Valence Bond Methods

Theory and Applications

By: Gordon A. Gallup

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Hardcover | 22 October 2002

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256 Pages

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Valence bond theory is one of two commonly used methods in molecular quantum mechanics, the other is molecular orbital theory. This book focuses on the first of these methods, ab initio valence bond theory. The book is split into two parts. Part I gives simple examples of two-electron calculations and the necessary theory to extend these to larger systems. Part II gives a set of case studies of related molecule sets designed to show the nature of the valence bond description of molecular structure. It also highlights the stability of this description to varying basis sets. There are references to the CRUNCH computer program for molecular structure calculations which is currently available in the public domain. The book will be of primary interest to researchers and students working on electronic theory and computation in chemistry and chemical physics.

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Preface	p. xiii
List of abbreviations	p. xv
Theory and two-electron systems
Introduction	p. 3
History	p. 3
Mathematical background	p. 4
Schrodinger's equation	p. 5
The variation theorem	p. 9
General variation functions	p. 9
Linear variation functions	p. 9
A 2 x 2 generalized eigenvalue problem	p. 14
Weights of nonorthogonal functions	p. 16
Weights without orthogonalization	p. 18
Weights requiring orthogonalization	p. 19
H[subscript 2] and localized orbitals	p. 23
The separation of spin and space variables	p. 23
The spin functions	p. 23
The spatial functions	p. 24
The AO approximation	p. 24
Accuracy of the Heitler-London function	p. 27
Extensions to the simple Heitler-London treatment	p. 27
Why is the H[subscript 2] molecule stable?	p. 31
Electrostatic interactions	p. 32
Kinetic energy effects	p. 36
Electron correlation	p. 38
Gaussian AO bases	p. 38
A full MCVB calculation	p. 38
Two different AO bases	p. 40
Effect of eliminating various structures	p. 42
Accuracy of full MCVB calculation with 10 AOs	p. 44
Accuracy of full MCVB calculation with 28 AOs	p. 44
EGSO weights for 10 and 28 AO orthogonalized bases	p. 45
H[subscript 2] and delocalized orbitals	p. 47
Orthogonalized AOs	p. 47
Optimal delocalized orbitals	p. 49
The method of Coulson and Fisher[15]	p. 49
Complementary orbitals	p. 49
Unsymmetric orbitals	p. 51
Three electrons in doublet states	p. 53
Spin eigenfunctions	p. 53
Requirements of spatial functions	p. 55
Orbital approximation	p. 58
Advanced methods for larger molecules	p. 63
Permutations	p. 64
Group algebras	p. 66
Some general results for finite groups	p. 68
Irreducible matrix representations	p. 68
Bases for group algebras	p. 69
Algebras of symmetric groups	p. 70
The unitarity of permutations	p. 70
Partitions	p. 70
Young tableaux and N and P operators	p. 71
Standard tableaux	p. 72
The linear independence of N[subscript i]P[subscript i] and P[subscript i]N[subscript i]	p. 75
Von Neumann's theorem	p. 76
Two Hermitian idempotents of the group algebra	p. 76
A matrix basis for group algebras of symmetric groups	p. 77
Sandwich representations	p. 79
Group algebraic representation of the antisymmetrizer	p. 80
Antisymmetric eigenfunctions of the spin	p. 81
Two simple eigenfunctions of the spin	p. 81
The [Xi] function	p. 84
The independent functions from an orbital product	p. 85
Two simple sorts of VB functions	p. 87
Transformations between standard tableaux and HLSP functions	p. 88
Representing [theta] NPN [Xi] as a functional determinant	p. 91
Spatial symmetry	p. 97
The AO basis	p. 98
Bases for spatial group algebras	p. 98
Constellations and configurations	p. 99
Example 1. H[subscript 2]O	p. 100
Example 2. NH[subscript 3]	p. 102
Example 3. The [pi] system of benzene	p. 105
Varieties of VB treatments	p. 107
Local orbitals	p. 107
Nonlocal orbitals	p. 108
The physics of ionic structures	p. 111
A silly two-electron example	p. 111
Ionic structures and the electric moment of LiH	p. 113
Covalent and ionic curve crossings in LiF	p. 115
Examples and interpretations
Selection of structures and arrangement of bases	p. 121
The AO bases	p. 121
Structure selection	p. 123
N[subscript 2] and an STO3G basis	p. 123
N[subscript 2] and a 6-31G basis	p. 123
N[subscript 2] and a 6-31G* basis	p. 124
Planar aromatic and [pi] systems	p. 124
Four simple three-electron systems	p. 125
The allyl radical	p. 125
MCVB treatment	p. 126
Example of transformation to HLSP functions	p. 129
SCVB treatment with corresponding orbitals	p. 132
The He[superscript + subscript 2] ion	p. 134
MCVB calculation	p. 134
SCVB with corresponding orbitals	p. 135
The valence orbitals of the BeH molecule	p. 136
Full MCVB treatment	p. 137
An SCVB treatment	p. 139
The Li atom	p. 141
SCVB treatment	p. 142
MCVB treatment	p. 144
Second row homonuclear diatomics	p. 145
Atomic properties	p. 145
Arrangement of bases and quantitative results	p. 146
Qualitative discussion	p. 148
B[subscript 2]	p. 149
C[subscript 2]	p. 152
N[subscript 2]	p. 154
O[subscript 2]	p. 157
F[subscript 2]	p. 160
General conclusions	p. 161
Second row heteronuclear diatomics	p. 162
An STO3G AO basis	p. 162
N[subscript 2]	p. 164
CO	p. 166
BF	p. 168
BeNe	p. 171
Quantitative results from a 6-31G* basis	p. 173
Dipole moments of CO, BF, and BeNe	p. 174
Results for 6-31G* basis	p. 174
Difficulties with the STO3G basis	p. 175
Methane, ethane and hybridization	p. 177
CH, CH[subscript 2], CH[subscript 3], and CH[subscript 4]	p. 177
STO3G basis	p. 177
6-31G* basis	p. 186
Ethane	p. 187
Conclusions	p. 189
Rings of hydrogen atoms	p. 191
Basis set	p. 192
Energy surfaces	p. 192
Aromatic compounds	p. 197
STO3G calculation	p. 198
SCVB treatment of [pi] system	p. 200
Comparison with linear 1,3,5-hexatriene	p. 203
The 6-31G* basis	p. 205
Comparison with cyclobutadiene	p. 208
The resonance energy of benzene	p. 208
Naphthalene with an STO3G basis	p. 211
MCVB treatment	p. 211
The MOCI treatment	p. 212
Conclusions	p. 213
Interaction of molecular fragments	p. 214
Methylene, ethylene, and cyclopropane	p. 214
The methylene biradical	p. 215
Ethylene	p. 215
Cyclopropane with a 6-31G* basis	p. 218
Cyclopropane with an STO-3G basis	p. 224
Formaldehyde, H[subscript 2]CO	p. 225
The least motion path	p. 226
The true saddle point	p. 227
Wave functions during separation	p. 228
References	p. 231
Index	p. 235
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Valence Bond Methods

Theory and Applications

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