Minimax rational approximation of the Fermi-Dirac distribution

TL;DR

This paper develops a minimax rational approximation method for the Fermi-Dirac distribution, significantly reducing the number of poles needed in electronic structure calculations, especially when the energy interval is large.

Contribution

It introduces a minimax approximation approach that decreases the pole count by a factor of four compared to existing methods, optimizing calculations over the occupied energy interval.

Findings

Reduces the number of poles needed for accurate approximation

Improves efficiency in electronic structure calculations with large basis sets

Achieves error tolerance with fewer computational resources

Abstract

Accurate rational approximations of the Fermi-Dirac distribution are a useful component in many numerical algorithms for electronic structure calculations. The best known approximations use $O( \log (\beta \Delta) \log (\epsilon^{-1}))$ poles to achieve an error tolerance $\epsilon$ at temperature $\beta^{-1}$ over an energy interval $\Delta$. We apply minimax approximation to reduce the number of poles by a factor of four and replace $\Delta$ with $\Delta_{\mathrm{occ}}$, the occupied energy interval. This is particularly beneficial when $\Delta \gg \Delta_{\mathrm{occ}}$, such as in electronic structure calculations that use a large basis set.