Rational approximation to the Fermi-Dirac function with applications in Density Functional Theory

Abstract

We are interested in computing the Fermi-Dirac matrix function in which the matrix argument is the Hamiltonian matrix arising from Density Function Theory (DFT) applications. More precisely, we are really interested in the diagonal of this matrix function. We discuss rational approximation methods to the problem, specifically the rational Chebyshev approximation and the continued fraction representation. These schemes are further decomposed into their partial fraction expansions, leading ultimately to computing the diagonal of the inverse of a shifted matrix over a series of shifts. We descibe Lanczos methods and sparse direct method to address these systems. Each approach has advanatges and disadvatanges that are illustrated with experiments