SIC-DFT: Correcting Self-Interaction Errors in DFT

• SIC-DFT is a collection of computational methods that correct self-interaction errors in standard DFT, improving the treatment of localized electrons.
• Techniques like Perdew–Zunger, generalized Slater-OEP, and FLO–SIC balance enhanced accuracy with computational efficiency across various systems.
• Its applications span molecules, surfaces, and bulk materials, enabling more reliable predictions of band gaps, defect energetics, and response properties.

SIC-DFT refers to computational methodologies that incorporate self-interaction correction (SIC) within the framework of density functional theory (DFT), addressing the inherent errors in standard local or semi-local exchange-correlation functionals. SIC-DFT removes the spurious self-repulsion in DFT functionals and improves predictions of electronic properties, energy levels, and molecular properties, particularly for systems where electron localization plays a decisive role. The SIC-DFT umbrella includes formalism such as the Perdew–Zunger SIC, generalized Slater and optimized effective potential (OEP) approaches, and orbital localization schemes, as well as practical implementations in ab initio calculations across condensed matter, molecular, and surface systems.

1. Motivation and Formal Basis for Self-Interaction Correction in DFT

Standard Kohn–Sham DFT with local (LDA) or GGA functionals suffers self-interaction error: each electron interacts with itself via the Hartree term and similarly in exchange-correlation approximations, leading to incorrect asymptotic potential decay and errors in band gaps, ionization energies, and electron localization. In exact theory, the self-Coulomb term is non-physical and eliminated in Hartree–Fock (via exact exchange), but not in LDA/GGA. The widely adopted Perdew–Zunger SIC functional explicitly subtracts single-electron contributions from the total energy:

$$E_{\rm SIC}[\{\psi_\alpha\}] = E_{\rm LDA}[n] - \sum_{\alpha} E_{\rm LDA}[|\psi_\alpha|^2]$$

where $n = \sum_\alpha |\psi_\alpha|^2$.

Orbital-dependent potentials follow from functional derivatives, leading to nonlocal, non-Hermitian mean-field Hamiltonians for each orbital (Messud et al., 2010). Restoration of correct asymptotic behavior and improved energetics makes SIC essential for systems with significant orbital localization (finite systems, defects, surfaces, molecules).

2. Generalized SIC-OEP and Slater Approximations

The OEP formalism for SIC replaces the nonlocal orbital-dependent mean field with a local potential optimized to reproduce the total energy shift under orbital variations. A "double-set" technique distinguishes between Kohn–Sham-like propagating orbitals ($\phi_i$) that diagonalize the single-particle Hamiltonian and localized minimizing orbitals ($\psi_\alpha$) that fulfill the symmetry condition for SIC minimization (Messud et al., 2010). The OEP integral equation relates the variation in the local potential to shifts between the orbital sets.

In practice, the Slater, KLI, and residual OEP terms decompose the local SIC potential:

$$U_{\rm SIC}^{\rm local}(r) = V_S(r) + V_K(r) + V_C(r)$$

Generalized Slater (GS) approximation operates when one localized orbital dominates the density locally, allowing the potential to retain much of the accuracy of full OEP-SIC while being computationally inexpensive:

$$U_{GS}(r) = \sum_\alpha \frac{|\psi_\alpha(r)|^2}{\rho(r)} U_{\rm LDA}[|\psi_\alpha|^2](r)$$

Numerical benchmarks confirm GS matches full SIC for potential energy surfaces and response properties in molecules and small clusters (Messud et al., 2010).

