Algebraic theory of molecules

Algebraic Theory of Molecules

TOPICS IN PHYSICAL CHEMISTRY A Series of Advanced Textbooks and Monographs Series Editor, Donald G. Truhlar

F. lachello and R.D. Levine, Algebraic Theory of Molecules P. Bernath, Spectra of Atoms and Molecules J. Simons and J. Nichols, Quantum Mechanics in Chemistry J. Cioslowski, Electronic Structure Calculations on Fullerenes and Their Derivatives

Algebraic Theory of Molecules F. Iachello R. D. Levine

New York Oxford OXFORD UNIVERSITY PRESS 1995

Oxford University Press Oxford New York Toronto Delhi Bombay Calcutta Madras Karachi Kuala Lumpur Singapore Hong Kong Tokyo Nairobi Dar es Salaam Cape Town Melbourne Auckland Madrid and associated companies in Berlin Ibadan Copyright © 1995 by Oxford University Press, Inc. Published by Oxford University Press, Inc., 200 Madison Avenue, New York, New York 10016 Oxford is a registered trademark of Oxford University Press 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 permission of Oxford University Press. Library of Congress Cataloging-in-Publication Data Iachello, F. Algebraic theory of molecules / F. Iachello and R.D. Levine. p. cm. — (Topics in physical chemistry series) Includes bibliographical references. ISBN 0-19-508091-2 1. Molecular spectroscopy—Mathematics. 2. Molecular dynamics—Mathematics. I. Levine, Raphael D. II. Title. III. Series. QD96.M65123 1995 539'.6— dc20 93-37702

9 8 7 6 5 4 3 2 1 Printed in the United States of America on acid-free paper

Preface

For over a dozen years we have been developing an algebraic approach to molecular energetics and dynamics. One of us came from nuclear physics, the other from molecular reaction dynamics, but we were both interested in algebraic methods when we met in 1981. This volume represents that part of our work that we regard as being ready to be published as a book. In this book we present an algebraic approach to molecular vibrotation spectroscopy. We discuss the underlying algebraic techniques and illustrate their application. We develop the approach from its very beginning so as to enable newcomers to enter the field. Also provided are enough details and concrete examples to serve as a reference for the expert. We seek not only to introduce the spirit and techniques of the approach but also to demonstrate its quantitative application. For this reason a compilation of results for triatomic molecules (both linear and nonlinear) is provided. (See Appendix C.) The approach we discuss emphasizes anharmonicity (and cross anharmonicity) even in zeroth order. Higher-order couplings can be introduced in a systematic and sequential fashion. As such, we consider the approach particularly appropriate for the many challenges of modern spectroscopy, as discussed in our introductory and concluding chapters. We are however equally interested in the application of the algebraic approach for the representation of higher accuracy spectroscopic data and especially so for larger molecules. The algebraic approach starts with a Hamiltonian and hence such a fit provides more than just a compact parametrization of the data. Rather, it determines some (or most) of the parametes in the Hamiltonian and so provides explicit predictions (including, as discussed in chapter 7, information on the potential). We invite readers to keep us informed of their work in these respects.

Preface

vi

The technical prerequisites for following our development are modest. Essentially, introductory quantum mechanics, say, on the level of Merzbacher (1961). Of particular use is familiarity with angular momentum. Otherwise, we have tried to make the book self-contained and tools are developed as needed. Appendices provide important but strictly technical aids. The book has a conventional sequential organization. Equations, figures, tables and footnotes are numbered sequentially within each chapter. Footnotes elaborate on the text. The references serve to provide a source of more details, to acknowledge the original source where appropriate, and to enable the interested reader to examine an alternative or a complementary point of view. We sincerely apologize if your favorite reference is not included. We did not mean (nor is it humanly feasible) to be exhaustive when such a broad area, with so many excellent papers, is covered. We hope that the work of others and that of our co-workers will soon enable us to complement this volume by at least one on electronic aspects and another on applications to dynamics. We wish in particular to mention the important work of our co-workers that contributed to this volume. Special thanks are due to Professors Y. Alhassid, R.D. Amado, I. Benjamin, I.L. Cooper, A. Frank, O.S. van Roosmalen, and C.A. Wulfman and to Drs. L. Ya. Baranov, R. Bijker, Y.M. Engel, S. Kais, R. Lemus, A. Leviatan, N. Manini, S. Oss, B. Shao, and L. Viola. Mrs. E.D. Guez typed the entire manuscript which has been typeset directly from her computer file. Only she knows how we kept track of our many changes. The people at Oxford University Press were always a pleasure to deal with and special thanks are due to R.L. Rogers, Dolores Oetting, Jacki Hartt, and Ellen Barrie. Last but not least, the Department of Energy, The Air Force Office of Scientific Research, and the Volkswagen Stiftung supported our research, represented herein, and while this book was written. It is not possible to spend a dozen tranquil years doing research and to crown it in book form without the constant support and encouragement of our families. Our gratitude to them is evident. New Haven Jerusalem

F. I. R. D.

Contents

Introduction,

Chapter 1

xiii

The Wave Mechanics of Diatomic Molecules,

3

1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9

Introduction, 3 The two-body Schrödinger equation, 3 Eigenvalues and eigenfunctions, 6 Angular momentum, 9 Emission and absorption of radiation: Infrared, 11 Emission and absorption of radiation: Raman, 14 Intensities of vibrational transitions, 15 Schrödinger equation in two dimensions, 16 The Schrödinger equation in one dimension and the quasidiatomic model, 17 1.10 Representation of molecular spectra by fitting formulas: Dunham expansion of energy levels, 79 1.11 Herman-Wallis expansion for intensities, 19

Chapter 2 Summary of Elements of Algebraic Theory, 2.1 Lie algebras, 21 2.2 Lie subalgebras, 22 2.3 Invariant (Casimir) operators, 23 2.4 Basis states (representations), 23 2.5 Eigenvalues of the Casimir operators, 24

21

2.6 Algebraic realization of quantum mechanics, 25 2.7 Dynamical symmetries, 27 2.8 One-dimensional problems, 27 2.9 Dunham-like expansion for one-dimensional problems, 35 2.10 Transitions in one-dimensional problems, 37 2.11 The harmonic limit, 38 2.12 The Hamiltonian in dimensions, 39 2.13 Dynamical symmetries for three-dimensional problems, 41 2.14 Energy levels: The nonrigid rovibrator, 43 2.15 Energy levels: The rigid rovibrator, 44 2.16 Dunham-like expansion for three-dimensional problems, 46 2.17 Infrared transitions, 48 2.18 Electrical anharmonicities, 50 2.19 Rotational-vibrational interaction, 52 2.20 Raman transitions, 54 Chapter 3 3.1 3.2 3.3 3.4

60

Triatomic molecules, 60 Polyatom Schrödinger equation, 65 One dimensional coupled oscillators, 66 Nonlinear classical dynamics, 67

Chapter 4 4.1 4.2 4.3 4.4 4.5 4.6 4.7 4.8 4.9 4.10 4.11 4.12 4.13 4.14 4.15 4.16 4.17 4.18 4.19 4.20 4.21

Mechanics of Molecules,

three

Three-body Algebraic Theory,

72

Algebraic realization of many-body quantum mechanics, 72 One-dimensional coupled oscillators by algebraic methods, 73 The local-mode limit, 75 The normal-mode limit, 76 Local-to-normal transition, 78 An example: Stretching vibrations of water, 79 Infrared intensities, 80 Three-dimensional coupled roto-vibrators by algebraic methods, 81 Local basis, 83 The normal-mode basis, 84 Expansion of the coupled basis into uncoupled states, 84 Linear triatomic molecules, 85 Local-mode Hamiltonian for linear triatomic molecules, 85 The normal-mode Hamiltonian for linear triatomic molecules, 88 l-dependent terms, 89 Linear XY2 molecules, 91 Majorana couplings (Darling-Dennison couplings), 91 Quantum number assignment, 96 Fermi couplings, 96 Bent triatomic molecules, 98 Local Hamiltonians for bent triatomic molecules, 99

4.22 4.23 4.24 4.25 4.26 4.27 4.28 4.29 4.30 4.31 4.32 4.33

Linear-bent correlation diagram, 101 The normal mode Hamiltonians for bent triatomic molecules, Bent XY2 molecules, 702 Majorana couplings, 702 Higher-order corrections. Linear molecules, 704 Higher-order corrections. Bent molecules, 706 Rotational spectra, 108 Higher-order corrections to rotational spectra, 770 Rotation-vibration interaction, 777 Diagonal rotation-vibration interactions, 777 Nondiagonal rotation-vibration interactions, 113 Properties of nondiagonal rotation-vibration interactions: Linear molecules, 775 4.34 Properties of nondiagonal rotation-vibration interactions: Nonlinear molecules, 117 Chapter 5 5.1 5.2 5.3 5.4 5.5 5.6 5.7 5.8 5.9 5.10

779

Tetratomic molecules, 779 Recoupling coefficients, 720 Linear tetratomic molecules, 123 Local Hamiltonian for linear tetratomic molecules, 723 Majorana couplings in linear tetratomic molecules, 726 Vibrational / doubling. Casimir operators, 727 Higher-order terms in tetratomic molecules, 729 Fermi couplings, 737 Amat-Nielsen couplings, 737 Summary of interbond couplings in linear tetratomic molecules,

Chapter 6 6.1 6.2 6.3 6.4 6.5 6.6 6.7 6.8 6.9 6.10 6.11 6.12 6.13 6.14 6.15

Four-Body Algebraic Theory,

Many-Body Algebraic Theory,

101

733

Separation of rotation and vibration, 733 Internal symmetry coordinates, 134 Quantization of coordinates and momenta, 134 Stretching vibrations, 735 Hamiltonian for stretching vibrations, 736 Higher-order terms, 737 Symmetry-adapted operators, 738 The benzene molecule, 138 Isotopic substitutions. Lowering of symmetry, 141 Infrared intensities, 743 Octahedral molecules, 746 Bending vibrations. The Pöschl-Teller potential, 148 Hamiltonian for bending vibrations, 750 Bending vibrations of benzene, 757 Complete spectroscopy, 752

737

6.16 Removal of spurious states, 752 6.17 Complete spectroscopy of benzene, Chapter 7 7.1 7.2 7.3 7.4 7.5 7.6 7.7 7.8 7.9 7.10 7.11 7.12 7.13 7.14 7.15 7.16 7.17 7.18 7.19 7.20 7.21 7.22 7.23 7.24 7.25

Classical Limit and Coordinate Representation,

756

Potential functions, 756 Exact results. One dimension, 757 Exact results. Three dimensions, 75$ Geometric interpretation of algebraic models, 759 One-dimensional problems, 759 Intensive boson operators, 767 One-dimensional potential functions, 762 Coupled one-dimensional problems, 164 Potential functions for two coupled one-dimensional problems, 765 Three-dimensional problems, 167 Intensive boson operators in three dimensions, 769 Three-dimensional potential functions, 1 70 Coupled three-dimensional problems, 7 77 Potential functions for two coupled three-dimensional problems, 772 Vibrations and the shape of the potential, 1 74 One-dimensional problems, 174 Three-dimensional problems, 777 Rotations and the equilibrium distance, 180 Coupled problems, 787 Vibrations and the shape of the potential in linear triatomic molecules, 182 Rotations and equilibrium positions, 186 Tetratomic molecules, 187 Higher-order terms, 187 Mean-field theory, 188 Epilogue, 188

Chapter 8

Prologue to the Future,

APPENDIX A A.1 A.2 A.3 A.4 A.5 A.6 A.7 A.8 A.9

153

190

Properties of Lie Algebras,

Definition, 797 Generators and realizations, 198 Cartan classification, 798 Number of operators in the algebra, Isomorphic Lie algebras, 799 Casimir operators, 200 Example of Lie algebras, 200 Representations, 207 Tensor products, 203

797

199

A. 10 A. 11 A. 12 A. 13

Branching rules, 203 Example of representations of Lie algebras, 204 Eigenvalues of Casimir operators, 204 Examples of eigenvalues of Casimir operators, 204

APPENDIX B

Coupling of Algebras,

206

B.I Definition, 206 B.2 Coupling coefficients, 206 B.3 Addition of angular momenta, SO(3), 207 B.4 Properties of Clebsch-Gordan coefficients, 207 B.5 Tensor operators, 209 B.6 Wigner-Eckart theorem, 209 B.7 Tensor products, 209 B.8 Recoupling coefficients, 270 B.9 Addition of three angular momenta, SO(3), 210 B.10 Properties of 6 - j symbols, 277 B. 11 Addition of four angular momenta, 272 B. 12 Reduction formulas, 213 B. 13 Coupling of SO(4) representations, 274 B. 14 Racah's factorization lemma, 275 B.15 Coupling coefficients of SO(4), 276 B.16 Recoupling coefficients of SO(4), 276 APPENDIX C References, Index,

239

Hamiltonian Parameters,

227

218

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Introduction

Molecular spectroscopy is undergoing an essential change. Seemingly the change is only quantitative; better initial-state preparation, improved light sources and specially designed pumping schemes, and more sensitive detection techniques are providing ever-improved resolution and a wider range of accessible final states (Figure 0.1). A closer examination suggests a qualitative change as well. New ideas, not only better results, are forthcoming. One example of the changing attitudes is the increasing concern with time evolution. The time-energy uncertainty relation and the pursuit of higher resolution means that traditional spectroscopy is implicitly equivalent to the study of the stationary states determined by the long-time limit of the intramolecular dynamics. The recent increasing interest in the role of anharmonicities and resonance couplings, made unavoidable by the study of higher-lying rovibrational states and the experimental reality of avoiding inhomogeneous broadening (e.g., using supersonic expansion, Quack, 1990) makes the entire time domain of direct interest to spectroscopists (Bitto and Huber, 1992). The very complementarity with the studies in the frequency domain (broad homogeneous spectral features = early time dynamics and vice versa) makes lower-resolution spectra of interest. On the other hand, the traditional concerns of spectroscopy (Herzberg, 1945, 1950; Barrow, 1962; King, 1964; Hollenberg, 1970; Herzberg, 1971; Bunker, 1979; Steinfeld, 1985) remain very much with us (Figure 0.2). Better determination of overtone and combination bands of familiar molecules and the spectroscopic characterization of new species [radicals (Shida, 1991; Bernath, 1990), ions (Miller and Bondybey, 1983; Leach, 1980), and van der Waals molecules in particular (Nesbitt, 1988; Saykally, 1989; Hutson, 1990; Heaven, 1992)] continue to receive wide attention (Figure 0.3).

XIV

Introduction

Figure 0.1 Stimulated emission pumping (SEP, Hamilton et al, 1986; Northrup and Sears, 1992) is a new experimental technique for accessing higher-lying vibrational levels of molecules in their ground electronic states. Shown is the SEP vibrational spectrum of SO2, where a pair of dips represent one vibrational level. (Adapted from Yamanouchi, Takeuchi, and Tsuchiya, 1990.) The stick spectrum at the bottom represents the position of the vibrational levels given by Equation (0.1) with the constants given in Table 0.1. The bright levels are represented by longer sticks. What is needed for modern spectroscopy is a formalism able to discuss both level structure beyond the harmonic limit and the corresponding dynamics. A Hamiltonian is thus unavoidable since it is the generator of time evolution. Yet there needs to be a practical method for the determination of the eigenvalues of this Hamiltonian. As in the traditional Dunham-like expansion, it will be useful if the spectra can be well approximated by a small number of constants. An

Figure 0.2 Direct overtone spectroscopy of C2H2 using Fourier transform spectroscopy. Here, at high resolution, the entire band of rotational transitions, which accompany a given vibrational transition, can be resolved. Here the band, in the visible range, corresponding to the direct excitation of v = 5 of the V3 stretch mode is shown. (Adapted from Herman et al., 1991. See also Scherer, Lehmann, and Klemperer, 1983, and Figure 8.4.)

Introduction

xv

Figure 0.3 The structure of the van der Waals molecule C6H6 • Ar as determined by very high-resolution spectroscopy. (Adapted from Weber, van Bargen, Riedle, and Neusser, 1990.) The potential along the C6H6 - Ar stretch motion is shown in Figure 1.4.

example of such an expansion is the fit of the vibration spectrum of SO2 in its electronic ground state (Figure 0.1) by the expansion

Here G(v l5 v 2 , v 3 ) is the level energy in wave number units (as far as possible we follow the notation of Herzberg, 1950) and the constants in Equation (0.1) are given in Table 0.1. As usual the vs are the vibrational quantum numbers of SO2 and rather high (above 10) values can be reached using the SEP technique. Equation (0.1) provides a fit to the observed levels to within an error below 10 cm"1, which is almost the experimental accuracy. We need, however, to be able to relate the parameters in this expansion directly to a Hamiltonian. The familiar way of doing this proceeds in two steps. First, the electronic problem is solved in the Born-Oppenheimer approximation, leading to the potential for the

xvi

Introduction Table 0.1 Vibrational constants" (cm"1) for the ground electronic state of SO2

CO!

C02 C03

x\\ X22

XK

Xl2 X\l *23

1167.84(15) 522.21(19) 1382.18(50) -3.655(23) -0.374(36) -5.36(20) -3.129(29) -14.277(71) -4.122(8)

y\\\ y222 ^333

y\n ym

yn2 ym ym V221

yw

-0.0061(10) -0.0014(18) -0.031(22) -0.0001(17) -0.1574(41) -0.0063(19) -0.0509(53) 0.255(11) 0.0214(50) -0.008(12)

a

A number in parentheses represents a standard error obtained by the least-squares analysis. The vibrational constants, CD,, x:j, and ytj, are the expansion coefficients of Eq. (0.1). (Adapted from Yamanouchi, Takeuchi, and Tsuchiya, 1990.)

motion of the nuclei. Then the Schrodinger equation for the eigenvalues of this potential is solved. Since for any but diatomic molecules the potential is a function of many coordinates, neither the first nor the second step is simple to implement. For a number of test cases this procedure has been carried out and for diatomic molecules of lower-row atoms it can challenge experiments in its precision. For larger molecules it is still not practical to compute the required potential with sufficient accuracy. It is therefore often approximated using convenient functional forms. Not too far from a deep equilibrium point, the potential can be expanded in the displacement coordinates relative to the equilibrium configuration. Such a "force field" representation is quite convenient but is of limited validity for higher-lying states due to the slow convergence of such a power-series expansion. More flexible functional forms that can describe the asymptotic dissociation plateaus (Murrell et al., 1984) require many parameters. The purely numerical solution of the Schrodinger differential equation for the eigenvalues of such a potential makes the optimization of the parameters in the potential, via a fit to the observed spectrum, a large-scale numerical problem complicated, as all such nonlinear problems are, by nonuniqueness and by local minima. In this book we present an alternative approach. Our discussion in this introductory volume will put particular emphasis on the traditional concerns, namely, determing the levels and intensities of the corresponding transitions. The approach we present retains, at least in part, the simplicity of a Dunhamlike approach in that, at least approximately, it provides the energy as an analytic function of the quantum numbers as in Equation (0.1). If this approximation is not sufficient, the method provides corrections derived in a systematic fashion. On the other hand, the method starts with a Hamiltonian so that one obtains not only eigenvalues but also eigenfunctions. It is for this reason that it can provide intensities and other matrix elements. The Hamiltonian used in our approach is an algebraic one and so are all the

Introduction

xvii

operations in the method, unlike the more familiar differential operators of wave mechanics. The technical advantage of an algebraic approach is the comparative ease of algebraic operations. Equally important, however, is the result, obtained by comparison with experiment, that there are generic forms of algebraic Hamiltonians and that entire classes of molecules can be described by a common Hamiltonian where only the (typically, linear) parameters are different for the different molecules. The algebraic (or matrix) formulation of quantum mechanics1 is less familiar than the differential (or wave) formulation. This is a disadvantage, and one purpose of the present volume is to show, by explicit examples, the benefits of the algebraic approach. The interested reader will have to judge if the benefits are sufficient to overcome the potential barrier to the understanding of a new approach. We intend to demonstrate that the algebraic formulation is indeed a viable alternative. The algebras one uses are Lie algebras. These algebras were introduced at the end of the nineteenth century by Lie, but it is only in the much more recent past2 that they are being used in physics. In the approach we follow, emphasis is put on starting with the Hamiltonian, with the main technical tool being the algebra, rather than on the corresponding group. In the appendices we give an account of the important properties of Lie algebras that are of interest for the applications in the study of molecules.3 Even if one restricts one's attention to vibrations and rotations of molecules, there are a variety of Lie algebras one can use. In some applications, the algebras associated with the harmonic oscillator are used. We mention these briefly in Chapter 1. We prefer, however, even in zeroth order to use algebras associated with anharmonic oscillators. Since an understanding of the algebraic methods requires a comparison with more traditional methods, we present in several parts of the book a direct comparison with both the Dunham expansion and the solution of the Schrodinger equation. In this book we shall write the Hamiltonian as an (algebraic) operator using the appropriate Lie algebra. We intend to illustrate by many applications what we mean by this cryptic statement. It is important to emphasize that one way to represent such a Hamiltonian is as a matrix. In this connection we draw attention to one important area of spectroscopy, that of electronically excited states of larger molecules,4 which is traditionally discussed in terms of matrix Hamiltonians, the simplest of which is the so-called picket fence model (Bixon and Jortner, 1968). A central issue in this area of spectroscopy is the time evolution of an initially prepared nonstationary state. We defer a detailed discussion of such topics to a subsequent volume, which deals with the algebraic approach to dynamics. In this book we deal mainly with stationary states, their energies, and matrix elements. Unless otherwise stated, we use the wave number (cm"1) as a measure of the energy. The conversion factors with other units are shown in Table 0.2. The present volume deals exclusively with rotation—vibrational spectra. Electronic excitations can also be described algebraically, but this description is still at too preliminary a stage for inclusion in a book format.

Introduction

XV111

Table 0.2. Energy conversion factors" Unit 1cm'1 leV 1 Hz (sec'1) 1 cal/mole K

cm '

eV

Hz (sec'1)

J/mole

K

1 8.06573(3) 3.3356(-ll) 0.34976 0.69503

1.2398(-4) 1 4.1356(-15) 4.3363(-5) 8.6170(-5)

2.997924(10) 2.41804(14) 1 1.04854(10) 2.0836(10)

1.1962(1) 9.6487(4) 3.9903(-10) 4.1840 8.1340

1.4388 1.16049(4) 4.7993(-ll) 0.50325 1

" The numbers in parentheses represent powers of ten. IA = ICT* cm = 0.1 nm = lO"4 p. m = 108/(cnr'). The speed of light in vacuum is c = 2.99792458 x 10s msec"1. The meter is now defined as the distance traveled by light in vacuum in 1/c sec, so that c is a defined constant and the second (as measured by atomic clocks) is the fundamental unit. The cal is still often used as an energy unit. For a complete account, see Mills (1988).

Notes 1. The matrix formulation of quantum mechanics was introduced as early as 1925 by Heisenberg (1925), Born and Jordan (1925), Dirac (1925), and Born, Heisenberg, and Jordan (1926). 2. It was only in the 1930s that Lie algebras were being used in physics (Weyl, 1931; Wigner, 1931, 1937; van der Waerden, 1931; Yamanouchi, 1937; Racah, 1942, 1949). Most of the early applications dealt with the algebra of rotations (Wigner-Racah algebra). An approach that starts from the algebra as the key tool for the construction of spectra originated in elementary particle physics in the 1960s (Dothan, Gell-Mann, and Ne'eman, 1965; Barut and Bohm, 1965) and had major applications in nuclear physics (Arima and lachello, 1975. For an update, see the books by lachello and Arima, 1987, and lachello and van Isacker, 1991, and the reviews in Bohm, Ne'eman, and Barut, 1988). In our discussion we shall emphasize the role of the algebra rather than that of the corresponding group. Specifically, we shall seek such a description that from the very beginning the spectrum is anharmonic. In technical terms, we shall emphasize an approach where even in zeroth order the Hamiltonian is a bilinear form in the operators of the algebra. We shall also try to forge a clear link with the geometrical point of view. There are many other important applications of Lie groups and algebras. In particular, we do not discuss time-dependent aspects (Alhassid and Levine, 1977; Wulfman, 1979; Levine and Wulfman, 1979; Levine, 1985) nor other applications [e.g., coherent states (Zhang, Feng, and Gilmore, 1990) or configuration interaction in electronic structure calculations (Judd, 1967; Pauncz and Matsen, 1986; Adams et al., 1987)] where the group structure is of central importance. 3. Lie groups and Lie algebras are discussed in many textbooks (Hamennesh, 1962; Gilmore, 1974; Wybourne, 1974; Barut and Raczka, 1986). We follow closely the notation of Wybourne (1974). There are also a number of mathematical texts (Miller, 1968; Talman, 1968; Vilenkin, 1968; Miller, 1977; Olver, 1986). 4. Hamiltonians expressed in matrix forms have been extensively employed in the theory of radiationless transitions of electronically excited states of larger molecules (Bixon and Jortner, 1968; Schlag et al., 1971; Freed, 1972; Nitzan et al., 1972; Avouris et al., 1977; Jortner and Levine, 1981; Felker and Zewail, 1988; Seel and Domcke, 1991).

Algebraic Theory of Molecules

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Chapter 1 The Wave Mechanics of Diatomic Molecules

1.1

Introduction

The spectroscopy of diatomic molecules (Herzberg, 1950) serves as a paradigm for the study of larger molecules. In our presentation of the algebraic approach we shall follow a similar route. An important aspect of that presentation is the discussion of the connection to the more familiar geometrical approach. In this chapter we survey those elements of quantum mechanics that will be essential in making the connection. At the same time we also discuss a number of central results from the spectroscopy of diatomic molecules. Topics that receive particular attention include angular momentum operators (with a discussion of spherical tensors and the first appearance of the Wigner-Eckart theorem which is discussed in Appendix B), transition intensities for rovibrational and Raman spectroscopies, the Dunham expansion for energy levels, and the Herman-Wallis expansion for intensities.

1.2

The two-body Schrodinger equation

Two-body quantum mechanical systems are conveniently discussed by transforming to the center-of-mass system1 (Figure 1.1). The momentum (differential) operator for the relative motion is

where V is the gradient operator whose square V • V is the Laplacian and, as usual, i2 = - 1 and ft is Planck's constant/271. The kinetic energy operator is 3

4

Chapter 1

Figure 1.1 The diatomic molecule of masses m, and m 2 . The transformation to the relative coordinate r, r = rl-r2 is useful when the potential depends on r only. The reduced mass, |j, is, as usual, (0, = m 1 m 2 /(m 1 + m 2 ).

p2/2(i so that for the Hamiltonian // = p2/2p, + V(r), where V(r) is the potential, we have the Schrodinger differential equation

The wave function, v|/(r), is a function of the vector position variable r. To determine it at every point in space it is convenient to take advantage of the fact that the potential V(r) depends only on the scalar interatomic distance r. In spherical coordinates (Figure 1.2), the Laplacian operator V2 has the form

The Schrodinger equation (1.2) can be solved by separating variables. Writing

Figure 1.2 The spherical coordinates r, 9, <|> corresponding to a position vector r. The center of mass is at the origin of the coordinate system.

The Wave Mechanics of Diatomic Molecules

5

one obtains the equations

The solutions of the first two equations are the spherical harmonics (Edmonds, 1960; Brink and Satehler, 1968; Zare, 1988). The orbital angular momentum, /, can take non-negative integer values / = 0,1,2,..., and the projection of the orbital angular momentum on the fixed z axis takes the integer values -I < m < +1. The angular part of the solution is always of this type, as long as the potential V(r) between the two particles depends only on the distance r and not on the angles 0, <|>. The last equation in (1.5) (radial equation) depends instead on the explicit form of the potential V(r). The potential curve V(r) of a bound molecule typically has the form of the full curve in Figure 1.3. It is

Figure 1.3 Potential curve of a molecule (ground state of HC1). The full curve is the Morse potential of Eq. (1.6). The dashed curve is the harmonic approximation. De is the dissociation energy, and re is the equilibrium separation.

6

Chapter 1

steeply repulsive at short interatomic distances and attractive at large separations, approaching asymptotically a finite plateau. These two aspects can be approximated in a variety of functional forms, such as by a Morse potential (Morse, 1929)

where p is the range parameter. The Morse potential reasonably describes the potential around the minimum but is not physically realistic at high r values. Improved descriptions can be obtained by expanding V(r) as a sum of exponential functions (Huffaker, 1978), where, with y = 1 — exp[-(3(r - r e )]

Many other useful forms have been proposed (Steele and Lippincott, 1962) and their parameters were related to spectroscopic constants as will be given for the Morse potential by Eq. (1.14). Quite often, the potential V(r) is expanded as a power series in the displacement from equilibrium (force field method)

The lowest-order term is the harmonic approximation

The advantage of the expression (1.8) is that it can be treated easily in perturbation theory. The disadvantage is that the expansion (1.8) diverges for large r, and thus the expansion suffers from convergence problems. At large interatomic distances an expansion in inverse powers of r is more realistic (Buckingham, 1967), and the variable (r - re)/r serves to provide an expansion that is physically reasonable both near equilibrium and asymptotically (Simons et al., 1973; Kryachko and Koga, 1985). The first term in such an expansion is known as the Kratzer potential (Fliigge, 1971).

1.3

Eigenvalues and eigenfunctions

The bound-state energies and eigenfunctions can be obtained by solving the Schrodinger equation with boundary conditions that the radial wave function vanishes at both ends

For the Morse potential, the / = 0 eigenvalues are labeled by the vibrational quantum number v and given by

The Wave Mechanics of Diatomic Molecules

where according to (1.6), Dehc = V0. The approximately equal sign arises from the fact that Eq. (1.11) has been obtained under the condition that V —> oo. By inspecting Eq. (1.6)

For typical values of p, re and V0 encountered in molecules, Eq. (1.11) is an excellent approximation to the exact solution (better than 1 part in 109). The Morse potential is the simplest member of a family of potentials that give rise to a vibrational spectrum of the functional form E(v) - coc(v +1/2) <&exe(v +1/2)2. This is quite realistic at lower levels of excitation. The vibrational spectrum does not however suffice, by itself, to specify the potential uniquely. The dependence of the eigenvalues on the rotational state is therefore important. For / ^ 0 (as well as for the / = 0) the energy eigenvalues are given by

The approximately equal sign here arises from the fact that Eq. (1.13) has been obtained under the rigid rotor approximation that the centrifugal term /(/ + I)1l2/2\ir2 in Eq. (1.5) is replaced by its value at equilibrium 1(1+iyh2/2[ir2,. This is a good (though not excellent) approximation. An improved approximation is (Pekeris, 1934; Fliigge et al., 1967; Elsum and Gordon, 1982)

The correction terms to Eq. (1.13) are particularly important for floppy

8

Chapter 1

molecules, an example of which is shown in Figure 1.4. The radial part of the wave function corresponding to the eigenvalue (1.13) is given by

where L(z) is an associated Laguerre polynomial (Abramowitz and Stegun, 1964)

In the rigid rotor approximation, the radial wave function is independent of /. The total wave function is

Figure 1.4 The very anharmonic energy levels of the C6H6 - Ar stretch motion (cf. Figure 0.3) (adapted from Neusser, Sussman, Smith, Riedle, and Weber, 1992). The computed values of the rotational constant Bv [the coefficient of 1(1 + } ) in the expression for the energy eigenvalues] are given in the figure, as are the vibrational spacings.

The Wave Mechanics of Diatomic Molecules

9

In Section 1.5 we shall use the wave function (1.17) and the machinery for handling angular momentum (Section 1.4) for the computation of the intensities of spectroscopic transitions. For the Kratzer potential, which is quadratic in the variable (r - re)/r, the eigenvalues for / # 0 can be obtained in closed form. This is also the case for the potential quadratic in (r-r 2 /r) (Gol'dman et al., 1960). This potential does not, however, tend to a finite value as r -» oo.

1.4

Angular momentum

In view of the central role played by the angular momentum in molecular physics, it is of interest to review briefly its properties.2 For the relative motion the angular momentum is

In quantum mechanics, 1 is an operator whose Cartesian components satisfy the commutation relations

This is, incidentally, the first example of a Lie algebra that we encounter. We will return to it later. Also we have set ~h = 1 in (1.19) to simplify the notation. Instead of Cartesian coordinates it is convenient to use spherical coordinates. Properties of physical operators can be characterized according to the way they behave under rotation of the axes. These properties can be cast into a simple mathematical form by giving the commutation relations with the angular momentum. It is convenient to introduce the linear combinations

In terms of / z , /+ and /_ the commutation relations (1.19) become

One defines2 a spherical tensor of rank k by the commutation relations

where K spans -k to k, 2k + 1 values in all. Under rotations, a spherical tensor of rank k keeps its rank but its components are converted into linear combinations of all other components. One can verify that if 7"^ is a spherical tensor, the operator

10

Chapter 1

is also a spherical tensor. Among the spherical tensors there are scalar operators (tensors of rank zero), vector operators (tensors of rank one), and quadrupole operators (tensors of rank two). When more than one particle is present, one needs to combine the angular momenta of each particle. For two particles, for example, the total angular momentum 1 is

Quantum mechanically the combination of two angular momenta is more complicated since the angular momenta are operators. (They are tensor operators of rank one.) The law of combination of angular momenta can be expressed, in general, by the so-called tensor product of two operators, indicated by the symbol x,

The possible values of k are all the integers included in the triangle inequality

The expansion coefficients on the right hand side of Eq. (1.25) are the Clebsch-Gordan coefficients.2 The eigenfunctions of the angular momentum, which can be written, abstractly, using Dirac notation \l,m>, satisfy the equations (h = 1)

In the Schrodinger equation, the abstract relations are realized in terms of differential operators

One can see then that the states I /, m > are the spherical harmonics Ylm. It is convenient to give also the realization of the operators / + and /_

The Wave Mechanics of Diatomic Molecules

11

Another important property of the angular momentum is the Wigner-Eckart theorem.2 This theorem states that the matrix elements of any tensor operator can be separated into two parts, one containing the m dependence and one independent of m,

The double bar matrix elements are called reduced matrix elements. The m dependence is given by a Clebsch-Gordan coefficient. The Wigner-Eckart theorem simplifies considerably the calculation of matrix elements, since it is enough to compute the matrix element for one particular value of m and then obtain all others by using Eq. (1.30). Instead of the Clebsch-Gordan coefficient, sometimes one uses in (1.30) a Wigner 3—j symbol, related to the ClebschGordan coefficient by

1.5

Emission and absorption of radiation: Infrared

Two ingredients are needed to compute the intensities of transitions: the wave functions of the initial and final states and the form of the transition operator (Ogilvie and Tipping, 1983). For infrared transitions the appropriate operator is the dipole operator, M(r, 0, (j>). This operator is a vector (tensor of rank 1) and thus can be written as

The transition matrix elements are given by

These matrix elements can be separated into a radial and an angular part

12

Chapter 1

The angular part depends only on properties of the angular momentum. Using the Wigner-Eckart theorem, one has

The reduced matrix elements of the spherical harmonics can be written in terms of a Wigner 3 - j symbol

The Wigner 3 - j symbol in Eq. (1.36) has the selection rules

and values

This gives

The radial part of the matrix elements depends on the form of M(r). A typical form is shown in Figure 1.5. It is customary to expand M(r) in a polynomial

and to keep only the first few terms

The first term in Eq. (1.41) is the dipole moment, while the second is the electric anharmonicity. The expansion (1.40) diverges for large r. More appropriate forms of expansions are

The Wave Mechanics of Diatomic Molecules

13

r(a<) Figure 1.5 Typical behavior of the dipole function M(r) as a function of r (in atomic units). The results shown are realistic for HF (adapted from Zemke et al., 1991; see also Annum etal., 1992).

