Expansions in Spherical Harmonics. IV. Integral Form of the Radial Dependence

Abstract

A function f(r_AB) $Y_L^M$ (θ_AB, φ_AB) of a vector r_AB = Σr_i can be expanded in spherical harmonics $Y_l^m$ (θ, φ) of the directions of the individual vectors. The radial coefficients satisfy simple differential equations which, in three previous papers, were solved in terms of series in $r_i^2{\mathrm{/r}}_j^2$ ; these were different in various regions, depending on the relative magnitudes of the r_i. In this paper the solutions are found as multiple integrals over the product of Legendre polynomials and of a function G(w), where w depends linearly on the r_i. The kernel G(w) is independent of the number of constituent vectors, their relative sizes, and the orders of their harmonics; it contains the Heaviside step function H(w) as a factor which takes care of the various regions. The precise form of G can be found from f and L by an integral equation which for L = 0, 1 is solved for arbitrary f, and for L > 1 for sufficiently large positive powers. The expressions of Milleur, Twerdochlib, and Hirschfelder for the bipolar angle average can be obtained simply by repeated integration of G(w) or directly from the differential equations. For the inverse distance between two points, G(w) becomes Dirac's delta function; the number of integrations is thereby reduced by one. Possible applications of the new approach to the evaluation of molecular many‐center integrals are outlined. Some corrections are given for the results of the previous papers in the series.