Table Of ContentCHEMICAL
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
