Ewald method for polytropic potentials in arbitrary dimensionality

TL;DR

This paper generalizes the Ewald summation method to power-law potentials in various dimensions, providing explicit formulas, optimization procedures, and applications for different geometries and interaction ranges.

Contribution

It introduces a comprehensive generalization of the Ewald method for arbitrary polytropic potentials across multiple dimensions with explicit formulas and optimization strategies.

Findings

Explicit Ewald sum formulas for 1/|r|^k potentials in 1D, 2D, and 3D.

A parameter optimization procedure for simulations.

Application example demonstrating the method's effectiveness.

Abstract

The Ewald summation technique is generalised to power-law 1/|r|^k potentials in three-, two- and one-dimensional geometries with explicit formulae for all the components of the sums. The cases of short-range, long-range and "marginal" interactions are treated separately. The jellium model, as a particular case of a charge-neutral system, is discussed and the explicit forms of the Ewald sums for such system are presented. A generalised form of the Ewald sums for a noncubic (nonsquare) simulation cell for three- (two-) dimensional geometry is obtained and its possible field of application is discussed. A procedure for the optimisation of the involved parameters in actual simulations is developed and an example of its application is presented.