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 $O( \log (βΔ) \log (ε^{-1}))$ poles to achieve an error tolerance $ε$ at temperature $β^{-1}$ over an energy interval $Δ$. We apply minimax approximation to reduce the number of poles by a factor of four and replace $Δ$ with $Δ_{\mathrm{occ}}$, the occupied energy interval. This is particularly beneficial when $Δ\gg Δ_{\mathrm{occ}}$, such as in electronic structure calculations that use a large basis set.

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