Local SIC

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Local self-interaction correction (LSIC) method is an SIC method proposed recently by our group and collaborators [see Zope, R. R. et al., J. Chem. Phys. 151, 214108 (2019)]. It is a one-electron self-interaction-correction (SIC) method that uses an iso-orbital indicator to apply the SIC at each point in space by scaling the exchange-correlation and Coulomb energy densities. The LSIC method is exact for the one-electron densities, also recovers the uniform electron gas limit of the uncorrected density functional approximation, and reduces to the well-known Perdew-Zunger SIC (PZSIC) method as a special case. Iso-orbital indicator used in LSIC help in identifying the one electron self-interaction regions in many-electron system and in determining the magnitude of SIC in the many-electron regions. LSIC was conceptualized while pruning dense numerical grids required for SIC calculations with meta-GGA functionals and was inspired by the regional SIC and the functionals that use iso-orbital indicators to identify single electron regions. LSIC works well for both equilibrium properties as well as for properties that involve a stretched bond, and it provides an improved performance with respect to standard PZSIC method for a wide range of electronic structure properties.

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

Molecules polarizability (132 molecules from data set of Hait and Head-Gordon)

https://pubs.rsc.org/en/content/articlelanding/2021/cp/d0cp06512a

Exchange coupling constant

https://pubs.acs.org/doi/full/10.1021/acs.jpca.1c10354

In exchange coupling calculations, PZSIC localizes densities responsible for magnetic interactions. Many DFAs and PZSIC-DFAs show a system-dependent, mixed performance whereas LSIC performance seems consistent.

Reaction barrier height

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

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

Anion

https://pubs.rsc.org/en/journals/journalissues/cp#!issueid=cp022007&type=current&issnprint=1463-9076

https://meetings.aps.org/Meeting/MAR20/Session/J58.10

LSIC method was also applied to determine the vertical detachment energies of water cluster anion. This was done in two ways: 1. by taking the energy difference of an anion and a neutral water cluster and 2. from the highest occupied orbital eigenvalues. We find perturbative LSIC performs better than MP2, and quasi-SCF LSIC shows comparable to MP2 level of theory.