Page 9 - Valence Bond Methods. Theory and Applications

Contents
viii
2.8.1
2.8.2
42
Effect of eliminating various structures
44
2.8.3
Accuracy of full MCVB calculation with 10 AOs
2.8.4 Two different AO bases 40
44
Accuracy of full MCVB calculation with 28 AOs
2.8.5 EGSO weights for 10 and 28 AO orthogonalized bases 45
3H 2 and delocalized orbitals 47
3.1 Orthogonalized AOs 47
3.2 Optimal delocalized orbitals 49
3.2.1 The method of Coulson and Fisher[15] 49
3.2.2 Complementary orbitals 49
3.2.3 Unsymmetric orbitals 51
4 Three electrons in doublet states 53
4.1 Spin eigenfunctions 53
4.2 Requirements of spatial functions 55
4.3 Orbital approximation 58
5 Advanced methods for larger molecules 63
5.1 Permutations 64
5.2 Group algebras 66
5.3 Some general results for finite groups 68
5.3.1 Irreducible matrix representations 68
5.3.2 Bases for group algebras 69
5.4 Algebras of symmetric groups 70
5.4.1 The unitarity of permutations 70
5.4.2 Partitions 70
5.4.3 Young tableaux and N and P operators 71
5.4.4 Standard tableaux 72
5.4.5 The linear independence of N i P i and P i N i 75
5.4.6 Von Neumann’s theorem 76
5.4.7 Two Hermitian idempotents of the group algebra 76
5.4.8 A matrix basis for group algebras of symmetric groups 77
5.4.9 Sandwich representations 79
5.4.10 Group algebraic representation of the antisymmetrizer 80
5.5 Antisymmetric eigenfunctions of the spin 81
5.5.1 Two simple eigenfunctions of the spin 81
5.5.2 The function 84
5.5.3 The independent functions from an orbital product 85
5.5.4 Two simple sorts of VB functions 87
5.5.5 Transformations between standard tableaux and HLSP
functions 88
5.5.6 Representing θ NPN as a functional determinant 91