Manuscripts - Postdoc - Paper VI

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

One-, two-, three- and four-centre electron repulsion integrals; Associated transformations over two and four indices

Abstract

Detailed formulae are developed for all the electron-repulsion integrals over a Slater basis in polyatomic molecules of arbitrary geometry, using the Barnett- Coulson expansion for multi-centre integrals; alternative approaches to the one- and two-centre integrals are reviewed. Corresponding formulae are derived for the vibrational derivatives of all such integrals. The data-driven transformation of both one- and two-electron molecular integrals is surveyed, and extended to various novel applications such as the generation of linearly- dependent integrals in symmetric molecules. Asymptotic formulae are derived for the long-range one- and two-electron integrals between polyatomic molecules with arbitrary relative orientation. The analytic approximation of electronic property curves, surfaces and hyper-surfaces is investigated.

Contents

Page
ABSTRACT1
CONTENTS2
1. INTRODUCTION3
2. ONE-CENTRE ELECTRON-REPULSIONS19
3. TWO-CENTRE ELECTRON-REPULSIONS28
4. MULTI-CENTRE ELECTRON-REPULSIONS35
4.1 COULOMB POTENTIAL43
4.2 EXCHANGE POTENTIAL53
5. GRADIENTS OF ELECTRON-REPULSIONS76
5.1 COULOMB POTENTIAL79
5.2 EXCHANGE POTENTIAL84
APPENDIX A: RADIAL TRANSFORMATIONS89
APPENDIX B: INTERACTION MATRIX ASSEMBLY97
APPENDIX C: 2- AND 4-INDEX TRANSFORMATIONS119
APPENDIX D: APPLICATION OF MOLECULAR SYMMETRY138
D1: TRANSFORMATION OF FUNCTIONS & OPERATORS144
D2: TRANSFORMATION OF ORBITALS164
D3: TRANSFORMATION OF MOLECULAR INTEGRALS171
D4: PRE-GENERATION OF MOLECULAR INTEGRALS186
D5: POST-GENERATION OF MOLECULAR INTEGRALS201
D6: GENERATION OF MOLECULAR PROPERTIES207
APPENDIX E: ASYMPTOTIC INTERMOLECULAR INTEGRALS212
APPENDIX F: LEAST-SQUARES INTERPOLATION226
ACKNOWLEDGEMENTS235
REFERENCES236