KohnSham
Kohn-Sham density functional theory (DFT) is a practical reformulation of the quantum many-electron problem. The Kohn-Sham approach, introduced by Walter Kohn and Lu Jeu Sham in 1965, recasts the interacting electron system as a set of non-interacting electrons moving in an effective potential, chosen so that the non-interacting system has the same ground-state electron density as the real, interacting system. This mapping enables the ground-state energy and density to be calculated variationally.
In the Kohn-Sham framework, the electron density is constructed from a set of occupied single-particle orbitals
[-1/2 ∇^2 + v_ext(r) + v_H[n](r) + v_xc[n](r)] φ_i(r) = ε_i φ_i(r),
where v_ext is the external potential (e.g., nuclei), v_H[n](r) = ∫ n(r')/|r − r'| dr' is the Hartree electrostatic
The exchange-correlation functional E_xc[n] encodes all many-body effects beyond classical electrostatics, and its exact form is
Kohn-Sham DFT is widely used in chemistry and materials science due to its favorable balance between accuracy