Non-monotonic recursive polynomial expansions for linear scaling calculation of the density matrix

TL;DR

This paper introduces non-monotonic recursive polynomial expansions for density matrix purification, significantly reducing computational effort and improving convergence speed in linear scaling electronic structure calculations.

Contribution

It proposes novel non-monotonic purification schemes that halve the matrix multiplications needed, enhancing efficiency regardless of chemical potential location or band gap size.

Findings

Requires only half the matrix multiplications compared to previous methods

Speedup is independent of chemical potential location

Efficiency increases with decreasing band gap

Abstract

As it stands, density matrix purification is a powerful tool for linear scaling electronic structure calculations. The convergence is rapid and depends only weakly on the band gap. However, as will be shown in this paper, there is room for improvements. The key is to allow for non-monotonicity in the recursive polynomial expansion. Based on this idea, new purification schemes are proposed that require only half the number of matrix-matrix multiplications compared to previous schemes. The speedup is essentially independent of the location of the chemical potential and increases with decreasing band gap.