Category:Exchange-correlation functionals: Difference between revisions

Revision as of 13:12, 6 April 2022

Theoretical Background

In the KS formulation of DFT^[1]^[2], the total energy is given by

{\displaystyle E_{\rm tot} = -\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\ne 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 orbitals $\psi_{i}$ and the electron density $n=\sum_{i}\left\vert\psi_{i}\right\vert^{2}$ that are used to evaluate $E_{\rm tot}$ 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 $E_{\rm tot}$ and in the KS equations that are not known exactly are the exchange-correlation energy functional $E_{\rm xc}$ and potential $v_{\rm xc}=\delta E_{\rm xc}/\delta n$ . Therefore, the accuracy of the calculated properties depends mainly on the approximations used for $E_{\rm xc}$ and $v_{\rm xc}$ . Several hundreds of approximations for the exchange and correlation have been proposed^[3]. They can be classified into several types, like the local density approximation (LDA), generalized gradient approximation (GGA), meta-GGA, and hybrid. Functionals that include van der Waals corrections have also been proposed. 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.

How to

LDA and GGA: GGA.
Meta-GGA: METAGGA.
Hybrids: Hybrid functionals and List of hybrid functionals.
L(S)DA+ $U$ : LDAUTYPE.
van der Waals methods: IVDW

References

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.

A

B

BPARAM

C

D

E

G

H

I

IVDW
IVDW NL

M

N

O

ODDONLY

P

R

RSMBJ

S

T

V

X

XC
XC C

Z

ZAB VDW

[hohenberg:pr:1964-1] P. Hohenberg and W. Kohn, Phys. Rev. 136, B864 (1964).

[kohn:pr:1965-2] W. Kohn and L. J. Sham, Phys. Rev. 140, A1133 (1965).

[libxc_list-3] ttps://libxc.gitlab.io/functionals/

[1]

[2]

[3]

@@ Line 4: / Line 4: @@
 :<math>
-E_{\rm tot} = -\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}\rho({\bf r})d^{3}r + \frac{1}{2}\int\int\frac{\rho({\bf r})\rho({\bf r'})}{\left\vert{\bf r}-{\bf r'}\right\vert}d^{3}rd^{3}r' + E_{\rm xc} + \frac{1}{2}\sum_{A\ne B}\frac{Z_{A}Z_{B}}{\left\vert{\bf R}_{A}-{\bf R}_{B}\right\vert}
+E_{\rm tot} = -\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\ne B}\frac{Z_{A}Z_{B}}{\left\vert{\bf R}_{A}-{\bf R}_{B}\right\vert}
 </math>
-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 orbitals <math>\psi_{i}</math> and the electron density <math>\rho=\sum_{i}\left\vert\psi_{i}\right\vert^{2}</math> that are used to evaluate <math>E_{\rm tot}</math> are obtained by solving self-consistently the KS equations
+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 orbitals <math>\psi_{i}</math> and the electron density <math>n=\sum_{i}\left\vert\psi_{i}\right\vert^{2}</math> that are used to evaluate <math>E_{\rm tot}</math> are obtained by solving self-consistently the KS equations
 :<math>
-\left(-\frac{1}{2}\nabla^{2} -\sum_{A}\frac{Z_{A}}{\left\vert{\bf r}-{\bf R}_{A}\right\vert} + \int\frac{\rho({\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})
+\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})
 </math>
-The only terms in <math>E_{\rm tot}</math> and in the KS equations that are not known exactly are the exchange-correlation energy functional <math>E_{\rm xc}</math> and potential <math>v_{\rm xc}=\delta E_{\rm xc}/\delta\rho</math>. Therefore, the accuracy of the calculated properties depends mainly on the approximations used for <math>E_{\rm xc}</math> and <math>v_{\rm xc}</math>. Several hundreds of approximations for the exchange and correlation have been proposed{{cite|libxc_list}}. They can be classified into several types, like the local density approximation (LDA), generalized gradient approximation (GGA), meta-GGA, and hybrid. Functionals that include van der Waals corrections have also been proposed. 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.
+The only terms in <math>E_{\rm tot}</math> and in the KS equations that are not known exactly are the exchange-correlation energy functional <math>E_{\rm xc}</math> and potential <math>v_{\rm xc}=\delta E_{\rm xc}/\delta n</math>. Therefore, the accuracy of the calculated properties depends mainly on the approximations used for <math>E_{\rm xc}</math> and <math>v_{\rm xc}</math>. Several hundreds of approximations for the exchange and correlation have been proposed{{cite|libxc_list}}. They can be classified into several types, like the local density approximation (LDA), generalized gradient approximation (GGA), meta-GGA, and hybrid. Functionals that include van der Waals corrections have also been proposed. 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.
 == How to ==