Local SIC

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PZSIC equation

$E^{PZSIC-DFA}=E^{DFA}[{\rho }_{\uparrow },{\rho }_{\downarrow }]-\sum _i\{U^{}[{\rho }_{i\sigma }]+E_{XC}^{DFA}[{\rho }_{i\sigma },0]\}.$

LSIC equation

$E_{XC}^{LSIC-DFA}=E_{XC}^{DFA}[{\rho }_{\uparrow },{\rho }_{\downarrow }]-\sum _{i,\sigma }^{occ}\{U^{LSIC}[{\rho }_{i,\sigma }]+E_{XC}^{LSIC}[{\rho }_{i,\sigma },0]\}$

where

$U^{LSIC}[{\rho }_{i,\sigma }]=\frac{1}{2}\int d\overrightarrow{r}\{z_{\sigma }(\overrightarrow{r}){\}}^k{\rho }_{i,\sigma }(\overrightarrow{r})\int d\overrightarrow{r^{\prime }}\frac{{\rho }_{i,\sigma }(\overrightarrow{r^{\prime }})}{|\overrightarrow{r}-\overrightarrow{r^{\prime }}|}$

$E_{XC}^{LSIC}[{\rho }_{i,\sigma },0]=\int d\overrightarrow{r}\{z_{\sigma }(\overrightarrow{r}){\}}^k{\rho }_{i,\sigma }(\overrightarrow{r}){ϵ}_{XC}^{DFA}([{\rho }_{i,\sigma },0],\overrightarrow{r}).$