Two-Center Expansion for the Powers of the Distance Between Two Points

Abstract

The powers rⁿ of the distance between two points specified by spherical polar coordinates relating to two different origins, or of the modulus of the sum of three vectors, are expanded in spherical harmonics of the angles. The radial factors satisfy simple partial differential equations, and can be expressed in terms of Appell functions F₄, and Wigner or Gaunt's coefficients. In the overlap region, first discussed by Buehler and Hirschfelder, the expressions are valid for integer values of n ≥ −1, but in the other regions, for arbitrary n. For high orders of the harmonics, individually large terms in the overlap region may have small resulting sums; as a consequence the two‐center expansion is of limited usefulness for the evaluation of molecular integrals. Expansions are also derived for the three‐dimensional delta function within the overlap region, and for arbitrary functions f(r), valid outside that region.