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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]

The Wigner-Seitz radius $r_s$ is the radius of a sphere containing one electron at the local uniform density:In atomic units, the unpolarized homogeneous electron gas has exchange energy per electron $-0.4582/r_s$. The local density approximation applies that uniform-gas result at each position using the local density, then integrates the result weighted by $\rho$. It assumes local uniform-gas behavior rather than resolving nonlocal exchange explicitly.

For typical atoms and molecules, the Becke 1988 exchange functional improves exchange energies and commonly improves molecular energetics relative to exchange-only local density approximation, because it accounts for density gradients and corrects important nonuniformity errors. This is a general trend, not a theorem for every observable or system.

A comparison of total molecular predictions also depends on the correlation functional, basis, spin treatment and geometry. B88 is an exchange approximation, not by itself a complete specification of an exchange-correlation calculation.