relation

0qi - ~)qi

= ~'~

'

with ~ = x, y or z. Thus, the dipole moment derivatives are evaluated from the gradient of the energy in the presence of a weak electric field [ 175] 0E(f)] = (~-qi) (Vl H - Pf I~P) 9 ~qi

(7.5)

If qi are standard internal vibrational coordinates, the intensity of an infrared band is readily obtained from relations (1.47) and (2.4). Pulay et al. [177] suggest that a complete and non-redundant set of local internal coordinates should be used in these

A B INITIO IR INTENSITIES

167

calculations. In such a coordinate space the Taylor expansion of the potential energy is unique V = V 0 - E ~i qi + i

E Fij qi q j + ~ Fijk qi q j qk + .... ij ijk

(7.6)

where 4~i= - c3E/~

(7.7)

Fijk... = o~n)E / ~ c3qj 8qk" 9-

(7.8)

and

at the reference geometry. The derivatives 0E/&li are components of the energy gradient. The 0E/0aa quantities are used in the process of geometry optimization in most current ab initio programs. Thus, to estimate all 0pg/0qi derivatives, it is only necessary to determine the energy gradient for a small number of field strengths, instead of performing 3N-6 calculations for distorted geometries. Basically, three calculations of the energy gradient along the Cartesian axes are needed in a weak constant electric field to obtain all c32E/0fx&li, 02E/Ofy&]i and ~E/c3fz&li derivatives. A variation of the perturbation treatment of Komomicki and McIver [178] has been suggested by Schaad et al. [ 180]. It clarifies further the physical significance of the approach. These authors have shown that the derivative of the energy gradient with respect to the strength of an external field can be determined from model ab initio calculations by placing one or more point charges at a large distance from the molecule.

C. Analytic Dipole Moment Derivatives Consistent analytic evaluation of first, second and third derivatives of the energy with respect to nuclear coordinates has been an essential development in ab initio molecular orbital calculations of molecular properties. The theoretical basis of these developments was created relatively early [166] and implemented first by Pople et aL [181]. Force constants are evaluated in a much faster and accurate manner compared with the earlier methods based wholly or partially on finite-difference techniques. The methodology is now developed for most types of molecular wave functions. The analytic determination of dipole moment derivatives by ab initio methods requires derivation of a complete formulation of coupled-perturbed Hartree-Fock equations by nuclear coordinate and electric field variations.

168

GALABOV AND DUDEV

(7.9)

Op/Oq = 02Esc F / 0q Of

where ESCF may be different types of SCF energies. The mathematical formalism has been reviewed by Amos [ 175] and Yamaguchi et al. [ 182] who have derived expressions for analytic evaluation of dipole moment derivatives for closed-shell, open-sheU restricted and open-shell unrestricted Hartree-Fock wave funcfiom. They have also studied the effect of basis sets and electron correlation on dipole moment derivatives. Here we shall briefly outline some physical aspects of the formulation as presented by Yamaguchi et al. [182]. The H_amfl"tonian operator perturbed by nuclear coordinate q and electric field f may be presented as follows [ 182] H = H 0 + ~H'q

+ XfH'f

(7.10)

.

In Eq. (7.10) H 0 is the unperturbed Hamiltonian while Lq and Xf are the respective perturbations. The perturbed integrals for basis set atomic orbitals can then be represented in a similar way. The respective formulas for one-electron integrals h~v, overlap integrals S~v and two-electron integrals (~tv/po) [~t(1)v(1)O(2)o(2)] are bhttv ~}hgv httv = h~v + ~.q . ~q + ~.f ~f +...

(7.11)

~S~v S~tv=S~v+~,q 3q +...

(7.12)

(~tV/ DO)= (~tV/ pO)0 + Zq

+ ....

~q

(7.13)

For a standard atomic orbital basis set the perturbation Hamiltonian H'q affects both oneand two-electron integrals while H'f influences one-electron integrals only. The coefficients of the ith molecular orbital are given by the expression Citt = C 0 g + ~ q

=cOg

~)Citt

~)q + ~ f

~}Citt

~}f +-'-

cog

all +~,q s Uqmi

all +~f s ufi

m

m

cog + . . . .

(7.14)

The summation is over all occupied and virtual orbitals. U q and U f are defined by the relations

A B INITIO I R INTENSITIES

169

~iix all /)q = ~ u q i C~mix

(7.15)

m

~)Ciix = ~ u f i COmix .

(7.16)

m

The superscript 0 refers to unpcratrbcd values in Eqs. (7.11) through (7.16). It is seen that Eqs. (7.14) through (7.16) represent the molecular orbitals of the perturbed system using the respective coefficients of the unperturbed MO's. Formulas for the evaluation of perturbed one- and two-electron integrals are derived [ 182] and the general approach is applied to various types of SCF wave functions [174,175,182]. As expected, the effect of electron correlation on computed intensities has been initially treated by evaluating dipole moment derivatives through numerical differentiation [182,183]. The finite difference method of Komornicki and McIver [178] offers another possibility. Simandiras et al. [ 184] have later developed a complete formulation for deriving analytic dipole moment derivatives at the second order Moller-Plesset (MP2) level. An alternative approach based on the configuration interaction (CI) gradient concept [ 185] has been put forward by Lengsfield et al. [ 186].

III. C A L C U L A T E D I N F R A R E D I N T E N S I T I E S As already underlined, no attempt to present a comprehensive account of ab initio calculations of infrared intensities will be made. A number of reviews on the subject are available [168,174-176,187,188]. Besides, the progress in theoretical approaches and computational facilities makes earlier results of lesser current interest no matter how important they were at the time. The discussion will be concentrated on two main themes associated with these calculations: (1) the influence of basis set and (2) the effect of electron correlation on computed intensities. We shall also limit our considerations to results coming mostly from analytic dipole moment derivative calculations.

A. Basis Set Considerations It is useful to define first the different basis sets of wave functions used in calculating analytic dipole moment derivatives and infzared intensities that will be discussed in the present section. These are described in Table 7.1.

170

GALABOV AND DUDEV

Yamaguchi et al. [ 182] have applied the equations derived by them for analytic simultaneous evaluation of vibrational frequencies and intensifies for closed-shell, openshell unrestricted and open-sheU restricted Hartree-Fock wave functions for a number of molecules using basis sets of different complexity. Their results iUuslrate very clearly the basis set dependence of calculated vibrational parameters. Some of their results are represented in Table 7.2. Several molecular quantities are included since the comparisons for the accuracy of predicted values are quite interesting. Even larger basis have been used by Amos in consistent analytic derivative calculations of harmonic frequencies, infrared and Raman intensifies for H20, NH 3 and CH4 as test molecules [ 175,189]. The results for H20, HF, CO, NH3, CH4 and C2H2 of Yamaguchi et al. [182] and Amos [175, 189] are summarized in Tables 7.3 and 7.4. Analytic derivative ab initio results for the linear triatomic CO 2 and HCN molecules are shown in Table 7.2. The DZ and TZ basis sets appear to perform pretty well for calculating bond lengths for these molecules too. There could be no theoretical justification as to why medium size basis sets should work better than very large basis sets. Besides, a final judgment on this matter may be given after calculations that include correlated wave functions. Vibrational frequencies are predicted with varying success at the different level of calculations, though the general trend is that frequencies are overestimated by about 10%. R is not surprising, therefore, that a constant scaling factor of 0.9 for ab mitio force constants has been used with such a success in many laboratories [21,168,174,176,209-211]. The dipole moment of HCN is very well predicted almost independently on the basis set used. Intensities of the infrared active modes of CO2 are overestimated by nearly 150-200% for all basis sets. HCN intensities are quite well predicted for extended basis sets, such as TZ+2P or 6-311++G(3d,3p), except for the C=N stretching mode which is very weak in the experimental spectrum. Theoretically evaluated SCF energies, geometries, vibrational frequencies and infrared intensities at different levels of restricted Hartree-Fock ab m i t w calculations for the water molecule are summarized in Table 7.3. The theoretical results are taken from Refs. [ 182] and [ 189]. In geometry predictions it is notable that 6-31G, DZ and TZ basis sets fail in predicting the value of the valence angle. Larger basis sets perform in a much better way, thus bringing more logic into ab m i t w geometry calculations. The small 321G basis set appears to work quite satisfactory in evaluating geometric parameters. These findings have been confirmed by numerous calculations on larger systems carried out by Durig and co-workers [209-213]. Vibrational wavenumbers are overestimated by approximately 5-7% at even very extended basis set RHF/SCF calculations. There is no clear correlation between the size of basis sets and agreement with experiment. Predicted infrared intensities for water are quite disappointing, differing very significantly from the measured values. Particularly

A B INITIO IR INTENSITIES

171 TABLE 7.1 Description of basis sets Primitive functions/ contraction (Hydrogen)

Symbol

Primitive functions/ contraction (First row atoms)

STO-3G

(6s 3p / 2s lp)

(3s / Is)

3-21G

(6s 3p / 3s 2p)

4-31G

(8s 4p / 3s 2p)

DZ

(9s 5p / 4s 2p)

TZ

(9s 5p / 5s 3s)

6-31G

(10s 4p / 3s 2p)

6-3 I+G

(1 ls 5p / 4s 3p)

6-3 I+-r

(1 ls 5p / 4s 3p)

6-3 l+G(d) 6-31G(d)

(lls 5p ld/4s 3p ld) (10s 4p ld/3s 2p ld)

6-31G(2d)

(lOs 4p 2d / 3s 2p 2d)

6-31G(3d)

(10s 4p 3d/3s 2p 3d)

6-31G(d,p)

(10s 4p ld/3s 2p ld)

6-31G(2d,2p)

(lOs 4p 2d/3s 2p 2d)

DZ+P

(9s 5p ld/4s 2p l d)

6-3 l++G(d,p) TZ+P TZ+2P

(1 Is 5p ld/4s 3p ld) (9s 5p l d/5s 3p l d)

6-311G

(lls 5p / 4s 3p)

6-311++G

(12s 6p / 5s 4p)

6-3 llG(d) 6-311G(d,p)

(1 Is 5p ld/4s 3p ld) (1 Is 5p ld/4s 3p ld)

6-31 l+G(d,p)

(12s 6p ld / 5s 4p ld)

(3s / 2s) (4s / 2s) (4s / 2s) (5s / 3s) (4s / 2s) (4s / 2s) (5s/3s) (4s / 2s) (4s / 2s) (4s / 2s) (4s / 2s) (4s lp / 2s lp) (4s 2p / 2s 2p) (4s lp / 2s lp) (Ss lp / 3s lp) (5s lp / 3s lp) (5s 2p / 3s 2p) (5s / 3s) (6s / 4s) (Ss / 3s) (5s lp / 3s lp) (5s lp / 3s lp)

6-3 ll++G(d,p)

(12s 6p ld / 5s 4p ld)

(6s lp / 4s lp)

6-31 l++G(2d, p)

(12s 6p 2d / 5s 4p 2d)

(6s lp / 4s lp)

6-31 l++G(3d, p)

(12s 6p 3d/5s 4p 3d)

(6s lp / 4s lp)

6-31 l++G(Ed,2p)

(12s 6p 2d/5s 4p 2d)

(6s 2p / 4s 2p)

6-3 l++G(3d, ap)

(12s 6p 3d/5s 4p 3d)

(6s 3p / 4s 3p)

(9s 5p 2d/5s 3p 2d)

w

4

TABLE 7.2

h,

A b initio geometry, vibrational harmonic frequencies, infirared intensities and dipole moments for C02 and HCNa STO-3G

3-21G

DZ

TZ

6-31G (d)

DZ+P

6-31ffi (d)

1.1879

1.1558

1,1652

1.1594

I. I433

1, I453

1.1434

1.1357

1.1395

1.1351

1.160b

2536

2463

2384

2302

2585

2590

2564

2576

2565

2550

2349b

1435

1428

1400

1381

1518

1513

1511

1518

1510

1508

1330b

6-311(d,p)

TZ+2P

6-3116 (3d,3p)

EXP.

566

659

717

608

746

766

750

768

775

769

667b

342.9

730.9

I133

1010

990.5

1 I69

1I04

1 I62

1066

1088

500b

0

0

0

0

0

0

0

0

0

0

0

68.1

130.9

146.7

149.6

138.2

139.1

138.7

137.2

123.0

125.4

47.8b

1.0699

1.0502

1.0542

1.0547

1.0590

1.0638

1.0600

1.0583

1.0571

1.0568

1.058c

1.1530

1.1371

1.1507

1.1375

1.1325

1.1366

1.1331

1.1271

1.1273

1.1246

1.157C

3917

369 1

3697

3626

3680

363 1

3666

3620

3624

3608

3311C

2541

2395

2327

2315

2438

2403

2426

2406

2407

2400

2097C

952

990

883

895

889

857

879

878

902

87 1

712C

k

164.6

74.6

66.8

81.3

60.9

70.6

66.9

72.1

71.5

71.1

1.35

6.60

12.8

11.4

11.8

12.5

11.9

12.2

10.1

11.0

79.6

64.2

p (Debye) 2.455 3.043 3.296 3.283 3.209 3.216 3.307 3.275 aAb initio results are taken from Ref. [1821 with permission. bBond lengths from Ref. [ 1901, frequencies from Ref. [191], intensities from Refs. [ 1921 and [ 193). CBond lengths from Ref. [ 1941, frequenciesfrom Ref. [195], intensitiesfrom Ref. (1961 and dipole moment from Ref. [ 1971.

3.262

69.0 3.270

50.2C 2.989

Ga

48.0

128.0

131.5

134.9

76.3

83.4

98.1

0

7

b

TABLE 7.3 Basis set dependence of SCF energies, geometries, vibrational harmonic fiequencies and infrared intensities of H2Oa

Energy (Hartree)

r OH (A) LHOH (degree)

DZ

TZ

STO-3G

3-21G

6-31G

-74.9659

-75.5860

-75.9854

-76.01 10

-76.0143

-76.0107

-76.0534

-76.0576

-76.0659

-76.0675

0.9894

0.9667

0.9496

0.9514

0.9527

0.9473

0.9412

0.9402

0.9399

0.9400

100.03

107.68

111.55

112.52

111.82

6-31G (d)

105.50

6-311(d,p)

106.22

6-311++G (3d,3p)

106.21

[8s6p4d] /[6s3p]

106.25

[8s6p4d2fl /[6s3p Id]

106.32

B

%

EX^.^

0.9572 104.52

4391

3946

4146

4204

4156

4189

4245

4229

4227

423 1

3943

WA1)

4140

3812

3989

4028

3990

4070

4143

4129

4129

4130

3832

w3(A1)

2170

1799

1737

1711

1723

1827

1726

1757

1759

1747

1649

30.0

9.19

A2

44.4

0.04

A3

7.24

P (D)

1.709

80.1 2.387

54 2.9 123

65.1 3.39 136.1 2.530

53.4 0.85 131.7 2.512

58.2

88.4

88.7

92

92

44.6

18.2

25.5

14.9

I5

I5

2.2

107.5

85.8

93.9

98

97

53.6

2.199

2.196

1.974

E5 W

wl(B2) (cm-1)

A1 ( M m o l )

3

1.85

aA6 initio results are reprinted with permission from Ref.[182]: basis sets STO-3G, 3-21G, DZ, TZ, 6-31G(d), 6-31 l++G(d,p), 6-31 l++G(3d,3p) and with permission from Ref. [189]: basis sets 6-31G, [8s6p4d]/[6s3pJand [8~6p4d2fj/[6~3pld]. bExperimental data:geometry and frequencies from Ref. [ 1981, mfrared intensities from Ref. [96] and dipole moment from Ref. (341.

8 IA

c

4

TABLE 7.4

P

6-3 1lteG(343p) a6 initio and experimental geometries (in A and degrees), vibrational frequencies (in cm-I), inErared intensities (in km mol-1) and dipole moments (in D) for HF, CO, NH3, CH4 and C2H2

_____-----_--------------------__-------_-~--------------------_---_--_---------------_~_---HF r V A co r V A ab initioa ExP.~.~

4486 158.5 1.908 1.103 2422 145.3 0.142 3962 99.8 1.819 1.128 2143 65.2 0.1 12 LHNH -?! __---_--------------------r ?!i9 _---V Z ~ ~ ~ - - - ~ ~ ~ E ~ _ - _ ~--_-_ ~ ~ Al2~ ----- - -A~L -----L A4 ---_-E--ab initioa 0.9990 107.62 3804 3683 1794 1128 11.2 1.35 36.7 180.7 1.582 1.0116 106.7 3577 Exp.d 3504 1691 1022 11.4e 31.7 138 1.472 r - _CH - A _______-_-___-----VJ1!3 ---- !2(P_d ----- YdEl-_-- J 4 W -_----A1------ A2 --__--43-----__ A4 ----_ ab initioa 1.0820 3244 3 147 1671 1455 114.0 0 0 28.4 Exp.f 1.0858 3 157 3026 1583 1367 65.5 0 0 3 1.8 --2_2 CH

0.897 0.917

_______________ JCC_____ ~ L

____

( ~ ) _ - - ~ 2 ~ ~ ~ - _ ~ 3 ~ ~ - A_ 2 ~ ~&_--& ~ ~ ~ ____ - _ L%-~ ~ ~ ~ ~ - - - A

ab initioa

3664 1.0540 1.1805 3551 2204 865 799 0 97.0 Exp3 1.0608 1.2031 3374 3289 1974 730 612 0 71 aReprintal from Ref. [ 1821 with permission. bExperimental data for HF from: Ref. [ 1951 (frequency), Ref. [ 1991 (bond length and dipole moment) and Ref. [200] (intensity). CExperimentaldata for CO from: Ref.[201] (bond length), Refs. [202] and [203] (frequency and intensity) and Ref.[204] (p). dExperimental data:geometry and frequencies from Ref. [205], intensities from Ref. [147] and p from Ref. [204]. eA1 + A 2 = 11.4kmm0l-~. fExperimental data:geometry and frequencies from Ref. [206] and infrared intensities from Ref. [207]. gExperimental data:geometry from Ref. [190], frequencies from Ref. [ 1951 and infrared intensities from Ref. [208].

0 0

225.5 175

0 0

AB INITIO IR INTENSITIES

175

overestimated is the intensity of the antisynmletric stretching mode which, at the largest basis sets, differs from the experimental value by a factor of seven. The predicted intensities of the other two vibrations are nearly a hundred percent higher than the experimental quantifies. The theoretically estimated dipole moment is, on the other hand, in good agreement with experiment. Infrared intensities for HF, CO, Nil 3, CH 4 and C2H2 are predicted with reasonable success at the 6-3 l l++G(3d,3p) level of SCF calculations (Table 7.4). The best agreement with experiment is obtained for NH 3 with deviations not exceeding 30%, while the absorption intensity of the CO stretching band is overestimated by about 120%. The carbon monoxide molecule is notable for difficulties in quantum mechanical calculations of dipole moment and its derivatives. As an example of ab initio calculations for a fairly large molecule we present in Fig. 7.1 the calculated and experimental gas-phase infrared spectrum of methyl ethyl ether [ 114]. The theoretical spectrum is as predicted from 6-31G(d,p) calculations. The simulated specmun represent an equilibrium mixture of s-tram and gauche conformers of the molecule. It can be seen that the discrepancies between computed and observed spectral curves are quite significant. The results presented so far show the potential of SCF level wave functions in predicting molecular properties. In general terms, medium and large basis set ab mitio calculations provide satisfactory results in calculating geometries, vibrational frequencies and, to a lesser extent, dipole moments. Smaller basis sets give in particular cases surprisingly good agreement with experimental values, especially with regard to geometry predictions. Some cancellation of errors is very possibly taking place. On the other hand, small non-polarized basis set ab mitio calculations may produce erroneous ordering of the frequencies of vibrational modes, thus hampering the important use of such calculations in vibrational assignments. It is, therefore, safer to employ extended basis sets with polarization functions on all atoms in force constant calculations. Infrared intensities are predicted with very limited success at SCF level ab mitio calculations. Fairly large basis sets with polarization functions on all atoms are needed to achieve qualitative similarity with the observed spectral profiles. Differences with experimental intensities by a factor of 2 and more are obtained with even the most sophisticated wave functions. Still, however, calculated intensities combined with predicted frequencies can be used in a quite productive way for the correct assignments of vibrational spectra. The predicted infrared intensities depend strongly on the basis sets employed. Smaller basis sets without polarization functions on all atoms produce results that should be considered, as stressed by Yamaguchi et al. [ 182], useless. Ab mitio calculations provide essential information that is very difficult to obtain from experiment: the signs of dipole moment derivatives. Thus, a formidable problem in

176

GALABOV AND DUDEV

""'~--~

"%. "\

75

\

/

/

/ !

A I--

~u

25

U

"~

//

",,.,, /'-~,,, '

l r-, /

""--'~\

/ ..._,,_/.

/,'" ~, .," ',,_,"

~~-.,...,...-..-"

I

l

i

\,/

z

I-I-t~

z

I.--

I 25

B

3o'oo z oo

14'oo

10'00

600

WAVENUMBER (cm "1)

Fig. 7.1. Calculated at 6-31G(d,p) level (A) and observed (B) infrared spectrum of methyl ethyl ether at 10 Torr partial pressure of the sample in 10 cm cell; theoretical spectrum is calculated for an equilibrium conformer mixture of 86% trans and 14% gauche methyl ethyl ether.

intensity analysis is solved in a straightforward manner. In very few cases SCF ab initio calculations have resulted in erroneous signs. An example is the HCN molecule [ 175]. In view of the limited success of ab initio calculations at the SCF level to predict vibrational absorption intensities, considerable attention has been focused on the development of appropriate methods that would account for the effect of electron correlation.

B. Influence of Electron Correlation on Calculated Infrared Intensities Consistent efforts have been focused over the past decade on the development of metho& for incorporation of electron correlation in evaluating molecular properties. Analytical derivative approaches have been formulated for multiconfiguration SCF (MC SCF) wave functions [168,214-218], configuration interaction (CI)wave functions [219-

A B INITIO I R INTENSITIES

177

221], coupled-cluster (CC) [222-224] and many body perturbation theories (MBPT) [151,225,226]. Dipole moment derivatives are most easily obtained from correlated wave functions by numerical differentiation or by using the finite field method of Komornicki and McIver [178]. Lengsfield et al. [218] have developed an efficient analytic method for calculating infrared intensities for general MC SCF wave functions. These authors have applied their mathematical approach in deriving correlated infrared intensities for H2CO. They have employed several Gaussian basis sets that have been proven to give satisfactory estimates of infrared absorption intensities. These are the DZP (DZ+P) basis with polarization functions on all atoms (Table 7.1), as well as the diffuse basis (BAC) of Bacskay et al. [227]. The BAC consists of (9sSp/4s3p) for first row atoms and (4s/3s) for hydrogen contractions augmented with diffuse s and p functions on all atoms and d polarization functions on the first row atoms and p polarization function on hydrogen. The authors have also used an additional basis set denoted by DZP+dSPD which contains both tight and diffuse polarization functions added to the standard DZ+P basis. The theoretical estimates of infrared intensities for H2CO as obtained by Lengsfield et al. [218] are given in Table 7.5. The MC SCF results are compared with SCF calculations using the same basis sets as well as with configuration interaction calculations obtained by the finite difference method. The experimental absolute infrared intensities as determined by Nakanaga et al. [228] are also given. The DMC 6/6 notation in Table 7.5 refers to MC SCF calculations consisting of all double excitations fi'om the six valence orbitals to the first six ~ orbitals. SDMC 6/6 refers to calculations involving all single and double excitations, while DMC 6/12 denotes all double excitations to the first 12 virtual orbitals. The configuration interaction calculations include single and double calculations except for the ls orbitals. The CI force constants are evaluated by numerical differentiation, while dipole moment derivatives by the finite field method [ 178]. The results presented in Table 7.5 show that higher level of SCF and electron correlation calculations improve in a quite systematic way the accord between predicted and observed in,areal intensities. A nice agreement with experiment is achieved from DZP+dSPD/DMC 6/6 calculations (last column of Table 7.5). Considering just one example does not, permit any definite conclusions. Moreover, as pointed out by Amos [ 175], MC SCF calculations account for only a fraction of the electron correlation. The many body perturbation theory has been applied to obtain higher accuracy in ab initio calculations of molecular properties. Pople et al. [ 181] have developed analytic derivative methods at second order perturbation theory level (MBPT(2)). Simandiras et al. [ 184] have derived specific expressions for analytic determination of dipole moment derivatives at MBPT(2). Dierksen and Sadlej [229] have shown by applying finite field MBPT in studying dipole and quadruple polafizabilities of the CO molecule that fourth and even higher level of MBPT is required to achieve satisfactory results.

TABLE 7.5. Theoretical and experimental Mared intensities of H2CO in km mol-1 a Mode

Exp.b

BAC SCF

DZP SCF

DZP SDMC

DZP DMC

BAC DMC

dTZP DMC

DZP DMC

616

616

616

616

6/12

DZP CI

DZP+dSPD DMC 616

"6

6.5

6.7

1.7

1.9

2.1

7.1

1.2

2.3

3.9

4.5

"5

9.9

17.2

19.3

13.4

13.6

10.9

12.5

14.0

14.8

13.6

v3

11

28.3

14.0

5.6

5.8

16.0

7.0

6.4

7.6

10.0

v2

74

105.4

157.9

107.8

106.3

65.9

112.7

103.6

93.1

107.1

V1

75

42.8

72.7

78.4

75.8

49.1

63.5

68.3

66.1

71.4

v4

87

59.8

112.1

128.5

125.0

68.6

114.0

115.5

106.2

95.1

aReprinted with permission from Ref. [2 181. bFrom Ref. [228].

A B INITIO IR INTENSITIES

179

TABLE 7.6 Correlated ab mitio calculations of dipole moment (It) and harmonic infrared intensities (A) of H20 (kt in units of Debye and A in km/mol)a. Normal modes are designated as follows: v l(Al) - OH sy~. stretching, v2 (AI) - HOH bending, v 3 (B2) - OH antisym, stretching. [5s4p3d]/[3s2p] Ix

A1

A2

A3

SCF

1.995

15.9

94.3

83.1

MBPT(2)

1.876

9.8

66.7

68.0

MBPT(3)

1.894

8.0

74.1

60.3

MBPT(4)

1.848

5.3

65.0

50.6

CCSD

1.878

6.2

70.6

54.1

CCSD+T(CCSD)

1.855

5.1

66.9

49.5

gt

A1

A2

A3

2.2

53.6

44.6

Experimentb Experimentc 1.85 aReprinted with permission from Ref. [224]. bFrom Ref. [96]. CFromRef. [34].

Yamaguchi et al. [182] in their comprehensive study on the theoretical calcdations of infrared intensities have used the finite difference technique to determine dipole moment derivatives at second order many body perturbation theory (MBPT(2) or MP2) and configuration interaction with single and double excitations (CISD). Higher order MBPT calculations and coupled cluster approach has been used by Stanton et at [224] in studying electron correlation effects on vibrational intensities. Their results for H20 are given in Table 7.6. Two types of couple cluster models are used: CCSD [230] with excitation operator consisting of cluster operators corresponding to first and second level (T = T 1 + T2) and CCSD+T(CCSD) in which the effect of connected triple excitation (T3) are considered approximately. The data for the two molecules considered show that incorporation of electron correlation reduces considerably the SCF estimates of infrared intensities, thus leading to much better agreement with experimental values. These results indicate, therefore, that more reasonable quantitative accord between predicted and experimental infrared intensities is achievable only through highly correlated wave function calculations.

180

GALABOV AND DUDEV

As we have seen, water molecule has been a subject of numerous studies. HF/SCF calculations at very large basis sets levels (Table 7.3) produce vibrational absorption intensities that differ by a factor of 2 or more (a factor of seven for the symmetric sCetching vibration) from the experimentally measured values. Even with highly correlated wave functions (Table 7.6) the disagreement between theory and experiment is more than 100 percent for the Vl(A1) vibration. The detailed theoretical study on infrared intensities of H20 carried out by Swanton et al. [231] is, therefore, of considerable interest. The authors have examined four major effects on predicted intensities: (1) the magnitude of electron correlation effect; (2) the applicability of Hellmann-Feynman theorem to the evaluation of dipole moment derivatives; (3) the role of basis sets; and (4) the accuracy of the double harmonic approximation. By attacking the problem from different sides they have been able to obtain theoretical results for the infrared intensities of H20 that are in very good agreement with the experimental data. The dipole derivatives are obtained via numerical differentiation. The dipole moments for the different basis sets levels have been obtained either as expectation levels (EV) for the respective operator or as energy derivatives (ED) using the finite field method [ 178]. The contracted basis sets employed are shown in Table 7.7. The configuration interaction calculations have been of CISD type. CEPA2 (CPA') and CEPA3 (CPA") [232-234] coupled electron calculations have also been employed. Dipole moment derivatives have been obtained by MC SCF using the complete active space self consistent field (CASSCF) formalism [235]. The calculated infrared intensities at different level of theory are shown in Table 7.7. It can be seen that for several levels of theoretical calculations the agreement between computed and experimental values is satisfactory and falls within the range of experimental uncertainties. With regard to basis set requirements Swanton et al. [231] concluded that both diffuse and compact polarization functions are required to produce satisfactory results. The ED approach in deriving dipole moment derivatives fIom CEPA2 and CEPA3 wave functions is preferable. Therefore, the use of Hellmann-Feynmaa theorem in calculating dipole derivatives is less accurate. The study of the effect of anhannonicity produced an important results: it is concluded that the double harmonic approximation yields results that are comparable with more rigorous calculations accounting for the effect of mechanical and electrical anharmonicity. It is also of interest to present the calculated dipole moment derivatives with respect to symmetry coordinates as obtained by Swanton, et al. [231]. The data are given in Table 7.8. The results show that the signs of dipole moment derivatives remain the same independently of the level of ab mitio MO calculations. Very similar results with CISD and CEPA-1 approaches are obtained by Botschwina [236] in calculating correlated infrared intensities for hydrogen cyanide (HCN), phosphaethyne (HCP) and cyanogen (C2N2). The author uses self-consistent field electron pairs theory (SCEP) [237] in solving the CI and CEPA-1 problems. Dipole moment derivatives are obtained from

AB INITIO IR INTENSITIES

181 TABLE 7.7

Calculated ab initio infrared absorption intensities for H20 (in km mol-l) a Basis set

[5s4p ld]/[4s lp] [6s4p 1d]/[4s lp] [Ss4p2d]/[4s2p]

[6s4p3 d]/[4s3p] [5s4p2dlfl/[3s2p ld] [4s3p3d]/[3s3p]

SCF CI-EV SCF CI-EV SCF CI-EV CI-ED CPA"-ED CPA'-ED CASSCF SCF CI-EV SCF CI-EV SCF CI-EV CI-ED CPA"-ED CPA'-ED

Experimental b aReprmted fTomRef. [23 l] with permission.

A1

A2

A3

16.66 7.25 4.63 0.08 12.89 3.86 3.75 2.52 1.42 5.20 13.71 5.11 13.06 4.83 13.62 4.94 3.45 3.97 3.50 2.24

87.09 69.70 99.67 76.03 94.65 75.32 74.69 72.37 70.12 72.76 92.28 73.53 92.60 75.52 94.25 75.97 81.08 74.47 74.31 53.6

63.62 43.67 61.31 34.15 82.06 55.05 56.14 53.13 48.22 67.17 83.42 60.01 81.95 53.94 84.02 54.53 48.92 50.39 42.31 44.6

bFrom Ref. [96].

electric dipole moment calculated as expectation values (EV) and as energy derivatives (ED). Again, as in the case of H20 for wave functions that do not obey the HellmannFeynman theorem, the ED approach produces more accurate results. In Fig. 7.2 the dipole moment variations with the C-H stretching coordinate in HCN and C-P coordinate in HCP are shown. It is seen that considerable difference between dipole moment slope at equilibrium for SCF and CEPA-1 (ED) results for the C-P stretching. The CEPA-1 (EV) is also quite different from CEPA-1 (ED). The CEPA-1 (EV) curve corresponds to a dipole derivative at equilibrium having opposite sign to the respective CEPA-1 (ED) value. The analysis of the effect of vibrational anharmonicity shows that it does not affect in any substantial way the predicted intensities.