In any case, denoting by M/ jV the radial matrix element, one has, for the intensity of transition v, / —> v', /',

the result

The line strength S, is given by

14 1.6

Chapter 1 Emission and absorption of radiation: Raman

A computation of Raman intensities can be done precisely in the same way as for infrared intensities. One needs here, in addition to the wave functions of the initial and final state, the polarizability tensor d(r,6, ())). This is a symmetric tensor of rank 2 that in Cartesian coordinates can be written as

It has six independent components. It is convenient to separate the trace of this tensor from the rest and thus introduce spherical tensors

where d(0) is the trace and d(2) is the traceless part. The matrix elements of the trace are simply given by

leading to a line strength

The matrix elements of d( are given by

Using the Wigner-Eckart theorem, one has

The reduced matrix elements of Y2 are given by

The Wave Mechanics of Diatomic Molecules

15

with selection rules /' = / ± 2, /. Inserting the appropriate values of the Wigner 3 - j symbols, one obtains the line strengths for Raman intensities

1.7

Intensities of vibrational transitions

The expressions of the Sections 1.5 and 1.6 are general and apply to any solution of the Schrodinger equation. In the special case of a Morse potential, the radial integrals in Eq. (1.34) can be evaluated, with some approximations, in closed form. The approximation consists in replacing the lower limit of integration by -oo. This approximation is similar to that used in Section 1.3 when obtaining the wave functions. Thus

Three types of matrix elements have been evaluated.3

16

Chapter 1

In these expressions y and \\im are the digamma function and its derivative and the symbol

The typical behavior of A/Q v is shown in Figure 1.6. One should note that, for the Morse potential, and in lowest approximation, the radial wave functions and thus M/ v are independent of /. This is no longer the case for more general potentials and for the exact solution of the Morse problem.

1.8

Schrodinger equation in two dimensions

In comparing the algebraic and differential approach in Chapter 2 it will turn out to be useful to consider also the case of two dimensions. Introducing polar coordinates in the plane,

Figure 1.6 Typical behavior of the radial matrix elements M0_,, as a function of v. The results shown are for HC1 and are given in SI units. Adapted from Ogilvie and Tipping (1983).

The Wave Mechanics of Diatomic Molecules

17

one can write the Schrodinger equation as

The Laplacian operator in these coordinates is

The Schrodinger equation (1.60) can be solved by separation of variables. Writing

one obtains the equation

For the special case of the Morse potential in the radial coordinate p

under the same conditions discussed in Section 1.2, one obtains the energy eigenvalues

to be compared with the expression (1.13) for three dimensions. One can see that the vibrational part is identical and that the only difference comes from the "rotational" part [three-dimensional rotations in (1.13) and two dimensional rotations in (1.65)].

1.9

The Schrodinger equation in one dimension and the quasidiatomic model

Even for higher energies one can sometimes describe a vibration of a polyatomic molecule as an uncoupled, diatomiclike mode. This is particularly true

18

Chapter 1

when the frequency of the vibration of interest is well separated from that of other motions. Examples include the overtones of higher-frequency (e.g., CH) stretching vibrations (Henry, 1977; Halonen, 1989). (The opposite extreme is that of a low-frequency van der Waals mode as in Figure 1.4). It is thus of interest to consider explicitly the Schrodinger equation in one dimension. Denoting by x the coordinate, the Schrodinger equation in one dimension is

In the particular case of the Morse potential

the energy eigenvalues are exactly given by

There are no approximations here, since the interval of definition of \\f(x) is now the entire interval -oo < x < + oo. Also any constant XQ could be added to x in (1.67) without changing the result. The wave function v|/(je) corresponding to (1.68) is

with

One may compare this result with that of Section 1.2. The vibrational part of (1.13) is again identical to Eq. (1.68). The "rotational" part is, however, missing in the one-dimensional problem. It is worth commenting on this special feature of the vibrational problem. It arises from the fact that molecular potentials usually have a deep minimum at r = re. For small amplitude motion (i.e., for low vibrational states) one can therefore make the approximation discussed in the sentence following Eq. (1.13) of replacing r by re in the centrifugal term. In this most extreme limit of molecular rigidity, the vibrational motion is the same in one, two and three dimensions.

The Wave Mechanics of Diatomic Molecules

19

1.10 Representation of molecular spectra by fitting formulas: Dunham expansion of energy levels Quite often, rotational-vibrational spectra of molecules are analyzed by means of empirical formulas. A convenient formula for diatomic molecules is the Dunham expansion (Dunham, 1932; Ogilvie and Tipping, 1983)

The first few coefficients of this expansion are traditionally denoted by

The rotation constant Bv is given by _y01 + y\\ (v+1/2) + ••• = Be-ae(v+1/2) + • • -. Even for such a weakly bound vibration as a van der Waals mode (Figure 1.4), ae is less than 10% of Be. For the more rigid, chemical bonds, ae/Be is typically below 0.05. Further discussion of rotational-vibrational separation is provided by Bunker (1970) and Lathouwers et al. (1987). Dunham obtained these eigenvalues using the semiclassical approximation for the potential (1.8) which is an expansion in powers of (r — re)/re. The results for the Morse potential [Eq. (1.14)J can also be written in this form, as can results for other potentials. One therefore often uses Eq. (1.71) as a convenient empirical form. A slightly different form of (1.71) is

The coefficients _y'y are related to the yi}-s in an obvious way.

1.11

Herman-Wallis expansion of intensities

Intensities of transition can be also analyzed by means of fitting formulas. An expansion quite often used for infrared transitions is the Herman—Wallis (1955) form

The quantity m in Eq. (1.74), which should not be confused with the projection of the angular momentum / on the z axis, is given by

20

Chapter 1

One can see by comparing with Eq. (1.45) that \m\ is the line strength S/. For the Morse potential, and in lowest approximation, Fv'tV(m)= 1 and Rv>rV is given by the formulas of Section 1.7.

Notes 1. The material in this section is discussed in all texts of quantum mechanics. A very useful compilation of explicitly solved examples is Flilgge (1971). See also the very good but not so easily available collection of problems of Gold'man et al. (1960). 2. More elementary introductions to the material in the rest of this section can be found in Messiah (1976) or Cohen-Tannoudji, Diu, and Laloe (1977). More detailed discussions are available in Fano and Racah (1959), Edmonds (1960), Brink and Satchler (1968), de Shalit and Talmi (1963), and Judd (1975). Zare (1988) is particularly useful on both the theory and the manner of its application. Special reference to diatomic molecules is made by Judd (1975) and Mizushima (1975). The close connection to Lie algebra is emphasized by Biedenharn and Louck (1981). A summary of the results we need is in Appendix B. 3. Evaluation of diagonal and off-diagonal matrix elements for the one-dimensional and also for the rotating (Chapter 2) Morse oscillator has been discussed by many authors. The results quoted in the text are from Matsumoto (1988).

Chapter 2 Summary of Elements of Algebraic Theory

2.1

Lie algebras

Algebraic theory makes use of an algebraic structure. The structure appropriate to ordinary quantum mechanical problems is that of a Lie algebra. We begin this chapter with a brief review of the essential concepts of Lie algebras.1 The binary operation ("multiplication") in the Lie algebra is that of taking the commutator. As usual, we denote the commutator by square brackets, [A, B] = AB - BA. A set of operators {X} is a Lie algebra when it is closed under commutation. That is, for every operator X in the algebra G (which we write as X e G)

together with the Jacobi identity The constants ccah, which characterize a given algebra, are called the Lie structure constants. A familiar example of a Lie algebra is the angular momentum algebra of Eq. (1.19), which, because of its importance, we repeat it here replacing / by /,

The algebra (2.3) is called the special orthogonal algebra in three dimensions, SO(3). Associated with each Lie algebra there is a group of transformation 21

22

Chapter 2

where the operators of the algebra are the generators of the group. The group associated with the algebra (2.3) is the group of real orthogonal transformations in three dimensions, that is, the rotation group. Groups are usually denoted by capital letters, here SO(3). In this book we do not make extensive use of groups. Therefore we limit our discussion to Lie algebras. It has become customary to denote both algebras and groups by the same capital letters. We shall follow this notation in this book as well. It is increasingly the case that one refers to the operators of the algebra as the generators even when the group is not the object of direct interest. In making the connection to the differential equations form of quantum mechanics we shall use a realization of the operators X as differential operators. One realization of the angular momentum operators was given already in Section 1.4. Many other realizations of the same SO(3) algebra are discussed in Miller (1968). All admissible Lie algebras were classified by Cartan in 1905. Cartan's classification is given in Appendix A, where many other properties and definitions are provided.

2.2

Lie subalgebras

A Lie subalgebra is a subset G' of operators of G, which, by itself, is closed with respect to commutation. In other words, the commutator of two elements is a linear combination of the same elements. In mathematical terms,

We shall use systematically the symbol z>, meaning "containing", to denote this situation. In some cases, the subset G is trivial. For example, it is clear that the single operator Jz, i.e. the component of the angular momentum on a fixed z axis, forms a subalgebra of the angular momentum algebra SO(3) since

The group associated with this algebra is that of real orthogonal transformations in two dimensions, i.e. rotations around the z-axis with Jz being the corresponding generator. This group and its algebra are denoted by SO(2). Thus

In this particular case, the algebra (2.5) is trivial, since the operator Jz obviously commutes with itself. Algebras formed by commuting operators are called Abelian.

Summary of Elements of Algebraic Theory

2.3

23

Invariant (Casimir) operators

For each Lie algebra, one can construct a set of operators, called invariant (or Casimir, 1931) operators after the name of the physicist who first introduced them in connection with the rotation group. These operators play a very important role since they are associated with constants of the motion. They are defined as those operators that commute with all the elements of the algebra

They are constructed from powers of the operators Xs and can be linear, quadratic, cubic, . . . . Quite often a subscript is attached to C in order to indicate the order. For example, C2 denotes a quadratic invariant. The number of independent Casimir invariants of an algebra is called the rank of the algebra. It is easy to see, by using the commutation relation (2.3) that the operator

commutes with the elements Jx,Jy, and Jz,

Thus the Casimir operator of SO(3) is the familiar square of the angular momentum (a constant of the motion when the Hamiltonian is invariant under rotation). One can show that SO(3) has only one Casimir operator, and it is thus an algebra of rank one. Multiplication of C by a constant a, which obviously satisfies (2.7), does not count as an independent Casimir operator, nor do powers of C (i.e., C2,...) count. Casimir operators can be constructed directly from the algebra. This construction has been done for the large majority of algebras used in physics.

2.4

Basis states (representations)

The next important problem in algebraic theory is the construction of the basis states (the representations) on which the operators X act. A particular role is played by the irreducible representations (Appendix A), which can be labeled by a set of quantum numbers. For each algebra one knows precisely how many quantum numbers there are, and a list is given in Appendix A. The quantum numbers are conveniently arranged in patterns (or tableaux), called Young tableaux. Tensor representations of Lie algebras are characterized by a set of integers

which can be displayed in a tableau, where the length of the first row is X1; that of the second row A,2, etc.

24

Chapter 2

The number v is given in Appendix A. For the algebras SO(n), the rule is v = (n - l)/2 if n is odd or v = n/2 if n is even. Thus for SO(3), v = 1. The quantum number here is / and has the meaning of the angular momentum. For SO(2), v = 1. The quantum number here is Mj and has the meaning of the projection of the angular momentum on a fixed axis. A complete basis is constructed by starting from the algebra G and considering its possible subalgebras. For example, for the rotation algebra SO(3), one considers SO(3) z> SO(2). The complete basis for this algebra is written as

This statement is equivalent, in the algebraic language, to the statement that the spherical harmonics, Yj Mj(Q, (j>), form a basis for the rotation group. A problem that arises in connection with the construction of the basis is that of finding what are the allowed values of the quantum numbers of the subalgebra G' contained in a given representation of G. For example, what are the allowed values of Mj for a given / in Eq. (2.12). In this particular case, the answer is well known from the solution of the differential (Schrodinger) equation satisfied by the spherical harmonics (see Section 1.4), that is,

The allowed values are all the integers between -/ and +/. However, in general, this is a rather complicated problem, and group theory offers a complete and straightforward solution. This problem is called the labeling problem. For the algebras used in this book, it turns out that the labeling problem is straightforward. However, in other cases, such as the description of the structure of atomic nuclei, the labeling problem is more complicated, in view of the so-called missing labels.2

2.5

Eigenvalues of the Casimir operators

The last problem of general interest in algebraic theory is the evaluation of the eigenvalues of the invariant operators in the basis discussed in Section 2.4. As mentioned before, the invariant operators commute with all the Xs. As a result, they are diagonal in the basis [^, A , 2 , . . . , X,v],

Summary of Elements of Algebraic Theory

25

The eigenvalues / have been evaluated for any Casimir operator of any Lie algebra, and a summary of the results is given in Appendix A. Using the expressions of the appendix, we find, for example, that the eigenvalues of the Casimir operator of SO(3), J2, in the representation I/ > is

a familiar result.

2.6

Algebraic realization of quantum mechanics

In Chapter 1 we have discussed the familiar realization of quantum mechanics in terms of differential operators acting on the space of functions (the Schrodinger wave function formulation, also called "wave mechanics"). A different realization can be obtained by means of creation and annihilation operators, leading to an algebraic formulation of quantum mechanics, sometimes called matrix mechanics. For problems with no spin, the formulation is done in terms of boson creation, b^, and annihilation, ba, operators, satisfying the commutation relations

The indices a, a' go from 1 to n + 1, where n is the number of spatial degrees of freedom.3 Thus, for one-dimensional quantum-mechanical problems, n = 2, for two-dimensional problems, n = 3, and for three-dimensional problems, « = 4. The operators of physical interest can be expanded as a power series in the bilinear products b\,b$ of the boson operators.4 Special cases include the Hamiltonian H,

whose eigenvalues give the energy levels, and the transition operators

whose matrix elements squared give the transition probabilities. The expansion coefficients can be related4 to the matrix elements of the operators, as will

26

Chapter 2

become clear by examples. Note also that b*aba can be regarded as the number operator for bosons of type a. The total number operator N

commutes with the Hamiltonian H. This would not be the case had we included terms of the type tfabl or bab$ in the expansions of the physical observables.5 In this realization, the states of the system are written as:

called the Fock space, where 91 is a normalization chosen usually in such a way as to have an orthonormal set. At this stage one notes that the bilinear products of creation and annihilation operators

satisfy the commutation relations

together with the Jacobi identities. They then constitute the elements of a Lie algebra, as defined in Section 2.1. This Lie algebra is the unitary algebra U(« +1). Hence, one can formulate quantum mechanics in n dimensions in terms of the unitary algebra U(n + I).6 When written in terms of the elements of the unitary algebra, the Hamiltonian (2.17) is

The diagonal elements Gaa are the number operators. The off-diagonal elements G a p, a*P can be thought of as "shift" or step-up and -down operators. Such operators "move" one boson of type P into a boson of type a. In operator form this follows from Eq. (2.21), which gives, for oc^p, G aa G a p = Gap(Gaa + 1). It can also be shown using the basis set (2.20), as is done in Eq. (2.56). For molecular problems, the appropriate algebra to begin with is U(4), since diatomic molecules live in a three-dimensional space, Section 1.2.7

Summary of Elements of Algebraic Theory

2.7

27

Dynamical symmetries

The Hamiltonian (2.23) represents the general expansion in terms of the elements G a p, and it corresponds to a Schrodinger equation with a generic potential. In some special cases, one does not have in Eq. (2.23) generic coefficients e 'ap> M apyS> but only those combinations that can be written as invariant Casimir operators of G and its subalgebras, G z> G' z> G" z> • • • . This situation

called dynamical symmetry is particularly useful for the analysis of experimental data, since in this case all quantities can be evaluated analytically in closed form. For example, since the Casimir operators are diagonal in the basis of G, their expectation values can be evaluated simply, and one has closed expressions for the energy E,

where the bracket denotes expectation values, examples of which will be given in the section to follow. These closed expressions are Dunham-like expansions that can be easily compared with experiment. Dynamic symmetries correspond to solvable potentials in the Schrodinger picture. One can thus anticipate that, since the Morse potential is, under certain conditions solvable, it will correspond to a dynamical symmetry of the fundamental algebra, U(4).

2.8

One dimensional problems

Although the treatment of rotations and vibrations requires the use of the full algebra U(4), we begin our discussion with the simpler case of one-dimensional problems, described by the algebra U(2). In addition to providing an introduction to the use of algebraic methods, this algebra can be used to describe stretching vibrations of molecules.8 To provide a realization for the algebra U(2) we take two boson creation and annihilation operators, which we denote by O^,t f and a, I. The algebra U(2) has four operators which can be realized as (Schwinger, 1965),

The three operators F+, F_, Fz are themselves closed under commutation and are elements of the algebra SU(2) which is a subalgebra of U(2). The SU(2) algebra

28

Chapter 2

is isomorphic to that of the angular momentum [cf. Appendix A and Eq. (1.21)], and thus the operators F+, F_, Fz are often denoted by /+, /_, Jz. However, in order to avoid confusion with the angular momentum, we denote them henceforth by F. Also we place a caret over the letter to indicate operators. Quantities with no caret denote eigenvalues. Like /+ and /_, the operators F+ and F_ will play the role of shift operators. A generic one-dimensional problem is obtained by expanding the Hamiltonian in terms of the operators of Eq. (2.26), or their linear combinations

where the dots imply terms like FxFy, or powers thereof. Dynamical symmetries for one-dimensional problems can be studied by considering all the possible subalgebras of U(2). There are two cases

We shall omit from here on the letter S in the orthogonal algebras since there is no difference in the algebraic structure of SO(«) and O(«). However, we will keep the letter S, if appropriate, in the unitary groups, since there is a difference in the algebraic structure of SU(«) and U(n). One-dimensional problems present on one hand the simplest (and most studied) example of algebraic theory, and on the other hand involve some subtle problems that are worthwhile elucidating. Chain (I). Basis states in this chain are characterized by the quantum numbers

The representations of U(2) are, in general, labeled by two quantum numbers. However, since we are considering only boson realizations, these representations must be totally symmetric and can thus be characterized by only one quantum number, that is, the first entry in the Young tableau

Summary of Elements of Algebraic Theory

29

The role of the quantum number N is discussed in the following. The values of the quantum number n^ characterizing the representations of U(l) are given by the reduction of U(2) to U(l), that is, the entire set of integers 0 < «T < N,

The representations are the eigenstates of the operator

Also, from Eq. (2.26),

The eigenvalues of Fz are thus

Introducing

one can label the representations (2.21) as

with

which makes the similarity with the usual algebra of the angular momentum clear. Dynamic symmetry corresponds to an expansion of the Hamiltonian in terms of Casimir operators. The Casimir operator of U(2) plays no role, since it is a given number within a given representation of U(2) and thus can be reabsorbed in a constant term EQ. The algebra U(l) has a linear invariant

Thus, to lowest order, a Hamiltonian with this dynamic symmetry is

30

Chapter 2

This Hamiltonian is obviously diagonal in the basis (2.31) with eigenvalues

The spectrum of Eq. (2.42) is shown in Figure 2.1. This is the spectrum of the one-dimensional truncated harmonic oscillator with a maximum vibrational quantum number equal to N. Thus N + I represents the number of bound states. When N —> oo one recovers the full oscillator spectrum. Since «T is an invariant, so is n\. One can thus write down the most general bilinear algebraic Hamiltonian with dynamic symmetry U(l) as

The eigenvalues are

This represents a truncated anharmonic oscillator with anharmonicity controlled by K. The basis states 17V, HT > or IF, Fz > can be written explicitly in terms of boson creation and annihilation operators

Figure 2.1 states.

The truncated harmonic potential and its spectrum of four (N = 3) bound

Summary of Elements of Algebraic Theory

31

Note how the finite number of bound states arises very naturally in the algebraic approach. This example also illustrates the role of the "extra" quantum number, N. All possible truncated oscillators are described by the same algebra, for different values of the quantum number N. In any given problem, the value of N is fixed, and «T plays the role of the vibrational quantum number. Chain (II). Basis states for this chain are characterized by the quantum numbers

with

Note the peculiarity of O(2), whose representations are characterized by both positive and negative numbers [see Appendix A, Eq. (A.22), and Hamermesh, 1962]. Also note that the quantum number M jumps by two units each time. Instead of the quantum numbers N, M we can introduce

as before, except that we have Fx instead of Fz, in order to emphasize the difference with the previous case. This difference is important for threedimensional problems, as discussed in the following sections. For onedimensional problems, since the subalgebras of U(2) [i.e., U(l) and O(2)] are isomorphic, the two cases (I) and (II) can be related to each other. Using Eq. (2.48), we can label the representations (2.46) as

where

Again note the similarity with the usual angular momentum. Dynamic symmetries for chain (II) correspond to an expansion of the Hamiltonian in terms of invariant operators of O(2). The linear invariant is

32

Chapter 2

Using Eq. (2.28), this can obviously be written as

The most general Hamiltonian with dynamic symmetry (II) again has a form similar to Eq. (2.43), with both linear and quadratic terms. This is a peculiar feature of one-dimensional problems. In order to simplify the discussion of three-dimensional problems, we prefer to consider a Hamiltonian with only quadratic terms

with

This Hamiltonian is again trivially diagonal in the representation (2.46), with eigenvalues

In view of Eq. (2.47), the states are doubly degenerate except for M = 0. This is a peculiarity of one-dimensional problems, as it will be commented in the following. We choose in Eq. (2.55) only the positive branch of M,

With a negative value of A, the spectrum (2.55) has the form shown in Figure 2.2. This is the spectrum of a one-dimensional anharmonic oscillator, and 1 + N/2 or [1 + (N - l)/2] represents the number of bound states.

Figure 2.2

Spectrum of states of the one-dimensional anharmonic oscillator, N = 6.

Summary of Elements of Algebraic Theory

33

It is convenient to introduce the quantum number v through

Then, the spectrum (2.55) can be rewritten as

This is precisely the spectrum of the one-dimensional Morse oscillator discussed in Section 1.9. The connection between the algebra of U(2)9 and the solutions of the Schrodinger equation with a Morse potential can be explicitly demonstrated in a variety of ways. One of these is that of realizing the creation and annihilation operators as differential operators acting on two coordinates x' and x",

Introducing the polar coordinates r, <)> as

and making the change of variable

one obtains from the equations

the following coordinate /Z WiM (£)exp(iAf<|>),

equation

for

the

function

vj/jv M (^, (|))

Introducing the appropriate dimensions, one can rewrite Eq. (2.63) in the form

=

34

Chapter 2

Also

Equation (2.64) gives

Comparing this equation with Eq. (1.68), one obtains

and from Eq. (2.65)

One can thus see how the algebraic parameters A and N are related to the potential. In terms of the usual spectroscopic constants of Eq. (1.72), one has

The familiar consistency relation V0 = w>e/4xe of the Morse oscillator is seen to be satisfied. Another manifestation of this relation is xe = l/(N +1). The double degeneracy of the O(2) case corresponds to the fact that the algebraic method describes in this case two Morse potentials related to each other by a reflection around x = 0. This is a peculiar feature of one-dimensional problems, and it does not appear in the general case of three dimensions. If one uses the O(2) basis for calculations, this peculiarity can be simply dealt with by considering only the positive branch of M. The simple examples discussed in this section illustrate the main properties of the algebraic method. By introducing the unitary algebra in 2 = (1+1) dimensions, one can simultaneously describe harmonic and anharmonic oscillators. Within the general description, two limiting cases can be solved exactly: (1) the purely harmonic oscillator V(x) = V0x2 corresponding to the subalgebras U(l) of U(2) and (2) the Morse oscillator V(x) = V0(l - e'^'^)2, corresponding to the subalgebra O(2) of U(2). For those two cases one can obtain explicit

Summary of Elements of Algebraic Theory

35

expressions for the energy eigenvalues. This is because the Hamiltonians used have a dynamical symmetry (Section 2.7). A general potential V(r) corresponds to a generic algebraic Hamiltonian (2.29). In the most general case the solution cannot be obtained in explicit form but requires the diagonalization of a matrix. The matrix is (N + 1) dimensional. An alternative approach, useful in the case in which the potential does not deviate too much from a case with dynamical symmetry, is to expand it in terms of the limiting potential. For the Morse potential, this implies an expansion of the type (1.7)

Within the algebraic approach, this corresponds to an expansion of the type

This expansion can be made as accurate as one wishes by including higher and higher-order terms. An important property of one-dimensional problems that is worth mentioning is the fact that since the two algebras U(l) and O(2) are actually isomorphic, there is no difference (from the topological and algebraic point of view) between the solutions of the two potential problems (harmonic oscillator and Morse). In fact there exists, within the group space, a transformation that takes one into the other. This transformation is actually a rotation that takes x into z. The difference between the oscillator and the Morse potentials does become central when considering problems in more than one dimension, as will be seen in the following sections.

2.9

Dunham-like expansion for one dimensional problems

It is convenient to take the U(2) ID O(2) symmetry of the preceding section as the starting point for approximations. Since it is unnecessary to carry the index z or x, the wave functions can be written simply as \N, m >. Denoting by C^ the Casimir operator of O(2) with eigenvalues

and rewriting the Hamiltonian H( ' as

one obtains

36

Chapter 2

As mentioned in the previous section this equation represents the energy eigenvalues of the Morse oscillator. In general one can write

The eigenvalues of Eq. (2.75) are

This is a Dunham-like expansion but done around the anharmonic solution. It converges very quickly to the exact solution if the potential is not too different from that of a Morse oscillator (Figure 2.3). This will not, however, be the case for the highest-lying vibrational states just below the dissociation threshold. The inverse power dependence of the potential suggests that fractional powers of n must be included (LeRoy and Bernstein, 1970).

Figure 2.3 A plot of the spacing, A£(v). between two adjacent eigenvalues versus v for H2 (lachello, 1981; lachello and Levine, 1982). For Eq. (2.74) such a plot should be linear (with an intercept at the quantum number of the highest bound state). This is a "Birge-Sponer" plot. The original application to H2 is due to Beutler (1934). Using Eq. (2.76), see also Eq. (2.123), one can account for deviations from linearity.

Summary of Elements of Algebraic Theory 2.10

37

Transitions in one-dimensional problems

In addition to energy eigenvalues it is of interest to calculate intensities of infrared and Raman transitions. Although a complete treatment of these quantities requires the solution of the full rotation-vibration problem in three dimensions (to be described), it is of interest to discuss transitions between the quantum states characterized by \N, m >. As mentioned, the transition operator must be a function of the operators of the algebra (here F x, Fy, Fz). Since we want to go from one state to another, it is convenient to introduce the shift operators F+, F_ [Eq. (2.26)]. The action of these operators on the basis \N, m>is determined, using the commutation relations (2.27), to be

These can be written in terms of the vibration quantum number v as

Thus, the shift operators can only induce near-neighbor transitions, that is, they change the vibrational quantum number V by ± 1. The typical behavior of intensities in molecules was discussed in Chapter 1. A realistic approximation of this behavior can be obtained in the algebraic framework by considering the operator

The matrix elements of the operator (2.79) can be calculated by making use of Eq. (2.78) and of the usual formula for expansion of exponential operators. An alternative is to recognize, using Eq. (2.28), that f can be thought of as a rotation operator so that its matrix elements can be computed (Levine and Wulfman, 1979) using the known results for the rotation matrices. When a
When the potential is of the Morse type, the first term in Eq. (2.80) provides an often-used rule of thumb for overtone (v' — v > 1) transitions

38

Chapter 2

On the other hand, one must recognize that quantal interference oscillations may result in transition probabilities that oscillate around the simple classicallike scaling laws (Levine and Wulfman, 1979). Figure 2.4 is an illustration of such a phenomenon for the overtone transitions in HE

2.11

The harmonic limit

The spectroscopic identifications in Eq. (2.69) enable us to take the harmonic limit where the anharmonicity vanishes, xe —> 0, and the well is deep (V0 -* °°) such that the harmonic frequency co<,, coe =4xeV0, remains finite. In our notation, this is the N -> oo, A -> 0, AN finite, limit. In the earlier days of the algebraic approach the harmonic limit (Levine, 1982) served as a useful guide to the connection with the geometrical picture. Since the harmonic limit is so well understood, taking it still provides a good intuitive link. The harmonic limit can be taken in a mathematically consistent way by the process of contraction (Gilmore, 1974). Returning to Eq. (2.26), one replaces the operators a and of by numbers, V#, and lets N -> oo. The algebra then becomes

with commutation relations

Figure 2.4 The transition probabilities I < vlf Iv + Av > I 2 , logarithmic scale, versus v for Av = l , 2 , and 3. Computed (Benjamin et al., 1987) for an HF molecule (N = 45, p=1.1741 a.u.~ 1 ,r = 4.56D/a.u.,a=1.6/JV. The scaling of ot with N is because of the harmonic limit where F+ + £_ tends to f>N(r - re) [Eq. (2.87)].

Summary of Elements of Algebraic Theory

39

This is seen to correspond to the usual commutation relation of the harmonic oscillator, since, replacing T by a, one has

One example when the harmonic limit provides a physical interpretation is that of the dipole operator (2.79). The limit of the operator F+ + F_ is

If one realizes the operators a* and a in terms of differential operators

one obtains

We shall return to the geometrical content of the algebraic approach in several places, and in particular in Chapter 7. See also note 5 therein.

2.12

The Hamiltonian in three dimensions

In order to be able to describe simultaneously rotations and vibrations, one needs to consider explicitly all three space dimensions. An algebraic treatment of this problem thus requires the use of the algebra corresponding to the unitary group U(4) (lachello, 1981; lachello and Levine, 1982). A boson realization of this algebra can be obtained by introducing four boson creation, hi, (a = 1,..., 4), and annihilation, ba(a = 1,..., 4), operators. The 4x4=16 bilinear products b^b^ span the algebra of U(4) [Eq. (2.21)]. This form of the algebra, called the uncoupled form, is not well suited for the analysis of the problem since one wants states of good angular momentum. The corresponding operators must then have definite transformation properties under rotations. This is equivalent to saying that Cartesian coordinates are not particularly useful to solve the Schrodinger equation with a spherically symmetric potential and that one prefers to use spherical coordinates. To this end, one divides the four boson operators, b^a, into a scalar, c^, and a vector nj (p, = 0, ±1). Here jo, denotes the spherical components of the vector. Thus the angular momentum and parities of a1^ and nj are Jp = 0+ and Jp = 1~, respectively.

40

Chapter 2

The spherical components of n^ are related to the Cartesian components by

Sometimes the operators o1',^ are denoted by

With these definitions the creation operators o f , TtJ transform as spherical tensors under rotation. The annihilation operators do not. However, it is easy to construct operators that do transform as spherical tensors [Eq. (1.23)]. These will be denoted by a tilde and written as

The algebra of U(4) can be written in terms of spherical tensors as in Table 2.1. This is called the Racah form. The square brackets in the table denote tensor products, defined in Eq. (1.25). Note that each tensor operator of multipolarity A, has 2X, + 1 components, and thus the total number of elements of the algebra is 16, as in the uncoupled form. Table 2.1

Elements of the algebra of U(4)

Explicit form

No. of components

Notation

-VSlVxTC]^

1

n

•N/^xn]^

J

[T^Xflf

3 5

[7I t XCT + O t XJl]^ )

3

i[7l t x6-a t X7l]^ ) 0

3 1

[a'-xa]' '

Q D D' "a

The operators /, Q, D, D' have the physical meaning of the angular momentum, quadrupole, coordinate, and momentum operators, respectively. The Hamiltonian operator for molecular rotational-vibrational spectra, which in the uncoupled form is written as in Eq. (2.17), can now be written in the coupled form as

Summary of Elements of Algebraic Theory

41

where we have written all terms up to second order in the bilinear products of creation and annihilation operators. By writing the Hamiltonian in this form [which is based on the notation of Eq. (1.25)], one ensures that H is a scalar operator, that is, that its eigenvalues do not depend on the coordinate system used to evaluate them. The parameters e0, en and H 0 » M 2 » v o > v i » M o characterize each molecule. Similarly, one can write the transition operators in terms of boson operators. For infrared transitions, which are induced by a tensor operator of rank one, the lowest-order expansion is

while for Raman transitions, which are induced by tensor operators of rank zero and two, the lowest-order expansion is

2.13

Dynamical symmetries for three dimensional problems

Dynamical symmetries for three-dimensional problems can be studied by the usual method of considering all the possible subalgebras of U(4). In the present case, since one wants states to have good angular momentum quantum numbers, one must always include the rotation algebra, O(3), as a subalgebra. One can show then that there are only two possibilities, corresponding to the chains

Chain (I). Basis states for this chain can be characterized by the quantum numbers

Since we are considering a boson realization of the algebra, the representations

42

Chapter 2

of U(4) are totally symmetric, corresponding to a Young tableau with only one row

Here N is the vibron number (related to the number of bound states, as shown in the sections to follow). The U(3) representations are characterized in general by three quantum numbers. However, in the reduction of totally symmetric states of U(4), only totally symmetric states of U(3) appear, characterized by a single quantum number, nK, which can take the values

In the further reduction from U(3) to O(3), the values of the angular momentum, /, contained in a given representation nn of U(3) are

Finally, in the further reduction from O(3) to O(2), one has the usual rule, discussed in Chapter 1,

Chain (II). Basis states for this chain can be characterized by the quantum numbers

Once more, although, in general, two quantum numbers characterize the representations of O(4), (o>i, co2), only the symmetric states appear, characterized by a single quantum number (co, 0) = co, which can take the values

The further reduction of O(4) to O(3) gives

and —J < Mj < + J as before. The procedure, by means of which the complete set of quantum numbers that characterize the states is derived, is called the branching rule and in general it can be done for any algebra U(n). One can also observe that for a fixed value of N (i.e., a fixed potential in the Schrodinger approach) there are always three quantum numbers that characterize the states

Summary of Elements of Algebraic Theory

43

of U(4). This statement is analogous to the statement that in the Schrodinger equation in three dimensions, there are always three quantum numbers characterizing the states. For problems with central potentials, two of these are always the angular momentum, /, and its z component, M}. The third is the radial quantum number. As mentioned before in connection with one-dimensional problems, the states (2.101) or (2.96) provide bases in which all algebraic calculations can be done. These bases are orthogonal bases for three-dimensional problems. They can be converted one into the other by unitary transformations that have been (Frank and Lemus, 1986) written down explicitly.

2.14

Energy levels: The nonrigid rovibrator

The eigenvalue problem for the Hamiltonian H [Eq. (2.92)] can be solved in closed form whenever H does not contain all the elements but only a subset of them, the invariant or Casimir operators. For three-dimensional problems there are two such situations corresponding to the two chains discussed in the preceding sections. We begin with chain (I). Restricting oneself only to terms up to quadratic in the elements of the algebra, one can write the most general Hamiltonian with dynamic symmetry (I) as

The coefficients e, K, and K' are linear combinations of the coefficients ec, en, u0, u2, v0, V[, M'O of Eq. (2.92). The eigenvalues of //(I) in the states (2.96) are given by

The energy levels corresponding to Eq. (2.105) are shown in Figure 2.5. These are the energy levels of the three-dimensional nonrigid rovibrator (Figure 2.6). For each oscillator quantum number, nn, the values of J are nn, nn — 2 , . . . , 1 or 0. If K = 0 and K' = 0, the spectrum is purely harmonic. Writing V(r) = -V0 + V0r2, one has

The terms with K and K' are anharmonic corrections that can still be dealt with analytically in the algebraic approach. An alternative form of Eq. (2.104) can be obtained by introducing the operators of Table 2.1. In terms of these operators

44

Chapter 2

Figure 2.5 Energy-level diagram of the nonrigid three-dimensional rovibrator [Eq. (2.105)]. Here N = 3 and e, K, K' > 0.

