Index

Abbott's Flatland, 61 Antisymmetry (cont.) Accidental symmetry, 26 operations, 190ff Acetanilide, 62ff Archimedean/semiregular polyhedra, 87ff Acetic acid, 96 Architecture, 41, 42, 60, 87, 106, 373 Acetylene, 130 Aromaticity, 322 Adamantane, 129, 131 Artistic expressions, 16, 18, 27, 30, 31, 34, Adamantanes, joint, 131 35, 37, 42, 44, 65, 66, 67, 69, 83, 87, Aesthetic appeal/Beauty/Harmony/Perfection, 90, 99, 150, 191, 194, 240, 324, 349, 1, 13, 16 355,356, 369, 382, 384, 405,457 Alkali halide crystals, 423 Asparagine, 75 Alkali sulfate molecules, 136-137 Asymmetry, 70 Alkanes, 356 Atomic sizes, 420ff Aluminum trichloride, 441ff Aulonia hexagona, 5 Aluminosilicates, 89 Aurelia insulinda, 35 Amino acids, 65, 67, 72ff Average structures, 151-152 Ammonia, 143, 254ff Avogadro's law, 3, 4 Ammonia-aluminum trichloride, 117 Amorphous materials, 456 Bach, J. S., 65 Analogies, 5, 7-8, 60, 67, 74-75, 98-99, Bader, R. E W., 289, 301 114, 139ff, 150, 240, 326, 344, 348, Bands 360ff, 428 one-sided, 342ff Animals, 22ff, 33, 72, 106, 357, 358 scheme to establish the symmetry of, 345 double-headed, 31 seven symmetry classes of, 343 Anisole, 429-430 two-sided bands, 346ff Antarafacial approach, 312, 322 Barlow, W., 405, 422 Antimirror symmetry, 189ff Bart6k, B., 1, 25, 29 Antiprisms, 89, 91, 128 Basic laws of crystals, 385ff Antisymmetry, 70, 189ff, 204 Basis for a representation, 176, 205, 210 elements, 190ff Belousov, B. P, 355 463 464 Index

Belousov-Zhabotinsky reactions, 355-356 Character (cont.) Belov, N. V., 81 tables (cont.) Bentley, W. A., 46, 50, 53 C2~, 201,211,249 Benzene, 260ff C3, 259 Benzene derivatives 63, ll5ff, 443 C3v, 186, 202,256 Bernal, J. D., 447 C4~, 246 Bernstein, J., 445 C6, 264 Berry, R. S., 154 D~h, 223 Bersuker, I. B., 284 D2h, 308 Berthelot, M., 2 D3h , 281 Beryllium dichloride, 348 D6h, 263 Bethe, H., 274 Chiral catalysts, 75 Bickart, P., 13 Chiral separation, 75 Bicone, 43 Chirality, 65ff Bicyclo[3.3.3]undecane, 107 of drugs, 74 Bilateral symmetry, 21ff heterochiral, 66-67, 70, 76 Biological macromolecules, 352ff homochiral, 66-67, 70, 76 Biphenyl, 440 cis-3,4-Dimethylcyclobutene, 317 Bipyramids, 43 Class, 167 Blech, I., 453 order of, 168 Bochvar, D. A., 8 Combined symmetries, 38 Bond length variations, 276ff Comparison of structural parameters, 273 Bonding orbital, 241ff, 250 Concerted reactions, 289, 309 Boranes, 122ff Conformational polymorphism, 445-446 Born-Oppenheimer approximation, 239 Conical symmetry, 26 Boron hydrides, 122ff Conrotatory reaction, 316-317, 319-320, 322, Braga, D., 436-437 325-326 Bragg law, 374 Consequences Brain, 24 of complex formation, 116 Bravais lattices, 402-403 of substitution, 115 Brisse, E, 367, 369 Conservation laws of physics, 9 Broken symmetries, 11 Copernical crystallography, 456 Buckminsterfullerene, 3-4, 114, 121 Copernicus, 86 Buerger, M. J., 359 Copper, 29 Bullvalene, 152, 154 Correlation diagrams, 300-301,303ff, 314ff Bunn, C., 456-457 Correspondence diagrams, 301, 311 Butadiene, 313ff Coulson, C. A., 2, 16, 237,300 Covalent radii, 135 (~apek, K., 