xii
Contents
4
Representations .............................. 51
4.1 Symmetry-Adapted Linear Combinations of Hydrogen Orbitals in
A m m o n i a .............................. 52
4.2 Character Theorems ......................... 56
4 . 3 C h a r a c t e rT a b l e s........................... 62
4.4 Matrix Theorem ........................... 63
4 . 5 P r o j e c t i o nO p e r a t o r s......................... 64
4.6 Subduction and Induction . . . ................... 69
4.7 Application: The sp 3 H y b r i d i z a t i o no fC a r b o n ........... 76
4.8 Application: The Vibrations of UF 6 ................. 78
4.9 Application: Hückel Theory . . ................... 84
Cyclic Polyenes ........................... 85
Polyhedral Hückel Systems of Equivalent Atoms . ......... 91
Triphenylmethyl Radical and Hidden Symmetry . ......... 95
4 . 1 0P r o b l e m s............................... 99
References .................................101
5
What has Quantum Chemistry Got to Do with It? ...........103
5.1 The Prequantum Era .........................103
5.2 The Schrödinger Equation . . . ...................105
5.3 How to Structure a Degenerate Space ................107
5.4 The Molecular Symmetry Group ..................108
5 . 5 P r o b l e m s...............................112
References .................................112
6
Interactions ................................113
6 . 1 O v e r l a pI n t e g r a l s...........................114
6 . 2 T h eC o u p l i n go fR e p r e s e n t a t i o n s ..................115
6.3 Symmetry Properties of the Coupling Coefficients .........117
6.4 Product Symmetrization and the Pauli Exchange-Symmetry ....122
6.5 Matrix Elements and the Wigner–Eckart Theorem .........126
6.6 Application: The Jahn–Teller Effect .................128
6.7 Application: Pseudo-Jahn–Teller interactions . . . .........134
6.8 Application: Linear and Circular Dichroism . . . .........138
Linear Dichroism ..........................139
C i r c u l a rD i c h r o i s m..........................144
6.9 Induction Revisited: The Fibre Bundle ...............148
6.10 Application: Bonding Schemes for Polyhedra . . . .........150
Edge Bonding in Trivalent Polyhedra ................155
Frontier Orbitals in Leapfrog Fullerenes ..............156
6 . 1 1P r o b l e m s...............................159
References .................................160
7
Spherical Symmetry and Spins .....................163
7.1 The Spherical-Symmetry Group ...................163
7 . 2 A p p l i c a t i o n :C r y s t a l - F i e l dP o t e n t i a l s ................167
7.3 Interactions of a Two-Component Spinor ..............170
Contents
4
Representations .............................. 51
4.1 Symmetry-Adapted Linear Combinations of Hydrogen Orbitals in
A m m o n i a .............................. 52
4.2 Character Theorems ......................... 56
4 . 3 C h a r a c t e rT a b l e s........................... 62
4.4 Matrix Theorem ........................... 63
4 . 5 P r o j e c t i o nO p e r a t o r s......................... 64
4.6 Subduction and Induction . . . ................... 69
4.7 Application: The sp 3 H y b r i d i z a t i o no fC a r b o n ........... 76
4.8 Application: The Vibrations of UF 6 ................. 78
4.9 Application: Hückel Theory . . ................... 84
Cyclic Polyenes ........................... 85
Polyhedral Hückel Systems of Equivalent Atoms . ......... 91
Triphenylmethyl Radical and Hidden Symmetry . ......... 95
4 . 1 0P r o b l e m s............................... 99
References .................................101
5
What has Quantum Chemistry Got to Do with It? ...........103
5.1 The Prequantum Era .........................103
5.2 The Schrödinger Equation . . . ...................105
5.3 How to Structure a Degenerate Space ................107
5.4 The Molecular Symmetry Group ..................108
5 . 5 P r o b l e m s...............................112
References .................................112
6
Interactions ................................113
6 . 1 O v e r l a pI n t e g r a l s...........................114
6 . 2 T h eC o u p l i n go fR e p r e s e n t a t i o n s ..................115
6.3 Symmetry Properties of the Coupling Coefficients .........117
6.4 Product Symmetrization and the Pauli Exchange-Symmetry ....122
6.5 Matrix Elements and the Wigner–Eckart Theorem .........126
6.6 Application: The Jahn–Teller Effect .................128
6.7 Application: Pseudo-Jahn–Teller interactions . . . .........134
6.8 Application: Linear and Circular Dichroism . . . .........138
Linear Dichroism ..........................139
C i r c u l a rD i c h r o i s m..........................144
6.9 Induction Revisited: The Fibre Bundle ...............148
6.10 Application: Bonding Schemes for Polyhedra . . . .........150
Edge Bonding in Trivalent Polyhedra ................155
Frontier Orbitals in Leapfrog Fullerenes ..............156
6 . 1 1P r o b l e m s...............................159
References .................................160
7
Spherical Symmetry and Spins .....................163
7.1 The Spherical-Symmetry Group ...................163
7 . 2 A p p l i c a t i o n :C r y s t a l - F i e l dP o t e n t i a l s ................167
7.3 Interactions of a Two-Component Spinor ..............170