Computation of Lattice Sums: Generalization of the Ewald Method

Abstract

The Ewald method was originally invented to compute the Madelung constant. In this paper we consider a lattice whose sites are associated with an arbitrary potential function. The “charge,” or the scale factor for these potential functions, need not be the same at each site. We consider the evaluation of the resulting lattice sum at an arbitrary point, not necessarily at a lattice site. The method involves two generalizations over previous work: (1) the displacement of the origin off a lattice site and (2) the handling of arbitrary periodic charge distributions by decomposing such distributions into simpler ones involving only +q and −q. The method should prove particularly useful for evaluating the expansion coefficients of the crystalline potential when this potential is expanded in the usual spherical harmonic series.