Preface ........................................................ xi
1 Introductory example: Squarene ............................... 1
1.1 In-plane molecular vibrations of squarene ............... 1
1.2 Reducible and irreducible representations of a group ... 12
1.3 Eigenvalues and eigenvectors ........................... 27
1.4 Construction of the force-constant matrix from the
eigenvalues ............................................ 30
1.5 Optical properties ..................................... 31
References .................................................. 34
Exercises ................................................... 35
2 Molecular vibrations of isotopically substituted АВ2
molecules ................................................... 39
2.1 Step 1: Identify the point group and its symmetry
operations ............................................. 39
2.2 Step 2: Specify the coordinate system and the basis
functions .............................................. 39
2.3 Step 3: Determine the effects of the symmetry
operations on the basis functions ...................... 41
2.4 Step 4: Construct the matrix representations for each
element of the group using the basis functions ......... 41
2.5 Step 5: Determine the number and types of irreducible
representations ........................................ 42
2.6 Step 6: Analyze the information contained in the
decompositions ......................................... 42
2.7 Step 7: Generate the symmetry functions ................ 43
2.8 Step 8: Diagonalize the matrix eigenvalue equation ..... 50
2.9 Constructing the force-constant matrix ................. 50
2.10 Green's function theory of isotopic molecular
vibrations ............................................. 52
2.11 Results for isotopically substituted forms of H2O ...... 60
References ............................................. 62
Exercises .............................................. 62
3 Spherical symmetry and the full rotation group .............. 66
3.1 Hydrogen-like orbitals ................................. 66
3.2 Representations of the full rotation group ............. 68
3.3 The character of a rotation ............................ 72
3.4 Decomposition of in a non-spherical environment ........ 75
3.5 Direct-product groups and representations .............. 76
3.6 General properties of D(l) direct-product groups and
representations ........................................ 79
3.7 Selection rules for matrix elements .................... 83
3.8 General representations of the full rotation group ..... 85
References .................................................. 88
Exercises ................................................... 88
4 Crystal-field theory ........................................ 90
4.1 Splitting of d-orbital degeneracy by a crystal field ... 90
4.2 Multi-electron systems ................................. 95
4.3 Jahn-Teller effects ................................... 116
References ................................................. 119
Exercises .................................................. 119
5 Electron spin and angular momentum ......................... 123
5.1 Pauli spin matrices ................................... 123
5.2 Measurement of spin ................................... 126
5.3 Irreducible representations of half-integer angular
momentum .............................................. 127
5.4 Multi-electron spin-orbital states .................... 129
5.5 The L-S-coupling scheme ............................... 130
5.6 Generating angular-momentum eigenstates ............... 132
5.7 Spin-orbit interaction ................................ 138
5.8 Crystal double groups ................................. 150
5.9 The Zeeman effect (weak-magnetic-field case) .......... 153
References ................................................. 155
Exercises .................................................. 156
6 Molecular electronic structure: The LCAO model ............. 158
6.1 N-electron systems .................................... 158
6.2 Empirical LCAO models ................................. 162
6.3 Parameterized LCAO models ............................. 163
6.4 An example: The electronic structure of squarene ...... 168
6.5 The electronic structure of H2O ....................... 182
References ............................................ 188
Exercises ............................................. 189
7 Electronic states of diatomic molecules .................... 193
7.1 Bonding and antibonding states: Symmetry functions .... 193
7.2 The "building-up" of molecular orbitals for diatomic
molecules ............................................. 198
7.3 Heteronuclear diatomic molecules ...................... 206
Exercises ............................................. 209
8 Transition-metal complexes ................................. 211
8.1 An octahedral complex ................................. 211
8.2 A tetrahedral complex ................................. 227
References ................................................. 237
Exercises .................................................. 237
9 Space groups and crystalline solids ........................ 239
9.1 Definitions ........................................... 239
9.2 Space groups .......................................... 244
9.3 The reciprocal lattice ................................ 246
9.4 Brillouin zones ....................................... 247
9.5 Bloch waves and symmorphic groups ..................... 249
9.6 Point-group symmetry of Bloch waves ................... 252
9.7 The space group of the k-vector, gks ................... 258
9.8 Irreducible representations of gks ..................... 259
9.9 Compatibility of the irreducible representations of
gk .................................................... 260
9.10 Energy bands in the plane-wave approximation .......... 265
References ................................................. 276
Exercises .................................................. 276
10 Application of space-group theory: Energy bands for the
perovskite structure ....................................... 280
10.1 The structure of the ABO3 perovskites ................. 280
10.2 Tight-binding wavefunctions ........................... 282
10.3 The group of the wavevector, gk ....................... 283
10.4 Irreducible representations for the perovskite
energy bands .......................................... 284
10.5 LCAO energies for arbitrary k ......................... 298
10.6 Characteristics of the perovskite bands ............... 300
References ................................................. 301
Exercises .................................................. 302
11 Applications of space-group theory: Lattice vibrations ..... 304
11.1 Eigenvalue equations for lattice vibrations ........... 305
11.2 Acoustic-phonon branches .............................. 309
11.3 Optical branches: Two atoms per unit cell ............. 314
11.4 Lattice vibrations for the perovskite structure ....... 320
11.5 Localized vibrations .................................. 327
References ................................................. 334
Exercises .................................................. 334
12 Time reversal and magnetic groups .......................... 337
12.1 Time reversal in quantum mechanics .................... 337
12.2 The effect of T on an electron wavefunction ........... 340
12.3 Time reversal with an external field .................. 341
12.4 Time-reversal degeneracy and energy bands ............. 342
12.5 Magnetic crystal groups ............................... 346
12.6 Co-representations for groups with time-reversal
operators ............................................. 350
12.7 Degeneracies due to time-reversal symmetry ............ 357
References ................................................. 361
Exercises .................................................. 361
13 Graphene ................................................... 363
13.1 Graphene structure and energy bands ................... 363
13.2 The analogy with the Dirac relativistic theory for
massless particles .................................... 368
13.3 Graphene lattice vibrations ........................... 369
References ................................................. 381
Exercises .................................................. 381
14 Carbon nanotubes ........................................ 383
14.1 A description of carbon nanotubes ..................... 384
14.2 Group theory of nanotubes ............................. 386
14.3 One-dimensional nanotube energy bands ................. 393
14.4 Metallic and semiconducting nanotubes ................. 401
14.5 The nanotube density of states ........................ 403
14.6 Curvature and energy gaps ............................. 406
References ................................................. 407
Exercises .................................................. 407
Appendix A Vectors and matrices ........................... 410
Appendix В Basics of point-group theory ................... 415
Appendix С Character tables for point groups .............. 430
Appendix D Tensors, vectors, and equivalent electrons ..... 442
Appendix E The octahedral group, О and Оh ................. 449
Appendix F The tetrahedral group, Td ...................... 455
Appendix G Identifying point groups ....................... 462
Index ......................................................... 465
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