MethfesselPaxton

class MethfesselPaxton(broadening, order=None)
Parameters:
  • broadening (PhysicalQuantity of type energy or temperature) – The broadening of the distribution.
  • order (int) – The order of the distribution. Must be an integer >= 0.
    Default: 1
broadening()
Returns:The broadening.
Return type:PhysicalQuantity of type energy
order()
Returns:The order.
Return type:int

Usage Examples

Use the Methfessel-Paxton occupation function with a broadening of 0.1 eV on an LCAOCalculator:

numerical_accuracy_parameters = NumericalAccuracyParameters(
    occupation_method=MethfesselPaxton(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 smearing scheme by Methfessel and Paxton [MP89] one replaces the delta function in the density of states by its finite expansion in Hermite polynomials

\[\tilde{\delta}_N(x) = \sum_{n=0}^N A_n H_{2n}(x) e^{-x^2},\]

where \(A_n = \frac{(-1)^n}{n! 4^n \sqrt{\pi}}\) and \(H_n(x)\) is the Hermite polynomial of order \(n\). The 0 order approximation is simply a Gaussian distribution.

The integer occupation numbers are therefore replaced by fractional occupations given by the distribution

\[f_N(x) = f_0(x) + \sum_{n=1}^N A_n H_{2n-1}(x) e^{-x^2},\]

with \(x = \frac{\epsilon - \mu}{\sigma}\).

In the Methfessel-Paxton smearing scheme the generalized entropy is

\[S_N = \frac{1}{2} A_n \sum_i H_{2n}(x_i) e^{-x_i^2}\]

where \(x_i = \frac{\epsilon_i - \mu}{\sigma}\) and \(i\) runs over all states.

The benefit of the Methfessel-Paxton smearing function is that for orders \(N>0\) the total free energy has no linear or quadratic dependence on the broadening. This means that total energies and forces are close to their zero broadening value even for large values of the broadening parameter. In fact, it is often not necessary to extrapolate the energy to zero broadening. However, an estimate of the zero broadening correction is given by the expression

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

While it is possible in ATK to use any order of the Methfessel-Paxton occupation function, there is typically no benefit of going to higher orders than \(N=1\), which is also the default.

Note

The order \(N=0\) Methfessel-Paxton occupation function is identical to the one in the Gaussian smearing scheme. This does therefore not have the favorable cancellation of the second order dependence of the broadening on the free energy.

[MP89]M. Methfessel and A. T. Paxton. High-precision sampling for Brillouin-zone integration in metals. Phys. Rev. B, 40(6):3616–3621, August 1989. doi:10.1103/PhysRevB.40.3616.