GaussianSmearing¶

class GaussianSmearing(broadening)¶

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

broadening()¶

Returns:: The broadening.
Return type:: PhysicalQuantity of type energy

Usage Examples¶

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

numerical_accuracy_parameters = NumericalAccuracyParameters(
    occupation_method=GaussianSmearing(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 Gaussian smearing scheme [1] one replaces the delta function in the density of states by a Gaussian distribution:

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

where \(\sigma\) is the broadening. This means that the integer occupation numbers are replaced by fractional occupations given by the distribution

\[f(\epsilon) = \frac{1}{2} \left[ 1 - \text{erf}\left( \frac{\epsilon - \mu}{\sigma}\right) \right],\]

where \(\epsilon\) is the energy of the state and \(\mu\) is the Fermi level.

In the Gaussian smearing scheme the generalized entropy is given by

\[S = \sum_i \frac{1}{2\sqrt{\pi}} \exp\left( -\left(\frac{\epsilon_i - \mu}{\sigma}\right)^2 \right)\]

The total energy at zero broadening can be estimated by adding the correction given by

\[\Delta E_{\sigma \to 0}(\sigma) = -\frac{1}{2} \sigma S(\sigma)\]