# GaussianSmearing¶

class GaussianSmearing(broadening)
Parameters: broadening (PhysicalQuantity of type energy or temperature) – The broadening of the distribution.
broadening()
Returns: The broadening. 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 [FH83] 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

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

 [FH83] C. -L. Fu and K. -M. Ho. First-principles calculation of the equilibrium ground-state properties of transition metals: Applications to Nb and Mo. Phys. Rev. B, 28(10):5480–5486, November 1983. doi:10.1103/PhysRevB.28.5480.