2.15

Energy levels: The rigid rovibrator

The more familiar rigid rovibrator has the dynamical symmetry associated with chain (II). The Hamiltonian up to quadratic terms is

with eigenvalues, in the states (2.101),

Figure 2.6 The potential V(r) that corresponds to the dynamical symmetry (I). The potential is "nonrigid" because [cf. Eq. (2.113)] the rotational spacings are comparable to the vibrational ones. In the harmonic limit V(r) is the potential of an isotropic harmonic oscillator.

Summary of Elements of Algebraic Theory

45

Figure 2.7 Energy-level diagram of the three-dimensional rovibrator [Eq. (2.109)]. (N = 3), A<0, B>0.

The energy levels corresponding to Eq. (2.109) are shown in Figure 2.7. These are the energy levels of the three-dimensional (re*&) rovibrator (Figure 2.8). The expression (2.109) can be rewritten in terms of the vibrational quantum number v,

One recognizes that these are the energy levels of the Morse rovibrator discussed in Chapter 1. In particular,

Figure 2.8

The potential V(r) that corresponds to the dynamic symmetry (II).

46

Chapter 2

These equations provide an explicit relationship between the parameters appearing in the Morse potential, re, p\ V0, and the reduced mass (i and the algebraic parameters E'0, A, B, and N. Particularly important is the relation N + 2 = 2oc, which shows explicitly how the vibron number N is related to the number of bound states in the potential of Figure 2.8. An alternative form of Eq. (2.108) is that in terms of the operators of Table 2.1,

The two chains (I) and (II) span the entire set of analytically solvable problems in three dimensions. Any solution for a generic potential can be expanded in terms of either of the two, since both provide an orthonormal set in three dimensions. An interesting quantity that characterizes the properties of the two solutions is the nonrigidity parameter R introduced by Berry (1980)

where Erot and Evib are the energies of the first rotational and vibrational energy level. One has for chain (I),

while for chain (II),

For molecules, N is usually large, while the other coefficients are of the same order of magnitude, so that

Thus, while chain (II) corresponds to rigid molecules where a separation between rotational and vibrational motion is possible, chain (I) corresponds to a softer, floppy structure.

2.16

Dunham-like expansion for three dimensional problems

The Morse potential results derived on the basis of the dynamic symmetry derived for chain (II) of U(4) provide a reasonably good description of spectra

Summary of Elements of Algebraic Theory

47

of rigid diatomic molecules. However, quite often, the actual situation deviates somewhat from that of a simple Morse rovibrator. The required refinement can be accommodated within the algebraic framework by adding higher-order terms. Introducing the operators

one expands the Hamiltonian as a series in these operators,

This Hamiltonian is still diagonal in the basis (2.91) with eigenvalues

The expression (2.119) is similar to the usual Dunham expansion

except that one expands around an anharmonic rather than harmonic limit. Since the situation in molecules is closer to the former rather than to the latter, fewer terms in Eq. (2.119) are sufficient. As an example, in Figure 2.3, the values of the quantity

for the H2 molecule are shown. With only one term, one obtains from Eq. (2.119)

that is, a straight line in the plot of AE versus v, with slope 8Fi0. One can see that the straight line approximation is a good one up to v = 10. With two terms one obtains

Chapter 2

48

A similar analysis can be performed for the rotational constant B, defined as the coefficient in front of the J(J + 1) term in E

Equation (2.119) gives, in lowest order,

and thus

For large N, this is a constant, -4(N + l)y u , with corrections which are lineai in v. Table 2.2 shows the experimental data on the molecule HC1. Table 2.2. Rotational constants of HC1 in the different vibrational levels of the electronic ground state V

B(v) (cm'1)

0

10.4400

1

10.1366

2

9.8329

3

9.5343

4

9.232

5

8.922

ABCvMcrrf 1 ) -0.3034 -0.3037 -0.2986 -0.302 -0.299

As already noted in Section 1.10, AB/B can be regarded as a measure of nonrigidity. The dependence of B(v) on v shown in Figure 0.4 is indeed more extreme than that for HC1.

2.17

Infrared transitions

We have seen that the wave functions of rigid diatomic molecules can be characterized by the wave functions \\\i > = \N, v, J, M, >. For a given molecular

Summary of Elements of Algebraic Theory

49

species the vibron number N is fixed and can be omitted. The goal of the calcu-

where f is the transition operator. We begin our discussion with the calculation of line strengths of infrared transitions. The transition operators, f , are, in the algebraic approach, expanded into elements of the algebra. For infrared transitions, the tensorial character of the operator is that of a dipole. Thus, in lowest order, the infrared transition operator must be written as

The matrix elements of this operator between the states (2.127) can be calculated. They have selection rules

and are given by (lachello, Leviatan and Mengoni, 1991)

It is convenient to introduce the standard Herman-Wallis notation (Section 1.11) and write

where

Equation (2.132) gives

Thus, Eq. (2.130) can be written as

and consequently the line strength is given by

50

Chapter 2

Thus with the operator (2.128), the Herman-Wallis factors are

One may recognize that, in the limit of large N, one recovers the Herman-Wallis factors for the harmonic limit Rv „ = const, /% iV (m) = 1.

2.18

Electrical anharmonicities

The algebraic transition operator of the preceding section corresponds to a dipole function, which in configuration space is a constant

For a more precise calculation of intensities of infrared bands it is necessary to take into account the variation of the dipole function with internuclear distance, discussed in Section 1.5,

This variation (electrical anharmonicity) can be taken into account, within the algebraic approach, by expanding the operator f ^ as

where n is the operator of Table 2.1 and the symmetrized form has been used since the operators n and D do not commute. One can also expand in more complicated functions, as, for example,

This expansion corresponds to the expansion (1.42) in configuration space. The matrix elements of the operator (2.140) must, in general, be evaluated numerically.9 However, when N is large (a situation that is almost always encountered in actual spectra), the matrix elements of the exponential operator can be evaluated in closed form. Since the operator « is a scalar, its matrix elements do not depend on / and one has

Summary of Elements of Algebraic Theory

51

where

From Eq. (2.141) one obtains the Herman-Wallis factor for the operator (2.140), in the large-TV limit,

A further improvement can be obtained by introducing even more complex electrical anharmonicities, for example, by replacing the operator D in Eq. (2.140) with

This leads to the most general expansion for the Herman-Wallis factor

where

Although the formula (2.145) is quite complex, often few terms are sufficient to describe the data. To lowest order in v(v/N « 1) and Z (ZVw« 1, Z = A,[/2), Eq. 2.145 gives

52

Chapter 2

The dependence on v given by Eqs. (2.147) is approximately followed by the experimental data. Figure 2.9 shows the situation observed in HC1.

2.19

Rotational-vibrational interaction

The rotational Herman-Wallis factor Fv>>v(w) of the operator (2.140) is still that of a rigid rotor. In order to describe rotational-vibrational interactions, one must introduce explicitly the angular momentum /. To lowest order, the dipole operator that includes rotational-vibrational interactions is

where the operator D' is given in Table 2.1. The matrix elements of the additional term in Eq. (2.148) are

Figure 2.9 Herman-Wallis factors observed in HC1 (dots) and compared with Eq. (2.145) (dotted lines); adapted from lachello, Leviatan, and Mengoni (1991).

Summary of Elements of Algebraic Theory

53

Combining these results with those of the previous section, one obtains, to lowest order in v/N « 1, ZV]V « 1,

with

The corresponding Herman-Wallis factor is

From Eqs. (2.149) one can see that the rotational Herman-Wallis factor is linear in m when v^v', and quadratic in m when v = v'. An example is shown in Figure 2.10.

Figure 2.10 Herman-Wallis factor F(m) for the transition v'= 1 -> v' = 0 in HC1. The solid line is the fit to Eq. (2.152).

54

2.20

Chapter 2

Raman transitions

As in the previous case of infrared transitions, one wants to calculate the line strengths S(v,J -» v',/') defined in Eq. (2.127). For Raman transitions there are two contributions, as discussed in Chapter 1. The so-called trace scattering is induced by the monopole operator

This operator is a constant and thus has selection rules

The corresponding line strength is

The operator (2.153) is the lowest-order approximation to the monopole term. Anharmonic terms can be introduced by considering the operator

The matrix elements of the exponential operator are given by Eq. (2.141). Only the selection rule A/ = 0 remains for this operator. It is convenient to write the line strength for trace scattering in general as

The operator (2.156) gives the following Herman-Wallis factors in the large-W limit

From these expressions, and to lowest order, v/N «1, ZV/V« 1, one obtains, for example,

The second, and most important, contribution to Raman intensities is the quadrupole contribution, induced by the operator

*('}'}

when Q^ is the operator defined in Table 2.1.

Summary of Elements of Algebraic Theory f> 121

The operator Q

has selection rules

Its nonzero matrix elements are given by (Leviatan, 1992)

55

56

Chapter 2

In the limit of large N, applicable to most molecules, the matrix elements in Eq. (2.162) can be written in the compact form

Summary of Elements of Algebraic Theory

57

where

It is convenient to introduce the standard notation for Raman intensities. We denote by S, O, and Q branches those with

as usual. We write the line strengths as

Inserting the appropriate values of the Clebsch-Gordan coefficients, one obtains

With the operator (2.160), the factors R(fiv and F(^v(J) are given, in the limit of large N, by

The result (2.167) is particularly important, since it is used to analyze experimental data. It is merely a consequence of the fact that the quadrupole operator is a tensor of rank 2. Sj is just the square of the Clebsch-Gordan coefficients in

58

Chapter 2

Eq. (2.164). For A/ = 0, one should add to the quadrupole term the monopole term. The full expression can be written as

where a0 = a^/a^ (cf. Herzberg, 1950, p. 128). Raman anharmonicities can be introduced as the infrared anharmonicities of the previous sections. The appropriate operator is

No symmetrization is necessary here since n and Q commute. In the limit of large N, one can evaluate explicitly the matrix elements of the operator exp(A,n)<2 . They are given by

The matrix elements of exp(kn) are given in Eq. 2.141.

Notes 1. References to the physics, chemistry and mathematics literature are given in the introductory chapter. An introductory book is that of Wybourne (1974). 2. The problem of missing labels is discussed in detail in connection with nuclear spectroscopy (lachello and Arima, 1987). 3. A problem in n-dimensional space is thus given n + 1 quantum numbers. The role of the additional quantum number is to distinguish between the different physical problems that are described by the same algebraic structure. This "extra" quantum number is not only not a drawback but is an important advantage of the algebraic approach in that it allows an entire family of problems to be treated under the same heading. Illustrations of this point will occur in abundance in the following chapters. If so, can one not use more than one additional quantum number so as to introduce subfamilies? The answer is "yes." A physical example is the familiar observation (sometimes known as "the law of corresponding states") that, to a good approximation, potential energies of diatomic molecules depend on two parameters. One would then want two "extra" quantum numbers so as to distinguish one potential from another, and this can be done (Wulfman and Levine, 1984). Other applications where two extra quantum numbers are used will be found in Wu, Alhassid, and Giirsey (1989). In this introductory discussion we limit our-

Summary of Elements of Algebraic Theory

59

selves to the most familiar and useful case where only one additional quantum number is used. 4. The (bilinear) expansion in the products of boson operators b^bt^ serves to ensure the correspondance with quantum mechanics. To see this explicitly, say A, B, and C are operators familiar from wave mechanics and let A,B, and C be their corresponding matrix representations. If [A, B] = C, then [A, B] = C. Now define A = Z Aap6^p <x,p

and similarly for B and C. Then one readily verifies that [A, B] = C. Working with bilinear products is simplified when one notes that the operation [X, ] satisfies the familiar chain rule [X, YZ} = [X, Y]Z + Y[X, Z]. 5. Expansions including terms of the type b^bl and bab^ are sometimes used. The corresponding algebraic structures are more complicated than those of the unitary algebras U(n + 1) of Eq. 2.21. The algebras constructed from b^b'^, bab$, b^b^ are the symplectic algebras Sp (2» + 2). 6. The same expansion can be done for quantum-mechanical problems with halfinteger spin, except that one needs fermion operators, a^a and ap. The bilinear products a^ap also generate the unitary group. This accounts for the useful applications of the unitary group for problems of electronic configuration interaction (Judd, 1967; Hinze, 1981; Pauncz and Matsen, 1986). 7. The U(4) or "vibron" model was introduced in lachello (1981) and lachello and Levine (1982). For reviews see lachello (1992) and Levine (1988). In comparing the results of the vibron model to those for one dimensional problems, Section 2.8, one should note that the eigenvalues of the Casimir operators are different for O(4) and O(2). See Section A.12 for more details. The eigenvalue spectrum is co2, co(co+l) and co(co + 2) in one, two, and three dimensions, respectively. 8. The Morse oscillator was discussed by Levine and Wulfman (1979) and by Berrondo and Palma (1980). Levine (1982) is a review of algebraic work on onedimensional anharmonic vibrations. Additional work on one dimensional motion is found in Alhassid, Giirsey, and lachello (1983a, 1983b) and Levine (1988). 9. Analytic expressions for matrix elements of the rotating Morse oscillator can be found in Heaps and Herzberg (1952); Elsum and Gordon (1982), Huffaker and Tran (1982), Requena et al. (1983), and Nagaoka and Yamabe (1988).

Chapter 3 Mechanics of Molecules

3.1

Triatomic molecules

The Hamiltonian of a triatomic molecule in the Born-Oppenheimer approximation is

where mf is the mass of the 1'th atom. Although in this equation there are nine coordinates and momenta, the number of relevant degrees of freedom is actually six, since one can remove the center-of-mass motion from the kinetic energy. In general, for a molecule composed of v atoms, there are 3v - 3 relevant coordinates. There are several ways in which the relevant coordinates can be chosen (Murrell et al., 1984). For triatomic molecules, a common choice is that of the symmetry coordinates

The irrelevant center-of-mass coordinate is

The Hamiltonian operators can be written in terms of the symmetry coordinates

60

Mechanics of Molecules

61

Another important point is that the potential of a triatomic molecule depends only on three scalar coordinates. The reason is that, in the Born-Oppenheimer approximation, the potential will be unchanged if we rotate the plane denned by the three atoms. Hence the potential is fully denned by any three scalar coordinates that specify the planar triangle formed by the three atoms. Such "intrinsic" coordinates are not necessarily those most convenient for expressing the kinetic energy. The route to intrinsic variables often begins with the bond variables TI,TI (Figure 3.1). The bond variables can then be organized into three internal (or intrinsic variables) rl, r 2 ,6 and three Euler angles a, (3, y characterizing the orientation in space of the molecule. The Hamiltonian operator written in the intrinsic set has the form1

The various terms in this formula have the meaning of the potential function (force field) y(r 1 ,r 2 ,6); the vibrational, f v ; rotational, fx, fy, fz; and rotational-vibrational, fvr; kinetic energy terms. The latter are differential operators acting in the space of wave functions \|/(r 1 ,r 2 ,6;a, (3,y). The potential function V(r,, r2,9) is either calculated ab initio or parametrized in a suitable fashion. A commonly used parametrization is that provided by the force-field method

where the harmonic terms provide the zeroth-order approximations. Figure 3.2 shows the typical behavior of the potential function V(ri,r2,Q). The Schrodinger equation

can be solved by expanding the wave function xj/ in the basis

where the D function is the Wigner function, characterized by the angular momentum, /, the projection of / on a laboratory fixed axis, M, and the projection of / on a body-fixed axis, K. The kinetic energy terms are differential operators that have a rather complex form. Introducing the quantities

one can write the kinetic terms as

Figure 3.1 Bond and intrinsic variables for triatomic molecules (a) and the Euler angles characterizing the orientation in space of the molecule (b).

Figure 3.2 Potential function of H2O: (a) the OH stretching potential; (b) the bending potential, § = 1/2 (rc -9-9,,). Adapted from Sorbie and Murrell (1975).

64

Chapter 3

when n^TTy,!!., are differential operators acting on a, (3, y. The Hamiltonian operator written in this form is best suited to describe bent molecules. For linear molecules, 9 —> n and the terms involving riz and f[zllx diverge. A different form must therefore be written for linear molecules. The Schrodinger equation (3.7) is usually solved by variational methods, expanding the intrinsic wave function as

with § ( r i ) and cj)(r2) taken usually as Morse functions, and (])(9) taken as associated Legendre polynomials P^(cos 0).

Mechanics of Molecules

3.2

65

Polyatom Schrodinger equation

The Hamiltonian operator for a molecule composed of v atoms is

From this expression one has to remove the center-of-mass motion. The corresponding Hamiltonian can then be rewritten in terms of the intrinsic variables and the Euler angles. The general form of the Hamiltonian operator is, in the Eckart frame,1

where y, 8 denote the three variables (x, y, z), (x', y', z'). In this equation, the intrinsic coordinates have been chosen as the normal coordinates and the k sum goes over them. 3K, represents the collective angular momentum of the molecule and can be expressed in terms of the Euler angles a, (3, y and their associated momenta Pa, P$, PT Also m, represents the vibrational angular momentum of the molecule and can be expressed in terms of the normal coordinates xk and their associate momenta Pk. Finally, |Xy§ are the inertial parameters and |i is their determinant. The eigenvalue problem for H as it stands in Eq. (3.13) can rarely be solved. However, one can do a series of approximations that brings the Hamiltonian H to a more manageable form: (1) One can neglect the dependence of \\. and |Xy5 on normal coordinates. This brings H to the form

(2) Choose x, y, z as the principal axis of inertia and use constant moments of inertia, I j . This brings Eq. (3.14) to the form

(3) Ignore the vibrational momenta, my,

66

Chapter 3

As we shall discuss in Chapters 4-6, this neglect is not quite reasonable if there are degenerate vibrational modes. With these approximations, the rotational and vibrational motion is completely separated and the eigenvalue problem

can be solved. In Eq. (3.17), the xks are the intrinsic normal coordinates and Pk the associated momenta. Q denotes the set of three Euler angles, called a, (3,y in the previous sections [not to be confused with the index j in Eq. (3.16)]. Also here the rotational term 9li^//° becomes singular for linear molecules, and thus Eq. (3.16) is appropriate only for bent molecules.

3.3

One dimensional coupled oscillators

In view of the complexity of the quantum-mechanical many-body problem, several approximate schemes have been suggested and developed to deal with vibrations of polyatomic molecules. The simplest scheme is that in which one neglects altogether rotations and rotational-vibrational couplings and reduces the problem to a system of coupled one-dimensional oscillators. Often, these are taken as the displacements of bonds from their equilibrium positions. Such a description offers a particularly useful starting point when there is a wide variation in the force constants so that a subset of modes can be selected (Henry, 1977; Child and Halonen, 1984; Halonen, 1989). As an example, consider again the problem of the water molecule (Figure 3.2). If one limits the discussion to purely stretching vibrations, one can model the situation to that of two coupled one-dimensional oscillators. Introducing the variables

one can expand the potential in these variables

and solve the eigenvalue problem

where

Mechanics of Molecules

67

Note that the variables defined in Eq. (3.18) are not normal modes so that the kinetic energy operator is not diagonal. As in the case of a single variable, x, discussed in Chapter 2, the expansion (3.19) has convergence problems. A better expansion is

where the zs are Morse functions,

The eigenvalue problem (3.20) can be solved and provides an excellent description of the problem particularly so in the case in which the mass, m2, of the atom in the middle is much larger than that of the two atoms at the end.

3.4

Nonlinear classical dynamics

It is not possible to discuss highly excited states of molecules without reference to the recent progress in nonlinear dynamics.2 Indeed, the stimulation is mutual. Rovibrational spectra of polyatomic molecules provides both an ideal testing ground for the recent ideas on the manifestation of chaos in Hamiltonian systems and in turn provides many challenges for the theory. The notion of "chaos" is interwoven with the discussion of time evolution, which we do not pursue in this volume. It is worthwhile, however, to note that it is, by now, well understood that a quantum-mechanical system with a finite Hamiltonian matrix cannot satisfy many of the purely mathematical characterizations of chaos. Equally, however, over long periods of time such systems can manifest many of the qualitative features that one associates with classically chaotic systems. It is not our intention to follow this most interesting theme. Instead we seek a more modest aim, namely, to forge a link between the elementary notions of classical nonlinear dynamics and the algebraic approach. This turns out to be possible using the action-angle variables of classical mechanics. In this section we consider only the nonlinear dynamics aspects. We complete the bridge in Chapter 7. In nonlinear classical dynamics it is convenient to express the Hamiltonian in action-angle variables. The total Hamiltonian H can then be resolved as

The zero-order Hamiltonian is a function of the actions alone. It therefore corresponds to uncoupled modes whose actions are conserved (since dljdt = - dH/dQj). From Section 7.5 on we will express the classical limit of algebraic Hamiltonians in terms of variables £,-, i = !,...,«. These are related to the action-angle variables by £, = 7 1/2 exp(/0),^* = /1/2exp(-;0). Loosely speaking, the action variables when measured in units of n correspond to the

68

Chapter 3

vibrational quantum numbers. The zeroth-order Hamiltonian (corresponding to an algebraic Hamiltonian, with a dynamical symmetry, which contains terms bilinear in the generators) has a Dunham-like expression

Here di = 1/2 except for bending modes in linear molecules, for which dl• = I . The frequencies of the zeroth-order modes are given by

and do therefore depend on the actions in all the modes. The separation of the total Hamiltonian in the form (3.24) is the classical analogue of writing the algebraic Hamiltonian as a zeroth-order (but already quite realistic) form that has a dynamical symmetry (and whose eigenvalues are analytic functions of the quantum numbers) and coupling terms that break the symmetry. In Chapters 4-6 we will encounter many examples of this type. A key point is that in the algebraic approach, the coupling terms can be arranged in a sequential order. This order is determined by which symmetries are broken and which remain. As will be emphasized, such symmetry-breaking terms do not simply destroy good quantum numbers. Rather, they typically introduce new but fewer good quantum numbers. This successive reduction in the number of conserved quantum numbers, by successive coupling terms, is readily discussed in the classical limit and is the central point of this section. We ask the reader to keep this point in mind while pursuing the developments in the following chapters. The proof depends on the notion of a resonance, to which we now turn. The "resonance condition" for a molecule of n modes is the (n - ^-dimensional surface in action space denned by

The vector m has the integers m, as components. (Note that in this section ;' is the index of the modes and the integers w, are not to be confused with the masses.) For Eq. 3.25 the resonance surface is a plane. The (n— ^-dimensional surface of constant unperturbed energy is H0(I) = E. The essential technical point is that by the very definition [Eq. (3.26)], of the frequencies, the vector 00 evaluated at the point I in action space, is the gradient to the surface of constant energy passing through that point. It follows that the resonance vector m, which defines the particular resonance through Eq. (3.27) is perpendicular to 0) and hence is tangent to the surfaces of constant (unperturbed) energy. An illustration is provided in Figure 3.3.

Mechanics of Molecules

69

Figure 3.3 The surface of constant energy H0(l) = 7600 cm ' shown versus the three vibrational actions of HCN (mode 2 is the bend for which <jf2 - 727 cm"1 is low compared to the two stretches). The lines are the intersections of the surface H0(l) = 7600 cm"1 with different resonance planes m • CO = 0. The vectors m are used to label the different intersections. Note that there are no "low-order" (mr • m low) resonances. Adapted from Engel and Levine (1989).

The corresponding zeroth-order quantum-mechanical results are obtainable by regarding the vector of actions I as having components which, in units of ~h, are integers. Thus, zero-order quantum-mechanical states that are compatible with the resonance condition (i.e., two separable states n and n' such that n' - n = m) are degenerate,

This quasidegeneracy is the analogue of the classical condition of the intersection of the resonance surface (3.27) with the constant (zeroth-order) energy surface. The essential point is that, starting with n good action variables, a resonance condition reduces this number by one. We will provide many examples of the corresponding result in the algebraic approach. Here we provide the classicalmechanics analogue. First, note that while in the absence of coupling the actions are constant, to leading order in the coupling, the motion in action space about I is in the direction m, the direction of the resonance vector at that point. The proof is by expanding the coupling for a bound motion as a Fourier series in the angle variables

70

Chapter 3

Here, the sum is over all possible resonance vectors. Since the leading contribution to 0,(f) is oo,f, it follows that near the particular resonance m of interest, other phase factors in Eq. (3.29) will be oscillatory functions of time while due to Eq. (3.27), exp(zm • 9) will be slowly varying. Retaining only the one resonance term in Eq. (3.29),

or -I = ikmVm. Hence, if I0 is the solution of Eq. (3.27), the deviation from I0 is in the direction of m, which lies in the surface of constant energy. While Eq. (3.30) is valid for any resonance, as a practical point only lowerorder resonances are typically important. The reason is that the Fourier coefficients Vm [cf. Eq. (3.29)] are expected to decline rapidly once the oscillations of exp(-z'm • 9) are more rapid than those of V(1,9). This result is very familiar in the semiclassical evaluation of matrix elements, where to zeroth order the decline is exponential with Iml. It follows that if m' is a vector orthogonal to the resonance vector m (m' • m = 0, and there are n - 1 such vectors), then m' • I is a constant of the motion

One can therefore transform away a resonance condition by going over to a new set of n action variables, n - 1 of which are conserved. Having eliminated a primary resonance [i.e., one having a large coefficient Vm in Eq. (3.29)], one can eliminate the next one in turn, etc. Of course, with each additional term in Eq. (3.29) that is included in the Hamiltonian, a larger range of actions become accessible to the dynamics. The books cited in Note 2 will all provide more details on this point. The point made in Eq. (3.31), namely, that the coupled, old, n action variables can be transformed to new, uncoupled, n - 1 conserved action variables is one to which we shall repeatedly return, in the quantum-algebraic context, in Chapters 4-6. Of course, we shall first discuss //0, which has n good quantum numbers, and which we shall call "a Hamiltonian with a dynamical symmetry." At the next order of refinement we shall introduce coupling terms that will break the full symmetry but that will still retain some symmetry so that new, good, but fewer quantum numbers can still be exactly defined. In particular, we shall see that this can be done in a very systematic and sequential fashion, thereby establishing a hierarchy of sets of good quantum numbers, each successive set having fewer members.

Mechanics of Molecules

71

Notes 1. By introducing Euler angles and intrinsic coordinates one seeks to separate the overall rotation of the molecular frame from the purely "internal" motions. Such a separation is typically not rigorously feasible so that the kinetic energy always contains interactions between vibrations and rotations. For rigid molecules the Eckart molecular frame minimizes this interaction. The "standard" reference is Wilson, Decius, and Cross (1955). Other important books include: Lister, Macdonald, and Owen (1978), Papousek and Aliev (1982), and Ezra (1982). Reviews include: Nielsen (1951, 1959), Mills (1972), Watson (1977), Carney et al. (1978), Bunker (1983), Jensen (1983), Tennyson (1983), Carter and Handy (1987, 1988), Fleming and Hutchinson (1988), Fried and Ezra (1988), Sibert (1988), Chang et al. (1988), Wierzbicki and Bowman (1988), Tucker et al. (1988), Ermler et al. (1988), Back and Light (1989), Hutson (1991), Schwenke (1991). Recent papers are numerous. A partial listing includes: Watson (1984), Wallace (1984), Harter and Patterson (1984), Clodius and Cade (1985), Leroy and Wallace (1987), Sadovskii and Zhilinskii (1988), Jensen (1988), Chapuisat et al. (1991, 1992), and Sutcliffe and Tennyson (1991). 2. The literature on nonlinear mechanics is very rich. Books include Lichtenberg and Lieberman (1992), Reichl (1992), Gutzwiller (1990), Tabor (1989), Sagdeev, Usikov, and Zaslavsky (1988). Reviews include: Gomez Llorente and Pollak (1992), Cerdeira et al. (1991), Uzer (1991), Heller (1991), Sibert (1990), Bohigas and Weidenmuller (1988), Reinhardt (1989), Taylor (1989), Pollack and Schlier (1989), Seligman and Nishioka (1986), Rice (1981), Brumer (1981), Zaslavsky (1981), Noid, Koszykowski, and Marcus (1981), Chirikov (1979), Percival (1977), and Ford (1973).

Chapter 4 Three-body Algebraic Theory

4.1

Algebraic realization of many-body quantum mechanics

In the previous chapter we discussed the usual realization of many-body quantum mechanics in terms of differential operators (Schrodinger picture). As in the case of the two-body problem, it is possible to formulate many-body quantum mechanics in terms of algebraic operators. This is done by introducing, for each coordinate rl,r2,... and momentum p,, p 2 , . . . , boson creation and annihilation operators, b'ia, bia. The index ;' runs over the number of relevant degrees of freedom, while the index a runs from 1 to n + 1, where n is the number of space dimensions (see note 3 of Chapter 2). The boson operators satisfy the usual commutation relations, which are for i ^ j,

In this realization, the basis states are written as

where 9li, 9lj,... are normalization constants usually chosen in such a way as to have an orthonormal set. All operators of the theory are then expanded in terms of the boson operators, b]a, bia. This expansion can be written in a more compact form by noting that 72

Three-body Algebraic Theory

73

the bilinear products of creation and annihilation operators form an algebra. One thus introduces the operators

and writes the Hamiltonian as

where

Since the operators bia, bia with different indices are assumed to commute, the algebraic structure of many-body quantum mechanics is the direct sum of the algebras of each degree of freedom

in other words the different degrees of freedom add. In view of the fact that the corresponding wave functions multiply

quite often the situation described here is denoted by a multiplication sign ®, meaning that the representations of G\, GI, ... are multiplied when constructing the total wave function

In this book we shall follow this common practice, and use the multiplication sign throughout.

4.2

One-dimensional coupled oscillators by algebraic methods

The formulation of the preceding section is very general. We are interested, however, in rotations and vibrations of polyatomic molecules. We therefore discuss now specific applications of the algebraic method beginning with the simple case of one-dimensional coupled oscillators, presented in Section 3.3 in the Schrodinger picture. In the algebraic theory, as mentioned, one associates to each coordinate, x, and related momentum, px = — in d/dx, an algebra. For

74

Chapter 4

one-dimensional problems, this algebra is U(2), as discussed in Section 2.8.' Thus if one wants to describe stretching vibrations of polyatomic molecules by one-dimensional coupled anharmonic oscillators, one introduces an algebra U(2) for each mode. For example, in triatomic molecules, there are two bonds, and thus the overall algebra is

Each algebra can be realized in terms of creation and annihilation operators, as discussed in Section 1 and in more detail in Chapter 2. The operators now have an index corresponding to the bond that they describe, and there are two creation and annihilation operators per bond: a{,T},(Ji,Ti; <s\,i\,O2,^2- The overall algebra is composed of

The Hamiltonian operator can be written in terms of these algebraic bond variables. It has the form

where H} and 7/2 describe the properties of bonds 1 and 2 and V12 denotes the bond-bond interaction. It is convenient to expand all interactions in terms of algebraic Morse functions. In the case of stretching vibrations, for which Morse functions are good approximations, the expansion is quickly convergent. As discussed in Sections 2.8-2.11, Morse functions are algebraically described by the group chain U(2) ^> O(2), with wave functions

where we have deleted the index y or z on m, since from now on we consider only U(2) r> O(2) states. When discussing coupled systems, a new problem appears, namely, how the oscillators are coupled. In the algebraic approach different couplings correspond to different ways in which the operators of bond 1 are summed with those generators of bond 2. This is indicated by the following pattern (called a lattice of algebras), which shows two possible chains

Three-body Algebraic Theory

75

For each of the two chains in Eq. (4.13) one can write a Hamiltonian operator that can be diagonalized analytically (van Roosmalen et al., 1984; Cooper and Levine, 1989).

4.3

The local mode limit

Consider first chain (I). A dynamical symmetry corresponding to this route implies that the Hamiltonian operator contains only invariant operators of the chain,

In order to simplify the notation, we introduce the operators

Hence,

The states of the chain (I) are characterized by the quantum numbers

The quantum number of O 1 2 (2),m, is not an independent number since it is given by

Introducing the vibrational quantum numbers

as in Eq. (2.57), one can denote the states (4.17) by \Ni,N2, va, vb >. The eigenvalues of the operators C [Eq. (4.15)] are

76

Chapter 4

In terms of the vibrational quantum numbers, va,vh , the eigenvalues of // (/) can be written as

One can see that these represent the eigenvalues of two local anharmonic oscillators. The spectrum of Eq. (4.21) when the two oscillators are identical, as in H2O, where they represent the stretching of the O-H bonds, is shown in Figure 4.1.

4.4

The normal mode limit

For two coupled oscillators, the second possibility is chain (II) of Eq. (4.13). A dynamical symmetry corresponding to this route implies that the Hamiltonian operator contains only invariant operators of this chain,

It has become customary to call the Casimir operator of U12(2), Majorana operator since it was introduced by Majorana in the 1930's within the context of other problems,

Figure 4.1 Spectrum of two coupled local anharmonic oscillators. Note the inherent degeneracies in the spectrum.

Three-body Algebraic Theory

77

The states of chain II are characterized by the quantum numbers

This result arises from the rules of multiplication of representations of the unitary groups. Note that the number of independent quantum numbers is the same in both cases, I and II. The eigenvalues of the Hamiltonian in Eq. (4.22) in the basis (4.24) are given by

It is convenient to introduce again vibrational quantum numbers. Denoting these quantum numbers Vj and v2 (not to confuse them with va and vh of the preceding section),

Eq. (4.25) can then be rewritten as

The spectrum corresponding to Eq. (4.27) is shown in Figure 4.2. One can see that this represents the usual spectrum of two normal anharmonic coupled oscillators.

Figure 4.2 Spectrum of two normal coupled anharmonic oscillators. Note how the different levels are almost equispaced.

78

4.5

Chapter 4

Local to normal transition

The fact that both the local- and the normal-mode limits are contained within the algebraic approach allows one to study in a straightforward way the transition from one to the other. It is convenient to use, for this study, the local basis [Eq. (4.17)] and diagonalize the Hamiltonian for two identical bonds

which contains operators of both chains. There is only one constant A = AI = A2 since the two bonds are assumed to be identical. The matrix elements of the operator M12 are given in the basis (4.17) by

The off-diagonal elements of the Majorana operator in Eq. (4.29) illustrate, for the first time, the appearance of a nonlinear resonance (Section 3.4) within the algebraic approach. For two identical modes, an m = (1,—1) resonance is expected to be very important. M 12 indeed couples such [nearly degenerate, cf. Figure 4.1 or Eq. (3.28)] states whose quantum numbers differ by m. Note furthermore that the 0^(2) quantum number is conserved, va + vh = const, as expected for a 1,-1 resonance because m' = (1,1) is orthogonal t o m = (l,-l), cf. Eq. (3.31). The transition from the local- to the normal-mode limit is described by the parameter Xi2/A. When this parameter is zero, the Hamiltonian (4.28) is in the local limit, when the parameter is large the spectrum approaches the normalmode limit. It is convenient to define the dimensionless locality parameter as

With this definition, due to Child and Halonen (1984), local-mode molecules are near to the ^ = 0 limit, normal mode molecules have ^ —> 1. The correlation diagram for the spectrum is shown in Figure 4.3, for the multiple! P = va + vh = 4. It has become customary to denote the local basis not by the quantum numbers v a , vh, but by the combinations

Three-body Algebraic Theory

79

Figure 4.3 Correlation diagram between the local- and normal-mode limits as a function of the parameter ^. Note how the degeneracies typical of the local-mode limit are split and as ^ —> 1 become the almost harmonic spacings characteristic of the normalmode limit.

