The Ewald sums for singly, doubly and triply periodic electrostatic systems

TL;DR

This paper derives Ewald summation formulas for electrostatic systems with one, two, and three periodic dimensions using Fourier analysis, facilitating efficient computation in various boundary conditions.

Contribution

It provides a unified derivation of Ewald sums for singly, doubly, and triply periodic systems, enhancing understanding and computational approaches.

Findings

Unified Fourier analysis framework for Ewald sums

Derivation of formulas for singly and doubly periodic cases

Facilitates FFT-based fast summation methods

Abstract

When evaluating the electrostatic potential, periodic boundary conditions in one, two or three of the spatial dimensions are often needed for different applications. The triply periodic Ewald summation formula is classical, and Ewald summation formulas for the other two cases have also been derived. In this paper, derivations of the Ewald sums in the doubly and singly periodic cases are presented in a uniform framework based on Fourier analysis, which also yields a natural starting point for FFT-based fast summation methods.