ColdSmearing

class ColdSmearing(broadening)
Parameters:broadening (PhysicalQuantity of type energy or temperature) – The broadening of the distribution.
broadening()
Returns:The broadening of the distribution.
Return type:PhysicalQuantity of type energy

Usage Examples

Use the cold smearing occupation function with a broadening of 0.1 eV on an LCAOCalculator:

numerical_accuracy_parameters = NumericalAccuracyParameters(
    occupation_method=ColdSmearing(0.1*eV))

calculator = LCAOCalculator(numerical_accuracy_parameters=numerical_accuracy_parameters)

Notes

Note

For comparison of different occupation methods and suggestions for which one to choose, see Occupation Methods.

In the cold smearing scheme [MVDVP99] one replaces the delta function in the density of states by the function:

\[\tilde{\delta}(x) = \frac{1}{\sqrt{\pi}} e^{-(x - 1/\sqrt{2})^2} (2 - \sqrt{2} x),\]

The integer occupation numbers are then replaced by fractional occupations determined by the distribution

\[f(x_i) = \frac{1}{2}\left[ \sqrt{\frac{2}{\pi}} e^{-x_i^2 - \sqrt{2} x_i - 1/2} + 1 - \text{erf}\left(x_i + \frac{1}{\sqrt{2}}\right) \right],\]

where \(x_i = \frac{\epsilon_i - \mu}{\sigma}\) with \(\sigma\) the broadening. In this context the generalized entropy becomes

\[S = \frac{1}{\sqrt{\pi}} \sum_i e^{-(x_i + 1/\sqrt{2})^2} (1 + \sqrt{2} x_i)\]

with \(x_i = \frac{\epsilon_i - \mu}{\sigma}\).

Like in the Methfessel-Paxton scheme (see MethfesselPaxton) the benefit of the cold smearing scheme is that contributions to the free energy from the broadening, lower than second order, are eliminated. Therefore, the total energy and forces are quite close to the real (zero temperature) value even for large values of the broadening parameter. Compared to the Methfessel-Paxton scheme, cold smearing has the additional benefit that negative occupations can not occur.

It is possible to extrapolate the total energy to zero broadening by including the term

\[\Delta E_{\sigma\to 0}(\sigma) = \frac{5}{6} \sigma S.\]

[MVDVP99]Nicola Marzari, David Vanderbilt, Alessandro De Vita, and M. C. Payne. Thermal Contraction and Disordering of the Al(110) Surface. Phys. Rev. Lett., 82(16):3296–3299, April 1999. doi:10.1103/PhysRevLett.82.3296.