Perdew and Zunger

Of course, this self-correction error is not limited to one electron systems, where it can be identified most easily, but applies to all systems. Perdew and Zunger, 1981, suggested a self-interaction corrected (SIC) form of approximate functionals in which they explicitly enforced equation (6-34) by substracting out the unphysical self-interaction terms. Without going into any detail, we just note that the resulting one-electron equations for the SIC orbitals are problematic. Unlike the regular Kohn-Sham scheme, the SIC-KS equations do not share the same potential for all orbitals. Rather, the potential is orbital dependent which introduces a lot of practical complications. As a consequence, there are hardly any implementations of the Perdew-Zunger scheme for self-interaction correction. 

The correlation energy is known analytically in the high-and low-density limits. For typical valence electron densities (1 < r, < 10) and lower densities (r, > 10), it is known numerically from release-node Diffusion Monte Carlo studies [33]. Various parametrizations have been developed to interpolate between the known limits while fitting the Monte Carlo data. The first, simplest and most transparent is that of Perdew and Zunger (PZ) [34] ... 

The term Vi in the above equation involves two-electron integrals over orbitals which can make the construction of Vxc rather expensive in molecular or solid state calculations. For an application of the KLI potential in bandstruc-ture calculations Li et al. [119] approximated this term using the SIC functional of Perdew and Zunger [87] by... 

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. 

Note that, within this approximation, the nuclear KS potential depends on the state it acts on. This is analogous to the SIC scheme of Perdew and Zunger... 

At present the most frequently used parameterization for exchange and correlation are those of Perdew and Zunger [40] or Vosko, Wilk and Nusalr [41]. These two functionals which are based on the Monte Carlo simulations by Ceperly and Adler [42] give rather similar results. There are deficiencies in the LDA schemes which are due to the wrong asymptotic behaviour of the exchange and correlation potential since it falls off exponentially and not as... 

There are a number of model exchange-correlation functionals for the ground-state. How do they perform for ensemble states Recently, several local density functional approximations have been tested [24]. The Gunnarsson-Lundqvist-Wilkins (GLW) [26], the von Barth-Hedin (VBH)[25] and Ceperley-Alder [27] local density approximations parametrized by Perdew and Zunger [28] and Vosko, Wilk and Nusair (VWN) [29] are applied to calculate the first excitation energies of atoms. 

Perdew and Zunger (1981), in the Xa-like equivalent of the Hartree approximation, advocate subtracting the total self-interaction of each electron in Xa-like models. This proposal would remove the m dependence of hydrogenic systems. Since the self-interaction of each electron (orthonormal orbital), as well as their sum, is not invariant under a unitary transformation among the orbitals, in contrast to the first-order density matrix and thus Xa-like models, Perdew and Zunger propose picking out a unitary transformation... 

Perdew and Zunger (PZ) have listed a number of inadequacies of the LSD approximation which they attributed primarily to the spurious selfinteraction terms. They then proposed an orbital self-interaction correction scheme. The list of LSD failures given by PZ is ... 

Comparing Equation (5) with (12), and (6) with (13), it is clear that while the SIC functional due to Perdew and Zunger [21] gives the HF energy for a two-electron closed-shell system, for an open-shell system this is no the case. We see that the difference found in the open-shell system, between hf and E, is related to the exchange contribution. Assuming that the SIC functional removes the self-interaction term in the KS equations, it follows that... 

For high density system, the enhancement factor becomes unity, and exchange effects dominate over the correlation effects. When the density becomes lower, the enhancement factor kicks in and includes correlation effects into the exchange energies. The enhancement factor is not unique, but can be derived differently in different approximations. The most reliable ones are parameterizations of molecular Monte-Carlo data. Some well known, and regularly used, parameterizations have been made by Hedin and Lundqvist [29], von Barth and Hedin [22], Gun-narsson and Lundqvist [30], Ceperly and Adler [31], Vosko, Wilk, and Nusair [32], and Perdew and Zunger [27]. 

There is one further orbital-dependent functional which can be mentioned at this point. In the nonrelativistic context it has been realized rather early [184] that, as a matter of principle, the self-interaction corrected (SIC) LDA of Perdew and Zunger [143] represents an implicit functional for which the 0PM should be used. A relativistic version of the Perdew-Zunger SIC has been proposed by Rieger and Vogl [185] as well as Severin et al. [186,187,46]. This functional, however, has not yet found widespread use, neither within the conventional,... 

Besides, self-interaction correction (SIC) is one of the most popular correction schemes. Perdew and Zunger suggested a scheme for the application of SIC to occupied orbitals where the self-interaction components of the Coulomb and exchange energies are simply subtracted from the total exchange-correlation energy [81]... 

Eqs. (44) and (45) were stressed in the self-interaction correction scheme of Perdew and Zunger [77]. 

Perdew and Zunger [77] (PZ81) suggested the following parametrization of the Ceperley-Alder data for the spin-compensated and spin-polarized cases... 

Another disadvantage of the LDA is that the Hartree Coulomb potential includes interactions of each electron with itself, and the spurious term is not cancelled exactly by the LDA self-exchange energy, in contrast to the HF method (see A1.3I. where the self-interaction is cancelled exactly. Perdew and Zunger proposed methods to evaluate the self-interaction correction (SIC) for any energy density functional [40]. However, full SIC calculations for solids are extremely complicated (see, for example [41. 42 and 43]). As an alternative to the very expensive GW calculations, Pollmann et al have developed a pseudopotential built with self-interaction and relaxation corrections (SIRC) [44]. 

These potentials as well the density in the Coulomb-term were expressed as linear combinations of Gaussians. For ex the Gaspar-Kohn-Sham (GKS)89,90 functional and the Kohn-Sham functional90 were used. For the correlation functional the electron-gas correlation results of Ceperley and Alder91 were analytically fitted by Perdew and Zunger (PZ).92... 

As has been widely accepted in Car-Parrinello (CP) simulations [38, 39], the exchange-correlation energy and potential are described in the local density approximation (LDA) [49]. A reasonable level of accuracy is achieved with the LDA including the correlation part by Ceperley and Alder [148] as parametrized by Perdew and Zunger [149], applied to... 

So far, the most frequently used is the Perdew-Zunger self-interaction correction (Perdew and Zunger 1981), which simply removes the self-interaction errors from total electronic energies. 

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