Algebraic Theory of Molecules (Topics in Physical Chemistry Series)

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) 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 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