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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.

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.

Lu Jeu Sham and Walter Kohn developed self-consistent equations that make density functional theory computationally useful. The Kohn-Sham equations introduce an auxiliary system of noninteracting electrons with the same ground-state electron density as the interacting system. An effective potential includes the external potential, electrostatic repulsion and an exchange-correlation functional. Solving the equations and updating the electron density iteratively gives a self-consistent state. The difficult many-electron effects remain in the exchange-correlation functional, so practical calculations still need an approximation to it.^[ref]

Density functional theory replaces the interacting many-electron ground state wavefunction by its spatial density $n(\mathbf{r})$. For fixed interaction and a nondegenerate ground state, the Hohenberg-Kohn theorem fixes the external potential up to an additive constant. If two distinct potentials gave the same density, their strict variational energy inequalities would sum to the contradiction $E_0+E_0^{\prime }<E_0+E_0^{\prime }$.

The density functional variational principle minimizesover admissible densities of the correct particle number. The universal $F$ contains the kinetic and interaction energies. This is exact in principle; an explicit useful functional is the practical difficulty.^[ref]

The Kohn-Sham equations introduce noninteracting orbitals with the same density and decompose $F=T_s+E_H+E_{\mathrm{xc}}$. With $n={\sum }_i{f}_i|{\varphi }_i{|}^2$,The exchange-correlation functional includes exchange, correlation and the interacting kinetic-energy correction relative to $T_s$. Guess a density, construct the potentials, solve and occupy orbitals, then mix and iterate to self-consistent field convergence. The local density approximation uses $E_{\mathrm{xc}}^{\mathrm{LDA}}=\int n{ϵ}_{\mathrm{xc}}^{\mathrm{uniform}}(n)d^{3}r$; a generalized gradient approximation also depends on density gradients.

Kohn–Sham orbital energies are not generally excitation energies, and their sum double counts interactions. The Kohn-Sham energy double-counting correction iswith ion–ion energy added for a total crystal energy. Approximate functionals can miss self-interaction cancellation, band gaps and strong correlation; convergence alone does not prove the global ground state.^[ref]

Kohn-Sham equations in density functional theory use an auxiliary one-electron Hamiltonian whose effective potential depends on the density. Its orbitals construct a density representing an interacting electronic system; this is distinct from explicitly diagonalizing a large many-electron determinant Hamiltonian.^[ref]