Self-interaction effects, Coulomb

Encl[p] is the non-classical contribution to the electron-electron interaction containing all the effects of self-interaction correction, exchange and Coulomb correlation described previously. It will come as no surprise that finding explicit expressions for the yet unknown functionals, i. e. T[p] and Encl[p], represents the major challenge in density functional theory and a large fraction of this book will be devoted to that problem. 

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. 

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... 

The method of choice has to be carefully evaluated prior to a simulation study to assess whether a chosen functional is appropriate for a particular system. Ionic solvates are strongly dominated by Coulombic interactions and significant polarization effects are observed resulting from the presence of the charged solute. The unphysical self-interaction inherent to DFT is a striking disadvantage in these cases. Furthermore, the parametrization... 

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... 

A different approach to treat correlation effects which are not well described within the LSDA consists in incorporating self-interaction corrections (SIC) [111-114] in electron structure methods for solids, Svane et al. [115-120]. In the Hartree-Fock (HF) theory the electron-electron interactions are usually divided into two contributions, the Coulomb term and the exchange term although they both are Coulomb interactions. The separation though, is convenient because simplifications of self-consistent-field calculations can be obtained by including in both terms the interaction of the electron itself. In the HF theory this has no influence on the solutions because these selfinteractions in the Coulomb and exchange terms exactly cancel each other. However, when the exchange term is treated... 

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... 

The LMWGs have in common the property that they self-assemble into fibrous aggregates a process that can be driven by different noncovalent interactions like coulomb interactions, hydrogen bonding, n-n interactions, van der Waals forces, and solvophobic effects. For most of the early examples of LMWGs, the gelation prop-... 

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. 

Here, w,y((w) = tpi w o)) ipj) is the matrix element for the extra field which would have been the only contribution if the electrons were non-interacting. But since the electrons interact through Coulomb, exchange and correlation effects, an extra term describing the linear response of the self-consistent field to the perturbation has to be included. This term contains fire quantities... 

---