Density functional theory

A framework describing electronic ground-state properties using the electron density. Practical approximations require an exchange-correlation functional; this does not mean that explicit electrons have been replaced by a classical atom-only force field.

The part of the exact density functional left after subtracting the auxiliary noninteracting kinetic energy and classical Hartree electrostatic energy. It includes fermionic exchange, correlation and the difference between interacting and noninteracting kinetic energies. Its functional derivative enters the Kohn-Sham equations. Practical approximations to this functional, rather than the existence of a density description itself, are a central source of error.

The constrained-search functional $F[n]={\min }_{\Psi \to n}⟨\Psi |T+W|\Psi ⟩$ is universal for a fixed interaction. Minimization of $F[n]+\int vn$ over admissible densities with the correct electron number gives the ground-state energy. Each trial density represents a constrained set of antisymmetric states, so minimizing over densities and their representing states recovers the ordinary variational principle. The admissible domain and possible degeneracy must be handled explicitly.

Effective one-electron equations for orbitals whose density represents that of the interacting electronic system. The effective potential contains nuclear attraction, Hartree repulsion and an exchange-correlation contribution.

Summing the Kohn-Sham equations over occupied normalized orbitals yields ${\sum }_i{f}_i{ϵ}_i=T_s+\int nv+2E_H+\int n{v}_{\mathrm{xc}}$. The exact ground-state energy functional is $T_s+\int nv+E_H+E_{\mathrm{xc}}$, so subtraction proves the correction in the title. Nuclear or lattice source self-energies must be added consistently. This accounting does not turn all auxiliary orbital eigenvalues into physical excitation energies.