A general approximation to the two-center exchange integral between complex STOS
| Autor: | R. E. Coffman |
|---|---|
| Rok vydání: | 2009 |
| Předmět: |
Simple (abstract algebra)
Product (mathematics) Neumann expansion Mathematical analysis Center (group theory) Physical and Theoretical Chemistry Term (logic) Differential (infinitesimal) Condensed Matter Physics Legendre polynomials Atomic and Molecular Physics and Optics Mathematics Volume integral |
| Zdroj: | International Journal of Quantum Chemistry. 24:207-216 |
| ISSN: | 1097-461X 0020-7608 |
| DOI: | 10.1002/qua.560240827 |
| Popis: | Starting from the formulation of the exchange integral between complex STOS due to Wahl, Cade, and Roothaan, one may obtain a rapidly calculable but approximate form for the two-center exchange integral by successive application of the mean-value theorem and an approximation to a resulting ratio of Legendre polynomials. Then keeping only the λ = M term which arises from integrating the Neumann expansion of 1/r12, we obtain a new approximation for two-center integrals which is relatively simple and general. For the case λ = M = 0, the new approximation reproduces the numerical results of the Mulliken approximation for exchange, and one can show that both approximations are equivalent in this case. The new approximate equation for two-center exchange integrals can be cast into a form proportional to an integral of the product of two differential overlap functions, and the computational time per exchange integral is found to be roughly equal to the calculational time of two overlap integrals. The new approximation has “chemical” accuracy which can be adjusted by means of a mean-value theorem parameter, which has a nonmultiplicative but fundamental effect on the exchange equation. |
| Databáze: | OpenAIRE |
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