Category:Exchange-correlation functionals: Difference between revisions

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\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{\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})
</math>
</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 <math>E_{\rm xc}</math> and potential <math>v_{\rm xc}=\delta E_{\rm xc}/\delta\rho</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 <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}=\delta E_{\rm xc}/\delta\rho</math>.
In the KS-DFT method. Therefore, the accuracy of the calculated properties depends mainly on the approximations used for <math>E_{\rm xc}</math> and <math>v_{\rm xc}=\delta E_{\rm xc}/\delta\rho</math>.


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

Revision as of 13:49, 18 January 2022

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

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. The orbitals and the electron density that are used to evaluate are obtained by solving self-consistently the KS equations

The only terms in and in the KS equations that are not known exactly are the exchange-correlation energy and potential . Therefore, the accuracy of the calculated properties depends mainly on the approximations used for and .

Theoretical Background

How to


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Pages in category "Exchange-correlation functionals"

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