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CHEMICAL APPLICATIONS —OF GROUP THEORY ¥. Albert Cotton Department of Chemistry, Tenas A&M University College Station, Yexas THIRD EDITION WILEY ‘A WILEV-INTERSCIENCE PUBLICATION ——JOHN-WILEY-& SONS: New York / Chichester / Brisbane / ‘Toronto / Singapore Capyrigh "1990 by F-Atbert Cotton Copyright 1965, 1971 by John Wiley & Sons, Ine AIL rights reserved: Published simullaneously in Canada. Repraviyction 01 transation of any pare af this work ‘eyeing Mar pera By NEeHON 17 oF TOR ETRE 19% United States Copyright Act without the permission ‘of dhe copyright over is unlawiul. Requests for permission o« further information should be wédeessed (0 the Peemissions Depactment, din Wiley & Sons. Te Library of Congres Cataloging in Publication Data Conon, F Albert (Frank Albee), 1230- (Chemies! applications of group theory J F. Allet Colton, —Jrd ed Pom. A Wiey-totersiones Publication.” Bibliography: p. ISBN 0471-51047 1. Molecular theery. 2. Group theory. 1. title ‘ODABT.CAS TH TE MaVZ2UIS 2220 op Printed in the United States of America 1918171615 14.3 PREFACE The aim, philosophy, and methodology of this edition are the same as they were in carlier editions, and much of the content remains the same. However, ‘there are changes and additions to warrant the publication of a new edition. “Phe most obvious change is the addition of Chapter 11, which deals with the symmetry properties of extended arrays—that is, crystals. My approach may not (or it may) please crystallographic purists; { did itot have them in mind as ] wrote it. Thad in mind severat generations of students J have taught 10-use crystallography as-a-chemist's tool. have tried to focus on-some of the bedrock fundamentals that | have so often noticed are not understood even by students who have learned to “do.a crystal structure.” Several of the chapters in Part I have been reworked in places and new ‘operators seem a bit more “uscr-friendly.” In Part II, besides adding Chapter 11, I have considerably changed Chapter 8 (0 place more emphasis on LCAO molecular orbitals and somewhat less ‘on hybridization. A section on the basis for electron counting rules for clusters Yas also been added. Finally, in response to the entreaties of many users, T have written an Answer Book forall of the exercises. This actually gives not only the “bottom line" imeach case, but an explanation af how to get these in many cases. The Answer Book will be available From the author at a nominal cost ‘am indebted to- Bruce Bursten. Richard-Adams, and Larry Falvello for helpful comments on several sections and to Mrs. Irene Casimiro for her E, ALgeRT Corron Coltege Savon, Tests Doverber 1989 PREFACE TO THE SECOND EDITION In the seven years since the first edition of Chemical Applications of Group Theory was written, | have continued to teach a course along the lines of this. book every other year. Steady, evolutionary change in the course finally led to a situation where the book and the course itself were no longer as closely related as they should be, Ihave, therefore, revised and augmented the book. “The new book has not lost the character of Rlavor of the old one—at least, T hope not. Ef aims to teach the use of symmetry arguments to the typical ‘experimental chemist in a way that he will fnd meaningful and useful. At the same time T have tried to avoid that excessive and unnecessary superficiality {arr unfortunate consequence of «misguided desire, evident in many books and articles on “theory for the chemist,” to shelter the poor chemist from ity attendant frestrations. Too brict or too superficial a tition in the use of symmnetty arguments is a waste of whatever time is devoted to it think that the subject needs and merits a student's attention for the equivalent of a onc: ___semester course. The student who masters this book will know what he is doing, why he is doing it, and Row to do it. The range of subject matter is that which, in my judgment, the great majority of organic, inorganic, and physical chemists are likely to encounter in their daily research activity This book differs from its ancestor in three ways. First, the amount of illustrative and exercise material has been enormously increased. Since the demand for a eacking textbook in this field far exceeds what ! had previously anticipated, | have tried now to equip the new edition with the pedagogic Paraphemalia- appropriate to moct this necd Second, the treatment of certain subjects has been changed—improved. £ hope-as a result of my continuing classroom experience. These improve- a vii PREFACE TO THE SECOND EDITION ments in_presentation-are-neither extensions in- coverage not-rigorizations; they are simply better ways of covering the ground. Such improvements will be found especially in Chapters 2 and 3, where the ideas of abstract group theory and the groups of highest symmetry are discussed Finally, new material and mote rigorous methods have been introduced in several places. The major examples ate (1) the explicit presentation of projection operators, and (2) an outline of the # and G matrix treatment of molecular vibrations, Although projection operators may seem a trifle for- Ddidding at the outsct, their potency and convenience and the nearly universal relevance of the symmetry-adapted linear combinations (SAI.C’s) of basis Tunctions which they genérate justily the effort of learning about them, The student who does so frees himself forever from the tyranny and uncertainty of “intuitive” and “'seat-of-the-pants” approaches. A new chapter which de- velops and illustrates projection operators has therefore becn added, and many changes in the subsequent exposition have necessarily been made. Because chemists seem to have become increasingly interested in employ- ing vibration