Local SIC

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PZSIC equation

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

This is one-electron self-interaction correction scheme and is exact self-interaction cancellation for one electron cases.

LSIC equation

Reference: https://aip.scitation.org/doi/figure/10.1063/1.5129533

In local SIC method, PZSIC energy densities are scaled down with a local scaling factor or iso-orbital indicator z(r).

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

where

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

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

A typical choice for the local scaling factor is

$z(\mathbf{𝐫})=\frac{{\tau }_w(\mathbf{𝐫})}{\tau (\mathbf{𝐫})}$

This quantity becomes unity at the single orbital regions and becomes zero at the uniform density regions. It cannot differentiate weak interaction regions and uniform density region.

Quasi-SCF approach

You can apply the local scaling to PZSIC potentials in place of PZSIC energy densities. This approach is referred as quasi-SCF. The resulting Hamiltonian is as follows,

$H_j=-\frac{1}{2}{\mathrm{\nabla }}^2+v_{ext}(\overrightarrow{r})+\int \frac{\rho (\overrightarrow{r^{\prime }})}{|\overrightarrow{r}-\overrightarrow{r^{\prime }}|}d\overrightarrow{r^{\prime }}+v_{XC}^{DFA}([\rho ],\overrightarrow{r})-f(\overrightarrow{r})(\int \frac{{\rho }_j(\overrightarrow{r^{\prime }})}{|\overrightarrow{r}-\overrightarrow{r^{\prime }}|}d\overrightarrow{r^{\prime }}+v_{XC}^{DFA}([{\rho }_j],\overrightarrow{r}))$

Successes of LSIC method

Static dipole polarizability

water cluster paper

polyacene paper

Mol pol paper