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An expansion for four‐center integrals over Slater‐type orbitals
Author(s) -
Salmons Lydia S.,
Ruedenberg Klaus
Publication year - 1972
Publication title -
international journal of quantum chemistry
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.484
H-Index - 105
eISSN - 1097-461X
pISSN - 0020-7608
DOI - 10.1002/qua.560060214
Subject(s) - laguerre polynomials , atomic orbital , spherical harmonics , center (category theory) , type (biology) , slater integrals , character (mathematics) , point (geometry) , mathematical physics , physics , electron , expression (computer science) , mathematics , mathematical analysis , quantum mechanics , chemistry , geometry , crystallography , ecology , computer science , programming language , biology
Abstract A new expression is given for the electron repulsion integral over Slater‐type orbitals on four different centers. It is based on the asymptotic expansion derived from the bipolar expansion of a previous paper. The expression has the form\documentclass{article}\pagestyle{empty}\begin{document}$$ I\,\mathop \sim \,\sum \limits_{q_1 }^\infty \mathop \sum \limits_{q_2 }^\infty \,F_{q_1 q_2 } (R_{{\rm PQ}})\sigma _{q_1 } (A,\,B)\sigma _{q_2 } (C,\,D) $$\end{document} where q p = { n p , l p , m p }. Both F and σ are closed expressions. The quantity F is a combination of incomplete gamma functions, Laguerre polynomials and spherical harmonics. It depends upon the relative coordinates of a point P on the AB axis and a point Q on the CD axis. The functions σ nlm ( A , B ) depend on the charge distribution (χ A χ B ); they have the character of overlap integrals and are of the form\documentclass{article}\pagestyle{empty}\begin{document}$$ \sigma _{nlm} (A,\,B)\, = \,\mathop \Sigma \limits_v \,\mathop \Sigma \limits_w \,F_{vw} (\zeta _A R_{AB},\,\zeta _B R_{AB})K_{vw} (R_{AB}) $$\end{document}