Self-interaction Corrected LDA

The first of these functionals is the self-interaction corrected LDA (SIC-LDA) [22]. It has been emphasized that the self-interaction error of the LDA and the GGA is a major source of problems. It is thus tempting to try to eliminate this self-interaction in a semi-empirical form. This is the main idea behind the SIC-LDA. The starting point is the spin-density dependent version of the standard LDA, nj,]. In contrast to the exact E c[n,  

In a many-particle system this self-interaction error is present for all particles in the system. In the SIC-LDA one eliminates the self-interaction component a posteriori by explicitly subtracting the erroneous terms for the individual 

The standard scheme for the application of this functional imphes the use of orbital-dependent Kohn-Sham potentials A separate KS equation has to be solved for each individual KS state. This procedure leads, in general, to non-orthogonal KS orbitals, so that an a posteriori orthogonal-ization is required [66]. However, it has been realized very early [67] that the SIC-LDA should be apphed within the framework of the OPM. For any orbital-dependent functional, the OPM produces the corresponding KS-type multiplicative potential, which automatically avoids the problem of nonorthogonality. 

On the other hand, the use of the OPM does not resolve the unitarity problem which is inherent to this functional [68,69,70] If one performs a unitary transformation among the KS orbitals, the individual orbital densities will change, even if the transformation only couples degenerate KS states. Consequently, also the value of changes. An additional prescription 

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SIC, or self-interaction corrected LDA [21, 22], and more recently the exact exchange (EXX) functionals [23], both of which include a dependence of the exchange functional on the occupied molecular or crystalline orbitals. 

Note that in the sum the term with i = y is not excluded. This diagonal represents the interaction of one electron with itself, and is therefore called the self-interaction term. It is clearly a spurious term, and is exactly canceled by the diagonal part of the exchange energy. It is easy to see that neither the LDA nor the GGA exchange energy cancel exactly the self-interaction. This is, however, not the case in more sophisticated functionals like the exact exchange or the self-interaction-corrected LDA. 

One obvious drawback of the LDA-based band theory is that the self-interaction term in the Coulomb interaction is not completely canceled out by the approximate self-exchange term, particularly in the case of a tightly bound electron system. Next, the discrepancy is believed to be due to the DFT which is a ground-state theory, because we have to treat quasi-particle states in the calculation of CPs. To correct these drawbacks the so-called self-interaction correction (SIC) [6] and GW-approximation (GWA) [7] are introduced in the calculations of CPs and the full-potential linearized APW (FLAPW) method [8] is employed to find out the effects. No established formula is known to take into account the SIC. 

As for the bulk modulus, the LDA well represents its trend across the 4f series, although the absolute errors are quite large (7 A careful analysis of the bulk modulus for the mixed valent metal TmSe indicates that the LDA error in that case can be corrected if the f levels are shifted down 40 mRy (4). This problem is connected to the fact that the LDA does progressively worse for higher angular momentum states (due to probable misrepresentation of the shape of the exchange-correlation hole). Self-interaction corrections may be able to explain this error (9-1OL... 

Note added After this manuscript was completed, a contribution presenting an orbital-dependent self-interaction correction for the relativistic LDA xc-energy functional was published [HI]. 

The linear photoresponse of metal clusters was successfully calculated for spherical [158-160, 163] as well as for spheroidal clusters [164] within the jellium model [188] using the LDA. The results are improved considerably by the use of self-interaction corrected functionals. In the context of response calculations, self-interaction effects occur at three different levels First of all, the static KS orbitals, which enter the response function, have a self-interaction error if calculated within LDA. This is because the LDA xc potential of finite systems shows an exponential rather than the correct — 1/r behaviour in the asymptotic region. As a consequence, the valence electrons of finite systems are too weakly bound and the effective (ground-state) potential does not support high-lying unoccupied states. Apart from the response function Xs, the xc kernel /xc[ o] no matter which approximation is used for it, also has a self-interaction error. This is because /ic[no] is evaluated at the unperturbed ground-state density no(r), and this density exhibits self-interaction errors if the KS orbitals were calculated in LDA. Finally the ALDA form of /,c itself carries another self-interaction error. 

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[... 

Similar experience has been made for other f-electron systems. Nevertheless, we should point out that by applying the LDA+U scheme we leave the framework of DFT. This does not apply to the SIC (self-interaction correction) formalism (Dreizler and Gross 1990), for which a proper relativistic formulation has been worked out recently (Forstreuter et al. 1997 Temmerman et al. 1997) and applied to magnetic solids (Temmerman et al. 1997). 

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,... 

In this context the HF Hamiltonian is of particular interest. HF theory is distinguished from the LDA in that it attempts to compute the ground state wave function, whereas LDA computes the ground state density and energy. The HF approximation therefore contains explicit interactions between the orbitals and is automatically self-interaction corrected. The on-site Coulomb interaction is therefore treated quite naturally and a qualitatively correct description of the ground states is obtained. The density of states of NiO projected onto the Ni Eg, J2g and O orbitals computed in the HF approximation is shown in Fig. 8.3. 

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]. 

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... 

Many interpretations of experimental measurements have been limited to the comparison with KS eigenvalues, based on the fact that for extended solids the KS band structure sometimes provides the correct qualitative features of photoemission experiments. Unfortunately, the deviation of the KS eigenvalues from physical excitation energies becomes stronger in going from extended to finite systems, because the yV-dependence of the exchange-correlation potential becomes crucial, and because some errors implicit in standard approximations (like the self-interaction in LDA) are amplified by the reduced extension of the KS orbitals. 

A third approximation which we tried is the self-interaction corrected (SIC) ALDA, which is simply the second functional derivative of the SIC-LDA energy ... 

One qualitative defect in LDA for example is the imperfect cancellation of the Coulomb self-interaction in the mean field Coulomb energy (Hartree energy Eh - see eq. 2.2.) and the corresponding potential Vh (eq. 2.5.), due to the approximate nature of Ex[c)- There are hints that this defect might have a significant influence on reaction barriers [29] - see also chapter 3.3. The self-interaction may be corrected in DFT by a self-interaction correction (SIC) [29, 30, 31]. However, these corrections are rather cumbersome and therefore they have been applied up to now only very rarely. 

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