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9.4 Partial occupancies, different methods


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


where tex2html_wrap_inline4953 is the Dirac step function. Due to our finite computer resources this integral has to be evaluated using a discrete set of k-points[34]:


Keeping the step function we get a sum


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 few number of k-points (this is the case for semiconductors and insulators).

For metals the trick is now to replace the step function tex2html_wrap_inline4955 by a (smooth) function tex2html_wrap_inline4957 resulting in a much faster convergence speed without destroying the accuracy of the sum. Several methods have been proposed to solve this dazzling problem.

Mon Mar 29 10:38:29 MEST 1999