This notation is used in Figure 4.3.

4.6

An example: Stretching vibrations of water

As an example, we consider the two equivalent stretching vibrations of the water molecule, H2O. The Hamiltonian is Eq. (4.28) and with E0 = 0 we have four parameters N = N\ = N2, A — A{ = A2, A 12 , and ?i12. By fitting these parameters to the experimental data one obtains the results of Table 4.1. The two stretching modes are called vl and v3 here in order to conform with standard notation (Herzberg, 1950; v 2 is the bending mode). Several other cases have been analyzed. Typical root-mean-square deviations for the lowest-order Hamiltonian of Eq. (4.28) are < 5 cm"1 up to the sixth overtone. For example, the calculation of water of Table 4.1 has a root-mean-square deviation of 4.0 cm"1. In addition to providing a calculation of stretching overtones, one is also able to determine, in a simple way, the nature of the spectrum. If one compares, for example, water, H2O, with sulfur dioxide, SO2, one observes the situation of Table 4.2. Thus SO2 is much closer to the normal limit than H2O. We shall

80

Chapter 4 Table 4.1

Vibrational energy levels of water" (in cm ') V,V2V3

Exp.

Calc.*

(100) (001) (200) (101) (0 0 2) (300) (201) (102) (003) (202) (301) (400) (103) (004) (302) (401) (500) (203) (104) (005)

3 658.82 3 749.13 7 205.41 7 247.04 7 438.49 10 604.46 10614.69 10 865.28 11031.58 13 832.77 13 834.12 14220.69 14316.17 14 546.49 16 895.22 16 895.34 17 459.57 17492.63 17 746.26 17971.71

3 657.05 3 755.93 7201.54 7 249.82 7 445.05 10 599.66 10613.41 10 868.86 11032.40 13 828.30 13 830.92 14221.14 14318.80 14536.87 16 898.40 16898.83 17 458.20 17 495.52 17 748.07 17970.90

"van Roosmalen, Benjamin, and Levine (1984); Higher overtones can be found in Benjamin and Levine, 1987. *W=44, /1=-18.9820 cm"1, A12 = 1.1319 cm*1, K12 = 1.0263 cm"1.

return to this question later when treating the full three-dimensional problem. See, in particular, Figures 4.3 and 4.19, which is very instructive in connection with Table 4.2. Table 4.2. Locality parameter, ^, of some triatomic molecules Molecule

H20 03 S02

4.7

$ 0.25 0.3 0.85

Infrared intensities

The study of intensities of transitions can be done in a way similar to that discussed in Section 2.10. In the lowest approximation the transition operator, f , is the sum of the transition operators of each bond,

Three-body Algebraic Theory

81

where each bond operator has the matrix element

In order to find the coefficients a, for infrared transitions (for which the operator is a dipole operator), one returns to the geometric structure of the molecule and introduces a set of Cartesian coordinates (in the plane x, y), shown in Figure 4.4. The operator f has two components, fx and f y , given by

where a is an overall scale. Intensities are then computed using

where v, v' denote generically vibrational states vj,v 2 and v\,v'2. Examples of these calculations are given in the following sections.

4.8

Three-dimensional coupled roto-vibrators by algebraic methods

In the preceding sections we have discussed the algebraic treatment of onedimensional coupled oscillators. We now present the general theory of two three-dimensional coupled rovibrators (van Roosmalen, Dieperink, and

Figure 4.4 Coordinates x, y appropriate to water superposed on the contours of the potential energy for the stretch motion. See also Lawton and Child (1980).

82

Chapter 4

lachello, 1982; van Roosmalen et al., 1983a). In the usual quantum mechanical treatment, this situation is now characterized by two vector coordinates, rl and r2, instead of two scalar coordinates x\ and x2. The general algebraic theory then tells us that we should use the algebra

that is, we have to describe each bond, i= 1,2, with U(4) rather than U(2). Apart from this difference, the analysis of this situation follows precisely the same scheme of the previous case. We introduce for each bond four boson operators

together with the corresponding annihilation operators a1; n^, O2, ft^. From the latter operators we form spherical tensors

The elements of the algebras 1^(4) and U2(4) are the same as those in Table 2.1 except that an index i = 1,2 is attached to each operator. The Hamiltonian operator is the same as before

where HI and H2 describe bond 1 and 2 and V12 is the bond interaction. A generic algebraic Hamiltonian is

and

Three-body Algebraic Theory

83

As discussed in Chapter 2, each bond can behave either rigidly or softly. Rigid bonds correspond to the group chain U(4) => O(4) z> O(3) => O(2) with wave functions given by Eq. (2.101). This group chain is the algebraic analog of the quantum-mechanical problem of a three-dimensional Morse rovibrator. In the following sections we consider only rigid molecules. When one combines two (or more) bonds, a new problem arises, as discussed in Section 4.2, namely, how the bonds are coupled. There are two main ways in which the bonds can be coupled,

which, as in the previous case, correspond to local and normal coupling, respectively. For these two situations the Hamiltonian operator can be diagonalized analytically.

4.9

Local basis

The local basis is characterized by the representations of chain I,

The numbers N{,N2 are the vibron numbers of each bond. As discussed in Chapter 2 they are related to the number of bound states for bonds 1 and 2, respectively. For Morse rovibrators they are given by Eq. (2.111); that is, they are related to the depth of the potentials. They are fixed numbers for a given molecule. The numbers co^a^Ti.Ta are related to the vibrational quantum numbers, as discussed explicitly in the following sections. We have written the Oj(4) representations as (co^O) and not simply as K>1, since for coupled systems one can have representations of O(4) in which the second quantum number is not zero. The values of o>i and co2 are given by the rule (2.102),

The values of (TJ , T2) are instead obtained from the direct product of representations, discussed in Appendix B. They are given by

84

Chapter 4

Finally for each representation (ii,T 2 ) the values of the allowed angular momenta and parities are

The z component of the angular momentum is given by the usual rule

4.10

The normal-mode basis

The normal basis is characterized by the representations of chain II,

The numbers N{,N2 are as before. The representations of U12(4) that one can obtain are given by the direct product rules (Appendix B) with

There is a complication here, when reducing the representations of U12(4) to those of O12(4). This step is not fully reducible, in the sense that a representation with the quantum numbers (11,12) can appear more than once in a given representation [A^ + N2 - n, n\. This problem, called the multiplicity problem, requires the introduction of an additional label, %, called the missing label. For U(4) 3 O(4) the problem has been solved by Hecht and Pang (1969). The solution is rather complex, and we do not give it here. The further decomposition from O12(4) to O12(3) is the same as in the previous case and given by Eq. (4.45). The decomposition O12(3) Z) O12(2) is again given by the usual rule (4.47).

4.11

Expansion of the coupled basis into uncoupled states

For explicit evaluation of matrix elements it is necessary to expand the coupled basis of the previous two sections in terms of uncoupled states. The general theory is discussed in Appendix B. The expansion of the local-mode basis, which is that used in most calculations, is given by

Three-body Algebraic Theory

85

In this expression the coefficients in brackets < > are the isoscalar factors (Clebsch-Gordan coefficients) for coupling two O(4) and two O(3) representations, respectively. They can be evaluated either analytically using Racah's factorization lemma (Section B.14) or numerically using subroutines explicitly written for this purpose.2

4.12

Linear triatomic molecules

In the next eight sections we discuss linear triatomic molecules. We begin with the simple situation of an exact dynamical symmetry, when a closed analytical expression for the energy in terms of the quantum numbers can be written down. The short summary is that even for the highest overtones currently experimentally accessible, the dynamical symmetry is typically satisfied to an rms accuracy of 10 cm"1. There are two possible dynamical symmetries, corresponding to the two chains in Eq. (4.42), which we refer to as the local- and normal-mode limits, respectively. More molecules are closer to the local-mode limit, which is also the basis in which computations are simpler. There is, however, a unitary transformation connecting the two bases. When resonances are possible and, in general, for higher accuracy, the exact symmetry will be broken. As will become evident, this does not mean that all previously good quantum numbers no longer apply. Rather, breaking the dynamical symmetry is a sequential process whereby the number of good quantum numbers decreases by one (or more) at a time in a well-prescribed way. The remaining good quantum numbers define not a single state (as in the case for a dynamical symmetry) but a group of states, sometimes called a multiplet. We have already encountered such multiplets in Section 5, where all states with a given value of P, the sum of the two stretch quantum numbers, form a multiplet (Figure 4.3). We shall have more examples in the following and further examples towards the end of this chapter when vibrational-rotational coupling is introduced.

4.13

Local-mode Hamiltonian for linear triatomic molecules

We have stated several times that whenever the Hamiltonian can be written in terms of invariant (Casimir) operators of a chain, its eigenvalue problem can be solved analytically. This method can be applied to the construction of both local and normal Hamiltonians. For local Hamiltonians, one writes H in terms of Casimir invariants of Eq. (4.43).

86

Chapter 4

where we have dropped the subscript 2 from the Casimir invariant operators in order to simplify the notation. The Hamiltonian (4.51) is diagonal in the local basis with eigenvalues

the last term representing the rotational part. One can now convert Eq. (4.52) into the usual form by introducing the local vibrational quantum numbers,

The quantum numbers v a , v c denote local stretching vibrations, while the quantum numbers, vl£, denote the doubly degenerate bending vibrations (Figure 4.5). Equation (4.52) can be converted, using Eq. (4.53), to

In this expression we have dropped the rotational part and written only the vibrational part. We also note that the spectroscopic notation

Figure 4.5 Local vibrational quantum numbers of a linear triatomic molecule. The arrows indicate the corresponding displacements.

87

Three-body Algebraic Theory

Figure 4.6 Schematic representation of a portion of the spectrum of linear XYZ local molecule. The scale is that appropriate to HCN. The energy levels are obtained using Eq. (4.54) with Nl = 144, N2 = 47, A} = -1.208 cm"1, A2 = -10.070 crrf1, An = -1.841 cm"1.

is quite often used to characterize the bending vibrations. A schematic representation of the local spectrum of XYZ molecules is shown in Figure 4.6. There are several molecules that can be very well approximated by a local Hamiltonian, for example, HCN and OCS. A comparison of the vibrational frequencies of HCN and those calculated in the strict local limit is shown in Table 4.3. Table 4.3. Vibrational energy levels of hydrogen cyanide" (in cm ') Vj

0 1 0 0

1

2 0

1

0 2 1 1

4 2°

0° 0° 4° 2° 0° 2° 0° 0° 2° 2° 0°

V3

Expt."

Calc.*

0 0

1411.43 2096.85 3311.48 2802.85 3501.13 4173.07 4684.32 5393.70 6519.61 5571.89 6761.33 8585.57

1413.30 2104.63 3299.38 2811.46 3502.79 4184.22 4697.54 5388.87 6503.50 5567.24 6771.89 8577.86

1

0 0 0

1

1

2 0

1

2

88

Chapter 4 Table 4.3 (continued) Vl

0

1

1

0 0 3 2 1 0

5 0

0 1 0 1 2

0 1 0 0 0

1 0

4 0° 2° 0°

0° 0° 0° 0° 0° 0°

0°1

I

^3

3 2

3 4 5 3 4 5 6 2 0

31 1] I1 31 I1 31 I1

0

I1 22

2

2

0 0

4 22 22

0

1 0 0

1

1

0

1

Expt."

Calc.*

9627.02 9914.41 11674.46 12635.90 15551.94 15710.53 16674.21 17550.39 18377.01 16640.31 711.98 2113.46 2805.58 4004.17 4201.29 4878.27 5366.86 6083.35 7194.75 1426.53 2818.16 3516.88 4699.21

9612.37 9945.75 11671.59 12625.99 15544.35 15714.92 16689.12 17573.30 18367.46 16624.91 706.65 2112.38 2803.71 3998.46 4194.30 4875.73 5389.05 6080.38 7195.01 1405.73 2803.89 3495.22 4689.97

"van Roosmalen et al. (1983); see also lachello, Oss, and Lemus (1991) and Cooper and Levine (1991). X

4.14

= 140, N2 = 47, AI = -1.234, A2 = -10.034, An = -1.889; A(™,s) = 12.3 cm"1.

The normal-mode Hamiltonian for linear triatomic molecules

Although not much used in actual calculations, we also quote the case of the strict normal-mode limit. In this case, H can be written in terms of Casimir invariants of Eq. (4.48)

where again we have dropped the subscript 2 of the invariant Casimir operators. The Hamiltonian (4.56) is diagonal in the basis (4.48) with eigenvalues

Three-body Algebraic Theory

89

The missing label % does not appear in the expression (4.57). One can convert Eq. (4.57) to the usual form by introducing the normal vibrational quantum numbers (Figure 4.7),

In terms of these quantum numbers, the vibrational part of the energy eigenvalues is

We have called the vibrational quantum numbers here V j , v22, v 3 in order to distinguish them from the local quantum numbers, va, v£, vc. Note that, in view of the presence of the missing label, %, the normal basis is not very convenient for calculations. The spectrum corresponding to Eq. (4.59) is shown in Figure 4.8. There are fewer examples of molecules for which the dynamical symmetry of the normal chain II, provides a realistic zeroth-order approximation. The normal behavior arises when the masses of the three atoms are comparable, as, for example in XY2 molecules with mx = m^. More examples are discussed in the following sections.

4.15

/-dependent terms

Before addressing the problem of XY2 molecules, we return to Eqs. (4.54) and (4.59) and note that the only / dependence arises from -Al2l2b or -A 12 /2- The coefficient of these terms is fixed from other considerations, since it is governed

Figure 4.7 Vibrational quantum numbers in the normal mode limit of a linear triatomic molecule.

90

Chapter 4

Figure 4.8 Schematic representation of the portion of the spectrum of an XYZ normal-mode molecule. The energy levels are computed using Eq. (4.59) with N\ = 144, N2 =47, }.12 = 2.078 cm"1, An = - 1.571 cm'1.

by the location of the bending modes vb and v 2 , respectively. In many molecules, one may wish to consider the explicitly /-dependent terms that are present in the Hamiltonian introduced in Section 4.8. The algebra of O(4) has two Casimir (invariant) operators. The first operator was already introduced in Chapter 2 and written in terms of generators of O(4) as

The second invariant has the form

The eigenvalues of this operator in a representation (il, T2) are given by

In diatomic molecules, T2 = 0, and thus the expectation value of C vanishes. This is the reason why this operator was not considered in Chapter 2. However, for linear triatomic molecules, T2 = / * 0, and the expectation value of C does not vanish. We note, however, that D • J is a pseudoscalar operator. Since the Hamiltonian is a scalar, one must take either the absolute value of C [i.e., IC(O(4 12 ))I or its square IC(O(412))I2. We consider here its square, and add to either the local or the normal Hamiltonians (4.51) or (4.56) a term /412IC(O(412))I2. We thus consider, for the local-mode limit,

Three-body Algebraic Theory

91

The eigenvalues of the added contribution, written in the local mode notation are

This causes a splitting of vibrational bands with different lb (Figure 4.9). A similar situation occurs in the normal mode limit with lh replaced by /2. / splitting is an important (at the level of accuracy of 10 cnT1) feature of linear triatomic molecules.

4.16

Linear XY2 molecules

Most XYZ molecules, especially those with mx very different from mz (for example, HCN) have local behavior. The situation is very different in XY2 molecules, for which the two bonds are identical. Thus, one must have

in Eq. (4.54). As a result, the states (10°0) and (00° 1) are now degenerate (Figure 4.10), and (resonant) nondiagonal couplings become important. These molecules generally display a level scheme intermediate between the local and normal limits. We therefore now proceed to study these couplings.

4.17

Major ana couplings (Darling-Dennison couplings)

We have already discussed in Section 4.5 the local-to-normal transition for two coupled oscillators. The situation is quite analogous for two coupled rovibrators. The local-to-normal transition can be described by combining the operators of the local chain with those of the normal chain. It is convenient to introduce the Majorana operator

Figure 4.9. Splitting caused by the second invariant of O12(4) (/ splitting).

92

Chapter 4

Figure 4.10 Local spectrum of XY2 molecules. The local quantum numbers (va, v'f, v c ) and the corresponding algebraic quantum numbers (co 1 ,co 2 ,T 1 ,T 2 ) are shown next to each level.

and consider the Hamiltonian

where we have introduced the operators

in order, again, to simplify the notation. The only term in Eq. (4.67) that is not diagonal in the local basis is the Majorana operator. This operator, being the Casimir operator of Uj 2 (4) plus a constant has the following selection rules

The nonzero matrix elements, deleting the rotational quantum numbers, are

Thrf^-Knrlv Alfrphrflip Thpnrv

Q^

When converted to the local vibrational quantum numbers of Eq. 4.53, the selection rules (4.69) imply that the Majorana operator conserves the quantity

Thus the secular matrix for Eq. (4.67) splits into blocks characterized by a given value of M (Figure 4.11). We call each block a multiple!. By diagonalizing Eq. (4.67), the degeneracies of the local spectrum of Figure 4.10 are lifted, and one goes more and more towards a normal spectrum as the parameter X J2 is increased. This situation is depicted in Figure 4.12, where the first multiplet is shown. In this figure, states are labeled by the local quantum numbers V O > V A > V C > by the normal quantum numbers, v 1 , V 2 , v 3 and by the geradeungerade species

Figure 4.11 Form of the secular equation for the Hamiltonian (4.67). Each block is a multiplet of states.

94

Chapter 4

Figure 4.12 Representation of the couplings induced by the Majorana operator in the first multiple!, n = \.N and give the order of magnitude of the couplings.

Here g means symmetric under interchange a < — > c (bond 1 < - > 2) and u means antisymmetric under interchange a < — > c (bond 1 < - > 2). It is instructive (van Roosmalen et al., 1983a; Cooper and Levine, 1989; lachello, Oss, and Lemus, 1991) to display explicitly the Hamiltonian matrix in the limit

For the first multiple!, n = 1, and similar expressions for the other multiplets. The states of the second multiplet, n = 2, which are coupled by the operator Mi 2 , are (04°0), (12°0), (02° 1) (20°0), (10°1), (00°2). From the structure of the matrices, one can see that the Majorana operator does two things simultaneously. It produces the local couplings that are needed to go from local to normal situations, and it introduces, when viewed from the normal-mode basis, Darling-Dennison (1940) couplings of the type < v t , v 2 2 , v 3 IVIv ( =p 2, v22, v 3 ±2 >. Note that there is a duality that stems from the two different ways one can view the Hamiltonian (4.67) (Lehmann, 1983; Levine and Kinsey, 1986). As written, the Majorana operator serves to couple the local-mode states. But the Majorana operator is [cf. Eq. (4.66)] the Casimir operator of U(4) and is a leading contributor to the Hamiltonian, Eq. (4.56) describing the exact normal-

95

Three-body Algebraic Theory

mode dynamical symmetry. Thus, an equivalent way of writing the Hamiltonian (4.67) is

Here C} and C2, defined in Eq. (4.68), are the Casimir operators for the two O(4) groups, one for each local mode, and it is now Ci and C2 that break the dynamical symmetry. From Chapter 2 [Eq. (2.47)], the magnitude of A determines the anharmonicity of the mode. Hence strong Darling-Dennison normalmode coupling is expected when the local modes are very anharmonic. This is one of the reasons why such coupling is so prominent in hydrides. Another generic case for strong Darling-Dennison coupling is when the local-mode potential supports a few bound states so that A = a£/4De is high. The local-to-normal transition is governed by the same parameter £,, of Eq. (4.30). The difference is that now the local-to-normal transition occurs simultaneously for the stretching and bending vibrations. The correlation diagram for stretching vibrations is the same as in Figure 4.3. The local-to-normal transition can also be studied for XYZ molecules, for which the Hamiltonian does not have the condition A) = A2 and is

For these molecules there are two locality parameters

corresponding to the two bonds. A global locality parameter for XYZ molecules can be denned as the geometric mean

Locality parameters of some linear triatomic molecules are given in Table 4.4. Table 4.4

Locality parameter, Jj, of some triatomic molecules in the full U(4) model0 Molecule 12

C 02 C I3 0 2 Molecule HCN OCS N20 a

$

0.93 0.93

$1

^2

t,

0.24(CN) 0.49(CS) 0.80(NN)

0.06(CH) 0.23(CO) 0.88(NO)

0.12 0.34 0.84

lachello, Oss, and Lemus (1991).

96

4.18

Chapter 4

Quantum number assignment

An important problem of molecular spectroscopy is the assignment of quantum numbers. Quantum numbers are related to conserved quantities, and a full set of such numbers is possible only in the case of dynamical symmetries. For the problem at hand this means that three vibrational quantum numbers can be strictly assigned only for local molecules (i; = 0) and normal molecules (^ = 1). Most molecules have locality parameters that are in between. Near the two limits the use of local and normal quantum numbers is still meaningful. The most difficult molecules to describe are those in the intermediate regime. For these molecules the only conserved vibrational quantum number is the multiple! number n of Eq. (4.71). A possible notation is thus that in which the quantum number n and the order of the level within each multiple! are given. Thus levels of a linear triatomic molecules can be characterized by

4.19

Fermi couplings

The Hamiltonians of the previous sections describe realistic vibrational spectra of linear triatomic molecules except when accidental degeneracies (resonances, cf. Section 3.3) occur. A particularly important case is that in which the bending overtone (02°0) is nearly degenerate with the stretching fundamental (10°0) of the same symmetry (Fermi, 1929, resonance). This situation occurs when the coefficient Xi 2 in Eq. (4.67) is nearly equal to -A (Figure 4.13). The Majorana

Figure 4.13 Schematic representation of the effects of the Majorana operator MI2, which removes the degeneracy of the local modes and of the Fermi operator 5F12, which splits the degenerate (when A,12 = - 1) normal multiplets.

Three-body Algebraic Theory

97

operator Mn induces both local-mode (or, equivalently, Darling-Dennison normal-mode) couplings and Fermi-type couplings, as is evident from Figure 4.12. However, the strength of both couplings is governed by one single parameter, A,12. In general, one needs to treat the two couplings separately. One therefore adds to the Hamiltonian (4.67) a Fermi coupling operator 512. The total Hamiltonian is then

The strength of the direct Fermi coupling is governed by the parameter *12. The Fermi operator 3\2 is defined through its matrix elements

It is instructive to display explicitly the matrix elements of the operator yn in the first multiplet n=\. They are given by

The total matrix, including both Majorana and Fermi couplings, is then

Thus A.12 determines the strength of the Majorana couplings and *i2A,12 that of the Fermi couplings. This situation is depicted in Figure 4.13. For strongly mixed Fermi multiplets the question of the appropriate notation to use reappears. A commonly used notation is that of indicating the number of the multiplet with the lowest number of bending quanta and an ordering number. Thus, the two members of the Fermi mixed multiplet (10°0), (02°0) are denoted by

98

Chapter 4 Table 4.5 VjV^S 02°0 10°0 00° 1 02°1 10°1 04° 1 12° 1 20°1 00°3 06° 1 14° 1 22° 1 30° 1 02°3 10"3

1+ Zj Z£ Z+ Z+ E+ Z+ E+ Z£ Z£ £+ Z£ Z+ Z; Z^

Vibron Model Fits of C12O7

Expt."

Fill*

8

Fit2 c

5

1285.4 1388.2 2349.2 3612.8 3714.8 4853.6 4977.8 5099.6 6972.6 6075.9 6227.9 6347.8 6503.1 8192.6 8294.1

1296.6 1393.3 2328.6 3611.0 3705.7 4862.9 4975.0 5099.9 6940.3 6080.9 6231.8 6344.7 6507.8 8191.9 8287.6

11.2 5.1 -20.6 -1.8 -9.1 9.3 -2.8 0.3 -32.3 5.0 3.9 -3.1 4.7 -0.7 -6.5

1286.6 1388.4 2348.3 3612.4 3714.4 4854.0 4978.3 5100.0 6971.5 6077.5 6230.8 6348.5 6504.4 8191.2 8293.3

1.2 0.2 -0.9 -0.4 -0.4 0.4 0.5 0.4 -1.1 1.6 2.9 0.7 1.3 -1.4 -0.8

Adapted from lachello, Oss, and Lemus (1991), where the explicit use of the Fermi operator S [cf. Eq. (4.78)] is introduced. See also van Roosmalen et al. (1983). Results for higher overtones can be found in these references. See Appendix C for the parameters. "All energies in cnT1. ft Using only terms linear in the Casimir operators. 'Using all the terms bilinear in the Casimir operators in Eq. (4.92).

(10°0)1 and (10°0)2. An example of strong Fermi mixing is provided by carbon dioxide (Table 4.5).

4.20

Bent triatomic molecules

For bent XYZ molecules there are several possible choices of geometrical variables, as discussed in Chapter 3. The two most useful sets are the bond displacements themselves, and the symmetry coordinates. The use of the latter leads naturally to a scheme in which the Hamiltonian for bent molecules is no longer diagonal in the total O(4) quantum numbers (tl, T2), and thus one loses the simple form of the secular equation (Figure 4.11). The secular equation must be now diagonalized in the full space with dimensions that become rapidly larger. This scheme, developed by Leviatan and Kirson (1988), can be implemented only if the vibron numbers N are relatively small, N < 10. An alternative approach, developed by van Roosmalen et al. (1982, 1983a), is based on the use of the bond coordinates, and treats bent molecules still keep-

Three-body Algebraic Theory

99

ing the Hamiltonian diagonal in the total O(4) quantum numbers (t 1 ,T 2 ) but introducing a new term, van Roosmalen's scheme can be implemented even for large vibron numbers, N ~ 100, typical of realistic molecules, and we therefore make use of this scheme in this book.

4.21

Local Hamiltonians for bent triatomic molecules

For bent triatomic molecules one can easily construct a local mode Hamiltonian whose eigenvalues reproduce the spectrum:

This Hamiltonian is diagonal in the local mode basis [Eq. (4.43)] with eigenvalues

A particularly simple case is that in which A 12 — 2A\2- The energy eigenvalues become then

This expression can be converted to the usual spectroscopic notation by introducing the local vibrational quantum numbers, va, vh, vc of Figure 4.14. These quantum numbers are related to the group quantum numbers by

where now one of the quantum numbers, T 2 > has been converted to the quantum number K describing the projection of the rotational angular momentum on the molecular fixed axis. The vibrational part of Eq. (4.84) is

The corresponding spectrum is schematically illustrated in Figure 4.15.

100

Chapter 4

Figure 4.14 Local vibrational quantum numbers of bent triatomic molecules. Also shown are the relative displacements of the atoms in the different modes.

Figure 4.15 Schematic representation of a portion of the spectrum of a bent XYZ local-mode molecule.

Three-body Algebraic Theory 4.22

101

Linear-bent correlation diagram

The particularly simple form of the Hamiltonian (4.82) allows one to construct a correlation diagram relating linear and bent triatomic molecules (Herzberg, 1950; Amar, Kellman, and Berry, 1980). The former correspond to the value of the parameter A12 = 0, while the latter correspond to An = 2Aj 2 . As one varies this parameter continuously from 0 to 2A 12 , one obtains the correlation diagram shown in Figure 4.16. This correlation diagram is identical to that given in Herzberg (1950). It allows one to study situations intermediate between rigid linear or bent (quasilinear molecules). The correlation diagram is best described in terms of a linearity parameter

When L, = 1 the molecule is linear; when ^ = 0 is bent. Intermediate cases (such as quasilinear molecules, Bunker, 1983) have 0 < ^ < 1.

4.23

The normal-mode Hamiltonians for bent triatomic molecules

Within van Roosmalen's scheme, it is not possible to construct simple diagonal Hamiltonians with the degeneracies required by bent normal-mode molecules. These molecules must therefore be dealt with by numerically diagonalizing the Hamiltonian matrix as discussed in the following sections.

Figure 4.16

Linear-bent correlation diagram for triatomic molecules.

102

4.24

Chapter 4

Bent XY2 molecules

As in the case of linear molecules, most XYZ bent molecules have local behavior. The situation is again different in XY2 molecules, where the two stretching modes (100) and (001) have the same energy and therefore couple strongly, leading to normal behavior. The local spectrum of bent XY2 molecules is shown in Figure 4.17, together with the algebraic quantum numbers that characterize the states.

4.25

Majorana couplings

Normal behavior is induced by Majorana couplings. The situation is similar to that discussed in Section 4.17 for linear molecules, the Hamiltonian being

where C12 = IC(O(4i2))l and A 12 = 2AU. The matrix elements of the Majorana operator are still given by Eq. (4.70), but now the conversion from algebraic to vibrational quantum numbers is different [Eq. (4.85)]. In view of this difference the states that are coupled are now those belonging to the multiplets with

This situation is depicted in Figure 4.18. In addition to the local-mode notation, (va, vb, vc), Figure 4.18 also shows the normal notation, (vj, v 2 , v-$), corresponding to the modes of Figure 4.19, and the A, B (Herzberg, 1950) notation characterizing the symmetry of the wave function under interchange of bonds 1 < - > 2,

Figure 4.17 Local-mode spectrum of bent XY2 molecules. The local quantum numbers (v a ,Vj,,v c .) and the corresponding algebraic quantum numbers are shown next to each level.

103

Three-body Algebraic Theory

Figure 4.18 Representation of the couplings induced by the Majorana operator in the first multiple!, n = 1. N and Vlv give the order of magnitude of the couplings.

The local-to-normal transition is again characterized by the locality parameter ^ of Eq. (4.75). Locality parameters of several bent triatomic molecules are shown in Table 4.6. Table 4.6

Locality parameter, ^, of some bent triatomic molecules Molecule

S02 D2O H2O16 H2O18 H2S

t 0.80 0.48 0.38 0.33 0.17

Figure 4.19 Normal-mode vibrational quantum numbers for a bent triatomic molecule. Contrast the results for water, which is (cf. Table 4.6) near the local-mode limit with that for SO2, which is near the normal-mode limit.

104

Chapter 4

An example of vibrational analysis of the bent H2S molecules in shown in Table 4.7. Table 4.7

Vibrational analysis of lower overtones 2S of H

VlV 2 V 3

010 100 001 020 110 Oil 200 101 021 210 111 012 300 201 102 003 211 301 103 311

Expt."

Calc.fc

Calc.— Expt.

1182.6 2614.4 2628.5 2354.0 3779.2 3789.3 5145.1 5147.4 4939.2 6288.2 6289.2 6388.7 7576.3 7576.3 7751.9 7779.2 8697.3 9911.1 10194.5 11008.8

1187.2 2616.1 2650.7 2341.9 3777.6 3808.8 5153.0 5166.6 4934.5 6287.7 6299.0 6382.4 7592.2 7594.8 7717.2 7782.0 8700.9 9927.1 10153.4 11006.3

4.6 1.7 22.2 -12.0 -1.6 19.5 7.9 19.2 -4.7 -0.5 9.8 -6.3 15.9 18.5 -34.7 2.8 3.6 16.0 -41.1 -2.5

"All energies in cm . 'lachello and Oss (1990); see also Cooper and Levine (1991) and Appendix C.

4.26

Higher-order corrections. Linear molecules

The Hamiltonian

provides a description of the vibrational spectra of linear triatomic molecules in terms of six parameters, the vibron numbers N^,N2, and the interaction strengths At, A2, A12, A,12.2 For XY2 the number of parameters is reduced to four since N\ = N2 and A\ = A2. Even in the presence of low-order resonances this description is usually realistic. However, in many cases, one needs more accurate descriptions (say, to order of 1 cm"1). This can be achieved, as in the case of the Dunham expansion discussed in Chapter 3, by adding higher-order terms to the Hamiltonian. One of these terms was already considered in Section 4.15. We now treat the inclusion of higher-order terms in general. To second order in the operators C\, C2, C12, M12, one has

Three-body Algebraic Theory

105

The following notation has been introduced in Eq. (4.92): As denote coefficients of terms linear in the Casimir operators, 'ks denote coefficients of terms linear in the Majorana operators, Xs denote coefficients of terms quadratic in the Casimir operators, 7s denote coefficients of terms containing the product of one Casimir and one Majorana operator, and Zs denote coefficients of terms quadratic in the Majorana operators. This notation is introduced here to establish a uniform notation that is similar to that of the Dunham expansion, where cos denote terms linear in the vibrational quantum numbers, xs denote terms that are quadratic in the vibrational quantum numbers and y's terms which are cubic in the quantum numbers (see Table 0.1). Results showing the improved fit using terms bilinear in the Casimir operators are given in Table 4.8. Terms quadratic in the Majorana operators, Z coefficients, have not been used so far. A computer code, prepared by Oss, Manini, and Lemus Casillas (1993), for diagonalizing the Hamiltonian is available.2 In the local basis, all terms involving Casimir operators are diagonal. For example, the term X u Cf has an expectation value of X,,, < C\ > = XL, { -4KJV, + l)v fl - v2a]}2 .

(4.93)

Terms involving Majorana operators are nondiagonal, but their matrix elements can be simply constructed using the formulas discussed in the preceding sections. The total number of parameters to this order is 15 in addition to the vibron numbers, NI and N2. This has to be compared with 4 for the first-order Hamiltonian (4.91). For XY2 molecules, some of the parameters are equal, ^1,1 = ^2,2. ^1,12 - ^2,i2> ^1,12 = ^2,12. ^i = ^2> reducing the total number to 11 plus the vibron number N = NI = N2. Calculation of vibrational spectra of linear triatomic molecules with second-order Hamiltonians produce results with accuracies of the order of 1-5 cm"1. An example is shown in Table 4.8. The algebraic vibrational analysis should be compared with the vibrational analysis carried out using the Dunham expansion. The quality of the fit of Table 4.8 is equivalent to that of a Dunham expansion with cubic terms

106

Chapter 4 Table 4.8

Lower overtones of OCS

VlV 2 2 V 3

Expt.fl

Fitlfc

5

Fit 2C

8

10°0

859.0 1047.0 2062 .1 1711.1 1892.2 2104.8 2918.1 3095.6 4101.4 2556.0 2731.4 2937.2 3170.6 3768.5 3937.4 4141.2 4953.9 5121.0 6117.6

853.2 1049.9 2058.7 1702.6 1896.9 2094.4 2905.8 3102.3 4098.8 2548.2 2740.0 2935.1 3133.5 3749.0 3943.1 4140.5 4939.7 5136.1 6120.4

-5.8 2.9 -3.4 -8.5 4.7 -10.4 -12.3 6.7 -2.6 -7.8 8.6

860.2 1045.7 2062.2 1713.2 1892.5 2104.3 2918.5 3094.4 4101.1 2558.6 2732.3 2937.3 3174.4 3767.8 3937.7 4139.6 4953.6 5120.3 6116.5

-1.3 0.1 2.1 0.3 -0.5 0.4 -1.2 -0.3 2.6 0.9 0.1 3.7 -0.7 0.3 -1.6 -1.6 -0.7 -1.1

02°0 00° 1 20°0

12°0 04°0 10° 1

02° 1 00°2 30°0 22°0

14°0 06°0

20° 1 12° 1 04° 1 10°2 02°2 00°3

-2.1

-37.1 -19.5 5.7 -0.7

-14.2 15.1 2.8

1.2

Adapted from lachello, Oss, and Lemus (1991); see also, Cooper, and Levine (1991). Results for higher overtones can be found in these references. All energies in cm~ ! . Using only terms linear in the Casimir operators. Using all the terms bilinear in the Casimir operators in Eq. (4.92).

where dt = 1 or 2 depending on whether i refers to a nondegenerate or degenerate vibration. However, in the latter case, there are 20 parameters, co(, Xy, y^The Hamiltonian (4.92) thus represents a more economical way of characterizing the energy levels of a triatomic molecule.

