Perdew-Zunger self-interaction

OPM approach, the Krieger-Li-Iafrate approach and the Perdew-Zunger self-interaction-correction (SIC) method [87] however yield correct neutral fragments. 

Vydrov OA, Scuseria GE (2005) Ionization potentials and electron affinities in the Perdew-Zunger self-interaction corrected density-functional theory, J Chem Phys, 122 184107... 

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. 

O. A. Vydrov and G. E. Scuseria,/. Chem. Phys., 121, 8187-8193 (2004). Effect of the Perdew-Zunger Self-Interaction Correction on the Thermochemical Performance of Approximate Density Functionals. 

Perdew J P and Zunger A 1981 Self-interaction correction to density-functional approximations for many-electron systems Phys. Rev. B 23 5048... 

Perdew J P and A Zunger 1981. Self-Interaction Correction to Density-Functional Approximations for Many-Electron Systems. Physical Review B23 5048-5079. 

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. 

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. 

The discrepancies between the calculated values of —eHOMO and experimental ionization potentials are frequently attributed to self-interaction error. Indeed, applying the Perdew-Zunger technique4 to correct this error of the LDA-, GGA-(PBE), and meta-GGA (TPSS) functionals improves the numerical values of —8HOmo-98 Interestingly, these studies showed that the Perdew-Zunger correction does not improve ionization potentials and electron affinities if calculated as energy differences (ASCF). 

To improve upon these defects, one has to go beyond the LDA The (modified) weighted density approximation [189] retains the correct asymptotic behaviour of and improves the response properties of metal clusters [162, 165]. A different route to improvement provides the self-interaction correction (SIC) of Perdew Zunger [37], where the spurious self-interaction of the LDA is compensated by additional terms in the ground-state potential [166] and in the effective perturbing potential as well [167] (Full-SIC[... 

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... 

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]. 

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]. 

The Perdew-Zunger or one-electron self-interaction error in the ASCF method is written as [63]... 

Perdew J, Zunger A (1981) Self-interaction cOTiection to density-functional t pioximations for many-electron systems. Phys Rev B 23 5048-5079... 

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