spectra quantitatively--ur at least scmiquantitativelyto obtain information on bond strengths. it seemed mandatory to augment the previou matrix method for condueting vibrational analyses. The fact thal the conve- nient projection operator method for setting up symmetry coordinates has also been introduced makes inclusion of this material particularly feasible and desirable. In view of the enormous impact which symmetty-based rules concerning the stereochemistry of concerted addition und cyclization renctions (Wood- ward-Hoffmann rules} have had in cecent years a detuiled introduction to this subject has been added, In conclusion, itis my pleasant duty to thank a new generation of students M: questi have stimulated me to seek better ways to present the subject. {aim especially grateful to Professor David-L Weaver, Drs. Marie D. LaPrade, Batty G. DeBoer and James Smith and to Messrs. J. G. Bullitt. 1. R. Pipal, C. M. Lukehast and J_G_N¢ Jr_for their g " " proof. Finally. Miss Marilyn Milan, by the spccd and excellence of her typing, did much to lighten the task of preparing a new manuscript. F Ausert Corton Cambridge, Massochusetis May 1930 PREFACE TO THE FIRST EDITION This book is the oulgrowth of a one-semester course which has bee for sever institute of Technol graduate students in chemistry. ‘The treatment of the subject matter is un- pretentious in that I have not hesitated to be mathematically unsophisticated, ‘occasionally unrigotous, or somewhat prolix, where I felt thal this really helps fo make the subject more meaningful and comprehensible for the average student, By the average student, { mean one who does not aspire to be a thcorctician but who wants to have a fecl for the strategy used by theoreticians in treating problems in which symmetty properties are important and ta have arworking knowledge of the more common and-well-establishct! techniques. feel that the great power and beauty of symmetry methods, not te mention mat ic principles ia P y it very worthwhile for all chemists to be acquainted with the ba and_rnain- applications of zroup theoretical methods. Despite the fact that there seems to be a'yrowing desire among chemists at large to acquire this knowledge, itis still tue that on! Few oths than professional theoreticians, have donc so. The reason is not hard to discover. There is, so far as [ know, no book available which is not likely to strike some terror into the hearts of all but those with an innate love of apparently esoteric theory. It scemed 10 me that ideas of the sort developed in this buok would not soon be assimilated by a wide community of chemists vuntil they were presented in as unpreicntious and down-to-earth a manner as possible. That is what I have tried to do here. [ have attempted ta make This the kind of book which “one can read in bed without # pencil,” 35 my a x PREFACE TO THE FIRST EDITION colleague, John Waugh, once aptly described another textbook which has found wide favor because of its down-to-earth character.* Perhaps the book may also serve as a first introduction for students in- tending to do theoretical work, giving them some averall perspective before they aim for depth. fam most grateful for help I have received from many quarters in writing, this book. Over the years students in the couse have offered much valuable criticism and advice. In checking the final draft and the proofs 1 have had very weleome and efficient assistance from Dr. A. B. Blake and Messrs, R.C. Eider, T. B, Haas, and J.T. Mague. I, af course assume sole responsibil- hry for al remaiaing ertors. Finally, T wish to thank Mrs. Nancy Blake for ex- pert secretarial assistance, F ALRERT Corton Cambridge, Massechuste Sarary 1963 1 This statemeot is actually (and intentionally) aot applicable to pais of Chapter 3 where Lave mate no concessions ta the Teader wha refuses 1 inspect steric models in conjunction with sty DF the text, CONTENT! Part I Principles 1 INTRODUCTION DEFINITIONS AND THEOREMS OF GROUP THEORY 24 22 24 24 ‘The Defining Properties of a Group, 6 Some Examples of Groups, 8 Subgroups, {2 Classes, 13 Exercises, 16 3 MOLECULAR SYMMETRY AND THE SYMMETRY GROUPS WW {General Remarks, 1 3.2 Symmetry Elements and Operations, 18 3.3 Symmetry Planes and Reflections, 18 3.4 The Inversion Center, 22 3.5. Proper Axes and Proper Rotations, 22 3.6 Improper Axes and Improper Rotations. 27 3.7 Products af Symmetry Operations, 29 3.8 Equivalent Symmetry Elements and Equivalent Toms, 32 3.9 General Relations Among Symmetry Elements and Operations, 33 XG CONTENTS 3:40 Symmetsy Elements-ond Optical Isomerism, 34 3.11 The Symmetry Point Groups. 3 3.12 Symmetries with Multiple High-Order Axes, 44 3.13. Classes of Symmetry Operations, 50 3.14 _A Systematic Procedure for Symmetry Classification of Molecules, 54 3.15. Hlustrative Exumples, 36 Exercises, 61 4, REPRESENTATIONS OF GROUPS 68 4.1 Prefactory Comments on Matrices and Vectors, 68 4.2 Representations of Groups, 78 4.3 The “Great Orthogonalily Theorem” and ts Consequences, 8! 44 Character Tables, 90 4.5 Representations for Cyeiie Groups, 93 Exercises, 99 3.1 Wave Functions as Bases for Irreducihic Representations, 100 5.2 The Direct Product, 105 5.3. Identifying Nonzero Matrix Elements, 109 Exercises. 113 4 SYMMETRY-ADAPTED LINEAR COMBINATIONS ua 6.1 Introductory Remarks, 114 62 Derivation of Projection Operators, 115 6.3 Using Projection Operators to Construct SALCs, 119 Exercises, 129 Part ff. Applications 7. MOLECULAR ORBITAL THEORY AND ITS APPLICATIONS IN ORGANIC CHEMISTRY 3 71 General Principles, 133 7.2 Symmetry Factoring of Secular Equations, 140 3 —Cartvoryclic Systems, 142 7.4 Move General Cases of LCAO-MO x Bonding, 159 7.5 A Worked Example: Naphthalene, 172 7.6 Eleetronic Exeitations of Naphthalene: Selection Rules and Configaration Interaction, 178