182

GALABOV AND DUDEV TABLE 7.8 A b imtio calculated dipole moment derivatives with respect

to symmetry coordinates for H2Oa,b (in D A-1 or D tad -1) [5s4p2d]/[4s2p] SCF CI-EV CI-ED CPA"-ED CPA'-ED CASSCF Experimental c

op/os I

Op#S2

Opx/OS3

-0.538 -0.294 -0.290 -0.237 -0.178 -0.342 -0.234

0.959 0.862 0.859 0.847 0.836 0.846 0.726

-1.346 -1.103 -1.114 -1.084 -1.033 - 1.218 -0.992

aReprinted fxomRef. [231] with permission. bSynnnetry coordinates are defined as follows: Sl=(Arl+Ar2)/,]2, S2:Ae and S3=(&rl-&r2)/ ~/2. Cartesian cxx)rdinatesystemand internal coordinates are shown in Fig. 3.6. eObtained from the absolute gas-phase infiared intensifies of Clough et al. [96] and the force field of Mills [23].

A

/

' 9

,/so,

,~

/,./~E

/./~//./~-'"

pCI--SD (EV) CI-SD(ED) PA-I(ED)

"',c~Pa-,c~vl

o30.

B

0.25/

/

/

/

/

/c,

o~o/

-~

t)

015 I;0

At 1.'5 -

0.15- ~ ~ , C E P A - I ( E V ) -o3

6

o'.3

Fig. 7.2. Variations of the dipole moments of (A) HCN with the C-H stretching coordinate and (B) of HCP with C-P stretching; ~t is in atomic units. (Reproduced with permission firom Ref. [236]).

AB INITIO IR INTENSITIES

183

A valuable theoretical contribution to the understanding of vibrational absorption intensities is the work of Willetts, Handy, Green and Jayatilaka [238]. Following earlier studies [239-246] the authors have used the contact transformation approach (Chapter 5) to derive complete second-order perturbation theory formulated for the dipole matrix element, the intensity of vibrational fundamentals, first overtone and combination transitions. Willetts et aL [238] have been able to investigate in detail the anharmonic and Coriolis contributions to the integrated intensities for H20 and H2CO. Theoretical calculations using several different basis sets at SCF and MBPT(2) level on these molecules have allowed to determine in quantitative terms the effect of vibrational anharmonicity and vibrational-rotational interactions on infrared intensities. DZ+P and TZ+2P basis set calculations at SCF and MBPT(2) levels as well as an extended 6-31G basis set at MBPT(2) level are performed. The 6-31Gext basis set [ 184] include two sets of polarization functions plus some diffuse polarization functions, a combination that proved successful in dipole moment calculations. In Table 7.9 the transition matrix elements for H20 for MP2 6-31Gext ab imtio calculations are presented, as given by Wiiletts et al. [238]. In general, the anhannonic contributions to intensities are small. The cubic dipole and mixed cubic potential and quadratic dipole terms have more significant contribution. In the case of H2CO the picture is complicated from the presence of Fermi resonances near some fundamentals. Thus, large terms appear in the anharmonic correction terms. The authors conclude that the prediction of overlapping resonances is a particularly difficult task in intensity analysis. The calculated anharmonic corrections to vibrational absorption intensities do not explain the differences between calculated and observed intensities at this level of theory. It appears that larger basis sets and more exact treatment of electron correlations are needed to obtain better agreement between theoretical predictions and experiment. This is confirmed in most convincing terms by recent high-level ab mitio studies carried out by Schaefer and co-workers [247, 248]. The authors apply self-consistent field (SCF), single and double excitations configuration interaction (CISD), single and double coupledcluster (CCSD) and single, double and perturbative triple excitations coupled -cluster [CCSD(T)] calculations to analyze systematically the effect of basis set and electron correlation on calculated equilibrium geometries, dipole moments, harmonic vibrational flequencies and infrared intensities of HCN, HNC, CO2, CH4, NH4 +, HCCH, H20, H2CO, NH 3 and FCCH. Three basis sets are employed: DZ+P, TZ+2P and TZ+2P attgmented with higher angular momentum functions [TZ(2df,2pd)]. In general, the highest levels of theoretical calculations produce best agreement with experiment for the various molecular properties considered. With DZ+P basis set the CCSD(T) method produces some harmonic flequencies below the experimental values. With TZ+2P basis set the average absolute errors in harmonic vibrational frequencies is found to be 9.9%

184

GALABOV AND DUDEV

TABLE 7.9 MP2 6-31Gext ab initio results for transition matrix elements of H20 (in Debye units) a mode

value

mode

value

1 2

-2.17x10- 2 0.1249

1 2

--6,34x 10-5 1.87x10- 3

1 2

Mixed (from P2 and cubic potential) 2.72x10 -3 3 -7.89• 10-4

2.40x10 -3

1 2

Mechanical (f~om quartic potential) 7.15x10 -4 3 9.64x10-4

-7.04x 10-4

1 2

Mechanical (1~om cubic potential) -5.05 x 10-4 3 1.61xlO -3

Double Harmonic 3

-7.77x10 -2

3

-1.20x 10-3

Electrical (fIom P3)

-l.90x 10-4

Coriolis 1 -5.00x 10-.6 3 -9.40x 10-5 2 -3.64x 10-4 aReprinted with permissionfromRef. [238]. Copyright[1990] AmericanChemicalSociety.

for SCF, 3.8% for CISD, 1.5% for CCSD and 2.3% for CCSD(T) calculations. With TZ(2d~2pd) basis set the average absolute errors obtained vary systematically from 10.3% for SCF, 6.3% for CISD, 3.7% for CCSD to 2.2% for CCSD(T) calculations. Equilibrium geometries are accurately predicted at coupled cluster levels of calculations. In Tables 7.10 and 7.11 the results for H20 and HCCH as obtained by Schaefer and coworkers [248] are presented. The results for H20 and HCCH and also for the other molecules investigated [247, 248] clearly show that infrared absorption intensities can be predicted with satisfactory accuracy ~om ab initio calculations employing TZ+2P or higher basis sets and theoretical methods producing highly correlated wave functions. In effect, the authors have found [248] that TZ+2P and TZ(2d~2pd) CCSD(T) methods provide the best balance between basis set and theoretical method quality.

b

B

z

TABLE 7.10 SCF, CISD, CCSD and CCSD(T) results for TZ+2P and TZ(2df,2pd) basis sets and experimental values for molecular properties of H20.a SCF TZ+ZP

E (Hartree)

re(0-H)

(4

%(HOW (O) Pe (D)

-76.061 0.9400 106.3 1.988

CISD

TZ(Zdf,Zpd)

-76.063 0.9402 106.4 1.984

TZ+2P

-76.312 0.9523 104.9 1.940

TZ(Zdf,Zpd)

-76.338 0.9530 104.8 1.939

CCSD TZ+ZP TZ(Zdf,Zpd)

-76.323 0.9558 104.7 1.932

-76.350 0.9568 104.5 1.931

5 5J CCSD(T)

TZ+ZP

-76.329 0.9582 104.4 1.922

TZ(Zdf,Zpd)

Experiment

-76.358 0.9594 104.2 1.920

0.9572b 104.5b

1.W

4139

4134

3943

3937

3883

3 874

3845

3835

3832b

1764

1745

1702

1676

1690

1662

1679

1650

1649b

0 3 032)

4238

4235

4042

4039

3987

3981

3951

3944

3943b

A1 (km mol-I)

14.8

16.5

6.2

8.4

4.4

6.2

3.2

4.7

2.2d

A2

96.4

96.5

75.7

76.3

72.1

72.6

69.2

69.5

53.6d

85.4 53.6 80.7 aAb initio results are taken from Ref. [248]with permission. bFrom Ref. [198]. CFromRef. [34]. dFrom Ref. [96].

60.3

47.2

52.9

42.9

48.4

44.6d

A3

B

A !.

(A1)(cm-1) a2 (A11 01

1

TABLE 7.1 1

SCF, CISD, CCSD and CCSD(T) results for TZ+2P and TZ(2dt2pd) basis sets and experimental values for molecular properties of HCCHa SCF

TZ+2P E (Hartree)

re(C-C)

(4

re(C-H)

-76.849

CISD

TZ(Zdf,Zpd) -76.8 53

TZ+2P -77.158

CCSD

TZ(Zdf,Zpd) -77.190

TZ+2P -77.184

CCSD(T)

TZ(2df,2pd) -77.218

TZ+2P -77.199

TZ(Zdf,Zpd)

Experiment

-77.234

1.1797

1.1799

1.1934

1.1907

1.2009

1.1980

1.2073

1.2042

1.203lb

1.0536

1.0542

1.0564

1.0559

1.0601

1.0595

1.0618

1.0611

1.0608b

o I (zg+)(crn-')

367 1

3668

3590

3613

3533

3560

3508

3536

3499

02(ZgC)

2208

2210

2098

2131

2039

2078

1991

2032

2008C

W3(Z,+)

3556

3556

3462

3503

3408

3453

3386

3434

34 I5C

w4(q3)

785

813

635

788

578

752

528

716

624c

o5(&)

855

870

787

839

755

813

734

794

747c

97

95

84

90

77

83

75

81

71d

236

23 1

199

194

191

186

186

182

17Sd

A3 (km mol-1) A5

aA6 initio results are taken from Ref. [248] with permission. bFrom Ref. [190]. CFrom Ref. [249]. dFrom Ref. [208].

G

ii

3

.4B INITIO IR INTENSITIES

187

IV. C O N C L U S I O N S Extended basis set ab initio calculations of HF/SCF levels are needed to obtain vibrational absorption intensifies that are in a qualitative accord with experiment. Polarization functions on all atoms are required as a minimum. With such basis sets weak absorptions are predicted as low intensity bands and strong bands usually correspond to higher ab mitio values. In contrast to other molecular properties such as geometry parameters and vibrational fTequencies, there is no definite tendency in overestimating or underestimating the respective quantities. Very large basis sets near the Hartree-Fock limit provide only partial improvement of predicted infrared intensities. Consideration of the effect of electron correlation is needed to arrive at predicted intensities comparable in quantitative terms with experimental values. Since the number of molecules treated in calculations accounting for large proportion of correlation energy is limited, definite conclusions as to what approach is best for quantitative IR intensity predictions are still to come. Analytical derivative methods for higher order perturbation theory approaches, configuration interaction treatment and, especially, coupled cluster theory, appear to be the best hopes. Whether such calculations would become a routine exercise is yet to be seen. Fortunately, the studies carried out show that the double harmonic approximation works quite well as far as ab initio intensity predictions are concerned.

This Page Intentionally Left Blank

CHAPTER 8

INTENSITIES

IN RAMAN

SPECTROSCOPY

Molecular Polarizability ....................................................................................

190

II.

Intensity of Raman Line ....................................................................................

199

11I.

Raman Intensities and Molecular Symmetry ...................................................... 205

IV.

Resonance Raman Effect ...................................................................................

V.

Experimental Determination of Raman Intensities ............................................. 211

207

A.

Absolute Differential Raman Scattering Cross Section of Nitrogen .......... 212

B.

Differential Raman Scattering Cross Sections of Gaseous Samples .......... 213

189

190

GALABOV AND DUDEV

I. M O L E C U L A R

POLARIZABILITY

When an electric field with strength f is applied to a molecule, an induced dipole ~t is created = ,x f .

(8.0

~t is an important molecular quantity called molecular polarizability. The electrons of the molecule oscillate in accord with the external field applied, thus becoming a source of a secondary radiation which is the scattered radiation. In the general case, the vectors Ix and f do not coincide and Eq. (8.1) can be written in the following form , x = a x x fx + a x Y fv + ~ x z fz

(8.2)

~Y = ~YX fx + ~YV fv + ~ v z fz ~ z = ,xzx fx + a z v fY + a z z fz or

~y

= Otyx

Otyy

Otyz

fy

~z

~zx

~zY

Cxzz

fz

(s.3)

In Eqs. (8.2) and (8.3) IXj and fj (J = X, Y, Z) are the Cartesian components of IX and f, respectively, in a space-fixed coordinate system and tXjK (J, K = X, Y, Z) are the elements of the second rank polarizability tensor or. ct is a symmetrical tensor [CtjK = CtKj (K~J)] and, therefore, has six independent components only. The elements of ct may, in the general case, all be different from zero. The components depend on the space orientation of the molecule but not on the electric field applied. As is known, for each symmetrical tensor a special set of Cartesian axes x', y' and z' exists, such as in respect to which the tensor assumes a unique diagonal form. a acquires the following structure: Otx,x, 0t=/ 0

0 r 0

0 / 0 .

(8.4)

Otz'z'

The induced dipole is parallel to the external field vector. The quantifies ~x'x', Oty,y,and Otz,z, are called principal values of the molecular polarizability, and the axes x', y' and z'

INTENSITIES IN RAMAN SPECTROSCOPY

191

Y

Io

/ ~

TTT

X

Direction of propagation

;I•

/••ection

of observation

Fig. 8.1. Principal scheme of scattering experiment

principal polarizability axes. In the general case, ax'x' # Oty,y, ~: Otz,z,. If the molecule possesses isotropic (spherical) polarizability, it is evident that ax,x, = o~y,y,= Otz,z,. For molecules with cylindrical symmetry with z' parallel to the main symmetry axis, Otx,x, = ay,y, ~: Otz,z,. The tensor a is characterized with two invariants with respect to any reorientations of the molecule in space. These are the mean polarizability ~ and the anisotropy 7. and y are defined as = ( a x x + a v y + otzz) / 3

(8.5)

v2 = [ ( a x x - a Y V ) 2 + ( a v Y - a z z ) 2 + ( a z z - a x x ) 2

+ 6 (otXy2 + otXZ2 + a y z 2 ) ] / 2 .

(8.6)

It can easily be seen that ~/vanishes for molecules possessing spherical polarizability. In ordinary scattering experiments a linearly polarized light is usually used to illuminate the sample. The incident beam propagates along one of the laboratory-fixed Cartesian axes (X in our case; Fig. 8.1) and the scattered light is collected in direction perpendicular to that of the incident beam (Z axis). The excitation light may be linearly

192

GALABOV AND DUDEV

polarized along the Y axis. The scattered light may then be depolarized in some extent and contain radiation polarized either parallel (with intensity Ill) or perpendicular (with intensity I_L)with respect to the polarization of the incident beam. Since the intensity of scattered radiation is proportional to the squares of the respective polarizability components [4,5,155,250,251], in accordance with the convention introduced, we have Iii ~ a y y 2 (8.7) Is

~Xy 2 .

Evidently, the intensity of the total scattered light IT = Ill + I L will depend on both a y y and a x y components of the polarizability tensor:

(8.8)

IT -~ (t~Xy2 + t~yy2) .

In gases, the molecules are free and randomly oriented with respect to the space-fixed set of Cartesian axes X, Y and Z. Since all molecules contribute to the intensity of the scattered radiation, it is necessary to average the components of a over all possible orientations of the molecule-fixed axes x, y and z with respect to the inertial (laboratory) Cartesian frame. In analytical form this condition reads [4,5] ajK = ~ ajk tjj tKk. j,k

(8.9)

In Eq. (8.9) tIi (I = X, Y, Z; i = x, y, z) are the direction cosines between the two Cartesian coordinate frames. With respect to the principal axes of the molecule, Eq. (8.9) transforms into (g = x', y' and z') t~jK -- ~ r g

tjg tKg .

(8.10)

Squaring and averaging operations executed upon Eq. (8.10) result in [4]

/2 (OtIK)2 =

tXg tJg tKg (8.11)

= E 0:2 t2g t~g + E egg r g g
tJg tKg tJr tKr.

INTENSITIES IN RAMAN SPECTROSCOPY

193

Since ag remains unchanged under the averaging process, the expressions t2g t ~ and tjg tKg tJr tKr need only be averaged. These are found to be [4]

t~gt

2g

{1/5 if J =K = 1/15 if J ~ K

(8.12)

:/15ifJ=K tjg tKg tjr tKr = _ /30 if J r K

(8.13)

Application of Eqs. (8.11), (8.12) and (8.13) to axY and ayy components of the polarizability tensor results in

:(3go+2g
(8.14)

Introducing the invariants of the molecular polarizability ~ and y into Eqs. (8.14) leads to 45 ~2 +4T2 ~ = 45 O~2 = 3,Y2 45 45 ~2 + 7 T2 t~ = 45

(8.15)

An important observation in every scattering experiment is the depolarization ratio p. By definition p is I• P =~Ill

(8.16)

194

GALABOV AND DUDEV

Combining Eqs. (8.7), (8.15) and (8.16) yields

P

=

3T 2

(8.17)

45~2 +4.t,2 "

It should be emphasized that the ~2 and ~,2 coefficients in Eq. (8.17) are evaluated for the case of a polyatomic non-linear molecule, linearly polarized incident fight and rectangular experimental setup as shown in Fig. 8.1. Alternative sets of coefficients are to be used for experiments with unpolarized (natural) exciting light and different experimental geometries. These are summarized in Table 8.1. The polarizability terms for 27 most commonly employed scattering geometries [252] are presented. Different coefficients are used in the case of diatomic and linear polyatomic molecules as well [253] 45 ~2 +72 45 iX2 = (3/4)T 2 45

(S.lS)

45 ~2 + (7 / 4)T 2 45 The changes in molecular polarizability during vibrational transitions determine the intensities of Raman lines. The ~ ( polarizability matrix element for a transition from a vibronic state m to a vibronic state n can be presented as follows [4,254-256] (Ct/K)mn = f ~

Ot/K Wm dl;= (n

l ajK I m) (8.19)

h

Ve -Vm - v 0

M(J)em M(K)ne ) . Ve - Vn + v0

In Eq. (8.19) ~m and ~n represent vibronic wave functions of states m and n, respectively, e denotes an intermediate state of the undistorted molecule, v 0 is the frequency of the incident light and h is the Planck's constant. M(J)i 1 and M(K)i I (i, l= m, n, e) are the respective transition moments ( i I ~ I 1 ) and ( i I ~tk I 1 ), with ~ and ~tk the components of the dipole moment operator. It is seen from the above relation that the polarizability of the molecule is frequency dependent through the denominators (% - vm - vo) and (% - v n + v0). It has been shown [255-258] that when the frequency of

INTENSITIES IN RAMAN SPECTROSCOPY

195

TABLE 8.1

Polarizability terms a 2 for the most commonly used scattering geometries a Incident fight Scattered light Polarizability term 2 45 • a i Direction of Direction of Direction of Direction of propagation polarization observation polarization Y X Y 4 5 ~ 2 + 4 y2 X (180 ~)

X

Z

Unpolafized

3~2

Y+Z

45 ~ 2 + 7 72

Y

X

3 ](2

(90 ~

Z

3 y2

X+Z

6 y2

Z

X

3 ](2

(90 ~

Y

45~ 2 + 4 ](2

X+Y

45 ~2 + 7 72

X

Y

3 ](2

(180 ~

Z

4 5 ~ 2 + 4 ](2

Y+Z

4 5 ~ 2 + 7 ~,2

X Z

3 y2 4 5 ~ 2 + 4 y2

X+Z

45 ~ 2 + 7 ](2

Z

X

3 ](2

(90 ~

Y X+Y

3 3,2 6,/2

X

Y

(45 ~2 + 7 ),2)/2

( 180~

Z Y+Z

(45 ~2 + 7 72)/2 45 ~2 + 7 ./2

Y

X

3 ](2

(90 ~

Z

(45~ 2 + 7 ](2)/2

X+Z

(45~ 2 + 13 y2)/2

Z

X

3]( 2

(90 ~

Y

(45~2 + 7 ](2)/2

X+Y

(45~ 2 + 13 ](2)/2

Y (90 ~

X

z

aReprmted from Ref. [252] with permission from Society for Applied Spectroscopy.

196

GALABOV AND DUDEV

exciting light v 0 approaches to, or coincides with, the fxequency of some electronic transition in the molecule, dramatic changes in the magnitude of molecular polarizability, and hence in the intensity of Raman lines, occur. The phenomena are known as preresonance and resonance Raman effects, respectively. The dependence of (aJK)nm on v0 becomes small and can be neglected in the case of ordinary (far-from-resonance) Raman experiment when the following conditions are met: (1) the electronic ground state of the molecule is nondegenerate, and (2) the frequency of exciting radiation v 0 exceeds by far the frequency of the Raman shift Vnm. At the same time v0 must be small as compared with any electronic absorption frequency of the molecule. These conditions are known as Placzek's conditions and in case these are satisfied, the Born-Oppenheimer approximation can be used to justify the separation of the vibronic wave function ~Pi =~F~ o~i vr .

(8.20)

We is the electronic wave function of the ith state and Wvr is the wave function describing the rotovibrational motion for state i. Within the above approximation the electronic wave functions can be excluded from further consideration. Thus, the rotovibrational wave functions of the electronic ground state will only be treated. The left-hand side of Eq. (8.19) assumes the form (O~JK)nm = [ (Wvr) * ~jK Wvr d'l:vr. Rotational and vibrational distortions in the molecule can also be treated separately which leads to a fitnher splitting of the wave functions: Wvr = ~i v .~Fir '

(8.22)

where superscripts v and r stand for vibrational and rotational wave functions, respectively. Eq. (8.21) transforms into (Ot/K)mn =~ (~Fn)* ( ~ r ) * ( Z j K ~ v ~ F r d f f d z r .

(8.23)

In order to assess the contributions of the two wave functions ~i v and ~Fir to the intensity of the Raman line, a molecule-fLxed Cartesian coordinate system (x, y, z) which is able to rotate together with the molecule is necessary to introduce. The CqK can then be expressed in terms of polarizability tensor components in the molecule-fixed coordinate

INTENSITIES IN RAMAN SPECTROSCOPY

197

system and the respective direction cosines between the two Cartesian coordinate systems. Combination of Eqs. (8.9) and (8.23) leads to (OqK)mn = ~ ~ (~FV)* (~Fr) * Otjk tjj tKk Wv ~Fr dxV dxr j,k =Z ~ j,k

(~r),

tjj tKk ~ r dxr I

(,,.i.,#),O~jk,.i.,vdxV

(8.24)

Thus, it is clear that the polarizability of a molecule, and hence the intensity of a Raman line, depend on the properties of the two integrals appearing in Eq. (8.24). These must not be zero if the transition m ~ n is to be Raman active. Examination of the first integral leads to the rotational selection rules which govern the appearance of rotational lines in the Raman spectra. These are discussed in detail elsewhere [4]. The properties of the vibrational integral

(".~")m,, =S (,IV)* '~k"I": <,.,:v

(8.25)

will be considered. For small molecular vibrations, and in case the Placzek's conditions (1) and (2) are satisfied, the polarizability of the molecule can be expanded in a Taylor series along the normal coordinates Qi

0~=0~ O+

~r

~176 0 Qr

~r

Z r,s

0

Qr Qs (8.26)

+ -- ~ ' ' Qr esQt + . . . . 6 r,s,t OQr~Qs~Qt 0 Here ot0 is the polarizability tensor in an equilibrium non-perturbed state and the subscript o in the polarizability derivative terms means that these are taken at the equilibrium geometry of the molecule. Combining Eqs. (8.25) and (8.26) and neglecting higher terms in the expansion (8.26), the following expression is obtained for the ith normal mode

(",~)~-(",~)o s (~v), ~v ~v +(c)(Zjk/c)Qi)0 S (~Fv)* Qi ~v dl:V.

(8.27)

198

GALABOV AND DUDEV

The vibrational wave functions ~Pmv and ~pv can be presented as a product of the wave functions of the individual normal coordinates tpv =WVl (QI) ~Pv2 (Q2) --. ~PV(3N_6)(Q3N-6)

(8.28)

~pv =~PnVl(Q1) ~PnV2(Q2) "--~Fn(3N_6)(Q3N-6) 9

(8.29)

Eq. (8.27) transforms to (O~Jk)mn = (O~jk)o {f [~nVl(Q1)]* tpv, (Q1) dQ1--X f [~pv (Q1)]* ~Pv i (Qi)dQi'" "}

(8.30) + (~O~jk/OQi )o {~ [Wnvl(QI)]* w v 1 (Q1)dQ1 "'" •

[~Pm (Qi)]* QiWVi(Qi)dQi-- } 9

The vibrational quantum numbers associated with vibrational levels mi and ni (i = 1 to 3N6, where'N is the number of atoms in the molecule) may be denoted as v mi and v ai, respectively. The first term in the fight-hand side of Eq. (8.30) is zero unless vml = v nl, v m2 = vn2, . . ., vmi = v n i , . . . , etc. because of the orthogonality of vibrational wave functions. In other words, this term does not vanish ordy for elastic light scattering when the molecule does not change its vibrational state. The first term in Eq. (8.30) is, therefore, responsible for the Rayleigh scattering. The second term that gives rise to an inelastic scattering will not vanish if: (a) the polarizability derivative (a~k//gQi)o is not zero (this condition requires ~k to change when the molecule undergoes a particular vibrational transition), and (b) the integral is not zero. In the framework of the harmonic approximation, the last condition is satisfied if the vibrational quantum numbers of the ith transition v mi and v ni only differ by a factor of +1 Av = v m i - v ni = +1 .

(8.31)

All other vibrational quantum numbers vml, vial, vm2, v n 2 , . . . , vm(i-l), vn(i-l), vm(i+l), vn(i+l), . . . , etc. must not change. Under far-from-resonance conditions there are six independent elements of the polarizability tensor: axx, axy, axz, ayy, O~yz and CZzz. Therefore, six equations analogous to Eq. (8.30) exist for each normal mode i. At least one of the six (CZjl0mn(j, k = x, y, z) components must be non-zero if the ith vibration is to

INTENSITIES IN RAMAN SPECTROSCOPY

199

be Raman active. According to Eq. (8.31) fundamental transitions between vibrational energy levels are only allowed. Due to anhannonicity effects, overtone and combination bands appear in the Raman spectra, though typically with lower intensity. In analyzing the anharmonicity effects, higher terms in the Taylor series expansion [Eq. (8.26)] need to be considered. The selection rule (8.31) was discussed in detail in Chapter 1 and we will not elaborate further here. The symmetry selection role will be discussed in Section 8.3. If the vibrational wave functions qJmiV(Qi) and q~niV(Qi) are substituted with the respective analytical expressions, the relation given by Eq. (1.29) (Chapter 1) is obtained. The first integral in Eq. (8.30) does not contribute to the intensity of a Raman band. The change in Ctjk due to vibrational transition i can, then, be expressed as (Ctjl0i = (0Ctjk/c3Qi)0 [b 2 (vi + 1)1 89,

(8.32)

bE=

(8.33)

where

h 8g2toi

is the zero-point amplitude.

II. I N T E N S I T Y O F R A M A N L I N E The intensity of ith Stokes-shifted Raman line is given by the relation [250,255, 259] 27x5 )4 2 Ii = c------32 ~- I0 Nvi (vo - v i X (~Jk)i j,k

(8.34)

where I0 and v 0 are the intensity and frequency of the incident light, respectively, v i is the vibrational frequency, c is the velocity of light, and Nvi is the number of molecules in vibrational state with quantum number vi. By combining Eqs (8.32) and (8.34) the following expression is obtained Ii = 27/1:5 32c4 I o ( v o - v i )4 Nvi (vi+ 1) b 29X (c)O~jk / t)Qi)20 9 j,k

(8.35)

200

GALABOV AND DUDEV

The selection rule for Raman intensities (b) discussed in the previous section requires that the vibrational quantum number changes by + 1 for the Stokes lines in the Raman spectra. This is, however, true for the fundamental lines, corresponding to transition 0--->1, and for the "hot bands" related to transitions of the type 1-->2, 2---~3, etc. These appear at the same frequency in the spectra as the fundamental bands. Thus, the frequencies associated with fundamental and "hot band" transitions are hardly discernible. All such transitions contribute to the intensity of a Raman band. A summation over all vibrational quantum numbers v i from zero to infinity is needed to account for all contributions to the intensity Ii. Following the considerations given in Section 1.1 [Eqs. (1.31) and (1.39)], the expression is obtained

Z N vi(Vi + 1 ) = ~ -i vi=0 vi=O

e(-hvi/kT)vi (vi + 1) (8.36)

N 1_ e-hvi/kT ' where N is the total number of scattering molecules per unit volume. Therefore, 27/I; 5 Ii=

32C4 IoN(v0-v i

)4

9b 2

1-

2

e_-~i/k T ~ (~0tjk/~Qi)0 . j,k

(8.37)

This equation can also be written in the following form Ii = I0 N o i ,

(8.38)

where 27~ 5 )4 b2 ffi = ~c--"--'~-32(v0 - vi 1 e-hVi/kT X (~~ j,k

2 (8.39)

-

is the total Raman scattering cross section for full solid angle 4~. If wave numbers are used instead of frequencies, Eq. (8.39) transforms into

ffi :

27/1:5 )4 32 (V0 -Vi

b2 2 - hc/kT X (OO~jk/oQi)0, 1 - e -vi j,k

(8.40)

where v0 and~i are the wave numbers of the incident light and of the vibrational transition, respectively. Since the number of scattering molecules N and the intensity of

INTENSITIES IN RAMAN SPECTROSCOPY

201

irradiation Io are quantifies that are difficult to measure experimentally, o i appears more appropriately observable than the Raman intensity Ii. The polarizability derivatives c3Ctjk/0Qiappearing in Eqs. (8.35), (8.37), (8.39) and (8.40) form a three dimensional tensor of size 3•215 For the sake of simplicity i t can be presented in rectangular form. Thus, the supertensor CtQ can be expressed as:

O~Q =

(

~)(Zxx/OQI OCtxy/()QI OCtxz/OQ1 ... ()~yy/t)Ql bCtyz/c)Q1 ... \symmetrical bazz / bQ1 ... (8.41) OCtxx / ~)Q3N-6 ~)Ctxy/ ~)Q3N-6 ~)(Zxz/ ~Q3N-6"~ ~O~yy/OQ3N_6 ~Ctyz/~Q3N_6] symmetrical 3azz / ~)Q3N-6

9

The last term in Eq. (8.40) containing elements of the supertensor aQ Can be expressed in terms of the invariants of the molecular polarizability ~ and ), with respect to normal coordinates: --' (ti

=

~

bet

bQi

=

-

l(bax x Oayy 0azz )

3 [ ~)Qi + ~)Qi

+

'

(8.42)

~)Qi

1 I(3axx t~Qi

+6

~,~i

/)Qi

OQi

~.~-"Qi ~)Qi )

OQi

21}

(8.43)

"

Unlike the equilibrium mean polarizabilities ~0 which are non-zero quantifies, their derivatives with respect to normal coordinates may equal zero. These quantifies vanish for non-totally symmetric vibrations. In such cases the anisotropic part of the tensor tZQ contributes only to the intensity of the Raman line. Introducing ~ and "/i' as well as the degree of degeneracy gi in Eq. (8.40) leads to [2601 27~ 5 )4 b2 [ 2 ,)2] t~i = 32 (V0-~i 1- e_~ihc/kT gi 3(~[) 2 +-~0'i 9

(8.44)

202

GALABOV AND DUDEV

Thus, the total Raman scattering cross section of the Stokes-shifted line into a full solid angle 4n is expressed in terms of derivatives of the molecular polarizability invariants with respect to normal vibrational coordinates. Only a portion of the scattered by the sample light is, however, collected. Usually the scattered radiation is detected into a fixed direction with a narrow collection angle. Another quantity, the differential Raman scattering cross section into a given direction (dt~/d~)i, is employed in Raman intensity measurements [254,260]. If the rectangular experimental setup, as shown in Fig. 8.1, is employed, the differential Raman scattering cross section is given by [260]

"~ i

24454 ( v 0 - v i ,4 l_e_~ihc/k T gi E45(~[ ,2 +7 (121 Ti -

(8.45)

The quantity (da/d~)i is called absolute differential Raman scattering cross section. It is often termed absolute Raman intensity as well. From Eq. (8.45) it is clear that (do/d.O)i depends on several factors such as ~0 and ~i and the absolute temperature T. In order to operate with comparable quantifies that are independent from the experimental conditions, the so-called "standard intensity" or "scattering coefficient" Si is commonly used [73,253,260-263] Si = gi [ 4 5 ( ~ ) 2+ 7(Ti')21 -

(8.46)

Eq. (8.45) assumes the following form

do) = "~ i

b2 24 g4 )4 Si " 4-'--~ (T0 - v i 1_ e_Vi-hc/kT

(8.47)

Another experimental observable in the Rarnan intensity experiment is the depolarization ratio Pi of the vibrational line 3(~) 2 Pi

4 5 ( ~ ) 2+4(T~) 2 -

(8.48/

Pi provides a unique possibility for examining the symmetry of vibrational transitions. Since ~ vanishes for distortions belonging to a non-fully symmetric species, it easily can be realized that Pmax = 3/4. Depolarization ratio assumes its minimum value for the totally-symmetric vibrations of spherical top molecules. In these cases the Raman line is fully polarized and Ti' and Pi are zero.

