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 φ_i(r): n(r) = ∑_i f_i |φ_i(r)|^2, with occupations f_i determined by temperature or degeneracy. The orbitals satisfy the Kohn-Sham equations

[-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 potential, and v_xc[n](r) = δE_xc[n]/δn(r) is the exchange-correlation potential. The ground-state density is obtained self-consistently by iterating these equations until convergence.

The exchange-correlation functional E_xc[n] encodes all many-body effects beyond classical electrostatics, and its exact form is unknown. Practical calculations use approximations such as the local density approximation (LDA), generalized gradient approximations (GGA), meta-GGA, or hybrid functionals that mix a portion of exact exchange.

Kohn-Sham DFT is widely used in chemistry and materials science due to its favorable balance between accuracy and computational cost, though known limitations include band-gap underestimation and self-interaction errors in some systems.