The Core-Valence Separation in Real Space

In writing Eq. (4.1), we assume that all particles, the molecule and the ions, are at rest and that the molecule is in its equilibrium geometry. The total energy of the latter, E = ( // ), is expressed in the Born-Oppenheimer approximation. The 

The second term represents the attraction between the nuclei and the electrons, where rn is the distance between electron i and nucleus k. The third term represents the repulsion between electrons i and j at a distance r. The last term represents the repulsion between the nuclei, where Rj i is the distance between nuclei with charges and Z/. 

The derivative (dE/dZ)p at constant electron density p with respect to the nuclear charge of one of the nuclei is obtained with the help of the HeUmann-Feynman theorem [74]. So we get the potential at the center k, namely, VjJZk = dEjdZ )p, and the corresponding potential energy 

The molecule is taken in its equilibrium geometry. So we apply the molecular virial theorem, 2E = Vne + Vee + Kn, where Vee is the interelectronic repulsion. 

On the other hand, in Hartree-Fock theory the total energy is E = ( ee Kn), where Vj is the occupation (0, 1, or 2) of orbital / with eigenvalue ,. Combining this result with (4.6) to get rid of its (Vee nn) part, we write 

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