Symmetry through the Eyes of a Chemist Second Edition Symmetry through the Eyes of a Chemist Second Edition Istv~.n Hargittai Budapest Technical University and Hungarian Academy of Sciences Budapest, Hungary Magdolna Hargittai Hungarian Academy of Sciences Budapest, Hungary Plenum Press • New York and London Library oF Congress Cataloging-In-Publication Data Harg~ttal, Istv~n. Symmetry through the eyes oF a chemist / Istv~n Harglttai, Magdolna Harglttai. -- 2nd ed. p. cm. Includes bibliographical references and Indexes. ISBN 0-306-44851-3 (hc>, -- ISBN 0-306-44852-1 (pbk.) 1. Molecular theory. 2. Symmetry (Physics) I. Hargtttalo Magdo|na. II, Title. QD461,H268 1995 541.2'2--dc20 95-30533 CIP ISBN 0-306-44851-3 (Hardbound) ISBN 0-306-44852-1 (Paperback) © 1995 Plenum Press, New York A Division of Plenum Publishing Corporation 233 Spring Street, New York, N. Y. 10013 10987654321 All rights reserved 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 othenNise, without written permission from the Publisher The first edition of this book was published by VCH, Weinheim, Germany, 1986 Printed in the United States of America Preface to the Second Edition We have been gratified by the warm reception of our book, by reviewers, colleagues, and students alike. Our interest in the subject matter of this book has not decreased since its first appearance; on the contrary. The first and second editions envelop eight other symmetry-related books in the creation of which we have participated: I. Hargittai (ed.), Symmetry: Unifying Human Understanding, Pergamon Press, New York, 1986. I. Hargittai and B. K. Vainshtein (eds.), Crystal Symmetries. Shubnikov Centennial Papers, Pergamon Press, Oxford, 1988. M. Hargittai and I. Hargittai, Fedezziikf6l a szimmetri6t! (Discover Symme- try, in Hungarian), Tank6nyvkiad6, Budapest, 1989. I. Hargittai (ed.), Symmetry 2: Unifying Human Understanding, Pergamon Press, Oxford, 1989. I. Hargittai (ed.), Quasicrystals, Networks, and Molecules of Fivefold Symme- try, VCH, New York, 1990. I. Hargittai (ed.), Fivefold Symmetry, World Scientific, Singapore, 1992. I. Hargittai and C. A. Pickover (eds.), Spiral Symmetry, World Scientific, Singapore, 1992. I. Hargittai and M. Hargittai, Symmetry: A Unifying Concept, Shelter Publica- tions, Bolinas, California, 1994. We have also pursued our molecular structure research, and some books have appeared related to these activities: vi Preface to the Second Edition I. Hargittai and M. Hargittai (eds.), Stereochemical Applications of Gas-Phase Electron Diffraction, Parts A and B, VCH, New York, 1988. R. J. Gillespie and I. Hargittai, The VSEPR Model of Molecular Geometry, Allyn and Bacon, Boston, 1991. A. Domenicano and I. Hargittai (eds.), Accurate Molecular Structures, Oxford University Press, Oxford, 1992. M. Hargittai and I. Hargittai (eds.), Advances in Molecular Structure Research, Vol. 1, JAI Press, Greenwich, Connecticut, 1995. For this second edition, we have revised both text and illustrative material. It gives us pleasure to acknowledge the kind assistance from several colleagues, including Lawrence F. Dahl (University of Wisconsin, Madison), Avitam Halevi (Technion, Haifa), Lionel Salem (University of Paris, Orsay), P6ter Surj~n (E6tv6s University, Budapest), and Richard Wiegandt (Mathematical Research Institute, Budapest). We are grateful to Istv~n F~ibri and Judit Szfics for their dedicated technical assistance. For over a quarter of a century, our research work in structural chemistry has been supported by the Hungarian Academy of Sciences. Scientific meetings and lecture invitations have taken us to many places, and these travels have helped us build up the illustrative material of this book. We have enjoyed the friendship and enthusiastic interest of our colleagues all over the world. Istv(tn and Magdolna Hargittai Budapest, Hungary From the Preface to the First Edition This book surveys chemistry from the point of view of symmetry. We present many examples from chemistry as well as from other fields, in order to emphasize the unifying nature of the concepts of symmetry. We hope that all those chemists, both academic and industrial, who take broader perspectives will benefit from our work. We hope that readers will share some of the excitement, aesthetic pleasure, and learning that we have experienced during its preparation. In the course of our work we have become ever more conscious of the diverse manifestations of symmetry in chemistry, and in the world at large. We believe that consciousness will also develop in the reader. Despite its breadth, our book was not intended to be comprehensive or to be a specialized treatise in any specific area. We would like especially to note here two