Exchange Energy for Two-Active-Electron Diatomic Systems Within the Surface Integral Method
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We have analyzed and reduced a general (quantum-mechanical) expression for the atom-atom exchange energy formulated as a five-dimensional surface integral, which arises in studying the charge exchange processes in diatomic molecules. It is shown that this five-dimensional surface integral can be decoupled into a three-dimensional integral and a two-dimensional angular integral which can be solved analytically using a special decomposition. Exact solutions of the two-dimensional angular integrals are presented and generalized. Algebraic aspects, invariance properties and exact solutions of integrals involving Legendre and Chebyshev polynomials are also discussed.
KeywordsSymbolic integration Numerical integration Molecular physics Special functions and sums Asymptotic series
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One of us (T.C.S.) would like to thank the members of Zentralinstitut für Angewandte Mathematik (ZAM - Central Institute for Applied Mathematics) and Arne Lüchow of the Institut für Physikalische Chemie, RWTH Aachen, and Georg Jansen of the Institut für Organische Chemie of the University of Essen for their wonderful hospitality and support for allowing this work to be possible. The author M.L.G. would like to thank the NSF for support under grant DMR0121146. We would also like to thank Juergen Hinze of the Theoretical Chemistry group at the University of Bielefeld (Faculty of Chemistry) and James Babb and Alexander Dalgarno of the Institute for Theoretical Atomic and Molecular Physics at the Harvard-Smithsonian Center for Astrophysics, for helpful discussions. Special thanks go to Bruno Salvy and Philippe Flajolet of INRIA-Rocquencourt (project ALGO) for their helpful hints with respect to asymptotic series expansions and Claude Gomez (project METALAU) for his assistance in the use of Macrofort. We would also like to thank Frédéric Desprez of the Ecole Normale Supérieure of Lyon and INRIA-Rhône-Alpes. Financial support for T.C.S. from the PSMN (ENS Lyon and Université Lyon 1) is gratefully acknowledged.
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