densityfunctional

A density functional is a functional that assigns a scalar value to the electron density n(r) of a many-electron system. In practice, density functionals are central to density functional theory (DFT), a widely used approach for calculating the electronic structure of atoms, molecules, and solids.

The theoretical basis rests on the Hohenberg-Kohn theorems. The first theorem establishes a one-to-one correspondence between the ground-state electron density and the external potential, implying that all ground-state properties are functionals of n(r). The second theorem states that the ground-state energy can be written as E[n] = F[n] + ∫ v_ext(r) n(r) dr, where F[n] is a universal functional of the density.

In the Kohn-Sham formulation, the interacting many-electron problem is mapped onto a fictitious non-interacting system with the same density. The energy functional is written as E[n] = T_s[n] + U[n] + E_xc[n], where T_s is the non-interacting kinetic energy, U is the classical Coulomb energy, and E_xc[n] is the exchange-correlation functional capturing all many-body effects.

Exchange-correlation functionals come in several families. Local density approximation (LDA) uses n(r) only; generalized gradient approximations (GGA, e.g., PBE) include density gradients; meta-GGA functionals add kinetic-energy-density terms; hybrids mix in a portion of exact exchange (e.g., B3LYP, PBE0, HSE06).

Time-dependent DFT (TDDFT) extends the framework to excited states and time-dependent phenomena.

Advantages of density functionals include computational efficiency and applicability to large systems; limitations involve the accuracy of the chosen functional, self-interaction errors, and challenges with band gaps and dispersion forces.