EFERMI_NEDOS
EFERMI_NEDOS = [integer]
Default: EFERMI_NEDOS = 21
Description: Number of Gauss–Legendre integration points 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 |
EFERMI_NEDOS sets the number of points in the Gauss–Legendre grid used to integrate the Fermi–Dirac distribution for determining the Fermi level. Larger values improve accuracy, especially at low temperatures or with sharp DOS features, but also increase computational cost. A brief convergence test is recommended.
Implementation details
At [math]\displaystyle{ T=0 }[/math], the integrated and differential densities of states are $$ n(\epsilon)=\sum_{n\mathbf{k}}\theta(\epsilon-\epsilon_{n\mathbf{k}}), \qquad g(\epsilon)=\sum_{n\mathbf{k}}\delta(\epsilon-\epsilon_{n\mathbf{k}}). $$
At finite temperature, $$ N_e(\epsilon_F,T)= \sum_{n\mathbf{k}}f(\epsilon_{n\mathbf{k}}-\epsilon_F,T) =\int g(\epsilon)f(\epsilon-\epsilon_F,T)\,d\epsilon. \tag{1} $$
With the substitution [math]\displaystyle{ x = 1 - 2f(\epsilon-\epsilon_F,T) }[/math], $$ \epsilon = k_BT\ln\!\frac{1+x}{1-x}+\epsilon_F, \qquad d\epsilon = -k_BT\,\frac{2}{x^2-1}\,dx, $$ Eq. (1) becomes $$ N_e(\epsilon_F,T)= \frac{1}{2}\int_{-1}^{1} n\!\left(k_BT\ln\!\frac{1+x}{1-x}+\epsilon_F\right)\,dx. $$
In practice, this integral is discretized as $$ N_e(\epsilon_F,T)\simeq \frac{1}{2}\sum_{i=1}^{N}w_i\, n\!\left(k_BT\ln\!\frac{1+x_i}{1-x_i}+\epsilon_F\right), $$ where [math]\displaystyle{ w_i }[/math] and [math]\displaystyle{ x_i }[/math] are Gauss–Legendre weights and abscissas. The step functions \(\theta(\epsilon-\epsilon_{n\mathbf{k}})\) entering \(n(\epsilon)\) are evaluated using the tetrahedron method, with the number of energy points [math]\displaystyle{ N }[/math] given by EFERMI_NEDOS.
Related tags and articles
ISMEAR, SIGMA, Smearing technique, K-point integration