Coulomb self-interaction

Do so, we use the formalism of the variational density fitting method [55, 56] where the Coulomb self-interaction energy of the error is minimized ... 

Ex involves a sum over orbital (a, a ) and spin (cr) quantum numbers, of a product of one-particle orbitals xjjaa with occupation numbers fa<, obeying Fermi statistics. 7[n] includes a spurious Coulomb self-interaction, as can be realized if Eq. (2) is applied to a single electron system. This Coulomb self-interaction is exactly cancelled out by the a = o terms in Ex (selfexchange term). 

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 factor of V2 allows the double sum to run over all electrons (it is easily seen from eqs (3.29) and (3.30) that the Coulomb self-interaction is exactly cancelled by the corresponding exchange element A u). 

The most familiar correction for functionals may be the self-interaction correction, which removes the self-interaction error of exchange functionals. In density functional theory, the self-interaction error indicates Coulomb self-interactions, which should cancel out with the exchange self-interactions but remain due to the use of exchange functionals as a substitute for the Hartree-Fock exchange integral in the exchange part of the Kohn-Sham equation. 

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. 

As another significant fundamental condition, there is a condition on the selfinteraction error. The self-interaction error is the Coulomb self-interaction, which should inherently cancel with the exchange self-interaction but remains due to the use of the exchange functional (see Sect. 6.2). Since one-electron systems... 

The basics of DFT are embodied in Eq. 14.54. The total energy is partitioned into several terms. Each term is itself a functional of the electron density. is the electron kinetic energy term (the Bom-Oppenheimer approximation is in place, so nuclear kinetic energy is neglected). The E potential energy term includes both nuclear-electron attraction and nuclear-nuclear repulsion. The term is sometimes called the Coulomb self-interaction term, and it evaluates electron-electron repulsions. It has the form of Coulomb s law. The sum of the first three terms (E + -I- ) corresponds to the classical energy of the charge distribution. 

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. 

The second term consists of the Coulombic self-interaction of the electron cloud and can be written in a perhaps somewhat more illuminating way as... 

Other features of the TDLDA are revealed in the total photoabsorption cross section of krypton just above threshold as illusr-trated in Figure 4, Unlike the RPAE result, the TDLDA cross section is seen to vanish at a threshold energy below the experimental onset. In the LDA, the Coulomb self-interaction of the atomic electrons is not explicitly subtracted away as in the HFA, Hence, all the electrons move in a Z-electron neutral field which decays ex,ponentially... 

Another way to deal with the Coulomb self-interaction error is to use a hybrid functional that combines exact Hartree-Fock exchange with standard LDA/GGA. Recently, the hybrid HSE functional has been reported to describe successfully the localization of a single 4f electron in Cc203 (Da Silva et al., 2007). Even though the hybrid functional approach in some cases exhibits better results than the DFT -F U approach, DFT -F U can stiU compete well in terms of computational cost. Therefore, all the reported results in this chapter were obtained using DFT -F U calculations. 

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