Linear scaling solution of the all-electron Coulomb problem in solids

TL;DR

This paper introduces a linear scaling method for solving the all-electron Coulomb problem in solids, enabling efficient large-scale quantum calculations with high accuracy and convergence.

Contribution

A novel linear scaling formulation that handles singular nuclear potentials without smearing, using local neutralizing densities and enriched finite elements for efficient real-space Poisson solutions.

Findings

Accurate comparison with Ewald sums for point charge lattices

Successful all-electron calculations on crystalline diamond

Method achieves systematic improvability and convergence

Abstract

We present a linear scaling formulation for the solution of the all-electron Coulomb problem in crystalline solids. The resulting method is systematically improvable and well suited to large-scale quantum mechanical calculations in which the Coulomb potential and energy of a continuous electronic density and singular nuclear density are required. Linear scaling is achieved by introducing smooth, strictly local neutralizing densities to render nuclear interactions strictly local, and solving the remaining neutral Poisson problem for the electrons in real space. While the formulation includes singular nuclear potentials without smearing approximations, the required Poisson solution is in Sobolev space $H^1$, as required for convergence in the energy norm. We employ enriched finite elements, with enrichments from isolated atom solutions, for an efficient solution of the resulting Poisson problem in the interacting solid. We demonstrate the accuracy and convergence of the approach by direct comparison to standard Ewald sums for a lattice of point charges, and demonstrate the accuracy in all-electron quantum mechanical calculations with an application to crystalline diamond.