381-382 Coxeter, H. S. M., 11, 81, 118 Carbides, 340 Creativity, 2 Carbon dioxide, 221ff Crowe, D. W., 345,364 Carboranes, 122, 124 Crystal field effects, 440ff Cartesian displacement vectors, 178-179, 205, Crystal field theory, 273ff 211,218 Crystal polarity, 63 Caspar, D. L. D., 414 Crystal, representation of the 32 point groups, Cesium chloride, 423 391ff Character Crystal structure predictions, 433 of a representation, 182ff Crystal systems, 401 tables, 184ff Crystallite, 451 C2~, 188, 218 Crystalloid, 451 Index 465

Crystals, 62ff, 381ff Donor-acceptor complexes, 117, 444 basic laws of, 385ff Double helix, 73, 352, 354 Cu(II) compounds, 284 Diirer, A., 10, 13, 384 Cubane, 114, 125-126, 128 Dynamic properties, 203 Cube, 80-81, 84-85 Cuboctahedron, 417 Effel, J., 44 Curie, M., 72 Eiffel tower, 41 Curie, P., 71-72 Einstein, A., 9 Curtin, D. Y., 62ff Electrocyclic reaction, 316ff, 321-322, 326 Cycloaddition, 302ff, 322, 326 Electron diffraction patterns, 12 Cyclobutene, 317ff Electronic configuration, 238, 247 Cyclopentane, 151 Electronic states, 247ff Cylinder, 43 Electronic structure, 227ff Cylindrical symmetry, 25 Electronic wave function, 227ff angular, 229-230, 232-233, 237 da Vinci, Leonardo, 5-6, 83 one-electron, 229 Dalton, 387, 390, 408-409, 411 radial, 229, 237 Decorations, 338, 340, 341, 342, 344, 348, total, 234-235 360ff, 369ff, 397,419,421,425 Ellipsoid, 43 side effects of, 372-373 Energy calculations, 434ff Degas, E., 98-99 Energy integral, 272 Degeneracy, 230-231,247 Environmental symmetry, 273ff Degrees of symmetry, 13 Equality concept of M6bius, 12 Democritos, 436-435 Equilibrium structure, 151-152 Dendrimer chemistry, 47-48 Equivalent points, 59 Dendritic snow crystals, 45 Escher, M. C., 367, 369, 371,384, 427 Deoxyribonucleic acid, 352, 354 Ethane, 100, 130 Depero, L., 435 Ethane-l,2-dithiol, 442 Descartes, 50 Ethylene, 130, 302ff, 313ff Dewar, M. J. S., 322 dimerization, 302ff Diamond, 407-408 Euler's equation, 80, 81 Dibenzenechromium, 132-133 Eyring, H., 292, 293 Dicarba-closo-dodecaboranes, 156-157 1,2-Dichloro- 1,2-dibromoethane, 56 Faraday, M., 17 1,2-Dichloroethane, 100 Fedorov, E. S., 11, 385,405 1,2-Dichloroethylene, 100 Ferrocene, 132-133 Diels-Alder reaction, 299-300, 313ff Flowers, 25, 33, 36, 37, 39, 41, 49 1,2-Dihaloethanes, 446 Fukui, K., 3, 289, 299, 302 Diimide, 218ff Fuller, B., 3ff, 414, 436 Dimensionality, 60-61 Fullerene tubular structures, 379 Dimethyl sulfate, 143-142 Fullerenes, 121, 125 Dimolybdenum tetraacetate, 134 Functional purpose, 24 Dirac, P. A. M., 9, 11 Direct product, 200 Gal'pern, E. G., 8 Disrotatory reaction, 316-317, 320,322, 325- Galaxy, 30, 357 326 Gardner, M., 69, 452 Dissymmetry, 70ff Gas/solid structural differences, 440ff DNA, 73 Generalized crystallography, 446ff Dodecahedrane, 118, 125-126, 128 Generalized Woodward-Hoffmann rules, 322 Dodecahedron, 80ff Geodesic Dome, 4-5 466 Index

