Coulomb operator density functional theory

The combination of the Dirac-Kohn-Sham scheme with non-relativis-tic exchange-correlation functionals is sometimes termed the Dirac-Slater approach, since the first implementations for atoms [13] and molecules [14] used the Xa exchange functional. Because of the four-component (Dirac) structure, such methods are sometimes called fully relativistic although the electron interaction is treated without any relativistic corrections, and almost no results of relativistic density functional theory in its narrower sense [7] are included. For valence properties at least, the four-component structure of the effective one-particle equations is much more important than relativistic corrections to the functional itself. This is not really a surprise given the success of the Dirac-Coulomb operator in wave function based relativistic ab initio theory. Therefore a major part of the applications of relativistic density functional theory is done performed non-rela-tivistic functionals. 

These are important points for any quantitative work, electron-electron interactions must be taken into account, and the theories underpinning computation of MOs do this at various levels of accuracy. Approaches such as Hartree-Fock or density functional theory adapt the Hamiltonian operator to include electron-electron terms in an averaged way so electrons see the Coulomb field of each other averaged over the calculated density associated with each MO (see the Further Reading section in this chapter). 

The difference between this Fock operator and the Hartree-Fock counterpart in Eq. (2.51) is only the exchange-correlation potential functional, Exc, which substitutes for the exchange operator in the Hartree-Eock operator. That is, in the electron-electron interaction potential, only the exchange operator is replaced with the approximate potential density functionals of the exchange interactions and electron correlations, while the remaining Coulomb operator, Jj, which is represented as the interaction of electron densities, is used as is. The point is that the electron correlations, which are incorporated as the interactions between electron configurations in wavefunction theories (see Sect. 3.3), are simply included... 

Clearly, J (l) just multiplies xi) by the value of the potential at point JTi due to electrons distributed according to the density function for group S in state s. On the other hand K (l) is an integral operator, of the kind used in Hartree-Fock theory (Sections 6.1 and 6.4). Tliese two operators are the coulomb and exchange operators for an electron in the effective field due to the electrons of group S. It is now possible to write the interaction terms in (14.2.2) in the form... 

The first term is the familiar one-electron operator, the second term represents the Coulomb potential, and the third term is called exchange-correlation potential. HF and DFT differ only in this last term. In HF theory there is only a nonlocal exchange term, while in DFT the term is local and supposed to cover both exchange and correlation. It arises as a functional derivative with respect to the density ... 

In addition, the functional must somehow cancel the fictitious repulsion energy between an electron and itself, which arises if the electron density, due to all the electrons, is used to compute the Coulombic energy of a single electron. As discussed in Section 3.2.1, in HF theory cancellation of the self-repulsion energy results from the presence of the exchange operator in T. If this effect of Kj, in the Fock operator is not mirrored exactly by the functional chosen, the cancellation of the self-repulsion energy will not occur. 

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