An expansion for four‐center integrals over Slater‐type orbitals

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}