4.27

Higher-order corrections. Bent molecules

The Hamiltonian

provides a description of the vibrational spectra of bent triatomic molecules in terms again of six parameters, A^, N2, A 1; A 2 , Al2,^i2- It is identical to Eq. (4.91), except for the term 2Ci2, and for the identification of the algebraic quantum numbers with the vibrational quantum numbers, given now by Eq. (4.85) instead of Eq. (4.53).2 Higher-order corrections can be introduced in a way similar to Eq. (4.92). To second order one has

Three-body Algebraic Theory

107

where

and the notation for the coefficients is identical to that of Eq. (4.92). The eigenvalues of H can be obtained by diagonalizing it in the local basis.2 Again, all terms except those involving the Majorana operator are diagonal in this basis. For example, the term Xn,i2 £"12 nas an expectation value of

The number of parameters is still the same as for linear molecules, except that the last term in Eq. (4.96), ^12,12 Cn, contributes now to rotational energies in view of the fact that

This term will be discussed later on. Calculations of vibrational spectra of bent triatomic molecules with second order Hamiltonians produce results with accuracies of the order of 1-5 cm"1. An example is shown in Table 4.9. These results should again be compared with those of a Dunham expansion with cubic terms [Eq. (0.1)]. An example of such an expansion for the bent SO2 molecule is given in Table 0.1. Note that because the Hamiltonian (4.96) has fewer parameters, it establishes definite numerical relations between the many Dunham coefficients similar to the socalled x — K relations (Mills and Robiette, 1985). For example, to the lowest order in l/N one has for the symmetric XY2 case the energies E(vltV2, v^) given by

108

Chapter 4

In view of Figures 4.12 and 4.18, such I/TV expansions should be particularly instructive, and we return to them in Chapter 7. Table 4.9 VlV 2 V 3

010 100 001 020 110 Oil 200 101 002 030 120 021 210 111 012 300 201 102 003

Lower overtones of F^O

vav*vb

Expt.a

Fit

5

00+1 01+0 01-0 00+2 01+1 01-1 02+0 02-0 11+0 00+3 01+2 01-2 02+1 02-1 11+1 03+0 03-0 12+0 12-0

1595.0 3657.0 3755.9 3151.4 5234.9 5331.2 7201.5 7249.8 7445.0 4667.0 6775.0 6871.5 8761.5 8807.0 9000.1 10599.6 10613.4 10868.8 11032.4

1598.5 3654.4 3758.2 3148.1 5233.7 5325.6 7199.0 7247.0 7452.7 4675.7 6779.4 6860.5 8763.5 8802.6 8999.0 10597.9 10609.8 10868.0 11046.4

-5.4 -2.5 2.3 -3.2 -1.1 -5.6 -2.5 -2.7 7.7 8.7 4.3

11.0 1.9 -4.3 -1.1 -1.8 -3.6 -0.8 14.0

Adapted from lachello and Oss (1990). Terms both linear and bilinear in the Casimir operators in Eq. (4.96) have been used in the fit. See Appendix C. States are designated both by normal-mode quantum numbers and by localmode quantum numbers. All energies in cm'1.

4.28

Rotational spectra

We have discussed up to now vibrational spectra of linear and bent triatomic molecules. We address here the problem of rotational spectra and rotation-vibration interactions.3 At the level of Hamiltonians discussed up to this point we only have two contributions to rotational energies, coming from the operators C(O(3]2)) and IC(O(4i 2 ))l 2 . The eigenvalues of these operators are

Three-body Algebraic Theory

109

and

The first expression is valid for linear molecules and the second for bent ones. There is also a small contribution to the rotational energies in linear molecules, arising from the operator C(O(412)) [Eq. (4.54)]. Using the reduction of representations of groups given by Eq. (4.46), one then finds that the rotational spectra of linear molecules contain the angular momenta

The rotational energies to this order are

where

The spectrum corresponding to Eq. (4.104) is shown in Figure 4.20. In this approximation, levels of bands with lb = l,2,... (IT,A,O,...) are doubly degenerate, and energies increase with / as /(/ + 1). The same rules for the reduction of representations of groups give the following content of angular momentum in each vibrational band of bent molecules

The rotational energies to this order are

where

The spectrum corresponding to Eq. (4.108) is shown in Figure 4.21. Levels

110

Chapter 4

Figure 4.20 (4.104).

Rotational spectrum of a linear triatomic molecule, according to Eq.

with K ^ 0 are doubly degenerate. If B' = 0 the spectrum has further degeneracies.

4.29

Higher order corrections to rotational spectra

Higher-order corrections to rotational spectra can be introduced in a way similar to that described in the previous sections for purely vibrational spectra. Denoting by

one can expand the purely rotational Hamiltonian as

Figure 4.21

Rotational spectrum of a bent molecules, according to Eq. (4.107).

Three-body Algebraic Theory

111

The lowest contributions to the energies are

for linear molecules and

for bent molecules.

4.30

Rotation-vibration interaction

The expansions reported in the previous sections describe separately rotational and vibrational degrees of freedom. One needs, however, to take into account also rotation-vibration interactions obtained by the coupling of operators describing vibrations and operators describing rotations (Viola, 1991). Within the algebra of U(4), operators describing coordinates and momenta are D and D', while operators generating rotations are the angular momenta, /. Rotation-vibration interactions are thus characterized by the powers of D (or D') and / contained in the operator. A classification of these operators is given in Table 4.10. In view of the fact that D (and D') are vector operators, they can appear in the algebraic Hamiltonian only in an even power. Table 4.10 Classification of algebraic operators according to powers of D and / Powers of D (or D')

4.31

Powers of /

f

f

/

D°

h0,o

Vi

h0,2

ho,3

D2

h2,o

*2.1

h2,2

h2,i

D4

^4,0

^4,1

^4,2

h4,i

D6

h6,o

h(,,i

h6,2

h(,,j,

f

J4

h0,4 h2,4 /z4,4 h

6A

Diagonal rotation-vibration interactions

We begin our study of rotation-vibration interactions by considering the class of operators h2^- In this class we can distinguish two types of operators: (1) diagonal and (2) nondiagonal operators.

112

Chapter 4

Type (J). Operators of type (1) must be of the form

where Ck are the Casimir operators, C(O(4)), described in Sections 4.13 and 4.21, respectively, for linear and bent molecules. To lowest order, and in the local limit of linear molecules, there are three terms in Eq. (4.113),

with eigenvalues

When added to the rotational energy (4.112), this expression produces a total rotational constant that depends on the local vibrational quantum numbers

For example, when vb = 0, lh = 0, v c = 0, one has

A more general expression can be obtained by adding the Majorana operators of Sections 4.13 and 4.21 and/or higher-order terms. The most general expression for type (1) rotation-vibration interactions is

where hRV is the Hamiltonian (4.92) with the coefficients A], A2, An, • • • replaced by A^v, A^v ,A^, — In this way, one accounts for all diagonal contributions of type (1) including /z 2> 2, h^2, ^6,2 > A similar treatment can be done for bent molecules in the scheme of van Roosmalen. The lowest-order, local-mode limit is given by

Three-body Algebraic Theory

113

with eigenvalues

The most general expression for type (1) rotation-vibration interactions is still given by (4.118) but with hRV given by Eq. (4.96) with the coefficients A}, A2, An,... replaced by Af 7 , A$v, A^,.... The introduction of type (1) rotation-vibration interactions allows one to analyze in a straightforward way the variation of the rotational constants with the vibrational quantum numbers va, To this end, one rediagonalizes the Hamiltonian H but with different coefficients, thus obtaining the AB values. Examples of this analysis are given in lachello, Oss, and Viola (1993).

4.32

Non-diagonal rotation-vibration interactions

In addition to diagonal contributions of type (1), there are also nondiagonal contributions. The diagonal contribution of Eq. (4.114) can be rewritten as

In general, all terms hi 2 are of the form

On account of the hermiticity of the operators, only even values of L are allowed. Since the D operators are tensor operators of rank 1, the only allowed values are L = 0,2. The L = 0 contribution has been treated in type (1). Hence, only the L = 2 contribution must be considered here. The matrix elements of the operators (4.122) with L = 2 are difficult to evaluate. Nonetheless, by making use of the angular momentum algebra, they can be evaluated in explicit form. In general, the matrix elements of the operator [f given by

x [/ x /]^2)](0) are

114

Chapter 4

where the symbol in braces is a Wigner 6-j symbol. In the special case in which ^Cy\

T

is given by the operators in Eq. (4.122), one has

The sums in Eq. (4.124a) are limited to /^ = l{ ± 1, /'2 = 12 ± 1, while those in Eq. (4.124b) are limited to /" = ^ ±1, /', = / l 5 /, ±2 and those in Eq. (4.124c) are limited to /"= / 2 ± 1, I'2 = h> '2 ± 2 - The coefficients in brackets < l > are Clebsch-Gordan coefficients for the group SO(4) and are given in Appendix B. The remaining coefficients are either Wigner 6-j or 9-j symbols. The matrix elements of the operators D t and D2 are given in general by

Three-body Algebraic Theory

115

Combining Eqs. (4.123), (4.124), and (4.125) one obtains the final result. For example,

Similar expressions hold for V2 n and V2,22- The most general nondiagonal rotation-vibration interaction can be written as

and, to this order, is described in terms of the parameters and qn-

4.33 Properties of nondiagonal rotation-vibration interactions: Linear molecules The rotation-vibration interaction of the previous section can be rewritten in terms of the usual quantum numbers of linear molecules \vav'hbvcJM > by making use of Eq. (4.53). By explicit evaluation, one can show that, for fixed values of va, vh,vc, the matrix elements of Eq. (4.127) have selection rules

the operators (4.127) give rise for each / to matrices of a special form, which we now analyze. We consider the simple case in which va = 0, vc = 0, and
116

Chapter 4

For vb = 2, the values of lb are lb = 0, ±2. The matrix for / > 2 is 3 x 3, with entries

There is no direct coupling between lb = + 2 and lb = -2 because of the selection rule (4.128). The values of a, and a2 are obtained by using the equations of the previous sections and are given in general in terms of qu, q12, q\iIn view of the special form of the rotation-vibration matrices, it is convenient to introduce a transformed basis (Wang's basis, 1929; see also Herman et al., 1991; Holland et al., 1992), denned as

In this basis, the rotation-vibration matrix has a block form. For example, the matrix (4.131) becomes

Of course, to the nondiagonal piece one must add the diagonal contribution of Section 4.31. Denoting these contributions by E(TT), E(L), and £(A), the two matrices are:

Diagonalization of the rotation-vibration interaction produces splittings of the individual (degenerate) levels, as shown schematically in Figure 4.22. It is interesting to note that the matrix elements of Eq. (4.124a) can be approximately written as

Three-body Algebraic Theory

117

e

Figure 4.22 Splittings of states with / = 3 belonging to different bending vibrational states vb = 0,1,2,3 (va = 0, vc = 0); adapted from Viola (1991).

4.34

Properties of nondiagonal rotation-vibration interactions: Nonlinear molecules

The rotation-vibration interaction of Section 4.32 produces different effects in nonlinear molecules than those discussed in the previous section. In nonlinear molecules the quantum numbers are \vavhvcKJM >. The connection between the group quantum numbers \(a^, co2, TI, 12, J,M > and the usual quantum numbers is given by Eq. (4.85). The different effect can be traced to the different nature of the rotational spectrum. In lowest order, the spectrum of a bent molecule is given by Eq. (4.107) and Figure 4.21. The rotation-vibration interaction introduces terms with selection rules

For fixed va,vb,vc and /, there are 27 + 1 initially degenerate states with K = 0, ±1,±2,..., ±/. The rotation-vibration interaction splits these states. The matrices to diagonalize are of the type:

The entries in these matrices are given by the formulas of Section 4.32, in which T2 = K and x'2 = K'. As a result of the diagonalization, levels are split as shown schematically in Figure 4.23. In this figure, the splitting of levels is shown as a function of the parameter q12 governing the strength of the interactions.

118

Chapter 4

Figure 4.23 Splitting of states with / = 3 belonging to the ground-state band, va = 0,vh= 0, vc = 0, in nonlinear triatomic molecules.

Notes 1. By using U(2) to describe a one-dimensional oscillator, we have the advantage that the one dimensional potential can be anharmonic with a finite number of bound states. If in zeroth order one uses a harmonic oscillator algebra, then one can start with a Hamiltonian in the form of Eq. (4.5). Representative papers of such an approach include Kellman (1982); Abram, de Martino, and Prey (1982); Lehmann (1983); Gray and Child (1984); Kellman (1985); Harter (1986); Farrelly (1986); Michelot and Moret-Bailly (1987); Englman (1988); Zhang et al. (1988); Ding and Yi (1990); Wu (1991). 2. A realistic first approximation even for high overtones of triatomic molecules [whether linear, Eq. (4.91), or bent, Eq. (4.95)] is obtained by adding the symmetrybreaking Majorana term to the zeroth-order Hamiltonian, which has a dynamical symmetry. A carefully documented program for diagonalizing this Hamiltonian has been described by Oss, Manini, and Lemus Casillas (1993). The program, with the routine for computing the Wigner coefficients is too lengthy to be reproduced in this volume. In any case, it is more efficient to obtain it by file transfer (FTP). Drs. Oss, Manini, and Lemus have arranged for this to be possible, as follows: Connect to the computer ITNCPL.CINECA.IT by the command ftp [enter] open 130.186.34.12. Login as anonymous. Use your own login name as the password. Then change directory (cd) to "download/physics/vibron." To copy the files therein, type get "filename." to copy all the files use mget *. Instructions for using the program as well as demonstration input/output files are available in this directory and also in the reference cited. 3. Additional useful references on the algebraic approach to rotational dynamics include Anderson et al. (1973); Bohm and Teese (1976); Gilmore and Draayer (1985); Martens and Ezra (1987); Harter (1986); Halonen (1987); Michelot (1989); Martens (1992).

Chapter 5 Four-body Algebraic Theory

5.1

Tetratomic molecules

In tetratomic molecules,1 there are three independent vector coordinates, r l 5 r2, and r 3 , which we can think of as three bonds. The general algebraic theory tells us that a quantization of these coordinates (and associated momenta) leads to the algebra

As in the previous case of two bonds, discussed in Chapter 4, we introduce boson operators for each bond

together with the corresponding annihilation operators 01,711^,02,^,03,7%. The elements of the algebras U,-(4) are the same as in Table 2.1, except that a bond index i = 1,2,3 is attached to them. The Hamiltonian is now

where //, is given by Eq. (4.40) and Vy is given by Eq. (4.41) with the indices 1,2 replaced by i, j. The discussion of Sections 4.8 and 4.9 applies here as well, 119

120

except that when three or more bonds are coupled a new problem arises. The problem is that one has to choose the order of coupling of the bonds (called a coupling scheme). If one does a full calculation, the order of coupling is not important, since the same final result is obtained in any coupling scheme. However, in many cases the bond-bond interactions are not all of similar strength. Say bonds 1 and 2 are more strongly coupled. The most convenient coupling scheme for this situation is thus that in which bonds 1 and 2 are coupled first and subsequently bond 3 is coupled to it. This coupling scheme is denoted by (12)3. The local basis for this coupling scheme is characterized by the quantum numbers

The values of the quantum numbers (T],T 2 ) contained in the product (co^O) ® (C02,0) are given by Eq. (4.44). The values of the quantum numbers (TILTI2 ) are given by

The values of the allowed angular momenta J are given by Eq. (4.46) with TJ and T2 replaced by v\\ and r|2. Equation (5.5) is obtained by making use of the formula (B.44) discussed in Appendix B.

5.2

Recoupling coefficients

When three or more bonds are present, there are, as mentioned in the previous section, several possible coupling schemes. In particular, for three bonds there are three possible coupling schemes, (12)3, (13)2, and (23)1. The wave functions of these coupling schemes are related by a transformation, called the recoupling transformation. The theory of recoupling of angular momenta is very well known (see Appendix B). Consider, for example, the coupling of three angular momenta,

Four-body Algebraic Theory

121

One can construct wave functions by first coupling 1 and 2 and subsequently 3 or in any other order (Figure 5.1). The angular momentum wave functions corresponding to the two coupling schemes are related by a recoupling coefficient [Eq. (B.24)l,

The recoupling coefficients have been calculated by Racah and are usually written in the form

The symbol W is called a Racah coefficient. Quite often, instead of the Racah coefficient W, one uses a Wigner 6 - j symbol

A similar procedure can be applied to the wave functions (5.4), and one can write recoupling transformations

Figure 5.1

Recoupling of angular momenta.

122

Chapters

The coefficients of recoupling transformations are given by

In these formulas the vibron numbers N l,N2,N33 have been omitted, as well as the quantum numbers that are zero, /, and M. Recoupling coefficients are important in computing matrix elements of operators. Consider, for example, the C operators defined, for triatomic molecules, in Eq. (4.68). For three bonds (tetratomic molecules) one has

The operators C\, C2, and C3 are diagonal in the basis (5.4) with eigenvalues

The operator C\i is also diagonal with eigenvalues

but the operators Ci3 and C23 are not. However, it is straightforward to compute their matrix elements since the operators C13 and C2s are diagonal in the

Four-body Algebraic Theory

123

coupling schemes (13)2 and (23)1, respectively. Their matrix elements are given by

and

In a similar way one can compute matrix elements of any interbond interaction. The use of recoupling techniques (Racah algebra) allows one to reduce calculations of properties of molecules with n bonds to those of molecules with 2 bonds.

5.3

Linear tetratomic molecules

For tetratomic molecules there are three possible geometric arrangements: (1) linear, (2) bent planar, and (3) aplanar, examples of which are shown in Figure 5.2. We discuss here only linear tetratomic molecules.

5.4

Local Hamiltonian for linear tetratomic molecules

The procedure for studying tetratomic molecules is identical to that followed in the study of diatomic and triatomic molecules. One begins with a local-mode Hamiltonian

The Casimir operators, C, and Cy have been defined in Eq. (5.12). The operators Cijk are given by

124

Chapter 5

Figure 5.2 Schematic representation of four atomic molecules: (from top to bottom) linear (acetylene), planar (formaldehyde), a-planar (ammonia).

In tetratomic molecules there is only one such operator, C123. The local Hamiltonian (5.16) is diagonal in the basis (5.4) with eigenvalues

Both in Eq. (5.16) and (5.18) we have deleted the rotational term, C(O(3123)). One can convert the algebraic labels to the usual local vibrational quantum numbers vavhvcv^v1^ by means of the relations

The quantum numbers va,vh,vc denote the three local stretching modes, while vj' and v'/ denote the two bending modes. For tetratomic molecules, it becomes

Four-body Algebraic Theory

Figure 5.3

125

Bond coordinates of linear acetylene.

important how the bonds are coupled. The coupling scheme of Eq. (5.4) is (12)3. This coupling scheme emphasizes the coupling (12). When studying a specific molecule, it is convenient to use as labels of the bonds those that reflect the coupling scheme. For example, in acetylene, C2H2, a convenient labeling is that shown in Figure 5.3. With this labeling, va and vc represent the C-H stretches, and vh the C-C stretch. Equation (5.18) can be converted, using Eq. (5.19), into

The spectrum of fundamental vibrations of acetylene, in the local-mode approximation (5.16) is shown in Figure 5.4. For acetylene, C2H2, bonds 1 and 2 are identical and thus one must have Nl — N2, A] = A2. The local mode Hamiltonian (5.16) includes only the operator C12; that is, interactions of the Casimir type between bonds 1 and 2. One may wish, in some cases, to include also interactions of this type between bonds 1 and 3, C^, and 2 and 3, C23. These can be included by diagonalizing the secular matrix obtained by evaluating the matrix elements of C13 and C23 in the basis (5.4). These matrix elements are given by (5.15a) and (5.15b).

Figure 5.4

Fundamental vibrations of acetylene in the local approximation.

126

5.5

Chapters

Majorana couplings in linear tetratomic molecules

We consider next Majorana-type couplings. These are introduced, as in the previous case of triatomic molecules (cf. Section 4.17), by the operators

where i, j, k = 1,2,3. Consider first the Hamiltonian

The matrix elements of the operator M12 are easy to construct since they are identical to those already encountered in triatomic molecules [Eq. (4.70)]. The corresponding secular equation can be diagonalized, yielding the results shown in Figure 5.5. The main effect of the Majorana term is splitting of the degenerate C-H stretching modes into g and u species, as in the previous triatomic case, Section 4.5. In the same way as discussed in the preceding sections, one can include Majorana operators, A/13 and M23. Since these are in the "wrong" coupling scheme, one must use the recoupling techniques of Section 2. The matrix elements of M13 and M23 are given by

Figure 5.5 Splitting of the degenerate C-H stretching modes of acetylene due to the Majorana interaction, M 12 .

Four-body Algebraic Theory

127

where we have kept the vibron quantum numbers Ni,N2,N^, since the matrix elements of the Majorana operators depend explicitly on them. The effect of Majorana couplings of the type M13 and M2j, is similar to that of the operator M12 except that they act on different algebraic coordinates. As a result of the introduction of Majorana operators one moves away from the local limit. In order to emphasize this point it is convenient to relabel the vibrational quantum numbers and to introduce the usual labeling IV] v 2 V3 v44 v55 >. This labeling is shown in Figure 5.5 and used in Table 5.2.

5.6

Vibrational /-doubling. Casimir operators

An important problem of molecules with four or more atoms is that of vibrational l-doubling. This problem appears in the combination modes of the bending modes v44 and v$. The simplest example is the combination mode v44 = I*1, v55 = I*1 (Figure 5.6). There are two states with /4 + /5 = 0, denoted by 2T, 1? in the figure. With the Hamiltonian of the previous sections, these two modes are degenerate. In the usual approach to this problem the combination modes are split by an interaction (Amat and Nielsen, 1958; Papousek and Aliev, 1982)

In the algebraic approach, there is a class of operators that leads naturally to / splittings. These are the operators already introduced in the previous sections to

Figure 5.6 molecule.

/-splitting scheme for (v 4 4 , v55) = (l :il , I*1) states of the acetylene

128

Chapter 5

account for (I-A) splittings in linear triatomic molecules, and linear-bent transitions in triatomic molecules (Section 4.15). These operators, denoted by C, are now of the form

Hence, the /-splitting algebraic Hamiltonian is

The first and fourth terms in this Hamiltonian are diagonal in the local basis, with eigenvalues

Converting these eigenvalues to the usual local quantum numbers one has

where the first term in the right-hand side is given by Eq. (5.20). Equation (5.28) leads to the following energies for the states of the combination mode of Figure 5.6,

The second and third term in Eq. (5.26) are not diagonal. However, their matrix elements can be constructed using the recoupling technique of Section 5.2. They are once more given by

Four-bodv Algebraic Theory

129

The matrix elements of C13 (and C23) are precisely of the type (5.24) (Amat-Nielsen couplings), except that they include anharmonic cutoff effects. This can be seen in Table 5.1. Table 5.1

Comparison between selected matrix elements -2

of the operator C]3 and those of Amat and Nielsen.

Z bands 1

(000; 1'r ) (000; 31!"1)

(000; 5' r1)

1

1

(000; T 1 ) (000; 3^1') (000; 5"1 1 1 )

1.000 1.953 2.861

1.000 2.000 3.000

11 bands 2

(000; 2 r') (000; 42!"1)

1

(000; 2° I ) (000; 4° I 1 )

1.398 2.364

1.414 2.449

A bands 3 1

(000; 3 !' ) 3

1

(000; 5 1" )

1

1

(000; 3 1 ) (000; 51 1 1 )

1.691 2.697

1.732 2.828

O bands 2

1

(000; 4 1 )

4

(000; 4 r')

1.930

2.000

Computed using Eq. (5.30a). 6

Computed using Eq. (5.24). From Viola (1991).

5.7

Higher-order terms in tetratomic molecules

Higher-order terms can be added as in all previous cases by including powers of the basic operators C, M, and C. The results of such an analysis are given in lachello, Oss, and Lemus (1991b), Table 5.2, and Figure 5.7. The fit at higher energies (measured by Scherer et al., 1983) is shown in Figure 5.7.

130

Chapters

Figure 5.7 £„ energy levels (local-mode notation) in C2H2 in the region of 15600-16000 cm"1, which was probed experimentally by Scherer, Lehmann, and Klemperer (1983).

Table 5.2

Sample vibrational fit (in cm"1 ) of C2H2 "

21 bands

(nv^vX5)

(Vlv2v3v;4v^

Expt.

Fit

K V, K % K K 3 % K % K

(OOOg;! 1 ! 1 ) (OOO,,;! 1 ! 1 ) (010g;0°0°) (010g; 2°0°) (OlO^l 1 ! 1 ) (001U;0°0°) (001K;0°0°) (001g;0°2°) (001U;0°2°) (002?; 0"0°) (002U;0°0°)

(000; I 1 1 1 ) (000; I 1 1 1 ) (010; 0°0°) (010;2°0°) (010; I 1 ! 1 ) (100;0°0°) (100;0°0°) (001;0°2°) (100;0°2°) (002; 0°0°) (101;0°0°)

1328.07 1340.55 1974.32 3179.90 3281.90 3294.84 3372.80 4800.90 4727.07 6502.33 6556.47

1328.10 1341.49 1970.82 3176.06 3288.07 3290.94 3367.52 4791.27 4721.42 6511.63 6556.00

Similar results are obtainable for the n, A, and
Four-body Algebraic Theory

5.8

131

Fermi couplings

The interactions of the previous sections describe linear tetratomic molecules with accuracies of 1-10 cm"1, except in some particular circumstances, when accidental degeneracies occur. One of these circumstances is that of Fermi resonances, already discussed in Section 4.19. These circumstances are dealt with by introducing additional operators. Generalizing the Fermi operators of Section 4.19 to the case of tetratomic molecules, one has S12, %, '?23 and 'J123. The operators y are defined through their matrix elements as in Eq. (4.79). The operator 512 can be dealt with easily since its matrix elements are given by the nondiagonal matrix elements of M12. The operators If 13 and IF23 are more difficult to treat since their matrix elements are the nondiagonal elements of M13 and M23. However, these can be computed by the recoupling techniques and are in fact given by Eqs. (5.23a) and (5.23b).

5.9

Amat-Nielsen couplings

The C operators of Section 5.6 provide a solution to the problem of / splitting. These operators have both diagonal and nondiagonal matrix elements in the local basis. The magnitude of those matrix elements is given in terms of a single strength, for example £> ]3 . This appears to describe / splitting in most molecules, for example, acetylene, C2H2. There are molecules, however, that show a rather unusual splitting pattern, for example, monofluoro acetylene, HCCF. For these molecules, it is convenient to introduce operators defined in a way similar to that of the Fermi operators, S. We call these operators Amat-Nielsen operators, (i. They are defined as leaving the same nondiagonal matrix elements as the operators C, but zero diagonal matrix elements,

Use of these operators allows one to treat very unusual situations like that encountered in HCCF, where the separation A/E in the combination modes is = 0, while it is ~ 4 cm"1 in the overtones of the bending mode v5 and = 20 cm"1 in the overtones of the bending mode v4 (Table 5.3).

5.10

Summary of interbond couplings in linear tetratomic molecules

The types of couplings needed to describe accurately vibrational spectra of linear tetratomic molecules are summarized in Table 5.4. In most molecules, only the C, M, and C operators are needed. The 5 and (i operators are necessary only in some exceptional cases.

132

Chapter 5 Table 5.3

Analysis of / splittings (in cm ') in HCCF

using the Amat-Nielsen operators & Vibr. level

Expt.

'1

0001 15T 0001 l4 'A 0001 'r

l 001 1 '1 5T 001 1 '1 A 001 1 '1 'X l 1

!

+

l

0101 'zr 0101 'A 0101 ^

'1 '1

Center CL+-T)/2

a

Splitting A/S Expt. Fit II

Splitting r/2T Expt. Fitll

949 .03 951 .20 952 .67

950.85

0.35

0.36

3.64

3.62

2011..30 2013..52 2014,,96

2013.13

0.39

0.32

3.67

3.60

3166..02 3168..96 3170..92

3168.47

0.49

0.34

4.90

3.60

Adapted from lachello, Oss, and Viola (1993a).

Table 5.4

Interbond couplings in linear tetratomic molecules Couplings

Type Casimir Majorana Fermi

Mn

Casimir (/ splitting)

y\2

Af, 3 Sis

C

L

Amat-Nielsen (/ splitting)0

C, 2

2 r12 -

Cl3

2 r13 T

Ct13

C23 M23 ?23

^123 ^123 'f!23

L

C

2 r23 2

^3

2 r123 -

The operators Ct12 and C^23 are identically zero.

Notes 1. Beyond the early work on acetylene (van Roosmalen et al., 1983a; see also van Roosmalen, Benjamin, and Levine, 1984, and Benjamin, van Roosmalen, and Levine, 1984, for the work on the stretch modes), much of the algebraic approach to tetratomic molecules is yet to be fully published. We specifically draw attention to the thesis work of Lemus (1988), which contains important details on the Clebsh-Gordan coefficients of O(4), and the theses of Viola (1991) and Manini (1991). The formalism necessary to describe linear and quasilinear molecules can be found in lachello, Oss, and Lemus (1991b); lachello, Manini, and Oss (1992); and lachello, Oss, and Viola (1993a,b). See also Bernardes, Hornos, and Hornos (1993).

Chapter 6 Many-body Algebraic Theory

6.1

Separation of rotation and vibration

For molecules with many atoms, the simultaneous treatment of rotations and vibrations in terms of vector coordinates r 1 ,r 2 ,r 3 ,..., quantized through the algebra

becomes very cumbersome. Each time a U(4) algebra is added one must go through the recoupling procedure using Racah algebra, which, although feasible, is in practice very time consuming. An alternative treatment, which can be carried out for molecules with any number of atoms, is that of separating vibrations and rotations as already discussed in Sections 4.2-4.5 for triatomic molecules.1 For nonlinear molecules, there are three rotational degrees of freedom, described by the Euler angles a, P, j of Figure 3.1, and thus there remain 3n - 6 independent vibrational degrees of freedom, where n is the number of atoms in the molecule. For linear molecules, there are two rotational degrees of freedom, described by the angles a, [3, and thus there remain 3n - 5 independent vibrational degrees of freedom, some of which (the bending vibrations) are doubly degenerate. In this alternative treatment, the algebraic theory of polyatomic molecules consists in the separate quantization of rotations and vibrations. Each bond coordinate is then a scalar, and the corresponding algebra is thatofU(2). 133

Chapter 6

134

6.2

Internal symmetry coordinates

In polyatomic molecules, the geometric symmetry of the molecule also plays a very important role. For example, the benzene molecule, which is the example we discuss in this book (Figure 6.1) has the point group symmetry D6h. A consequence of the symmetry of the molecule is that states must transform according to representations of the appropriate symmetry group. In terms of coordinates, this implies that one must form internal symmetry coordinates. These are linear combinations of the internal coordinates. For example, denoting in Fig. 6.1 by s\, s2, s3, s4, ss, s6 the stretching coordinates of the six C-H bonds, the internal symmetry coordinates are linear combinations

with coefficients that depend on the symmetry species (A1)?, A2g, B\U,E\U in this case). A full discussion of how to construct symmetry coordinates in the geometrical theory is given in Wilson, Decius, and Cross (1955). We shall present in the subsequent sections the corresponding algebraic theory.

6.3

Quantization of coordinates and momenta

The algebraic treatment of polyatomic molecules proceeds in the same way as described previously. Each one-dimensional degree of freedom is quantized with the algebra of U(2),

Figure 6.1 Symmetry of the benzene molecule for small displacements. The z axis is perpendicular to the plane of the figure.

Many-body Algebraic Theory

135

as discussed in Section 4.2. The operators, o, i have now an index corresponding to the coordinate they describe

The construction of the algebra, Hamiltonian operator, etc., is straightforward. The difficulties that arise are: (1) in linear molecules, the bending vibrations are intrinsically doubly degenerate and (2) while for stretching vibrations the algebraic coordinates (o, T) correspond to the stretching coordinates and momenta (s, ps), for bending vibrations a new interpretation of the algebraic coordinates (o, T) must be given, since the corresponding geometrical coordinates are the angle 9 and angular momentum pe. Both of these problems will be discussed in the following sections.

6.4

Stretching vibrations

For these vibrations, the quantization scheme of Section 4.2 can be carried over without any modification (lachello and Oss, 1991 a). The potentials in each stretching coordinate 5 are in an anharmonic force field approximation represented by Morse potentials. The boson operators (o^,t^) correspond to the quantization of anharmonic Morse oscillators, with classical Hamiltonian

For each oscillator i, states are characterized by representations of

with rrij = Nh N,: - 2,..., 1 or 0 (Nt = odd or even). The Morse Hamiltonian [Eq. (6.5)] can be written, in the algebraic approach, simply as

where C,- is the invariant operator of O,(2), with eigenvalues

Introducing the vibrational quantum number v, = (Nt - mf)/2, as in Eq. (4.19), one has

136

Chapter 6

For noninteracting oscillators the total Hamiltonian is

with eigenvalues

If there are n uncoupled oscillators (all equivalent), states have degeneracies that can be simply seen from Eq. (6.11). They are depicted in Figure 6.2. It is convenient to represent the wave functions using the notation of Halonen and Child (1983). 12,0,0,0,0,0 > denotes a state with two quanta in the same oscillator, while I I , 1,0,0,0,0 > denotes two states with one quantum in one oscillator and another in another oscillator.

6.5

Hamiltonian for stretching vibrations

The local modes introduced in the previous section can interact. The interaction potential can be written as

which reduces to the usual harmonic force field when the displacements are small

Interactions of the type (6.12) can be taken into account in the algebraic approach by introducing two terms, as done previously in Chapters 4 and 5.

Figure 6.2

Degeneracies of states for six uncoupled anharmonic oscillators.

Many-body Algebraic Theory

137

One of these terms is the Casimir operator, Cy, of the combined O,(2) ® O;(2) algebra. The matrix elements of this operator in the basis (6.6) are given by

The operator Cy is diagonal and the vibrational quantum numbers v, have been used instead of m f . In practical calculations, it is sometime convenient to subtract from Cy a contribution that can be absorbed in the Casimir operators of the individual modes / and j, thus considering an operator Cy whose matrix elements are

The second term is the Majorana operator, My. This operator has both diagonal and off-diagonal matrix elements

The Majorana operators My annihilate one quantum of vibration in bond / and create one in bond j, or vice versa. The total Hamiltonian for n stretching vibrations is

If A,,y = 0 the vibrations have local behavior. As the X,,yS increase, one goes more and more into normal vibrations.

6.6

Higher-order terms

As in the previous cases, one can improve the description by adding higherorder terms to Eq. (6.17). These are powers (or products) of the C,yS and M,yS. One can also introduce couplings in which any number of quanta in the i bond are created and any number of quanta in the j bond are annihilated. For example,

138

Chapter 6

The matrix elements of the operator My can be easily evaluated. Such terms are needed when higher-order resonances (cf. Section 3.4) are important.

