exchangecorrelation

Exchangecorrelation, in the context of density functional theory (DFT), refers to the exchange-correlation energy functional E_xc[n] that captures all many-body effects beyond the classical Hartree term. It combines the effects of the Pauli exclusion principle (exchange) and the correlated motion of electrons due to their Coulomb repulsion (correlation). In spin-polarized systems, E_xc depends on the spin densities n_up and n_down, reflecting how exchange and correlation differ for each spin channel.

In the Kohn-Sham formulation of DFT, the ground-state energy is expressed as E[n] = T_s[n] + ∫ v_ext(r) n(r) dr + 1/2 ∫∫ n(r)n(r')/|r−r'| dr dr' + E_xc[n], where T_s[n] is the non-interacting kinetic energy and the last term accounts for exchange and correlation. The exchange-correlation potential v_xc(r) is the functional derivative δE_xc/δn(r) and drives the Kohn-Sham equations used to obtain self-consistent electron densities.

E_xc[n] is not known exactly and must be approximated. Common approximations include the local density approximation (LDA), which uses data from a uniform electron gas, and generalized gradient approximations (GGA), which incorporate density gradients (examples include PBE and PW91). Meta-GGA functionals incorporate additional information such as kinetic energy density. Hybrid functionals mix a portion of exact exchange from quantum chemistry with a DFT exchange-correlation term.

The choice of E_xc[n] influences predicted properties such as lattice constants, reaction barriers, and electronic structure. While standard functionals perform well for many systems, they often struggle with van der Waals forces, strongly correlated electrons, or dispersion interactions, prompting ongoing development of improved functionals and nonlocal corrections.