The Jahn-Teller Effect and Vibronic Interactions in Modern Chemistry

MODERN INORGANIC CHEMISTRY

Series Editor: John P. Fackler, Jr. Texas A&M University

METAL INTERACTIONS WITH BORON CLUSTERS Edited by Russell N. Grimes

HOMOGENEOUS CATALYSIS WITH METAL PHOSPHINE COMPLEXES Edited by Louis H. Pig no let

THE JAHN-TELLER EFFECT AND VIBRONIC INTERACTIONS IN MODERN CHEMISTRY I. B. Bersuker

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The Jahn-Teller Effect and Vibronic Interactions in Modern Chemistry

I. B. Bersuker Institute of Chemistry Moldavian Academy of Sciences Kishinev, USSR

Plenum Press • New York and London

Library of Congress Cataloging in Publication Data

Bersuker, I. B. (Isaak Borisovich) The Jahn-Teller effect and vibronic interactions in modern chemistry.

(Modern inorganic chemistry) Bibliography: p. Includes index. 1. Jahn-Teller effect. 2. Chemical reactions. I. Title. n. Series.

QD461.B46 1983 541'.22

ISBN-13: 978-1-4612-9654-6 DOl: 10.1007/978-1-4613-2653-3

© 1984 Plenum Press, New York

e-ISBN-13: 978-1-4613-2653-3

Softcover reprint of the hardcover 1 st edition 1984

A Division of Plenum Publishing Corporation 233 Spring Street, New York, N.Y. 10013

All rights reserved

83-16070

No part of this book may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, microfilming, recording, or otherwise, without written permission from the Publisher

Preface

The first half of the title of this book may delude the uninitiated reader. The term '"Jahn-Teller effect," taken literally, refers to a special effect inherent in particular molecular systems. Actually, this term implies a new approach to the general problem of correlations between the structure and properties of any molecular polyatomic system, including solids. Just such a new approach, or concept (in some sense, a new outlook or even a new way of thinking), which leads not to one special effect but to a series of different effects and laws, is embodied in the many ( ~ 4000) studies devoted to the investigation and application of the Jahn-Teller effect. The term "vibronic interactions" seems to be most appropriate to the new concept, and this explains the origin of the second half of the title.

The primary objective of this book is to present a systematic develop­ment of the concept of vibronic interactions and its applications, and to illustrate its possibilities and significance in modern chemistry. In the first three chapters (covering about one-third of the book) the theoretical background of the vibronic concept and Jahn-Teller effect is given. The basic ideas are illustrated fully, although a comprehensive presentation of the theory with all related mathematical deductions is beyond the scope of this book. In the last three chapters the applications of theory to spectro­scopy, stereochemistry and crystal chemistry, reactivity, and catalysis, are illustrated by a series of effects and laws. Greatest consideration is given to those applications which initiate new trends in appropriate areas of investigation.

Both the content and structure of this book are in principle different from those of monographs in chemistry devoted to a definite class of chemical compounds or a certain method of research. The concept of vibronic interactions, aimed at improving the bridge between the structure and properties of substances, is to a certain degree concerned with all kinds of molecular systems and solids, and affects all methods of investigation employed in chemistry. This universality of aims and methods is a characteristic feature of the vibronic approach to the problems in modern

v

vi Preface

chemistry demonstrated in this book. Joined by a common concept, the various phenomena arising in different systems (and observable by differ­ent methods of study) may serve as a source of new ideas: the reader may find quite unexpected similarities between very different systems united by the concept of vibronic interactions.

The author has tried to present the material in a way suitable for a large number of chemists and physicists (such as scientific workers, university professors and students, and high s-chool teachers), as well as engineers engaged in investigating the structure and properties of matter. For this purpose most attention is paid to the physical meaning of the results. A limited number of mathematical expressions is analyzed briefly (mainly in the first three chapters) in order to maintain the quantitative level of the theory. More detailed deductions can be found easily in the original papers cited in the References. On the other hand, applications of the theory are given in such a form that they can be used without a detailed study of the theory. Nevertheless, for a full understanding of this book the reader is expected to know the basic ideas of quantum mechanics and the theory of symmetry.

I should like to express my thanks to my co-workers and colleagues, especially to S. A. Borshch, S. S. Budnikov, A. S. Dimoglo, M. D. Kaplan, I. Ya. Ogurtsov, Yu. E. Perlin, V. Z. Polinger, Yu. B. Rosenfeld, B. S. Tsukerblat, and B. G. Vekhter, with whom research on, and discussion of, vibronic phenomena (with most of them during a long period of more than twenty years) made possible the general presentation of the problem treated in this book. I am grateful to A. Abragam, J. Ammeter, M. Bacci, C. J. Ballhausen, F. Basolo, C. A. Bates, H. Bill, L. A. Boatner, J. Brickmann, L. S. Cederbaum, R. E. Coffman, T. M. Dunn, J. Duran, R. Englman, J. Gazo, G. L. Hofacker, B. Hoffman, O. Kahn, H. Koppel, S. Kirschner, D. I. Khomskii, R. Lacroix, M. Lambert, S. Leach, A. D. Liehr, K. A. Muller, M. C. M. O'Brien, R. Pearson, O. E. Polansky, A. Ranfagni, M. Ratner, D. Reinen, J. S. Slonczewski,M. Thomas, M. Wagner, and their co-workers for cooperation and discussions. Many thanks are due to Professor J. P. Fackler, Jr., who read the manuscript thoroughly and made important suggestions. Also I apologize to those readers whose contribu­tion to works on vibronic problems are not cited here. A comprehensive list of references can be found in the specially prepared Bibliographic Review. *

I. B. Bersuker

Kishinev

• The Jahn- Teller Effect: A Bibliographic Review, I. B. Bersuker, IFIIPlenum, New York, 1984.

