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

${\displaystyle \sum_n \frac{1}{\Omega_{\mathrm{BZ}}} \int_{\Omega_{\mathrm{BZ}}} \epsilon_{n\bold{k}} \, \Theta(\epsilon_{n\bold{k}}-\mu) \, d \bold{k}, }$

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^[1]^[2]^[3]:

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

Keeping the step function we get a sum

${\displaystyle \sum_{\bold{k}} w_{\bold{k}} \epsilon_{n\bold{k}} \, \Theta(\epsilon_{n\bold{k}}-\mu), }$

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).

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

Linear tetrahedron methods

Within the linear tetrahedron method, the term $\eps_\nk$ is interpolated linearly between two k-points. Bloechl~\cite{blo94} has recently revised the tetrahedron method to give effective weights $f(\{\eps_\nk\})$ for each band and k-point. In addition Bloechel was able to derive a correction formula which removes the quadratic error inherent in the linear tetrahedron method (linear tetrahedron method with Bloechel corrections). The linear tetrahedron is more or less fool proof and requires a minimal interference by the user.

The main drawback is that the Bloechels method is not variational with respect to the partial occupancies if the correction terms are included, therefore the calculated forces might be wrong by a few percent. If accurate forces are required we recommend a finite temperature method.

References

[baldereschi:prb:1973-1] A. Baldereschi, Phys. Rev. B 7, 5212 (1973).

[chadi:prb:1973-2] D.J. Chadi and M.L. Cohen, Phys. Rev. B 8, 5747 (1973).

[monkhorst:prb:1976-3] H.J. Monkhorst and J.D. Pack, Phys. Rev. B 13, 5188 (1976).

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