Local self-interaction-correction

Local self-interaction-corrected local spin density approximation... 

Goedecker, S., Umrigar, C. J., 1997, Critical Assessment of the Self-Interaction-Corrected-Local-Density-Func-tional Method and its Algorithmic Implementation , Phys. Rev. A., 55, 1765. 

Hamada, N. and Ohnishi, S. (1986) Self-interaction correction to the local-density approximation in the calculation of the energy band gaps of semiconductors based on the full-potential linearized augmented-plane-wave method, Phys. Rev., B34,9042-9044. 

While the LSD exchange-correlation hole is accurate for small interelec-tronic separations (Sect. 2.3), it is less satisfactory at large separations, as discussed in Sect. 2.5. For example, consider the hole for an electron which has wandered out into the classically-forbidden tail region around an atom (or molecule). The exact hole remains localized around the nucleus, and in Sect. 2.5 we give explicit results for its limiting form as the electron moves far away [19]. The LSD hole, however, becomes more and more diffuse as the density at the electron s position gets smaller, and so is quite incorrect. The weighted density approximation (WDA) and the self-interaction correction (SIC) both yield more accurate (but not exact) descriptions of this phenomenon. 

DPT schemes, which allow to calculate the electron affinities of atoms are based on the exact [59,60] and generalized (local) [61,62] exchange self-interaction-corrected (SIC) density functionals, treating the correlation separately in some approximation. Having better asymptotic behavior than GGA s, like in the improved SIC-LSD methods, one should obtain more... 

A number of different methods have been proposed to introduce a self-interaction correction into the Kohn-Sham formalism (Perdew and Zunger 1981 KUmmel and Perdew 2003 Grafenstein, Kraka, and Cremer 2004). This correction is particularly useful in situations with odd numbers of electrons distributed over more than one atom, e.g., in transition-state structures (Patchkovskii and Ziegler 2002). Unfortunately, the correction introduces an additional level of self-consistency into the KS SCF process because it depends on the KS orbitals, and it tends to be difficult and time-consuming to converge the relevant equations. However, future developments in non-local correlation functionals may be able to correct for self-interaction error in a more efficient manner. 

M. A. Whitehead and S. Manoli, Phys. Rev. A, 38,630 (1988). Generalized-Exchange-Local-Spin-Density-Functional Theory Self-Interaction Correction. 

Hartree s original idea of the self-consistent field involved only the direct Coulomb interaction between electrons. This is not inconsistent with variational theory [163], but requires an essential modification in order to correspond to the true physics of electrons. In neglecting electronic exchange, the pure Coulombic Hartree mean field inherently allowed an electron to interact with itself, one of the most unsatisfactory aspects of pre-quantum theories. Hartree simply removed the self-interaction by fiat, at the cost of making the mean field different for each electron. Orbital orthogonality, necessary to the concept of independent electrons, could only be imposed by an artificial variational constraint. The need for an ad hoc self-interaction correction (SIC) persists in recent theories based on approximate local exchange potentials. 

We have added SIC capabilities to the DFT package in Q-Chem [20]. Initially, this involves localizing the KS canonical orbitals with the Boys procedure [78] and using these to evaluate the self-interaction correction. Thus, the present scheme simply applies the correction perturbatively to the KS energy. Table 9 lists the corrected DFT barriers, evaluated at the re-optimized SIC geometries. 

The energy gap obtained in such band-structure calculations is the one called HOMO-LUMO gap in molecular calculations, i.e., the difference between the energies of the highest occupied and the lowest unoccupied singleparticle states. Neglect of the derivative discontinuity A, defined in Eq. (65), by standard local and semilocal xc functionals leads to an underestimate of the gap (the so-called band-gap problem ), which is most severe in transition-metal oxides and other strongly correlated systems. Self-interaction corrections provide a partial remedy for this problem [71, 72, 73, 74],... 

An interesting aspect of the density functional calculations of Penzar and Ekardt is that these include self-interaction corrections. It is well known that the local density approximation (LDA) to exchange and correlation effects is not sufficiently accurate to give reliable electron affinities of free atoms or clusters [47,48]. This defidency is due to the fact that, in a neutral atom for instance, the LDA exchange-correlation potential Vif (f) decays exponentially at large r, while the exact behavior should be — 1/r. As a consequence, some atomic and cluster anions become unstable in LDA. The origin of this error is the incomplete cancellation of the self-interaction part of the classical coulomb energy term... 

Harrison, J. G. (1987). Electron afilnities in the self-interactions-corrected local spin density approximation. J. Chem. Phys. 86,2849-2853. 

Note that this correction has the problem that the Kohn-Sham equation is not invariant for the unitary transformation of occupied orbitals, even after the correction, differently from the Hartree-Fock equation. In the Hartree-Fock equation, the variations of the Coulomb self-interaction energy and its potential for the unitary transformations of occupied orbitals cancel out with those of the exchange self-interaction, while these are not compensated, even after the correction in the Kohn-Sham equation. Therefore, the effect of the self-interaction correction depends on the difference in occupied orbitals before and after the unitary transformation. For removing this difference, it is usual to localize the orbitals before the self-interaction correction (Johnson et al. 1994). Note, however, that there are various types of orbital localization methods, and the effect of the selfinteraction correction inevitably depends on them. Combining with the optimized effective potential (OEP) method (see Sect. 7.5) may be one of the most efficient ways to solve this problem. This combination enables us to consistently obtain localized potentials with no self-interaction error. 

Manoh, S. D., Whitehead, M. A. (1988). Generalized-exchange local-spin-density-functional theory calculation and results for non-self-interaction-corrected and self-inter-action-corrected theories. Phys. Rev. A 38, 3187-3199. 

A local formulation of the self-interaction-corrected (LSIC) energy functionals has been proposed and tested by Luders et al. (2005). This local formulation has increased the functionality of the SIC methodology as presented in Section 3.5. The LSIC method relies on the observation that a localized state may be recognized by the phase shift, tj , defined by the logarithmic derivative ... 

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