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Improved functional form for $ f$ -- method of Methfessel and Paxton

Table 2: Typical convenient settings for $ \sigma $ for different metals: Aluminium possesses an extremely simple DOS, Lithium and Tellurium are also simple nearly free electron metals, therefore $ \sigma $ might be large. For Copper $ \sigma $ is restricted by the fact that the d-band lies approximately 0.5 eV beneath the Fermi-level. Rhodium and Vanadium posses a fairly complex structure in the DOS at the Fermi-level, $ \sigma $ must be small.
  Sigma (eV)
Aluminium $ 1.0 $
Lithium 0.4
Tellurium 0.8
Copper, Palladium 0.4
Vanadium 0.2
Rhodium 0.2
Potassium 0.3

The method described in the last section has two shortcomings:

These problems can be solved by adopting a slightly different functional form for $ f(\{\epsilon _{n{\bf k}}\})$. This is possible by expanding the step function in a complete orthonormal set of functions (method of Methfessel and Paxton [36]). The Gaussian function is only the first approximation (N=0) to the step function, further successive approximations (N=1,2,...) are easily obtained. In similarity to the Gaussian method, the energy has to be replaced by a generalized free energy functional

$\displaystyle \vspace*{1mm}
F =E - \sum_{n{\bf k}}w_{\bf k}\sigma S(f_{n{\bf k}}).

In contrast to the Gaussian method the entropy term $ \sum_{n{\bf k}}w_{\bf k}\sigma S(f_{n{\bf k}})$ will be very small for reasonable values of $ \sigma $ (for instance for the values given in table 2). The $ \sum_{n{\bf k}}w_{\bf k}\sigma S(f_{n{\bf k}})$ is a simple error estimation for the difference between the free energy $ F$ and the 'physical' energy $ E(\sigma \to 0)$. $ \sigma $ can be increased till this error estimation gets too large.

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