A periodic Green function for calculation of coloumbic lattice potentials

Abstract

The modified Green function appropriate for solution of interior boundary value problems of Laplace's equation in a three-dimensional rectangular parallelepiped, subject to periodic boundary conditions, is developed. This allows the determination of the potential due to an arbitrary continuous charge distribution and its periodic replications in three dimensions. Summation of the eigenfunction expansion by application of the Poisson-Jacobi formula gives a Ewald sum, while application of the Poisson summation formula results in a two-dimensional potential that is perturbed by a rapidly converging Fourier cosine series involving K0 Bessel functions. The latter constitutes a generalization of formulae described by Lekner. Numerical results show that the K0 expansion is more rapidly convergent than the Ewald sum, and could therefore substantially reduce the computational effort involved in the molecular simulation of ionic and polar fluids. The Green function is also shown to be related to the asymptotic behaviour of lattice sums for the screened Coulomb potential, in the limit as the screening constant tends to zero.