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

Static dipole polarizabilities in DFA are often overestimated because of notorious delocalization errors. Application of PZSIC localizes density, but the polarizabilities are then underestimated. Here, PZSIC potentials are locally scaled down (quasi-SCF)

Water clusters

https://aip.scitation.org/doi/10.1063/5.0025601

Polyacenes

https://aip.scitation.org/doi/10.1063/5.0041265

Mol pol paper

Reaction barrier height

https://aip.scitation.org/doi/10.1063/5.0070893

LSIC works very well for hydrogen transfer types of reaction barriers.