6.7

Symmetry-adapted operators

In polyatomic molecules, the geometric point group symmetry of the molecule plays an important role. States must transform according to representations of the point symmetry group. In the absence of the Majorana operators My, states are degenerate, as shown in Figure 6.2. The introduction of the Majorana operators has two effects: (1) it splits the degeneracies of Figure 6.2 and (2) in addition it generates states with the appropriate transformation properties under the point group. In order to achieve this result the A,,y must be chosen in an appropriate way that reflects the geometric symmetry of the molecule. The total Majorana operator

is divided into subsets reflecting the symmetry of the molecule

The operators S(I), S(II), S (III) ,... are the symmetry-adapted operators (lachello and Oss, 1991). The construction of the symmetry-adapted operators of any molecule will become clear in the following sections where the cases of benzene (D6h) and of octahedral molecules (O/,) will be discussed.

6.8

The benzene molecule

The construction of the symmetry-adapted operators and of the Hamiltonian operator of polyatomic molecules will be illustrated using the example of the benzene molecule. In order to do the construction, draw a figure corresponding to the geometric structure of the molecule (Figure 6.1). Number the degrees of freedom we wish to describe. By inspection of the figure, one can see that there can be three types of interactions in benzene, C6H6: 1. First-neighbor couplings [for example, (12), (23),... ]; 2. Second-neighbors couplings [for example, (13), (24),... ]; 3. Third-neighbor couplings [for example, (14), (25) . . . ]. The symmetry-adapted operators of benzene with symmetry D6h are those corresponding to these three couplings, that is,

Many-body Algebraic Theory

139

with

The total Majorana operator S is the sum

Diagonalization of S produces states that carry representations of S6, the group of permutations of six objects, while diagonalization of the other operators produces states that transform according to the representations A}g, £ IH , E2g, Biu of D(,h. This result can be verified by computing the characters of the representations carried by the eigenstates of S1®, as shown for example in Wilson, Decius, and Cross (1955). The Hamiltonian operator that preserves the symmetry of the molecule can now be constructed. Since all the bonds in Figure 6.1 are equivalent, the most general lowest order Hamiltonian for C-H stretching vibrations of C6H6 is

where

In addition, since all bonds are equivalent, the vibron numbers Nf must be all equal, Nt = NH. Thus, the symmetry of the molecule imposes the following conditions on the coefficients in Eq. (6.17):

140

Chapter 6

C-H stretching vibrations of C6H6 are therefore characterized by five quantities A H , A HH , A,®H, XHH, and A^'H • If. instead of C6H6, one wishes to study C6D6, which has the same symmetry D6h of C6H6, the same Hamiltonian (6.24) will apply except that now the five quantities A H , A HH , ^®H, ^'H, ^H

an

&

tne

vibron numbers NH are replaced by AD, ADD, A,^,, A^Q, A,QD , and ND. The results of some sample calculations are shown in Table 6.1.

Table 6.1

Experimental and Calculated^ Frequencies* and Infrared Intensities'^ in C6H6 and C6D6 (C-H and C-D stretches). Frequency

C6H6 E2g(v1) Biu(vl3) £ig(v2o) Aig(v2)

Elu Elu

E:u Eiu

Calc.

Obs.ft

3056.93 3057.50 3065.15 3073.93 6004.05 8827.53 11530.41 14113.33

3056.6 3057 3064.367 3073.94 6006 8827 11498 14072

2275.23 2283.27 2286.04 2304.89 4497.75 6643.78 8719.57 10725.38

2272.5 2285 2289.3 2303.44 4497 6644 8734 10763

IR intensity Calc.

Obs.c

16.0

16.0

1.05 5.1 (-2) 2.2(-3) 9.2(-5)

0.58 3.5(-2) 3.0(-3) 3.7(-4)

6.4

6.4

4.2(-l) 2.2(-2) 1.0(-3) 4.3(-5)

4.2(-l) 2.2(-2) 1.0(-3) 4.3(-5)

C6D6 £2g(V 7 ) 5l«(Vl 3 )

£i«(v2o) Aig(v2) Elu

E\u E\u E\u

Adapted from lachello and Oss (1991). In cm^1. From Brodersen and Langseth (1956); Takhur, Goodman, and Ozkabak (1986); Pliva and Pine (1987), and Page, Shen, and Lee (1988). 'in 106 b/cm. From Reddy, Heller, and Berry (1982).

It is instructive to analyze the effect of the interaction terms (Majorana operators) in Eq. (6.24). These terms split the degeneracies of the multiplets of Figure 6.1, as shown in Figure 6.3. Thus, the Majorana terms remove the degeneracies of the local modes and bring the behavior of the molecule towards the normal limit, precisely in the same way as in tri- or tetratomic molecules.

141

Many-body Algebraic Theory

Figure 6.3 Splitting of the local C-H and C-D modes in C6H6 and C6D6. The total splitting is 17 cm"1 in C6H6 and 31 cm"1 in C6D6.

6.9

Isotopic substitutions. Lowering of symmetry

The same method of construction of symmetry-adapted operators can be used for any isotopic substitution in which a H atom is replaced by D, for example, the molecule 2,4,6-CgH3D3 of Figure 6.4. This molecule has Dj,h symmetry (i.e., the symmetry is lowered). The symmetry-adapted operators can be obtained by inspection of Figure 6.4. The nearest-neighbor interactions are all identical (and are of the H-D type). The same is true for the third neighbor interactions. Thus the symmetry-adapted operators 5® and S^ are the same as

Figure 6.4

The molecule 2,4,6-C6H3D3.

142

Chapter 6

in Section 8. The second-neighbor interactions are of two types (H-H and D-D). The symmetry-adapted operator S(II) is split into two pieces (lowering of symmetry)

The Hamiltonian for this molecule is given by Eq. (6.17) with

The vibron numbers are obviously

For this molecule one needs, in addition to the H-H and D-D interactions, the H-D interactions. These can be obtained from experiments, if available, or can be estimated from scaling arguments. Introducing the reduced masses, \i, of the C-H and C-D bonds, and defining p by

one expects on the basis of the considerations of Chapter 7

Scaling arguments then suggest that

143

Many-body Algebraic Theory

Diagonalization of the Hamiltonian H is straightforward and produces the results shown in Table 6.2. The symmetry species are here A'\, A'2, E'. The same procedure can be applied to all 13 isotopic substitutions of benzene, 9 of which are shown in Figure 6.5. The calculated frequencies of the fundamental vibrations of benzene and some of its substituted form are shown in Table 6.3.

6.10

Infrared intensities

Infrared intensities (IR) in polyatomic molecules can be calculated in the same way as in diatomic or triatomic molecules. One introduces a bond dipole operator ij with matrix elements given by Eq. (2.62), which we now rewrite as

Figure 6.5

Nine isotopomers of benzene.

144

Chapter 6 Table 6.2 Calculated frequencies" and Infrared Intensities of the fundamental vibrations of benzene and its substituted formsc

Symm.

Mode

Computed energy

Computed IR int.

E2g

V7

B\u

V 13

Elu

V20

Alg

V2

3056.91 3057.51 3065.13 3073.93

0.16(+2)

AI B^ A, At BI AI

Mode

Symm.

Vl3,7 V

70,20i>

V2,20 Vl3,7 V V

20i,7a

2,20a

Mode

Computed IR int.

2280.96 2282.57 2298.10 3053.03 3056.49 3064.80

0.25(+0) 0.31(+1) 0.30(+1) 0.43(+0) 0.80(+1) 0.75(+1)

v?

B3u B2u

v

A

8

V2,7*

B

3u

V

Ag

V7

2291.26 3050.50

0.21(+2) 0.19(40)

V

3053.97

0.32(+1)

3057.06 3058,93 3070.31

0.54(+1) 0.16(+2) 0.21(+1)

Computed energy

Computed IR int.

£' A\

2282.06 2295.39 3056.36 3061.17

0.33(+1)

Computed energy

Computed IR int.

Computed energy

Computed IR int.

2274.37

0.14(-2)

13, 20a

V20

2276.77 2288.43 2291.28

0.12(+1) 0.22(+1) 0.62(+1)

20a.l3

V 7 ft,2

2301.08

0.68(+0)

3054.00

0.22(-4)

V

76,20u

7A.13

V

13,20o

V7 V2

Mode

2,4,6- C6H3D3 V

7a,206

Vl3,2

E'

V

A\

V 2 ,12

Symm.

Mode

20a,7fc

0.80(+1)

C6HD5

1,2,4,5 -C6H2D4

B lg

Symm.

Computed energy

1,2,6- C6H3D3

A, B, A! AI BI AI

Computed IR int.

Mode

C6H5D

C6H6

Symm.

Computed energy

Symm.

2276.51 2290.52 2293.41 2297.14

0.21(+1) 0.64(+1)

3053.11

0.11(+2)

3053.25

BI AI AI BI AI AI

V7

V7fc,l3 V

13,20a

V20 V2 * 20a,7ft

V

Adapted from lachello and Oss (1992). All values in cm"1. M\ values in 106 b/cm. '"Wilson numbering is used to denote the vibrational modes in C5H^ and C^D^; for the other species the "scrambling" notation of Thakur, Goodman, and Ozkabak (1986) is used; states with more than two components are denoted by an asterisk (*). Some symmetry species differ from those of Thakur et al., because of the choice of axis of Figure 6.1 (Z axis out of plane). b

Many-body Algebraic Theory

145

The infrared transition operator is

Since the wave functions of the vibrational states have already been calculated by diagonalizing the Hamiltonian H, one can compute the matrix elements of f without further assumptions. They are given in terms of a, and P,. These quantities must be chosen according to the symmetry of the molecule. In the case of in-plane stretching vibrations, the transition operator f , has two components, fx and fy (choice of axis as in Figure 6.1). The intensities of infrared transitions are given by

The absorption intensities are given by

where v is the frequency of the absorbed radiation. In benzene and its substituted forms, one has two types of bonds, C-H and C-D, and thus parameters aH, PH and dtp, PD. The explicit form of the transition operator can be obtained once more by inspection of Figure 6.1. For CgHg, the transition operator has symmetry E\u and the explicit form

The same formulas apply to C6D6 with the replacement of aH with OCD and (3H with PD. The results of a sample calculation are shown in Tables 6.1 and 6.2. In the case of the molecule 2,4,6-C6H3D3, the in-plane dipole operator transforms like £", and its explicit form is

The results of a sample calculation are shown in Table 6.2. Similar results can be obtained for other isotopomers (Table 6.2).

146

6.11

Chapter 6

Octahedral molecules

As a second example of construction of symmetry adapted operators, we consider the case of octahedral molecules, XY6 (Figure 6.6). The symmetryadapted operators can be constructed by inspection. There are two types of interactions: 1. Adjacent bond couplings [for example, (12), (14),... ]; 2. Opposite bond couplings [for example, (13), (24),... ]. The symmetry-adapted operators for octahedral symmetry, Oh, are thus:

with

The total Majorana operator, S, is the sum

Figure 6.6

Schematic representation of an XY6 octahedral molecule.

147

Many-body Algebraic Theory

Diagonalization of S produces states that carry representations of S6, while diagonalization of the other operators produces states that transform according to the representations Eg, Alg and Flu of Oh. This result can be again verified by computing the characters of the representations carried by the eigenstates of Sm and S(ll). The Hamiltonian operator that preserves the symmetry of octahedral molecules can now be constructed. For XY6 molecule it is

In addition, since all bonds are equivalent, the vibron numbers Nf must be all equal, N; = 7VY. Thus, the symmetry of the molecule imposes the following conditions on the coefficients in Eq. (6.17):

X-Y stretching vibrations of octahedral molecules are thus characterized by four quantities, A Y , A YY , ^YY, ^ YY - The results of some sample calculations are shown in Table 6.3. Isotopic substitutions can also be considered, using the method of Section 6.9. Table 6.3 Comparison" between calculated and observed energy levels of SFg, WFg, and UFg WF6

SF6 IV1V2V3 >

1010 > I100> 1001 > 1020 >

Symm. E

s

Alg Flu Alg Eg

F-calc

F-obs

F-calc

645.36 774.09 948.19

643.35 774.54 948.10

678.00 772.14 712.60

1288.18 1289.48

1354.07 1354.83 1448.25 1543.02

UF6 F-calc

E0bs

678.20 772.10 712.40

533.52 666.37 625.72

534.10 667.10 625.50

1354.00 1354.00*

1065.86 1066.33

1066.50

1198.51 1331.85

1197.00

E0bs

1066.30

I110> 1200 >

E

s Aig

1416.72 1546.75

1011 >

Flu

1588.31 1593.66

1588.10

1387.17 1390.21

1387.10

1157.11 1159.00

1156.90

F2u

Fiu

1719.65

1719.59

1482.76

1482.80

1290.74

1290.90

E

1890.91 1890.91 1896.49

1889.05 1889.05 1896.53

1422.29 1422.42 1424.81

1422.40 1422.40* 1422.40*

1249.41 1249.44 1251.19

1101 > 1002 >

s Aig F

2g

148

Chapter 6 Table 6.3 (continued)

SF6 IViV 2 V 3

>

1201 >

Symm.

Flu Fiu

1003 >

F2u

F2u A2u

N A A' ^ X'

UF6

WF6

*^ calc

E obs

F i '-'calc

2489.66

2488.40

2828.14 2839.31 2839.31 2844.90

2827.55 2840.35 2840.35 2845.28

F , L-'obs

^ calc

& obs

2251.65

1954.89

1955.00

2129.22 2134.10 2134.20 2136.63

1871.15 1874.65 1874.67 1876.42

1874.60

180

200

250

-0.915 -0.017 -0.119 +0.722

-0.289 -0.068 -0.078 +0.008

-0.141 -0.053 -0.089 +0.096

Adapted from lachello and Oss (1991a) where the source of the data is given. All values in cm which is dimensionless.

except N,

Not used in the fit.

6.12

Bending vibrations. The Poschl-Teller potential

For bending vibrations, the representation of the potential function, V(s), in terms of Morse potentials [Eq. (6.5)] is not appropriate. The coordinate i is now s = aQ (Figure 6.7). The potential must be symmetric under s —> - s. However, one can make use of another correspondence between potentials and algebras. Consider the Poschl-Teller (1933) potential of Figure 6.8,

with A, an integer. The energy levels of this potential are given by

The Poschl-Teller potential can be put into a correspondence with the algebra U(2). The procedure is identical to that discussed in Section 2.8. Consider the chain (2.31)

149

Many-body Algebraic Theory

Figure 6.7

C-H bending coordinates of benzene.

with mz = N, N - 2,..., 1 or 0 (N = odd or even) and the Hamiltonian

This Hamiltonian is diagonal in the basis (6.46) with eigenvalues

Figure 6.8

Pbschl-Teller potential.

150

Chapter 6

Replacing

one can rewrite Eq. (6.48) as

This is identical to Eq. (6.45) with A = D' and A, - 1 = N/2. One also notes that the spectrum of the Poschl-Teller potential in one dimension is identical to that of the Morse potential in one dimension. These two potentials are therefore called isospectral. This identity arises from the fact that, as mentioned in Chapter 3, the two algebras O(2) and U(l) are isomorphic. The situation is different in three dimensions, where this is no longer the case.

6.13

Hamiltonian for bending vibrations

Having established the correspondence between the Poschl-Teller potential and the algebra U(2), one can proceed to a quantization of bending vibrations along the lines of Section 4.2. We emphasize once more that the quantization scheme of bending vibrations in U(2) is rather different from that in U(4) and implies a complete separation between rotations and vibrations. If this separation applies, one can quantize each bending oscillator i by means of an algebra U,(2) as in Eq. (6.6). The Poschl-Teller Hamiltonian

where we have absorbed the A,(A,- 1) part into D, can be written, in the algebraic approach, as

This Hamiltonian is identical to that of stretching vibration [Eq. (6.7)]. The only difference is that the coefficients A, in front of C, are related to the parameters of the potential, D and a, in a way that is different for Morse and Poschl-Teller potentials. The energy eigenvalues of uncoupled Poschl-Teller oscillators are, however, still given by

One can then proceed to couple the oscillators as done previously and repeat the same treatment of Sections 6.5, 6.6, and 6.7.

Many-body Algebraic Theory

6.14

151

Bending vibrations of benzene

As an example of an application of the method of the previous sections, consider the in-plane bending vibrations of benzene (Figure 6.7). The symmetry adapted operators are still given by Eq. (6.21). However, in view of the difference in the orientation of the displacements for bending vibrations as compared to stretching vibrations, the diagonalization of the symmetry-adapted operator produces states with species Elu, B2u, E2g, and A l g . The Hamiltonian for C-H in-plane bending vibration of C6H6 is still given by Eq. (6.24), but the parameters now refer to C-H bends rather than C-H stretches,

The study of bending vibrations is, however, complicated for two reasons: (1) The vibron number Nt, which characterizes the anharmonicity of the corresponding degree of freedom cannot be taken directly from that of the free diatomic molecule (C-H in the case of benzene). In the treatment in terms of U(4) discussed in Chapter 4, the anharmonicity of bending of two bonds with individual anharmonicities N\ and A^ was given by (A^ + 7V2). If this prescription is used here, it gives vibron numbers for C-H bends that are A7, =43 + 137=180, that is, the sum of the anharmonicities of the C-H and C-C stretches. (2) The couplings between bending vibrations are larger than the corresponding ones for stretching vibrations, and thus it is not simple to assign bending modes. Nonetheless, some information on in-plane bending vibrations of benzene is available (Table 6.4). On the basis of this information, one can determine the parameters A, A', ?i(I), ?i(II), ^(III) of Eq. (6.54) and calculate the corresponding spectra.

Table 6.4 Experimental" and calculated* frequencies'^ of in-plane C-H bending vibrations of C6H6 Sym.

Calc.

Obs.

Obs.-calc.

n=\

Elu B2u E2g A2g

1037.7 1149.1 1177.9 1349.9

1038.3 1148.5 1177.8 1350.0

+0.6 -0.6 -0.1 +0.1

n =2

Elu E]u Eiu

2214.5 2325.4 2386.1

2214.0 2326.0 2386.0

-0.5 +0.6 -0.1

a Brodersen and Langseth (1956). b Adapted from lachello and Oss (1993a). c —1 All values in cm

152

6.15

Chapter 6

Complete spectroscopy

The introduction of the algebra of U(2) both for stretching and for bending vibrations allows one to do a complete study of all molecular vibrations within this framework.2 Consider again the case of benzene. The total number of vibrational degrees of freedom is 3n - 6 = 30, while the total number of coordinates is 3« = 36. It is convenient to introduce the coordinates of Wilson, Decius, and Cross (1955), shown in Table 6.5. Since there are 36 of these coordinates but only 30 vibrational degrees of freedom, six coordinates must be spurious. The set of spurious coordinates can be determined using the method of Wilson, Decius, and Cross (1955), and is also shown in Table 6.5. Table 6.5

Coordinates and symmetry species of benzene

Coordinates

Number

Species

C-H stretch C-C stretch C-H in-plane bend C-H out-of -plane bend C-C in-plane bend C-C out-of-plane bend

6 6 6 6 3+3 3+3

£2, + Blu + E IK + Alg

Alg + B2u + E i« + E2« E\u + B2u + E 2j? + A2s A2u + B2g + E ig + E2u BIU + E2* B2g + E2u

Spurious

A H + E i« AIU + E is

A calculation of the complete spectrum of benzene can now proceed as follows. First calculate the spectra of each block of six vibrations (C-H stretches, C-C stretches, . . . ) and subsequently couple the blocks. Each block is characterized by the Hamiltonian

with parameters A, A', X(I), A,(II), and A,(III), which change from block to block. The only problem that one encounters is the occurrence of spurious states. These states must be removed, and the procedure for their removal is discussed in the following section.

6.16

Removal of spurious states

In order to remove the spurious states, one takes advantage of the fact that the symmetry-adapted operators S1®, 5(1I\ 5(II1), and their sum S are linear combinations of the projection operators on the symmetry species. If one rewrites Eq. (6.55) in terms of these operators, putting A' = 0, one has for the C-C in-plane bend

153

Many-body Algebraic Theory

where P denotes the projection operator into the appropriate state. By choosing large values of \\." and JJ,'" one can remove the spurious states. In practice the unwanted species are placed at energies > 10 times the energies of the nonspurious species. The projection operators for C-C in-plane bends are:

This method of removal is exact for harmonic vibrations and acquires a small error for anharmonic vibrations. The error becomes larger and larger as A' increases.

6.17

Complete spectroscopy of benzene

The method described in the previous sections provides a way to do a complete calculation of medium-size and large molecules. We mean by "complete" a calculation in which all vibrational modes are included. For benzene there are 30 nonspurious vibrational degrees of freedom, which give rise to 20 vibrational modes since some are doubly degenerate. The vibrational modes are first divided into 6 blocks, as shown in Table 6.6. While the assignment to a block is in some cases obvious (for example in the case of C-H stretching vibrations), in some other cases it is not. (There are several ambiguities in the assignments of Table 6.6.) A separate calculation is then done for the fundamentals and overtones of vibrations in each block. This calculation gives the values of the parameters A, A', A,(I\ , X(III) for each block. With these parameters one can calculate any overtone and combination block including combinations between modes belonging to different blocks. Table 6.6 Block C-H stretch

Fundamental vibrations of benzene

Wilson numbering

Frequency0

Alg Eiu E2g BU

3073.9 3064.4 3056.6 3057

V[ V 19

Aig

Vg

E2g B2u

993.1 1484 1601 1309.8

V2 V20 V7 Via

C-C stretch

Species

Vl4

EIU

154

Chapter 6 Table 6.6 (continued) Block

Frequency"

Wilson numbering

Species

Vis V9

£i«

1038.3 1177.8 1350 1148.5

A2u Elg B2g E2u

674 707 967

E2, B\u

605.6 1010

B2/, E2u

990 398

C-H in-plane bend

E2g A2K B2u

V3 V5

C-H out-ofplane bend

Vll V]0 V4 V 17

C-C inplane bend

V6 V] 2

C-C out-ofplane bend

V5

Via

848.9

« .All 1 I values , • cm -1 in

Once this calculation is completed, one can then examine each spectral region bounded by intervals of energy of the order of AE = 100cm"1 and couple the states of a given species that fall into that region. Table 6.7 shows, for example, states up to three quanta of vibration of total species £ l w that fall in the region 5950-6050 cm"1. These states are subsequently coupled by residual interblock interactions of the Majorana type [Eq. (6.16)]. A complete account of this type of calculations is given in lachello and Oss (1993). Table 6.7 Vibrational states of species Elu in Benzene with up to three quanta that fall in the region of 5950-6050 cm" Calculated energy"

State V2 + V20

6004.4

V8+V3+V13 V8+V3+V20

6009.6 6017.1

v8 + v14 + v7

5957.1 5973.9 6028.4 6028.7 6035.9

V8+V14+V2

(2v19)£2^ + v13 (2v 19 )A lg +v 20 (2v 1 9 )E 2 £+v 2 0 aAll values in cm -1

Many-body Algebraic Theory

155

Figure 6.9 Observed opto-thermal spectrum of the Avcw = 3 overtone of benzene (points with error bars) and a stick spectrum calculated by means of the algebraic theory. Labels indicate the most important states involved in borrowing intensity from the CH overtone. Adapted from Bassi et al. (1993).

An example of an application to actual spectroscopy is shown in Figure 6.9. This shows the states computed in the range of the 0 —> 3 CH spectral transition. The intensities were computed as a Franck-Condon overlap of the optically bright CH overtone state (Holme and Levine, 1989; Levine and Berry, 1989) with the relevant eigenstate.

Notes 1. The entire approach presupposes the separation of electronic degrees of freedom. As already noted, for the higher electronic states of polyatomic molecules, there can be important couplings with both spectroscopic and dynamic implications. The vibronic spectroscopy of benzene is reviewed by Ziegler and Hudson (1982). 2. Beyond the work on stretching vibrations reported in Sections 6.4-6.11, much of the work on the complete spectroscopy has not been published yet. An account of the method and the results of calculations for benzene are given in lachello and Oss (1993b). Complete calculations are also available for other molecules, such as methane (CF^) and ethylene (C2H4).

Chapter 7 Classical Limit and Coordinate Representation

7.1

Potential functions

In this chapter we return to the question of the geometrical interpretation of the algebraic approach. Specifically, we need to make contact with the concept of the potential function which is central to the geometrical point of view. For example, in three dimensions, one has the Schrodinger equation (1.2)

One may therefore wish to know what are the potential functions V(r) that correspond to a given algebraic model. The general answer to this question is provided by the solution of the inverse Schrodinger problem: Since one knows the spectrum of the algebraic model, one finds the potential that reproduces the spectrum.1 A simple approach consists in expanding the potential V(r) into a set of functions with unknown coefficients, say

If one includes in the sum as many terms as bound states, one obtains a (not necessarily unique1) potential which exactly reproduces the algebraic spectrum. This method is cumbersome, and in order to carry it out one usually stops the expansion after the first few terms, thereby obtaining a potential that only approximately describes the algebraic spectrum. The accuracy of the approximation depends on how many terms are kept and also on the set of functions 156

Classical Limit and Coordinate Representation

157

used in the expansion.2 In this chapter, we describe faster and more efficient ways to extract potential functions from algebraic models.

7.2

Exact results. One dimension

We begin with a brief summary of exact results. For one-dimensional problems we have used the algebraic structure of U(2), with two subalgebra chains

As mentioned already in Chapter 2, the algebras U(l) and O(2) are isomorphic (and Abelian). A consequence of this statement is that in one-dimension there is a large number of potentials that correspond exactly to an algebraic structure with a dynamical symmetry. Of particular interest in molecular physics are: 1. The Morse potential

As shown explicitly in Section 2.8 the solutions of the Schrodinger equation with this potential can be mapped into the representations of U(2) ^) O(2) (Levine, 1982). 2. The Poschl-Teller potential

The solutions of the Schrodinger equation with this potential can also be mapped into representations of U(2) z> O(2) (Alhassid, Gursey, and lachello, 1983b). 3. The cutoff harmonic oscillator potential

The solutions of the Schrodinger equation with this potential are related to the representations U(2) z> U(l). In the case in which the quantum number N characterizing these representations goes to infinity, the cutoff harmonic oscillator potential of Figure 2.1 becomes the usual harmonic oscillator potential. Situations in which the algebraic Hamiltonian is

158

Chapter?

correspond therefore exactly to solutions of the Schrodinger equation with either the Morse or the Poschl-Teller potentials. These potentials have the same bound-state spectrum (isospectral potentials). They differ in their scattering behaviors (unbound spectra) and their spectra in three dimensions. Situations in which the algebraic Hamiltonian is

correspond exactly to solutions with the harmonic oscillator potentials (with or without cutoff). Situations in which the Hamiltonian does not have a dynamic symmetry (i.e., it contains Casimir operators of both chains), as, for example,

correspond to generic potentials that are intermediate between the different cases.

7.3

Exact results. Three dimensions

In three dimensions the algebraic structure we have used is U(4), with two subalgebra chains

The potentials corresponding to these two chains are now distinct. Situations in which the algebraic Hamiltonian is

correspond exactly to solutions of the three-dimensional Schrodinger equation with the harmonic oscillator potential

with and without cutoff for N finite and infinite, respectively. As mentioned in the previous chapters, the three-dimensional Schrodinger equation with the Morse potential

Classical Limit and Coordinate Representation

159

cannot be solved in closed form. A good approximation is given by Eq. (1.13). The algebraic Hamiltonian

corresponds exactly to the solution of the Schrodinger equation with the Morse potential in the approximation (1.13). The relation between the parameters A, B, and N appearing in the algebraic Hamiltonian and V0, p, and re appearing in the potential (7.13) is given by Eq. (2.111). Situations in which the Hamiltonian does not have a dynamic symmetry (i.e., it contains Casimir operators of both chains)

correspond to generic potentials that are intermediate between the two cases [i.e., a combination of Eqs. (7.12) and (7.13)].

7.4

Geometric interpretation of algebraic models

In the previous sections the correspondence between the Schrodinger picture and the algebraic picture was briefly reviewed for some special cases (dynamical symmetries). In general the situation is much more complex, and one needs more elaborate methods to construct the potential functions. These methods are particularly important in the case of coupled problems. This leads to the general question of what is the geometric interpretation of algebraic models. The connection between algebra and geometry can be made precise by a well-defined mathematical method, since associated with any algebra, G, there is a geometry. The geometric space of an algebraic structure, G, is called, in mathematical terms, coset space? Since the required theory of coset spaces goes deeply into the mathematical structure of groups, which is not the purpose of this book, we shall not discuss it here in detail, but rather present a simplified version of it.

7.5

One-dimensional problems

We begin with the simple case of one-dimensional problems described algebraically by U(2). The coset space for this case is just a single complex variable, which we call ^. We denote the complex conjugate by ^ . These variables can be interpreted in terms of the position (q) and momentum (p) variables in phase space. Equivalently the J; variables can be related to the action-angle variables /, 0 introduced in Section 3.4. To be more precise

160

Chapter?

or

Any algebraic operator, written in terms of the boson operators a, T of Chapter 2, can be converted into a classical operator, written in terms of the variables £, £ * (or p, q). There are several (equivalent) ways of deriving the classical limit of boson operators. We describe here that due to van Roosmalen (1982) and van Roosmalen and Dieperink (1982). A classical limit corresponding to any algebraic operator is obtained by considering its expectation value in the state

called a group intrinsic or coherent state. If the operator is the Hamiltonian H, this gives the classical Hamiltonian

This is a function of the complex variable £ (and £*) or equivalently of p and q, which can be evaluated explicitly. If the values of ^ and ^* are related to p and q by Eq. (7.16), the classical Hamiltonian Hd satisfies the classical equations of motion

One must note that, in general, the evaluation of Eq. (7.18) gives a classical Hamiltonian, Hd, which is a generic function of (q, p) and not necessarily of the Cartesian type

There are several ways in which the Hamiltonian Hcl(p, q) can be converted to an equivalent Hamiltonian where the kinetic energy has the simple form with the mass being independent of coordinates and momenta.4 These ways differ in order I/A', where N is the vibron number. We present first a particularly simple construction.

Classical Limit and Coordinate Representation

7.6

161

Intensive boson operators

The expectation value of H in the coherent state (7.17) can be evaluated explicitly for any Hamiltonian. However, an even simpler construction of Hd (valid to leading order in N) can be done (Cooper and Levine, 1989) by introducing intensive boson operators (Gilmore, 1981). In view of its simplicity, we report here this construction. If one divides the individual creation and annihilation operators by the square root of the total number of bosons, the relevant commutation relations become

In the limit N -» oo, these commutators vanish, and the corresponding operators commute. By making the identifications

and introducing

the conservation of total boson number, N, becomes

However, since the overall phase is arbitrary, one can choose the phase such that T] is real and non-negative, so that

The classical limit, Hch can then be obtained by simply replacing the operators CT, a", T, n1' according to the prescription

which is equivalent, to order l/N, to taking the expectation value of H in the coherent states. The method produces, in a straightforward way, a classical Hamiltonian, Hci(p, q). This Hamiltonian is in general of the form

162

Chapter 7

that is, it corresponds to a Schrodinger equation with a coordinate dependent "effective mass"

However, quite often, one is only interested in the potential function, V(q). The potential function, V(q), can be defined as the value of Hd(p, q) when p - 0; that is,

7.7

One-dimensional potential functions

The method of Section 7.6 can be used to find the potential functions corresponding to the boson Hamiltonians of Chapter 2. According to Eq. (2.30), one has in this case two possible chains

making the replacement (7.29), one obtains

Using Eq. (7.16), one then has

This is the classical Hamiltonian of the harmonic oscillator. By letting p = 0, one has

Classical Limit and Coordinate Representation

163

One must note that q and p are coordinates and momenta up to a scale transformation. A more general form of the Hamiltonian //® is

which gives

that is, a potential with quartic anharmonicity. (II) Potentials corresponding to O(2). If one takes as boson Hamiltonian

and uses Eqs. (7.29) and (7.16), one obtains

Hence, the classical potential becomes

This potential is symmetric under the transformation q —> - q and has minima at q = ±1. One also sees that Eq. (7.38) is of the type (7.27) with

We also note that had we not used the method of intensive boson operators but rather evaluated Eq. (7.18) exactly, we would have obtained (van Roosmalen, 1982)

This differs from Eq. (7.38) by terms of order 1/N, N2 -> N(N - 1). The potential (7.39) can be brought into a form more convenient for molecular structure by the change of variable

Then

164

Chapter 7

This is simply the Morse potential function with

where D is the dissociation energy. This result must be compared with the exact result obtained directly from the Schrodinger equation and given in Section 2.8. By combining Eqs. (2.65) and (2.67), we obtain

Once more the difference is of order l / N , N2 —> (N + I) 2 . The change of variables (7.42) allows one to estimate in a very simple way the strength of the Morse potential VQ (dissociation energy D) in terms of the algebraic parameters A and N. It does not tell us anything concerning the value of (3 (in onedimension the value of re is irrelevant). In order to extract the value of (3 from the algebraic parameters one must either consider explicitly the kinetic energy term in Eq. (7.27) or go beyond the evaluation of Eq. (7.18), which is the expectation value of the Hamiltonian in the ground state. This method will be discussed in the following sections. Finally consider the case in which the algebraic Hamiltonian is a mixture of operators of O(2) and U(l). An interesting example is

The classical limit of this Hamiltonian gives

This result is equivalent, with the change of variable (7.42) to a generalized Morse function (Fliigge et al., 1967).

7.8

Coupled one-dimensional problems

The method discussed in Sections 7.5-7.7 is particularly useful for coupled problems. We begin the discussion by considering two coupled onedimensional degrees of freedom described algebraically by U[(2) ® U2(2) (Section 4.2). The coset space is here composed of two complex variables, £,\ and ^ 2 > describing the coordinates and momenta of the two bonds

Any algebraic operator written in terms of the creation and annihilation operators, CTJ,T]',(TI,T I ; <^2,^2,a2,^2 can be converted into a classical operator

Classical Limit and Coordinate Representation

165

written in terms of the variables £1,^1,^2,^2 or> equivalently, Pi,qi,p2><}2One introduces the "group" coherent states

and considers the expectation value of the various operators in Eq. (7.49). If the operator is the Hamiltonian, H, one obtains the classical Hamiltonian,

This is a function of the complex variables ^, E,2- Once more, by making use of intensive boson operators, one can easily obtain Hc!(pi,q[, p2,q2). The potential functions can then be denned as

7.9

Potential functions for two coupled one-dimensional problems

The Hamiltonian H for this case is

We consider first the local limit, in which

The classical Hamiltonian corresponding to Eq. (7.53) can be easily calculated and is given by

The potential function is

One can see that the coupling between the two oscillators is proportional to q\q2, and that the global minimum of Eq. (7.55) occurs when q,• = ±1. The potential function (7.55) can be rewritten in a variety of ways by making appro-

166

Chapter?

priate coordinate transformations. If one makes the transformation (7.42) for each bond,

one obtains

where

The potential function V(ri,r2) describes two coupled Morse oscillators with couplings involving the square root of the product of Morse oscillators. As in the previous case of a single oscillator, the analysis of the ground-state energies does not provide information on the values of (3, and r el (i= 1,2), but only relates the depths of the potentials to the algebraic parameters. These depths (or dissociation energies) are given by

and are good to order l/N^ and \/N2. In order to obtain relations between the algebraic parameters and Pi,p2 one must consider excited states. This will be done in subsequent sections. We consider here instead the realistic case in which the Hamiltonian is given by

This case is discussed in Section 4.5. The Majorana term

introduces couplings with classical limit

The potential function becomes

Classical Limit and Coordinate Representation

167

By making the transformation (7.42), one can rewrite Eq. (7.63) as

One can compare this expression with the so-called Morse oscillator-rigid bender Hamiltonian of Jensen (1988), where powers of products of Morse potentials appear. The stretching overtones of molecules such as H2O and SO2 were described in Chapter 4 by Hamiltonians of the type (7.60). If one uses the parameters determined from a fit to the data (Table 4.1), one can then calculate from Eq. (7.64) the corresponding potential function. Two examples are shown in Figures 7.1 and 7.2.

