Implicit Purification for Temperature-Dependent Density Matrices

TL;DR

This paper introduces an implicit purification method for efficiently computing temperature-dependent density matrices in electronic structure calculations, with computational complexity scaling logarithmically with inverse temperature.

Contribution

It presents a novel implicit purification scheme that improves efficiency in calculating finite-temperature density matrices using sparse matrix algebra.

Findings

Computational complexity scales logarithmically with inverse temperature.

The method is efficient when combined with linear scaling electronic structure theory.

Compared favorably to explicit purification methods at zero temperature.

Abstract

An implicit purification scheme is proposed for calculation of the temperature-dependent, grand canonical single-particle density matrix, given as a Fermi operator expansion in terms of the Hamiltonian. The computational complexity is shown to scale with the logarithm of the polynomial order of the expansion, or equivalently, with the logarithm of the inverse temperature. The system of linear equations that arise in each implicit purification iteration is solved efficiently by a conjugate gradient solver. The scheme is particularly useful in connection with linear scaling electronic structure theory based on sparse matrix algebra. The efficiency of the implicit temperature expansion technique is analyzed and compared to some explicit purification methods for the zero temperature density matrix.