The optimal P3M algorithm for computing electrostatic energies in periodic systems

TL;DR

This paper presents an optimized P3M algorithm tailored for highly accurate electrostatic energy calculations in 3D periodic charged systems, including a new influence function and error estimation methods.

Contribution

It introduces an optimal influence function for energy accuracy, a new real-space correction term, and an analytical RMS error estimate for improved parameter selection.

Findings

Achieved maximal accuracy in electrostatic energy calculations.

Derived a new real-space cut-off correction term.

Provided an analytical RMS error estimate for parameter optimization.

Abstract

We optimize Hockney and Eastwood's Particle-Particle Particle-Mesh (P3M) algorithm to achieve maximal accuracy in the electrostatic energies (instead of forces) in 3D periodic charged systems. To this end we construct an optimal influence function that minimizes the RMS errors in the energies. As a by-product we derive a new real-space cut-off correction term, give a transparent derivation of the systematic errors in terms of Madelung energies, and provide an accurate analytical estimate for the RMS error of the energies. This error estimate is a useful indicator of the accuracy of the computed energies, and allows an easy and precise determination of the optimal values of the various parameters in the algorithm (Ewald splitting parameter, mesh size and charge assignment order).