INTENSITIES IN RAMAN SPECTROSCOPY

203

By combining Eqs. (8.46) and (8.48) expressions for the absolute values of ~ and ~'i' are obtained:

I :1=

1

Si0-40i)] 1/2

3g (iu

SiPi 11/2 ['f:[=

3gi(l+pi)J

(8.49)

(8.50)

As in the case of dipole moment derivatives with respect to normal coordinates, absolute values of ~ and )'i' only can be evaluated from the experiment. Extended basis set ab mitio MO calculations of polarizability derivatives are usually employed in solving the sign ambiguity problem. Eq. (8.45) expresses the dependence of the absolute Raman differential cross section on ( ~ ) 2 and (~'i')2. As already mentioned, the scattered light consists of two components: III -~ 4 5 ( ~ ) 2 + 40,i')2

(8.51)

I_1_-~ 3('/i')2 .

(8.52)

and

These quantifies can be measured separately. Thus, the so-called trace and anisotropy spectra can be obtained from experiment [264]. The trace spectra are, as a rule, simpler and are due to fully symmetric vibrational transitions only. The lines are narrow, well separated and easy to analyze. Anisotropy spectra originate from non-totally symmetric vibrational distortions in the molecule and usually represent a complicated superposition of strongly overlapped bands. The interpretation of spectra in the gas-phase is also hampered by the presence of a complex rotational structure of the vibrational bands. Trace (It) and anisotropy (Ia) spectra can be obtained after careful polarization experiments using the following equations [264]: It = Ill - (4/3) I_L

(8.53)

Ia = (7/3) l_t- .

(8.54)

204

GALABOV AND DUDEV

Thus, the respective absolute trace and anisotropy Raman differential cross sections are functions of a single polarizability invariant derivative only (-~)

i

=2484 ( V o - v i ) 4

b2 (~)2 1 - e -'~i'ihe/kT

(8.55)

drya ) = 7 2484 (Vo-Vi) 4 b? )2 d~Ji 4-5 1_ e-~i hc/kT gi()'~ "

(8.56)

Equations (8.45), (8.55) and (8.56) are derived for the CGS system of units" (d6/d.Q)i is expressed in cm2/sr, the molecular polarizability ct in cm 3, 0~OQi in cm2/g 89 and the scattering coefficients Si in cm4/g. Martin and Montero [263] first suggested using SI units in Raman intensity measurements. In adapting Eq. (8.45) to the new system of units the factor (47re0)2 has to be introduced in the denominator of the equation where to = 8.8542• C V -1 m -1 is the permitivity of vacuum (C - Colomb, V - Volt). Following the considerations given in Ref. [263] Eq. (8.45) can be further simplified if a dimensionless normal coordinate qi instead of the mass-weighted Qi is used [263] (8.57)

qi = (47t 2 c ~ i / h ) 8 9

where ~i is the harmonic vibrational wavenumber. Eq. (8.45) transforms into [263]

(do/

82

(~0 -~i) 4

" ~ i =9~"~'2 1-e_~i-ffc~ T gi

[45(~)2 + 7(~/~)2

].

(8.58)

In Eq. (8.58) (da/d.Q)i is expressed in m2/sr, ct and t3ntx/aqin in C V -1 m 2 and Si in C 2 V -2 m 4. The conversion factors between different unit systems for these quantifies are as follows 1 cm2/sr = 10--4 m2/sr 1 ~3 = 10-24 cm 3 = 1.112644• 10--40 C V -1 m 2 .

(8.59)

The invariants ~ and ~i' represent combinations between the six independent components a~k/0Qi (j, k = x, y, z) of the polarizability tensor t:tQ [Eqs. (8.42) and (8.43)]. These quantifies can be determined from experiment for very few small and highly symmetric molecules only. Typically, the number of intensity parameters to be evaluated exceeds by far the number of experimental observations. The XY 2 bent

INTENSITIES IN RAMAN SPECTROSCOPY

205

molecule is a good example in this respect. It has three normal vibrations, both infrared and Raman active, grouped as follows: FV = 2A 1 + B 2.

(8.60)

Suppose the molecule is oriented along its principal axes with x axis coinciding with the C 2 symmetry axis and z axis lying in the plane of the molecule. The supertensor CtQ has then the following structure:

Q1 o tZQ=

0

Q2 o

o

o

o

1 {Xzz

0

d.r

0

Q3 o

o

o

o

o

o

o.

2 Otzz

ct 3

0

0

(8.61)

~ik (i = 1, 2, 3; j, k = x, y, z) denotes the derivatives of the polarizability components ff-jk with respect to the ith normal mode. Since for such a molecule r • ctlyy ~ Ctlzz and O~Zxx ~ r , ~2zz, the number of independent polarizability derivatives is 7. The number of experimental quantifies, however, is just 5 ( ~ , ~'1', ~ , ~'2', ~'3'). It is evident that even for such simple and symmetric molecules the analysis meets serious challenges. In comparison, infi'ared intensities for C2v molecules can, in principle, be transformed into dipole moment derivatives since there is only one non-zero component of the dipole moment derivative vector for each normal mode. The great complexity in acquiring experimental polarizability derivatives as compared with dipole moment derivatives is probably the reason for the relatively limited use of Raman intensities in analyzing molecular properties.

III. R A M A N I N T E N S I T I E S A N D M O L E C U L A R

SYMMETRY

According to the harmonic oscillator selection rule, as briefly discussed in Section 8.1, fundamental transitions are only allowed in Raman spectra. In practice, it means that the fundamental lines will appear with higher intensity than the overtone and combination lines. In this section the relationship between molecular symmetry and the intensity of the Raman line will be considered. Eq. (8.25) shows that the intensity of each fundamental transition in the Raman spectrum is governed by integrals of the following type

206

GALABOV AND DUDEV

( n I ajk I m )

(j, k =x, y, z) .

(8.62)

Since, in the ordinary Raman experiment, the polarizability tensor has six independent components, six integrals with such a structure exist ( n l CXxxI m )

(nl~lm)

( n [ cxzzl m )

( n l ~ y I m>

(nl~lm>

( n I Cry, I m>

.

(8.63)

A vibrational transition is allowed if at least one of these integrals differs from zero. The symmetry selection rule states that the integral ( n I ~k I m ) is not zero if the direct product of representations for the ith vibrational mode F i = F n x F~k • F m

(j, k =x, y, z)

(8.64)

belongs to the fully symmetric species of the molecular point group. Since the ground state wave function is always fully symmetric, the behavior of the product F i will depend on the symmetry properties of the polarizability tensor components and of the vibrational wave function of the first excited state. Hence, if that wave function belongs to the same symmetry species as the respective polarizability component, the direct product F i will be fully symmetric. In this case, the ith transition will be Raman active. An interesting case are molecules possessing a center of symmetry. According to the symmetry selection rule, a given vibrational transition is infrared active ff the direct product between the two vibrational wave function representations and that of the dipole moment vector component is totally syaunetric (Section 1.2.2). The ground state vibrational wave function, as mentioned, is always totally symmetric. It belongs to even or gerade representation since it is not affected by inversion operation with respect to the center of symmetry. The dipole moment vector, however, changes sign when the inversion operation is executed. It, consequently, belongs to odd or ungerade representation. Thus, it is easy to deduce that the representation of the first vibrational excited state wave function must be odd (ungerade)(belonging to A u, B u, etc. synunetry species) if the fundamental transition is to be infrared active. The molecular polarizability tensor ct is defined in relation with two vectors: the induced dipole ~t and the electric field vector f [Eq. (8.1)]. Hence, the representation of ct can be expressed as a direct product of the representations of the respective vectors [265] F a = F~t • I f .

(8.65)

INTENSITIES IN RAMAN SPECTROSCOPY

207

Since the inversion operation alters the sign of the two vectors, according to Eq. (8.65), ct remains unaffected. It, therefore, belongs to an even symmetry species. It is evident that the excited state wave function must possess the same symmetry properties as ct if the transition is to be Raman active. Thus, for molecules with center of synanetry fundamental transitions to excited states belonging to even syrmnetry species (Ag, Bg, etc.) are only active. The above discussion outlines the so-called rule of mutual exclusion in vibrational spectroscopy which reads: for molecules with a center of synanetry vibrations that are Raman active are infrared inactive and v i c e v e r s a .

IV. R E S O N A N C E

RAMAN EFFECT

The relations and conclusions drawn in the previous sections were deduced for the non-resonance Raman experiment performed with frequency of the exciting light laying far from the frequency of any electronic transition in the molecule. In such a case the differential Raman scattering cross section (do/dD)i depends on the incident light frequency Vo through the term (v0 - v i ) 4 [Eqs. (8.45) and (8.58)]. If the Placzek's conditions (section 8.1) are satisfied, the dependence of molecular polarizability on Vo is negligible. In the treatment presented in the preceding sections, vibrational wave functions of the ground electronic state were only considered. If, however, the excitation radiation frequency approaches to Ore-resonance conditions) or coincides with (resonance conditions) the frequency of some electronic transition in the molecule, the application of the ground-state approach is not justifiable any more. The dependence of molecular polarizability on the incident light frequency has to be taken into account. As was already mentioned, under pre-resonance and resonance conditions some Raman lines are enhanced hundreds and thousands of times. In this section a brief outline of the basic theory of resonance Raman effect is presented. Before proceeding further, it is necessary to introduce the following notation: (a) the ground electronic state is denoted by g; vibrational levels belonging to this state are designated as i (initial vibrational state) and f (final vibrational state). The respective vibronic wave functions will be labeled as gi and gf; (b) the higher-energy electronic levels assume symbols r s and t; transitions g--,e and g--,s are considered allowed and g--,t - forbidden; and (c) vibrational levels belonging to e, s and t are denoted by v, I and u, respectively. In the subsequent derivations all electronic and vibrational states are considered as non-degenerate.

208

GALABOV AND DUDEV

The theory of intensities of resonantly-enhanced Raman lines is based on the Kramers-Heisenberg-Dirac dispersion equation [266,267]. The problem is analyzed in terms of vibronic interactions in the molecule. The JKth matrix component of the molecular polarizability for the gi~gf transition in the Raman spectrum can be expressed as follows [256,257]

(0:JK)gi,gf= E / evg:gi

Eev-Egi-E0

(evlMj ] E[g- i)eEgf (gflvM - E0 Klev)

(8.66) "

Mj and M K are the respective dipole moment operators and E 0 is the excitation light energy. In resonance conditions the first term in Eq. (8.66) becomes dominant since the denominator (Eev - Egi - E0) rapidly decreases. In this case, a phenomenological damping constant F is introduced in the denominator expression. Thus, retaining the resonant term in Eq. (8.66) only, we obtain

( .(ev[M-K!gi)(-gflMJ lev)) . )gi,gf = evg:gi E ~, Eev - Egi - E0 + iF

(8.67)

This equation can be further developed if the electronic wave function is expanded in a Taylor series along the normal coordinates Qa of the molecule. This is known as the Herzberg-Teller expansion [268] and its application to Eq. (8.67) yields [255,256] (ctji0gi,gf = A + B + C ,

(8.68)

A = ~ ~ (Mj)ge ( MK)eg (gf[ ev) (ev Igi) e~g v Eev - Egi - E0 + iF

(8.69)

where

INTENSITIES IN RA_MAN SPECTROSCOPY

X X

E

209

(gflQ~lev) (ev Igi)

~., ( (MJ)gshae(MK)eq

e~g v s~e a

E e - Es

Eev - Egi - E0 + iF (8.70)

(Mj)ge heas(MK)sg

E e ~ Es

C

_

Eev - Egi - E0 + iF

(M j )ge (M K )et h~g

(g~Qalev) (ev[gi)

Eg-Et

Eev - Egi - E0 + iF

_

~ ~v ~a e~g tag

(gfl ev)(eqQalgi) /

J

(8.71)

h~ (Mj)te (M K )eg Eg - E t

(gfl ev)(eqQ~lgi) / Eev - Egi - E0 + iF

J

i

In these equations (Mp)s~ = [~0] Mpl e0] (p = J, K; ~, ~ = g, e, s, t) are the electric transition dipole moments at the equilibrium molecular geometry and lk~a= [~0[ (aH/0Qa)ol so]. (aH/SQa)0 is the vibronic coupling operator for the normal mode a. H is the electronic Hamiltonian of the molecule. The contribution of each term A, B and C to the intensity of a Raman line is considered as follows. A term. One excited electronic state e is included in this term. With v 0 approaching v e the Raman line associated with a totally symmetric mode derives its intensity from this term through the Franck-Condon mechanism. This term does not depend on vibronic mixing of state e with other electronic states of the molecule. If the two electronic states g and e lie far below the next electronic levels and, hence, the probability of vibronic coupling between state e and the other excited electronic states is very low, the A term dominates in magnitude over the other two terms in Eq. (8.68). B term. This term comprises allowed transitions between the ground electronic state and two electronic states e and s (E e < Es). It determines to a great extent the intensity of a Raman line associated with a non-fully symmetric transition. In analyzing the properties of the B term the role of vibronic coupling in generating Raman intensifies becomes clear. Analysis shows that those normal vibrations which can couple two excited electronic states (e and s) undergo a striking intensity enhancement if the frequency of incident radiation is tuned on the frequency of the lowest electronic transition g---~. The nearer the state s to state e, the more pronounced is the effect of interaction. C term. It considers transitions to two electronic states e and t. Transition g~t is

not necessarily allowed. Since, in the general case, E e - E s < E g - E t, the magnitude of

210

GALABOV AND DUDEV

TABLE 8.2 Intensities of some Raman lines of pyrazine relative to the intensity of 940 cm-1 Raman line of benzene-d6 obtained at different excitation fight wavelengths (Reprinted from Ref. [274] with permission) Relative intensity a

Excitation wavelength

(in.m)

703cm-1 v 4 (b2g)

754cm- 1 v5 (b2g)

925cm -1 Vl0a (big)

514.5

0.38

0.041

0.046

457.9

0.47

0.082

0.12

363.8

0.57

0.20

1.14

aIn liquid phase.

this term will be small compared with the B term [Eqs. (8.70) and (8.71)]. Therefore, the contribution of C term to the overall intensity of the Raman line is not expected to be significant. This term is important for molecules possessing low-lying forbidden electronic states. Eqs. (8.68) through (8.71) were first derived by Albrecht [255,256]. His predictions have proved correct in many cases [269-277]. An example will demonstrate the role of vibronic coupling in generating resonance Raman intensities as derived in Albrecht's theory 03 term). It has been shown in a series of publications of Ito et al. [270, 271,274] that the Vl0a non-totally symmetric vibration of pyrazine (C-H out-of-plane bending) demonstrates a remarkable enhancement in Raman spectrum when the exciting radiation frequency approaches the frequency of the lowest-lying electronic transition at 323 nm. The Vl0a vibrational transition belongs to blg symmetry species and appears at the 925 cm-1 in the Raman spectnma. Results presented in Table 8.2 illustrate quite clearly the resonance effect on the intensity of this line. Analysis of the data accumulated reveals that the coupling between the lowest-lying allowed electronic state IB3u (n--+n*) and the next excited electronic state IB2u (rc---~*) through non-fully syuunetric big vibrational mode is responsible for the enhancement of 925 cm-1 Raman line. Low energy separation between the two excited electronic states, as seen from Fig. 8.2, favors this process. Vibronic coupling between IB3u and the two 1Blu electronic states will be less effective due to the higher energy separation. Therefore, vibrational modes (v 4 and v5) that are able to couple these electronic states are not expected to be significantly enhanced. The data collected in Table 8.2 illustrate this conclusion. Recently Okamoto [257] evaluated additional second-order terms to Eq. (8.68), thus expanding the applicability of Albrecht's theory. The emphasis is laid on the role of forbidden electronic transitions in generating resonance Raman intensities. It has been shown that: (1) the Raman line can derive intensity from a forbidden electronic transition

INTENSITIES IN RAMAN SPECTROSCOPY

I

211

60700

IBlu (n-~r*), IB2u (~-->x*)

50900

1Blu (n--~r*)

37800

1B2u (a:-~*)

30900

1B3u (n-~x*)

z:

0

Ag

Fig. 8.2. Energy diagram of the low-lying electronic states of pyrazine.

even if the excitation radiation is tuned in resonance with an allowed electronic transition, and (2) enhancement may be detected if the excitation is in resonance with a vibronically allowed, though electronically forbidden, transition.

V. E X P E R I M E N T A L D E T E R M I N A T I O N OF RAMAN INTENSITIES Eq. (8.45) shows that for an ordinary Raman experiment the absolute differential Raman scattering cross sections can be expressed in terms of derivatives of the molecular polarizability invariants ~ and V with respect to normal coordinates. These derivatives contain valuable information about the variation of molecular polarizability with vibrational motion. Gas-phase Raman scattering cross sections are most suited for intensity analysis since at low partial pressure of the sampling gas these quantities are not influenced by effects of intermolecular interactions, thus reflecting properties of individual molecules.

212

G.M.,ABOV AND DUDEV

Direct determination of absolute Raman differential cross sections is quite difficult and tedious work often leading to incorrect results. It is easier to measure cross sections relative to some standard. The absolute differential Raman scattering cross sections of the sample can then be straightforwardly obtained.

A. Absolute Differential Raman Scattering Cross Section of Nitrogen The absolute scattering cross section of the Q-branch of the Raman band of nitrogen at 2331 cm-1 has been chosen as a standard in Raman intensity experiments. The special role of this molecule in gas-phase Raman spectroscopy is based on several reasons: (1) nitrogen is a relatively inert gas and does not react with the gas sample; (2) it quickly forms mixtures with the sampling gas; (3) the region of the vibrational specmma where the nitrogen Raman line appears is very low populated and, for many gases, this line does not overlap with sample gas lines; and (4) since the absorption band of nitrogen lies in the far ultraviolet, laser excitation beams with wavelengths corresponding to near ultraviolet or visible light do not cause resonance enhancement of the nitrogen Raman line. Its intensity depends on the excitation frequency through the term (V 0 - v i ) 4 only. Thus, the differential scattering cross section of nitrogen can be used as a standard for a wide range of excitation laser lines. The absolute differential scattering cross section of the standard needs to be determined as precisely as possible. A number of measurements has been performed over the past forty years [260,278-287]. The introduction of lasers in Raman spectroscopy and of computer processing of spectral data has improved highly the accuracy of Raman intensity measurements. As a result, the absolute differential Raman scattering cross section of nitrogen reported from different laboratories deviates within a few percent only [260,284,285,287]. One of the possible approaches in determining (dtr/d.Q)Q,N2 is to use the absolute differential cross section of the strongest purely rotational Raman line J = 1--->3 (J the rotational quantum number) of hydrogen at 587 em-1 as a standard. It is given by [260] __~] rot,H2

= 24 g4 )4 3(J + 1)(J + 2) 7,y20 4---~ (~0 -Vrot 2(2J + 3)(2J + 1) "

(8.72)

70 is the anisotropy at the equilibrium geometry. The rectangular experimental setup, as shown in Fig. 8.1, is considered. Both experiment and theory have provided a reliable value for the hydrogen anisotropy. Moreover, its dependence on the excitation wavelength has been thoroughly established. As a result, the absolute differential cross section of the hydrogen rotational line (do/d~)rot, H2 has been accurately determined. The estimated value has been employed in determining the absolute differential cross

INTENSITIES IN RAMAN SPECTROSCOPY

213

section of the Q-branch of the vibrational Raman line of nitrogen. The following value has been obtained [260] (do/d.Q)Q,N2 = (5.05 4- 0.1)x 10-48 (~0 - 233 lcm-l) 4 cm6 sr-1.

(8.73)

It can be applied for a wide range of laser excitation wavelengths coveting the visible as well as the near ultraviolet up to 330 nm [260].

B. Differential Raman Scattering Cross Sections of Gaseous Samples The differential Raman scattering cross section of the ith line of a gas sample relative to that of the 2331 cm -1 line of nitrogen is given by [260] (do

/ d.Q)i

(v0 - vi )4

2331 cm -1

(~0 - 2331 cm-1)4

Vi (1- e-~ihe/kT)

~_

(do / m)Q,N 2

(8.74) gi [45(~)2 + 7gi ('Y[)2 ]

I45 (~)22

+7ZN2 (Ylq2

)2 I

"

In Eq. (8.74) li is the portion of the anisotropic scattering localized in the Q-branch of the respective Raman line. Its value has been determined for a number of molecules. In the case of linear molecules with small rotational constants gi is 0.25 [280,281]. Since the Bolzmann factor for the nitrogen molecule is very small at room temperature, it has been neglected in deriving Eq. (8.74). It is seen fxom the above expression that the relative differential scattering cross section depends on the excitation light wavenumber and on the absolute temperature. To make the measured quantifies comparable, a relative normalized differential Raman scattering cross section Ei has been defmed [260] (d~ / d'Q)i

.

(~0 -Vi) -4

Ei = (do / dn)Q,N2 (% _ 2331 r

(1-e-Vihr (8.75)

2331 cm-1 Vi

gi [45(~) 2 + 7gi (,y~)2] 7%N2('~N2

]

214

GALABOV AND DUDEV

The integrated intensity of the ith Raman line of a gaseous sample for a rectangular experimental setup can be expressed as I i = I0 p (do/d.O)i ,

(8.76)

where p is the partial pressure of the sampling gas. Combining Eq. (8.76) with the respective expression for the integrated intensity of the 2331 cm -1 Raman line of nitrogen

IN2 = IO

(8.77)

(da/d ) z

the following relation is obtained Ii

~2

=

(do/d.O)i

P

(do/d.O)N2

PN2

.

(8.78)

The relative differential Raman scattering cross section of the ith line of the sample can, therefore, be determined experimentally by employing the expression: (do / d.Q)i (do / d.O)N2

Ii

IN2

PN2 P

(8.79)

The absolute differential Raman scattering cross section of the ith line of the sample can be obtained l~om the relative value by using the absolute scattering cross section of nitrogen as given in Eq. (8.73).

CHAPTER 9

PARAMETRIC FOR INTERPRETING

MODELS RAMAN

INTENSITIES

Rotational Corrections to Polarizability Derivatives .......................................... 216

H.

A.

Zero-Mass Method ................................................................................... 218

B.

Heavy-Isotope Method ............................................................................. 219

C.

Relative Rotational Corrections ................................................................ 223

Valence-Optical Theory of Raman Intensities .................................................... 223 A.

Theoretical Considerations ....................................................................... 224

B.

Valence Optical Theory of Raman Intensities: An Example

C.

Compact Formulation of VOTR ............................................................... 235

D.

Compact Formulation of VOTR: An example of Application ................... 239

of Application ..........................................................................................

232

llI.

Atom Dipole Interaction Model (ADIM) ........................................................... 245

W.

Atomic Polarizability Tensor Formulation (APZT) ............................................ 249 A.

V.

APZT: An Example of Application .......................................................... 253

Relationship Between Atomic Polarizability Tensors and Valence Optical Formulations of Raman Intensities ..................................................................... 258

VI.

Effective Induced Bond Charges From Atomic Polarizability Tensors .............. 261 A.

Theoretical Considerations ....................................................................... 261

B.

Applications .............................................................................................

C.

Discussion of Effective Induced Bond Charges ........................................ 266

215

263

216

GALABOV AND DUDEV

As was shown in Chapter 8, the experimental gas-phase differential Raman scattering cross sections are directly related to the molecular polarizability derivatives with respect to normal coordinates forming the supertensor r [Eq. (8.41)]. In intensity analysis the 0oJ0Qi derivatives are usually further transformed into different types of parameters. The eventual goal is to transform the experimental observables into molecular quantifies reflecting electro-optical properties of simple molecular sub-units. Several formulations for parametric interpretation of Raman intensities have been put forward. In this chapter the basic principles and characteristics of the theories developed will be discussed. The mathematical formalism inherent of each theoretical approach will be illusu'ated with examples. It should be emphasized here that the models for analyzing Raman intensities have seen limited appLication, especially if compared with infrared intensity theories. Several factors are responsible for the relatively slow progress in the field of Raman intensity analysis: (1) Polarizability is a second-rank tensor quantity with six independent components in far-from-resonance conditions. Thus, the number of intensity parameters, if compared with those in the infrared, is doubled. This leads to indeterminacy of the inverse electro-optical problem for even very small and symmetric molecules. (2) Relatively few reliable experimental gas-phase Raman intensity data are available to date. (3) Analytical derivative, expressions allowing efficient and reliable evaluation of molecular polarizability parameters, were introduced in ab mitio quantum mechanical calculations few years ago. The impact of ab mitio methods in Raman intensity analysis is still to come.

I. ROTATIONAL CORRECTIONS TO POLARIZABILITY DERIVATIVES The transformation of vibrational intensities in Raman spectra into molecular parameters involves several calculation stages. An essential initial step is the reduction of intensity data to polarizability derivatives with respect to symmetry vibrational coordinates. As pointed out in previous chapters, the inverse electro-optical problem of vibrational intensities can be performed with success only for molecules possessing sufficient symmetry. The transformation between O~OQi and O~/OSj derivatives is carried out with the aid of the normal coordinate transformation matrix L S according to the expression:

PARAMETRIC MODELS OF KA_MAN INTENSITIES

cO(x/cT~Sj = E

217

(&~cOQi)L~l ,

(9.1)

i

or in a matrix form: as

:aQ

(9.2)

L-~~.

a s is a supertensor with dimensions [3x3(3N-6)] and has the following structure: 3ctxx/t)S1 (zS=

t)ctxy/t)Sl Octxz/t)S1 t)ctyy / t)Sl t)ctyz / t)S1

L syaunmeWical

ONtzz / ~$1

.-. ... ...

(9.3) Octxx / t)S 3N-6 ~)ctxy / t)S3N_6 i)ctxz / ~)S3N-6 ~)ctyy / ~)S3N-6 ~)ctyz / ~$3N_6 symmetrical 0ctzz / ~)$3N-6

J

Polarizability derivatives with respect to symmetry coordinates obtained from Eqs. (9.1) and (9.2) are not always purely intramolecular quantifies since contributions from the compensatory molecular rotation accompanying some vibrations may be present. Such contributions arise in the cases of non-totally symmetric modes of molecules having a non-spherical polarizability eUipsoid. Polarizability derivatives corrected for contributions from molecular rotation can be obtained according to the relation ctSc~

aS - PS ,

(9.4)

where PS denotes the tensor comprising the rotational correction terms to the polarizability derivatives. It has the same structure as a S. The rotational correction to a /~/i3Sj derivative is a 3x3 matrix given by the following expression [35] psj : wj • a 0 - a o • w j .

(9.5)

wj is the compensatory rotation arising when a molecule undergoes particular vibrational distortion, and a 0 is the static molecular polarizability tensor. If the vector wj is presented in a pseudo-tensor form [Eq. (3.7)], Eq. (9.5) can be rewritten as psi : ~ ((wj)) - ((wj)) a0.

(9.6)

The absolute compensatory molecular rotation can be evaluated, as already discussed in Section 3.II.A, by employing the hypothetical isotope approach [34-36]. The hypothetical species obtained by setting the masses of some appropriately chosen atoms equal to zero [35,36] or weighted by factors of 1000 or more [34] are incorporated in the

GALABOV

218

A N D DUDEV

calculations. These isotopes have negligibly small compensatory rotation. A formulation adapting the hypothetical isotope method in evaluating rotational corrections to polarizability derivatives was recently put forward by Dudev and Galabov [288]. The elements of wj can be derived according to Eq. (3.9) (Section 3.H.A). Thus, combining Eqs. (3.9) and (9.6) we obtain PSj

=

tx0 ((~ ~ A j

A j )) r 0

(9.7)

.

The superscripts r and a stand for the reference and actual molecule, respectively. More details about the matrices 13 and A are given in Chapters 2 and 3. Some examples of application of the hypothetical isotope approach in deriving rotational corrections to polarizability derivatives are given below.

A.

Zero-Mass Method

Within the zero-mass approach the reference hypothetical isotope contains atoms with zero masses. Typically, the respective atoms do not lie on the main molecular symmetry axis. As an example, the CH3C1 molecule will be considered. In this case, it is appropriate to set all hydrogen-masses equal to zero. Thus, the C-CI bond will maintain fixed direction during vibrational motion and this hypothetical isotope species will have negligibly small compensatory rotations. Methyl chloride has Raman-active vibrations belonging to A1 and the doubly degenerate E, symmetry species. The E-vibrations are non-totally symmetric and contain contributions from compensatory molecular rotation. Geometry parameters, definition of symmetry coordinates and the orientation of the molecule in Cartesian space are given in Table 3.8 and Fig. 3.7. The equilibrium molecular polarizability tensor of CH3C1 employed in the calculations has the following form [289]:

tt 0 =

/4i o Oo/ 4.03 0 558

A3 .

(9.8)

Application of Eq. (3.9) yields (in units of A -1 or rad-1) S4a

S5a

S6a

~P a / 0 0 0 Aj = 0.038 0.037 -0.070 0 0 0

S4b

S5b

S6b

0.038 0 . 0 3 7 - O ! 7 0 / x 0 0 y 0 0 z

(9.9)

PARAMETRIC MODELS OF RAMAN INTENSITIES

219

a

After Iransforming the vector wj = [~ A j into pseudotensor form and applying Eq. (9.7) for each symmetry coordinate, the PS array is obtained (in units of A2 or A3 rad-1)

(

L

~

0.059

0 0 0

S4a 00.059 0 0 0 0

0 0 0.057

S5a 00.057 0 0 0 0

0 0 -0.108

S6a 0-0.108 0 0 0 0 (9.10)

S4b 0 0 0 -0.059 -0.059 0

no

0 0 0

S5b 0 0 0 -0.057 -0.057 0

0 0 0

S6b 0 0 "~ 0 0.108 0.108 0

J

Heavy-Isotope Method

The heavy-isotope approach to evaluate rotational contributions to polarizability derivatives [288] will be illustrated with calculations on a series of molecules consisting of acetonitrile (C3v symmetry), dichloromethane (C2v symmetry) and acetone (C2v symmetry). Structural parameters and polarizability tensors employed in the calculations are summarized in Table 9.1. Since the axes of the Cartesian reference systems (Fig. 9.1) are chosen to coincide with the respective inertial axes, the static polarizability tensors acquire simple diagonal form. The symmetry coordinates corresponding to vibrations which may contain contributions from compensatory molecular rotation for the three molecules are given in Tables 9.2, 9.3 and 9.4, respectively. The following heavy isotopes are employed: 1. Acetonitrile: The masses of the two carbon atoms for acetonitrile are multiplied by a factor of 104. 2. Dichloromethane: The dichloromethane molecule possesses C2v symmetry and has two groups of vibrations belonging to B 1 and B2 symmetry classes that contain contributions from compensatory molecular rotation. Two heavy isotopes are created in this case: C*H2C12" in evaluating rotational corrections to B 1 class (weighting factor of 103); and C*H2*CI2 in the case orB 2 vibrations (weighting factor of 106). The asterisks mark the heavy atoms in the isotopes. 3. Acetone: The masses of oxygen and central carbon atoms for acetone were weighted 104 times. Rotational correction terms to polarizability derivatives for the three molecules obtained by following the procedure described above are given in Tables 9.2, 9.3 and 9.4, respectively.

