Pseudopotential contributions to the quadrupole moment in charged periodic systems

TL;DR

This paper revises the Makov-Payne correction for charged periodic systems to align with modern DFT practices, addressing discrepancies in electrostatic energy divergence corrections.

Contribution

It demonstrates how to adapt the Makov-Payne correction to be consistent with current density functional theory implementations.

Findings

Revised correction reduces errors in charged system calculations.

Aligns classical correction with modern DFT energy treatments.

Improves accuracy of electrostatic energy computations in periodic systems.

Abstract

There has been much interest over many years in studying charged systems after the artificial imposition of periodic boundary conditions, and correcting for the resulting divergence of the electrostatic energy density. A correction for cubic cells was derived by Makov and Payne in 1995, and its leading error term is of the form $L^{-5}$ for a cube of side $L$. Most modern Density Functional Theory codes use a different treatment of the `Z$\alpha$' energy term to that used by Makov and Payne, resulting in an error term of the form $L^{-3}$ if their correction is used unmodified. This paper shows how the Makov and Payne result can be made consistent with modern practice.