Convergence rates of supercell calculations in the reduced Hartree-Fock model

TL;DR

This paper analyzes how quickly supercell calculations in the reduced Hartree-Fock model converge to the true periodic crystal energy, showing exponential convergence for insulators and semiconductors.

Contribution

It provides a rigorous proof of exponential convergence rates of supercell calculations in the reduced Hartree-Fock model for insulators and semiconductors.

Findings

Supercell energy converges exponentially fast to the periodic energy.

Convergence rate depends on the insulating or semiconducting nature of the crystal.

Results are relevant for improving computational efficiency in crystal simulations.

Abstract

This article is concerned with the numerical simulations of perfect crystals. We study the rate of convergence of the reduced Hartree-Fock (rHF) model in a supercell towards the periodic rHF model in the whole space. We prove that, whenever the crystal is an insulator or a semi-conductor, the supercell energy per unit cell converges exponentially fast towards the periodic rHF energy per unit cell, with respect to the size of the supercell.