ISMEAR

EFERMI_NEDOS = [integer]
Default: EFERMI_NEDOS = 21 

Description: Choose the number of points in the Gauss–Legendre integration grid used to evaluate the Fermi–Dirac distribution and determine the Fermi level at finite temperature using the tetrahedron method. Only relevant when ISMEAR = −15 or −14.

Mind: Available as of VASP 6.5.0

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During the self-consistent solution of the electronic structure, the Fermi level is obtained by integrating the electronic density of states (DOS) weighted by the Fermi–Dirac occupation function. By performing a variable transformation, this integral can be efficiently evaluated using Gauss–Legendre quadrature. The parameter EFERMI_NEDOS controls the number of quadrature points used in this integration.

Increasing the number of integration points generally improves the precision of the computed Fermi level, particularly at low temperatures or in systems with sharp features in the DOS near the Fermi energy. However, very high values may lead to unnecessary computational overhead without a significant change in the resulting Fermi level. A short convergence test is recommended to find an optimal balance between accuracy and cost.

Computation of the number of electrons

The integrated and differential densities of states at [math]\displaystyle{ T=0 }[/math] are given by $$ n(\epsilon)=\sum_{n\mathbf{k}}\theta(\epsilon-\epsilon_{n\mathbf{k}}), \qquad g(\epsilon)=\sum_{n\mathbf{k}}\delta(\epsilon-\epsilon_{n\mathbf{k}}). $$

The total number of electrons can be written either as a sum over Fermi occupations at finite [math]\displaystyle{ T }[/math] or as an integral over the DOS: $$ N_e(\epsilon_F,T)= \sum_{n\mathbf{k}}f(\epsilon_{n\mathbf{k}}-\epsilon_F,T) =\int_{-\infty}^{\infty}g(\epsilon)f(\epsilon-\epsilon_F,T)\,d\epsilon. \tag{1} $$

Making the substitution $$ x = 1 - 2f(\epsilon-\epsilon_F,T), $$ we obtain $$ \epsilon = k_BT\ln\!\frac{1+x}{1-x}+\epsilon_F, \qquad d\epsilon = -k_BT\,\frac{2}{x^2-1}\,dx. $$

Using this change of variable, the integral in Eq. (1) becomes $$ N_e(\epsilon_F,T) =\frac{1}{2}\int_{-1}^{1} n\!\left(k_BT\ln\!\frac{1+x}{1-x}+\epsilon_F\right)\,dx, $$ which is the expression evaluated numerically in VASP using Gauss–Legendre quadrature with EFERMI_NEDOS points.

In practice, this integral is discretized as a weighted sum over [math]\displaystyle{ N }[/math] energy grid points: $$ N_e(\epsilon_F,T) \simeq \frac{1}{2}\sum_{i=1}^{N} w_i\, n\!\left(k_BT\ln\!\frac{1+x_i}{1-x_i}+\epsilon_F\right), $$ where \(w_i\) and \(x_i\) are the Gauss–Legendre weights and abscissas. The step functions \(\theta(\epsilon-\epsilon_{n\mathbf{k}})\) entering \(n(\epsilon)\) are evaluated using the tetrahedron method. The number of energy points [math]\displaystyle{ N }[/math] is defined by EFERMI_NEDOS.

Related tags and articles

ISMEAR, SIGMA, Smearing technique, K-point integration