Manuscripts - Postdoc - Paper V

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New developments in the Barnett-Coulson method
for calculating molecular integrals

One-, two- and three-centre integrals of the regular and irregular solid cubic harmonics; Cartesian fields and multipole moments

Abstract

General formulae are provided for the evaluation of integrals for both the regular and the irregular solid cubic harmonic operators over real Slater orbitals in polyatomic molecules, using the Barnett-Coulson-Hobson expansion. The effect of radial singularity in the irregular case is investigated analytically. Careful attention is paid to the influence of angular selection rules; appropriate linear combinations are developed and tabulated for the treatment of the analogous cartesian operators. The Sack expansion is examined in detail and suitable formulae deduced for the real phaseless cubic harmonics with arbitrary radial exponent. Matrix elements for nuclear displacement one-electron operators are described in terms of the solid cubic harmonics.

Contents

Page
ABSTRACT1
CONTENTS2
1. INTRODUCTION4
2. REGULAR SOLID CUBIC HARMONICS8
2.1 One-Centre Coulomb12
2.2 Two-Centre Coulomb13
2.3 Two-Centre Exchange14
2.4 Three-Centre Exchange16
3. IRREGULAR SOLID CUBIC HARMONICS17
3.1 One-Centre Coulomb21
3.2 Two-Centre Coulomb22
3.3 Two-Centre Exchange24
3.4 Three-Centre Exchange25
4. CARTESIAN MULTIPOLE MOMENTS30
4.1 One-Centre Coulomb32
4.2 Two-Centre Coulomb33
4.3 Two-Centre Exchange34
4.4 Three-Centre Exchange35
5. OTHER ONE-ELECTRON OPERATORS36
5.1 The Sack Expansion37
5.2 Displacement Perturbations50
APPENDIX A: OFF-CENTRE CONTACT TERMS55
APPENDIX B: CARTESIAN FIELD PROPERTIES67
APPENDIX C: THE a-COEFFICIENTS69
APPENDIX D: ULTRAHEXADECIMAL STORAGE79
ACKNOWLEDGEMENTS89
REFERENCES90