Category:Exchange-correlation functionals

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

{\displaystyle 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} }

where $T_{\rm s}$ is the non-interacting kinetic energy of the electrons, $J$ the Hartree energy, the third term is the energy of the electrons-nuclei attraction interaction, and $V_{\rm nn}$ is the nuclei-nuclei repulsion energy.

{\displaystyle 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 }

{\displaystyle V_{\rm en}[\rho] = \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', }

{\displaystyle U_{\rm H}[\rho] = \int v_{\rm ext}({\bf r})\rho({\bf r})d^{3}r }

Failed to parse (Conversion error. Server ("cli") reported: "[INVALID]"): {\displaystyle V_{\rm nn} = \frac{1}{2}\sum_{A\noteq B}^{M-1} \frac{Z_{A}Z_{B}}{\left\vert{\bf R}_{A}-{\bf R}_{B}\right\vert} }