3. FLO–SIC and Orbital Localization Strategies

The Fermi–Löwdin Orbital SIC (FLO–SIC) approach constructs localized orbitals from prescribed Fermi orbital descriptors (FODs), followed by orthonormalization via the Löwdin scheme (Liebing et al., 2022). Starting from the Perdew–Zunger SIC energy, FLO–SIC derives orbital-dependent corrections added to the Kohn–Sham Hamiltonian in each spin channel:

$$v_{\rm SIC}^\sigma(r) = \sum_i^{N^\sigma} \left[ v_{\rm H}[n_i^\sigma](r) + v_{\rm XC}[n_i^\sigma,0](r) \right] |\Phi_i^\sigma(r)|^2$$

Optimization cycles adjust both FOD positions and the density matrix, yielding highly localized corrective potentials. FLO–SIC reliably increases calculated dipole moments by 0.1–0.4 D over standard DFT (which itself matches experiment within 0.1 D), and slightly reduces polarizabilities, with chemical sensibility ensured by appropriate FOD selection (e.g. Linnett double-quartet for nitromethane) (Liebing et al., 2022).

4. SIC-DFT in Bulk, Surfaces, and Polytypes

SIC-DFT is vital for accurate treatment of surface and defect energetics in semiconductors. For H-in-jellium models, Perdew–Zunger SIC reduces LSDA overbinding by a factor of two and aligns immersion energies closer to quantum Monte Carlo results, though it does not correct the Friedel oscillation profile (0705.2702). In solid-state applications such as silicon carbide, advanced DFT codes (VASP, CP2K, FHI-aims) with LDA, GGA, meta-GGA, and hybrid functionals model the subtle energy differences among SiC polytypes. Bulk thermodynamics slightly favors hexagonal 4H/6H over cubic 3C polytypes, but surface energies crucially stabilize 3C nucleation, underscoring the need for surface-corrected DFT for growth and stability modeling (Ramakers et al., 2022).

5. SIC-DFT and Quantum Defects, Surfaces, and Vibrational Properties

Accurate defect energetics and spin properties in SiC require hybrid-functional DFT with SIC components. For silicon vacancy-related EPR centers in 4H-SiC, DFT/HSE06 plus PBE ZFS calculations predict formation energies, spin splittings, and zero-phonon lines matching experimental signals (Csóré et al., 2021). In surface studies, such as β-SiC(100) reconstructions, GGA-based DFT reveals metallization upon selective hydrogenation of the second Si layer, with the emergence of distinct vibrational modes and a kinetic barrier that stabilizes metallic coverage under continuous H dosing (Westover et al., 2011).

6. Limitations, Extensions, and Practical Considerations

SIC-DFT's effectiveness depends on the localization of electronic states, the choice of SIC functional, the numerical scheme for orbital construction (double-set, FLO, etc.), and the exchange-correlation parameterization. Hybrid functionals (e.g. HSE06) further improve gap predictions but increase computational cost. Full OEP-SIC remains expensive, but GS and FLO–SIC methods offer practical efficiency. For accurate response properties, force thresholds and basis/grid convergence are critical (Liebing et al., 2022). SIC does not fully recover correlation effects (e.g. in screening oscillations), nor does it address strong correlation beyond one-electron self-interaction error.

Table: SIC-DFT Formalisms and Application Domains

Formalism	Key Features	Application Domains
Perdew–Zunger SIC	Orbital-dependent correction, subtracts self-energy	Atoms, jellium, molecules, semiconductors
Generalized SIC-OEP/GS	Double-set orbitals, local potential optimization	Static/dynamic DFT, clusters, time-dependent response
FLO–SIC	Fermi orbital descriptors, Löwdin orthonormalization	Molecular dipoles, polarizability, chemical bonding
Hybrid DFT + SIC	Exact exchange + SIC for defect levels, optical response	SiC defects, band gaps, quantum emitters

Implementations require convergence in orbital localization, careful accounting of normalization in response calculations, and selection of functionals respective to the problem domain.

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SIC-DFT represents a collection of rigorous methodologies for correcting self-interaction error in DFT. By restoring physical accuracy in the asymptotics and energetics of localized electronic states, SIC-DFT enables reliable prediction of molecular properties, defect spectra, and surface phenomena. Its variants (Perdew–Zunger, generalized Slater/OEP, FLO–SIC) offer tuned accuracy and practicality for both static and dynamic quantum simulations in condensed matter and chemical physics.