7.10

Three-dimensional problems

The algebraic description of three-dimensional problems is in terms of U(4). The coset space for this case is a complex three-dimensional vector, £. We denote its complex conjugate by ^ . One can then introduce the canonical position and momentum variables q and p by the transformation

Any algebraic operator, written in terms of the boson operators a, n of Chapter 2 can be converted into a classical operator, written in terms of the variables ^, £ (or p, q). We describe here again the derivation of van Roosmalen 1982. One introduces a "group" coherent state

The classical Hamiltonian, Hc;, is then the expectation value of the algebraic Hamiltonian, H, in the coherent state (7.66),

This is a function of £ (and £ ) or, equivalently, of p and q. The classical Hamiltonian satisfies the classical equations of motion, as in Eq. (7.19). The evaluation of Hci is best done in spherical coordinates. In these coordinates the boson operators are given by Eqs. (2.63) and (2.64). The complex vector ^ can be written in spherical coordinates as ^(u. = 0, ±1). Using Eq. 7.65 it can be written in terms of spherical coordinates (q, 0, (J>) and momenta (p, pe, p^).

168

Chapter 7

Figure 7.1 Potential energy contour plot for H2O in U t (2) ® U2(2) (Benjamin and Levine, 1985), plotted as a function of r/re. The energy contours are 0.5 eV apart. N 44, A = -18.96 cm"1, \ = 1.025 cm"1, (3 = 2.36 A'1.

Figure 7.2 Potential energy contour plot for SO2 in U](2)®U 2 (2) versus r/re (Cooper and Levine, 1989). The energy contours are 1 eV apart. N = 156, A = -1.615

cm~', X = 0.67 cm"1, (5 = 2.20 A~'.

Classical Limit and Coordinate Representation 7.11

169

Intensive boson operators in three dimensions

The expectation value of H can again be simply evaluated if one introduces intensive boson operators such that

where i, j = x,y,z (Cartesian coordinates). By making the identification

and introducing

the conservation of total boson number, N,

becomes

One can then choose

The classical Hamiltonian, Hct, is obtained by replacing the operators o, a f , K, n^ by

This procedure yields ffd(p, q). The classical potential is the value of Hct(p, q) when p = 0,

170

7.12

Chapter 7

Three-dimensional potential functions

According to Eq. (2.70), one has in this case two dynamic symmetry chains

(I) Potentials corresponding to chain (I), U(3). If one takes as boson Hamiltonian

the procedure described in the previous section gives

This is the classical Hamiltonian of the three-dimensional harmonic oscillator. By letting p —> 0, one has

A more general form is

giving

The Casimir operator of O(3), C2(O3), makes no contribution to the potential, since it has a classical limit

(II) Potentials corresponding to chain II, O(4). If one considers the most general Hamiltonian with O(4) symmetry given, up to quadratic terms, by Eq. (7.83),

Classical Limit and Coordinate Representation

171

and notes that

with

by using the procedure of the previous section one obtains

Hence the potential function is

By making the change of variable

one obtains

This is the three-dimensional Morse function with strength

This result must be compared with the result of Eq. (2.Ill), which gives

One can see again that one obtains the exact result up to order I/TV,

yv2 -> (N + 2)2.

7.13

Coupled three-dimensional problems

We consider next the case of two three-dimensional coupled problems described by U(1)(4) ® U(2^(4). The coset space is here composed of two com-

172

Chapter 7

plex vectors, £, and £ 2 > describing the coordinates and momenta of the two bonds

Any algebraic operator written in terms of vibron operators aj, n{^ (<Ji, n^), O2, i4(i (°2> ^2|a) can be converted into a classical operator, written in terms of the variables £j, £1; ^ 2 > ^2 or conversely p l 5 q l 5 p2, q2. This is done precisely in the same way as in Section 7.8, by introducing the "group" coherent states

The classical Hamiltonian is

and the potential functions are

7.14 Potential functions for two coupled three-dimensional problems There are two cases of interest here described by the chains of algebras

We begin with chain I, the local limit. The Hamiltonian for this chain is

The potential function corresponding to this chain can be easily evaluated and is given by

Classical Limit and Coordinate Representation

173

In this evaluation one uses the results of Section 11 and the fact that

The terms involving the angular momenta do not contribute to V(q l5 q2). We come next to the normal limit, chain II. The Hamiltonian for this chain is

Instead of the Casimir operator of U12(4), it is convenient to introduce the Majorana operator of Eq. (4.66)

The classical limit of this operator when p j = p2 = 0 is

The potential corresponding to the Hamiltonian

is

The intermediate situations can be obtained by combining appropriately the results for chain I and chain II. The potentials in Eqs. (7.98) and (7.104) can be transformed into Morse-like potentials by introducing the transformation

174

Chapter 7

where a t and a2 are unit vectors in the direction of QJ and q2. The potentials (7.98) and (7.104) then become functions of r l 5 r 2 and the angle 0 between al and a2

For example, the potential (7.98) becomes

where VM(r) is the Morse potential function. This expression is similar to Eq. (7.57) and reduces to it when 0 = 0 and 7t (linear molecules). We note that, if the unit vectors aj and a2 are chosen in a different direction, one obtains the same expression but with 0 replaced by (0 - 00), where 00 is the angle formed by aj and a2 in their undisplaced position.

7.15

Vibrations and the shape of the potential

The method discussed in the previous sections provides information on the strength, V0, of the potential

but not on its range parameter (3. The information on V0 was obtained by equating the expectation value of the algebraic Hamiltonian, H, in the lowest state (the ground state) with the corresponding classical expression. In order to obtain information on (3, one must proceed further and impose additional equalities. The next simple equalities are obtained from the energies of the fundamental modes of vibration. A general theory has been developed by Leviatan (1986) and Leviatan and Kirson (1988) and refined by Shao, Walet, and Amado (1992, 1993).

7.16

One dimensional problems

The general theory of classical limits of algebraic models is formulated not in terms of the "group" coherent states of Eq. (7.17) but rather in terms of projectile coherent states. The ground-state projective coherent state is

Classical Limit and Coordinate Representation

175

The coherent state (7.109) is simply related to that in Eq. (7.17). We have called a the coordinate in (7.109) in order not to confuse it with £ of Eq. (7.17). The relation between a and £ (or rather q) is

The operator

is called the condensate boson operator, since the ground state projective wave function can be written as a condensate of c bosons

The expectation value of H in the ground state gives its energy as a function of a

The expectation value of the Hamiltonian (7.37) is given, to order l/N, by

By the change of variable (7.110) one can see that Eq. (7.114) reduces precisely to Eq. (7.39). The energy of the ground state is obtained by minimizing E(N; a) with respect to a, that is, imposing the condition

The minimum occurs at a = ± 1 and is given by

This result is identical to that obtained in the previous sections. If one assumes a potential of the type

its minimum value is at r = re and it is V0. Thus

as before.

176

Chapter 7

However, one can now proceed further and construct coherent states for the excited states. The first (vibrationally) excited state can be written as (Leviatan and Kirson, 1988)

where

is called the vibration (or fluctuation) boson. Evaluation of the expectation value of the Hamiltonian in this state gives, at the minimum a = ± 1,

The first vibrational state (v = 1) has thus an energy -4AN above the ground state. But, for the Morse potential (7.117), the excitation energy of the first excited state is given by

Thus, one obtains the relationship

Taking the square of Eq. (7.123) and using Eq. (7.118) one can obtain the parameters of the Morse potential VQ and (3 from the algebraic parameters A and N,

The relations (7.124) are valid to order l/N. The procedure discussed in the preceding paragraphs can be extended to any vibrational state, written in the form

This procedure is sometimes referred to as the \/N expansion.5

Classical Limit and Coordinate Representation 7.17

177

Three-dimensional problems

The situation for three-dimensional problems is similar to that discussed in Section 7.15. One begins by constructing the ground state. The coordinate is now a vector, a, and the intrinsic state is

The condensate boson operator is

The expectation value of H in the ground state gives its energy as a function of a. It is convenient to write a in spherical coordinates, a = (a, 0, <|>). The expectation value of H depends only on a. For the Hamiltonian

the expectation value of H is given, to order l/N, by

To this order, the rotational term Z?C2(O(3)), does not contribute. See also Section 7.12. The minimum of E(N, a) occurs at a = ± 1 and is given by

as before. The exact expectation value of H for the Hamiltonian (7.128) is

from which one can see the approximation involved when Eq. (7.131) is replaced by Eq. (7.129). The intrinsic state (7.126) describes the ground state of a diatomic molecule. The orientation of the axis of the molecule in space can be chosen arbitrarily. It is convenient to choose it along the z direction (Figure 7.3). The coherent state (7.126) depends then only on the magnitude of a, and can be written as

where n^ is the creation operator for the z component of it. The expectation values of H depend only on a [Eqs. (7.129) and (7.131)], since upon rotation of the axes it must remain invariant. The rotated wave functions can be obtained

178

Chapter 7

Figure 7.3 Choice of axes for a diatomic molecule and for a linear triatomic molecule. The case of the triatomic molecule is discussed in Section 7.20.

from Eq. (7.132) by applying to it the rotation matrix R(Q, <j), 0) (Leviatan and Kirson, 1988). The energy functional given by Eq. (7.129) is shown in Figure 7.4. In view of its potential application to the description of floppy molecules,6 it is interesting to quote the result for the expectation value of H in the case where the Hamiltonian is given by

The energy functional corresponding to Eq. (7.133) is

This functional has a minimum at a = 0 and it is shown in Figure 7.5. Returning to the determination of the potential that corresponds to the algebraic Hamiltonian, one can see that using the method of this section one still obtains (in the limit of large AO

To proceed further and determine the shape of the potential, one must evaluate

Figure 7.4

The energy functional E(tt)(N; a) as a function of ex.

Classical Limit and Coordinate Representation

Figure 7.5

179

The energy functional E(i\N; a) as a function of ot.

the expectation value of the Hamiltonian in the excited states. The condensate boson was given by Eq. (7.127)

For three-dimensional problems, there are in total four boson operators. There must be three more boson operators in addition to Eq. (7.136). They are given by (Leviatan and Kirson, 1988)

The boson operator b\ is called the vibration (or fluctuation) boson since, when applied to the ground state, it generates the vibrational modes

The bosons b\. induce (spurious) rotational modes. Also

Evaluation of the expectation value of H in the state (7.138) gives, when a = 1,

We thus find once more that, if we use a Morse potential, its range parameter p is given by Eq. (7.124),

180

7.18

Chapter?

Rotations and the equilibrium distance

By using the method discussed in the previous section we have been able to obtain the strength of the potential, V0, and its range parameter p. We still need, however, to determine what is the equilibrium distance re that corresponds to the algebraic Hamiltonian. A generic potential will in fact be of the form

In order to be able to determine re we must consider rotations. A straightforward procedure is a generalization of that discussed in the previous section, and it is called the cranking method. The method has been used extensively in nuclear physics (Schaaser and Brink, 1984). It consists in evaluating the expectation value of the modified Hamiltonian

where wx is called the cranking frequency and Jx is the x component of the angular momentum operator (Jy can also be used), in the generalized intrinsic state

Minimizing the expectation value of H' leads to a determination of the moment of inertia as a function of a2. We do not report here the results of these calculations, but rather quote the lowest order, obtained at the minimum value a = +1. One obtains, when the algebraic Hamiltonian H is given by Eq. (7.128), the moment of inertia

But, for the Morse potential, the moment of inertia can be written as

We thus obtain the equilibrium position

We summarize in Table 7.1 the results in leading order.

Classical Limit and Coordinate Representation Table 7.1

181

Correspondence between the algebraic and geometric Hamiltonians

Algebraic Hamiltonian Geometric Hamiltonian

7.19

Coupled problems

We limit ourselves to a discussion of coupled three-dimensional problems, since the one-dimensional problems can be obtained from the three-dimensional ones by deleting some terms. The study of coupled systems is done precisely in the same way as that of single systems. We consider in particular the case of linear triatomic molecules described by the algebra

with the Hamiltonian

This Hamiltonian describes a generic linear triatomic molecule, as discussed in Chapter 4. When Xn = 0 we have a local molecule, and, as Xn increases, we move from the local to the normal limit. As in the previous case of a single U(4), we introduce projective coherent states, constructed with condensate boson operators (Leviatan and Kirson, 1988; Shao, Walet, and Amado, 1992, 1993)

The ground state is

The expectation value of the Hamiltonian in the condensate is

182

Chapter 7

This expectation value depends only on the magnitudes of a t and a2 and on the angle between a, and a2, which we call 9. It is again convenient to choose o^ and a2 along the z direction, as shown in Figure 7.3. The energy of the ground state is obtained by minimizing

with respect to a{, oe2, and 6.

Solving with respect to the angle 0, one finds two roots

For typical values of the parameters An and Kn, the second root is imaginary and we thus find only

Indeed, with the choice of axes of Figure 7.3, one finds 9 = n. The minimum in a t and «2 is at OC[ = a2 = 1. Equating the ground-state energy so obtained with that resulting from the solution of the Schrodinger equation with interatomic potentials, one can obtain information on these potentials. One may also note that, with the change of variables

where Hi and a2 are unit vectors in the direction of q, and q2, Eq. (7.152) reduces to Eq. (7.98).

7.20

Vibrations and the shape of the potential in linear triatomic molecules

A representation of the potential function for linear triatomic molecules has been given in Section 7.14. The potential is constructed in terms of Morse func-

Classical Limit and Coordinate Representation

183

tions, VM(r\) and VM(r2), with range parameters p[ and (32. The range parameters P, and (32 can be determined easily in the local limit, described by the Hamiltonian (7.97). When X12 * 0, the determination of the range parameters Pi and p2 is more complicated. We describe here this more complex situation and then specialize to the case with X12 = 0. As we have mentioned before, in order to obtain information on p! and p2, we must consider the excited vibrations. The methods of many-body theory discussed in the previous sections allow one to determine easily the vibrational energies. We introduce, in addition to the condensate bosons, bli, the vibrational bosons, ftj,-, and the bosons, b\t, b^,

where we have chosen the axes as in Figure 7.3. The generic Hamiltonian of Eq. (7.149) can then be written, up to order 1/N, as (Shao, Walet, and Amado, 1992)

We have deleted from Eq. (7.159) all rotational terms which will be discussed in the following section. In Eq. (7.159) the boson operators b\(bB) create (destroy) a bending mode

When A,12 = 0, one can immediately read off Eq. (7.159) the vibrational frequencies. The two stretching frequencies are given by

184

Chapter 7

while the (doubly degenerate) bending frequency is given by

The value of the constant «i = a2 = 1. cos 9 = 1, is

E0

in Eq.

(7.159)

at

the minimum

If the potential function is represented by

by using Eqs. (7.161) and (7.162), we obtain

where (i, and u.2 are the reduced masses for bonds 1 and 2. We note that, in this approximation, the frequency of bending vibrations is given only in terms of A, 2 and that the range parameter of bending vibrations is fixed since the potential in terms of the angle 9 is simply cos 6. For small 0,

The analysis leading to Eqs (7.165) can be repeated when A,12 * 0. Everything remains the same except that now the two stretching modes are coupled by the term -'ku^NiN2(b\lb^ + b\2b^). In order to obtain the stretching frequencies one must diagonalize a 2 x 2 matrix with entries

Classical Limit and Coordinate Representation

185

These matrices have been discussed in detail in Section 4.17. As a result of these Majorana couplings there is a modification to the shape parameters of Eq. (7.165). Also when Fermi couplings are present, there is a modification to the shape parameters of Eq. (7.165). We do not report these modifications here but rather show in Table 7.2 how good the l/N method is in determining the vibrational frequencies (and thus the range parameters p\ and (J2). As one can see from the table, the error is of the order is 1%. Since N is of order 100, this error reflects the l/N accuracy mentioned.

Table 7.2 Comparison between the vibrational frequencies of linear triatomic molecules obtained by exact diagonalization of the Hamiltonian and the l/N (mean field) result." Level

Exact

MFA

Error

HCN(A' 1 = 140,7V 2 =47) Ol'O 10°0 00° 1

712.4 2095.8 3312.6

707.8 2092.1 3318.0

0.65% 0.18% 0.16%

OCS (#1 = 190, #2 = 159) Ol'O 10°0 00° 1

519.4 860.2 2062.2

518.3 861.8 2064.9

0.21% 0.19% 0.13%

N2O (Ni = 163, N2 = 134) Ol'O 10°0 00° 1

588.9 1284.9 2223.6

586.9 1270.7 2224.9

0.34% 1.1% 0.06%

C 1 2 0 2 (#i=Af 2 = 153) Ol'O 10°0 00° 1

666.8 1388.4 2348.3

664.0 1384.0 2350.7

0.42% 0.32% 0.10%

C 1 3 0 2 (/V,=Ar 2 = 154) Ol'O 10°0 00° 1

648.6 1370.1 2283.2

645.0 1362.7 2284.7

0.56% 0.53% 0.07%

Adapted from Shao, Walet and Amado, 1992. For CO2, Fermi couplings have been included. The quantum numbers for each vibrational level are in their usual order, v,v 2 2 V3.

186

7.21

Chapter?

Rotations and equilibrium positions

The Hamiltonian (7.149) contains also a rotational term, SC(O(312)). This term can be treated in the same way as the vibrational terms. The complete Hamiltonian is obtained by adding to the vibrational Hamiltonian of Eq. (7.157) three more pieces

where A is a shorthand notation for

and

J x , J y , J z are the three components of the total angular momentum. The term Bj(J2x + J2y) is the contribution to the rotational energy, while the other two terms contribute to the bending modes and couplings between bendings and rotations. The expectation value of these terms in the ground state I/V,, a.\, N2, a2 > is zero. In order to study the contribution of the rotational terms, one must use more elaborate methods, as discussed in Section 7.17. The rotational terms contribute in lowest l/N order a rotational energy

The parameter B is thus related to the moment of inertia of the molecule

which in turn is related to the masses of the constituent atoms and their equilibrium distances. For example, for a linear XY2 molecule, the moment of inertia is given by

Classical Limit and Coordinate Representation 7.22

187

Tetratomic molecules

The methods discussed in Sections 7.14-7.20 can also be used for polyatomic molecules. Shao, Walet, and Amado (1993) have analyzed linear tetratomic molecules and found that even for these molecules the error introduced by the 1/N expansion is of order 1%. Their results are shown in Table 7.3. Table 7.3 Comparison between vibrational frequencies of linear tetratomic molecules obtained by exact diagonalization of the Hamiltonian and the 1/N (mean field) result. Exact

MFA

Error

HCCH(JV i = N2 = 43, N3 = 137) 100;0°0° 001;0°0° 010;0°0° 000;l'0° 000;00l'

3286.93 3366.88 1975.80 617.12 724.50

3286.73 3368.42 1983.68 611.20 717.21

0.00% 0.05% 0.40% 0.96% 1.00%

DCCD(Ar1 = N2 = 61, JV3 = 137) 100;0°0° 001;0°0° 010;0°0° 000;l'0° OOOjO0!1

2435.58 2706.35 1769.57 513.55 534.08

2430.44 2704.44 1769.18 510.80 528.79

0.21% 0.07% 0.02% 0.05% 0.99%

Level

Adapted from Shao, Walet, and Amado (1993). The quantum numbers for each vibrational level are in their usual order, V j , V 2 , V3; V44, vs5.

7.23

Higher-order terms

In order to obtain spectroscopic accuracy it was found necessary, as discussed in Chapter 4, to introduce higher-order terms, for example, products and powers of Casimir operators. These higher-order terms can be dealt with easily within the approach discussed here. For example, a Hamiltonian of the type

which is obtained from Eq. (7.128) by adding the term [C2(O(4))]2 leads to an expectation value in the ground state

If one transforms this expression to Morse potential functions, one then finds

188

Chapter?

that the introduction of higher-order terms, corresponds to the introduction, in the Schrodinger equation, of powers of the Morse functions. Thus one may view the algebraic method as a systematic expansion of the potential functions into a set of functions, such as the Morse functions.

7.24

Mean-field theory

The techniques discussed in this chapter are examples of a general theory, called mean field theory? Mean-field theory is particularly well suited to deal with algebraic models. Its role is twofold: (1) It can be used to elucidate the connection between algebras and geometry, as shown in this chapter, where the connection between algebraic Hamiltonians and potential models has been discussed. (2) It can be used as a computational tool to evaluate physical observables. In this context, it is sometimes called \/N expansion. We have discussed here only the lowest order in 1/W, but techniques exist to evaluate \/N2 and higher-order corrections. The technique is particularly useful when N is large (as in molecules where N ^ 100), and when the situation is so complicated that numerical diagonalizations are difficult to perform. In this respect the \/N expansion can be a viable alternative to the replacement of U(4) by U(2), when going from small to large molecules.

7.25

Epilogue

Spectroscopists are traditionally concerned with states. Our intuition, which to this very day is fashioned by classical mechanics (i.e., by the properties of the world as we see them) makes us more familiar with a geometrical point of view. Wave mechanics, in its Schrodinger-like approach, provides a bridge between the two. At the atomic level of description we have no reason to question the validity of this bridge. But for larger molecules and at higher levels of excitation, the route from a simple geometrical picture to the observed spectra and/or dynamics is no longer as simple as for diatomic molecules. In this volume we have systematically developed an algebraic approach where the concept of the state, suitably generalized, is the start. One goes from the algebra to the geometry, as in Chapter 7. It is possible to object to an approach where the notion of the geometrical structure of the molecule emerges only in a mean-field sense. Such an objection will, however, be misdirected. It is quantum mechanics as such rather than the algebraic approach that deprives us of simple classical concepts, and it is the interest in low-amplitude vibrations that can be described in the harmonic limit that enabled us to maintain a classical intuition in a quantal problem. In the algebraic approach, where the finite anharmonicity (i.e., a finite vibron number N) is incorporated from the very beginning, it is unavoidable that the classical geometrical picture is well defined only to order \/N.

Classical Limit and Coordinate Representation

189

Notes 1. In general, a vibrational bound spectrum does not determine a unique potential V(r). One also needs the rotational spectrum and scattering data at positive energies. There is a very rich literature on the inverse problem (e.g., Newton, 1982), made richer by the recent connection with the theory of solitons (Calogero and Degasperis, 1982). The problem of classification of the potentials in one dimension that can be represented by Hamiltonians bilinear in the generators of a Lie algebra has been discussed by Kamran and Olver (1990). See also the thesis of Wu (1985) for many examples. 2. Practical methods of going directly from experimental spectroscopic or scattering data to the potential function are often discussed. Representative references include: Sorbie and Murrell (1975); Shapiro and Gerber (1976); Hoy et al. (1972); McCoy and Sibert (1992). 3. The concept of a coset space is discussed in detail in books on group theory (Gilmore, 1974) and is reviewed in Chapter 3 of lachello and Arima (1987). The coset spaces of interest for algebraic models with structure U(n) are the spaces U(w)/U(« — 1) ® U(l). These spaces are complex spaces with (n — 1) complex variables (coordinates and momenta). 4. When it comes to polyatomic molecules, there are two problems that complicate the issue, as already discussed in Note 1 of Chapter 3. One is the separation of the overall rotation of the molecule (Jellinek and Li, 1989). The other is that, depending on the choice of internal coordinates, certain coupling terms can be assigned to be kinetic or potential terms. A simple and familiar case is a linear triatomic, when one uses bond coordinates versus Jacobi coordinates. The case for Fermi coupling for a bending motion is discussed in Sibert, Hynes, and Reinhardt (1983). 5. Both the classical and the harmonic limit correspond to N —> oo. Yet the two limits are quite different. In the harmonic limit N —> oo and A —> 0 such that AN is finite. In this limit, the spacing between adjacent energy levels [cf. Eq. (7.122)], is constant, and the dissociation energy, -AN2 [cf. Eq. (7.124)] tends to infinity. In the classical limit, #->oo and A -»0, such that -A/H2 = fi2/2\i [Eq. (7.124)] is finite and 2 ftN = (2mV0)1/2/P is finite. The dissociation energy V0 = -(Afh2)(hN)2 as well as the 2 harmonic frequency -4ANfh and the anharmonicity Afh are finite in the classical limit. The harmonic limit is |J -> 0, V0 —» oo, such that Vo/2P 's finite. See also Wulfman and Levine (1984). 6. Group chains suitable for the nonrigid limit have been discussed by van Roosmalen etal. (1983a). 7. The mean-field approximation has been extensively applied in many-body physics. Its application to molecular algebraic Hamiltonians and the connection with the coherent-states expectation method was begun by van Roosmalen (1982). See also, van Roosmalen and Dieperink (1982), and van Roosmalen, Levine, and Dieperink (1983). For applications in the geometrical context see Bowman (1986) and Gerber and Ratner (1988).

Chapter 8 Prologue to the Future

At this point, we hope to have demonstrated that the algebraic approach provides a viable method for the quantitative description of molecular vibrotational spectra. Chapters 4 (triatomic molecules, both linear and bent) and 5 (linear tetratomic molecules) and Appendix C provide extensive documentation for the quantitative applications, while Chapter 6 shows that larger molecules can also be treated. Throughout, but most particularly in Chapter 7, we have sought to forge a link with the more familiar geometrical approach. It is precisely our requirement that even in zeroth order the Hamiltonian with which we start describes an anharmonic motion, which makes this link not trivial. The advantage of our approach in providing, even in zeroth order, high overtone spectra that are typically more accurate than 10 cm"1, should not be overlooked. Yet much remains to be done. In this chapter we look to the future: Where and why do we think that the algebraic approach will prove particularly advantageous? Of course, what we really hope for is to be surprised by unexpected new developments and applications. Here, however, is where we are certain that some of the future progress will be made, with special reference to the spectroscopy of higher-energy states of molecules. One area of spectroscopy where the Hamiltonian in matrix form is the route of choice is that of large poly atomics, particularly so when in an electronically excited state (see Note 3 of the Introduction). Such states are isoenergetic with very high vibrational overtones of the ground electronic states so that a fully geometrical approach is impractical. Even at lower energies, the exceedingly high density of vibrational states strongly favors an alternative approach, and the use of model Hamiltonian matrices is not uncommon. Such model matrices are introduced in order to account for the regularities that often survive in the observed spectrum. One such striking feature is often referred to as a "clump" 190

Prologue to the Future

191

(Hamilton, Kinsey, and Field, 1986). Consider a pure vibrational progression of states, as can be observed in stimulated emission spectroscopy (SEP, Figure 0.1). A clump is a spectral feature that appears as a broad line at a given spectral resolution but that is revealed to be a set of overlapping lines when a higher resolution is used. To qualify as a "clump," the width needs to be inherent rather than instrumental. By inspection of Figure 0.1 one can imagine that the spectrum will exhibit clumps at a lower resolution. It can even be that there are clumps within a clump (Figure 8.1).

Figure 8.1 (a) A spectral clump and (b),(c) its two tiers of fine structure. [The sharp feature at -8770 cm"1 in (b) is the envelope of the set of lines shown in (c)]. When states are coupled in a sequential manner there can often be a separation of time scales in the temporal evolution of a nonstationary state or a corresponding separation of frequency scales in the spectrum (Remacle and Levine, 1993). Algebraic Hamiltonians provide a convenient framework for the discussion of sequential coupling.

192

Chapter 8

For small polyatomics (e.g., NO 2 , C2H4) one can think of the coupling of different electronic states from a geometric point of view (Englman, 1972; Koppel, Domcke, and Cederbaum, 1984; Whetten, Ezra, and Grant, 1985). Even then a transformation to a matrix Hamiltonian is typically very useful and is often implemented particularly when only a very limited number (two or three) electronic states are coupled. By using a Hamiltonian expressed in terms of generators for the electronic degrees of freedom one can describe a multistate electronic spectrum (Frank, lachello, and Lemus, 1986; Frank, Lemus, and lachello, 1989; Holme and Levine, 1988; Lemus et al., 1992; Lemus and Frank, 1991b, 1992). Moreover, one can couple the electronic motion to anharmonic motion of the nuclei. In the future one can confidently expect to see many more such applications. An algebraic approach to electronic degrees of freedom will be particularly advantageous when the density of electronic states is high and the coupling to the rovibrational motion is strong. Such is the case for very high Rydberg states that lie just below the ionization continuum. For larger molecules there is a very facile energy exchange between the electronic and nuclear manifold of states. There is currently a revival of interest in such problems (Schlag and Levine, 1992). For smaller molecules (NO, H2O) the relevant couplings are currently being delineated by spectroscopic techniques (e.g., Bryant, Yiang, and Grant, 1992; Gilbert and Child, 1991). Next, we turn to dynamics proper. Spectroscopy itself is intimately related to intramolecular dynamics, that is, to the time evolution of a nonstationary state of the Hamiltonian. Detailed applications of algebraic methods to such problems have so far been mostly limited to problems for which the Hamiltonian is a linear expression in terms of the generators of the algebra. The reason for this being the case can be understood even without an appeal to the group concept: Let X at time t = 0 be a generator whose time evolution X(t) is of interest. Then, in the Heisenberg picture: ih dX(t)/dt = [X(t), H]. It follows from the closure property of Lie algebras [Eq. (2.1)] that if X(t) and H are in the algebra so is the time rate of change of X(t) and therefore so is X(t + dt). Hence, if X at t = 0 is an operator in the algebra and the Hamiltonian is any linear combination of generators, X(t) remains in the algebra. A key open problem1 is how to proceed in the more realistic case when the Hamiltonian is bilinear rather than linear in the generators of the algebra. The second set of problems in dynamics are those of scattering theory where the Hamiltonian is of the form H = H0 + V and the interaction V vanishes when the colliding particles are far apart. It is usually assumed that the //0 part is already solved and that the interesting or the hard part is to account for the role of V. For realistic systems, which are anharmonic, even the role of H0 can be quite significant. An example that has received much recent attention is the reaction of vibrationally excited HOD with H atoms (Sinha et al., 1991; Figure 8.2). The large difference in the OH and OD vibrational frequencies means that the stretch overtones of HOD are primarily local in character (cf. Section 4.21). It follows that one can excite HOD to overtones localized preferentially on either one of the two bonds and that an approaching H atom will abstract prefer-

Prologue to the Future

193

Figure 8.2 Schematic illustration of the preferential reactivity of HOD with an H atom when the stretch excitation is localized in either one of the two bonds.

entially either an H or a D atom (Figure 8.2). Even in the lowest stretch states this selectivity is maintained (Bronikowski et al., 1991). As in scattering theory in general, one can treat the role of V in either a time independent or a time dependent point of view. The latter is simpler if the perturbation V is either explicitly time dependent or can be approximated as such, say by replacing the approach motion during the collision by a classical path. Algebraic methods have been particularly useful in that context,2 where an important aspect is the description of a realistic level structure for HQ. Figure 8.3 is a very recent application to electron-molecule scattering.

Figure 8.3 The vibrational elastic and inelastic differential cross sections for electron scattering off LiF at £ = 5.44 eV (Alhassid and Shao, 1992b, where the source of the data is given). Solid lines: with an improved dipole interaction [which breaks the O(4) symmetry]. Long dashed lines: the calculations by Bijker and Amado (1986). The short dashed lines are the Born approximation.

194

Chapter 8

In time dependent collision theory, the asymptotic states "after" or "before" the collision are defined by the limits of t —» ±00. In time-independent theory one can either use a Lippmann-Schwinger-like approach (Alhassid and Levine, 1985) or define scattering through a coordinate representation (Alhassid, Giirsey, and lachello, 1983b; Wulfman and Levine, 1983; Frank and Wolf, 1984, 1985; Alhassid, lachello, and Wu, 1986; Ojha, 1986; Wu, lachello, and Alhassid, 1987; Wu, Alhassid and Giirsey, 1989). The latter has been more extensively studied, since for scattering by a potential the geometrical connection is well understood (cf. Chapter 7). Applications to systems with internal structure are still in their early stages.3 In its initial stage of development, the algebraic approach has sought to show why and how it provides a framework for the understanding of large-amplitude, anharmonic motion. Much of our concern in this volume has been with this stage, including, of course, the even simpler problem of uncoupled motions. In its current stage the approach will seek to keep pace with the new developments of modern spectroscopy, including the interest in floppy molecules. The better understanding of the overtones of polyatomics including the higher-energy regimes, where the nonrigidity is sufficient to allow for isomerization, will, no doubt, continue to be of central concern. This will include not only vibration-rotation coupling but also rovibration-electronic coupling. In terms of the connection to the geometrical picture, this means, of course, that one is interested in the shape of the potential farther and farther away from its global minimum and, in particular, in the possibility of local minima. Moreover, at increasing coupling, the very concept of the potential becomes less useful. The algebraic approach offers a special advantage under such circumstances. At higher light intensities, it is no longer possible to treat the coupling to the field in lowest order in perturbation theory (George et al., 1977), and algebraic techniques again become particularly useful. The emphasis of a time domain interpretation of spectroscopy is very likely to increase further. This is not only because of spectroscopic experiments in the time domain (Khundkar and Zewail, 1990; Pollard and Mathies, 1992) but also because of the "time window" that is revealed by experiments in the frequency domain (Imre et al., 1984), a time window that covers the very range of interest to chemists. The emphasis on the time domain has also revived the interest in coherence effects in optical spectroscopy (Steinfeld, 1985). Then there are larger molecules up to and including molecules of biological interest, clusters, etc. One has both experimental and theoretical reasons to expect that increasing the size does not necessarily imply an exponential increase in the complexity and unwieldiness of the spectrum (Figure 8.4). Finally, one expects spectroscopy to have an increasing impact not only on intramolecular but also on intermolecular dynamics. This includes a better understanding of the reactants, as in Figure 8.2, as well as the application of spectroscopy to the very collision process (Brooks, 1988; Neumark, 1992). At lower light intensities, spectroscopy acts in its usual role as a probe except that it is probing very transitory events. At higher intensities one can guide, not just monitor, the evolution of the system (Jortner, Levine and Pullman, 1991; Rabitz

Prologue to the Future

195

Figure 8.4 The v = 6 C-H overtone spectrum as determined by photoacoustic absorption (Hall, 1984) for increasing size alkynes. Each panel contains the molecular formula and the density of vibrational states per cm'1. The band origin does shift some with size but is roughly at 18450 cirT1 for all the molecules shown (see also Kerstel et al., 1991).

196

Chapter 8

and Shi, 1991; Rice, 1992; Brumer and Shapiro, 1992). The discussion of control of quantum systems is one more area to which algebraic techniques can and will usefully contribute.

Notes 1. The problem of time evolution for a Hamiltonian bilinear in the generators (Levine, 1982) has been extensively discussed. The proposed solutions include the use of variational principles (Tishby and Levine, 1984), mean-field self-consistent methods (Meyer, Kucar, and Cederbaum, 1988), time-dependent constants of the motion (Levine, 1982), and numerous others, which we hope to discuss in detail in a sequel to this volume. 2. Algebraic methods have been effectively employed in time dependent collision theory (Alhassid and Levine, 1977, 1978). In the sudden approximation (Levine and Wulfman, 1979) the use of an H0 that is bilinear in the generators presents no real problems. This has been very effectively employed in electron-molecule scattering (Bijker, Amado, and Sparrow, 1986; Bijker and Amado, 1986, 1988, 1992; Bijker, Amado, and Collins, 1990; Mengoni and Shirai, 1988, 1991; Alhassid and Shao, 1992), in which, due to the high velocity of the incident electron, one can neglect the change in its path due to energy transfer to the molecule. 3. Algebraic methods can be used to treat quantum mechanically both scattering and internal structure (lachello, 1987; Levine, 1985; Alhassid and lachello, 1989). For scattering even resonances can be described (Alhassid, lachello, and Levine, 1985; Benjamin and Levine, 1986).