Giger, H., 379 International or Hermann-Mauguin notation, Glass, 448 101-102 Glassy material, 455 Intramolecular cyclization, 316ff Glide reflection, 339ff, 399 Intramolecular motion, 149 Glide symmetry planes, 399, 401 Intramolecular nonbonded 1,3 radii, 134ff Glyceraldehyde, 65, 96 Intramolecular vibrations, 151 Golden mean, 350 Inversion, 56ff, 186-187, 197 Golubitsky, M., 72 Iodine heptafluoride, 155 Goniometers, 387 Iron dendrites, 29 Graphite, 368 Isolobal analogy, 326ff Grepioni, E, 436, 437 Isomerism, 96ff Group multiplication tables, 166-167 Group theory, 163ff, 205 Jahn-Teller effect, 280ff Groups, 163ff cooperative, 285 Grfinbaum, B., 371 dynamic, 284 Guinier, A., 448-449 first-order, 298 second-order, 298 H~ickel, E., 40, 81-82 static, 285 Halevi, E A., 301 Jahn-Teller stabilization energy, 284 Hamilton operator, 228 Jahn-Teller transition, 285 Handedness, 69ff Jellyfish, 33, 35 Hauptman, H., 416 Jones, D. E. H., 5 Hatiy, 386, 388 Heartfield, J., 33, 35 Helical symmetry, 352ff Kaolin, 421 Helices, 69 Kekul6, 17 a-Helix, 353 Kelvin, Lord/Thomson, W. H., 66 Heraldic symmetry, 30 Kepler, J., 10, 22, 49, 81-82, 84, 387, 389, Hexaprismane, 127, 129 411 Hierarchy of isomers, 97 Kitaigorodskii, A. I., 427, 430ff, 440 Hodgkin, D., 408 Klug, A., 414 Hoffmann, R., 3,289, 299, 300ff Koestler, A., 2, 82 HOMO, 299ff, 313ff, 326 Koptsik, V. A., 373,406, 413 Homonuclear diatomics, 249, 250 K6r6s, E., 355 Honeycomb, 368 Kroto, H. W. 5-6 Houk, K. N., 292 Htickel system, 322-323, 325-326 La coupe du roi, 75, 77-78 HiJckel-M6bius concept, 322, 324, 326 reverse, 77, 79 Human body, 21, 23, 72 Law of rational indices, 386 Human face, 23, 27, 28 Law of rational intercepts, 389 Hund, F., 300 LCAO, 239 Hydrogen, 249-250 Left-and-right symmetry, 9, 21ff Hypersymmetry, 438-439 Lehn, J.-M., 429 Hypostrophene, 153-154 Levine, D., 453-454 Levine, R. D., 273 Iceane, 129-130 Lewis's cubical atom, 148-149 Icosahedral packing, 413ff Lewis's theory, 148 Icosahedron, 80-81, 84ff, 413ff Ligand field theory, 273ff Insulin, 408,410 Lipscomb model, 156 Internal coordinates, 205, 213, 219 Lipscomb, W. N., 156 Index 467

Literary examples, 2, 21-22, 47, 49, 57, 61, Molecular symmetries, 103ff 86, 381,384, 391,414, 448,456ff Molecular vibrations, 207ff Loeb, A. L., 13 Muetterties, E., 118-119 Logos, 32, 34, 36, 67, 105, 194 Mulliken, R. S., 300 Longuet-Higgins, H. C., 300 Mulliken symbols, 187-188 Luca Pacioli, 5-6, 83 Multiplicity, 59 Lucretius, 2 Music, 1, 25, 29, 30, 65,339 LUMO, 299ff, 313ff, 326 N-(p-chlorobenzylidine)-p-chloroaniline, 445 MacGillavry, C., 367, 369, 371,427 Nakaya, U., 50ff Machinery parts, 33, 35, 36, 37, 38, 358 Nautilus, 357 Mackay polyhedron, 415 Nonbonded distances, 133ff Mackay, A. L., 9, 12, 391,415-416,446ff Nonbonded interactions, 136 Mackay, R. H., 450 Noncrossing rule, 301 Mamedov, K., 371-372 Noncrystallographic symmetries, 8, 396, 446ff Mann, T., 21, 47 Normal modes of vibration, 207ff, 281ff Marginal stability model, 44 Nucleic acids, 72 Mate selection, 1 Nucleotides, 72-73 Mathematical groups, 163ff rules for, 164ff Octahedron, 80-81, 84-85,417ff Matisse, H., 150-151 Orbitats Matrices, 168ff antibonding, 241ff, 250 block diagonal, 180 atomic, 205,236, 239ff