K-point integration

In this section we discuss partial occupancies. A must for all readers.

First there is the question why to use partial occupancies at all. The answer is: partial occupancies help to decrease the number of k-points necessary to calculate an accurate band-structure energy. This answer might be strange at first sight. What we want to calculate is, the integral over the filled parts of the bands

Failed to parse (Conversion error. Server ("cli") reported: "[INVALID]"): {\displaystyle \sum_n \frac{1}{\Omega_{\mathrm{BZ}}} \int_{\Omega_{\mathrm{BZ}}} \eps_{n\bold{k}} \, \Theta(\eps_{n\bold{k}}-\mu) \, d \bold{k}, }

where $\Theta(x)$ is the Dirac step function. Due to our finite computer resources this integral has to be evaluated using a discrete set of k-points\cite{bal73}:

${\displaystyle \frac{1}{\Omega_{\mathrm{BZ}}} \int_{\Omega_{\mathrm{BZ}}} \to \sum_{\bold{k}} w_{\bold{k}}. }$

Keeping the step function we get a sum

Failed to parse (Conversion error. Server ("cli") reported: "[INVALID]"): {\displaystyle \sum_{\bold{k}} w_{\bold{k}} \eps_{n\bold{k}} \, \Theta(\eps_{n\bold{k}}-\mu), }

which converges exceedingly slow with the number of k-points included. This slow convergence speed arises only from the fact that the occupancies jump form 1 to 0 at the Fermi-level. If a band is completely filled the integral can be calculated accurately using a low number of k-points (this is the case for semiconductors and insulators).