Controlling Errors in Recursive Fermi–Dirac Operator Expansions with Applications in Electronic Structure Theory

Abstract

Recursive Fermi–Dirac operator expansion is an efficient way to compute one-electron density matrices in electronic structure theory. The convergence is rapid and depends only weakly on the conditioning of the problem and, for many systems, the computational cost increases only linearly with system size. In this article, errors introduced when evaluating the recursive expansion are analyzed and schemes to control the forward error are proposed. The error has previously been analyzed for explicit schemes working at zero electronic temperature [J. Chem. Phys., 128 (2008), 074106]. Here, implicit schemes [Phys. Rev. B, 68 (2003), 233104] working at zero or finite temperature are treated. The proposed schemes for error control are demonstrated by tight-binding as well as density functional theory electronic structure calculations on several test systems. Condition numbers for the problem of computing the density matrix are derived, giving quantitative insight into under what circumstances a temperature dependent formulation results in better conditioning. It is shown that for the considered recursive expansions, the number of matrix-matrix multiplications needed to compute the density matrix increases only with the squared logarithm of the condition number of the problem.