Stochastic linear scaling for metals and non metals

TL;DR

This paper introduces deterministic and stochastic algorithms that achieve linear or near-linear scaling in electronic structure calculations, significantly improving computational efficiency for metals and non-metals.

Contribution

It presents new algorithms leveraging the localization of the density matrix to enable linear or quadratic scaling, including a stochastic method for system size scaling.

Findings

Deterministic algorithms scale linearly in 1D and quadratically in higher dimensions.

A stochastic algorithm achieves linear scaling with system size.

Numerical tests confirm effectiveness for metallic and non-metallic systems.

Abstract

Total energy electronic structure calculations, based on density functional theory or on the more empirical tight binding approach, are generally believed to scale as the cube of the number of electrons. By using the localisaton property of the high temperature density matrix we present exact deterministic algorithms that scale linearly in one dimension and quadratically in all others. We also introduce a stochastic algorithm that scales linearly with system size. These results hold for metallic and non metallic systems and are substantiated by numerical calculations on model systems.