Lekner summations and Ewald summations for quasi-two dimensional systems

TL;DR

This paper compares three methods for calculating long-range Coulomb interactions in quasi-two-dimensional systems, demonstrating the accuracy and efficiency of the Ewald method and providing guidelines for implementing Lekner summations.

Contribution

It provides a systematic comparison of Ewald, Hautman-Klein, and Lekner methods for Coulomb interactions in quasi-2D systems, highlighting the accuracy and implementation details.

Findings

Ewald method is exact and efficient for the studied system.

Lekner and Hautman-Klein methods are systematically compared to Ewald.

Guidelines for accurate implementation of Lekner summations are provided.

Abstract

Using the specific model of a bilayer of classical charged particles (bilayer Wigner crystal), we compare the predictions for energies and pair distribution functions obtained by Monte Carlo simulations using three different methods available to treat the long range Coulomb interactions in systems periodic in two directions but bound in the third one. The three methods compared are: the Ewald method for quasi-two dimensional systems [D.E. Parry, Surf. Sci. $\bm{49}$, 433 (1975); \it{ibid.}, $\bm{54}$, 195 (1976)], the Hautman-Klein method [J. Hautman and M.L. Klein, Mol. Phys. $\bm{75}$, 379 (1992)] and the Lekner summations method [J. Lekner, Physica A$\bm{176}$, 485 (1991)]. All of the three method studied in this paper may be applied to any quasi-two dimensional systems, including those having not the specific symmetry of slab systems. For the particular system used in this work, the Ewald method for quasi-two dimensional systems is exact and may be implemented with efficiency; results obtained with the other two methods are systematically compared to results found with the Ewald method. General recommendations to implement with accuracy, but not always with efficiency, the Lekner summations technique in Monte Carlo algorithms are given.