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New developments in the Barnett-Coulson method
for calculating molecular integrals
Expansions and transformations
Abstract
A generalisation of the Barnett-Coulson method is developed whereby the energy and other electronic properties of a polyatomic molecule can be succinctly formulated and efficiently computed. A description and analysis of the fundamental expansion theorems is presented in a form immediately applicable to the evaluation of multicentre integrals over real Slater-type atomic orbitals for a wide variety of operators. Underlying properties of the rotational parameters, and transformations of the cubic harmonics, are derived in detail. An introductory description is given of the techniques of angular integration and radial quadrature that are required, and the necessary auxiliary functions are summarised.
Contents
| Page | |
| ABSTRACT | 1 |
| CONTENTS | 2 |
| 1. INTRODUCTION | 3 |
| 2. CONVENTIONS | 10 |
| 3. TRANSFER OF ANGULAR FACTOR | 19 |
| 4. DETERMINATION OF EULER ANGLES | 24 |
| 5. EULERIAN ROTATION OF CUBIC HARMONICS | 28 |
| 6. TRANSFER OF RADIAL FACTOR | 34 |
| 7. COMPUTATION OF ZETA FUNCTIONS | 41 |
| 7.1 Derivative Method | 46 |
| 7.2 Quadrature Method | 48 |
| 7.3 Series Method | 50 |
| 7.4 Recursion Method | 53 |
| 7.5 Polynomial Method | 54 |
| 8. ASYMPTOTIC BOUNDS FOR ZETA FUNCTIONS | 55 |
| APPENDIX A: CUBIC HARMONICS | 57 |
| APPENDIX B: BESSEL FUNCTIONS | 63 |
| TABLES | 66 |
| ACKNOWLEDGEMENTS | 74 |
| REFERENCES | 75 |