Appendix A

Properties of Lie Algebras

A.I

Definition

A set of operators Xa (a = 1,..., r) satisfying the commutation relations

with

and the Jacobi identity

is said to form a Lie algebra G

A subset Yt of the Xas is said to form a subalgebra G' of G if

This situation is denoted by

197

198

Appendix A

An algebra (or subalgebra) is said to be Abelian if all its elements commute

Much of the terminology to be introduced is motivated by the group structure that can be associated with a given algebra. In this volume we shall, however, be primarily concerned with the algebraic structure itself rather than with the corresponding groups of transformations. It is in the discussion of time evolution and of collision processes that the group concept comes to the fore.

A.2

Generators and realizations

For the purpose of the definition of an algebra it is not necessary to specify any explicit form of the operators. If the operators are written down as differential operators, they are said to be the generators of the corresponding group of transformations. The resulting algebraic structure is said to be a realization (of the abstract algebra). One can realize also the abstract algebraic structure with a set of matrices or with products of creation and annihilation operators.

A.3

Cartan classification

Cartan's classification of all the admissible, semisimple, Lie algebras is given in Table A.I. 1 Properly speaking, algebras should be denoted by lower case letters, and their associated groups of transformations by capital letters. Thus so(3) denotes the algebra of special (s) orthogonal (o) transformations in three (3) dimensions, while SO(3) denotes the associated group. It has become customary, however, to denote both groups and algebras by capital letters. Table A.I Name [Special] Unitary [Special] Orthogonal [Special] Orthogonal Symplectic Exceptional

Admissible Lie algebras Label [S]U(«) [S]O(«), n = odd [S]O(n), n = even Sp(n), n = even G 2 ,/ 7 4, E6, Ej,E%

Cartan label •A(n-l) B

(n-l)/2

D(n/2) C(n/2)

G 2 ,F 4 , E6,E-,, E8

The letter S denotes special transformations, that is, transformations with determinant +1. When dealing with algebras, the letter S is not important for orthogonal algebras, B and D types, while it is important for unitary algebras, since U(«) and SU(n) differ in the number of operators (Section A.4). Both orthogonal and special orthogonal algebras have the same number of operators.

Properties of Lie Algebras

199

To avoid unnecessary burdening of the notation, the letter S is deleted in the text when dealing with orthogonal algebras.

A.4

Number of operators in the algebra

For any admissible Lie algebra, one knows the number of operators in the algebra, denoted by r in Section A. 1. This number is called the order of the algebra and is given in Table A.2. In this book, we make extensive use of U(4), with 16 operators, U(3) with 9, U(2) with 4, U(l) with 1 and of the orthogonal algebras SO(4) with 6 operators, SO(3) with 3 and SO(2) with 1. Table A.2

Number of the operators in Lie algebras Algebra

U(/i) SU(n)

n2

S0(«)

«2-l 1 -«(«-!)

Sp(n)

1 -«(«+!) 14 52 78 133 248

G2 F4 E6 E7 E«

A.5

Number

Isomorphic Lie algebras

Some algebras have identical commutation relations. They are therefore called isomorphic algebras. A list of isomorphic algebras (of low order) is shown in Table A.3. In this table, the sign © denotes direct sums of the algebra, that is, addition of the corresponding operators. There is also the trivial case U(l) ~ S(¥2\ Table A.3 Cartan notation A^B.^C, 52 = C2 D2 ~ A,®A, A3~£>3

Isomorphic Lie algebras Isomorphic algebras

SU(2) » S0(3) = Sp(2) SO(5) ~ Sp(4) S0(4) = SU(2)eSU(2) « SO(3)0SO(3) « Sp(2)®Sp(2) SU(4) « SO(6)

200

A.6

Appendix A

Casimir operators

For any Lie algebra, one can construct a set of operators, called invariant or Casimir operators, C, such that

that is, the operators C commute with all the elements of the algebra.2 They are therefore invariant operators. The operators C can be linear, quadratic, cubic,..., in the X&,

A Casimir operator containing p operators X, is called of order p. Only unitary algebras U(n) have linear Casimir operators. All other algebras have Casimir operators that are quadratic, cubic, If C commutes with all the X's, so does the product of C by any constant ocC and any power of C. These do not count as new operators. The number of independent Casimir operators of a Lie algebra is called the rank of the algebra.

A.7

Example of Lie algebras

The simplest example of a Lie algebra is the angular momentum algebra discussed explicitly in the text. This algebra, which is a realization of SO(3), has three elements, the three components of the angular momentum

with commutation relations

SO(3) has a (trivial) subalgebra, SO(2), composed only by one component, say J2,

satisfying the (trivial) commutation relation

The algebra of SO(3) has only one independent Casimir invariant

It is therefore an algebra of rank 1, called B I , by Cartan.

Properties of Lie Algebras

201

The subalgebra SO(2) also has a (trivial) Casimir invariant, that is, Jz itself,

It is a trivial algebra of rank 1, called D{ by Cartan. Invariant operators are important because they are related to conserved quantities, as the example of the angular momentum discussed in the text makes evident.

A.8

Representations

In applications to problems in physics and chemistry, one needs also to construct representations of the algebras. These are linear vector spaces over which the group elements act. Representations of Lie algebras are characterized by a set of numbers (quantum numbers) that can take either integer or half-integer values. Those which take integer values are called tensor representations, while those that take half-integer values are called spinor representations. In this book, we consider only tensor representations. Also important is the concept of irreducible representations, that is, vector spaces that transform into themselves by the operations of the algebra, and cannot be further reduced. The irreducible representations of unitary algebras, U(«), are characterized by a set of n integers, corresponding to all possible partitions of an integer s,

The integers A-s are often arranged into a diagram, called a Young diagram (or tableau). The first row in the diagram is A,], the second is A, 2 ,...

Another notation is [X1; A - 2 , . . . , A.J. For example, the diagram (A.17) is [5,3,2]. The irreducible representations of special unitary algebras, SU(«), are characterized by a set of integers, as in the case of U(«), but with one fewer; that is,

The irreducible representations of SO(«) are also characterized by a set of integers, but corresponding to the partition

202

Appendix A

where

The irreducible representations of Sp(n) are also characterized by a set of integers

The situation is summarized in Table A.4, where the results for the exceptional algebras are also given. One may note that the number of integers that characterize the representations is also equal to the rank of the algebra. Table A.4 Number of integers that characterize the tensor representations of Lie algebras Algebra

Number

U(») SU(n) SO(«), « = even SO(n),n = odd Sp(n)

n n-\ nil

G2 ^4

£6 £7 E*

(n - 1)12

n/2 2

4 6 7 8

There is a complication that arises only when dealing with orthogonal algebras in an even number of dimensions, SO(«), n - even. The complication is that the partition (A. 19) is not sufficient to characterize uniquely the representations since there are, when the last quantum number, |0.v, is different from zero, two equivalent representations (Hamermesh, 1962). This is denoted either by writing explicitly

or by writing simply

Properties of Lie Algebras

203

and remembering that there are two such states. Since in molecular physics SO(4) and SO(2) play an important role, the complication (A.22) cannot be overlooked. The complication (A.22) has also an explicit physical meaning, since it is associated, for example, with the double degeneracy of Ji, 8,... orbitals or the Ft, A , . . . vibrational states in linear molecules.

A.9

Tensor products

With the representations of Section A. 8 one can form tensor products. Tensor products are usually denoted by the symbol ®,

There are definite rules on how to multiply representations of which we give here one (Hamermesh, 1962). Consider the product of any representation, for example,

by a one-row representation, for example,

Draw the pattern for the first factor, using a symbol, for example, a,

Assign another symbol, for example, b, to the second pattern. Apply b to a in all possible ways subject to the rule that no two bs appear in the same column,

Example:

A.10

Branching rules

For any given quantum mechanical problem one needs to find the complete set of quantum numbers that characterize uniquely the states of the system. This corresponds to finding a complete chain of subalgebras

204

Appendix A

There is a definite mathematical procedure for solving this problem (called the branching problem). In applications, one needs also to deal with the following question. What are the representations of the subalgebra G' contained in a given algebra G (branching rules). This problem is also completely solved, and there exist tables of branching rules. The branching rules for the cases of interest in the molecular problem are reported in the text. Those for the cases of interest in nuclear physics are reported in lachello and Arima (1987).

A. 11

Example of representations of Lie algebras

We return to the simple example of the angular momentum algebra, SO(3). Its tensor representations are characterized by one integer (Table A.4), that is, the angular momentum quantum number /. Similarly, the representations of SO(2) are characterized by one integer (Table A.4); that is, M the projection of the angular momentum on the z axis. The complete chain of algebras is

The complete set of quantum numbers is thus

We use the bracket notation of Dirac, following standard practice. The ket \J,M> corresponding to (A.32) is also called a basis state. Accroding to the branching rules for SO(3) z> SO(2), in the representation / of SO(3) the values of M are all the integers between -/ and +/. Note again the complication mentioned at the end of Section A.8, due to the fact that SO(2) is in an even number of dimensions.

A. 12

Eigenvalues of Casimir operators

Another ingredient one needs in the application of algebraic methods to problems in physics and chemistry is the eigenvalues of Casimir operators in the representations of Section A.8. The known solution is given in Table A.5.

A. 13

Examples of eigenvalues of Casimir operators

Table A.5 gives the eigenvalues of the Casimir operator of SO(3) in the representation / as

Properties of Lie Algebras Table A.5 Algebra

205

Eigenvalues of some Casimir operators of Lie algebras Labels

U(n)

Order

Eigenvalue

1 2

SU(«)

2

SO(2n + 1)

2

S0(2n)

2

Sp(2«)

2

Adapted from lachello and Arima (1987).

a well-known result. Similarly, one can obtain the eigenvalue of the Casimir operator of order 2 of SO(4) in the representation TI , i^ as

Once more a complication arises when dealing with orthogonal algebras in an even number of dimensions, since often these algebras have two Casimir operators of order two. In the text we distinguish the two operators by placing a bar over the second operator. The eigenvalues of this operator are given by

Notes 1. The material in Sections A.3-A.11 is reviewed in Wybourne (1974), and lachello and Arima (1987), Chapter 2. 2. Invariant operators were introduced by Casimir (1931) for SO(3). Racah (1950) generalized them to all orders.

Appendix B Coupling of Algebras

B.I

Definition

If Xia(a =l,...,r) and X2b(b = 1,..., r) are a set of operators forming two isomorphic Lie algebras

the set Xia ® X2b forms a Lie algebra, called a direct sum G\ 0 G 2 . If Aj and A2 denote generically representations of GI and G 2 , the product I A! > IA2 > is called the direct product of IAj > and IA2 >

B.2

Coupling coefficients

If IAX, > denotes generically a basis for the representations of G z> G',

one can expand the product IAj > <8> IA2 > as 206

Coupling of Algebras

207

The expansion coefficients are called coupling coefficients.

B.3

Addition of angular momenta, SO(3)

The simplest example of coupled algebras is provided by the angular momenta (cf. note 2 of Chapter 1). The direct sum of two angular momenta, Jj and J2, is just

The basis of the angular momentum algebra is given by Eq. (A.32). The product (B.4) is then

The coefficients in (B.6) are called Clebsch-Gordan

B.4

coefficients.

Properties of Clebsch-Gordan coefficients

We define the Clebsch-Gordan coefficients with the usual (Condon and Shortley, 1967) phase convention

The selection rules are

The Clebsch-Gordan coefficients satisfy the orthogonality relations

Instead of Clebsch-Gordan coefficients it is often convenient to use the Wigner 3 - j symbols

208

Appendix B

The Wigner 3 - ;' symbols satisfy the symmetry properties

The orthogonality relations are:

There exist extensive tables of 3 - j symbols (Rotenberg et al., 1959) and computer subroutines for their calculation (Schulten and Gordon, 1976; Zare, 1988). The calculation is done using Racah's formula

with (-m)\ = oo when m > 0, t = integer, 0! = 1. Thus t is bounded by

Coupling of Algebras

B.5

209

Tensor operators

A tensor operator under the algebra G z> G', T£, is defined as that operator satisfying the commutation relations

with the operators of G. In the case of the angular momentum algebra, SO(3), Eq. (B.I5) is

B.6

Wigner-Eckart theorem

The celebrated Wigner-Eckart theorem states that the matrix elements of any tensor operator of an algebra G can be split into two pieces, a coupling coefficient and a piece that depends only on A; that is,

where the double bar matrix element is referred to as a reduced matrix element. For the angular momentum algebra one has

B.7

Tensor products

The tensor product of two tensor operators is defined as1

This is conventionally denoted by

210

Appendix B

Conversely, one has

B.8

Recoupling coefficients

When more than two representations of G 3 G' are coupled, there is more than one way to do the coupling. All of them are, however, equivalent, and one can go from one to the other by means of recoupling coefficients. In the case of three algebras one has genetically

where the coupling of A] and A2 to A12 and subsequently of A3 with A12 is expressed in terms of the coupling of A2 and A3 to A23 and subsequently of AJ with A23 to form the total A. The coefficients in brackets in Eq. (B.22), < I >, are called recoupling coefficients.

B.9

Addition of three angular momenta, SO(3)

Equation (B.22) can be used in the case of coupling of three angular momenta

In this case one has

The recoupling coefficients are written as

The coefficient in bracket is called a Wigner 6 - j symbol. Tables of these coefficients exist (Rotenberg et al., 1959).

Coupling of Algebras

B.10

211

Properties of 6-7 symbols

One has

(Jl,J2,J3),(Ji,L2,L3),(Ll,J2,L3),(L),L2,J3) tion

satisfy the triangular condi-

The 6 — j symbols have the following symmetry properties:

The orthogonality relations are

6 — j symbols are related to the 3 — j symbols

They can be computed using Racah's formula

212

Appendix B

where

The number t > 0 is bounded by

Instead of the 6- j symbols, sometimes another symbol is used, called Racah symbol and related to the 6 — j symbol by

B.ll

Addition of four angular momenta

The process of addition of angular momenta can be continued. For four angular momenta

the recoupling coefficients are

The symbol in parentheses is called Wigner 9 - j symbol. For more angular momenta one can define 12 - j , . . . symbols.

Coupling of Algebras B.12

213

Reduction formulas

When evaluating matrix elements of operators for coupled systems, it is often convenient to make use of reduction formulas. These formulas reduce the evaluation of products of operators to matrix elements of individual operators. Two situations can occur. 1. Type I. The operator whose matrix elements one wants to evaluate is of the type

that is, the product of an operator of system 1 times an operator of system 2. The reduction formula for SO(3) is

2. Type II. The operator whose matrix elements one wants to evaluate is of the type

that is, a product of two operators of system 1. In this case, one has

for the matrix element of T^ for system 1. This result can then be inserted in the preceding one

The 9 - j symbol in Eq. (B.40) is related to a 6 - j symbol by

Appendix B

214

B.13

Coupling of SO(4) representations

The algebraic theory of molecules, discussed in Chapters 2, 4, and 5, makes use of the algebra of SO(4). In order to do calculations, one therefore needs the Racah calculus of SO(4) (Lemus, 1988). Fortunately, in view of the isomorphism of Table A.3,

the Racah calculus of SO(4) can be reduced to that of SO(3). Thus, despite the formidable-looking formulas that appear, any calculation can be done in a very straightforward way making use of the angular momentum algebra, SO(3). According to Section A.8, the representations of SO(4) are in general labelled by two integers, (tj^), with TJ > lT 2 l, T2 = i^- The quantum numbers (Ti,T 2 ) of SO(4) can be converted to those of SO(3) © SO(3) = SU(2) © SU(2) by

where s and t denote the quantum numbers of SU(2) © SU(2) ~ SO(3) © S0(3). The Clebsch-Gordan series for SO(4) can then be simply constructed and is given by

where w is the lesser of a + b and c + d, while u is the lesser of a - b and c — d. This is the analogue of the Clebsch-Gordan series of SO(3) [Eq. (B.8)],

where w is the lesser of 2j\ and 2j2. An important example is given in Section 4.9. The product of two symmetric representations

Coupling of Algebras

215

gives

Another important example is given in Section 5.1. Both examples can be obtained from Eq. (B.44).

B.14

Racah's factorization lemma

In the preceding sections, the coupling coefficients of G z> G' were considered. In the more general case, one needs the coupling coefficients of a chain of algebras

where lA^jo, > denotes generically a basis for the representation G z> G' ID G". One can expand, as before, the product IA t > ® IA2 > as

Racah (1942) showed that the coupling coefficients in Eq. (B.49) can be factorized into products of coupling coefficients for G :D G' and G' z> G",

This celebrated factorization lemma (Wybourne, 1974; Biedenharn and Louck, 1981) allows one to simplify considerably the calculations in the molecular case. The algebras of interest are

The expansion of the coupled wave function is

216

Appendix B

where the coefficient < LiM\L2M2\JM > is the usual Clebsch-Gordan coefficient of SO(3) 13 SO(2). The coupling coefficients are sometimes written as

in order to emphasize the similarity with the usual 3 - j symbols. They have been written in this form in Eq. (4.50).

B.15

Coupling coefficients of SO(4)

In view of the isomorphism (B.42), the coupling coefficients of SO(4) can be written in terms of recoupling coefficients of SO(3). The explicit expression is

where the coefficient in curly brackets is a 9 - j symbol [Eq. (B.35)]. The evaluation of the coupling coefficients of SO(4) is then straightforward, and it can be done using the same methods discussed in the previous sections.

B.16

Recoupling coefficients of SO(4)

The recoupling coefficients of SO(4) can also be written in terms of recoupling coefficients of SO(3). The explicit form (for recoupling of symmetric representations, which are the only ones needed in molecular physics) is

Coupling of Algebras

217

This recoupling coefficient is used in Chapter 5, where it is rewritten in a different form by making use of the W coefficients of Eq. (B.33).

Notes 1. The basic texts on tensor products are de Shalit and Talmi (1963) and Fano and Racah (1959).

Appendix C Hamiltonian Parameters

This appendix provides a summary of the functional form of the algebraic Hamiltonian used in the text for tri- and tetratomic molecules. Values of the parameters are reported both for a low-order realistic representation of the spectrum and for accurate fits using terms quadratic in the Casimir operators. 1. Linear triatomic molecules

Here the last term is the Fermi coupling discussed in Section 4.19. To higher order

2. Bent triatomic molecules

To higher order 218

Hamiltonian Parameters

219

3. Linear (and quasilinear) tetratomic molecules. For linear and quasilinear tetratomic molecules we use a notation similar to that of Eqs. (C.2) and (C.4). Written explicitly, the Hamiltonian of linear tetratomic molecules is

where

Many of these coefficients are not used in actual fits. The actual Amat-Nielsen parameters are denoted by £12,12. £13,13. £2323- fr> some of the fits, for convergence reasons, the Casimir operators Cj 2 , C 13 , C23, Ci23 are divided by their respective Ns, that is, the operators

and similar expressions for C13 and C23 are used. The operators in Eq. (C.7) are called reduced operators. Since the vibron numbers are typically of order 100, the numerical values of the reduced coefficients are typically a factor 100 larger than those of the unreduced coefficients. The values of the vibrational parameters (see note 2, Chapter 4) for some triatomic molecules are given in Tables C.I andC.2.

220

Appendix C Table C.I

#1

N2 AI

A2 An A.,2

HCN

HCP

140 47 -1.14 -9.84 -1.97 -0.185

155 26 0.63

-1.84 -0.06

Vibrational parameters for triatomic molecules" DCS

190 159 0.31 -1.53 -0.79 -0.07

N2O

C02

SO2

H20

D2O

H2S

165 136 -2.20 -1.0 -0.5 0.97

153 153 -3.204 -3.204 0.5 3.20

215 215 -1.211 -1.211 -0.06 0.5

36 36 -15.82 -15.82 -4.81 1.77

56 56 -7.74 -7.74 -2.08 1.17

41 41 -9.13

-9.13 -3.41 0.43

"All values in cm ' apart from the vibron numbers.

Table C.2

N, =N2

A,=A2 An %1,1

=

"^2,2

*,,2 ^12,12

X12

^1,12

=

^12,12

^2,12

Higher-order vibrational parameters" for some triatomic molecules

H20

D20

H2S

SO2

C026

39 -14.700 -4.374 -8.22(-4) 2.89(04) 1.39(-4) 1.409 1.43(-4) 5.67(-5)

57 -7.655 -2.034 -1.67(-4) 8.23(-5) 1.98(-5) 1.073 1.19(-4) -4.50(-5)

39 -9.323 -3.708 7.94(-5) 9.98(-4) 2.13(-5) 0.120 1.41 (-4) -1.46(-4)

205 -1.298 -0.048 2.36(-6) -8.03(-6) -4.62(-8) 0.533 1.15(-5) -2.97(-6)

153 -3.324 0.568 7.99(-5) 1.40(-4) -8.66(-5) 3.306 -4.24(-5) 3.74(-5)

"All values in cm l, except N, which is dimensionless. A Fermi parameter (Section 4.19) xl2 - -0.424 is also used.

A

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Index

action-angle variables, 67 algebra Abelian, 22, 198 angular momentum, 9, 21, 207 Cartan classification, 198 Casimir operators, 23, 200, 204 coupling of, 206-217 isomorphic, 199 Lie, 21, 197 order, 199 properties, 197—205 quantum mechanics, 25 rank, 23, 200 realization, 22, 198 representation, 23, 204 spinor representation, 201 subalgebra, 22, 197 tensor representation, 201, 202 Amat-Nielsen couplings, 129, 131 angular momentum, 9 addition, 207 algebra, 9, 21,200 anharmonicity, 30, 33 electrical, 50 bending vibrations, 86, 100, 148, 150 C6H6, 151 bent triatomic molecules, 101, 103 Hamiltonian, 99 Hamiltonian parameters, 220 linearity parameter, 101 locality parameter, 103 Birge-Sponer plot, 36 boson annihilation operators, 25, 40

boson condensate operator, 175, 183 boson condensate operator: three dimensional, 177, 183 boson fluctuation, 176, 179 boson operators, 25, 40 intensive, 161, 169 branching rules, 203, 204 Cartan classification, 198 Casimir operators, 23, 24, 127, 200 eigenvalues, 204 examples, 204 C2D2,79 locality parameter, 79 mean field approximation, 187 vibrational frequencies, 187 C6D6, 140-145 frequencies, 140 intensities, 140, 144 C2H2, xiv, 79, 124, 125, 126, 127 Fermi coupling, 131 /-doubline, 127 local modes, 125 Majorana couplings, 126 mean field theory, 187 vibrational energy levels, 130 vibrational frequencies, 130, 187 C6H3D3, 144 C6H2D4, 144 C6HD5, 144 C6H5D, 144 C6H6, 138, 144, 152 bending vibrations, 151 coordinates and symmetry species, 152

239

240 intensities, 140, 144 overtone spectra, 155 vibrational frequencies, 153 C6H6 • Ar, xv, 8 C2HF Amat-Nielsen coupling, 131 /-splittings, 132 CO2, 80, 95 Fermi resonance, 98 Hamiltonian parameters, 220 locality parameter, 95 vibrational energy levels, 98 vibrational frequencies, 185 chain of subalgebras, 203 chaos, 67 C-H overtone spectrum, 155, 195 classical limit, 160, 189 Clebsch-Gordan coefficient, 11, 85, 114, 207 clump, 191 coherent states, xviii nl, 160, 165, 172, 177, 180 intrinsic, 177 projective, 174 commutator, 21, 197 condensate boson operators, 175 configuration interaction, xviii nl, 59 constants of the motion, 23 Casimir operators, 23, 200 correlation diagram, 78, 101 coset space, 159, 189 coupled one dimensional oscillators, 66, 73, 134,165 coupling Amat-Nielsen, 129, 131, 132 Casimir, 127, 132 coefficients, 206, 207 Darling-Dennison, 91, 194 Fermi, 96, 131,132 interbond, 132 Majorana, 91, 97, 102, 126, 132 normal mode-local mode, 78, 91, 101, 126 recoupling coefficients, 120, 210 recoupling in polyatomic molecules, 133 recoupling of angular momenta, 121 tetratomic molecules, 126 /-type, 89, 127, 132 coupling scheme, 120 cranking frequency, 180 cranking method, 180 Darling-Dennison couplings, 91, 94 D2O, 103 Hamiltonian parameters, 220 locality parameter, 103 dipole function, 13, 37,48, 81, 143 many-body, 145 direct product, 83, 206 direct product of representations, 83, 201 dipole operator, 11, 13, 50, 145 direct sum, 73, 206

Index dynamical symmetries, 27, 41 tetratomic molecules, 123 three-dimensional problems, 41 triatomic molecules, 86, 88, 91, 99 Dunham expansion, xvi, 19, 35, 46, 105 Eckart frame, 65 eigenvalues of Casimir operators, 24, 205 electrical anharmonicities, 50 electron-molecule scattering, 193 Euler angles, 62 l/N expansion, 188 factorization lemma, 215 Fermi coupling, 96, 97, 131 resonance, 96, 98, 131 floppy molecules, xiv, xv, 8, 43, 46, 111, 178, 189, n6 fluctuation (boson), 176 Fock space, 26 force field method, 6 force field potential, 61 generators, 22, 198 group coherent state, 167 group intrinsic state, 160 Hamiltonian coordinate representation, 157-188 diagonalization: computer program, n2, 118 for bending vibrations, 150 for bent triatomic molecules, 99 for benzene, 139, 151 for linear tetratomic molecules 123, 126 local modes, 85 normal modes, 88 for octahedral molecules, 147 for stretching vibrations, 136 for triatomic molecules, 104, 106 bent, 99 with Darlin-Dennison coupling, 91 with Fermi coupling, 96 linear, 86, 87 parameters, 218 rotational, 110 harmonic limit, 38, 189 H2O, 79, 80, 103 Hamiltonian parameters, 220 locality parameter, 79, 80, 103 lower overtones, 108 normal modes, 103 potential, 63, 81, 168 potential energy, 168 stretching vibrations, 79 vibrational energy levels, 80, 108 HC1, 16, 52 Herman-Wallis factors, 52, 54 Morse potential, 5 rotational spectrum, 48

241

Index H 2 S,103

Hamiltonian parameters, 220 locality parameter, 103 vibrational energy levels, 104 HCN, 87 Hamiltonian parameters, 220 locality parameter, 95 vibrational energy levels, 87 vibrational frequencies, 185 HCP Hamiltonian parameters, 220 Herman-Wallis expansion, 19 Herman-Wallis factor, 49, 51 large N limit, 54 HF, 13, 38 dipole, 13 overtone transitions, 38 infrared transitions, 11,81, 145 intensities of vibrational transitions, 15, 37, 48, 143, 149 intensive boson operators, 161, 169 interbond coupling, 132 internal symmetry coordinates, 134 invariant operators, 23 irreducible representation, 23, 201 isomorphic Lie algebras, 199 isoscalar factors, 85 Jacobi identity, 21, 197 Kratzer potential, 6 large-TV limit, 189 Herman-Wallis factors, 54 mean-field theory, 188 one dimensional, 38, 161-167 tetratomic molecules, 187 three-dimensional, 169-174 lattice of algebras, 74 Lie algebras, xviii nl, 21 admissible, 198 Cartan classification, 198 isomorphic, 198, 199 representations, 198, 201 subalgebras, 22 Lie groups, xviii nl Lie structure constants, 21 linearity parameter, 101 line strength, 13,49,57 /-coupling, 89 /-splitting, 91, 127, 132 local-mode Hamiltonian, 85 local-mode limit, 75 local modes, 75 basis, 83 locality parameter, 78, 80, 95, 103 Majorana couplings, 91, 102, 126 linear tetratomic molecules, 126

Majorana operator, 76, 91, 97 linear tetratomic molecules, 126 matrix mechanics, xviii nl, 25 mean field approximation, 188, 189 missing labels, 58 Molecules. See also under individual molecules C2D2,79 C2HF, 132 C2H2, xiv,79, 125-127 C6HD5, 144 C6Y2D4, 144 C6H3D3, 144 C6H,D, 144 C6H6, 138, 144, 151-153 C6H6 • Ar, xv, 8 CO2, 95, 98 D2O, 103 HC1, 16, 52 HF, 13, 38 H2O, 79, 80 H2S, 103, 104 HCN, 87, 185 HCP, 220 N2O, 185,220 O3, 80 OCS, 95, 106 SF6, 147 SO2, xiii, 71,80 UF6, 147 WF6, 147 Morse oscillator, 33, 59 Morse oscillator-rigid bender Hamiltonian, 167 Morse potential, 6, 33, 157 multiple!, 85, 93 multiplicity problem, 84 N2O Hamiltonian parameters, 220 locality parameter, 95 vibrational frequencies, 185 nonlinear dynamics, 67 nonrigid rovibrator, 43. See also floppy molecules nonrigidity parameter, 46 normal-mode Hamiltonian, 88 normal-mode limit, 76 basis, 84

O(2), 31,41, 163 O3, 80 O(/i), 198 OCS, 95 Hamiltonian parameters, 220 locality parameter, 95 vibrational energy levels, 106 vibrational frequencies, 185 octahedral molecules, 146 one-dimensional problems, 27

242 operators Amat-Nielsen, 131 angular momentum, 9, 200, 207-212 boson, 25, 40, 175, 176, 177 Casimir, 23, 24, 200, 218 condensate boson, 175, 177 dipole, 13,37,48,81, 143 eigenvalues of Casimir, 24, 205 Fermi, 96, 131 intensive, 161, 169 invariant, 23 Majorana, 76, 91, 126 optical transitions, 15, 37, 48, 54, 66, 80, 143 Raman transitions, 54 realization, 22 shift, 26, 28, 37 step down, 26 step up, 26 symmetry adapted, 138 tensor, 209 transition, 49 orbital angular momentum, 5 order of the algebra, 199 overtone spectroscopy, xiv overtone transitions, 18, 37, 155 C6H6, 155' HF, 38 picket fence model, xvii polarizability, 14 polyatomic molecules recoupling procedure, 133 Poschl-Teller potential, 148, 157 potentials C6H6 • Ar, 8 coupled one-dimensional problems, 66, 134, 165 coupled three-dimensional problems, 172 diatomic, 6, 30, 35, 44, 162 equilibrium position, 174, 182, 186 force field, 6, 61 H2O,63, 81, 168 harmonic limit, 38, 189 Kratzer, 6 local modes, 66 Morse, 6, 7, 33, 167 O(2), 163 Poschl-Teller, 148, 157 potential functions, 165, 172 coupled one-dimensional problems, 81, 165 coupled three-dimensional problems, 172 quasidiatomic model, 17 U(l), 162 U(2)®U(2), 164, 168 U(4), 111, 167, 170 U(4)®U(4), 172 three-dimensional, 170

Index potential energy. See potentials projective coherent states, 174 quantum numbers extra, 58 missing, 58 quasidiatomic model, 17 Raccah coefficient, 121 Racah's factorization lemma, 85, 215 Racah's formula, 208, 211 Racah symbol, 212 radiationless transitions, xviii nl Raman spectra, 14 anharmonicities, 58 polarizability, 14 transitions, 41 rank, 23, 200 realization, 198 realization of operators, 22 recoupling coefficients, 120, 121, 210 recoupling of angular momenta, 121 recoupling scheme, 120 recoupling transformation, 120 reduced matrix elements, 11, 209 reduction formulas, 213 representations, 23, 201 irreducible, 23,201 spinor, 201 tensor, 201 resonance, 68, 96, 131 condition, 68 Darling-Dennison, 91 Fermi, 96, 98, 131 rigid rotor approximation, 8 rigid rovibrator, 44 rotational constant, 8, 48 rotation group, 22 rotational spectra, 43^8, 108, 110, 115, 117, 186 HC1, 48 higher order corrections, 110-118 rotation-vibration interactions, 52, 111, 113, 117 linear triatomic molecules, 115 nonlinear triatomic molecules, 117 Wang's basis, 116 rovibrator nonrigid, 43 rigid, 44 selection rules, 49 sequential coupling, 191 SF6, 147 Hamiltonian parameters, 148 shift operators, 28, 37 SO2, 71,79, 80 Dunham expansion, xvi Hamiltonian parameters, 220 locality parameter, 79, 103 normal modes, 103

243

Index SO2 (continued) potential energy, 168 spectrum, xiii SO(2), 22, 199, 204 SO(3), 21, 200, 204, 207, 210, 216 SO(4), 205, 214, 216 SO(n), 198, 199, 202 spectral clump, 191 special orthogonal algebra, 21 spectrum C2H2, xiv C6H6, 155 C 6 H 6 -Ar,xv overtone, 155 rotational HC1, 48 SO2, xiii stimulated emission pumping, xiii spherical harmonics, 5 spherical tensor, 9 spinor representations, 201 spurious states, 152 stimulated emission pumping, xiii stretching vibrations, 73 Hamiltonian, 136 H2O, 79 infrared transitions, 80 local mode limit, 75 normal mode limit, 76 potential, 165 subalgebras, 22 sudden approximation, 196 6-j symbols, 211,212 9-j symbol, 212 symmetry-adapted operators, 138 symmetry coordinates, 60 symplectic algebras Sp(2« + 2), 59, n5 tensor operators, 10, 209 tensor product, 10, 203, 209 tensor representations, 23, 201 three-dimensional dynamical symmetries, 41 potential functions, 170 transition operator, 11, 13, 37, 48, 81, 143 truncated anharmonic oscillator, 30 truncated harmonic oscillator potential, 157

U(l), 29, 162 U(2), 27, 29 U(3),41 U(4), 27,41, 170 U(«), 198-205 U(n + 1), 26 UF5, 147 Hamiltonian parameters, 148 U(2)®U(2), 74-81,164 U(4)®U(4), 83-104, 172

unitary algebra, 26 unitary algebras U(n + 1), 59 unitary group, 39, 59 van der Waals molecules, xiv, xv, 8 vibrational angular momentum, 65 vibrational energy levels. See also under individual molecules computer program, n2, 118 Dunham expansion, 19, 35, 46, 105 Hamiltonian parameters, 220 C2H2, 130 C2HF, 132 C6D6, 140 C6H6, 140, 151 C6H6 • Ar, xv, 8 CO2, 98 H2O, 80 H2S, 104 HCN, 87 DCS, 106 SF6, 147 SO2, xiii UF6, 147 WF6, 147 vibrational frequencies Hamiltonian parameters, 220 C2D2, 187 C2H2, 187 C6H2D4, 144 C6H3D3, 144 C6H5D, 144 C6H6, 144, 153 C6HD5,144 CO2, 185 HCN, 185 N2O, 185 DCS, 815 vibrational /-doubling, 89, 91, 94, 101, 127, 131, 132 Amat-Nielsen coupling, 131 Casimir operators, 127 vibron model, 59 n7 vibron number 42 Wang's basis, 116 WF6, 147 Hamiltonian parameters, 148 Wigner-Eckart theorem, 11, 12, 209 Wigner coefficients. See Wigner symbols computer program, n2, 118 Wigner 3-y symbol, 11 Wigner 6-j symbol, 114, 121, 212 x-K relations, 107 Young tableaux, 23, 28, 201