The pseudo-self-interaction correction (pseudo-SIC) of Filippetti and Spaldin [Phys. Rev. B 67, 125109 (2003)] removes the spurious self-interaction of the local spin-density approximation from localized orbitals by recasting it as a pseudopotential-like non-local projector scaled by self-consistent atomic-orbital occupation numbers, at a computational cost close to plain LSDA. The source paper itself identified four unresolved issues: the total-energy expression is not variational, so Hellmann-Feynman forces and stresses are unavailable; the self-consistent updating of the occupation numbers can oscillate and slow convergence; the comparison against LDA+U, SIC-LSDA, and model GW rested on a deliberately limited set of systems; and the transferability choices of the scheme (cutoff of the atomic SIC potential tails, placement of semicore states, a linear-scaling approximation for the exchange-correlation self-interaction term) were unvalidated across chemical environments. The forces/variationality issue has since been addressed by direct follow-up work — approximate force equations in 2009, the fully variational reformulation of the scheme (VPSIC, variational pseudo-self-interaction correction) in 2011, and a complete force implementation for the original non-variational formulation (pSIC) in the plane-wave DFT code Quantum ESPRESSO in 2011 — while the convergence stabilization, comprehensive cross-class benchmarking, and systematic transferability validation of the occupation-number-dependent pseudo-SIC family remain open; those three remaining components constitute the objective of this task.

Background

In Kohn-Sham density functional theory (DFT), the local spin-density approximation (LSDA) approximates the exchange-correlation energy by that of a uniform electron gas. Because this approximate functional is not exactly cancelled by the Hartree term for a one-electron region of charge, each electron spuriously interacts with its own charge density: the self-interaction error, which overly delocalizes electrons, underestimates band gaps, and misplaces localized 3d3d and 4f4f levels in transition-metal and rare-earth compounds. The Perdew-Zunger self-interaction correction (PZ-SIC) subtracts, orbital by orbital, the Hartree and exchange-correlation self-terms of each occupied one-electron density; applied fully self-consistently (SIC-LSDA), it works well for localized states but is demanding for solids because the functional depends on the choice of localized orbitals, breaks translational symmetry, and introduces a rugged landscape of alternative solutions. LDA+UU instead adds a Hubbard-like term for designated correlated orbitals, and model GWGW (a self-energy correction built from screened Coulomb interactions) addresses quasiparticle energies non-self-consistently.

The pseudo-SIC scheme of the source paper takes a different route: the atomic self-interaction correction is embedded in the pseudopotential itself. A pseudopotential replaces the chemically inert core and the strong valence-core Coulomb singularity by a smooth effective potential, and in the separable Kleinman-Bylander form a pseudopotential acts on a wavefunction through a small set of projector functions. Filippetti and Spaldin built, from an isolated self-interaction-corrected pseudo-atom, a non-local SIC projector of exactly this Kleinman-Bylander type, then scaled its strength in the crystal by occupation numbers piσp_i^{\sigma} (the projection of the spin-σ\sigma Bloch states onto the atomic orbital ii, averaged over occupied bands). These piσp_i^{\sigma} are recomputed at every self-consistent iteration, so the correction automatically switches off for itinerant states and acts at full strength for localized ones. The atomic SIC potentials are cut off at a radius where the orbital charge vanishes so the correction stays local, strongly localized semicore states (e.g. Y 4s/4p4s/4p, Ga 3d3d) may be moved into the core, and the exchange-correlation part of the SIC is evaluated with a linear-scaling assumption that is exact for the dominant Hartree term but introduces a non-linearity error of order pi1/3−pip_i^{1/3}-p_i. Tested on ZnO, GaN, NiO, MnO, and the multiferroic YMnO3_3, the scheme improved gaps and magnetic moments dramatically over LSDA at near-LSDA cost.

The same paper was explicit that the method, as implemented, left four practically and scientifically important issues unresolved. First, the adopted total-energy expression is not related to the pseudo-SIC Kohn-Sham equations by a variational principle (unlike LSDA or SIC-LSDA), so forces and stresses cannot be obtained from the simple Hellmann-Feynman theorem (the statement that, for a variational energy, the force on a nucleus equals the negative derivative of the energy with respect to its position) and require explicit wave-function response terms; the authors deferred forces and stresses to future publications. Second, the self-consistent updating of the occupation numbers piσp_i^{\sigma} can oscillate and increase the number of iterations, and robust mixing or stabilization schemes remained to be developed. Third, the encouraging but deliberately limited comparisons with LDA+UU, SIC-LSDA, and model GW fell short of a systematic assessment across broad classes of materials, including regimes where many-body self-energy effects rather than self-interaction dominate the residual error. Fourth, the transferability choices — the cutoff radius of the atomic SIC tails, the core-versus-valence placement of semicore states, and the linear-scaling approximation for the exchange-correlation self-interaction term — needed validation across diverse chemical environments, with their quantitative effect on marginal cases unquantified.

Problem Statement

The problem, as posed by the source, is to complete the methodological maturation of the pseudo-self-interaction-corrected pseudopotential scheme so that it becomes a robust, transferable, and systematically validated first-principles tool, comprising four components. (1) Supply a total-energy functional whose derivatives give consistent forces and stresses, restoring the Hellmann-Feynman route (or an equivalent complete response-term formulation) for geometry optimization; (2) develop and test robust, efficient mixing or stabilization schemes for the self-consistent updating of the occupation numbers piσp_i^{\sigma}, so that oscillations no longer inflate the number of self-consistent iterations; (3) perform a comprehensive, systematic assessment of the scheme's accuracy across broad classes of materials, in controlled comparison with LDA+U, SIC-LSDA, and model GW and against experiment, including the identification of regimes where many-body self-energy effects beyond self-interaction dominate the residual error; (4) validate the transferability choices of the scheme — the radial cutoff of the atomic SIC potential tails, the core-versus-valence treatment of semicore states, and the linear-scaling approximation for the exchange-correlation self-interaction term — across diverse chemical environments, and quantify their effect on quantitative accuracy in marginal cases. The open core of the problem as it stands today consists of components (2), (3), and (4) — stable occupation-number self-consistency, systematic cross-class benchmarking, and transferability validation for the occupation-number-dependent pseudo-SIC method family and its descendants; component (1) is included only to record the full set of issues the source raised, having been addressed in the literature. The three components to be resolved are: (2) an occupation-number mixing or stabilization scheme is specified and demonstrated on a stated test set to remove the oscillation of piσp_i^{\sigma} without inflating the number of self-consistent iterations, while the correction remains active and the run converges to the same self-consistent solution as an unstabilized reference, with the converged total energy, band gaps, magnetic moments, and occupation numbers agreeing within a stated tolerance; (3) the scheme's band gaps and magnetic moments are assessed across the material classes it was built on — non-magnetic semiconductors (ZnO, GaN), magnetic insulators (NiO, MnO), and the multiferroic YMnO3_3 — extended to at least one further class, in controlled comparison with LDA+U, SIC-LSDA, model GW, and experiment, with an explicit statement of where the residual error ceases to be dominated by self-interaction; (4) each of the three transferability choices — the SIC tail cutoff radius, the core-versus-valence placement of semicore states, and the linear-scaling approximation for the exchange-correlation self-interaction term — is varied in a controlled way over the environments of (3), with its effect on the accuracy of the marginal cases quantified. Components (2)-(4) remain open research problems rather than questions already settled in the literature; the cited works serve as background and comparison data.

References

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