classics in the literature of symmetry which have strongly influenced us: Weyl's Symmetry and Shubnikov and Koptsik's Symmetry in Science and Art. Our book has a simple structure. After the introduction (Chapter 1), the simplest symmetries are presented using chemical and nonchemical examples (Chapter 2). Molecular geometry is then discussed in qualitative terms (Chap- ter 3). Group-theoretical methods (Chapter 4) are applied in an introductory manner to the symmetries of molecular vibrations (Chapter 5), electronic structure (Chapter 6), and chemical reactions (Chapter 7). These chapters are followed by a descriptive discussion of space-group symmetries (Chapter 8), including the symmetry of crystals (Chapter 9). The general perception of symmetry that most people have is sufficient vii viii From the Preface to the First Edition for reading Chapters 1, 2, 3, 8, and 9. However, in order to appreciate Chapters 5, 6, and 7, the introduction to group theory given in Chapter 4 is necessary. Chapter 4 also deals with antisymmetry. We express our thanks to those distinguished colleagues who have read one or more chapters and helped us with their criticism and suggestions. They include James M. Bobbit (University of Connecticut), Russel A. Bonham (Indiana University), Arthur Greenberg (New Jersey Institute of Technology), Joel E Liebman (University of Maryland), Alan L. Mackay (University of London), Alan P. Marchand (North Texas State University), Kurt Mislow (Princeton University), Ian C. Paul (University of Illinois), P6ter Pulay (Uni- versity of Arkansas), Robert Schor (University of Connecticut), and Gy6rgy Vars~inyi (Budapest Technical University). We thank those authors and copyright owners who gave us permission to use their illustrations in our book. We made all efforts to identify the sources of all illustrative materials and regret if, inadvertently, we missed anything in doing so. Most of the final version was compiled during our stay at the University of Connecticut, 1983-85, and we greatly benefited from the school's creative and inspiring atmosphere. We express our gratitude to Dean Julius A. Elias, to IMS Director Leonid V. Azaroff, and to our colleagues in the departments of Chemistry and Physics. We dedicate this book to the memory of J6zsef Poll~k (1901-1973), who was the stepfather of one of us (IH). He was an early and decisive influence in stimulating the interests which eventually led to the creation of this book. Istv~n and Magdolna Hargittai Storrs, Connecticut, and Budapest, Hungary Contents 1. Introduction .......................................... 1 References ............................................ 18 . Simple and Combined Symmetries ....................... 21 2.1. Bilateral Symmetry ............................... 21 2.2. Rotational Symmetry .............................. 32 2.3. Combined Symmetries ............................. 38 2.3.1. A Rotation Axis with Intersecting Symmetry Planes ................................... 38 2.3.2. A Rotation Axis with Intersecting Symmetry Planes and a Perpendicular Symmetry Plane .... 43 2.3.3. Snowflakes ............................... 43 2.4. Inversion ........................................ 56 2.5. Singular Point and Translational Symmetry ............ 59 2.6. Polarity ......................................... 61 2.7. Chirality ........................................ 65 2.7.1. Asymmetry and Dissymmetry ............... 70 2.7.2. Relevance to Origin of Life .................. 72 2.7.3. La coupe du roi ........................... 75 2.8. Polyhedra ....................................... 79 References ............................................ 91 ix X Contents 3. Molecules: Shape and Geometry ......................... 95 3.1. Formulas, Isomers ................................ 96 3.2. Rotational Isomerism .............................. 98 3.3. Symmetry Notations .............................. 101 3.4. Establishing the Molecular Point Group ............... 102 3.5. Examples ....................................... 103 3.6. Consequences of Substitution ....................... 115 3.7. Polyhedral Molecular Geometries .................... 118 3.7.1. Boron Hydride Cages ...................... 122 3.7.2. Polycyclic Hydrocarbons .................... 124 3.7.3. Structures with Central Atom ................. 130 3.7.4. Regularities in Nonbonded distances ........... 133 3.7.5. The VSEPR Model ......................... 137 3.7.6. Consequences of Intramolecular Motion ........ 149 References ............................................ 158 . Helpful Mathematical Tools ............................. 163 4.1. Groups ......................................... 163 4.2. Matrices .......................................