Group Theoretical Methods and Applications to Molecules and Crystals

Table of Contents......Page 6
Preface......Page 16
List of symbols......Page 18
1.1 Vectors......Page 19
1.2 Linear transformations and matrices......Page 20
1.2.1 Functions of a matrix......Page 23
1.2.3 Direct products of matrices......Page 24
1.2.4 Direct sums of matrices......Page 25
1.3.1 Functions of a matrix (revisited)......Page 26
1.4 The characteristic equation of a matrix......Page 27
1.4.1 Diagonalizability and projection operators......Page 28
1.5 Unitary transformations and normal matrices......Page 30
1.5.1 Examples of normal matrices......Page 31
1.6 Exercises......Page 32
2.1 Involutional transformations......Page 35
2.2.1 The Dirac γ-matrices......Page 38
2.2.2 The Dirac plane waves......Page 39
2.2.3 The symmetric Dirac plane waves......Page 41
2.3 Intertwining matrices......Page 42
2.3.1 Idempotent matrices......Page 44
2.4 Matrix diagonalizations......Page 45
2.5 Basic properties of the characteristic transformation matrices......Page 48
2.6 Construction of a transformation matrix......Page 49
2.7 Illustrative examples......Page 52
3.1 Group axioms......Page 55
3.1.2 Examples of groups......Page 56
3.2 Group generators for a finite group......Page 58
3.2.1 Examples......Page 60
3.3 Subgroups and coset decompositions......Page 61
3.3.2 Lagrange\'s theorem......Page 62
3.4.1 Normalizers......Page 63
3.4.2 The centralizer......Page 64
3.4.4 Classes......Page 65
3.5 Isomorphism and homomorphism......Page 66
3.5.1 Examples......Page 67
3.5.2 Factor groups......Page 68
3.6 Direct products and semidirect products......Page 69
4.1.1 Basic properties......Page 71
4.1.2 The exponential form......Page 72
4.2.1 Basic properties......Page 73
4.2.2 Improper rotation......Page 74
4.2.3 The real orthogonal group O(n, r)......Page 75
4.3.1 Basic properties of rotation......Page 76
4.3.2 The conjugate rotations......Page 80
4.3.3 The Euler angles......Page 81
5.1 Introduction......Page 84
5.1.2 Multiaxial groups. The equivalence set of axes and axis-vectors......Page 85
5.1.3 Notations and the multiplication law for point operations......Page 86
5.2 The dihedral group D[sub(n)]......Page 91
5.3 Proper polyhedral groups P[sub(o)]......Page 93
5.3.1 Proper cubic groups, T and O......Page 95
5.3.2 Presentations of polyhedral groups......Page 97
5.3.3 Subgroups of proper point groups......Page 100
5.3.4 Theorems on the axis-vectors of proper point groups......Page 101
5.4 The Wyle theorem on proper point groups......Page 103
5.5.1 General discussion......Page 104
5.5.2 Presentations of improper point groups......Page 106
5.5.3 Subgroups of point groups of finite order......Page 108
5.6.1 General discussion......Page 109
5.6.2 Icosahedral group Y......Page 111
5.6.3 Buckminsterfullerene C₆₀ (buckyball)......Page 115
5.7 Coset enumeration......Page 116
6.1.1 Hilbert spaces......Page 120
6.1.2 Linear operators......Page 121
6.1.3 The matrix representative of an operator......Page 123
6.2.1 Homomorphism conditions......Page 125
6.2.2 The regular representation......Page 127
6.2.3 Irreducible representations......Page 128
6.3.1 The carrier space of a representation......Page 130
6.3.2 The natural basis of a matrix group......Page 132
6.4.1 General discussion......Page 133
6.4.2 The group of transformation operators......Page 135
6.4.3 Transformation of operators under G = {[omitted]}......Page 137
6.5 Schur\'s lemma and the orthogonality theorems on irreducible representations......Page 138
6.6.1 Orthogonality relations......Page 143
6.6.2 Frequencies and irreducibility criteria......Page 144
6.6.3 Group functions......Page 145
6.7 Irreducible representations of point groups......Page 146
6.7.2 The group D[sub(n)]......Page 147
6.7.3 The group T......Page 149
6.7.4 The group O......Page 151
6.8.1 The orthogonality of basis functions......Page 153
6.8.2 Application to perturbation theory......Page 154
6.9.1 Generating operators......Page 156
6.9.2 The projection operators......Page 159
6.10 Selection rules......Page 162
7.1 Introduction......Page 167
7.2.1 Equivalent point space S⁽ⁿ⁾......Page 168
7.2.2 The correspondence theorem on basis functions......Page 170
7.2.3 Mathematical properties of bases on S⁽ⁿ⁾......Page 171
7.2.4 Illustrative examples of the SALCs of equivalent scalars......Page 173
7.3.1 The general expression of SALCs......Page 176
7.3.2 Two-point bases and operator bases......Page 178
7.3.4 Alternative elementary bases......Page 179
7.3.5 Illustrative examples......Page 180
