A hybrid approach to basis set independent Poisson solver for an arbitrary charge distribution

TL;DR

This paper introduces a hybrid numerical scheme combining finite-difference and Green's function methods to solve the Poisson equation for arbitrary charge distributions, improving accuracy and efficiency over existing methods.

Contribution

A novel hybrid approach for solving the Poisson equation that reduces errors and enhances convergence for arbitrary charge distributions.

Findings

Hybrid method outperforms individual schemes in accuracy.

Efficient convergence with standard radial grid parameters.

Applicable to density functional theory calculations.

Abstract

We review two common numerical schemes for Coulomb potential evaluation that differ only in their radial part of the solutions in the spherical harmonic expansion (SHE). One is based on finite-difference method (FDM) while the other is based on the Green's function (GF) solution to the radial part of the Poisson equation. We analyze the methods and observe that the FDM-based approach appears to be more efficient in terms of the convergence with the number of radial points, particularly for monopole (l=0). However, as a known issue, it suffers from error accumulation as the system size increases. We identify the source of error that comes mainly from l=1 (and sometimes l=2) contribution of SHE induced by the charge partitioning. We then propose a hybrid scheme by combining the two methods, where the radial solution for l=0 is obtained using the FDM method and treating the remaining terms using GF approach. The proposed hybrid method is subsequently applied to a variety of systems to examine its performance. The results show improved accuracy than earlier numerical schemes in all cases. We also show that, even with a generic set of radial grid parameters, accurate energy differences can be obtained using a numerical Coulomb solver in standard density functional studies. ~