The generating integral Il(λ,λ′;E) =∫d3r d3r′ exp(−λr−λ′r′)(rr′)l−1 ×Ȳl,m(̄θ̄,̄φ̄)̄G(r,r′;E) ×Yl,m(θ′,φ′) is evaluated, where the Coulomb Green’s function G is the inverse (E−H)−1 with H the hydrogenic Hamiltonian −(1/2) ∇2−Zr−1 and Yl,m is a spherical harmonic. The result can be used for the evaluation of matrix elements of G with respect to wave functions of the form Yl,m(θ,φ)f(α)l,p(r) where f(α)l,p(r) =N(α)l,p ×exp(−ar/2)(ar)lL(α)p (ar) with L(α)p a Laguerre polynomial and N(α)l,p a normalizing factor. For general E the result is given in terms of a special case of the hypergeometric function which satisfies an inhomogeneous linear first-order ordinary differential equation. For E=En where En=−Z2/(2n2) is a hydrogenic bound state energy, G is replaced by the generalized Green’s function (generalized inverse) and the results are given in closed form in terms of elementary functions.
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