Table of 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
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
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 Hückel-Möbius Concept 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 Moirés 408
References 410
9 Crystals 413
9.1 Basic Laws 417
9.2 The 32 Crystal Groups 423
9.3 Restrictions 424
9.4 The 230 Space Groups 432
9.4.1 Rock Salt and Diamond 438
9.5 Dense Packing 440
9.5.1 Sphere Packing 442
9.5.2 Icosahedral Packing 446
9.5.3 Connected Polyhedra 449
9.5.4 Atomic Sizes 453
9.6 Molecular Crystals 456
9.6.1 Geometrical Model 457
9.6.2 Densest Molecular Packing 466
9.6.3 Energy Calculations and Structure Predictions 470
9.6.4 Hypersymmetry 474
9.6.5 Crystal Field Effects 476
9.7 Beyond the Perfect System 483
9.8 Quasicrystals 489
9.9 Returning to Shapes 494
References 496
Epilogue 505
Other Titles by the Authors 507
Index 509
