# Category:Exchange-correlation functionals

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

${\displaystyle E_{\rm {tot}}^{\rm {KS-DFT}}=-{\frac {1}{2}}\sum _{i}\int \psi _{i}^{*}({\bf {r}})\nabla ^{2}\psi _{i}({\bf {r}})d^{3}r-\sum _{A}\int {\frac {Z_{A}}{\left\vert {\bf {r}}-{\bf {R}}_{A}\right\vert }}n({\bf {r}})d^{3}r+{\frac {1}{2}}\int \int {\frac {n({\bf {r}})n({\bf {r'}})}{\left\vert {\bf {r}}-{\bf {r'}}\right\vert }}d^{3}rd^{3}r'+E_{\rm {xc}}+{\frac {1}{2}}\sum _{A\neq B}{\frac {Z_{A}Z_{B}}{\left\vert {\bf {R}}_{A}-{\bf {R}}_{B}\right\vert }}}$

where the terms on the right-hand side represent the non-interacting kinetic energy of the electrons, the electrons-nuclei attraction energy, the classical Coulomb electron-electron repulsive energy, the exchange-correlation energy, and the nuclei-nuclei repulsion energy, respectively. The KS orbitals ${\displaystyle \psi _{i}}$ and the electronic density ${\displaystyle n=\sum _{i}\left\vert \psi _{i}\right\vert ^{2}}$ that are used to evaluate ${\displaystyle E_{\rm {tot}}^{\rm {KS-DFT}}}$ are obtained by solving self-consistently the KS equations

${\displaystyle \left(-{\frac {1}{2}}\nabla ^{2}-\sum _{A}{\frac {Z_{A}}{\left\vert {\bf {r}}-{\bf {R}}_{A}\right\vert }}+\int {\frac {n({\bf {r'}})}{\left\vert {\bf {r}}-{\bf {r'}}\right\vert }}d^{3}r'+v_{\rm {xc}}({\bf {r}})\right)\psi _{i}({\bf {r}})=\epsilon _{i}\psi _{i}({\bf {r}}).}$

The only terms in ${\displaystyle E_{\rm {tot}}^{\rm {KS-DFT}}}$ and in the KS equations that are not known exactly are the exchange-correlation energy functional ${\displaystyle E_{\rm {xc}}}$ and potential ${\displaystyle v_{\rm {xc}}=\delta E_{\rm {xc}}/\delta n}$. Therefore, the accuracy of the calculated properties depends strongly on the approximations used for ${\displaystyle E_{\rm {xc}}}$ and ${\displaystyle v_{\rm {xc}}}$.

Several hundreds of approximations for the exchange and correlation have been proposed[3]. They can be classified into families: the local density approximation (LDA), generalized gradient approximation (GGA), meta-GGA, and hybrid. There is also the possibility to include van der Waals corrections or an on-site Coulomb repulsion using DFT+U on top of another functional. More details on the different types of approximations available in VASP and how to use them can be found in the pages and subcategories listed below.

## Subcategories

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

## Pages in category "Exchange-correlation functionals"

The following 97 pages are in this category, out of 97 total.