Self-interaction correction, Hartree

Hartree s original idea of the self-consistent field involved only the direct Coulomb interaction between electrons. This is not inconsistent with variational theory [163], but requires an essential modification in order to correspond to the true physics of electrons. In neglecting electronic exchange, the pure Coulombic Hartree mean field inherently allowed an electron to interact with itself, one of the most unsatisfactory aspects of pre-quantum theories. Hartree simply removed the self-interaction by fiat, at the cost of making the mean field different for each electron. Orbital orthogonality, necessary to the concept of independent electrons, could only be imposed by an artificial variational constraint. The need for an ad hoc self-interaction correction (SIC) persists in recent theories based on approximate local exchange potentials. 

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... 

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]. 

This explicit removal of the self-interaction amongst the electrons is a great strength of the (algebraic approximation to the) Hartree-Fock equations. We shall see later that separate approximations to parts of the total interaction energy of a system of electrons do not have this convenient property and may often include spurious energies of interaction between different parts of a given electron . The so-called self-interaction correction (SIC) must be invoked in such... 

In other words, the existence of the so-called Self-Interaction-Correction (SIC) makes the method reminiscent of the Hartree (product) wavefunction rather than the Hartree-Fock (determinant) case. 

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. 

The self-interaction correction was first developed for atoms by Hartree in the early days of quantum mechanics (Hartree 1928). Then, Fermi and Amaldi suggested a self-interaction correction for the Thomas-Ferml theory (see Sect. 4.1) as a correction for the Coulomb interactions (Fermi and Amaldi 1934). The well-known form of this correction is that for the Coulomb potential. 

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. 

These are the Hartree-Fock equations. The first summation term (the coulomb potential) is the repulsive potential experienced by an electron in orbital j at ri due to the presence of all the other electrons in orbitals k at r2. Note however that this summation also contains a term corresponding to an electron s interaction with itself (i.e., when j=k) and this self-interaction must be compensated for. The second summation is called the exchange potential. The exchange potential modifies the interelectronic repulsion between electrons with like spin. Because no two electrons with the same spin can be in the same orbital j, the exchange term removes those interactions from the coulombic potential field. The exchange term arises entirely because of the antisymmetry of the determinental wavefunctions. The exchange term also acts to perform the self-interaction correction since it is equal in magnitude to the coulomb term when j =k. 

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 essential contribution to the KS-energy comes from the so-called exchange correlation energy xc- It incorporates corrections to the kinetic energy due to the interacting nature of the electrons of the real system, aU non-classical corrections to the electron-electron repulsion, as well as electron self-interaction corrections. If xc is ignored, the physical content of the theory becomes identical to that of the Hartree approximation. Thus, within the KS-formaUsm,... 

This expression excludes self-interaction. There have been a number of attempts to include into the Hartree-Fock equations the main terms of relativistic and correlation effects, however without great success, because the appropriate equations become much more complex. For a large variety of atoms and ions both these effects are fairly small. Therefore, they can be easily accounted for as corrections in the framework of first-order perturbation theory. Having in mind the constantly growing possibilities of computers, the Hartree-Fock self-consistent field method in various... 

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