7.4 The general classification of SALCs......Page 184
7.4.1 D[sup((A))] SALCs from the equivalent orbitals ࢠ D[sup((A))] x Δ⁽ⁿ⁾......Page 185
7.5 Hybrid atomic orbitals......Page 188
7.5.1 The σ-bonding hybrid AOs......Page 189
7.5.2 General hybrid AOs......Page 191
7.6 Symmetry coordinates of molecular vibration based on the correspondence theorem......Page 192
7.6.1 External symmetry coordinates of vibration......Page 193
7.6.2 Internal vibrational coordinates......Page 195
7.6.3 Illustrative examples......Page 197
8.1 Subduced representations......Page 206
8.2 Induced representations......Page 207
8.2.3 The irreducibility condition for induced representations......Page 209
8.3.1 Conjugate representations......Page 211
8.3.2 Little groups and orbits......Page 212
8.3.3 Examples......Page 213
8.4.1 Solvable groups......Page 215
8.4.2 Induced representations for a solvable group......Page 216
8.4.3 Case I (reducible)......Page 217
8.4.4 Case II (irreducible)......Page 219
8.4.5 Examples......Page 220
8.5.1 Induction and subduction......Page 221
8.5.2 Small representations of a little group......Page 223
8.5.3 Induced representations from small representations......Page 224
9.1 Introduction......Page 227
9.1.1 Mixed continuous groups......Page 228
9.2 The Hurwitz integral......Page 229
9.3 Group generators and Lie algebra......Page 233
9.4 The connectedness of a continuous group and the multivalued representations......Page 237
10.1.1 The generators of SU(2)......Page 242
10.1.2 The parameter space Ω′ of SU(2)......Page 244
10.1.4 Quaternions......Page 246
10.2 The homomorphism between SU(2) and SO(3, r)......Page 247
10.3 Unirreps D[sup((j))](Ǝ) of the rotation group......Page 250
10.3.1 The homogeneity of D[sup((j))](S)......Page 251
10.3.3 The irreducibility of D[sup((j))](Ǝ))......Page 252
10.3.5 Orthogonality relations of D[sup((j))](Ǝ)......Page 253
10.3.6 The Hurwitz density function for SU(2)......Page 254
10.4.1 The generalized spinors......Page 255
10.4.2 The transformation of the total angular momentum eigenfunctions under the general rotation E;[sub(j)]......Page 256
10.4.3 The vector addition model......Page 258
10.4.4 The Clebsch-Gordan coefficients......Page 259
10.4.5 The angular momentum eigenfunctions for one electron......Page 263
11.1.1 The projective set of a point group......Page 265
11.1.2 The orthogonality relations for projective unirreps......Page 266
11.2.1 Defining relations of double point groups......Page 268
11.2.2 The structure of the double dihedral group [omitted]......Page 269
11.2.3 The structure of the double octahedral group O′......Page 271
11.3.1 The uniaxial group C[sub(ÝE;)]......Page 275
11.3.3 The group D[sub(ÝE)]......Page 276
11.3.4 The group D[sub(n)]......Page 278
11.3.5 The group O......Page 279
11.3.6 The tetrahedral group T......Page 281
12.1 Basic concepts......Page 284
12.2 Projective equivalence......Page 286
12.2.1 Standard factor systems......Page 287
12.2.2 Normalized factor systems......Page 288
12.2.3 Groups of factor systems and multiplicators......Page 289
12.2.4 Examples of projective representations......Page 290
12.3 The orthogonality theorem on projective irreps......Page 292
12.4.1 Covering groups......Page 294
12.4.2 Representation groups......Page 296
12.5.1 Representation groups of double proper point groups P′......Page 297
12.5.2 Representation groups of double rotation—inversion groups [omitted]......Page 299
12.6 Projective unirreps of double rotation—inversion point groups [omitted]......Page 301
12.6.1 The projective unirreps of [omitted]......Page 303
12.6.2 The projective unirreps of [omitted]......Page 304
12.6.3 The projective unirreps of [omitted]......Page 305
13.1 The Euclidean group in three dimensions E[sup((3))]......Page 307
13.2 Introduction to space groups......Page 311
13.3.1 Primitive bases......Page 313
13.3.2 The projection operators for a Bravais lattice......Page 316
13.3.3 Algebraic expressions for the Bravais lattices......Page 317
13.4.1 The hexagonal system H (D[sub(6i)])......Page 320
13.4.2 The tetragonal system Q (D[sub(4i)])......Page 321
13.4.3 The rhombohedral system RH (D[sub(3i)])......Page 324
13.4.4 The orthorhombic system O (D[sub(2i)])......Page 325
13.4.5 The cubic system C (O[sub(i)])......Page 326
13.4.6 The monoclinic system M (C[sub(2i))......Page 327
13.4.8 Remarks......Page 329
13.5 The 32 crystal classes and the lattice types......Page 331
13.6.1 Introduction......Page 333
13.6.2 The space groups of the class D₄......Page 334
