Minimax rational approximation of the Fermi-Dirac distribution

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.