EFERMI NEDOS: Difference between revisions
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==Implementation details== | ==Implementation details== | ||
At <math>T=0</math>, the integrated and differential densities of states are | At <math>T=0</math>, the integrated and differential densities of states are | ||
$$ | |||
n(\epsilon)=\sum_{n\mathbf{k}}\theta(\epsilon-\epsilon_{n\mathbf{k}}), \qquad | n(\epsilon)=\sum_{n\mathbf{k}}\theta(\epsilon-\epsilon_{n\mathbf{k}}), \qquad | ||
g(\epsilon)=\sum_{n\mathbf{k}}\delta(\epsilon-\epsilon_{n\mathbf{k}}). | g(\epsilon)=\sum_{n\mathbf{k}}\delta(\epsilon-\epsilon_{n\mathbf{k}}). | ||
$$ | |||
At finite temperature, | At finite temperature, | ||
$$ | |||
N_e(\epsilon_F,T)= | N_e(\epsilon_F,T)= | ||
\sum_{n\mathbf{k}}f(\epsilon_{n\mathbf{k}}-\epsilon_F,T) | \sum_{n\mathbf{k}}f(\epsilon_{n\mathbf{k}}-\epsilon_F,T) | ||
=\int g(\epsilon)f(\epsilon-\epsilon_F,T)\,d\epsilon. | =\int g(\epsilon)f(\epsilon-\epsilon_F,T)\,d\epsilon. \tag{1} | ||
$$ | |||
With the substitution <math>x = 1 - 2f(\epsilon-\epsilon_F,T)</math>, | With the substitution <math>x = 1 - 2f(\epsilon-\epsilon_F,T)</math>, | ||
$$ | |||
\epsilon = k_BT\ln\!\frac{1+x}{1-x}+\epsilon_F, \qquad | \epsilon = k_BT\ln\!\frac{1+x}{1-x}+\epsilon_F, \qquad | ||
d\epsilon = -k_BT\,\frac{2}{x^2-1}\,dx, | d\epsilon = -k_BT\,\frac{2}{x^2-1}\,dx, | ||
$$ | |||
Eq. (1) becomes | Eq. (1) becomes | ||
$$ | |||
N_e(\epsilon_F,T)= | N_e(\epsilon_F,T)= | ||
\frac{1}{2}\int_{-1}^{1} | \frac{1}{2}\int_{-1}^{1} | ||
n\!\left(k_BT\ln\!\frac{1+x}{1-x}+\epsilon_F\right)\,dx. | n\!\left(k_BT\ln\!\frac{1+x}{1-x}+\epsilon_F\right)\,dx. | ||
$$ | |||
In practice, this integral is discretized as | In practice, this integral is discretized as | ||
$$ | |||
N_e(\epsilon_F,T)\simeq | N_e(\epsilon_F,T)\simeq | ||
\frac{1}{2}\sum_{i=1}^{N}w_i\, | \frac{1}{2}\sum_{i=1}^{N}w_i\, | ||
n\!\left(k_BT\ln\!\frac{1+x_i}{1-x_i}+\epsilon_F\right), | n\!\left(k_BT\ln\!\frac{1+x_i}{1-x_i}+\epsilon_F\right), | ||
$$ | |||
where <math>w_i</math> and <math>x_i</math> are Gauss–Legendre weights and abscissas. | where <math>w_i</math> and <math>x_i</math> are Gauss–Legendre weights and abscissas. | ||
The step functions <math>\theta(\epsilon-\epsilon_{n\mathbf{k}})</math> entering <math>n(\epsilon)</math> are evaluated using the tetrahedron method, with the number of energy points <math>N</math> given by {{TAG|EFERMI_NEDOS}}. | The step functions <math>\theta(\epsilon-\epsilon_{n\mathbf{k}})</math> entering <math>n(\epsilon)</math> are evaluated using the tetrahedron method, with the number of energy points <math>N</math> given by {{TAG|EFERMI_NEDOS}}. | ||
Revision as of 12:14, 15 April 2026
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 with ISMEAR = −14 or -15 .
| 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 within the tetrahedron method when ISMEAR = −14 or -15 . Larger values improve accuracy, especially at low temperatures or with sharp features in the electronic DOS, but also increase computational cost. A brief convergence test is recommended in case very accurate occupancies are required, e.g., in the context of transport calculations.
| Mind: ELPH_FERMI_NEDOS is a valid alternative way of writing this tag. |
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 [math]\displaystyle{ \theta(\epsilon-\epsilon_{n\mathbf{k}}) }[/math] entering [math]\displaystyle{ n(\epsilon) }[/math] are evaluated using the tetrahedron method, with the number of energy points [math]\displaystyle{ N }[/math] given by EFERMI_NEDOS.