Selective orbital scaling self-interaction correction (SOSIC)

Selective orbital scaling self-interaction correction (SOSIC) is a new SIC method that eliminates self-interaction errors. The SOSIC method is based on Perdew, and Zunger's SIC (PZSIC) proposed on 1981 (https://journals.aps.org/prb/abstract/10.1103/PhysRevB.23.5048). PZSIC calculation eliminates the one electron self-interaction error (SIE) produced by an improper cancellation of the self-Coulomb and self-Exchange. SOSIC is one of the approaches to scaling down the excessive SI correction seen in PZSIC, where scaling down is applied in the occupation space. The idea of SOSIC is similar to the orbital scaling (OSIC) approach; the difference is that SOSIC applies SIC to those orbitals where the correction is essential while leaving the rest of the orbitals at the DFA level. Unlike PZSIC, which is exact for the one-electron case, SOSIC (and OSIC) may not necessarily be exact for the one-electron case when an orbital scaling factor ($\ne 1$) is used.

PZSIC theory

Density functional approximations (DFAs) suffer from self-interaction error (SIE) due to the incomplete cancellation of self-Coulomb and self-Exchange energies. One way to eliminate SIE is to use the one electron self-interaction-correction (SIC) method (e.g., Perdew and Zunger SIC).

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

This is an orbital-dependent theory. In earlier PZSIC applications, Kohn Sham orbitals were used. But PZSIC used this way suffers from size extensivity problem (cite size extensivity papers here) and consequently requires a localized set of orbitals, e.g., Fermi Loewdin orbitals (FLOs) which are Loewdin orthogonalized Fermi orbitals (FO). An FO is constructed as follows,

$F_{j\sigma }(\overrightarrow{r})=\frac{\sum _i{\psi }_{i\sigma }({\overrightarrow{a}}_{j\sigma }){\psi }_{i\sigma }(\overrightarrow{r})}{\sqrt{{\rho }_{\sigma }({\overrightarrow{a}}_{j\sigma })}}$

Here $i$ represents the KS occupation index (KS-orbitals) and $j$ is the FO index (local orbital), ${\psi }_i$ is the KS-orbital, $\rho$ is the total electron spin density, and ${\overrightarrow{a}}_j$ is the so-called Fermi orbital descriptor (FOD) position.

PZSIC improves many properties, e.g., barrier heights for chemical reactions, and predicts stable atomic anions. But for some properties, especially thermochemical properties, PZSIC performs poorly, mainly when used with GGAs, and meta-GGAs functionals. PZSIC incorrectly predicts spin crossover (SCO) even with LSDA, and this is typically due to excessive PZSIC correction.

Scaling methods for PZSIC

PZSIC overcorrecting tendency inferred a decrease in the SIC amount for the PZSIC method (cite papers for scaled SIC). The most successful scaling scheme is the locally scaled self-interaction correction (LSIC) proposed principally by Drs Zope, Yamamoto, and Baruah from UTEP electronic structure lab (ESLAB) (cite LSIC or website) showed an outstanding performance for the chemical reaction of barrier heights, atomization energies, highest occupied molecular (HOMO) eigenvalues (ionization energies), polarizabilities, and some other applications (cite those).

Orbital scaled SIC (OSIC) total energy is defined as,

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

where each local orbital-wise scaling factor $X_{i\sigma }^k$ is defined as,

$X_{i\sigma }^k=\int z_{\sigma }^k(\overrightarrow{r}){\rho }_{i\sigma }(\overrightarrow{r})d^3\overrightarrow{r}.$

Here, $i$ indicates the orbital index, $\sigma$ is the spin index, $z_{\sigma }$ is the iso-orbital indicator and $k$ is an integer power. The original LSIC iso-orbital indicator is the ratio of Von Weizaker kinetic energy density ${\tau }_{\sigma }^W(\overrightarrow{r})=\frac{|\overrightarrow{\mathrm{\nabla }}{\rho }_{\sigma }(\overrightarrow{r}){|}^2}{8{\rho }_{\sigma }(\overrightarrow{r})}$ and the non-interacting kinetic energy density ${\tau }_{\sigma }(\overrightarrow{r})=\frac{1}{2}\sum _i|\overrightarrow{\mathrm{\nabla }}{\psi }_{i\sigma }(\overrightarrow{r}){|}^2$, this being defined as follows,