220

GALABOV AND DUDEV

xI,'

A

R1 a 2 (~C

@

R2

132

I

Z

~

R1

R2

S

131 RI rl

B

C

R2

r4 C

q

Fig. 9.1. Cartesian reference systems and definition of internal coordinates for (A) acetonitrile, 03) dichloromethane and 03) acetone.

PARAMETRIC MODELS OF R A M A N INTENSITIES

221

TABLE 9.1 Structural parameters and gas-phase static polarizabilities for acetonitrile, dichloromethane and acetone Molecule

Geometry

axx

0%

CZzza,b

(A 3) CH3CN

r c c = 1.460 A,

rCN = 1.158 A,

3.98

3.98

5.49

rCH = 1.092 A,
rCH = 1.068 ~ r c c 1 = 1.7724 A,
5.36

8.81

6.30

CH3COCH 3

r c c = 1.515 A, r c o = 1.215 A,

7.16

4.88

7.14

rCH = 1.086 A,
T A B L E 9.2 Symmetry coordinates and rotational correction terms to polarizability derivatives with respect to symmetry coordinates for acetonitrile (Reprinted from Ref. [288] with permission of John Wiley & Sons, Ltd., Copyright [1993] John Wiley & Sons, Ltd.) Symmetry coordinate a E'

E"

Rotational correction (A 2 or A 3 rad - I )

S4a = (2At 1 - Ar 2 - Ar3)/x/-6

XZ

S5a = (2Act 1 - Act2 - Atx3) /

XZ

0.051

S6a = (2A~l - a132 - AI33)/xf6

XZ

-0.096

S7a = A~x

XZ

0.739

S4b = (Ar2 - Ar3)/xf2

YZ

0.052

S5b = (Atz2 - ao~3) /

YZ

0.051

S6b = (AI32 - AI33)/x/2

YZ

-0.096

S7b = A~y

YZ

0.739

alnternal e o o r ~

are defined in Fig. 9.1.

0.052

222

G A L A B O V AND DUDEV

T A B L E 9.3 Symmetry coordinates and rotational correction terms to polarizability derivatives with respect to symmetry coordinates for dichloromethane (Reprinted from Ref. [288] with permission of John Wiley & Sons, Ltd., Copyright [ 1993] John Wiley & Sons, Ltd.) Symmetry coordinatea B1

B2

Rotational correction (A 2 or A3 r a d - l )

S 6 = (Ar 1 - ,~r2)/,r

XZ

-0.057

$7 = (Aal3 + AtZl4 - Atz23 - Atx24) / 2

XZ

0.198

S$ = (ZLRl - AR2)/,~/'2

YZ

0.509

59 = (Actl3 - AtzI4 + Act23 - Act24) / 2

YZ

2.547

alntemal coordinates are defined in Fig. 9.1.

T A B L E 9.4 Symmetry coordinates and rotational correction terms to polarizability derivatives with respect to symmetry coordinates for acetone (Reprinted from Ref. [288] with permission of John Wiley & Sons, Ltd., Copyright [1993] John Wiley & Sons, Ltd.) Symmetry coordinate a B1

Rotational correction (A2 or A 3 rad -1)

S13 = (zMR1 - zMR2)/.,~

XZ

S14 = (/M 1 + ,d~t"2 + zM"3 - ,d~"4 - zM'5 - ~r 6)/.,r

XZ

S15 = (2,d~rI - z~t"2 - ,du"3 - 2,~r4 + zXr5 + ,d~t"6 ) / . ~ S 16 = a (Act I + Ao~2 + Act3 - Ao~4 - Aa 5 - Ao~6)

xz XZ

0 0 --0.001 0

- b (A~I + All 2 + Alia - A l l 4 - A~5 - All6)

a = 0.292782,

B2

b = 0.284503

S 17 = (2AOtl - A~2 - A~3 - 2Ao~4 + Aa5 + A a 6 ) / ' ~

XZ

0

S18 = (2All 1 - All 2 - A~3 - 2A134 + All5 + A ~ 6 ) / x f i 2

XZ

0.001

S19 = (A# 1 - Ate2)/'42

XZ

0.011

S20 = (Ar 2 - Ar 3 + Ar 5 - Ar 6) / 2

YZ

0.052

S21 = (Act2 - At~3 + Atz5 - Atz6) / 2

YZ

0.052

$22 = (All 2 - Aft3 + Aft5 - All 6) / 2 SEa = (A1;1 - A X E ) / ~ b

YZ

--0.095

YZ

0.173

YZ

0.956

S24 = A0 b aDefinition of internal coordinates is given in Fig. 9.1. bx, torsion; 0, out-of-plane angle.

PARAMETR/C MODELS OF RAMAN INTENSITIES

C.

223

Relative Rotational Corrections

Rotational correction terms are mass-dependent quantities and are, therefore, different for different isotopes with identical symmetry. This affects the values of the respective polarizability derivatives with respect to symmetry coordinates which will also vary in the respective series. In such a case it is convenient to use the polarizability derivatives of a given (reference) molecule ~om the series as a standard as proposed by Escribano, et al. [71]. Thus, the intensity analysis is carried out in a uniform way. The polarizability derivatives of each molecule i from the series are related to those of the reference species through the equation [71 ] ot[ =ot~ -PS',

(9.11)

where superscripts i and 0 stand for the ith and reference molecules, respectively. PS' is an array containing rotational correction terms to a~aSj derivatives of the ith molecule taken relative to the reference species. PS' can be calculated according to Eq. (9.6) where the compensatory rotation vector wj is expressed as [71] w j = 1-1 ~ ~ ' (Am a / m 0) ( r a • Sk,a)(GO) -1 . k a

(9.12)

In Eq. (9.12) I is the inertial tensor of the ith isotopic derivative with respect to center of masses, r a is the position vector of the atom a in an inertial Cartesian coordinate system, Sk,a are the Wilson's s-vectors [4], G is the kinematic coefficients matrix [Eq. (2.15)] in a symmetrized basis, and Am a = m a - m 0 .

(9.13)

m a and m 0 are the masses of atom a in the ith and the reference molecule, respectively. The inner summation in Eq. (9.12) concerns those atoms which have been isotopically substituted. It should be pointed out that if a non-rotating (hypothetical-mass) isotope is chosen as a reference molecule, the procedure proposed could be used for calculating the absolute rotational correction terms to polarizability derivatives.

II.

VALENCE-OPTICAL

THEORY OF RAMAN INTENSITIES

The valence-optical theory of Raman intensities (VOTR) is the first comprehensive theoretical formalism for interpreting vibrational intensities in the Raman

224

GALABOV AND DUDEV

spectra. It has been introduced by Volkenstein [294,295]. A number of authors have contributed in developing the theory over the past fifty years [73,76,77,155,296-299]. In spite of problems encountered in its application, the valence-optical scheme is stiU the most commonly used theoretical approach in the field of gaman intensities.

A.

Theoretical Considerations

Each bond k of the molecule is assigned a polarizability tensor a(k) in a moleculefixed Cartesian coordinate system. The molecular polarizability tensor is then represented as a sum of the polarizability tensors of the constituent bonds 0t = X r k

(9.14)

Each bond polarizability tensor is a 3 • square matrix which, in the case of an ordinary Raman experiment, has six independent elements. If, however, a special set of local bond Cartesian axes is chosen, the bond polarizability tensor can be presented in a simple diagonal form. Thus, the number of parameters characterizing electro-optical properties of each molecular bond can be reduced to three. A local bond Cartesian system can be defined as follows: (1) One of the coordinate axes is directed along the bond; its unit vector is denoted by el(k); (2) The other two axes are chosen in such a way that they are perpendicular to the longitudinal vector and, at the same time, are at right angle one to another. Their particular orientation in space depends on the bond site symmetry. The respective unit vectors can be designated as e2(k) and e3(k). In the local bond coordinate system a(k) acquires the following form: a (a(k))o =

)

0

0

ot2(k)

0

0

iX3(k)

(9.15)

The indices 1, 2 and 3 correspond to the numbering of unit vectors as def'med above. At this stage important approximations are introduced. It is required that the unit vector el(k) lies always along the bond k and that the bond polarizability tensor preserves its simple representation when the bond participates in vibrational motion. These approximations are kept in order to reduce the number of intensity parameters. A generalized version of the valence-optical theory of Raman intensities which considers off-diagonal elements of the bond polarizability tensor has been discussed by Rupprecht [299].

PARAMETRIC MODELS OF RAMAN INTENSITIES

225

Combining Eqs. (9.14) and (9.15) and recalling the general rules for transforming second-rank tensors between two Cartesian coordinate flames, the following relation is obtained [ 155] (k) e2e(k) e3e(k) /~i~(k) e2~(k) e3~(k)

a=~

~.el~(k) c2~ (k) e3~(k) (9.16) Ctlik)

0 0 a2 (k) 0 0 a3 (k)

(ele k) el~ (k) el~ (k) e2e(k) e2~(k) e2~ (k) e3e (k) e3~(k) e3~(k)

~

or

/

~xx r (Xxz/ tXyy tXyz =

sym.

azz ) (9.17)

Ii=~lO~i(k)[eie(k)] 2 i=~ltZi(k)eie(k)ei~ (k) i=~ltZi(k)eie(k)ei~ (k) i~l ~i(k)[ei~(k)]2 i~l ai(k)ei~(k)ei~(k)

In Eqs. (9.16) and (9.17) ein(k) (i = 1, 2, 3 11= ~, ~, ~) are the direction cosines between the unit vectors el(k), e2(k) and e3(k) and the coordinate axes s, ~ and ~ of the molecule-fLxed Cartesian coordinate system. Two types of polarizability tensor elements exist: a diagonal Ot~ - ~ ~ cxi(k)[ei (k)] 2 , k i

(9.18)

and off-diagonal o ~ = ~ ~ a i (k) eie(k) ei~(k). k i

(9.19)

226

GALABOVAND DUDEV

Expressions (9.18) and (9.19) can be expressed in a more detailed form if the interrelations between direction cosines are used: [el~tk)]2 + [e2s(k)]2 + [e36(k)]2 = 1

(9.20)

ele(k) el~(k)+ e2~(k)e2~(k)+ e3~(k)e3~(k) = 0

(9.21)

The following expressionsfor the diagonal and off-diagonalelements are obtained [ 155]: Otez= ~ {O~l(k)+[Ot2(k)-txl(k)] [e2~:(k)]2 k

(9.22)

OCt= k~{[o~2(k)-Ocl(k)]e2e(k)e2~(k) + [r

(9.23)

txl(k)] e3e(k)e3~(k) }.

Differentiation of both sides of Eqs. (9.22) and (9.23) with respect to the ith normal coordinate yields: t)(X~i= ~k [IoOcl(k!()Qi+ I ~)Oc2(k)t}Qi()(~l(k) ] 2 ( ) Q1 [e2e(k) i OOt3(k) OOt'l(k)[e3e(k)]2+[~176 + ~}Qi ()Qi '

(9.24)

+[O~3(k)- O~l(k)]2e3E(k) Oe~eQ[)} . ~0~ fI00~2(k)OO~l(_k)] [00~3(k) OO~l(k)] (k) ~)Qi =k~ [L ~Qi ~-Qi e2e(k)e2~(k) + t)Qi ~(~i e3e(k) e3~ ] oc2(k)-Ocl(k)][e2~(k)()e2e(k) + e2e(k) ~)e2OQ~(k) i

(9.25)

~)Qi

-

Ol,k,l[e3,k,e3'k'e3'k'l} o-Q-ii + e3e(k) ~)Qi

"

PARAMETRIC MODELS OF RAMAN INTENSITIES

227

Taking into a c c o u n t the transformation between polarizability derivatives with respect to normal coordinates and polarizability derivatives with respect to internal vibrational coordinates

i-iz

Lli,

Eqs. (9.24) and (9.25) can be expressed in terms of bond polarizability derivatives with respect to internal coordinates. Thus, for a molecule with K bonds and M internal coordinates we obtain [ 155] 3Rl(l)

"~}0~l(l) ~)RI . . . . . .

3Qi

~)RM

[1, ... ,l] . . . . . . . . . . . . . . . . . . . . ~)0tl(K) ~)O~I(K) ~)R1

~)RM

~)R2(1) )2,

....

)2

3R1

0(t 1(1)

0s 2 (1) 3o~1(1)

~)R1 "'" 3RM

] ................................... 3~2 (K) _ 3R 1

3~1 (K) 3R1

3~2 (K) 9

o

bCt3(l) t)R1(l) +[(e3 (1))2,...,(e3E(K))2]

~)Rl

~

3RM _

bOt3(l) 3s t)R M

3~3 (K)

3(/.1(1)

3RM

~)RM

3R1

Ie2e(

~al (l)

3RM

t)R 1 "'" ~)RM

bo~3(K) ~O~I(K) . 3R l

~)RM

1) .... 0

+ 2[ct2(1)- O~l(l), . . . , (z2 (K) _ (xI(K) ]9 . . . . . . . . . . . L0 . . . . . . . . e2e

(K)

228

GALABOV AND DUDEV

i3e2e(1)

i)e2e(1)

aR 1

aR M

ae2e(K) aR 1

ae3e(l) OR1

. . . . . . . . . . .

acto~

aQ---~ =

ae3e(l) 3R M

I

(9.27)

4-

e3e(K)

ae3e (K) bR l

ae3e (K) aR M

[e2e(1)e2~(1) ,..., e2e(K) e2 ~(K) ]

~)O~2(l) t)Ctl(1) aR l X

+ 2[ot3(I)-o~ 1(1),... ,~3(K) _ ~I(K) ]

~)e2e(K) ,aRM .

e3e (1) .... 0 0 ........

.

aR 1

ao~2(1) a(gl (1) 9

.

.

aR M

aRM

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

~)Ol;2(K) i30~l(K) _ ~RI aR1

...

ao~2(K) OOtl(l) i)R M ,aRM

act30) aOtl(1)

+ 31(I) e3~(1), ... e

aRl

,e3e (K)

aR~

aRM

aRM

e3~(K)] ...................................

~)0~3(K) aCzI(K) bR l

+

atx3 (I) aO~l(I) ~

aR 1

o2K,_o,,K,] le2 l' LO........

a0~3(K) a0~l(K)

.

.

,

~ e2 ~(K)

aR M

aR M

PARAMETRIC MODELS OF RAMAN INTENSITIES ~)eEe(l) OR1 ~)e2e(K) ~)R1

be2E(1) ~R M ~)e2e(K) ORM

o~e2~,(l)

Oe2~,(1)

~)R1

~)RM

~)e2~(K) _ ~)Rl

i iii.~

229

"

0".'.'.'.'.... eze (K) 1

+ [0~3(1)- 0~1(1),... ,~3 (K) - O~I(K) ]

~)e2~(K) /)R M .

~)e3e(l) bR 1 0 . . . . . . . . e3~(Kj

+ [ e~ili':):''~

[0 . . . . . . . . e3e(K)

I +

~)e3e(K) ~)R1

Oe3g(1) ~R1 ~e3~(K)

/)e3e(l) ORM ~)e3e(K) ~)RM .

Oe3~,(i) ~RM

1 (9.28)

Oe3~(K) !

_ ~RI

~RM _

It is more convenient to use polarizability derivatives with respect to symmetry coordinates instead of derivatives with respect to normal coordinates. Recalling that internal vibrational coordinates are connected with synunetry coordinates through the orthogonal U matrix [Eq. (2.6)] and using more compact notation, we can write

230

GALABOV AND DUDEV

~

~)Sj { L~)RJ "

~

"t-

~

+

e31~ ~-R (9.29)

+ 2[r176

OR

+2[~3-C~l]E3eL OR

Uj

and

OSj

e2ee2~ OR

~)R + e3ee3~[ ~-R

~)R

(9.30)

+

[

- xl] E3g L

1

+r3 L

Uj

where Uj is the jth column of the transpose of the U matrix. Thus, changing indices for the molecular axes a set of linear equations are obtained. From these relations the bond polarizabilities ~ (i = 1, 2, 3) and their derivatives with respect to all natural coordinates 0cq/0Rl (i = 1, 2, 3; I = 1 to M) are to be evaluated. These quantifies are called electrooptical parameters. In the following the abbreviation reop will be applied (Raman electro-optical parameters). The other matrices appearing in Eqs. (9.29) and (9.30) can be evaluated employing molecular structural data. The formulation will be illustrated with an example of application in the succeeding section. A serious problem in solving the set of linear equations appears; however, there are too many reop to be determined from much less experimental data. Application of the group theory, thus, considering the symmetry properties of the molecule, si~ificanfly reduces the number of unknowns. The problem still remains undetermined. As is shown in the following section, the linear set of equations has not a unique solution even for such a small and symmetric molecule like SO2. The problem is much tougher compared with that in the infrared (Chapter 3). This is due to the complex nature of the polarizability tensors and their derivatives. For example, in solving the reverse electrooptical problem for ethane in the infrared, 13 reop have to be evaluated from 5 experimental data. At the same time, 30 reop have to be determined from intensity data for only six Raman active bands [300]. Various approaches are used to overcome the indeterminacy problem: (1) Parameters reflecting long-term interactions are neglected;

PARAMETRIC MODELS OF RAMAN INTENSITIES

231

(2) The theory is applied in zero-order approximation. All polarizability derivatives are constrained equal to zero except bond polarizability derivatives with respect to stretching of the same bond, i.e. &t(k)/0Rl ~ 0 for R 1= rk only, where rk is the stretching of the kth bond. These restrictions result in considerable reduction of the number of reop, although their physical justification is questionable. (3) Evaluating combination of reop instead of individual quantities; (4) Performing a least squares fitting of the set of reop. Calculated Raman intensities and depolarization ratios are compared with experimental observations and the set of reop ensuring the best fit is determined. Usually the best-fit procedure is carried out when Raman intensity data for large series of isotopically substituted molecules are available. Eqs. (9.29) and (9.30) can be further developed if explicit expressions for the matrices [3e2rl/3R] and 13e3rl/3R] (rl = E, ~) are introduced [ 155] ] = - E l f I (E 2 r -1 A A)

(9.31)

[3e3q/3R ] = - E1TI (E 3 r -1 A A).

(9.32)

[3r

E 2 and E 3 are matrices containing the respective unit vectors arranged in an appropriate way. r-l, A and A have the same meaning as explained in Chapter 3. We arrive at the final form of the VOTR formulas:

3Sj

[ [ {)R .] + e2e L 3R

3R

3R

3R

+ 2[al-a2lEleE2e(E2 r-I AA) + 2[~I-~3]EIeE3e(E3 r-1 AA)

= e2ee2~

+ e3ee3~

(9.33) Uj

L

+ [Ctl-~2I(EIEE2~ +EI~E2eXE2 r-1 AA)

+ [~~-~3] (EI~E3~+EI~E3e)(E3

r-1 AA) ) Uj.

(9.34)

232

GALABOV AND DUDEV

It should be noted that the zero angular momentum condition is automatically considered through the last two terms in Eqs. (9.33) and (9.34). The Eckart-Sayvetz conditions are implicitly introduced in the VOTR equations thus avoiding the necessity for correcting 0a/0Sj derivatives for rotational contributions. If the symbols [oq~] and [o~] are used to replace the expressions inside the brace of Eqs. (9.33) and (9.34), respectively, we can write 0~ = [txee] U j 0Sj

(9.35)

~

(9.36)

OSj

= [o~] Uj ,

or in a tensor form

t xx3 [ xy] aS=

[,yy]

ksymm

[tZyz] U=([tt])U.

(9.37)

[azz

In these equations each [oq~] (z, ~ = x, y, z) is a linear array of rank M.

BO

Valence Optical Theory of Raman Intensities: An Example of Application

In this section the general equations (9.33) and (9.34) of VOTR are applied in interpreting Raman intensities of SO2. As was pointed out in Section 8.11, the Raman intensity experiment for bent XY 2 molecules is not favorable in determining a complete set of molecular polarizability derivatives with respect to normal coordinates. That is why, in order to overcome the indeterminacy problem, a set of polarizability derivatives, with respect to symmetry coordinates for SO2 evaluated by means of ab initio MO calculations, is used [301]. The 0ct/OSj derivatives forming the a S matrix are computed by applying the numerical differentiation approach. Other entries needed in solving the problem are taken from the same source [301]. Structural parameters for the sulfur dioxide molecule are given in Table 9.5. The Cartesian reference system and definition of internal coordinates and unit bond vectors are shown in Fig. 9.2. The a S tensor employed in the calculations is as follows (in units of A 2 or A 3 rad -1) [301].

PARAMETRIC MODELS OF RAMAN INTENSITIES

233

TABLE 9.5

Structural parameters for SO 2 Geometry: a

rso = 1.432 A,

Static molecular polarizability:a c4.700 ot 0 =

A3

2.747 3.069

Symmetry coordinates b A1 B2

S 1 = (Ar 1 + Ar2)/,r S2 = Ar $3 = (Ar l _ At2)/~r

aFrom Ref. [301]. blntemal coordinat,~ are defined in Fig. 9.2.

1(2)~s

2(2)

e2(1)

Z

e

)

r2

Fig. 9.2. Cartesian reference frame, numbering of atoms, and definition of internal coordinates and of bond unit vectors for SO 2.

LOo (6.208

OrS=

S1 0 0 0.884 0 0 1.527

$2 0.867 0 0 0 0.252 0 0 0 -0.676

S3 0 0 L908"~ 0 0 1.908 0

Ooj

(9.38)

The bond unit vectors are defined as follows: (1) The longitudinal vectors el(l) and el (2) are directed along the respective bonds;

234

GALABOV AND DUDEV

(2) The transverse vectors e2(1) and e2(2) are perpendicular to the longitudinal vectors and lie in the xz symmetry plane of the molecule; and (3) The other couple of transverse vectors e3(1) and e3(2) are chosen to be perpendicular to the plane xz. The respective direction cosine matrices are as follows: x

y

0"864 -10 bond 1 (~-0~04

x

y

(--0.864 bond 2

0 0 -1

z (9.39)

: . ! : / / e2 e3 el(l' (1, (1)

z 0304/ el(2) e2 (2) e3 (2)

(9.40)

The two S=O bonds possess non-cylindrical symmetry and, therefore, no further simplifications can be introduced in Eqs. (9.33) and (9.34). Substituting the respective quantifies in Eqs. (9.33) and (9.34) with their numerical values yields the following set of linear equations: 1.055(c3ct1/o~r)+ 1.055 (tgot1/tgr')+O.359(&x2/~)+0.359(&x2/~')=6.208 1.414(o~3/0r)+ 1.414(&x3/~')=0.884 0.359(tgot 1/tgr)+0.359(aa 1//9r')+1.055 (&z2/tgr)+1.055( aa2/igr')= 1.527 1.492(0ct 1/tg~)+0.508(tgct2/tgr

l-a2)=0.867

(9.41)

2(&t3/a~)-~.252 0.508(i9ot 1/ar

1.492(0a2/&~)-0.435(a l-Ct2)--0.676

0.615(,9a 1/,gr)--O.615(,9a 1/,gr')--O.615 (,ga2/&')+O.615 (,ga2/,gr')-O. 121 (a l-a2) = 1.908. In these equations 0cti/0r (i = 1, 2, 3) denote bond polarizability derivatives with respect to the stretching of the same bond (the so-called zero-order reop) and aai/&' and &ti/a ~ are derivatives with respect to the other internal coordinates: the adjacent SO stretching and the OSO bending, respectively (first-order parameters). The term -0.121(al-a2) in the last equation represents the rotational contribution to aaxz/aS 3. Since the two S=O bonds

PARAMETRIC MODELS OF RAMAN INTENSITIES

235

are equivalent, the equilibrium bond polarizabilities can be derived by applying Eq. (9.17) and using the data presented in Table 9.5. The result is 2.770 a(0sO) =

/ 1.114

A3 .

(9.42)

1.374 The number of unknowns in the set of equations (9.41) is, thus, reduced. Still, however, the number of parameters (nine) exceeds the number of equations (seven). It is evident that some parameters have to be constrained equal to zero in order to obtain a defmite set of equations. If 3a2/3r' and 3a3/&' parameters are set equal to zero, the following set of electro-optical parameters is obtained (in units of A 2 or A 3 rad -1)

c3tXl/tgr = 4.449 C3al/tgr' = 1.650 cna2/tgr = ---0.628 3a2/3r' = 0

tga3/tgr = 0.625 0a3/o~r' = 0 OCtl/O{ = 0.100 aa21ad~ = -0.005 cga3/3d~ = 0.126

(9.43)

The indeterminacy of the set of linear equations evaluated in the VOTR leads to divergent solutions. There is no solid ground in selecting a plausible solution.

CQ

CompactFormulationofVOTR

As can be seen from the preceding sections, the evaluation of reop for a given molecule is quite a cumbersome and tedious process. In order to simplify the calculation procedure, Montero and Del Rio have put forward a compact formulation of the valenceoptical theory of Raman intensities known as bond polarizability model [296,297]. The bond polarizability model has been applied in extracting Raman intensity parameters for a number of molecules [263,302-311]. To each bond k of a molecule a local Cartesian coordinate system with z-axis directed along the bond direction is assigned. The other two bond coordinate axes are perpendicular to the z-axis and their particular orientation depends on the local symmetry of the bond. A bond polarizability tensor is presented in the form

a(k)=/a [3 ~,1

(9.44)

236

GALABOV AND DUDEV

(z, 13 and T represent the equilibrium polarizability components of the bond in the bond axis system. These correspond to (z2, o~3 and (zI quantifies, respectively, used in the previous sections. In the general case, (z r 13r T. For bonds possessing cylindrical local symmetry, r = 13. In the bond polarizability model the changes of bond length and bond direction during vibrational distortions are described by introducing a displacement vector ~k. It is defined in terms of a set of polar coordinates, namely: (1) Az, the changes in the bond length; and (2) A0y and A0x, the rotations of the bond around y and x axes, respectively. The ~k vector is presented in the form [296]

=[ ( AYx~. T AYa)/ r k )k k

(AOy~ ((Axb - Axa)/r k /

-i Oxi C

'

(9.45)

where Axi, Ayi and Azi (i = a, b) are the projections of displacements of the initial (a) and terminal (b) atoms of bond k along the local Cartesian axes. r k is the equilibrium bond length. If synunetry vibrational coordinates are employed, the displacement components can be calculated according the expression [296]

/(AXb / (AYAbT.AYa) 7 / rk -AZa

with

AYa

/

~AZa j

AXb - A X a ] = Tk AYb AYa

kj

= Aaj and AYb ttAZb

(9.46)

t
/ j

- A bj.

(9.47)

In Eq. (9.46) T k is a 3• transformation matrix between the molecule-fixed Cartesian coordinate flame (X, Y, Z) and the bond axis system (x, y, z). Aaj and Abj are the three elements of the jth column of the Crawford's A s matrix [Eq. (4.125)] referring to atoms a and b, respectively. In zero-order approximation of the bond polarizability model the following three types of electro-optical parameters representing derivatives of the bond polarizability with respect to bond displacement coordinates are defined [296,297]

PARAMETR/C MODELS OF RAMAN INTENSITIES

237

0 0 "t-or / 0 0 0 y - tx 0 0 k

Otx = ('~0~(k)/t)0y)=

(9.48)

0/

/i ~ (! o Oo)

~,-15

~'-13

0

(9.49)

k

0 7' k

/

(9.50)

~y/~z k

These expressions reflect the changes of bond polarizability a(k) with vibrational displacements of the bond k. In the general case, ct(k) may depend on the bond surroundings as well. The changes in bond polarizability inflicted by vibrational distortions of neighboring bonds should, therefore, be taken into account. First-order reop are then introduced [296,297]

(~).(k)/~)R1)=

//

0 ~i'

0 0

0

~ k

~)ct/~)R 1

=

~)I3/~)Rl

/

.

(9.51)

0Y/0R1 k

Usually, derivatives with respect to internal coordinates that include at least one atom of the bond k (for instance, stretching of the bond adjacent to the bond k or a bending motion including bond k) are retained only. All other long-distance interactions are neglected. The introduction of first-order electro-optical parameters in intensity analysis leads to more complete and realistic interpretation of the experimental observables. A price is, however, paid: the number of intensity parameters increases significantly and the problem often becomes mathematically undetermined. Evidently, each set of reop evaluated represents a compromise between the desire to describe the intramolecular interactions as fully as possible and the limitations imposed by the mathematical formalism employed. The general formula of the bond polarizability model relates molecular polarizability derivatives with respect to symmetry vibrational coordinates with the set of electro-optical parameters [296,297]: (9.52)

238

GALABOV AND DUDEV

TABLE 9.6 Zero-order electro-optical parameters for CH4, C2H6, C2H4, C2H2, C6H6, CHEC12 and CHCI 3 (in units of 42) Parameter

CH4a

a' 0.45 ~/' 2.51 aFromRef. [302]. bFromRef. [263]. CFromRef. [303]. dFromRef. [304]. eFromRef. [305]. fFrom Ref. [306]. gFromRef. [307].

C 2 H 6 b C2H4C C 2 H 2 d 0.443 2.421

0.48 2.32

0.482 2.342

C6H6e CH2CI2f 0.28 2.99

0.484 2.390

CHCI3g 0.36 2.00

where {ax', ay', az'}k. ~kj

=

ax' (AOy)j + ay' (AOx)j + a z' (Az)j.

(9.53)

Tx is the transpose of the Tx matrix and U matrix has its usual meaning. The first summation in Eq. (9.52) comprises the zero-order electro-optical parameters. The fluctuations of bond polarizability a(k) with changes in length and spatial orientation of the same bond are only considered. The second term is responsible for the first-order intensity parameters. All internal coordinates are considered less the stretching of bond k. In the framework of the compact formulation of bond polarizability theory the computational procedure is significantly simplified and appropriate for computer programming. A package of computer programs has been developed [312]. The approach developed allows for factorization of the physical properties of the molecule. Thus, {ax' , Cry', az' } and 0ct(k)/0Rl depend mainly on electro-optical properties of the molecule while the ~-vectors - on the molecular geometry and atomic masses. The contributions of zero-order reop and first-order parameters to 0ct/0Sj are clearly separated, as well. The application of bond polarizability approach to gas-phase Raman intensities for different molecules has resulted in evaluating sets of electro-optical parameters for various bonds [263,302-307,310]. It has been found that some zero-order intensity parameters for C-H bonds in the series CH 4, C2H6, C2H4 and C2H2 differ in narrow limits and can be considered transferable. These are summarized in Table 9.6. reop for some other molecules are also given. It can be seen that C-H parameters for polar molecules such as CH2C12 and CHC13 are not too different from the hydrocarbon intensity parameters. The analysis of data implv that the C-H bond polarizabilities, at least in the series of the molecules treated, are not very sensitive to environment. This

PARAMETRIC MODELS OF RAMAN INTENSITIES

239

finding has been confirmed by theoretical calculations of polarizability derivatives as well [313]. It is interesting to note here that Raman intensity parameters appear to be more transferable than the respective quantifies employed as parameters in infrared intensity analysis.

