Calculation of the multipole potential from an arbitrary localized charge distribution

A method for solving Poisson's equation as a set of finite-difference equations is described for an arbitrary localized charge distribution expanded in a partial-wave representation. The procedure is an extension of the widely used technique developed by Loucks for spherically symmetric charge densities. In the present development, the higher partial-wave components of the density lead to an $l$ dependence in the boundary conditions and the finite-difference equations. The potential components for $l>\mathrm{}0\mathrm{}$ can be calculated recursively with the same degree of accuracy and efficiency as for the $l=0\mathrm{}$ case. This procedure requires the additional computational step of evaluating a one-dimensional integral of each density component to determine the multipole moment, required by the boundary condition at large $r$. Specific modifications required to adapt the Loucks technique for this more general charge density are described, and illustrative results are given for a model density with components ${\rho }_l\mathrm{}\left(r\right)=C_l{r}^l{e}^{-a_{l}r}$, which can also be treated analytically.