column, 170 frontier, 299 dimension of, 169 molecular, 205, 238ff square, 169 construction of, 238ff unit, 169 Orbital correspondence analysis, 310-311 Mazurs, E. G., 15 Orbital splitting, 274, 276ff, 282ff Mendeleev, D. I., 14 Osawa, E., 7 Mendeleev's periodic system, 14-15 Oscillating reactions, 355-356 Mermin, D., 456 Overlap integral, 241 Metal halides, 152 Metal-carbonyl clusters, 157-158 p-Acetanilide, 64 Metal-metal multiple bonds, 133-134 p-Chloroacetanilide, 62 Methane, 119-120 p-Methyl-N-(p-methylbenzylidene)aniline, 445 derivatives, 96, 115 Packing Michelangelo, 240 dense, 408ff Mirror-rotation symmetry, 58 densest molecular, 430ff Mislow, K., 13 dove-tail, 425 M6bius, equality concept of, 12 head-to-tail, 425, 426 M6bius system, 323ff Paddlanes, 133 Moir6s, 374ff [2.2.2.2]Padlane, 134 Molecular crystals, 424ff para-tert-butylcalix[4]arene, 429-430 geometrical model for, 424ff Paracrystal lattice, 449 Molecular geometry, 95ff Parallelepiped, 57 Molecular point group, scheme to establish, Pasteur, L., 68, 71, 73,453 102-103 Pasteur's models, 68 Molecular polarity, 63 Paul, I. C., 62ff Molecular recognition, 273 Pauli principle, 238 Molecular shape, 95ff Pauling, L., 14 468 Index

Pearson, R. G., 289, 301 Quasicrystals, 8-9, 452ff Penrose, R., 8,452 Quilts, 33, 35 Penrose tiling, 11, 12, 452-453 Pentagonal tiling, 10 Radial symmetry, 25-26 Pentaprismane, 127, 129 Radiolarians, 82, 111 Periodic tables, 14ff, 238 Reaction coordinate, 294ff, 317ff Periodicity, 60-61, 337 symmetry rules for, 295ff Permutational isomerism, 154 Reducing a representation, 198ff Phenol crystal, 438 Reduction formula, 199 Phyllotaxis, 349-350 Religion, 23, 27 Planar networks/lattices, 359ff, 397 Representation, 176ff scheme to establish symmetry of, 364 basis for, 176, 205,210 Plants, 25-26, 33, 36ff, 49, 72, 106, 109, 111, character of, 182ff 141, 349, 351,357, 428 irreducible, 182ff, 196, 210ff, 244ff, 272 Plato, 79 symbol of, 187 Platonic solids, 79-80, 82, 85, 124 reducible, 182ff, 196, 250, 252, 255-256, Point groups, 59ff, 95ff, 163ff, 210ff, 228- 260 229,235-236,240ff, 294, 296ff, 337,366 Reston, J., 57 Points-on-the-sphere model, 140 Rods, 347ff Polanyi, M., 293 Rotational antisymmetry, 193, 195 Polar axis, 61 Rotation axis, 32, 34, 38ff, 57-58, 84ff, 98, Polar crystals, 63-64 100ff, 163, 192, 343,345, 398 Polar plane, 61 Rotational isomerism, 98 Polarity, 61 Rotational symmetry, 32ff Polycyclic hydrocarbons, 118, 124ff Royal section, 75 Polyethylene, 352 Ruskin, J., 456 Polygons, 79-80, 396 regular, 79-80 Saddle point, 291-292, 296 Polyhedra, 79ff SALCs, 245, 271 connected, 416ff Salem, L., 290 regular convex, 79ff Salt columns, 25, 28 regular star, 86-87 Scattered leaf arrangement/phyllotaxis, 349- semiregular, 87ff 350 Polyhedral boron chemistry, 118-119 Schneer, C. J., 389 Polyhedral molecular geometries, 118 Schoenflies, 405 Polyhedrane molecules, 128 Schoenlties notation, 101-102, 163 Polymorphism, 445-446 Schrrdinger equation, 228, 238 Potassium tetrafluoroaluminate, 131-132 Scoresby, 50-51 Potential energy