Dg

Compact Formulation of VOTR: An example of Application

In this section the predictive power of the bond polarizability model is tested in calculating Raman intensifies of propyne by transferring parameters from other molecules. Propyne has four different types of chemical bonds: methyl C-H, C-C, C~--C and =-C-H bonds. The set of propyne bond polarizability parameters is constructed by taking the values needed from ethane (methyl C-H and C-C bonds) [263] and acetylene (C=-C and -=C-H bonds) [304]. Both molecules have been treated in terms of bond polarizability approach in analyzing experimental gas-phase Raman intensities. Cylindrical symmetry has been assumed for all chemical bonds. The full set of parameters employed are given in Table 9.7. The theory is applied at zero-order level. Internal coordinates, bond directions and Cartesian coordinate reference system for propyne are shown in Fig. 9.3. Geometry parameters and synunetry coordinates used in the calculations are given in Table 9.8. The normal mode transformation matrices L S for A 1 and E symmetry classes needed to convert polarizability derivatives with respect to symmetry vibrational coordinates into polarizability derivatives with respect to normal coordinates are evaluated by using the harmonic force field of Duncan, McKean and Nivellini [316]. The L s matrices are given in Table 9.9. ~-vectors for each bond and the predicted scattering activities and depolarization ratios are calculated with the aid of the computer programs "XIVEC" and "IRAM", respectively, belonging to a program package for Raman intensity analysis [312]. Calculated depolarization ratios and differential scattering cross sections (dc/d.Q)i for propyne are presented in Table 9.10. The simulated Raman spectrum is compared with the experimental gas-phase spectral curve [317] in Fig. 9.4. The band half-widths are taken from the experiment. The lines of A 1 transitions have sharp features, while Evibrations are characterized with much broader bands. Since no quantitative intensity data for this molecule exist, a qualitative assessment of the results obtained can be done only. Fig. 9.4 reveals that the overall shape of the Raman specmun is reproduced correctly. The most intense Raman lines are calculated to be those positioned at 2941, 2142 and 930 cm-1 with intensities decreasing in the same order in agreement with the experimental spectrum. These lines are highly polarized. The other vibrational transitions giving rise to low- or medium-intensity lines in the specmun are predicted to have intensities of the same order. The most significant difference between calculated and

240

GALABOV AND DUDEV

TABLE 9.7 Zero-order polarizability parameters for methyl C-H, C-C, C-C and - C - H bonds used in predicting Raman intensities of propyne Bond polarizability parameters Bond Methyl C-H a C-C a C-C b -C-H b a From Ref. [263]. b From Ref. [304].

(~-a)

a'

~'

(A3)

(A2)

(A2)

-0.144 1.276 1.740 0.025

0.443 0.273 2.506 0.482

2.421 2.582 5.480 2.342

X

Cx

%

cz3 r2

')

\

--

I

)

133

Fig. 9.3. Definition of internal coordinates, bond directions and Cartesian reference system for propyne

PARAMETRIC MODELS OF RAMAN INTENSITIES

241

T A B L E 9.8

Molecular geometry and symmetry coordinates for propyne Geometry: a rC_H(Me) = 1.096 A , rc_ c = 1.4596/~, r=_c_H = 1.060/~,
rc_ c = 1.2073/~,

Symmetry coordinates:b, c AI

E

S1 = (Arl + Ar2 + Ar3)/~f3 S2 = A [ a (Act 1 + Act2 + Act3) - b (AI31 + AI32 + A133) ] a = 0.416755 , b = 0.399560 $3 - AR S4 = AT S5 = Ad S6a = (2Arl - A r 2 - Ar3)/'f6 S7a = /tt (2Atx 1 - Atx2 - Aoc3) /~f6 SSa = /~ (2A131 - A[32 - A133)/~/6 S9a = A AOx Sl0a = AAW x

S6b = S7b = S8b = S9b = St0b =

( r2 - at3)/4 fit (Act2 - Atx3)/-f2 A (A[32 - A133)/~f2 2~ A~y AAWy

aFrom Ref. [314]. bFrom Ref. [315]. Clntemal coordinates are defined in Fig. 9.3.

experimental spectra is observed in the region of the methyl antisymmetrical stretching vibration (3001 cm-1). A value of 140x 10-36 m2/sr for dcr/d~ is predicted. A very weak line is, however, detected in experimental conditions [317]. Overestimated to some extent are the intensities of the - C - H and C - C stretching vibrations centered at 3335 and 930 cm -1, respectively (Fig. 9.4). Nevertheless, in view of the approximations introduced in calculations, namely: (1) The hypothesis of full transferability of intensity parameters, and (2) The application of bond polarizability model at zero-order level, the results obtained are satisfactory. It can be concluded that the set of polarizability parameters transferred from ethane and acetylene is close to the actual set characterizing Raman intensities of propyne. Minor refinement of parameter values is required to obtain a very good fit between observed and calculated spectral curves. The predicted value of the equilibrium molecular anisotropy of propyne of 3.176 A 3 agrees very well with an experimental magnitude of 3.298 A 3 reported by Alms et al. [318].

242

GALABOV AND DUDEV

TABLE 9.9 L S matrices for propyne (in amu- 89 a,b A1 S1

QI -0.0016

S2

0.0010

S3

0.0283

Q2 1.0111

Q3 -0.0112

Q4 0.0288

Q5 0.0050

-0.1485

-0.0266

1.4406

-0.0814

-0.0582

-0.2593

0.1286

0.2805

$4

-0.1296

0.0034

0.3816

0.0111

0.0643

S5

1.0290

0.0029

0.1286

0.0028

0.0143

S6

Q6 1.0489

Q7 0.0203

Q8 --0.0196

Q9 -0.0111

Qlo 0.0251

S7

0.0984

1.4440

--0.3269

-0.0025

-0.0356

S8

-0.1047

0.4100

0.9073

0.0494

0.0822

$9

0.0453

0.0134

-0.2281

--0.1549

0.4571

SIO 0.0147 0.0017 0.0164 1.0894 aObtained fromthe forcefield of Duncan, McKeanand Nivellini [316]. bSymmetrycoordinatesare definedin Table 9.8.

-0.1201

E

Calculated spectral characteristics for prop)me obtained by transferring bond polarizability parameters are compared with those evaluated by RHF/6-31G(d,p) ab initio MO calculations [311] in Table 9.10 and Fig. 9.4. It is seen that the predicted spectrum obtained in applying the bond polarizability model is in better agreement with experiment than the ab mitio estimated spectrum. More advanced quantum mechanical computations are, evidently, needed to satisfactorily calculate the Raman intensities of propyne. The computations employing transferable sets of polarizability derivatives are simple and give good results. On the basis of transferability properties for intensity parameters in the hydrocarbon series, Martin [309] has predicted depolarization ratios, scattering coefficients and absolute differential Raman scattering cross sections for propane in the gas-phase employing electro-optical parameters determined for ethane. A good correspondence between the predicted and experimental data has been achieved. A survey of calculated Raman spectra of other hydrocarbons is presented in the book of Gribov and Orville-Thomas [155]. Gussoni and co-workers [300,319] have predicted the Raman spectrum of polyethylene and perdeutero-polyethylene by transferring electrooptical parameters evaluated for methane and cyclohexane.

TABLE 9.iB Experimental and calculated spectral parameters for propyne in the gas phase (wavenumbers in cm-l, absolute differential Raman scattering cross sections (du/dQ)i in 10-36 m*sr-1, the depolarization ratios pi are dimensionless) (Reprinted fiom Ref [3 I I] with permission of John Wiley & Sons, Ltd. Copyright [1995] John Wiley & Sons, Ltd.) vexpa

A1

~3

E

v7

3335 3001 2941 2142 1407

A1 E A1

v4

1385

v8 ~5

1045 930

A1 E A1

vl ~6 v2

=C-Hstr. CH3 as. str. CH3sym. str.

CH3 sym. def CH3 rock C-Cstr.

CzCstr. CH3 as. def

Pexpa

Pcalcb

P

0.234 0.75 0.027 0.028 0.75 0.252 0.75 0.133

dP

P P dP P dP

P

(doWcalcb’C 61.13 140.0 321.2

208.3 12.68 25.32 1.61 170.1 23.33 74.38

PMOd

(dddR)MOC.d

0.196 0.75 0.001 0.301 0.75 0.585 0.75 0.233 0.75 0.75

40.58 3 12.1 308.2 255.1 202.1 67.73 0.27 53.73 67.07 83 1.6

0.75 E v9 0.75 E vln aFrom Ref. [3 171; Abbreviations used are: str. - stretching, as. - antisymmetric,sym. - symmetric, def. - deformation, p - polarized, dp - depolarized. bCalculated by transferring polarizability parameters from ethane and acetylene (see text). CCalculated for laser excitation wavelength A. = 488 nm. dResults from RHF/6-3 IG(d,p) a b inilio MO calculations. 649 338

C=C--Hbend C-CXbend

dP dP

h)

P W

244

GALABOV AND DUDEV

A

| =.:i

'

'

'

'

'

I

" "

3000

'

'

I

'

'

'

2500 RAMAN

'

I

'

'

2000 SHIFT

'

'

I

.

.

1500

.

.

I

'

1000

,~

.

'

500

(cm "1)

Fig. 9.4. Comparison between Raman spectra of propyne in the gas-phase obtained (A) experimentally (Ref. [316]); (B) by transferring bond polarizability parameters from ethane and acetylene; and (C) by employing RHF/6-31G(d,p) ab initio calculations. Laser excitation wavelength is ~ = 488 rim. Asterisks mark overtone and combination lines (Reproduced from Ref. [311] with permission of John Wiley & Sons, Ltd. Copyright [ 1995] John Wiley & Sons, Ltd.).

PARAMETRIC MODELS OF RAMAN INTENSITIES

III.

ATOM DIPOLE

INTERACTION

245

MODEL

(ADIM)

The atom dipole interaction model has been proposed by Silberstein [320] and revived in the be~nning of the 1970's by Applequist, Carl and Fung [289]. Later ADIM has been further developed and applied in solving different spectroscopic problems [321329]. The atom dipole interaction model is an alternative expression of the additive approximation for vibrational intensities. It is based on the assumption that the molecular polarizability is a sum of polarizabilities of the constituent atoms. The interactions between atoms in the molecule are realized by the electromagnetic fields of induced atomic dipoles. An isotropic polarizability a i centered at the nucleus is assigned to each atom i of an N-atomic molecule. When an external electric field fi is applied to atom i, an induced dipole Ixi is created: [289]

Bi = r

N fi - ? 1 Tij ~tj .

(9.54)

Tij is the induced dipole field tensor and I~g and fg (g = 1 to N) are column vectors. The last term in Eq. (9.54) accounts for the contribution of the induced dipole fields to the overall field strength. In explicit form the dipole field tensor Tij is given by [289] x 2 - (1 / 3)ri~ 3 Tij = - r-~jS,

xy xz

xy

xz

y2 _ (1 / 3)ri~

yz

yz

z

.

(9.55)

2 (l/3)ri~ -

In this equation rij is the length of a vector pointing fi'om atom i towards atom j, and x, y and z are the respective vector components in a molecule-fixed Cartesian coordinate system. Expressions analogous to Eq. (9.55) can be written for each atom of the molecule. Thus, after rearranging the components of Eq. (9.54), the following set of N matrix equations is obtained

246

GALABOV AND DUDEV

N ( a l ) -1 I l l + ~ TIj "j = f l j=2 (9.56)

N-I (a N)-I gN + ~-, TNj I.tj = fN j=l or in a matrix form

I

(~l) -I

TI2

T21

(-2) -l

~, TN1

9. .

TIN

~tl

...

T2N

~t2

~

o

f!

=

f2

.

(9.57)

9

TN2

Eq. (9.57) can be written also in the form Kg = f,

(9.58)

where the square array K has dimensions 3N• and ~t and f are column vectors of size 3N. Each element of K is a second-rank tensor. Eq. (9.58) can be rearranged if the inverse of K matrix W = K -1

(9.59)

is introduced. Thus, ~t = W f.

(9.60)

W is a squire matrix of rank 3N called "relay tensor" [289]. If the applied field is assumed to be uniform for all atoms of the molecule, the following expression for the induced dipole can be written

(9.61)

If this expression is compared with Eq. (8.1) (Chapter 8), it can be seen that the term Y~ Wij plays a role of an effective polarizability of atom i. Thus, for the molecular induced dipole moment the following equation holds:

PARAMETRIC MODELS OF RAMAN INTENSITIES

la=E"i i=l

=

E Wij f i=l j=l

247

(9.62)

where N N

(9.63)

E wij

i=l j=l

is the polarizability of the molecule. It is seen from the above expressions that the molecular polarizability is represented as a function of atomic polarizabilities and interatomic distances. In this way, provided that molecular geometry and atomic polarizabilities are known, values for the individual components of the molecular polarizability tensor can be easily obtained. Using the experimental sodium D-line (589.3 nm) refractive index data for a large series of organic molecules Applequist et al. [290] have obtained an optimized set of atomic polarizabilities for a number of atoms. These quantifies reproduce quite well the mean molecular polarizabilities of many molecules. At the same time, however, molecular anisotropies predicted are too large. In order to overcome this difficulty, Thole [325] and Birge [326] introduce improvements in the ADIM approach that have resulted in better fit between observed and calculated molecular polarizabilities. Better transferability of atomic parameters has been achieved as shown in the cases of H20, ethylene oxide, HCONH 2, CH3CONH2, C3H7OH and CH 4 molecules [325]. Eqs. (9.62) and (9.63) have been derived for molecules in the gas-phase where each atom experiences the influence of the induced dipole fields of other atoms only. The interactions become much more complex in condensed state. The effect of intermolecular interactions needs to be considered [327]. The theoretical approach discussed so far describes physical properties of a vibrationally undistorted molecule. The effect of molecular vibrations is considered by differentiating both sides of Eq. (9.63) with respect to normal coordinate Qt- Expressions for c3ot/c3Qt derivatives are evaluated [321]: N

~)ot/aQt = - 2

N

N

2

~, 2 wij (~)Kjk/aQt)Wkl

N

(9.64)

i=l j=l k=l l=l

or

or' = - W K' W ,

(9.65)

248

GALABOV AND DUDEV

where prime denotes derivative with respect to Qt. The matrix K’ has the following structure [321,323] :

(9.66)

It consists of two types of elements: (1) diagonal elements that reflect the changes in atomic polarizability with the normal vibration

and (2) off-diagonal elements

that arise fiom changes of the dipole interactions with molecular vibration. Taking into account relation (9.26), the following expression for the diagonal terms is obtained: (9.69)

In the last equation M is the number of internal coordinates &, and L b is an element of the normal coordinate transformation matrix L. The off-diagonal term can be rearranged in the following manner: (9.70)

A is a topological matrix having +1 for the final atom j and -1 for the initial atom i of each atomic couple and zeros for all other atoms which are not included in the interaction. The derivatives aX/a& are elements of the Crawfords A matrix [Eq. (2.16)]. An explicit expression for aTij/&ij is given elsewhere [321]. Eqs. (9.69) and (9.70) express the derivatives with respect to normal coordinates in terms of derivatives with respect to internal vibrational coordinates. Since, however, the number of ao;/aRk parameters exceeds by far the number of experimental observables

PARAMETRIC MODELS OF RAMAN INTENSITIES

249

even for simple molecules, some approximations are necessarily introduced in applying the ADIM formulation: (1) The method is often used in zero-order approximation. Atomic polarizability derivatives with respect to stretching of bonds containing atom i are retained only, i.e. igcti/aRml, 0 if either m or 1 equals i. These parameters are considered to a great extent independent of the atomic environment. (2) Usually a single numerical value is given to each tensor quantity tgtxi/c3Rk thus allowing for further decrease in the number of parameters to be evaluated. Within this approximation all diagonal terms are assumed identical and the atomic tensor preserves its spherical shape with changes in bond length and spatial orientation. Using the ADIM formulation Applequist and Quicksall [321] have made an m attempt to reproduce the experimental t9 c t / ~ t and c3y/~t quantifies in the series of methane and its halogenated derivatives employing five atomic polarizability parameters only. These are &XH/aRCH, aaF/aRcF, &zCl/aRcc l, aaBr/C3RcB r and &xC/c3RC. One parameter for the carbon atom is only used regardless of the type of bond containing the atom. The correspondence between experimental and calculated polarizability derivatives is less satisfactory if compared with equilibrium molecular polarizability calculations.

IVo

ATOMIC POLARIZABILITY TENSOR FORMULATION (APZT)

The atomic polarizability tensor formulation (APZT) represents the Raman counterpart of the atomic polar tensors theory (APT) of infrared intensities (Section 4.I). The idea of transforming absolute Raman intensities into molecular polarizability derivatives with respect to atomic Cartesian displacement coordinates was first put forward by Bogaard and Haines [330]. More complete formulation of APZT approach was given later [299,331]. Unlike its infrared analog, the atomic polarizability tensor calculations have attracted little interest. Nevertheless, atomic polarizability tensors are standard output of the current ab initio quantum mechanical program packages. In the APZT approach the changes in molecular polarizability ct of a molecule having N atoms are represented as a function of atomic displacement vectors ra: N AtX= ~ r a--1

with

r a,

(9.71)

250

GALABOV AND DUDEV

r a = Axai + AYaj +Azak.

(9.72)

Axa, Aya and Aza are the Cartesian displacement coordinates of atom a in a space-fixed Cartesian coordinate system, and i, j and k are the respective unit vectors. Atomic polarizability tensors a x a are third-rank tensor quantifies which can be written as 3x9 rectangular arrays:

( t)Otxx / ~)xa

tkZxy / ~)xa t)O~xz/ ~)xa

tt)O~zx /t)xa

t)O~zy/~xa

c)O~zz/ ~ a

aaxx / aya 3axy / aya 3axz / aya 3ar,,: / aya 3ayy / aya aayz / aya aa= / ~ a aa~y / aya aa= / ~

(9.73)

Baxx / ~)za 3axy / Oza Oaxz / ~)za 0Ctyx / t)za ~)ayy / ~)za ~)ayz / t)za / " t)azx / Oza 3Ctzy / Oza t)azz / ~)za As can be seen from Eq. (9.73) 27 intensity parameters are assigned to each atom of the molecule. If the frequency of incident fight lies far from any electronic absorption frequency of the molecule, the number of independent parameters for each atom a reduces to 18 since the polarizability tensor is symmetric under the far-from-resonance conditions. Arranged in a row all a x a tensors give the atomic polarizability tensor a X of the molecule

tzX = ( a X 1 r

tXX3..,

txxa..,

ctXN) ,

(9.74)

with dimensions 3• The elements of the atomic polarizability tensor can be evaluated by applying an expression analogous to that used for dipole moment derivatives [Eq. (4.12)] [299] a X = a Q L s - 1 B S + ap13

(9.75)

a x = a s BS + ap 13.

(9.76)

or

PARAMETRIC MODELS OF RAMAN INTENSITIES

251

In these equations OtQ and a s are the matrices containing molecular polarizability derivatives with respect to normal and symmetry vibrational coordinates, respectively [Eqs. (8.41) and (9.3)], and cxp is an array comprising polarizability derivatives with respect to molecular translations and rotations. The matrix product (xS B S represents the so-called vibrational atomic polarizability tensor Va accounting for the changes in molecular polarizability with molecular vibrations. The V a tensor for the entire molecule can be expressed as a juxtaposition of individual atomic tensors: otSBS = V a = ( v a l Va 2 V a 3 . . .

Vaa...

vN).

(9.77)

All elements of the Va tensor are taken with respect to a molecule-fLxed Cartesian coordinate frame. The second term in Eqs. (9.75) and (9.76), the rotational atomic polarizability tensor R ~ reflects the contribution of molecular translation and rigid-body rotation to (zX. The inclusion of the six external molecular coordinates in those equations - the three translations Xx, Xy and Xz, and the three rotations Px, Py and Pz, completes the set of molecular coordinates up to 3N. In this way polarizability derivatives are transformed into quantifies corresponding to a space-fixed Cartesian coordinate system. As already pointed out in section 4.I, the great advantage of such a step is that the intensity parameters defined in terms of a space-fixed coordinate system are independent on isotopic substitutions provided the symmetry of the molecule is preserved. This will be illustrated with an example in the succeeding section. By analogy with Eq. (9.77), the rotational polarizability tensor can be represented as Otp[3 = P~x = (R(xl Ra 2 Ra 3- - - R~ a- 9 9 l~N)

9

(9.78)

Eq. (9.76) assumes the form (zX : Vet + R a .

(9.79)

Ra acquires zero value for molecules possessing a spherical polarizability ellipsoid. In that case, ctX = V ~ .

(9.80)

The 13 matrix in Eq. (9.78) contains, as pointed out earlier, derivatives of the six external molecular coordinates with respect to atomic Cartesian displacements. Expressions for calculating the elements of 13 are given in Chapter 2 [Eq. (2.22)]. The elements of ctp tensor can be evaluated if the second-rank molecular polarizability tensor is transformed into third-rank pseudo-tensor. This can be done by considering the effect

252

GALABOV AND DUDEV

of infinitesimal rotations on molecular polarizability [129,299]. expression is obtained: l

The following

0

-axz axy 03x 9 - 2 a y z ayy - azz sym. 2ay z

OCp=

(9.81) 2axz ay z azz - axx -2axy 0 -r sym. -2axz sym.

axx - ayy - a y z - 8 9 I 2axy r 0

J

In this expression 03x9 is a 3• null matrix containing the polarizability derivatives with respect to the three translational coordinates. The next three 3x3 blocks in the matrix comprise the polarizability derivatives with respect to the three rotations Px, Py and Pz, respectively, a ~ (~, g = x, y, z) represent the equilibrium molecular polarizability tensor components and I is the principal inertia tensor of the molecule. Multiplication of both sides of Eq. (9.76) by the Crawford's A s matrix [Eq. (4.125)] results in a X A S = a s B S A S + ap13A S.

(9.82)

Taking into account that B s A s = E and 13AS = 0 [Eq. (2.15)] this is reduced to a x AS = a s .

(9.83)

The relation is particularly useful in evaluating polarizability derivatives with respect to symmetry coordinates from atomic polarizability tensors obtained through ab mitio MO calculations. As mentioned, the APZT appear as a standard output from programs for ab mitio quantum mechanical calculations employing analytical derivative methods. The elements of atomic polarizability tensors are interrelated by the following equations N y_, tX~( = 03x9

(9.84)

a=l

N

a~,((ra))=((aO)) .

(9.85)

a=l In these equations 03• 9 is a zero matrix with dimensions 3x9 and ra is the position vector of atom a taken with respect to the center of mass of the molecule. The pseudotensor

PARAMETRIC MODELS OF RAMAN INTENSITIES

253

((ra)) is constructed in a form that satisfies the matrix multiplication rules. Because of relations (9.84) and (9.85) the interpretation of the atomic polarizability tensors is not trivial.

AQ

APZT: An Example of Application

In this section an example of calculations employing the APZT formalism is presented. Atomic polarizability tensors for formaldehyde-d 0 and formaldehyde-d 2 are evaluated. The initial data are taken from RHF/6-31G(d,p) ab initio MO calculations [332]. Molecular geometry, static polarizability tensor and definition of symmetry coordinates are given in Table 9.11. The orientation of the molecule in the Cartesian space, definition of internal coordinates and numbering of atoms are shown in Fig. 3.1. The a s matrices comprising molecular polarizability derivatives with respect to symmetry coordinates for H2CO and D2CO are given below. H2CO

-0!62 IXS =

SI 0 0 0.339 0 0 3.158

3.016 0 0

$2 0 0.195 0

0 0 1.093

0.280 0 0

$3 0 0.042 0

0 0 -0.473 (9.86)

0 0 -1.657

$4 0 0 0

-1.657 0 0

0 0 --0.404

$5 0 0 0

-0.404 0 0

0 0 0

3.016 0 0

$2 0 0.195 0

0 0 1.093

0.280 0 0

$6 0 0 0 -0.125 --0.125 0

J

D2CO

~S =

--0!62

$1 0 0 0.339 0 0 3.158

$3 0 0.042 0

0 0 -0.473 (9.87)

0 0 -1.645

$4 0 0 0

-1.645 0 0

0 0 -0.458

$5 0 0 0

--0.458 0 0

0 0 0

$6 0 0 0 --4).027 -0.027 0

J

254

GALABOV AND DUDEV

TABLE 9.11 Molecular geometry, static polarizability and definition of symmetry vibrational coordinates for formaldehydea Geometry:

rco = 1.1844 A ,

rCH(D) = 1.0933 A ,

Static polarizability (in A 3) ~~

1.939

et O = 0.995

o~0= 2.641

Symmetry coordinates b A1

B1

S1 S2 S3 $4

= = =

AR (Ar I + Ar2)/,r (2Aet - A[31 - A[32)/,q~ (Ar 1- Ar2)/~f2

s5 = (a13]- a ~ 2 ) / 4 ~ B2 $6 = A0 aFrom RHF/6-3 ICKd,p)ab mitio calculations [332]. bThe internal coordinates are defined in Fig. 3.1; 0 is an out-of-plane angle.

It can be seen from Eqs. (9.86) and (9.87), that polarizability derivatives with respect to symmetry coordinates belonging to non-totally symmetric modes are different for the two isotopic species. This is expected since the 0a/0Sj derivatives are mass-dependent quantifies. The ap tensor is evaluated from Eq. (9.81): 0 0 Ctp = 03x 9 0 0 0-1.646

0 0 0 0.702 0 0.944 0~ -L646 0 0 0 0.944 0 ~/ I"A 0 0.702 0 0 0 0

(9.88)

The atomic polarizability tensors for H2CO and D2CO are obtained from Eq. (9.76). The elements of B S and fl matrices need to be rearranged to match the dimensions of a s and CZp,respectively. The a s B s, otp [3 and ctx tensors obtained for the two species are given in Tables 9.12 and 9.13. It can be seen that the atomic polarizability tensors etx for formaldehyde-d o and formaldehyde-d2 are, as expected, identical.

PARAMETRIC MODELS OF RAMAN INTENSITIES

255

T A B L E 9.12 a s B S, ap 13 and aX tensors for formaldehyde-do (in units of A2) a

as BS 0 0 1.223

0 0 0

1.223 0 0

0 0 0

C1 0 0 0.320

0.320 0

2.599 0 0

0 -0.272 0

0 0 -1.439

0 0 0.482

0 0 0

0.482 0 0

0 0 0

02 0 0 -0.106

0 -0.106 0

-0.862 0 0

0 0.339 0

0 0 3.158

1.973 0 -0.854

0 0.142 0

-0.854 0 0.373

0 0 0

H3 0 0 -0.108

0 -0.108 0

-0.869 0 0.844

0 -0.034 0

0.844 0 -0.859

-1.973 0 -0.854

0 -0.142 0

-0.854 0 -0.373

0 0 0

0 0 -0.108

0 -0.108 0

-0.869 0 0.844

0 -0.034 0

-0.8440 )

0 0 -0.348

0 0 0

-0.348 0 0

0 0 0

C1 0 0 -0.928

0 -0.928 0

0 0 0

0 0 0

0 0 0.464

0 0 0

0.464 0 0

0 0 0

02 0 0 1.238

0 1.238 0

0 0 0

0 0 0

0 0 -0.058

0 0 0

-0.058 0 0

0 0.510 0

H3 0.510 0 -0.155

0 -0.155 0

0 0 -0.046

0 0 0

o

0 0 0

-0.058 0 0

0 -0.510 0

-0.510 0 -0.155

0 -0.155 0

0 0 0. 046

0 0 0

H4

-0.859

ap 13

0 -0.058

H4

-0.046 0 0

0.04600 )

GALAEiOV AND DUDEV

256

TABLE 9.12 (continued) ‘LX C1 0 0 -0.608

0.875

0 0 0

0.875 0 0

0 0 0

0 0 0.946

0 0 0

0.946 0 0

0 0 0

1.132

1.973 0 -0.912

0 0.142 0

-0.912 0 0.373

0 0.510 0

0.510 0 -0.263

-1.973 0 -0.912

0 -0.142 0

-0.608 0

2.599 0 0

0 -0.272 0

0 0 -1.439

0 1.132 0

-0.862 0 0

0 0.339 0

0 0 3.158

0 -0.263 0

-0.869 0 0 -0.034 0.798 0

0.798 0 -0.859

0 -0.510 0 -0.912 0 -0.510 0 -0.263 -0.373 0 -0.263 0

0 -0.869 0 -0.034 -0.798 0

-0.789

0

02

0

H3

H4

-0.859

Tattesian reference system and the numbering of atoms are shown in Fig. 3.1.

TABLE 9.13 as Bs. a, f3 and ax tensors for formaldehyde-d2 (in units of @)a

as BS 0 0 0

1.108 0 0

0 0

0

C1 0 0 0.068

0.546

0 0 0

0.546 0 0

0 0 0

0 0 0 -0.023 -0.023 0

1.973 0 -0.828

0 0.142 0

-0.828 0 0.373

0

-1.973 0 -0.828

0 -0.142 0

-0.828 0 -0.373

1.1 08

0 0

2.599 0.068 0

02

D3

0 0

0 -0.272 0

0 0 -1.439

-0.862 0 0

0 0.339 0

0 0 3.158

0

0 0 -0.023

0 -0.023 0

-0.869 0 0 -0.034 0.869 0

0.869 0 -0.859

0 0 0

0 0 0 -0.023 -0.023 0

0 -0.869 -0.034 0 0 -0.869

-0.869

0

D4

-0.859

PARAMETRIC MODELS OF RAMAN INTENSITIES

257

T A B L E 9.13 (continued)

% 13 0 0 -0.234

0 0 0

-0.234 0 0

0 0 0

C1 0 0 -0.675

0 -0.675 0

0 0 0

0 0 0.401

0 0 0

0.401 0 0

0 0 0

02 0 0 1.154

0 1.154 0

0 0 0

0 0 0

0 0 -0.084

0 0 0

-0.084 0 0

0 0.510 0

D3 0.510 0 -0.240

0 -0.240 0

0 0 -0.070

0 0 0

0 0 -0.084

0 0 0

-0.084 0 0

0 -0.510 0

D4 -0.510 0 -0.240

0 -0.240 0

0 0 0.070

0 0 0.874

0 0 0

0.874 0 0

0 0 0

C1 0 0 -0.607

-0.607 0

2.599 0 0

0 -0.272 0

0 0 -1.439

0 0 0.947

0 0 0

0.947 0 0

0 0 0

02 0 0 1.131

0 1.131 0

-0.862 0 0

0 0.339 0

0 0 3.158

1.973 0 -0.912

0 0.142 0

-0.912 0 0.373

0 0.510 0

D3 0.510 0 -0.263

0 -0.263 0

-0.869 0 0.799

0 -0.034 0

0.799 0 -0.859

-1.973 0 -0.912

0 -0.142 0

-0.912 0 -0.373

0 -0.510 0

D4 -0.510 0 -0.263

0 -0.263 0

-0.869 0 -0.799

0 -0.034 0

-0.7990 )

-0.070 0 0

0.07000 /

aX

aCartesian reference system and the numbering of atoms are shown in Fig. 3.1.

-0.859

258

V.

GALABOV AND DUDEV

RELATIONSHIP BETWEEN ATOMIC POLARIZABILITY T E N S O R S AND VALENCE O P T I C A L F O R M U L A T I O N S OF RAMAN INTENSITIES

Since different laboratories apply different approaches in interpreting vibrational intensities it is quite essential to define the interrelations between the theoretical formulations developed. Formulas defining the relationship between VOTR and APZT theories of Raman intensities were recently derived by Dudev and Galabov [331]. In this section we discuss these dependencies. A numerical example is provided. Multiplying both sides of Eq. (9.37) by B s results in:

o.sB S =

[Oty~]J0 B s = [Ctyy] [%~]JB. [a=]) k sym. [,x=])

[Ot,yy]

(9.89)

The matrix product ap B is added to both sides of Eq. (9.89) to obtain

c~sUs + % P =

/

[s~l [axyl [a=]]

[C~yy] [%~]J B +=pl] 9 [o~=l)

(9.90)

or

ax =

[~yy] [(zy~ 9

13 +~pB.

