Electron Coulomb repulsion energy difference

To put it crudely, this correlation ensures that electrons of the same spin cannot be in the same place at the same time. Therefore this type of correlation makes the Coulombic repulsion energy between electrons of the same spin smaller than that between electrons of opposite spin. This is the reason why Hand s rule states that, an electronic state in which two electrons occupy different orbitals with the same spin is lower in energy than an electronic state in which the electrons occupy the orbitals, but with opposite spins. 

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 a singlet state a or p carbene, the electron-electron coulomb repulsion would be severe, since two electrons are constrained to the same small MO. On the other hand, the triplet configuration is stabilized by relief of the coulomb repulsion and exchange repulsion however, the separation of electrons into different MOs does not come without a cost. Thus, the magnitude of the energy difference between the triplet and singlet states (the singlet-triplet split-... 

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