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 $O( \log (\beta \Delta) \log (\epsilon^{-1}))$polestoachieveanerrortolerance$\epsilon$attemperature$\beta^{-1}$overanenergyinterval$\Delta$. We apply minimax approximation to reduce the number of poles by a factor of four and replace $\mathrm{\Delta }$ with ${\mathrm{\Delta }}_{\mathrm{o}\mathrm{c}\mathrm{c}}$, the occupied energy interval. This is particularly beneficial when $\mathrm{\Delta }\gg {\mathrm{\Delta }}_{\mathrm{o}\mathrm{c}\mathrm{c}}$, such as in electronic structure calculations that use a large basis set.