Contents

Preface. . . . . . . . . . . . . . . . . . v

Mathematical Notation . . . . . . . . . . . . xi

Introduction - The Concept of Vibronic Interactions 1

Chapter 1. Theoretical Background . . . . . . 7

1.1. The Adiabatic Approximation . . . . . . . 7 1.2. Vibronic Interactions. Linear Vibronic Constants . 10 1.3. Force Constants and Quadratic Vibronic Constants 20 1.4. Orbital Vibronic Constants as Parameters of Molecular

Dynamic Structure . . . . . . . . . . . 1.5. Anharmonicity and Instability of Molecular Systems . 1.6. The Jahn-Teller Theorem. .

Chapter 2. Adiabatic Potentials.

2.1. General Considerations. The Orbital Doublet (E Term) 2.2. Triplet and Quadruplet Terms 2.3. Pseudodegenerate States 2.4. The Multimode Problem

Chapter 3. Energy Spectra. Dynamic Distortions of Nuclear Configurations . . . . . .

3.1. Some Limiting Cases. Free and Hindered Rotations of the Distortion . . .. ..... .

3.2. Tunneling Splitting. Pulsating Deformations . 3.3. Vibronic Reduction Factors 3.4. Numerical Solutions . . .

vii

23 27 32

41

41 52 61 69

73

73 82 91 96

viii Contents

Chapter 4. Applications to Spectroscopy

4.1. Electronic Spin Resonance and Related Methods The Method of ESR (103) . Vibronic Reduction in ESR Spectra (105) . Influence of Tunneling Splitting. Transition from Dynamic to Static JTE (108) . Low­Symmetry External Perturbations (111)· The Role of Random Strain in the ESR E-e Problem (111)· Relaxation Effects (114) . Tunneling Splitting Plus Random Strain and Relaxation in ESR (117) . Tunneling Splitting in Mi:issbauer Spectra (128) . Absorption of Microwaves and Ultrasonics (131)

4.2. Vibronic Infrared and Raman Spectra . Vibronic IR Spectra. Selection Rules (133) . The Pure Rotational Vibronic Spectra (138) . Vibronic Raman Spectra (144)

4.3. Electronic Spectra Vibronic Fine Structure of the Spectra in Gas Phase or in Matrixes (148) . Structure of the Zero-Phonon Line (155) . Band Shape of Electronic Absorp­tion. Semiclassical Approach (161)· Polarized Luminescence (170)· The Method of Moments in the Analysis of Band Shapes (173)

Chapter 5. Stereochemistry and Crystal Chemistry

5.1. Stereochemistry Rules with JTE and PJTE . Notion of Nuclear Configuration. Semiclassical Approach (177) . Relativity to the Means of Observation (178) . Stabilization of Static Distortions by Weak Perturbation. Vibronic Amplification (179) . Dipolar Instability and Dipole Moments (182) . Some Qualitative Features of Vibronic Stereochemistry (186) . Displacement of the Central Atom in Coordination Compounds (192) . Geometry of Small Ligand Coordination to the Central Atom (198) . Examples of Molecular Systems in Which Vibronic Effects Were Studied (203)

5.2. Cooperative Vibronic Effects in Crystals. Structural Phase

103

103

133

148

177

177

Transitions 203 Cooperative JTE (203) . Rare Earth Zircons (210) . Spinels, Perovskites, and Other Crystals (218) . K2PbCu(N02)60 Incommensurate Phases (221) . Structural­Magnetic Transitions. Spin and Orbital Orderings (224) . Ferroelectric Phase Transitions (231)

5.3. Vibronic Crystal Chemistry. Plasticity and Distortion Isomerism 235 Plasticity of the Coordination Sphere around Vibronic Centers (235) . Distor-tion Isomers (243) . General Validity of the Vibronic Model in Stereochemistry, Crystal Chemistry, and the Theory of Structural Phase Transitions (247)

Chapter 6. Activation Mechanisms in Chemical Reactions and Catalysis . 251

6.1. Nature of Instability of the Activated State of a Chemical Reaction and the Possibility of Its Experimental Observation. 251 Activated States of Chemical Reactions (251) . Stable Excited States of Activated Complexes (253) • Existence of Bonding States of Arbitrary (Neutral)

Contents

Polyatomic Groups (255) . Possible Experimental Information about Activated Complexes (257)

ix

6.2. Orbital Symmetry Rules in Mechanisms of Chemical Reactions 259 The Vibronic Approach (259) . Comparison with the Classical Woodward­Hoffmann Description (265) . Example with Catalyst Participation (268)

6.3. Vibronic Activation in Elementary Acts of Chemical Reactions and Catalysis 269 Vibronic Structure as a Basis for a New Approach to the Problem of Chemical Transformations (269) . Change of Reactivity as a Result of Electronic Rear­rangement (274) . Parametrization (278) . Examples: Activation of Carbon Monoxide, Nitrogen, and Nitrogen Oxide by Coordination (280) . Mechanisms of Chemical Reactions in Systems with the JTE (286)

References.

