Hartree-Fock calculation exchange potential

Here, h is the one electron matrix in the non-orthogonal Gaussian basis and G (P ) is the two electron matrix for Hartree-Fock calculations, but for DFT it represents the Coulomb potential. The term Exc is the DFT exchange-correlation functional (for Hartree-Fock Exc = 0), while Vmn represents the nuclear repulsion energy. In the orthonormal basis, these matrices are h = etc., where the overlap... 

One should bear in mind that originally the RMF model was formulated within the Hartree and no-sea approximations. Implementing the Dirac sea may require serious revision of the model and inclusion of additional terms. Hartree calculations including the Dirac sea and Hartree-Fock calculations including exchange terms lead to smaller nucleon potentials in normal nuclei. Shallower potentials will produce smaller attraction for antinucleons, but the qualitative effect that the presence of antiprotons reduces repulsion and enhances attraction for nucleons will remain valid. We expect that the additional binding and compression of the nucleus will appear even for an antinucleon potential as low as 200 MeV. 

Table 8 Energies (in eV) of the highest (HOMO), second-highest (HOMO-1), and third-highest (HOMO-2) occupied orbital as well as of the lowest (LUMO) and second-lowest (LUMO+1) unoccupied orbital for H2O, N2, and CO from exact-exchange (EXX), Hartree-Fock (HF), and local-spin-density (LSDA) calculations in comparison with the experimental first ionization potentials (IP). For N2, the ordering of the HOMO and the HOMO-1 is not correctly reproduced by the Hartree-Fock calculations. The results are from ref. 82...

The exchange-correlation energy density can be split into two parts exchange component Ex n) and correlation component e Cn). The explicit expression for the exchange component is known from Hartree-Fock theory but the correlation component is known only numerically. Several parametrisations exist for the exchange-correlation energy and potential of a homogeneous gas system which can be used for the LDA calculations within DFT. 

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