13.7 Equivalence criteria for space groups......Page 336
13.8.1 Notations......Page 339
13.8.2 Defining relations of the crystal classes......Page 340
13.9 The space groups of the cubic system......Page 341
13.9.1 The class T......Page 344
13.9.2 The class T[sub(i)] (=T[sub(h)])......Page 345
13.9.3 The class O......Page 347
13.9.4 The class T[sub(p)] (=T[sub(d)])......Page 348
13.9.5 The class O[sub(i)] (=O[sub(h)])......Page 349
13.10 The space groups of the rhombohedral system......Page 350
13.10.3 The class D[sub(3)]......Page 351
13.10.4 The class C[sub(3v)]......Page 352
13.10.5 The class D[sub(3i) (=D[sub(3d)])......Page 353
13.11 The hierarchy of space groups in a crystal system......Page 354
13.11.3 The rhombohedral system......Page 355
13.12 Concluding remarks......Page 356
14.1 The unirreps of translation groups......Page 358
14.2.1 General discussion......Page 360
14.2.2 Reciprocal lattices of the cubic system......Page 362
14.2.3 The Miller indices......Page 363
14.2.4 The density of lattice points on a plane......Page 364
14.3.1 General construction of Brillouin zones......Page 365
14.3.2 The wave vector point groups......Page 366
14.3.3 The Brillouin zones of the cubic system......Page 367
14.4.1 The wave vector space groups C;[sub(k)]......Page 370
14.4.2 Small representations of C;[sub(k)] via the projective representations of G[sub(k)]......Page 372
14.4.3 Examples of the small representations of C;[sub(k)]......Page 374
14.5 The unirreps of the space groups......Page 376
14.5.1 The irreducible star......Page 377
14.5.2 A summary of the induced representation of the space groups......Page 378
15.1 Energy bands and the eigenfunctions of an electron in a crystal......Page 380
15.2 Energy bands and the eigenfunctions for the free-electron model in a crystal......Page 383
15.2.2 Example 1. A simple cubic lattice......Page 386
15.2.3 Example 2. The diamond crystal......Page 389
15.3.1 General discussion......Page 395
15.3.2 The small representations of the wave vector groups C;[sub(k)] based on the equivalent Bloch functions......Page 397
15.4.1 General discussion......Page 400
15.4.2 Construction of the symmetry coordinates of vibration......Page 403
16.1.1 General introduction......Page 409
16.1.2 The time correlation function......Page 411
16.1.3 Onsager\'s reciprocity relation for transport coefficients......Page 412
16.2.1 General introduction......Page 414
16.2.2 The properties of the time-reversal operator B8;......Page 418
16.2.3 The time-reversal symmetry of matrix elements of a physical quantity......Page 419
16.3.1 General discussion......Page 421
16.3.2 The classification of ferromagnetics and ferroelectrics......Page 425
16.4.1 General discussion......Page 427
16.4.2 Three types of co-unirreps......Page 428
16.5 Construction of the co-unirreps of anti-unitary point groups......Page 431
16.5.1 [omitted]......Page 432
16.5.2 [omitted] and [omitted]......Page 433
16.5.3 [omitted] and [omitted]......Page 435
16.5.4 The cubic groups......Page 436
16.6 Complex conjugate representations......Page 437
16.7 The orthogonality theorem on the co-unirreps......Page 440
16.8.2 Irreducibility criteria for co-unirreps......Page 442
16.8.3 The type criterion for a co-unirrep......Page 443
17.1 Introduction......Page 446
17.2 Anti-unitary space groups of the first kind......Page 448
17.2.1 The cubic system......Page 450
17.2.5 The orthorhombic system......Page 451
17.2.7 The triclinic system......Page 453
17.3 Anti-unitary space groups of the second kind......Page 456
17.3.2 Concluding remarks......Page 459
17.4 The type criteria for the co-unirreps of anti-unitary space groups and anti-unitary wave vector groups......Page 460
17.5 The representation groups of anti-unitary point groups......Page 463
17.6 The projective co-unirreps of anti-unitary point groups......Page 468
17.6.1 Examples for the construction of the projective co-unirreps of H[sup(z)]......Page 476
17.7 The co-unirreps of anti-unitary wave vector space groups......Page 478
17.7.1 Concluding remarks......Page 481
17.8.1 General discussion......Page 482
17.8.2 Transitions between states belonging to different co-unirreps......Page 483
17.8.3 Transitions between states belonging to the same co-unirrep......Page 485
17.8.4 Selection rules under a gray point group......Page 488
Appendix. Character tables for the crystal point groups......Page 490
References......Page 501
C......Page 505
G......Page 506
L......Page 507
P......Page 508
S......Page 509
W......Page 510