$z_{\sigma }=\frac{{\tau }_{\sigma }^W(\overrightarrow{r})}{{\tau }_{\sigma }(\overrightarrow{r})}$

Though OSIC succeeded in improving SIC performance, this approach leads to parameter $k$ dependent performance. (One exception is OSIC-PBE, where the k~1/2 works well for both barrier heights and atomizations, etc.). It also provides an incorrect asymptotic potential as $-X_{HO}/r$ instead of the correct $-1/r$ and results in an inaccurate dissociation energy curve. This $-X_{HO}/r$ behavior of OSIC was mentioned in Scuseria's article, and from there, we got an idea to tune the orbital scaling factor to make the HOMO eigenvalue match experimental IPs. Another misbehavior is that many-electron SIE and fractional-charge dissociation behavior of positively charger dimers reemerge with OSIC.

The later proposed LSIC from our lab members (Zope, Yamamoto, and Baruah) showed an outstanding performance for properties such as chemical reactions of barrier heights, atomization energies, dissociation energies, dipole moments, polarizabilities, etc., by reducing the PZSIC overcorrecting tendency fixed deep highest occupied molecular (HOMO) eigenvalues when compared to experimental ionization energies. We found recently that PZSIC overcorrecting tendency completely misdescribed the spin crossover (SCO) of Fe(II) octahedral systems when compared to the diffusion Monte Carlo (DMC) approach, and astonishingly LSIC reaches an accuracy comparable to or better to local-hybrid functionals.

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

where,

$U^{LSIC}[{\rho }_{i\sigma }]=\frac{1}{2}\int d^{3}r{z}_{\sigma }(\overrightarrow{r}){\rho }_{i\sigma }(\overrightarrow{r})\int d^3{\overrightarrow{r}}^{\prime }\frac{{\rho }_{i\sigma }({\overrightarrow{r}}^{\prime })}{|\overrightarrow{r}-{\overrightarrow{r}}^{\prime }|},$

$E_{XC}^{LSIC}[{\rho }_{i\sigma },0]=\int d^{3}r{z}_{\sigma }(\overrightarrow{r}){\rho }_{i\sigma }(\overrightarrow{r}){ϵ}_{XC}^{DFA}([{\rho }_{i\sigma },0],\overrightarrow{r}).$

The greatest difference between LSIC and any of the previous scaling schemes is that LSIC reduces SIC amount according to an iso-orbital indicator which interpolates within uniform electron gas ($z_{\sigma }=0$) and single electron region ($z_{\sigma }=1$) for the original $z_{\sigma }$ iso-orbital indicator. Our lab has also proposed a simpler iso-orbital indicator inspired by Vydrov and co-workers' OSIC(w) work defined as $w_{i\sigma }={\rho }_i/\rho$

We have also proposed another scaling SIC scheme inspired by OSIC with more freedom to treat each orbital correction, which is called selectively orbital scaling self-interaction correction (SOSIC), and its energy definition is,

$E_{XC}^{SOSIC-DFA}=E_{XC}^{DFA}[{\rho }_{\uparrow },{\rho }_{\downarrow }]-\sum _{i\sigma =1}^P{X}_{i\sigma }^k(U[{\rho }_{i\sigma }]+E_{XC}^{DFA}[{\rho }_{i\sigma },0])-\sum _{i\sigma =P+1}^{occ}{Y}_{i\sigma }^k(U[{\rho }_{i\sigma }]+E_{XC}^{DFA}[{\rho }_{i\sigma },0])$

Yamamoto and co-workers showed that applying SIC where required improves the properties of higher functionals such as meta-GGA SCAN, but that work wasn't atomized yet. Note that P occupation is not straightforward and can be more complicated and beneficial for specific applications.