Expansion algorithm for the density matrix

TL;DR

This paper introduces an efficient purification algorithm for the density matrix that reduces computational effort and is effective for systems with varying occupancy, including metals, by minimizing matrix multiplications.

Contribution

The proposed expansion algorithm significantly decreases matrix multiplications needed for density matrix expansion, especially at extreme occupancy levels, and improves upon grand canonical McWeeny purification.

Findings

Requires less than half the matrix multiplications of existing methods at low and high occupancy.

Computational complexity is independent of system size, even for metallic materials.

Applicable with a fixed chemical potential, generalizing and improving previous purification methods.

Abstract

A purification algorithm for expanding the single-particle density matrix in terms of the Hamiltonian operator is proposed. The scheme works with a predefined occupation and requires less than half the number of matrix-matrix multiplications compared to existing methods at low (<10%) and high (>90%) occupancy. The expansion can be used with a fixed chemical potential in which case it is an asymmetric generalization of and a substantial improvement over grand canonical McWeeny purification. It is shown that the computational complexity, measured as number of matrix multiplications, essentially is independent of system size even for metallic materials with a vanishing band gap.