Self-interaction terms

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. 

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. 

The last term is introduced within the self-consistent Hartree approximation (within the functional up to one vertex), //// = 10/9 accounts different coefficients in functional for the self-interaction terms (ri4 - for the given field d and dla)2 d )2 terms), cf [20], We presented da = 2k d t>ke lkfix, ... 

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 form of this self-interaction term for CPCM seems very plausible if we consider an extended CPCM system of equations, analogous to that in Equation (1.86), collecting both the exposed and the shadow charges q. Starting from the CPCM equation (switching from q to q) ... 

We note the presence of the direct or Hartree term and the exchange term, third term inside the brackets and first term outside, respectively. The term i — j can be included in the direct term since it is canceled by the corresponding inclusion into the exchange term. Therefore, we have the well-known result that the HF approximation does not include self-interaction terms. 

The generalized two-particle HF equations are seen to have a structure equivalent to their single-particle counterparts, exhibiting the presence of a direct term, written in terms of the density, and an exchange term. As the canonical HF equations, the present expressions do not contain spurious self-interaction terms. However, unlike the single-particle equations, they allow the determination of fully correlated two-particle states removing to this extent the most basic objection to the HF method. 

Equation (83) includes self-interaction terms. That is, those contributions which remain finite at infinite separation must be subtracted from (83) to obtain only the interaction potential. At the same level of approximation these terms are given by [80]... 

In molecules and clusters, genuine exchange (as well as correlation) among identical nuclei is very small because, at typical internuclear separations, the overlap of nuclear wave functions is rather small. However, the exact xc functional also contains self-exchange contributions which are not small and which cancel the self-interaction terms contained in the Hartree potentials in Eqs. (71) and (72). Hence it will be a very good approximation to represent Fjc by the self-exchange terms alone. This leads to... 

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... 

The evaluation to the desired numerical accuracy of the density functional total energy has been a major obstacle to such calculations for many years. Part of the difficulty can be related to truncation errors in the orbital representation, or equivalently to basis set limitations, in variational calculations. Another part of the difficulty can be related to inaccuracies in the solution of Poisson s equation. The problem of maximizing the computational accuracy of the Coulomb self-interaction term in the context of least-squares-fitted auxiliary densities has been addressed in [39]. A third part of the difficulty may arise from the numerical integration, which is unavoidable in calculating the exchange and correlation contributions to the total energy in the density functional framework. 

The self-interaction term has not appeared and so is not corrected for by the diagonal exchange term. If we had used the (MO) single term standard expression we might have expected... 

In the full Hamiltonian the electron repulsion is exact , it is simply the sum of all the inter-particle repulsions. In the HF Hamiltonian this is replaced by Coulomb and exchange terms. The Coulomb term is simply the net average repulsion field due to all the electrons in the molecule and the exchange term removes the self-interaction term included in this average sum plus some further small corrections. 

The ion-molecule interaction term, kg, is the one that is most often calculated. Long and McDevit felt that the nonelectrolyte self interaction term could safely be ignored only when the nonelectrolyte solubility was vei low as in the case of the nonpolar electrolytes hydrogen, oxygen and benzene. For the more soluble polar nonelectrolytes, such as ammonia, carbon dioxide and phenol, the self interaction term is of much greater importance and should be determined where data is available. 

The self interaction term is independent of solution concentration and relates only to the nonelectrolyte therefore it is considered equal in binary and multicomponent solutions. ku = ky. The log yu term in equation (7.12) is replaced by the right hand side of equation (7.13) and subtracted from both sides of the equation ... 

Distribution data is particularly good where the nonelectrolyte is miscible in water. The experimental method is quite simple and the nonelectrolyte concentration in the aqueous phase can be kept low in order to avoid the self interaction term of equation (7.15). 

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