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2.1. Bilateral Symmetry Symmetry through the Eyes of a Chemist Magdolna Hargittai · Istvan´ Hargittai Symmetry through the Eyes of a Chemist Third Edition 123 Magdolna Hargittai Istvan´ Hargittai Budapest University of Technology and Economics P. O. Box 91 H-1521 Budapest Hungary [email protected] [email protected] ISBN: 978-1-4020-5627-7 e-ISBN: 978-1-4020-5628-4 DOI 10.1007/978-1-4020-5628-4 Library of Congress Control Number: 2008926635 c Springer Science+Business Media B.V. 2009 No part of this work 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, with the exception of any material supplied specifically for the purpose of being entered and executed on a computer system, for exclusive use by the purchaser of the work. Printed on acid-free paper 987654321 springer.com Preface It is gratifying to launch the third edition of our book. Its coming to life testifies about the task it has fulfilled in the service of the commu- nity of chemical research and learning. As we noted in the Prefaces to the first and second editions, our book surveys chemistry from the point of view of symmetry. We present many examples from chem- istry as well as from other fields to emphasize the unifying nature of the symmetry concept. Our aim has been to provide aesthetic plea- sure in addition to learning experience. In our first Preface we paid tribute to two books in particular from which we learned a great deal; they have influenced significantly our approach to the subject matter of our book. They are Weyl’s classic, Symmetry, and Shubnikov and Koptsik’s Symmetry in Science and Art. The structure of our book has not changed. Following the Intro- duction (Chapter 1), Chapter 2 presents the simplest symmetries using chemical and non-chemical examples. Molecular geometry is discussed in Chapter 3. The next four chapters present group- theoretical methods (Chapter 4) and, based on them, discussions of molecular vibrations (Chapter 5), electronic structures (Chapter 6), and chemical reactions (Chapter 7). For the last two chapters we return to a qualitative treatment and introduce space-group symme- tries (Chapter 8), concluding with crystal structures (Chapter 9). For the third edition we have further revised and streamlined our text and renewed the illustrative material. We have expanded the sections dealing with biopolymers and quasicrystals in particular. We have added an Epilogue. We dedicated the first edition to the memory of Jozsef´ Pollak´ (1901–1973), who was the stepfather of one of us (IH). The third edition we dedicate to our children and grandchildren, and to the memory of our parents. v vi Preface We note with pleasure our joint interest in both chemistry and symmetry for the past forty years that is behind this book. Our joint writing efforts have been an important facet of our professional as well as married life for these forty years. Budapest, January 2008 Magdolna Hargittai Istvan´ Hargittai Acknowledgments We thank our colleagues and our students who have helped the creation of this book over the years, especially at the Univer- sity of Connecticut (Storrs); Eotv¨ os¨ Lorand´ University (Budapest); University of Texas (Austin); University of Hawaii (Honolulu); Budapest University of Technology and Economics; University of North Carolina (Wilmington); and elsewhere. We also appreciate the helpful comments and suggestions from the reviewers of our book and from its users and readers from all over the world. We would like to express our gratitude to Professor Emil Molnar´ for helpful consultations. In connection with the preparation of the third edition of this book, we owe special thanks to Ms. Maria´ Kolonits, Ms. Judit Szucs,¨ and Mr. Zoltan´ Varga for their most careful technical help in many aspects of the work. For over forty years our research in structural chemistry has been generously funded by the Hungarian Academy of Sciences and lately also by the Hungarian National Scientific Research Funds. We are also grateful for support to Eotv¨ os¨ University and especially to the Budapest University of Technology and Economics. vii Contents 1 Introduction ........................................ 1 References.......................................... 20 2 Simple and Combined Symmetries .................... 25 2.1. Bilateral Symmetry . ........................... 25 2.2. Rotational Symmetry . ........................... 33 2.3. Combined Symmetries . .................... 37 2.3.1. A Rotation Axis with Intersecting Symmetry Planes.................................. 37 2.3.2. Snowflakes . ........................... 39 2.4. Inversion ...................................... 53 2.5. Singular Point and Translational Symmetry ......... 