(9.91)

[O~ZZ

Eq. (9.91) reveals the possibility of expressing molecular polarizability derivatives with respect to atomic Cartesian displacement coordinates in terms of electro-optical parameters. The validity of this relation will be checked in the case of SO2 molecule. Cartesian reference frame, geometrical data, definition of internal and symmetry coordinates and a s matrix for sulfi~ dioxide were already given in Section 9.II.B. The application of valence-optical theory of Raman intensities results in the set of electrooptical parameters given by Eq. (9.43). If these quantifies are substituted inside the brace of Eqs. (9.33) and (9.34), the elements of ([ct]) array are obtained ([4.390, 4.390, 0.867]

to, /

o,

~[1.349,-1.349,

0] 0]

[o,

o,

o]

[0.625, 0.625, 02521

[o,

o,

o]

[1.349,-1,349,

0]

/

[0, 0, 01 . (9.92) [1.081, 1.081, -0.677]

PARAMETRIC MODELS OF RAMAN INTENSITIES

259

After multiplying the matrices ([et]) and B (the elements of B matrix are appropriately arranged to match the dimensions of (Ict])) and subsequent stunmation with the etp 13 tensor the atomic polarizability tensor for SO 2 is obtained (in units A2):

t J t%1t l) t%1

[a=]) S1

0

0 0 0

-2002

-2.002 0 0

0 0 -3376 0 --0.446 0 --0.446 0 0

0 0 0 --0326 0 0 0 -1905 --0.473 (9.93)

O2 4.097 0 tool

0 tool 0.628 0 0 0.696

-4.097 0 t001 0 --0.628 0 t001 0 -0.696

0 0.789 0

0.789 0 0223

0 0223 0

0 --0.789 0 --0.789 0 0223 0 0.223 0

1.689 0 1244

0 0.163 0

1244 0 0953

O/

1689 0 -1.244 0 0.163 " -1244 0 0.953)

The atomic polarizability tensor a x for SO 2 evaluated by employing Eq. (9.76) is shown as follows together with the vibrational and rotational polarizability tensors (in units A2):

r

-2.331

0 0 0

-2.331 0 0

0 0 0

S1 0 0 0

0 0 0

4.097 0 1-165

0 0.629 0

1.165 0 0.695

0 0 0

02 0 0 0

0 0 0

1.688 0 1.680

-4.097 0 1_165 0 -0.629 0 1.165 0 -0.695

0 0 0

0 0 0

1.688 0 -0.680

BS =

03 0 0 0

-3.376 0 0 0 -3.376 0 0 0 -1.904

0 0.163 0

0.680 0 0.952

o/

0 --0.680 0.163 0 0.952)

(9.94)

26O

GALABOV AND DUDEV

apl~=

0

S1 0 0

0 -0.446

0 0

0 0

0 0

0

- 0.446

0

0

0

0

0

0

0.328

0

0

0

0

0.328

0

0

0

0 -0.164

0

0

0

-0.164 0

0

0 0 -0.164 0 0 0 -0.164 0 0

0 0 0

4.097 0 LO01

-2002 0 0

0 tOO1 0.629 0 0 0.695

--4.097 0 LO01 0 -0.629 0 1.OOl 0 -0.695

0 0.789 0

0 -0.789 0

0 0 0

0 0.789 0

02 0.789 0223

0 0.223 0

03 - 0.789 0 0223

0 0223 0

0

0 0 0 -0.446 -0.446 0

0.789 0 0 0223 0223 0

0 --0.789 0 ---0.789 0 0.223 0 0.223 0

0 0 0 0 0.564 0

0.564 0 0

(9.95)

0 0 -0_564 0 0 o -0_564 0

-3.376 0 0 0 -0326 0 0 0 -1.904

1688 0 1.244

L688 0 -1.244

0 0.163 0

0 0.163 0

1244 0 0.952

(9.96)

-1244 0 0.952

It can be seen that the atomic polarizability tensors obtained following the two alternative approaches as given in expressions (9.93) and (9.96) are equivalent. The equivalency is determined by Eq. (9.91).

PARAMETRIC MODELS OF RAMAN INTENSITIES

VI. E F F E C T I V E

261

INDUCED BOND CHARGES FROM ATOMIC POLARIZABILITY TENSORS

In this section a method for interpretation of Raman intensities based on further transformations of atomic polarizability tensors is presented. The formulation was recently proposed by Dudev and Galabov [333]. A new molecular quantity - effective induced bond charge, O k - is introduced. The effective induced bond charges are obtained from rotation-free atomic polarizability tensors following the strategy as outlined by Galabov, Dudev and rlieva [146] in the infrared case (Section 4.IV). The o k parameters are expected to be associated with polarizability properties of valence bonds.

A. Theoretical Considerations The atomic polarizability tensor of an atom a is defined by expression (9.73). Arranged in a row all cxXa tensors form the polarizability tensor of the molecule, ctx [Eq. (9.74)]. Its elements can be obtained via relations (9.75) and (9.76). As was discussed in Section 9.IV, atomic polarizability tensors are sum of two arrays: (i) vibrational and (ii) rotational polarizability tensors [Eq. (9.79)]. The elements of atomic polarizability tensors are interconnected by the dependency conditions (9.84) and (9.85). The presence of such relations hampers the physical interpretation of these quantifies. In vibrational analysis it is of particular importance to operate with independent quantifies that are associated with vibrational motions of the molecule only. Atomic polarizability tensors [Eq. (9.76)] can be corrected for non-vibrational contributions following the procedure outlined below. The first step is to eliminate the contribution from the rotational atomic polarizability tensor R~ [Eq. (9.78)]. After subtracting Ra from both sides of Eq. (9.76) the vibrational atomic polarizability tensor is obtained V= = ots B s = r x - (tol3 .

(9.97)

The resulting VQt tensor refers to a molecule-fixed Cartesian coordinate system. The respective polarizability derivatives are still not purely vibrational quantifies since they may contain rotational contributions from the compensatory molecular rotation accompanying some vibrational modes. As discussed in Section 9.I, these rotations arise from the zero-angular momentum condition [31,32]. Such terms appear in the case of the non-fully symmetrical distortions in molecules possessing non-spherical polarizability ellipsoid. The contributions from the compensatory molecular rotation form the PS array [F.~s. (9.4) - (9.7)]. A procedure for evaluating rotational contribution terms to polarizability derivatives employing the hypothetical-mass-approach is described in

262

GALABOV AND DUDEV

Section 9.I. After removing the polarizability tensor associated with the compensatory molecular rotation PS BS, Eq. (9.97) reads ax(v) = V a -

psBs = aX-

ap13 - p s B s

-

(9.98)

In this equation ax(v) is the atomic polarizability tensor free from any rotational contribution. Its elements are, however, stiU interrelated through the dependency condition (9.84). The problem can be solved if a set of bond displacement coordinates [Eqs. (4.96) and (4.97)] instead of atomic displacement coordinates is used. A rotationfree bond polarizability tensor is defined as

a~{k)(v)

=

a,~/ax~e) a~/ax~OO a~z~/a,qoo

aa~jax~OO a~,jax~OO a~/a,qoo a,~2~oo a~ax6OO a,~jaxaOO

aX,,x/ayaOO o~/OyaOO a~z,,/o3.aoo

a,~/ayae) oo~ayoOO a~/ayaOO

a,~.JOyaO,) a~/ayaOO ~dayaOO

0~x/0Zs~k)

~~{k)

~~k) (9.99)

Arranged in a row all as(k)(v) form the bond polarizability tensor of the molecule

tZS(V) = [tzS(1)(v)aS(2)(v)...as(k)(v)... {zs(N-I)(v)].

(9.100)

The elements of as(v) are independent quantifies that do not contain any rotational or translational contributions and can be considered as purely intramolecular parameters. These quantifies are defined with respect to a molecule-fixed Cartesian reference frame. The o~(v) tensor is easily obtained from the ax(v) tensor through the relation aa(v ) - aX(V) C -1 .

(9.101)

The structure of C -1 matrix is given in Section 4.IV. An invariant of the bond polarizability tensor with respect to reorientations of the Cartesian reference frame is now defined as the trace of the product

a a (k)(v).~ a (k) (v).

PARAMETRIC MODELS OF RAMAN INTENSITIES

Ok2 = Tr[ctG(k)(v).~i(k)(v)].

263

(9.102)

In Eq. (9.102) ~i(k)(v) is the transpose of a~i(k)(v). The quantity o k, termed effective induced bond charge, has dimensions of electric charge per electric field strength [C/(V/m)]. Ok are expected to be associated with the polarizability properties of chemical bonds. It is anticipated that crk will vary for bonds with different polarizabilities.

B. Applications A representative series of molecules is selected to determine the trends of changes of the effective induced bond charges as defined by Eq. (9.102). The formulation developed has been applied in interpreting atomic polarizability tensors evaluated by HF/6-311+G(d,p) ab initio MO calculations [333]. A series of 17 molecules containing various bonds in different environment have been studied. The molecules are grouped as follows: (a) simple hydrides: CH 4, NH 3, H20, HF, Sill4, PH3, H2S and HC1; (b) organic molecules containing single C-X bonds (X = C, N, O, F): CH3CH3, CH3NH 2, CH3OH and CH3F; (c) organic molecules with double C=X bonds (X = C, N, O, S): CH2CH 2, s-transbutadiene, CH2NH, CH20 and CH2S. The calculation procedure for evaluating crk is illustrated in detail for the water molecule. Calculated O-H bond length and HOH bond angle for H20 are 0.9413A and 106.277o, respectively [333]. The non-zero static molecular polarizability components are (in units of A3): axx = 1.096 , cbS,= 0.813 , azz = 0.863.

(9.103)

The symmetry coordinates employed have their usual form. Cartesian reference system and numbering of atoms and bonds are shown in Fig. 3.6. The atomic polarizability tensor a X obtained as a standard output of the ab initio calculations is as follows (in units of 10-30 C.m/V):

264

GALABOV AND DUDEV

q

~l

0 0 1.413

0 0 0

1.413 0 0

0 0 0

0 0 0.098

0 1378 0 0.098 0 0365 0 0 0

0 0 1.688

n2

-1_459 0 -0.706 0 -0209 0 -0.689 0 -0.702 0 0.083 0 -0209 0 -0.049 0 -0.182 0 -0.706 0 -0.451 0 -0.049 0 -0.702 0 -0.843

(9.104)

H3 t459 o --0.7090 0209 0 --0.689 0 0.702) 0 --0.083 0 0.209 0 --0.049 0 --0.182 0 / --0.706 0 0.451 0 -0.049 0 0.702 0 -0.843)

As already mentioned, aX comprises contributions from two rotational atomic polarizability tensors: apl3 and 9sBs which have to be subtracted. These are given below (in units of 10-30 C.m/V):

ch

153

0 0 0.076

0 0 -0.076

o-o1 0 0

0 0 0

0 0 0

0 0

o 0 0

o o 0 O.O98 0.098 0

o 0 0

o 0 0

H2

0.076 0 -0209 0 0 0 0 -0209 0 --0.049 0 0 0 0 -0.049 0 --0.115 0

-0.076 0 0209 0 0 0 0209 0 -0.049 0 0 0 --0.049 0 0.115

0 0 0

o 0 0

-0.115 0 0

0.HS~

~

(9.105)

PARAMETRIC MODELS OF RAMAN INTENSITIES

0 PsBs = 0 -0.012

0 0 0

O1 0 0 0

-0.012 0 0

0 0 0

H2 0 0 0

0 0 0

0 0 0

265

0 0 0

0 0 0.004

0 0 0

0 0 0

0 0 0.006

0 0 0

0.006 0 0

0 0 0

0 0 0

0.004 0 0

0

0

0.006

0

H3 0

0

0

0

-0.004")

0

0

0

0

0

0

0

0

0

0.006

0

0

0

0

0

-0.004

0

(9.106)

/

In evaluating the compensatory molecular rotation terms to polarizability derivatives, forming the PS array, the hypothetical-mass-isotope method [34-36,288] has been employed. According to the prescriptions given in the method, a non-rotating heavy isotope is constructed by multiplying the mass of the oxygen atom by a factor of 1000. Compensatory molecular rotation arises in the case of the third (antisymmetrical stretching) vibration only. The rotation-free atomic polarizability tensor ctx(v) obtained via Eq. (9.98) is (in units of 10-30 C.m/V): O1 0

tXx(V)= 1.578

0 0

0 0

780

0

0

78

0 0

0 0

0 0

0 0

-1.459 0 -0.788 0 0 0.083 0 0 --0.788 0 -0.451 0

HE 0 0 0

0 0 0

0

0

0365 0

0 1688

-0.689 0 -0.591 0 -0,182 0 -0.591 0 -0.843

(9.107)

H3 1.459 0 -0.788 0 0 0 -0.689 0 0 - 0 . 0 8 3 0 0 0 0 0 - 0 . 1 8 2 o o, 1o o o o

0_591~ O[

Finally, the rotation-free bond polarizability tensor is evaluated with the aid of Eq. (9.101) (in units of 10-30 C.m/V):

266

GALABOV AND DUDEV

~(v)=

r-7 0.083 ~ ~,--0.~o8 o

-0.788 0 -0.451

0 0 0

1.459 0 -0.788 0 0 -0.083 0 0 --(1788 0 0.451 0

rl 0 0 0

r2 0 0 0

0 0 0

-0689 0 -0591 0 -0.182 0 -0591 0 -0.843 (9.108)

0 -0.689 0 0591~ 0 0 -0182 0 0591 0 --0.843J

0]"

The effective induced charge for the O-H bond is calculated [Eq. (9.102)] to be 2.345• 10-30 C.m/V. It is included in Table 9.14. The same approach has been followed in analyzing the atomic polarizability tensors for the other molecules of the series [333]. Effective induced charges for X-H (X = C, N, O, F, Si, P, S, C1), C-H, C-X (X = C, N, 0, F) and C=X (X = C, N, 0, S) bonds have been evaluated. These are given in Tables 9.14, 9.15, 9.16 and 9.17, respectively. Average static atomic polarizabilities for the respective X atoms (AX) [334,335] and the ab mitw calculated bond lengths (rk) are included as well.

C. Discussion of Effective Induced Bond Charges A survey of the effective bond charges obtained for the X-H bonds in the series of eight simple hydrides (Table 9.14) reveals close relationship between these quantifies and the respective bond lengths. The length of a given bond is determined by several factors such as the electronegativity of constituent atoms, the atoms radii, the atoms valence state, conjugation with other bonds and other effects. Thus, bond length appears quite significant characteristics of the bond, reflecting its intrinsic properties. Calculated fix_H parameters are correlated with the respective bond lengths in Fig. 9.5. As can be seen, a satisfactory linear dependence is present. In accordance with expectations, the increase of bond lengths leads, in general, to higher c x - n values. It is interesting to compare the oX_H values with the heavy atom static polarizabilities tabulated in the last column of Table 9.14. The data show quite regular change in OX_H with variations in AX. Comparisons of OX_H parameters between molecules of the same symmetry (CH4 and Sill 4, NH 3 and PH 3, H20 and H2S, and HF and HC1) show that higher AX correspond to higher 6X-H value in the respective couple. This finding is in full accord with expectations since the heavier and bulkier is the atom of a given group of the Periodic table, the higher is the polarizability of the respective bonds.

PARAMETR/C MODELS OF RAMAN INTENSITIES

267

TABLE 9.14 Effective induced charges for X-H bonds (X = C, N, O, F, Si, P, S, C1) from HF/6-31 l+G(d,p) ab initio MO calculations Bond

Molecule

Effective induced bond charge a k (x 10-30 C.m/V)

Bond length rk (• 10-10 m)a

Average atomic polarizability A x (• 10--40 C.m2/V)b

C-H

CH 4

3.202

1.0844

1.958

N-H

NH 3 CH3NH 2

2.795 2.877

1.0003 0.9995

1.224

O-H

H20 CH3OH

2.345 2.598

0.9413 0.9397

0.892

F-H

HF

1.654

0.8973

0.620

Si-H

Sill 4

4.255

1.4773

5.986

P-H

PH 3

4.948

1.4076

4.039

S-H

H2S

4.285

1.3309

3.227

1.2695

2.426

C1-H HC1 3.368 aFrom HF/6-31l+G(d,p) ab initio results [333]. bFrom Ref. [334].

o k for N - H and O-H bonds in CH3NH 2 and CH3OH are quite close to the respective effective induced charges evaluated for NH 3 and H20 molecules, as can be seen from Table 9.14. o k parameters for the C-H bonds in different surroundings are collected in Table 9.15. Data presented show that crk vary in a narrow interval with an average value of 3.291 x 10-30 C.m/V. It can be concluded that the C-H effective induced bond charges are not very sensitive to the environment and reflect mostly local properties of the C-H bonds.

In the low symmetry molecules of the series, such as CH3NH2, CH2NH and

268

GALABOV AND DUDEV

TABLE 9.15 Effective induced charges for C-H bonds from HF/6-31 l+G(d,p) ab imtio MO calculations Bond

Molecule

Effective induced

Bond

bond charge tsk (x 10-30 C.m/V)

length rk (x 10-I0 m)a

C-H

CH 4

3.202

1.0844

Methyl C-H

CH3CH 3

3.159

1.0863

CH3NH2b

3.629trans 3.158 eis

1.0909 1.0846

CH3OHb

3.289trans 3.061 r

1.0878 1.0817

CH3F

3.004

1.0823

CH2CH 2

3.054

1.0766

CH2CHCHCH2b

3.923 trans 3.286 cis

1.0752 1.0772

CH2NHb

3.250trans 3.106 cis

1.0855 1.0814

3.380

1.0942

Methylene C-H

CH20

CH2S 3.578 1.0798 aFromHF/6-31l+G(d,p) ab initio results [333]. bin the case of CH3NH2, CH3OHand CH2NHthe respective C-H bonds are in trans or cis position with respect to the N or O lone pair; in the ease of s-trans-butadiene, methylene C-H bonds are defined with respect to the single C--C bond.

CH3OH, the C-H bonds have different lengths depending on their position with respect to the N and O lone pairs. The C-H bonds lying trans to the lone pair are calculated to have longer lengths. It is interesting to note that these have higher oC_n values.

PARAMETRIC MODELS OF RA_MAN INTENSITIES

269

/

s. ill,

4.2 E

"7

HC!

CH4

3.2 NH3e

=, uX

2.2 ItF 1.2

-lJ

0.8

|

|

,

l,

,

J l , , , , l l

| ~ ,

1.0

i , , , , , , , ,

,i

1.2

rx_n (•

, j | l l w l

1.4

I T

m)

Fig. 9.5. Plot of the dependence between the effective induced bond charges, 6x_ H, and the bond lengths, rx_ n (X = C, N, O, F, Si, P, S, C1) in the series of simple hydrides.

CH3CH3

5.0-

4.s= E

"7 >.

x

u

4.0--

3.s3.0_

CH~

CH3OH

_

2.5

,l

~ i l l W | W l

1.35

I

'

~

1.40

l

l

l

|1

|

I

I

l

1.45

'

'

|

l

|

l

t

l

I | f l | l l l ' i

1.50

rc_ x (x 10-Io m )

Fig. 9.6. Plot of the dependence between the effective induced bond charges, 6C_X, and the bond lengths, rc_ x (X = C, N, O, F) for ethane, methylamine, methanol and methyl fluoride.

270

GALABOV AND DUDEV

TABLE 9.16 Effective induced charges for C-X bonds (X = C, N, O, F) from HF/6-31 l+G(d,p) ab initio MO calculations Average atomic Bond polarizability Ax length rk (xl0-10m) a (x 10-40 C.m2/V)b

Bond

Molecule

Effective induced bond charge Ok (x 10-30 C.m/V)

C-C

CH3CH 3

4.900

1.5271

1.958

C-N

CH3NH2

3.625

1.4536

1.224

C-O

CH3OH

3.148

1.3998

0.892

1.3651

0.620

2.849 C-F CH3F aFrom I-IF/6-31l+G(d,p) ab mitio results [333]. bFrom Ref. [334].

TABLE 9.17 Effective induced charges for C=X bonds (X = C, N, O, S) from HF/6-31 l+G(d,p) ab initio MO calculations Average atomic Bond polarizability AX length rk (x 10-10 m)a (x 10--40 C.m2/V)b

Bond

Molecule

Effective induced bond charge Ok (x 10-30 C.m/V)

C=C

CH2CH 2

7.375

1.3184

CH2CHCHCH2

12.032

1.3235

C=N

CH2NH

5.897

1.2492

1.001

C=O

CH20

4.250

1.1797

0.946

C=S

CH2S

7.627

1.5959

3.494

aFromHF/6-31l+G(d,p) ab mitio results [333]. bThe values referto atom X participatingin a doublebond [335].

1.491

PARAMETRIC MODELS OF RAMAN INTENSITIES

271

In Table 9.16 the effective induced charges for the single bonds between carbon and some first row elements are tabulated. Again, good correspondence with the respective calculated bond lengths (Fig. 9.6) and atomic polarizabilities is observed. Double bonds, due to the higher n-electrons flexibility, are expected to have higher polarizabilities as compared with the respective single bonds. It is, therefore, of particular interest to examine effective induced bond charges for some double bonds. Calculated values are given in Table 9.17. Juxtaposing aC=X with the respective ac_ X (X = C, N, O) quantifies shows that, indeed, the double bonds have larger ak values. Thus, crC=C = 7.375x 10-30 C.m/V and aC_ C = 4.900• 10-30 C.m/V, aC=N = 5.897x 10-30 C.m/V and aC_N = 3.625x10 -30 C.m/V, aC=O=4.250• C.m/V and 6C_O= 3.148• 10- 30 C.m/V. A definite trend of lowering ac= X values with shorter bond lengths and lower atomic polarizabilities is found in the series CH2CH 2, CH2NH and CH20. Comparing the structural analogs CH20 and CH2S, it can be seen that the aC= S parameter value (7.627>(10-30 C.m/V) is much higher than the effective induced bond charge for the C=O bond (4.250• 10-30 C.ndV). This observation can be easily explained in view of the lower electronegativity of sulfur, its higher atom radius and atomic polarizability, and the longer C=S bond length. In Table 9.17 the C=C effective induced bond charge for s-trans-butadiene is also presented. This is the simplest hydrocarbon containing conjugated double bond system. As known, the conjugation results in a significant electron charge redistribution across the molecule and increase in the n-electrons flexibility. Higher carbon-carbon bond polarizabilities are expected in such a case. As can be seen from Table 9.17, the conjugation leads to a drastic increase in the aC= C value which jumps from 7.375• 10-30 C.m/V in ethylene to 12.032x10 -3~ C.m/V in s-trans-butadiene. Again, 6 k is in good accord with the general expectations. The discussion presented above outlines some definite trends of changes in the effective induced bond charges evaluated in analyzing ab initio calculated atomic polarizability tensors, a k are closely related with the polarizability properties of the respective bonds and depend strongly on bond lengths, atomic polarizabilities, bond multiplicity and conjugation with other bonds.

This Page Intentionally Left Blank

C H A P T E R 10

AB INITIO CALCULATIONS OF RAMAN INTENSITIES

Computational Methods ..................................................................................... 274

IIo

A.

Finite Field Calculations of Raman Intensifies .......................................... 274

B.

Polarizability Derivatives from the Energy Gradient ................................ 275

C.

Analytic Gradient Methods ...................................................................... 275

Calculated Raman Intensities ............................................................................. 276 A.

Basis Set Dependence ofAb lnitio Raman Intensities ............................... 276

B.

Influence of Electron Correlation on Quantum Mechanically Predicted Raman Intensities ..................................................................... 278

273

274

GALABOV AND DUDEV

Experimental complete polarizabilitytensor of a molecule of even small size is usually very difficultto obtain. This is a major obstacle in any attempt of consistent nonapproximate interpretationof experimental data. That is, very possibly, the main reason for the predominant application of simple zero order additive models for analyzing or predicting Raman intensities. It has been, therefore, of great importance to derive complete polarizability tensors and compute Raman intensities by ab mitw MO calculations. The remarkable efforts of scientists fzom several leading research groups have resulted in successful realization of these goals [166-168,175,177,178,181,336-341]. Current program systems for ab imtio calculations provide these quantifies as a standard output resulting from application of analytic derivative methods [ 153,170-172].

L COMPUTATIONAL

METHODS

A. Finite Field Calculations of Raman Intensities Hush et al. [336,337,342] have employed finite field perturbation treatment to derive molecular polarizabilities and polarizability gradients from SCF and correlated wave functions. In an uniform electric field f the induced dipole moment of a molecule Pind(f) is given by [336,343]

Pind(O = P(f) - Po = ct f + (1/2) [3 f 2 + (1/6) 7 f 3 + . . . .

(10.1)

f is the field strength, ct - the molecular polarizability,and Po - the dipole moment in absence of electricfield. [3 and ~/are the firstand second hyperpolarizabilitytensors. In the frmnework of the finitefield formalism the perturbation to the Hamiltonian by an uniform electricfield is expressed as [175] H(f) : H o + f ~ .

(10.2)

Ho is the Hamiltonian for the unperturbed system and ~t is the dipole moment operator. From the perturbed wave functions evaluated at SCF or CI level the dipole moment value p(f) is obtained. From the Taylor series expansion [Eq. (10.1)] the coefficients ~ [3 and 7 can be derived from calculations at a number of distortions along particular coordinate and field directions. Molecular polarizability is then obtained from the expression

AB INITIO CALCULATIONS OF RAMAN INTENSITIES

a = [p(f)- Po] / f -

(1/2)]3f2 _ (1/6) 7 f 3

275

.

(10.3)

B. Polarizability Derivatives from the Energy Gradient Komornicki and McIver [178] have provided an efficient method for deriving polarizability derivatives from the energy gradient. Following expressions (7.2)-(7.4) ~[~/O~ are presented as 3ot~,c; aqi

=

a

alan

aqi

afg

=

03 apn ar~

aqi

=

~)

03

~)E

af~ ar n aqi

.

(10.4)

In Eq. (10.4) Pn is a component of the dipole moment, E - the energy, 8E/c3qi- the potential energy gradient, f[ - a component of an external electric field, q i - an appropriate nuclear coordinate, and ~, ~, n = x, y, z in a Cartesian axis system. Again, the same result can be achieved by the procedure prescribed by Schaad et al. [180] by performing standard ab initio calculations with one or more point charges placed at sufficient distance from the molecule so that the electric field is weak. It is seen from Eq. (10.4) that the method of Komornicki and McIver incorporates double numerical differentiation to obtain the respective component of the polarizability derivative tensor.

C. Analytic Gradient Methods Raman intensities are proportional to the square of the derivatives of molecular polarizability with respect to nuclear coordinates. As can be seen from Eq. (10.4), a is a second derivative of the energy with respect to an external electric field. Raman intensities are, therefore, an important third energy derivative property. The development of analytic first [166,168], second [344,345] and third derivative [346,347] methods has been a major breakthrough in ab initio calculations. The creation of efficient ab initio program systems employing analytic derivative techniques has had, as already emphasized, a great effect on vibrational spectroscopy in general. Vibrational assignments and force field calculations have become much more reliable through the combined use of experimental and theoretical data. The solving of the problem with sign indeterminacy for dipole and polarizability derivatives have greatly contributed to all studies associated with the interpretation of vibrational absorption and Raman scattering intensities. These achievements have greatly contributed to our understanding of the intrinsic physical phenomena determining the main parameters of vibrational spectra of isolated or interacting molecules.

276

GALABOV AND DUDEV

The paramount difficulties in the experimental determination of accurate absolute infrared intensities and differential Raman scattering cross sections seem, however, to persist in the theoretical evaluation of these quantifies as well. As we will see, predicted Raman intensities appear to be more consistent with experiment as compared to infrared intensities. Analytic derivative methods for evaluating polarizability derivatives have been developed simultaneously by the theoretical chemistry groups in Cambridge [340] and Berkley [341]. According to Frisch et al. [341] the analytic evaluation of polarizability derivatives is achieved using the expression

aa~ = a3E =(aD ~ ap /

=

~-~

oq/ pl/( 2p oq

~8~ ~q

-~-

+

+

8~

a~

8q

aq

/

(lo.5)

-

In Eq. (1.0.5) Day~ = ( ~tl ~[ v> are dipole moment integrals, P is the electron density matrix, Gq ( ) denotes the contraction of integral derivatives with a matrix, po is a non-perturbed density matrix, S is matrix of overlap integrals, W is energy-weighted density matrix, h is the core Hamiltonian, ( ) denotes a trace of a matrix, and q is a nuclear coordinate (usually in Cartesian space). As can be seen from expression (10.5) no third derivative integrals appear in evaluating polarizability derivatives. No second derivative for the two-electron integrals are also needed. Thus, polarizability derivative calculations do not require much additional time. Second order couple-perturbed Hartree-Fock (CPHF) equations are solved with respect to the six pairs of electric field variables.

II. CALCULATEDRAMAN INTENSITIES A. Basis set Dependence ofAb

Initio

Raman Intensities

Analytic derivative calculations of Raman intensities and Raman depolarization ratios for H20, NH 3, CH 4 and C2H2 have been reported by Amos [ 175,189]. The author

A B INITIO CALCULATIONS OF RAMAN INTENSITIES

277

has used a variety of basis sets including calculafons near the Hartree-Fock limit. Intensities and depolarization ratios for H20 are presented in Tables 10.1 and 10.2 and calculated Raman intensities for CH4 and C2H2 are given in Table 10.3. Overview of the results presented in Tables 10.1-10.3 shows that SCF calculations provide quite satisfactory results for Raman intensi6es. This is in contrast to vibrational absorption intensifies (Chapter 7), where SCF calculations at the highest level could not match in any quantitative terms the experimental values. Acceptable quantitative agreement with experiment for Raman intensities is achieved, however, only with very large basis sets near the Hartree-Fock limit. This is, perhaps, the reason why the results for H20 and CH4 are in much better agreement with measured Raman intensifies than the ab mitio results for C2H2 employing smaller basis sets. Again, as with the case of infrared intensities, there is no general trend in predicted Raman intensities. Some values are overestimated while others are underestimated. The gradual improvement of basis sets shows very slow convergence towards the Hartree-Fock limit values. High level HF/SCF calculations on ethylene using analytic derivative methods have been reported by Frisch et al. [341]. Their results are presented in Table 10.4. Vibrational frequencies and infrared intensities are also given. Interestingly, even at this high level of SCF theory the frequencies for Vl0(Blu) and Vll(B2g) are predicted with reversed order. There are, however, no difficulties in assigning these bands for the correct normal mode on the basis of symmetry properties of the respective normal modes and the fact that B2g mode is Rmnan active while B lu -infrared active. Both infrared and Raman intensities are relatively well predicted at the SCF large basis set calculations. There are no specific theoretical reasons to expect that polarizability derivatives would be less sensitive to correlation effects. Nevertheless, the data of Amos for H20 and CH4 and to lesser extent for C2H2 [175,189] show that near Hartree-Fock SCF calculations may provide quite acceptable values for the intensities in Raman spectra. In a recent study Souter eta/. [352] have obtained experimental Raman spectra of Ga2H6 and Ga2D6 in low-temperature matrices and compared the experimental frequencies and relative intensities with theoretical ab mitio estimates obtained from extended basis set SCF calculations. 14sllpSd contracted to 10s8p2d basis set is used for gallium [353] and two alternative basis sets for hydrogen: (1) DZP contracted Gaussians (4s/2s) augmented with p-functions on the hydrogens; (2) TZP, (Ss/3s) contracted basis set augmented with p-functions on the hydrogens. The calculations have been performed using analytical third derivatives methods using the program PSI developed by Schaefer et a/. [ 171]. A good agreement between calculated and observed Raman intensities is obtained.

