Effective way to sum over long range Coulomb potentials in two and three dimensions

TL;DR

This paper introduces a simple, accurate method for summing long-range Coulomb and logarithmic potentials in 2D and 3D periodic systems, avoiding complex parameters and error functions, with faster convergence than previous methods.

Contribution

A novel, parameter-free approach for calculating Coulomb interactions in periodic systems that simplifies derivation and improves convergence speed.

Findings

Convergence of sums is exponential for most of the unit cell.

Method requires only a few dozen terms for accurate results.

Expressions for Madelung constants of CsCl and NaCl are provided.

Abstract

I propose a method to calculate logarithmic interaction in two dimensions and coulomb interaction in three dimensions under periodic boundary conditions. This paper considers the case of a rectangular cell in two dimensions and an orthorhombic cell in three dimensions. Unlike the Ewald method, there is no parameter to be optimized, nor does it involve error functions, thus leading to the accuracy obtained. This method is similar in approach to that of Sperb [R. Sperb, Mol. Simulation, 22, 199 (1999).], but the derivation is considerably simpler and physically appealing. An important aspect of the proposed method is the faster convergence of the Green function for a particular case as compared to Sperb's work. The convergence of the sums for the most part of unit cell is exponential, and hence requires the calculation of only a few dozen terms. In a very simple way, we also obtain expressions for interaction for systems with slab geometries. Expressions for the Madelung constant of CsCl and NaCl are also obtained.