surface, 29Off Screw axis, 346ff, 398ff Prelog, V., 2, 73 Scutptt~res, 7, 18, 21, 22, 33, 39, 52, 54, 66, Primitive organisms, 40 68, 83, 108, 111, 114, 240, 293, 324, Prismane molecules, 129 355, 390, 425 Prisms, 43, 89, 91, 128 Selection rules, 216-217 Projection operator, 202 Semiregular/Archimedean polyhedra, 87ff Propylene, 290 SF6 derivatives, 115-116 Pseudorotation, 151, 155 Shechtman, D., 9, 12, 453 Pseudosymmetry, 290 Shephard, G. C., 371 Shubnikov, A. V., 10, 12, 70ff, 373,406, 413 Quantum chemical calculations, 272-273 Silatranes, 444 Quartz, 65, 71 Similarity symmetry, 347 Index 469

Similarity transformation, 167 Symmetry operators, 38 Singular point, 59 Symmetry planes/mirror planes, 23, 34, 38ff, Smalley, R. E., 5, 6 58, 65, 84ff, 163-164, 190, 302ff, 338, Snow crystals, classification of, 54, 55 343ff, 407-408, 431 Snowflakes, 43-47, 112 Symmetry transformations, 38, 58ff pentagonal, 450 Symmetry-adapted linear combinations, see Sodium chloride, 406ff, 422,441 SALCs Space groups, 61,337ff one dimensional, 61,342ff Temari, 7-8 two dimensional, 61, 359ff 1-tert-Butyl-4-methylbenzene, 63 the 230 three dimensional, 61,402ff Tetra-tert-butyltetrahedrane, 124, 126 Sphere, 86 Tetraarsene, 119-120 Sphere packing, 409ff Tetrahedrane, 124, 126, 128 Spiral symmetry, 352ff Tetrahedron, 80-81, 84-85, 417ff Spirals, 347ff, 357 Tetr alithiotetrahedrane, 132-133 Stalactites, 25, 28 Thomson, W. H./Lord Kelvin, 66 Stalagmites, 25, 28 trans-3,4-dimethylcyclobutene, 317 Stamps, 32, 66, 67, 81, 84, 108, 112, 134, Trans-trans-trans perhydro-triphenylene, 110 370, 382, 437 Transition state/transition structure, 292ff, 321 Starfish, 40 Translational symmetry, 59, 337 State correlation, 305ff Trees, 26, 38, 49 Statistical symmetry, 13 1,3,5-Triphenylbenzene, 425 Steinhardt, P., 452ff Triprismane, 127 Steno, 386 Twistane, 110 Stereographic projections, 394ff Stewart, I., 72 van der Waals radii, 135 Structural isomerism, 97 Vasarely, V., 83, 191 Subgroup, 167 Vectors, 170ff Sulfones, 136-137 Violation of parity, 73 Sulfur dichloride, 144 Viruses 8,414 Sulfur difluoride, 143-144 VSEPR model, 137ff Sulfuric acid, 136-137 Suprafacial approach, 313-314, 322 Water, 210ff, 252ff, 409 Supramolecules, 429 Wells, A. E, 410, 411-412 Swastika, 33 Weyl, H., 13, 30, 80 Symmetry center/center of symmetry/inversion Wickham, A., 448 center, 56ff, 406-407 Wigner-Witmer rules, 289 Symmetry classes, 40, 58ff, 69, 82ff, 100-101, Williams, I. H., 293-294 342ff, 359ff, 391,396 Witschi, W., 378-379 Symmetry coordinates, 214ff Woodward, R. B., 289, 299ff Symmetry elements, 38ff, 56, 70ff, 84ff, 101ff, Woodward-Hoffmann rules, 303ff, 323ff 186, 190-191,245,254, 301,303ff, 338ff, 391,399, 403ff Yang, C. N., 9 Symmetry groups, 39, 60, 62, 170, 173, 211, Yin yang, 32, 49, 193 241, 245-246, 274 Symmetry notations, 101 Zachariasen, W. H., 448-449 Symmetry of opposites, 189ff Zeolites, 89-90 Symmetry operation, 38-39, 70, 84, 119, Zhabotinsky, A. M., 355 163ff, 210, 214, 221,228, 232, 235- Zimmermann, H. E., 322 236, 245ff, 304 Zirconium borohydride, 120