FermiDirac

class FermiDirac(broadening)
Parameters:broadening (PhysicalQuantity of type energy or temperature) – The broadening of the Fermi-Dirac distribution.
broadening()
Returns:The broadening of the Fermi-Dirac distribution.
Return type:PhysicalQuantity of type energy

Usage Examples

Use the Fermi-Dirac occupation function with a broadening of 0.1 eV on an LCAOCalculator:

numerical_accuracy_parameters = NumericalAccuracyParameters(
    occupation_method=FermiDirac(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 Fermi-Dirac smearing scheme one effectively considers the system at finite temperature. This means that one replaces the integer occupation numbers when calculating e.g. the electron density in DFT by fractional occupation numbers given by the Fermi-Dirac distribution,

\[f(x) = \frac{1}{e^{(\epsilon - \mu)/\sigma} + 1},\]

where \(\epsilon\) is the energy of a given state, \(\mu\) is the chemical potential/Fermi level and \(\sigma\) is a broadening parameter. For finite temperature calculations \(\sigma=k_\text{B} T\) with \(T\) the temperature.

The Fermi-Dirac smearing scheme also corresponds to replacing the Dirac delta-function in the density of states by a smeared function given by

\[\tilde{\delta}(x) = \frac{1}{4\sigma} \frac{1}{\cosh^2(x/2\sigma)}\]

In the Fermi-Dirac smearing scheme the contribution to the generalized entropy from a state with occupation \(f\) is [Mer65]

\[S(f) = -[f \ln f + (1 - f) \ln(1 - f)].\]

The total energy can extrapolated to zero broadening by adding to the total internal energy the following correction term

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

where \(i\) runs over all states.

[Mer65]N. D. Mermin. Thermal properties of the inhomogeneous electron gas. Phys. Rev., 137:A1441–A1443, Mar 1965. doi:10.1103/PhysRev.137.A1441.