A note on accurate pressure calculations of Coulomb systems with periodic boundary conditions

TL;DR

This paper clarifies the correct formulas for pressure calculation in Coulomb systems with periodic boundary conditions, correcting previous misconceptions and demonstrating the importance of background interactions in non-neutral systems.

Contribution

It proves the equivalence of Ewald-based pressure formulas to ensemble averages and introduces necessary corrections for non-neutral systems, addressing prior inaccuracies.

Findings

Ewald summation formulas reduce to ensemble averages of potential energy over volume.

Corrected pressure expressions include background interactions in non-neutral systems.

Numerical results confirm the corrected formulas' accuracy and independence from Ewald splitting parameter.

Abstract

In this note, we address some issues concerning the accurate pressure calculation of Coulomb systems with periodic boundary conditions. First, we prove that the formulas for the excess part of the pressure with Ewald summation also reduce to the ensemble average of one-third of the ratio between the potential energy and the volume so that the comments on our previous work in a recent paper by [Onegin~\emph{et al}.,~J. Phys. A: Math.~Theor.~57 (2024) 205002] are incorrect. Second, we demonstrate that in charge non-neutral systems, the pressure expression must be corrected to include interactions with the neutralizing background. This addresses the issues about pressure computation in LAMMPS raised in the paper by Onegin {\it et al.}. Numerical experiments are performed to verify that the pressure obtained via Ewald summation with corrected terms agrees with the average pressure using thermodynamics for the non-neutral OCP system, and are independent of the splitting parameter in the Ewald summation.