278

GALABOV AND DUDEV

TABLE 10.1 Ab mitio HF/SCF Raman scattering coefficients for H2Oa

(in units of A 4 a.m.u.-1) Basis set

vl(B2)

v2(Al)

v3(Al)

4-31G 6-31G DZ 4..31G(2d,2p) 6-31G(2d, Ep) DZP 6-31G extended [5s4p2d]/[3 sEp] [8s6p3d]/[6s2p] [8s6p4d]/[6sap] [Ss61Md lf]/[6s3p 1d] [8s6p4d2f]/[6s3p 1d] Experiment b

39 25 42 18 24 37 27 29 28 24 25 24 19.2

100 71 91 83 87 75 81 71 74 85 85 85 108

10 l0 12 12 4.5 6.8 2.0 3.5 1.3 0.67 0.67 0.68 0.9

aThe theoretical results are taken from Ref. [189] with permission. bFrom Refs. [348] and [349].

B. Influence of Electron Correlation on Quantum Mechanically Predicted Raman Intensities The relative success of SCF ab initio calclflations in predicting Raman intensities seems to be the main reason for the absence of recent detailed studies on the effect of electron correlation on theoretically estimated Kaman intensities. Most correlated ab imtio calculations have been performed sometime ago [337,354-361]. Amos [354] has used pair replacement MCSCF wave function to calculate electric dipole polarizability and polarizability derivatives for I-IF. Double excitations from the single determinant (Hartree-Fock) wave functions from the valence orbital is used. The MCSCF calculations yielded the following values for the mean polarizability and polarizability anisotropy (in units of A 3) = 0.720 tZzz - tZxx = 0.213.

(10.6)

AB INITIO CALCULATIONS OF RAMAN INTENSITIES

279

TABLE 1 0 . 2 Ab miao HF/SCF Raman depolarization ratios for H20a Basis set

vl(B2)

v2(A1)

v3(Al)

4-31G 6-31G DZ 4-31G(2d, Ep) 6-31G(2d,2p) DZP 6-31G extended [5s4p2d]/[3s2p] [Ss6p3d]/[6s2p] [8561Md]/[653 p] [8s6p4d 1f]/[6s3p 1d] [ 8s6p4d2f]/[ 6s3 p I d] Experiment b

0.75 0.75 0.75 0.75 0.75 0.75 0.75 0.75 0.75 0.75 0.75 0.75 0.75

0.720 0.074 0.211 0.669 0.667 0.164 0.080 0.119 0.102 0.069 0.069 0.069 0.03

0.403 0.637 0.412 0.291 0.662 0.521 0.558 0.581 0.670 0.750 0.749 0.75 0 0.74

aThe theoretical results are taken from Ref. [ 189] with permission. bFrom Refs. [348] and [349].

TABLE 10.3 Ab mitio HF/SCF Raman scattering coefficients for CH4a and

C2H2b (in units of A 4 a.m.u. -1) 6-31G(2d,2p)

[5s4p2d]/[3s2p]

[8s6p4d2q/[6s3pld]

Exp.

v 1(F2) v2(Al) v3(E ) v4(F2)

186 138 61 7.2

154 160 35 2.2

151 228 6.6 0.06

128 230 7.0 0.25

Vl(Eg)

53

41

75

v2(Eg )

64

98

125

v4(I-Ig)

6.5

15

4.1

CH4

C2H2

aThe theoretical results for CH4 are taken from Ref. [ 189] with permission and experimental data - from gel. [ 3 0 2 ] . bThe theoretical results for C2H2 are taken from Ref. [175] with permission of John Wiley & Sons, Ltd. Copyright [1987] John Wiley & Sons, Ltd. Experimental data are from Ref. [304].

TABLE 10.4 Analytic inflared intensities and Raman scattering coefficients for ethylene molecule

k0i

from RHF/6-3 1l++G(343p) calculationsa Mode

v I(B2u) v2(B I d v3(Ald v4(B3u) V5(Alg) V6(B3u) v7(Alg) vs(B I d v9(Alu) v lo@ lu) vlI(B2g) VlZ(B2u)

Vibrational frequencies (cm-1)

M a r e d intensities (km/mol)

Theory

Experimentb

Theory

ExperimentC

3367 3338 3285 3263 1813 1590 1472 1343 1137 1080 1097 89 1

3 106 3 103 3026 2989 1623 1444 1342 1236 1023 949 943 826

25.6

26.0

19.4

14.3

12.6

10.4

0 118.1

aThe theoretical results are taken from Ref. [338] with permission. bFrom Ref. [350]. CFrom Ref. (3511. dFrom Ref. [303].

0

Raman scattering coefficients (As/a.m.u.) Theory 113.2 183.8

66.8 172.2

55.8

17.5

61.2 0.34

26.0 2.92 0

0

84.4 7.7

0.004

Experimentd

0.03

1.57

&0

1

82

AB INITIO CALCULATIONS OF RAMAN INTENSITIES

281

The RHF/SCF results are slightly different = 0.715 (10.7) CXzz - axx = 0.180. Experimental value for the anisotropy is a z z - ~ x = 0.22 A 3 [362]. Polarizability derivatives have been also calculated at MCSCF level. Extended basis set SCF and configuration interaction (CISD) calculations for CO have been carried out by Amos [359] with the aim to assess the effect of electron correlation on calculated molecular polarizability and polarizability derivatives. The basis set employed has been of the type (Ss4p2d) contracted Gaussian function on each atom. The configuration interaction function consisted of single and double excitations of the valence shell orbitals. Polarizabilities have been calculated by the finite field method [ 179]. Polarizability derivatives are evaluated by numerical differentiation. The results are given in Table 10.5. In general terms, both SCF and CI results do not differ drastically. It should be remembered that the SCF results are not very close to the Hartree-Fock limit. Thus, the correlated wave functions also contain inaccuracies from the use of limited basis set. Similar calculations have been carried out on the nitrogen molecule [357]. The data are presented in Table 10.6. It can be seen that the CI results are in better agreement with experiment and differ by approximately 20% from the values estimated at SCF level. Marlin et al. [355] have applied multireference CI calculations to evaluate polarizability derivatives for H2S. The basis set for the sulfur atom consisted of 1 ls7p primitive Gaussian orbitals augmented with an additional diffuse functions on the s and p spaces and with three d type polarization functions. The primitive functions have been contracted to (7s/6p). Three s functions and two sets of p polarization functions have formed the contracted basis for the hydrogen. It is important to note that the multireference CI calculations produce values for the molecular polarizability components approximately 25% higher than the CISD estimates. Analytical derivative ab initio calculations on Raman scattering activities provide currently results that are in fairly good accord with the available experimental data. The impact of the theoretical calculations is even greater in view of the difficulties in obtaining full polarizability derivative tensors from experimental measurements.

282

GALABOV AND DUDEV

T A B L E 10.5 Molecular polarizability and polarizability derivatives for CO from SCF and CI calculations a Method

Equilibrium value (A 3)

First derivative (A 2)

at.= %cx

SCF

1.539

0.756

CI

1.637

0.669

all = tgzz

SCF

1.750

2.674

CI

1.872

2.943

SCF

1.609

1.395

1.715 1.93 8b

1.427 1.501 c 1.551 d

CI Experiment

aThr theoretical results arc taken from Rcf. [359] with permission. bFrom gcf. [363]. CFrom R~. [364]. dFrom Rcf. [279].

T A B L E 10.6 Theoretical a and experimental mean polarizability and anisotropy and their derivatives for N 2 (in units of A 3 and A 2) Method

Equilibrium value

SCF

1.681

.... First derivative b 2.184

CI Experiment c

1.720 1.741

1.752 1.752,1.815

SCF

0.779

2.463

CI

0.640

1.932

Experiment c

0.660

2.067

aT~ tla~mtic.al results are taken from Ref. [357] with permission. bThr first derivative is takga with respect to g = (R - Re)/Rr Re is the internuclear distance at equilibrium. CExpcrinamtal data arc taken flom Refs. [365] and [318] for the equilibrium ~ and '5 from Refs. [278] anti [3641 for ~' and from Rcf. [357] for f.

REFERENCES

1. .

3. .

5.

W. Heitler, The Quantum Theory of Radiation, Clarendon Press, Oxford, 1954. H.F. Hameka, Advanced Quantum Chemistry, Addison-Wesley, Reading, Mass., 1965. S. Califano, Vibrational States, Wiley, New York, 1976. E.B. Wilson, Jr., P.C. Cross and J.C. Decius, Molecular Vibrations, McGraw-Hill, New York, 1955. D. Steele, Theory of Vibraaonal Spectroscopy, Saunders, Philadelphia, 1971. M.V. Volkenstein, M.A. Elyashevich and B.L. Stepanov, Vibrations of Molecules (in Russ.), Vol. 2, State Publishers of Technical-Theoretical Literature, Moscow, 1949.

7.

D. Steele, J. Mol. Structure, 117, 163 (1984).

8.

B.L. Crawford, Jr. and H.L. Dinsmore, Jr. Chem. Phys., 18, 1682 (1950).

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B.R. Henry, in Vibrational Spectra and Structure, Vol. 10, (J.R. Durig, ed.), Elsevier, Amsterdam, 1981, p. 269.

10.

I.M. Mills, in Molecular Spectroscopy: Modern Research, Vol. I, (K.N. Rao and C.W. Mathews, eds.), Academic, New York, 1972, p. 115.

11.

E.B. Wilson, Jr. and A.J. Wells, 3". Chem. Phys., 14, 578 (1946).

12.

S.S. Penner and D. Weber, Jr. Chem. Phys., 19, 807 (1951).

13.

D.G. Bourgin, Phys. Rev., 29, 794 (1927).

14.

E. Bartholome, Z. Phys. Chem., B23, 131 (1933).

15.

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301.

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302.

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303.

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304.

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305.

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306.

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307.

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308.

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309.

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310.

S. Montero and T. Dudev, to be published.

311.

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312.

S. Montero, Package of Computer Programs for Raman Intensity Analysis.

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J.L. Duncan, Spectrochim. Acta, 20, 1197 (1964).

316.

J.L. Duncan, D.C. McKean and G.D. Nivellini, J. MoL Struct., 32, 255 (1976).

317.

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318.

G.1LAlms, A.K. BumhamandW.H. Flygare, J. Chem. Phys., 63,3321 (1975).

319.

S. Abbate, M. Gussoni, G. Masetti and G. Zerbi, J. Chem. Phys., 67, 1519 (1977).

320.

L. Silberstein, Phil Mag., 33, 92, 215, 521 (1917).

321.

J. Applequist and C.O. Quicksall, Jr. Chem. Phys., 66, 3455 (1977).

322.

K.IL Sundberg, J. Chem. Phys., 66, 1475 (1977).

323.

P.L. Prasad and D.F. Burow, J. Am. Chem. Soc., 101, 800, 806 (1979).

324.

P.L. Prasad and L.A. Nafie, J. Chem. Phys., 70, 5582 (1979).

325.

B.T. Thole, Chem. Phys., 59, 341 (1981).

326.

R.IL Birge, J. Chem. Phys., 72, 5312 (1980).

327.

T. Keyes, G.T. Evans and B.M. Ladanyi, Jr. Chem. Phys., 74, 3779 (1981).

328.

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329.

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330.

M.P. Bogaard and R. Haines, Mol. Phys., 41, 1281 (1980).

331.

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332.

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333.

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334.

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335.

L. Pauling, General Chemistry, Freeman & Co., San Francisco, 1970.

300

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336.

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337.

J.E. Gready, G.B. Bacskay and N.S. Hush, Chem. Phys., 24, 333 (1977).

338.

P. Pulay, J. Chem. Phys., 78, 5043 (1983),

339.

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342.

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343.

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344.

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345.

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346.

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347.

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359.

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360.

J. Kendrick, Jr. Phys. B., 11, L601 (1978).

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362.

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363.

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301

This Page Intentionally Left Blank

AUTHOR INDEX

Numbers in brackets are reference numbers and indicate that an author's work is referred to although his name is not cited in the text. Underlined numbers give the page on which the complete reference is listed.

-----

Amy, J. W., 50 [70], 286

A-----

Andrews, L., 277 [352], 300

Abbate, S., 55 [80], 63 [80], 142, 143, 144 [151], 224 [298], 242 [319], 287, 290, 29.8, 299

Applequist, J., 221 [290], 245 [289, 321], 247, 248 [321], 249, 297, .299

Ablichs, R., 180 [234], 29_..55 Adamowits, L., 177 [222], 294

Art~ken, G., 82 [119], 83 [119], 110 [119], 119 [1191, 289

Akopian, S. H., 37 [58], 286

Asfle, M. J., 47 [69], 50 [69], 286

Albrecht, A. C., 194 [255, 256, 258], 199 [255], 208 [255, 256], 210, 296

Averbukh, B. S., 78, 82 [118], 106, 109 [129, 130, 131], 110, 111 [118], 132 [129], 142 [131], 164 [129], 252 [129], 288, 289

Aldous, J., 66 [102], 288 Aleksanyaa, V. T., 68, 70, 82, [110], 142 [ 107, 110], 288

B

Bacskay, G., 177 [227], 180 [231, 233], 181 [231], 182 [231], 232 [301], 233 [301], 274 [336, 337, 342], 278 [337], 278 [356], .2.94, 29.5., 298, 300

Aliev, M. IL, 183 [245, 246], 295 Alms, G. R., 241,282 [318], 299 Allen, G., 202 [262], 296

Badawi, H. M., 170 [213], 294

Allen, W. D., 164 [171], 274 [171], 277 [171],291

Bader, R. F. W., 37 [50], 59 [50, 90], 112 [133], 285, 289

Amos, R_ D., 142 [1541, 150 [157], 164 [1541, 165 [175], 166 [1751, 168, 169 [175, 184], 170, 173 [189], 176 [175], 177 [184, 225], 183 [184], 274 [175, 340], 276 [175, 189, 340], 277, 278 [189, 354, 357, 359], 279 [175, 189], 281, 282 [357, 359], 290, 291, 292, 294, 300, 301

Bakshiev, M. G., 37 [56, 57, 58], 286

Amot, G., 152 [162], 153 [162], 159 [1621,291

Barbe, A., 150 [158], 152 [158], 155 [ 158], 183 [239], 291,295

Bailey, W. P., 114 [137], 290 Baird, R. C., 282 [365], 301 Baker, J., 142 [153], 274 [153], 290

Bartholome, E., 18 [ 14], 21,283 303

304

AUTHOR INDEX

Bartlett, R. J., 164 [1721, 177 [222, 223, 224, 226], 179 [224, 2301, 274 [ 172], 291, 29._44

Brandmuller, J., 199 [259], 296

Beers, Y., 63 [96], 64 [96], 173 [961, 179 [961, 181 [96], 182 [961, 185 [96], 288

Brewer, W. E., 170 [212], 29..__44

Behfinger, J., 199 [259], 199 Benedict, W. S., 64 [99], 288 Berckmans, D., 183 [244], 295 Bermejo, D., 235 [302, 307], 238 [302], 2791302], 298 Bemsteirg H. J., 194 [253], 202 [253, 262], 212 [279, 281], 213 [281], 282 [279], ~ 29_.27 Beyer, W. n., 47 [69], 50 [69], 286 Biarge, J. F., 68 [108], 72, 78, 83, 87, 99, 142 [108], 164 [108], 288 Binldey, J. S., 142 [1531, 167 [181], 168 [1821, 169 [1821, 170 [1821, 172 [182, 191], 173 [182], 174 [182], 175 [182], 177 [181], 179 [182], 274 [153, 182, 341], 276 [341], 277 [341], 290, 292. 293, 300 Birge, R. IL, 245 [326, 328, 329], 247, 299 Blanchette, P. P., 68 [105], 83 [105], 85 [ 1051, 87 [ 1051, 288 Bociaa, D. F., 245 [328, 329], 299 Bode, J. H. G., 174 [207], .293. Bogaard, M. P., 249, 282 [363], 299, 301 Boggs, J. E., 165 [ 177], 166 [ 177], 274 [ 1771, 292

Breeze, J. C., 174 [203], 293

Bridge, N. J., 218 [289], 245 [289], 246 [289], 297 Brooks, B. R., 164 [171], 176 [219, 220], 274 [171], 277 [171], 29.__!,294 Bruns, R. E., 37 [431, 55 [431, 57 [43], 285 Buckingham, A. D., 218 [289], 245 [2891, 246 [289], 274 [343], 282 [363], 297, 390, 301 Burnham, A. K., 241 [318], 282 [318], 299 Bttrow, D. F., 245 [323], 248 [323], 29__29

C Califano, S., 2 [31, 5 [3], 7 [31, 15 [3], 26 [3], 27 [3], 29 [3], 31 [3], 32 [3], 36 [31, 113 [3], 150 [3], 152 [3], 283 Carl, J. R_, 221 [290], 245, 247 [290], 297 Carsky, P., 165 [ 1741, 169 [174], 170 [ 174], 292 Castiglioni, C., 55 [81], 58 [81, 85], 62 [81], 63 [80], 102 [126, 127], 103 [127], 287, 289 Chang, R. K., 212 [283, 284], 297 Chantry, G. W., 192 [250], 199 [250], 29.._5.5 Cheam, T. C., 75 [115], 142 [115], 288

Botschwina, P., 180, 182 [236], 295

Cherlow, J. M., 212 [287], 29___77

Bourgin, D. G., 18 [131, 28_..~3

Clifford, A. A., 37 [55], 285

AUTHOR INDEX

305

Clough, S., 63, 64 [96], 173 [96], 179 [961, 181 [96], 182, 185 [96], 2.8..8

[41, 193 [41, 194 [4], 197 [4], 223 [4], 283, 288, 289

Cohen, H. D., 166 [179], 281 [179], 292

Defrees, D. J., 142 [153], 274 [153], 290

Cole, S. J., 164 [172], 274 [172], 291

DeLeeuw, B. J., 183 [247, 248], 184 [247, 248], 185 [248], 186 [248], 295

Colvin, M. E., 169 [186], 176 [218], 177 [2181, 178 [2181, 292, 294 Costain, C. C., 221 [293], 298 Coulson, C. A., 55, 287 Crawford, Jr., B. L., 11 [8], 31 [35, 36], 32, 34, 37 [55], 41 [35], 42, 43 [35, 36], 73 [35, 36], 98, 104 [35, 36], 119 [35, 36], 212 [280], 213 [280], 217 [35, 36], 265 [35, 36], 283, 284, 285, 297 Crawford, M. F., 212 [2781, 282 [278], 297 Crawford, T. D., 164 [171], 183 [248], 184 [248], 185 [248], 186 [248], 274 [171], 277 [171], 291, 295 Cremer, D., 112 [ 133], 289 Cross, P. C., 4 [4], 5 [4], 6 [41, 7 [41, 8 [4], 26 [4], 27 [4], 29 [4], 31 [4], 32 [4], 36 [4], 86 [4], 113 [4], 150 [4], 192 [4], 193 [4], 194 [4], 197 [4], 223 [4], 283 Cyvin, S. J., 27 [17], 36 [17], 283

-----D Daeyaert, F. F. D., 170 [211], 29..3 Davidson E. 1L, 278 [355], 281 [355], 30,,0 Decius, J. C., 4 [4], 5 [4], 6 [4], 7 [4], 8 [4], 26 [4], 27 [4], 29 I4], 31 [4], 32 [4], 36 [4], 68, 70 [109], 72, 82 [117], 86 [4], 88, 90, 91 [117], 113 [4], 143 [109, 117], 145 [117], 150 [4], 158, 192

Dellepiani, D., 210 [276], 29__77 del Olmo, A., 235 [303], 280 [303], 298 del Rio, G., 51, 223 [71], 224 [296], 235 [296], 236 [296], 237 [296], 286, 298 Dementiev, V. A., 38 [60, 61, 62], 55 [60, 61, 62], 61 [60, 61, 62], 286 Dennison, D. M., 68, 288 Depuis, M., 164 [ 170], 291 Dickson, A. D., 31 [36], 34 [36], 42 [36], 43 [36], 73 [36], 98 [36], 104 [36], 119 [36], 217 [36], 265 [36], 285 Dierksen, G. H. F., 177, 278 [361], 294, 30! Dinsmore, H. U, 11 [8], 283 Dinur, U., 37 [51], 59 [51], 103 [51], 285 Dirac, P. A. M., 208, 296 Doggett, G., 59 [88], 287 Domingo, C., 235 [303, 304, 3061, 238 [303, 304, 306], 239 [304], 240 [304], 279 [304], 280 [303], 29_~8 Downs, A. J., 277 [352], 300 Dudev, T., 64 [100], 66 [1001, 75 [112, 114], 95 [124], 118 [141], 119 [142], 120 [142, 143], 127 [141], 130 [143, 144, 145], 131 [146], 133 [146], 136 [148], 137 [148], 138 [148], 139 [148],

306

AUTHOR INDEX

140 [148], 142 [100, 124], 143 [100, 124], 175 [114], 218, 219 [2881, 221 [288], 222 [288], 235 [310, 311], 238 [310], 242 [311], 243 [311], 244 [311], 249 [331], 258, 261, 263 [333], 265 [288], 266 [333], 267 [333], 268 [333], 270 [333], 288, 289, 290, 297., 298, 299 Duncan, J. L., 27 [19], 28 [24], 36 [19, 24], 65, 66 [241, 126 [241, 174 [2051, 239, 240 [263], 241 [314, 315], 242, 244 [316], 284, 293, 299 Dunning, T. H., 277 [3531, 300 Dupuis, M., 169 [ 183], 29__.22 Durig, D. T., 170 [213], 294 Durig, J. 1L, 27 [221, 36 [221, 75 [1141, 118 [141], 120 [1431, 127 [1411, 130 [143], 170, 175 [114], 253 [332], 254 [332], 284, .288, 290, 293, 294, 299

Eugster, C. H., 210 [277], 297 Evans, G. T., 245 [327], 247 [327], 299 Ewig, C. S., 167 [ 180], 275 [ 180], 29__22 Exner, O., 51 [75], 287

F Feng, F. S., 170 [209], 293 Fenner, W. R., 212 [286, 287], 297 Ferigle, S. M., 52 [77], 224 [77], 287 Fermann, J. T., 164 [171], 274 [171], 277 [171], 29! Femandez-Sanchez, J. M., 235 [305], 238 [305], 298 Ferrisco, C. C., 174 [203], 293 Figeys, H. P., 183 [244], 295

------

E

-----

Ebenstein, W. L., 172 [ 197], 293 Eckardt, G., 202 [261], 29__fi6 Eckart, C., 29, 40, 54, 72, 261 [31 ], 284 EdgeU, W. F., 50 [70], 28__.fi6 Eggers, D. F., 278 [355], 281 [355], 390 Elkins, J. W., 174 [200], 293 Elyashberg, M. E., 38 [621, 55 [621, 61 [621, 28___fi6 Elyashevich, M. A., 7 [6], 26 [6], 27 [6], 29 [61, 32 [61, 36 [61, 40 [63], 51 [63], 52 [6, 63], 113 [6], 150 [6], 283, 286 Escdbano, R., 51, 223, 235 [306], 238 [306], 286, 298

Fitzgerald, G. B., 164 [171, 172], 177 [223, 226], 274 [ 171, 172], 277 [ 171], 29 !, 294 Fletcher, W. F., 32, 42 [38], 28.5 Flygare, W. H., 241 [318], 282 [318], 299 Fogarasi, G., 27 [21], 28 [21], 36 [21], 165 [21, 177], 166 [177], 170 [21], 274 [177], 284, 292 Fontal, B., 282 [364], 30_.._21 Foresman, J. B., 142 [153], 274 [153], 29.0 Fouche, D. G., 212 [283,284], 297 Fox, D. J., 142 [153], 274 [1531, 2..90 Franzosa, E. S., 210 [272], 297 Fried, A., 174 [200], 293

AUTHOR INDEX

307

Frisch, M. J., 142 [153], 164 [153], 168 [182], 169 [1821, 170 [182], 172 [182], 173 [182], 174 [182], 175 [182], 179 [182], 274 [153, 341], 276, 277, 290, 292 Fung, K. K., 221 [290], 245, 247 [290], 297 Furer, V. L., 38 [59], 286

G

,,,

Gailar, N., 64 [99], 288 Galabov, B., 37 [42, 44, 48], 51 [48], 55 [42, 44, 481, 57 [42, 44, 481, 58 [44], 60 I42], 61 [94], 64 [100], 66 [1001, 72 [111], 75 [112, 113, 114], 95 [124], 115, 118 [141], 120 [1431, 126 [48], 127 [141], 130 [143, 144, 145] 131, 133 [146], 136 [148], 137 [1481, 138 [148], 139 [148], 140 [148], 142 [100, 124], 143, 164 [140], 175 [114], 218, 219 [288], 221 [288], 222 [288], 249 [3311, 253 [332], 254 [332], 258, 261, 263 [333], 265 [288], 266 [333], 267 [333], 268 [333], 270 [333], 285, 287, .2.88, 289, 290, 297, 299 Gans, P., 27 [20], 36 [201, 284 Gatti, C., 37 [50], 59 [50], 285 Gaw, J., 168 [182], 169 [182, 187], 170 [182], 172 [182], 173 [182], 174 [182], 175 [182], 179 [182], 274 [341], 275 [345, 347], 276 [341], 277 [341], 292, 300 Georgieva, G., 75 [ 112], 288 Gerratt, J., 164 [ 166], 167 [ 166], 274 [ 1661, 275 [ 1661, 291 Geerlings, P., 183 [244], 295 Girin, O. P., 37 [56, 57], 286

Goddard, J. D., 164 [169], 165 [169], 176 [220], 291,294 Golden, D. M., 212 [280], 213 [280], 297 ,, Golden, W. G., 183 [243], 295 Goldman, L. M., 212 [285], 297 Goldsmith, M., 152 [162], 153 [162], 15911621,291 Gonzalez, C., 142 [153], 274 [153], 290 Gordy, W., 46 [671, 221 [2911, 286, 297 Gough, K. M., 115, 239 [313], 290, 29.8 Gounev, T., 75 [113], 288 Gray, D. L., 174 [206], 293 Gready, J. E., 274 [336, 337, 342], 278 [337], 300 Green, W. H., 150 [157], 183, 184 [238], 291, 29_..55 Greene, T. M., 277 [352], 300 Gribov, L. A., 38 [60, 61, 62], 51 [72], 52 [72, 78], 53 [72], 54 [72, 79], 55 [60, 61, 62, 72], 59, 61 [60, 61, 62, 72], 63 [72], 150 [155], 151 [155], 160, 161 [155], 162, 164 [72], 192 [155], 224 [155], 225 [155], 226 [155], 231 [155], 243 [155], 286, 287, 291 Guirgis, G. A., 170 [212], 294 Gussoni, M., 55 [80, 81], 58 [81, 85], 62 [81], 63 [80, 81, 95, 97], 102, 103 [127], 142, 143, 144 [151], 224 [298], 230 [300], 242, 287, 288, 289, 290, 298

308

AUTHOR INDEX

----

He=berg, G., 172 [190, 194], 174 [190, 199], 186 [190], 208, 293, 296

H - - - -

Hagler, A. T., 37 [51], 59 [51], 103 [51], 28.5 Haines, 1L, 249, 299

Hester, R. E., 192 [251], 296

Hameka, H. F., 2 [2], 283 Hamilton, T. P., 164 [171], 274 [171], 277 [171], 29__! Hamaguchi, H., 210 [275, 277], 297 Handy, N. C., 150 [157], 169 [184, 185, 187], 176 [216, 219], 177 [184, 225], 183 [184, 238], 184 [238], 274 [339], 291, 292, 294, 295, 300 Hanson, H., 152 [163], 291 Harada, L., 28 [29], 210 [275], 284, 297 Harris, IL R., 59 [89], .287 Harrison, IL J., 164 [172], 177 [223, 226], 274 [172], 291., 294 Havriliak, S., 169 [186], 176 [218], 177 [218], 178 [218], 292, 294 Hay, P. J., 278 [358], 301 Head-Gordon, M., [153], 29_.._0.0

Hess, Jr., B. A., 165 [174], 167 [180], 169 [ 174], 170 [ 174], 275 [ 180], 292

142 [153], 274

Hirschfeld, T., 194 [252], 195 [252], 29__fi6 Holzer, W., 212 [281], 213 [281], 297 Hornig, D. F., 51 [74], 54 [74], 172 [196], 287, 293 Hoy, A. R., 173 [198], 185 [198], 293. Huber, K. P., 174 [ 199], 293 Hurley, A. C., 180 [233], 295 Hush, N. S., 180 [231, 233], 181 [231], 182 [231], 232 [301], 233 [301], 274 [336, 337, 342], 278 [337, 356], 395, 298, 300 Hutley, M. C., 194 [258], 296 Hyatt, H. A., 212 [286, 287], 297 Hyde, G. E., 172 [196], 293 Hylden, J. L., 150 [156, 161], 152 [161], 155 [161], 156, 158, 160 [161, 165], 291

Hehre, W. J., 172 [ 191], 293 Heisenberg, W., 208, 296 Heitler, W., 2 [ 1], 28__.33 Helgaher, T., 176 [217], 294 Henry, B. R., 14 [91, 113 [9], 283 Herratm, J., 68 [108], 72, 78, 83 [108], 87 [1081, 99 [108], 142 [1081, 164 [108], 288

Ilieva, S., 72 [111], 75 [113, 114], 95 [124], 131 [146], 133 [146], 136 [1481, 137 [148], 138 [148], 139 [148], 140 [148], 142 [124], 143 [124], 175 [114], 261, 288, 289, 290 Inagaki, F., 210 [269], 296 Innes, K. K., 210 [272], 297. Ito, M., 210 [270, 271,274], 296, 297

AUTHOR INDEX

_.__

J._~_

309

Kindness, A., 113 [ 136], 114 [ 136], 115 [136], 289

Jalsovsky, G., 37 [41], 55 [411, 57 [41], 285

King, G. W., 206 [265], 296

Janssen, C. L., 164 [171], 274 [171], 2771171],291

King, W. T., 37 [45], 58 [45], 68 [45, 1051, 78, 83, 85, 87, 101, 102, 285, 288

Jaquet, R., 174 [201 ], 293

Klein, G., 63 [96], 64 [96], 173 [96], 179 [96], 288

Jayatilaka, D., 150 [157], 183, 184 [238], 291,295

Klein, M. L., 37 [40], 2.8.5

John, I. G., 278 [356], 300

K16ckner, H.W., 201 [260], 202 [260], 212 [260], 213 [260], 235 [307], 238 [307], 296, 298

Jorgensen, P., 176 [217], 294

Knowles, P. J., 176 [216], 294

Jouve, P., 150 [158], 152 [158], 155 [158], 183 [239], 291,295

Koga, Y., 65 [101], 288

Jensen, H. J., 176 [217], 294

Kahn, L. IL, 142 [1531, 274 [1531, 290 Kalantar, A. H., 210 [272], 297 Kalasinshy, V. F., 239 [317], 241 [3171, 243 [3171, 299 Kaplan, L., 47 [68], 286 Karasev, Y. Z., 38 [62], 55 [621, 61 [621, 286 Kato, S., 176 [214], 294 Kaya, K., 210 [270, 271], 296, 297 Kellam, J. M., 212 [286], 297 Kemble, D. E. C., 18 [ 15], 283 Kendrick, J., 278 [360], 301 Kessler, M., 221 [291], 297 Keyes, T., 245 [327], 247 [327], 299

Komornicki, A., 166, 167, 169, 177 [ 178], 180 [ 178], 274 [ 178], 275, .292 Kondo, S., 65, 93, 177 [228], 178 [228], 280 [351], 288, 289, 294, 300 Koops, Th., 61 [93], 135 [147], 174 [147, 208], 186 [208], 287, 290, 293 Kotov, S. V., 54 [79], 287 Kovner, M. A., 51 [73], 52 [73], 55, 59 [73], 60 [73], 67 [73], 202 [73], 224 [73], 287 Krainov, E. P., 51 [73], 52 [73], 55, 59 [73], 60 [73], 67 [73], 202 [73], 224 [73], 287 Kraka, E., 112 [ 133], 289 Kramers, H. A., 208, 296 Kfimm, S., 75 [115], 142 [115], 288 Krislman, R., 167 [181], 177 [181, 221], 274 [181], 292, 294 Kubulat, K., 58 [86], 68 [106], 103, 116,287

310

AUTHOR INDEX

Kuchitsu, K., 93 [122], 94 [122], 289 Kutzelnigg, W., 174 [201], 293 ---~

L------

"'

M

Ma, Buyong, 277 [352], 300 Magers, D. H., 164 [ 172], 274 [ 172], 2..9.!

