The Kohn-Sham MO model

The basic postulate in Kohn-Sham DFT is that we can apply a one-electron formulation to the system of N interacting electrons by introducing a suitable local potential V_XC(r), in addition to any external potentials V_ext(r) and the Coulomb potential of the electron cloud V_C(r), and solving:

[ T + V_ext(r) + V_C(r) + V_XC(r)] φ_i (r) = ε_i φ_i (r)

Here T is the kinetic energy operator. The potential V_XC is the functional derivative with respect to the density ρ of E_XC[ρ], the exchange-correlation energy functional. The one-electron molecular orbitals (MOs) φ_i with corresponding orbital energies ε_i define the exact electronic charge density and give, in principle, access to all properties because these are expressible as functional of the density, in particular the energy. Moreover, they provide an intuitively appealing view of the system as being built from independent-electron orbitals with all ensuing interpretations. The exact formof the exact energy density E_XC(r), representing and incorporating all exchange and correlation (XC) effects is unknown. From general principles one can formulate conditions on what E_XC(r) should look like, and several, more and more advanced expressions have been advocated for it in the literature. Their application to real systems has been impressively successful, and it seems likely that a further increase of accuracy is a matter of time.