Variational calculation of the multipole potential from an arbitrary localized charge distribution

Abstract

Very recently Painter has developed a method for solving Poisson’s equation as a set of finite-difference equations for an arbitrary localized charge distribution ρ(r) that is expanded in a partial-wave representation as ρ(r) = 𝒥LρL(r)YL(r̂), where L denotes l and m. In the present work a variational principle is established, and a possible approach is outlined, for obtaining approximate partial-wave coefficients VL(r) of the potential V(r) = 𝒥LVL(r)YL(?).