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

$\sum _{n}{\frac {1}{\Omega _{\mathrm {BZ} }}}\int _{\Omega _{\mathrm {BZ} }}\epsilon _{n{\mathbf {k}}}\,\Theta (\epsilon _{n{\mathbf {k}}}-\mu )\,d{\mathbf {k}},$

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}:

${\frac {1}{\Omega _{\mathrm {BZ} }}}\int _{\Omega _{\mathrm {BZ} }}\to \sum _{\mathbf {k}}w_{\mathbf {k}}.$

Keeping the step function we get a sum

$\sum _{\mathbf {k}}w_{\mathbf {k}}\epsilon _{n{\mathbf {k}}}\,\Theta (\epsilon _{n{\mathbf {k}}}-\mu ),$

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).

For metals the trick is now to replace the step function $\Theta (\epsilon _{n{\mathbf {k}}}-\mu )$ by a (smooth) function $f(\{\epsilon _{n{\mathbf {k}}}\})$ resulting in a much faster convergence speed without destroying the accuracy of the sum. Several methods have been proposed to solve this dazzling problem.