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:

• The forces calculated by VASP are a derivative of the free electronic energy F (see section 7.5). Therefore the forces can not be used to obtain the equilibrium groundstate, which corresponds to an energy-minimum of $E(\sigma \to 0)$. Nonetheless the error in the forces is generally small and acceptable.
• The parameter $sigma$ must be chosen with great care. If $sigma$ is too large the energy $E(\sigma \to 0)$ will converge to the wrong value even for an infinite k-point mesh, if $sigma$ is too small the convergence speed with the number of k-points will deteriorate. An optimal choice for $sigma$ for several cases is given in table 2. The only way to get a good $sigma$ is by performing several calculations with different k-point meshes and different parameters for $sigma$.

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

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

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.