Bipolar expansion of screened Coulomb potentials, Helmholtz' solid harmonics, and their addition theorems

Abstract

Helmholtz' solid harmonics (spherical Bessel functions X spherical surface harmonics) are generated from fundamental spherical waves through a ladder procedure using raising and lowering operators. The addition theorem for them and the bipolar expansion formula of a screened Coulomb potential are derived. A method for evaluating two-center integrals with a general potential is given in the last section. This is new and useful in practical calculations. A key function Jlnm (x) = 1/2 ∫-11 dμ eiμx Ρlm(μ) Ρnm(μ), which appears in all the essential results, is studied in detail.