Category:Exchange-correlation functionals: Difference between revisions

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:<math>
:<math>
E_{\rm tot}^{\rm KS} = -\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}\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}
</math>
</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. The orbitals <math>\psi_{i}</math> and the electron density <math>\rho=\sum_{i}\left\vert\psi_{i}\right\vert^{2}</math> are calculated by solving 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. The orbitals <math>\psi_{i}</math> and the electron density <math>\rho=\sum_{i}\left\vert\psi_{i}\right\vert^{2}</math> are calculated by solving the KS equations

Revision as of 13:15, 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 are calculated by solving the KS equations

Theoretical Background

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Pages in category "Exchange-correlation functionals"

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