Expansion of a function about a displaced center for multicenter integrals: A general and closed expression for the coefficients in the expansion of a Slater orbital and for overlap integrals

Abstract

Following a method analogous to one pointed out earlier, a simplified expression for the expansion of a general function about a point displaced from its center is described. Utilizing this, a general and closed expression useful for multicenter integrals in quantum mechanics is derived for the coefficients in the expansion of a general Slater-type orbital (including special cases). The derived expression for the Slater orbital contains terms only of the form ${r}^{k\ensuremath{-}1}\mathrm{exp}(\ifmmode\pm\else\textpm\fi{}\ensuremath{\eta}r)$ (where $k$ is an integer \ensuremath{\geqslant} 0; $l$ is the order of the coefficient and $\ensuremath{\eta}$ is the exponent in the Slater orbital)---very convenient and useful for the analytical evaluation of multicenter integrals. The expression is equivalent to the ones obtained by Silverstone and by Rakauskas and Bolotin but based on a completely different approach. The asymptotic forms for small as well as large values of $r$ are also presented. The importance of the expression is demonstrated by undertaking an example of overlap matrix elements involving Slater orbitals and deriving easily a simple and closed form applicable for all quantum numbers concerned. The ease with which one can write readily the overlap formulas in various cases starting from the general formula is indicated, and some numerical examples are given to support the usefulness of the expressions. The advantages associated with the expression (for the expansion coefficients) for large-scale calculations of multicenter integrals are discussed.