Local SIC

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The local self-interaction correction (LSIC) method is a new SIC method for addressing the limitation of PZSIC. Although PZSIC is an exact theory for one-electron SIE removal, it often worsens the good performance of density functional approximations. LSIC corrects this by scaling down SIC in many-electron regions.

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 a one-electron self-interaction correction scheme and is exact self-interaction cancellation for one electron cases. PZSIC is nearly many-electron SIE-free.

LSIC equation

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

In the 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 regions.

Quasi-SCF approach

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

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

The quasi-SCF approach gives better HOMO (closer agreement with NIST IPs) than the original LSIC. But have problems in the spin crossover systems(?).

Gauge problem

With LSIC with PBE or SCAN, the XC energy gauge differs from the Coulomb gauge.

Successes of LSIC method

Static dipole polarizability

water cluster paper

polyacene paper

Mol pol paper