FLO-SIC: Correcting Self-Interaction in DFT

• FLO-SIC is a method that corrects self-interaction errors in density functional approximations by constructing localized Fermi–Löwdin orbitals.
• It uses iso-orbital indicators to selectively apply corrections, ensuring accurate predictions of frontier orbital and ionization energies.
• Gradient-based optimization of Fermi orbital descriptors achieves quasi-particle-level accuracy with moderate computational cost.

The Fermi-Löwdin Orbital Self-Interaction Correction (FLO-SIC) scheme is an advanced approach for eliminating the self-interaction error (SIE) inherent in standard Kohn–Sham density functional approximations (DFAs). FLO-SIC implements the Perdew–Zunger self-interaction correction (PZ-SIC) by constructing localized orbitals—Fermi–Löwdin orbitals (FLOs)—which enable a size-extensive, unitarily invariant, and computationally efficient scheme for correcting one-electron self-interactions. This method is especially effective when coupled with local scaling functions based on iso-orbital indicators, resulting in accurate predictions of frontier orbital energies, ionization energies, and related electronic properties in molecular and extended systems (Adhikari et al., 2020).

1. Self-Interaction Error in Semilocal Density Functional Approximations

Semilocal DFAs (including LDA, GGA, and meta-GGA) fail to cancel the spurious Coulomb self-repulsion of each occupied orbital. For a one-electron density $n_{i\sigma}(r) = |\psi_{i\sigma}(r)|^2$, the Hartree self-energy

$$U[n_{i\sigma}] = \frac{1}{2}\int\int\frac{n_{i\sigma}(r)n_{i\sigma}(r')}{|r - r'|}dr\,dr'$$

is not fully canceled by the corresponding exchange–correlation functional: $$U[n_{i\sigma}] + E_{xc}^{\rm DFA}[n_{i\sigma},0] \neq 0.$$ This leads to incorrect Kohn–Sham (KS) eigenvalues, particularly for the highest-occupied orbital (HOO), rendering underestimation of vertical ionization energies—$I\approx -\varepsilon_{\rm HO}$—by ∼2 eV on average for standard semilocal functionals (Adhikari et al., 2020).

2. Perdew–Zunger Self-Interaction Correction and the FLO-SIC Formalism

Perdew–Zunger SIC remedies SIE by an orbital-wise subtraction: $$E^{\rm PZ-SIC}[n_\uparrow, n_\downarrow] = E^{\rm DFA}[n_\uparrow, n_\downarrow] - \sum_{i,\sigma}\left[ U[n_{i\sigma}] + E_{xc}^{\rm DFA}[n_{i\sigma}, 0] \right].$$ This formalism guarantees exact cancellation of one-electron SIE but introduces dependence on the choice of orbital representation and tends to overcorrect in many-electron regions, such as in extended π-systems. PZ-SIC by construction is not invariant to unitary rotations of the occupied subspace, thus motivating the use of Fermi–Löwdin orbitals for a consistent localization scheme (Adhikari et al., 2020, Yang et al., 2017).

3. Construction and Optimization of Fermi–Löwdin Orbitals

The FLO-SIC methodology constructs a set of localized orbitals parametrized by a set of Fermi orbital descriptors (FODs) $\{a_{i\sigma}\}$. The Fermi orbital for spin $\sigma$ at $a_{i\sigma}$ is defined as

$$\phi^{\rm FO}_{i\sigma}(r) = \frac{ \sum_j \psi_{j\sigma}^*(a_{i\sigma}) \psi_{j\sigma}(r) }{ \sqrt{ \sum_k |\psi_{k\sigma}(a_{i\sigma})|^2} },$$

where $\{\psi_{j\sigma}\}$ are occupied KS orbitals. These Fermi orbitals (FOs) are then symmetrically Löwdin orthonormalized, producing a set of N FLOs for N occupied states. The energy is minimized with respect to FOD positions using gradient-based optimization, achieving well-localized orbitals suited for the application of PZ-SIC (Adhikari et al., 2020, Schwalbe et al., 2019, Yang et al., 2017).

Optimization follows a nested-loop algorithm:

• Inner loop: SCF solution of orbital-dependent KS equations for a given set of FODs.
• Outer loop: Variational optimization of FODs using analytic energy gradients.

Self-consistency is achieved when both orbital energies and FOD positions are converged, typically in 4–5 cycles, with energy and force thresholds on the order of $10^{-7}$ Ha and $5\times10^{-4}$ Ha/Bohr, respectively (Adhikari et al., 2020, Yang et al., 2017).

