EFERMI NEDOS: Difference between revisions

Revision as of 11:27, 15 October 2025

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.

@@ Line 2: / Line 2: @@
 {{TAGDEF|EFERMI_NEDOS|[integer]|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.
+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 {{TAG|ISMEAR}} = −15 or −14.
 {{Available|6.5.0}}
 ----
-During the self-consistent solution of the electronic structure, the Fermi level is obtained by integrating the electronic density of states weighted by the Fermi–Dirac occupation function.
+'''EFERMI_NEDOS''' sets the number of points in the Gauss–Legendre grid used to integrate the Fermi–Dirac distribution for determining the Fermi level.
-By performing a variable transformation, this integral can be efficiently evaluated using Gauss–Legendre quadrature.
+Larger values improve accuracy, especially at low temperatures or with sharp DOS features, but also increase computational cost.
-The parameter '''EFERMI_NEDOS''' controls the number of quadrature points used in this integration.
+A brief convergence test is recommended.
-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 density of states near the Fermi energy.
+==Implementation details==
-However, very high values may lead to unnecessary computational overhead without a significant change in the resulting Fermi level.
+At <math>T=0</math>, the integrated and differential densities of states are
-A short convergence test is recommended to find an optimal balance between accuracy and cost.
+$$
+n(\epsilon)=\sum_{n\mathbf{k}}\theta(\epsilon-\epsilon_{n\mathbf{k}}), \qquad
+g(\epsilon)=\sum_{n\mathbf{k}}\delta(\epsilon-\epsilon_{n\mathbf{k}}).
+$$
+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>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>w_i</math> and <math>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>N</math> given by {{TAG|EFERMI_NEDOS}}.
 ==Related tags and articles==
@@ Line 21: / Line 54: @@
 [[K-point integration]]
-[[Category:INCAR tag]][[Category:Electronic occupancy]][[Category:Electronic minimization]][[Category:Density of states]]
+<!--[[Category:INCAR tag]]
+[[Category:Electronic occupancy]]
+[[Category:Electronic minimization]]
+[[Category:Density of states]]-->