Supercell calculations in the reduced Hartree-Fock model for crystals with local defects

TL;DR

This paper analyzes how quickly supercell reduced Hartree-Fock models approximate the full-space model for crystals with local defects, providing convergence rates and correction terms.

Contribution

It establishes the convergence rate of supercell rHF models to the full model for charged defects and identifies the correction term in isotropic cubic crystals.

Findings

Convergence rate of $L^{-1}$ for defect energy in supercell models.

Identification of Makov-Payne correction in isotropic cubic crystals.

Extension of results to non-isotropic crystals.

Abstract

In this article, we study the speed of convergence of the supercell reduced Hartree-Fock~(rHF) model towards the whole space rHF model in the case where the crystal contains a local defect. We prove that, when the defect is charged, the defect energy in a supercell model converges to the full rHF defect energy with speed $L^{-1}$, where $L^3$ is the volume of the supercell. The convergence constant is identified as the Makov-Payne correction term when the crystal is isotropic cubic. The result is extended to the non-isotropic case.