Category:Exchange-correlation functionals: Difference between revisions

From VASP Wiki
No edit summary
No edit summary
Line 4: Line 4:
E_{\rm tot}^{\rm KS}[\rho] = T_{\rm s}[\{\psi_{i}\}] + U_{\rm H}[\rho] + E_{\rm xc} + V_{\rm en}[\rho] + V_{\rm nn}
E_{\rm tot}^{\rm KS}[\rho] = T_{\rm s}[\{\psi_{i}\}] + U_{\rm H}[\rho] + E_{\rm xc} + V_{\rm en}[\rho] + V_{\rm nn}
</math>
</math>
where <math>T_{\rm s}</math> is the non-interacting kinetic energy of the electrons, <math>J</math> the Hartree energy, the third term is the energy of the electrons-nuclei attraction interaction, and <math>V_{\rm nn}</math> is the nuclei-nuclei repulsion energy.
where
 
:<math>
:<math>
T_{\rm s}[\{\psi_{i}\}]=-\frac{1}{2}\sum_{i=1}^{N}\int
T_{\rm s}[\{\psi_{i}\}]=-\frac{1}{2}\sum_{i=1}^{N}\int
\psi_{i}^{*}({\bf r})\nabla^{2}\psi_{i}({\bf r})d^{3}r
\psi_{i}^{*}({\bf r})\nabla^{2}\psi_{i}({\bf r})d^{3}r
</math>
</math>
is the non-interacting kinetic energy of the electrons,
:<math>
:<math>
U_{\rm H}[\rho] = \int v_{\rm ext}({\bf r})\rho({\bf r})d^{3}r
U_{\rm H}[\rho] = \int v_{\rm ext}({\bf r})\rho({\bf r})d^{3}r
</math>
</math>
is the Classical Coulomb Hartree term,
:<math>
:<math>
V_{\rm en}[\rho] =
V_{\rm en}[\rho] =
Line 18: Line 19:
{\left\vert{\bf r}-{\bf r'}\right\vert}d^{3}rd^{3}r',
{\left\vert{\bf r}-{\bf r'}\right\vert}d^{3}rd^{3}r',
</math>
</math>
is the electrons-nuclei attraction energy and
:<math>
:<math>
V_{\rm nn} = \frac{1}{2}\sum_{A B}^{M-1}
V_{\rm nn} = \frac{1}{2}\sum_{A B}^{M-1}
\frac{Z_{A}Z_{B}}{\left\vert{\bf R}_{A}-{\bf R}_{B}\right\vert}
\frac{Z_{A}Z_{B}}{\left\vert{\bf R}_{A}-{\bf R}_{B}\right\vert}
</math>
</math>
is the nuclei-nuclei repulsion energy.


== Theoretical Background ==
== Theoretical Background ==

Revision as of 10:52, 18 January 2022

In Kohn-Sham density functional theory (DFT)[1][2], the total energy is given by

where

is the non-interacting kinetic energy of the electrons,

is the Classical Coulomb Hartree term,

is the electrons-nuclei attraction energy and

is the nuclei-nuclei repulsion energy.

Theoretical Background

How to


Subcategories

This category has the following 5 subcategories, out of 5 total.

Pages in category "Exchange-correlation functionals"

The following 118 pages are in this category, out of 118 total.