Correction terms to the density-functional ground-state energy arising from occupation number broadening

In density-functional calculations of total energies and forces, the fractional occupation numbers used for the discrete sampling of the Brillouin zone are often generated by “finite-temperature” broadening schemes with a finite broadening parameter $\sigma .$ It is shown that if the broadening parameters are chosen properly, the results obtained with different broadening schemes, e.g., with Fermi-Dirac broadening or with Gaussian broadening, will differ only by terms of the order ${\sigma }^4.$ As a further result, it is demonstrated that if the ground-state energy for $\sigma =0\mathrm{}$ is approximated by $\frac{1}{2}\left[F\right(\sigma )+E(\sigma \left)\right],$ where $E\left(\sigma \right)$ and $F\left(\sigma \right)$ denote the energy and a suitably defined “free energy,” the correction terms are of order ${\sigma }^4$ rather than of order ${\sigma }^3$ as is commonly assumed.