Method for Rapid Convergence of Lattice Sums by the Use of Fourier Transforms

Abstract

A general and practical technique is proposed for lattice sum of long-range pair interaction potentials. As Ewald did, a pair potential is expressed by a sum of two terms; the one is a function of a vector r in real crystal space and the other is that of a vector q in reciprocal space and these functions converge rapidly to zero as r →∞ and q →∞, respectively. The present method finds appropriate transforms by the use of usual mathematical tables of Fourier transforms. For coulomb potential, the method derives many practical variations of the traditional Ewald transform. For pair potentials discussed in the pseudopotential theory, the method gives transforms useful for rapid convergence of lattice sums. In addition the method gives a simple estimation of errors in lattice sums due to cut-off at a finite term.