Exact exchange energy density functional from Fock operator

Functionals. The difference between the Fock operator, in wave function based calculations and the analogous operator in DFT calculations is that the Coulomb and exchange operators in T are replaced in DFT by a functional of the electron density. In principle, this functional should provide an exact formula for computing the Coulombic interactions between an electron in a KS orbital and all the other electrons in a molecule. To be exact, this functional must include corrections to the Coulombic repulsion energy, computed directly from the electron density, for exchange between electrons of the same spin and correlation between electrons of opposite spin. 

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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