Minimax rational approximation of the Fermi-Dirac distribution

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.

6 pages, 4 figures

References in corpus (2)

• A partial fraction decomposition of the Fermi function
• Cubic-scaling algorithm and self-consistent field for the random-phase approximation with second-order screened exchange

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