LaBoda, M. L, 183 [241], 295

Mallinson, P. D., 241 [314], 299

Laidig, W. D., 164 [172], 176 [219, 220], 177 [222, 223, 226], 274 [172], 291, 29.4

Marcott, C., 183 [243], 295

Laadanyi, B. M., 245 [327], 247 [327], 299 Lapp, M., 212 [285], 297 Larouche, A., 37 [50], 59 [50], 285 Lee, Min Joo, 170 [213], 294

Martin, J., 202 [263], 204, 235 [263, 3091, 238 [263], 239 [2631, 240 [2631, 242, 296, 298 Martin, R. L., 142 [153], 274 [153], 278 [355], 281, 290 Masseti, G., 210 [276], 242 [319], 297, 299

Lengsfield, B. H., 169, 176 [218], 177, 178 [218], 292, 294

Mast, G. B., 37 [45], 58 [45], 68 [45, 105], 78, 82 [ 117], 83 [ 1051, 85 [ 105], 87 [105], 88, 90, 91 [117], 101, 102, 143, 145 [117], 285, 288, 289

Leonard, D. A., 212 [282], 297

Matsmtra, H., 28 [28, 29, 30], 284

Libov, N. G., 37 [56], 286

Mayants, L. S., 78, 106, 109 [129, 130, 131], 132 [129], 142 [129], 164 [129], 252 [129], 288, 289

Lee, Y. S., 164 [172], 274 [172], 291

Lipscomb, W. N., 177 [224], 179 [224], 294 Little, T. S., 170 [212], 239 [317], 241 [317], 243 [317], 294, 29.9 Liu, Jie, 170 [210], 293 Livington, 1L L., 47 [68], 286 Lokshin, B. V., 68, 70 [107], 142 [107], 288 Long D. A., 52 [76], 224 [76], 28.7 Loytsyanski, L. G., 40 [64], 286 Lurie, A. I.,40 [64], 286

Mayers, D. H., 177 [224], 179 [224], 294 McClellan, A. L., 174 [204], 293 McCullosh, R. D., 241 [314], 299 McDougall, P. J., 37 [50], 59 [50], 285 Mclver, J. W., 166, 167, 169, 177, 180 [178], 274 [ 178], 275, 292 McKean, D. C., 28 [24], 36 [24], 51 [74], 54 [74], 65 [24], 66 [24], 113, 114 [136], 115 [136], 126 [24], 239, 241 [314], 242, 244 [316], 284, 287, 289, 299

AUTHOR INDEX

Melius, C. F., 142 [153], 274 [153], 290 Meyer, W., 180 [232, 237], 295 Miehalska, D., 167 [180], 275 [180], 292 Mikami, N., 210 [270, 271], 296, 297 Mildaailov, V. M., 183 [245], 295 Miller, F. A., 50 [70], 286 Mills, I. M., 14 [10], 28 [231, 31 [36], 34 [36], 36 [23], 42 [361, 43 [36], 63, 66 [102], 73 [36], 98 [36], 104 [36], 113 [110], 119 [36], 164 [166], 167 [166], 173 [198], 174 [205], 182, 185 [198], 186 [249], 217 [36], 265 [36], 274 [166], 275 [166], 283, 284, 285, 288, 291, 293, 295

Mink, J., 28 [261, 36 [261, 284 Miyazawa, T., 210 [269], 296 Moeia, 1L, 275 [346], 300 Montero, S., 202 [263], 204, 224 [296, 297], 235 [263, 296, 297, 302, 303, 304, 305, 306, 307, 308, 310], 236 [296, 297], 237 [296, 297], 238 [263, 310, 312], 239 [263, 304, 312], 240 [263, 304], 279 [302, 304], 280 [303], 296, 298 MorciUo, J., 68 [ 108], 72, 78, 83 [108], 87 [108], 99 [108], 142 [108], 164 [ 1081, 288 Morokuma, K., 176 [214], 294 Morrison, M. A., 278 [358], 301 Muenter, J. S., 172 [197], 281 [362], 293, 301 Mttlliken, R. S., 59 [87], 101, 287

311

Murphy, W. F., 115, 203 [2641, 212 [281], 213 [281], 235 [304], 238 [304], 239 [304], 240 [304], 278 [348,349], 279 [304, 348, 349], 290, 296, 297, 298, 300 ------

N

~

Nafie, L. A., 245 [324], 299 Nakagawa, I., 93, 289 Nakanaga, T., 65 [101], 177, 178 [228], 280 [351], 288, 294, .300 Newall, A. C., 282 [365], 301 Newton, J. H., 31 [33], 33 [33], 68 [33], 78, 79, 80 [33], 85, 86 [33], 87 [33], 142 [33], 164 [33], 284 Nielsen, H. H., 152 [162, 163], 153 [162], 159 [162], 29.1 Nikolova, B., 37 [48], 51 [48], 55 [48], 57 [48], 115 [140], 130 [145], 164 [ 140], 285, 290 NiveUini, G. D., 239, 242, 244 [316], 299 Novoselova, O.V., 38 [61], 55 [61], 61 [61], 286 O

Ogawa, Y., 28 [28, 29], 284 Oka, T., 44 [65], 286 Okamoto, H., 194 [257], 208 [257], 210 [257, 277], 296 Orduna, M. F., 235 [303, 304], 238 [303, 304], 239 [304], 240 [304], 279 [304], 280 [303], 298 Orr, B. J., 274 [343], 300

312

AUTHOR INDEX

Orville-Thomas, W. J., 37 [41, 42, 44, 48, 54], 51 [48], 55 [41, 42, 43, 44, 48], 57 [41, 42, 44, 48], 58 [44], 60 [42], 61 [94], 115 [140], 118 [141], 126 [48], 127 [141], 130 [144, 145], 150 [155], 151 [155], 160 [155], 161 [155], 162, 164 [140], 192 [155], 224 [155], 225 [155], 226 [155], 227 [155], 231 [155], 242 [155], 285, 287, 290, 291 Oz-z~ J. M., 51, 223 [71], 235 [306], 238 [3061, 286, 298 Osamma, Y., 164 [ 169], 165 [ 169], 275 [345], 291., .300 Overend, J., 21 [ 16], 22 [ 16], 23, 116, 150 [156, 159, 160, 161], 152 [159, 160, 161], 155 [161, 164], 156, 158, 159, [164], 160 [161, 165], 183 [240, 24 I, 242, 243], 283, 291, 295 p .,..._...._

Pal, S., 164 [ 172], 274 [ 172], 291 Pang, F., 165 [177], 166 [177], 274 [ 177], 292

Phan, H. V., 75 [ 114], 170 [209], 175 [ 114], 288, 293 Pierens, R. K., 282 [363], 30 ! Pine, A. S., 174 [200], 293 Placzek, G., 194 [254], 202 [254], 296 Plyer, E. K., 64 [99], 288 Politzer, P., 59 [89], 287 Pople, J. A., 142 [153], 167, 172 [191], 177 [181, 221], 274 [153, 181], 29__Q0 Porto, S. P. S., 212 [286, 287], 297 Pulay, P., 27 [21], 28 [21], 36 [21], 164 [167, 168], 165 [21, 168, 177], 166 [177], 169 [168], 170 [21, 168], 176 [168], 274 [167, 168, 177, 338], 275 [168], 280 [338], 284, 291,292, 300 Prasad, P. L., 245 [323, 324], 248 [323], 299 Purvis, G. D., 164 [172], 179 [230], 274 [ 172], 291,295

Pao, C., 47 [68], 286 Parisean, M. A., 155 [164], 159 [164], 291

Q

......_._.._

QuicksaU, C. O., 245 [321], 248 [321], 249, 299

Patat, F., 172 [ 194], 29..3 Pauling~ L., 140 [149], 266 [335], 270 [335], 29.0, .299 Penner, S. S., 17, 23, 283 Penney, C. M., 212 [285], 297 Person, W. B., 31 [33], 33 [33], 37 [43, 531, 38 [531, 51 [531, 55 [43, 53], 57 [43], 58 [86], 63, 68 [33, 106], 78, 79, 80 [33], 85, 86 [33], 87 [33], 103, 105, 116, 142 [33], 164 [33], 165 [53, 176], 169 [1761, 170 [176], 183 [240], 284, 285, 287, 288, 292, 295

---~

Raghabachari, [ 153], 290

R----

K.,

142 [153], 274

Ramos, M. N., 58 [85], 102 [126, 127], 103 [1271, 287, 289 Ratajczak, H., 37 [54], 285 Reiehle, Jr. H. G., 172 [ 193 ], 293 Remington, R., 164 [171], 274 [171], 277 [171], 291

AUTHOR INDEX

313

Rendell, A. P. L., 232 [301], 233 [301], 298

Samvelyan, S.Kh., 68, 70 [107, 110], 82 [ 110], 142 [ 107, 110], 288

Rice, J. E., 142 [1541, 164 [154], 177 [2251, 290, 294

Saxe, P., 176 [219], 275 [344, 345], 294, 300

Riley, G., 37 [441, 55 [441, 57 [441, 58 [44], 285

Sayvetz, A., 29, 40, 54, 72, 261 [31], 284

Ring, H., 221 [2911, 297

Schaad, L. J., 165 [174], 167, 169 [ 174], 170 [ 174], 275, 292

Rittby, M., 164 [172], 274 [172], 291 Robb, M., 142 [153], 176 [215], 274 [153], 290, 294 Robiette, A. G., 174 [206], 293 Rocks, L., 47 [68], 286 Rogers, M. T., 55 [83], 287 Roos, B. O., 180 [235], 295 Roothaan, C. C. J., 166 [179], 281 [ 179], 292 Rothman, L., 63 [96], 64 [96], 173 [96], 179 [96], 288

Schaefer III, H. F., 164 [169, 171], 165 [169], 168 [182], 169 [182, 185, 186, 188], 170 [182], 172 [182], 173 [182], 174 [182], 175 [182], 176 [218, 219, 220], 177 [218], 178 [218], 179 [182], 183, 184, 185 [248], 186 [248], 274 [171, 339, 341], 275 [344, 345, 347], 276 [341], 277 [171, 271, 341, 352], 291,292, 294, .295, 3.00 Schick, G.A., 245 [328, 329], 299 Schlegel, H. B., 142[153], 167 [181], 176 [215], 177 [181, 221], 274 [153, 181], 290, 292, 294

Rousseau, D. L., 210 [273], 297

Schr6tter, H.W., 201 [260], 202 [260], 212 [260], 213 [260], 296

Rupprecht, A., 31 [37], 106, 108 [37], 110 [37], 142 [37], 224 [299], 249 [299], 250 [299], 252 [299], 285, 298

Secroun, C., 150 [158], 152 [158], 155 [158], 183 [239], 291,295

------

S----

Sadlej, A. J., 177, 278 [361], .294, 301 Saebo, S., 177 [227], 294 Sa~ki, S., 65 [ l01], 93, 177 [228], 178 [2281, 221 [2921, 280 [351], 288, 289, 294, 298, 300 Saito, S., 2 l0 [277], 297 Salters, E. A., 164 [172], 274 [172], 291 Sambe, H., 99, 2.8.9

Seeger, R., 142 [153], 274 [153], 290 Segal, G. A., 37 [40], 285 Seidl, E. T., 164 [171], 274 [171], 277 [171], 291 Sexton, G. J., 176 [216], 294 Shaffer, W. H., 152 [ 163], 291 Sherrill, C. D., 164 [171], 274 [171], 2771171],291 Shimanouchi, T., 27 [18], 28 [28, 29], 36 [18], 93, 172 [195], 174 [195], 210 [275], 284, 289, 293,297

314 Siesbahnn, P. E. M., 180 [235], 295 Silberstein, L., 245, 299 Simandiras, E. D., 169, 177 [225], 183 [ 184], 292, 294 Slee, T. S., 112 [133], 289 Stair, W. M. A., 31 [34], 33 [34], 42, 45, 48 [341, 58 [84], 60, 61 [931, 65 [34], 73 [34], 96 [34], 97 [34], 104 [34], 119, 121 [34], 135 [34, 147], 142 [152], 173 [34], 174 [147, 207, 2081, 179 [34], 185 [34], 186 [208], 217 [341, 265 [34], 284, 287, 290

AUTHOR INDEX Stewart, J. J. P., 142 [153], 274 [153], 290 Strey, G., 173 [198], 185 [198], 186 [249], 293, _295 Sufxa, S., 210 [276], 29__27 Sullivan, J. F., 170 [213], 294 Sundberg, K. R., 245 [322], 299 Sutton, L. E., 46 [66], 286 Suzuka, I., 210 [270, 271, 274], 296, 297

Smith, W. V., 46 [67], 286

Suzuki, E., 210 [275], 297

Snyder, R. G., 28 [27], 39, 72, 73, 74, 75, 284

Suzuki, I., 155 [164], 159 [164], 29.._.21

Sosa, C., 164 [ 1721, 274 [ 1721, 291 Sourer, P. E., 277, 300 Speirs, G. K., 28 [241, 36 [241, 65 [241, 66 [24], 126 [241, 284 Spiro, T. G., 282 [364], 301

Suzuki, S., 37 [44], 55 [44], 57 [441, 58 [441, 285 Sverdlov, L. M., 51 [73], 52 [73], 55, 59, 60 [73], 67 [73], 202 [73], 224 [73], 287 . . . . .

Swalen, J. D., 221 [293], .298

Staemmler, V., 174 [201], 293

Swanton, D. J., 180 [231], 181 [231], 182 [231], 295

Stansbury, E. J., 212 [278], 282 [278], 297

Szczepaniak, K., 165 [ 176], 169 [ 176], 170 [ 176], 292

Stanton, J. F., 177 [224], 179 [224], 294.

-----

T

------

Takagi, K., 44 [65], 286.

Steele, D., 5 [5], 7 [5], 11, 26 [5], 27 [5], 29 [51, 32 [51, 36 [5], 37, 38 [53], 51 [53], 55 [53], 75 [113], 113 [5], 150 [5], 165 [53, 173], 192 [5], 283, 285, 288, 292

Tanabe, K., 221 [292], 298

Stepanov, B. L., 7 [61, 26 [6], 27 [61, 29 [6], 32 [6], 36 [61, 52 [61, 113 [61, 150 [ 6 ] , 283

Tasumi, M., 28 [301, 142 [1501, 210 [269, 277], 284, 290, 29.._.66

Stewart, D., 113 [136], 114 [136], 115 [1361, 289

Tang, J., 194 [256], 208 [256], 210 [256], 296

Taylor, P. R., 177 [227], 180 [233, 235], 294, .295

AUTHOR INDEX

315

Teller, E., 208, 29....fi6 Thole, B. T., 245 [325], 247, 29__.29 Thomas, J. R., 183 [247, 248], 184 [247, 248], 185 [248], 186 [248], 295

Volkenstein, M. V., 7 [6], 26 [6], 27 [6], 29 [6], 32 [6], 36 [6, 39], 40 [39, 63], 51 [39, 63], 52 [6, 63], 113 [6], 150 [6], 224, 283, 285, 286

- - - -

W

------

Todorovski, A. T., 38 [60], 55 [60], 61 [601, 286

Waggoner, J., 152 [ 163], 29_...!

Topiol, S., 142 [153], 274 [153], 290

Wagner, W. G., 202 [261], 296

Torii, H., 142 [1501, 29___0.0

Waiters, V. A., 114 [ 137], 29_....0

Trambarulo, R. F., 46 [67], 221 [291], 28_..66,297

Wang, A., 27 [22], 36 [22], 170 [209], 284, 293

Trucks, G. W., 142 [153], 164 [172], 274 [ 153, 172], 29..._00,291

Warsop, P. A., 235 [308], 298

Tubbs, L. D., 172 [192], 174 [202], 29__~3

Watson, J. K. G., 183 [246], 295 Weast, R. C., 47 [69], 50 [69], 286

U

,,

Udagawa, Y., 210 [270, 271], 296, 297

Weber, A., 52 [77], 224 [77], 28_.!7 Weber, D., 17, 23,283 Welsh, H. L., 212 [278], 282 [278], 29.7

----

V-----

Vacex, G., 164 [171], 183 [247, 248], 184 [247, 248], 185 [247, 2481, 186 [248], 274 [ 171], 277 [ 171], 291,295

Wendolski, J. J., 37 [46, 47, 49], 51 [47], 57 [46, 47, 49], 58 [46], 59 [47, 49], 61 [46], 126 [49], 134, 169 [183], 285, 2.9.2

van Dam, T., 58 [84], 287

Wells, A. J., 17, 19, 21,283

van der Kveken, B. J., 170 [210], 293

White, A. H., 282 [363], 301

van Straten, A. J., 31 [34], 33 [34], 42, 45, 48 [34], 60, 65 [34], 73 [34], 96 [34], 97 [34], 104 [34], 119, 121 [34], 135 [34], 142 [152], 173 [34], 179 [34], 185 [34], 217 [34], 265 [34], 284, 290

Whiteside, R. A., 142 [153], 274 [153], 290

Verleger, H., 172 [ 194], .293

Wiberg, K. B., 37 [46, 47, 49, 50], 51 [47], 57 [46, 47, 49], 58 [46], 59 [47, 49, 50], 61 [46], 66 [103], 114, 126 [49], 134, 285, 288, 290

Vincent, M. A., 275 [345], 300

WiUets, A., 150 [157], 2.91

Visser, T., 135 [147], 174 [147, 208], 186 [208], 290, 293

Willetts, S., 183, 184 [238], 295 Williams, D., 172 [192], 174 [202], 293

316

AUTHOR INDEX

Williams, P. F., 210 [273], 297 Wilson, Jr., E. B., 4 [4], 5 [4], 6 [4], 7 [4], 8 [4], 17, 19, 21, 26 [4], 27 [4], 29 [4], 31 [4], 32 [4], 36 [4], 86 [4], 113 [4], 150 [4], 192 [4], 193 [4], 194 [4], 197 [4], 223 [4], 283 Wooster, W. A., 110 [132], 199 [132], 289 X

Xie, Y., 164 [1711, 274 [171], 277 [171], 291

____

y___

Yamaguchi, Y., 164 [169, 171], 165 [169], 168, 169 [182, 188], 170 [182], 172 [lg2], 173 [182], 174 [182], 175, 176 [220], 179, 183 [248], 184 [248], 185 [248], 186 [248], 274 [171, 341], 275 [344, 345, 347], 276 [341], 277 [ 171, 341], 291, .292, 294, 295, 3..00 Yao, S., 150 [159], 152 [1591, 155 [159], 183 [2421, 291, 295 Yoshino, T., 212 [279], 282 [279], 297

Young, C., 172 [ 193], 293 - - -

Z----

Zahradnik 1L, 165 [1741, 169 [174], 1701174],292 Zerbi, G., 55 [80, 81], 58 [81, 85], 62 [811, 63 [80, 81, 971, 102 [126, 127], 103 [127], 210 [276], 224 [298], 242 [319], 287, 288, 289, 297, 298, 299 Zilles, B., 63, 288

SUBJECT INDEX

- - -

anisotropy spectra, 203

A - - -

argon, Ar, 23

ab mitio MO calculations, 40, 66, 112,

135, 142, 164, 203, 242

atom dipole interaction model, 245

ab mitio Raman intensifies, 276

atomic Cartesian displacement coordinates, 29, 30, 69, 249, 262

absolute differential Raman scattering cross section, 202, 214

atomic charge--charge flux model, 68

absolute differential Raman scattering cross section of nitrogen, 212

atomic dipoles, 58 atomic effective charge model, 142

absolute trace and anisotropy Raman differential cross sections, 204

atomic polar tensors, 79, 88, 93, 98, 100, 105, 109, 142, 143, 144, 145, 146

absorption coefficient A, 19 absorption probability, 2

atomic polarizabilities, 247, 258

acetimide, CH3CONH2, 247

atomic polarizability tensor, 249

acetone, 219, 220, 221, 222 ----

acetonitrile, 219, 220, 221

B----

basis sets, 171

acetylene, C2H2, 136, 139, 126, 127, 170, 174, 175, 183,238, 244, 277, 279

benzene, C6H6, 126 Boltzmann distribution, 5, 8

Albrecht's theory, 210

bond charge parameter formulation, 60

alkylacetylenes, 130

bond charge tensors, 106, 142

aUene, CH2=C=CH2, 126, 127

bond direction angles, 52

ammonia, NH3, 45, 135, 136, 136, 139, 170, 174, 175, 183, 263, 266, 267

bond direction cosines, 52 bond displacement coordinates, 132, 262

analytic derivative methods, 167, 177 analytic gradient methods, 275

bond displacement vectors, 106, 110

angular momentum, 41

bond moment, 51, 52, 54, 58, 59, 63, 65,

anharmonic vibrational transition moment, 151 317

318

SUBJECT INDEX

bond moment model 51, 55, 59, 158

CEPA2 (CPA'), 179

bond polar parameter, 111, 126, 128, 130, 143, 146

CEPA3 (CPA"), 179

bond polarizability model 235, 236 bond polarizability tensor, 224 Born-Oppenheimer approximation, 196 trans-butadiene, CH2CHCHCH2, 263, 268, 270, 271 2-butyne, CH3-C~-C-CH3, 127

charge densities, 112 charge flow model, 158 charge flux, 57, 72, 143, 158 charge flux parameters, 145 charge-charge flux overlap model, 100 chloroform, CHC13, 38, 114 CI, 281,282

----

C-----

CISD, 183, 281

calculated infrared intensities, 167, 176

combination bands, 14

calculated Raman intensities, 274, 276

compact formulation of VOTR, 235, 238

carbon dioxide, CO2, 19, 20, 170, 172, 183 carbon monoxide, CO, 70, 100, 170, 174, 175, 281, 282

compensatory molecular rotation, 40, 41, 42, 49, 217, 262 configuration interaction, 176

carbon tetrachloride, CC14, 38

coupled-cluster, 177

carbon tetrafluoride, CF4, 105

cyanogen, C2N2, 179

carbonic dichloride, C12C0, 127, 136, 139, 139

cyclohexane, c-C6H12, 114, 242

carbonic difluoride, F2CO, 127, 128, 136, 139 CASSCF, 179 CCSD, 179, 183

cyclopropane, 112 D

,

depolarization ratio, 193, 202

CCSD(T), 183

dich/oro methane thial, C12CS, 127, 128, 136, 139

CCSD+T(CCSD), 179

dichloromethane, 219, 220, 221,222

CEPA- 1 (ED), 181

differential Raman scattering cross sections, 213

CEPA- 1 (EV), 181

SUBJECT INDEX

319

difluoromethane thial, F2CS, 127, 128, 136, 139

electro-optical anharmonic parameters, 162

digallium hexahydride-do, Ga2H6, 277

electro-optical parameters, 52, 54, 59, 61, 142, 144, 158

dimensionless normal coordinate, 204 dimethyl ether, (CH3)20, 114 dipole moment derivatives, 12, 16, 26, 37, 38, 42, 64, 66, 166 DMC 6/12, 177 DMC 6/6, 177 DZ, 170 DZ and TZ basis sets, 170 DZ+P, 183 DZP, 277 DZP (DZ+P), 177 DZP+dSPD, 177 DZP+dSPD/DMC, 177

electron correlation, 176, 278 energy gradient method, 275 ethane, C2H6, 93, 114, 124, 127, 136, 238, 242, 244, 268, 270, ethylbromide, CH3CH2Br, 114 ethylchloride, CH3CH2C1, 114 ethylene oxide, c-C2H40, 247 ethylene, CH2CH2, 126, 127, 139, 238, 263, 268, 270, 271,277 experimental IR band intensities, 37, 39, 41 experimental determination of infrared intensities, 17 external coordinates, 31, 40

1,1-dichloroethylene, 47, 48, 49 ---~ E - - - -

F----

first-order reop, 237

Eckart-Sayvetz conditions, 29, 30, 34, 54, 72

fluorinated methanes, 130

Eckart-Sayvetz equations, 40

fluoroacetylene, FCCH, 183

effective atomic charge, 99, 132, 145

fluoroform, CHF3, 114, 127, 128

effective bond charge, 60, 132

force constants, 16, 28, 29, 61

effective charges, 72, 143, 158

force field, 26, 36, 39, 63, 65,

effective induced bond charge, 261, 263, 266, 271

formaldehyde, H2CO, 43, 44, 136, 139, 177, 178, 183,254, 268, 270, 271,

Einstein coefficient, 3, 6

formamide, HCONH2, 247

electric charge, 16, 40, 84, 107

fundamental transition, 14, 150

320

SUBJECT INDEX

a

....,,..,-..

hypothetical-mass-isotope method, 261, 265

gas-phase Raman scattering cross sections, 211 group dipole derivatives, 72 group moment derivatives, 74, 75 group symmetry coordinates, 39, 73 ,

H----

harmonic approximation, 8, 13, 26, 133

iso-butane, (CH3)3CH, 114 induced bond charge, 263 induced dipole, 190 integrated absorption coefficient, 6, 10, 16, 18

harmonic force constants, 26

integrated intensity of the infrared absorption band, 6, 8

harmonic force field, 239

intensity sum rule, 85

harmonic frequency, 8, 10

internal coordinates, 26, 44, 45, 46, 47, 52, 55, 64, 66, 94

harmonic oscillator selection rules, 12, 13, 16, 205 heavy isotope method, 45, 219 homopolar moment, 56

hot band, 200 hydrocarbons, 136, 242 hydrogen chloride, HC1, 70, 100, 263, 266, 267 hydrogen cyanide, HCN, 170, 172, 179, 181, 182, 183 hydrogen fluoride, HF, 70, 99, 170, 174, 175, 187, 263,266, 267

intramolecular charge distribution, 15 intramolecular charge reorganization, 37 invariants of atomic polar tensors, 83 ----

M----

many body perturbation theories, 177 mass-weighted Cartesian displacement coordinates, 31 MBPT(2), 177, 179, 183 MC SCF calculations, 177, 278, 281

hydrogen isocyanide, HNC, 183

mean polarizability, 191

hydrogen sulfide, H2S, 263,266, 267, 281

methane, CH4, 21, 114, 127, 136, 139, 170, 174, 183, 238, 242, 247, 263, 266, 277, 279

hypothetical isotope method, 42, 43, 51 hypothetical non-rotating isotope, 42

methanethial, CH2S, 263, 268, 270, 271

SUBJECT INDEX

methanimine, CH2NH, 263, 268, 270, 271 methanol, CH3OH, 114, 263,267, 268, 270 methyl amine, CH3NH2, 114, 263, 267, 268, 270

321

n-pentane, 128, 130 natural vibrational coordinates, 26 nitrogen, N2, 22, 23, 282 normal coordinate analysis, 26, 28, 32, 39

methyl amine-d2, CHD2NH2, 115

normal coordinate transformation matrix, 27, 36

methyl bromide, CH3Br, 114, 127, 128, 136, 139

normal coordinates, 7, 13, 14, 16, 36

methyl chloride, CH3C1, 65, 67, 95, 114, 121, 127, 128, 136, 139, 218

numerical differentiation, 165 ------

methyl ethyl ether, CH3OCH2CH3, 176 methyl fluoride, CH3F, 105, 114, 127, 128, 136, 139, 263, 268, 270

O - - -

overlap moment, 55 overtone and combination band intensities, 150

methyl iodide, CH3I, 114, 127, 139

overtone and combination bands, 16

methylene chloride, CH2C12, 37, 38, 114

overtone transitions, 14

methylene fluoride, CH2F2, 105, 114, 127, 128 methylidyne phosphine, HCP, 179, 182 molecular dipole moment, 51 molecular polarization, 190 molecular principle axes, 41 molecular rotation, 41 MP2, 179, 183 multicontiguration SCF, 176

'"

N

'

n-alkanes, 72, 75, 120, 130 n-butane, 130

____

p _ _ _

perdeutero-polyethylene, 242 perturbation theory formulation, 152 phosphine, PH3, 263, 266, 267 Placzek's conditions, 196, 207 point charge, 56 point charge approximation, 55 point charge model, 59 polar angles, 52 polar tensors, 158 polarizability derivatives, 201

322

SUBJECT INDEX

polarizability derivatives with respect to symmetry coordinates, 217 polyethylene, 242 propane, C3H8, 114, 127, 242 propanol, C3H7OH, 247 propyne, HCCCH 3, 126, 127, 136, 239, 240, 241, 244, pyrazine, 211 ----

rotational correction term matrix, 45 rotational correction terms, 41, 42, 40, 46, 50, 217 rotational correction terms to polarizability derivatives, 22, 216 rotational corrections to dipole moment derivatives, 40 rotational polar tensor, 80, 97 rotational polarizability tensor, 251

R-----

roto-translational coordinates, 79

Raman depolarization rations, 279 Raman electro-optical parameters, 230 Raman line intensity, 199 Raman scattering coefficients, 279 Raman scattering cross section, 200 rational correction terms to polarizabilty derivatives, 221 rational quantum numbers, 5

scattering coefficient, 202 SCF, 183, 281 SDMC 6/6, 177 second order perturbation theory, 177 semi-classical approaches, 40 silane, Sill 4, 263,266

Rayleigh scattering, 198

simple hydrides, 269

relative differential Raman scattering cross section, 214

static bond charges, 60 structure of atomic polar tensors, 91

relative rotational corrections, 223

sulfer dioxide, SO2, 230, 232, 259

reop, 230 resonance Raman effect, 207

symmetry and normal coordinates, 28

RHF/SCF calculations, 170

symmetry coordinates, 27, 33, 38, 40, 44, 50, 52, 65, 94

rotation-flee atomic polar tensor, 131

symmetry selection rules, 14

rotation-flee atomic polarizability tensor, 261, 265 rotation-flee bond polarizability tensor, 265

---T theorem for momen~.m conservation, 40

SUBJECT INDEX

trace spectra, 203 transition dipole, 15 transition dipole matrix, 4, 8

323

- - -

W - - - - -

water, H20, 19, 20, 63, 97, 120, 135, 139, 170, 173, 179, 181247, 263, 266, 277, 279

transition dipole moments, 157, 209 transition probability, 3 trifluorobromo methane, CF3Br, 128 trifluorochloro methane, CF3CI, 128 trifluoroiodo methane, CF3I, 128 trimethyl amine, (CH3)3N, 114 TZ basis sets, 170

Z

~

zero angular momentum condition, 52, 118,261 zero-mass approach, 43, 218 zero-order approximation, 54 zero-order approximation of the bond polarizability model, 236 zero-order electro-optical parameters, 238

TZ+2P, 170 TZ+2P, 183

zero-order polarizability parameters, 240

TZP, 277 1,1,1-trifluoroethane, 48, 49, 50 - - -

-----

V------

valence optical theory of Raman intensities, 223, 232, 258 valence-optical theory, 51, 63 variation method formulation, 151 vibrational atomic polar tensors, 80, 103 vibrational atomic polarizability tensor, 25 I, 261 vibrational intensity analysis, 41 vibrational quantum number, 13, 198 vibronic coupling operator, 209

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