Finite temperature approaches -- smearing methods

In this case the step function is simply replaced by a smooth function, for example the Fermi-Dirac function[33]

$$\vspace*{1mm} f(\frac{\epsilon-\mu}{\sigma}) = \frac{1}{{\rm exp}(\frac{\epsilon-\mu}{\sigma})+1}.$$

or a Gauss like function[34]

$$\vspace*{1mm} f(\frac{\epsilon-\mu}{\sigma}) = \frac{1}{2} \... ...erf} \left [ \frac{\epsilon - \mu}{\sigma} \right ] \right ).$$ (50)

is one used quite frequently in the context of solid state calculations. Nevertheless, it turns out that the total energy is no longer variational (or minimal) in this case. It is necessary to replace the total energy by some generalized free energy

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

The calculated forces are now the derivatives of this free energy $F$ (see section 7.5). In conjunction with Fermi-Dirac statistics the free energy might be interpreted as the free energy of the electrons at some finite temperature $\sigma=k_{\rm B} T$, but the physical significance remains unclear in the case of Gaussian smearing. Despite this problem, it is possible to obtain an accurate extrapolation for $sigma \to 0$ from results at finite $sigma$ using the formula

$$E(\sigma \to 0) =E_0= \frac{1}{2} ( F +E) .$$

In this way we get a 'physical' quantity from a finite temperature calculation, and the Gaussian smearing method serves as an mathematical tool to obtain faster convergence with respect to the number of k-points. For Al this method converges even faster than the linear tetrahedron method with Bloechel corrections.