55 2.6. Polarity . ...................................... 57 2.7. Chirality . ...................................... 60 2.7.1. Asymmetry and Dissymmetry . ......... 66 2.7.2. Vital Importance . .................... 69 2.7.3. La coupe du roi ......................... 74 2.8. Polyhedra. ............................... 76 References.......................................... 91 3 Molecular Shape and Geometry ....................... 97 3.1. Isomers . ...................................... 98 3.2. Rotational Isomerism . ........................... 100 3.3. Symmetry Notations. ........................... 104 3.4. Establishing the Point Group . .................... 105 3.5. Examples ...................................... 107 3.6. Consequences of Substitution . .................... 115 3.7. Polyhedral Molecular Geometries . ................ 119 ix x Contents 3.7.1. Boron Hydride Cages .................... 123 3.7.2. Polycyclic Hydrocarbons . ................ 125 3.7.3. Structures with Central Atom . ......... 133 3.7.4. Regularities in Nonbonded Distances . ..... 136 3.7.5. The VSEPR Model . .................... 139 3.7.6. Consequences of Intramolecular Motion. 152 References.......................................... 161 4 Helpful Mathematical Tools .......................... 169 4.1. Groups . ...................................... 169 4.2. Matrices . ...................................... 176 4.3. Representation of Groups . .................... 183 4.4. The Character of a Representation . ................ 189 4.5. Character Tables and Properties of Irreducible Representations . ........................... 191 4.6. Antisymmetry . ............................... 197 4.7. Shortcut to Determine a Representation . ......... 204 4.8. Reducing a Representation . .................... 206 4.9. Auxiliaries . ............................... 208 4.9.1. Direct Product . .................... 209 4.9.2. Integrals of Product Functions . ......... 209 4.9.3. Projection Operator . .................... 211 4.10. Dynamic Properties . ........................... 212 4.11. Where Is Group Theory Applied? . ................ 213 References.......................................... 214 5 Molecular Vibrations ................................ 217 5.1. Normal Modes . ............................... 217 5.1.1. Their Number . .................... 218 5.1.2. Their Symmetry . .................... 220 5.1.3. Their Types . ........................... 224 5.2. Symmetry Coordinates. .................... 225 5.3. Selection Rules . ............................... 227 5.4. Examples ...................................... 229 References.......................................... 237 6 Electronic Structure of Atoms and Molecules ........... 239 6.1. One-Electron Wave Function . .................... 241 6.2. Many-Electron Atoms ........................... 249 Contents xi 6.3. Molecules . ............................... 252 6.3.1. Constructing Molecular Orbitals . ......... 252 6.3.2. Electronic States . .................... 261 6.3.3. Examples of MO Construction . ......... 263 6.4. Quantum Chemical Calculations . ................ 287 6.5. Influence of Environmental Symmetry . ......... 290 6.6. Jahn–Teller Effect . ........................... 294 References.......................................... 308 7 Chemical Reactions .................................. 313 7.1. Potential Energy Surface . .................... 315 7.1.1. Transition State, Transition Structure . ..... 316 7.1.2. Reaction Coordinate . .................... 319 7.1.3. Symmetry Rules for the Reaction Coordinate 320 7.2. Electronic Structure . ........................... 324 7.2.1. Changes During a Chemical Reaction . ..... 324 7.2.2. Frontier Orbitals: HOMO and LUMO . ..... 325 7.2.3. Conservation of Orbital Symmetry ......... 326 7.2.4. Analysis in Maximum Symmetry . ......... 327 7.3. Examples ...................................... 328 7.3.1. Cycloaddition . .................... 328 7.3.2. Intramolecular Cyclization ................ 343 7.3.3. Generalized Woodward–Hoffmann Rules . 350 7.4. Huckel–M¨ obiusConcept.........................¨ 350 7.5. Isolobal Analogy ............................... 356 References.......................................... 364 8 Space-Group Symmetries ............................ 371 8.1. Expanding to Infinity . ........................... 371 8.2. One-Sided Bands ............................... 375 8.3. Two-Sided Bands ............................... 378 8.4. Rods, Spirals, and Similarity Symmetry . ......... 381 8.5. Two-Dimensional Space Groups . ................ 395 8.5.1. Simple Networks . .................... 401 8.5.2. Side-Effects of Decorations . ......... 406 8.5.3. Moires.................................´
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