Self-interaction effects, Coulomb energy

Here ip is an orbital of an electron with Mg = 1/2(t), e is its one-electron energy, is the classical Coulomb potential (including electron self-interaction terms), and represents the effects of electron exchange. In Slater s model, this is related to p h, the local density of electrons of the same spin... 

In summary, the use of approximate functionals can lead to errors for several reasons the neglect of correlation effects on electronic kinetic energy, the incorrect cancellation of the self-interaction involved in the Coulomb... 

The work Wnfr) is retained in the equation to ensure there is no self-interaction). In contrast to the Kohn-Sham equation, this differential equation can in practice be solved because the dependence of the Fermi hole p, (r, r ), and thus of the work W (r), on the orbitals is known. Furthermore, since the solution of this equation leads to the exact asymptotic structure of vj (r), and the fact that Coulomb correlation effects are generally small for finite systems, the highest occupied eigenvalue should approximate well the exact (nonrelativistic) removal energy. This conclusion too is borne out by results given in Sect. 5.2.2. 

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... 

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. 

Since the effective potential is a function of the density, which is obtained from Eq. (2.79) and hence depends on all the single-particle states, we will need to solve these equations by iteration until we reach self-consistency. As mentioned earlier, this is not a significant problem. A more pressing issue is the exact form of [n(r)] which is unknown. We can consider the simplest situation, in which the true electronic system is endowed with only one aspect of electron interactions (beyond Coulomb repulsion), that is, the exchange property. As we saw in the case of the Hartree-Fock approximation, which takes into account exchange explicitly, in a uniform system the contribution of exchange to the total energy is... 

Yang Y et al (2007) Extension of the self-consistent-charge density-fimctional tight-binding method third-order expansion of the density fimctional theory total energy and introduction of a modified effective coulomb interaction. J Phys Chem All 1 10861-10873... 

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