Author Index

Subject Index

291

307

315

Mathematical Notation

Symbols used more than once, but with another meaning, have local concern only in the corresponding chapter or section indicated in paren­theses A,B A

- rotational constants - hyperfine constants (§4.1) - LCAO coefficients - anharmonicity correction coefficients (§6.3) - barrier height - barrier width

E - doubly degenerate representation (term) E - energy EJT - JT stabilization energy e - strain components e* - effective charge (§5.1) 'if; - electric field intensity F - linear vibronic constant F(D) - form function for spectroscopic transition band shapes

(§4.3) / - linear orbital vibronic constant / - angular-dependence function in ESR spectra (§4.1) /0., - molecular field intensity (§5.2) G - quadratic vibronic constant Gij - cross-section of light scattering (§4.2) g - quadratic orbital vibronic constant (§ 1.3) g - EPR g-factor (§4.1) g I _ statistical weight (§4.2) H - Hamiltonian 'Je - magnetic field intensity 'Jen - Hermite polynomials (§1.2)

xi

xii

I J KF Kr(T) K(il) k~ k(i)

L I,m,n I, m,n M\2 M m n P Pa

P p P = K E (A 2 )

p Q q = KE(E) qi ql,2

R ri

Sij

S T1,2 T T Ux

V v W W X,Y,Z

Z Z"

f3 f3 r

Mathematical Notation

- nuclear spin - quantum number - force (elastic) constant - vibronic reduction factor - coefficient of light absorption (Ch. 4) - orbital force constant - force constant coefficients (§§ 1.4, 6.3) - sound propagation factor (§4.1) - quantum numbers of vibronic energy levels (Ch. 3,4) - direction cosines (§4, 1) - transition moment - nucleus mass (Ch. 1) - electron mass (Ch. 1) - vibrational quantum number - quadrupole interaction constant (§4.1) - amplification coefficient (§5.1) - electron-strain interaction - dipole moment - vibronic reduction factor in E-e problem - ratio of () and E components of vibration frequency (§3.2) - normal (symmetrized) nuclear coordinates - vibronic reduction factor in E-e problem - MO occupation numbers - transformed nuclear coordinates (§2.4) - nuclear coordinates, interatomic distance - electron coordinates - overlap integral - electron spin - triply degenerate representation - temperature - period of vibration (§5.1) - nondiagonal matrix element of Hamiltonian - electron-nuclear plus nuclear-nuclear interaction - sound velocity - vibronic interaction term - energy of external and relaxational perturbations (Ch. 4) - Cartesian coordinates - statistical sum - effective charge (§§ 1.5, 4.2) - magnetic field direction - Bohr magneton - anharmonicity correction (§6.3) - irreducible representation (term)

E

(), E

~,7], ? 7]

? K

Ar A Aj

f.L ,,/

w 1[1

l/I p, cf> p a­T

cp

Mathematical Notation xiii

- tunneling splitting (§4.1) - line of degenerate irreducible representations - correlation parameter in CJTE (§5.2) - anharmonicity coefficient (§6.3) - electron energy gap - mean value of random strain splitting - change in X - tunneling splitting - electronic energy - two components of doubly degenerate E representation - three components of triply degenerate T2 representation - asymmetry parameter (§5.3) - Corio lis constant (§4.2) - vibrational quantum number - dimensionless vibronic constant - spin-orbit interaction constant (§4.1) - power of atomic function exponent (§1.5) - correlation parameter in CJTE (§5.2) - quantum number of hyperfine interaction - incident irradiation frequency, electron transition

frequency - vibrational frequency - total wave function - electronic MO wave function - nuclear polar coordinates - light depolarization (§4.2) - ordering parameter (§5.2) - lifetime, relaxation time - angular wave function - approximate wave function for local state at AP

minimum (§3.2) - one-electron atomic or MO wave function - random strain direction (§4.1) - spherical top molecule rotational wave function

Abbreviations Commonly Used in Text

JT JTE PJTE CJTE CPJTE

- Jahn-Teller - Jahn-Teller effect - pseudo-Jahn-Teller effect - cooperative Jahn-Teller effect - cooperative pseudo-Jahn-Teller effect

xiv

AP VI ve ove MO HOMO LUMO IR UV

Mathematical Notation

- adiabatic potential(s) - vibronic interaction(s) - vibronic constant(s) - orbital vibronic constant(s) - molecular orbital(s) - highest occupied molecular orbital(s) - lowest unoccupied molecular orbital(s) - infrared - ultraviolet