4. Iso-Orbital Indicators and Interior Scaling

To prevent SIE overcorrection in many-electron or delocalized regions, an iso-orbital indicator $z_\sigma(r)$ is introduced: $$z_\sigma(r) = \frac{ \tau_\sigma^W(r) }{ \tau_\sigma(r) }, \quad \tau_\sigma^W = \frac{|\nabla n_\sigma|^2}{8 n_\sigma}, \quad \tau_\sigma = \frac{1}{2}\sum_i |\nabla\psi_{i\sigma}|^2.$$ $z_\sigma(r)\to 1$ in one-electron-like regions and $z_\sigma(r)\to 0$ in uniform-gas-like regions. The SIC energy density is scaled by a local function $f(z)$, e.g., $f(z) = z$ (LSIC), to apply full SIC only where $z \approx 1$. Polynomial variants LSIC(+) and rLSIC(+) offer further refinements. These "interior scaling" schemes ensure that correction is applied only in genuinely self-interacting regions, reducing overcorrection in delocalized systems (Adhikari et al., 2020).

The general form of the scaled SIC energy is: $$E_{\rm SIC}^{\rm scaled} = -\sum_{i,\sigma} \int dr\, f(z_\sigma(r)) \left\{ n_{i\sigma}(r) \int \frac{n_{i\sigma}(r')}{|r - r'|} dr' + \epsilon_{xc}^{\rm DFA}[n_{i\sigma}, 0](r) \right\}.$$

5. Algorithmic and Computational Properties

• FLO-SIC scales with $O(N_{\rm orb}^3)$ per SCF step due to the Löwdin orthonormalization, but interior scaling does not add to the computational load (Adhikari et al., 2020).
• Implementations (e.g., UTEP/NRLMOL or PyFLOSIC) use automatic FOD initialization, analytic gradients, and interfaces to standard geometric optimizers (Schwalbe et al., 2019).
• Self-consistent FLO-SIC is necessary for strict variational consistency, geometry optimizations, and cases involving significant changes in the occupied manifold (e.g., charge transfer, magnetic molecules) (Yang et al., 2017).
• With scaling functions, the total computational cost remains similar, while convergence is robust and overhead relative to standard FLO-SIC is negligible (Adhikari et al., 2020).

6. Performance for Ionization Energies and Molecular Properties

Benchmark results on a diverse set of 14 organic molecules (acenes, quinones, heterocycles) demonstrate the effectiveness of interior-scaled FLO-SIC. The observed mean errors (ME) for vertical ionization energies (IEs), computed as $I \approx -\varepsilon_{\rm HO}$, are:

• Uncorrected LDA/PBE: ME $\approx$ –2.0 eV (underestimation).
• Full SIC: ME $\approx$ +2.0 eV (overestimation).
• LSIC: ME $= +1.1$ eV.
• LSIC(+): ME $= +0.6$ eV.
• rLSIC(+): ME $= +0.4$ eV.
• $G_0W_0$@PBE (reference): ME $= +0.3$ eV (Adhikari et al., 2020).

All interior-scaling schemes outperform full SIC and uncorrected DFAs, reaching accuracy levels on par with $G_0W_0$ but at a fraction of the computational cost. For the G2-1 test set, similar trends are observed.

For molecular design and organic electronics, such sub-eV accuracy in HO eigenvalues is critical for quantitative prediction of charge-injection barriers and transport levels (Adhikari et al., 2020).

7. Implications, Best Practices, and Outlook

• FLO-SIC with interior scaling functions provides a robust, self-interaction-free framework that preserves the one-electron limit and size-extensivity while preventing overcorrection in delocalized or metallic systems (Adhikari et al., 2020).
• The use of iso-orbital indicators as switching functions enables pragmatic interpolation between "full SIC" and the uncorrected parent DFA, optimizing accuracy for both localized and extended states.
• For large π-conjugated systems and organic molecular materials, interior-scaled FLO-SIC offers a computationally efficient alternative to hybrid or many-body methods for reliable calculation of first-ionization energies.
• The approach is systematically improvable, modular (allowing for alternative scaling functions or hybridizations with other functionals), and well-supported in modern quantum chemistry software frameworks (Schwalbe et al., 2019).
• Ongoing developments include further refinement of scaling functions, extension to meta-GGAs and periodic systems, and algorithmic advances targeting enhanced scaling and automation.

In summary, FLO-SIC with iso-orbital-indicator-driven interior scaling achieves quasi-particle–level accuracy for HOMO eigenvalues and vertical ionization energies of organic molecules at moderate computational cost, providing a critical tool for the accurate prediction of spectroscopic and transport properties in complex molecular systems (Adhikari et al., 2020).