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Axel D. Becke develops practical approximations for density functional theory. His 1988 exchange functional uses the local electron density and its gradient to improve on a purely local density approximation. Its construction imposes the proper long-distance behavior of the exchange-energy density and fits one parameter using atomic exchange energies. This supplies a concrete example of building a useful approximate energy functional from physical constraints and reference calculations. The asymptotic statement concerns an energy density, and does not automatically establish the asymptotic behavior of its functional derivative.^[ref]

A generalized gradient approximation models the exchange and correlation contribution in density functional theory using both the local electron density $n(\mathbf{r})$ and its spatial gradient $\nabla n(\mathbf{r})$. Its schematic form is $E_{\mathrm{xc}}[n]=\int f(n,\nabla n)d^{3}r$. It extends the local density approximation, which uses only the density at each point.

Two regions with equal density can therefore receive different energy contributions if one is nearly uniform and the other varies rapidly. This makes the approximation useful for chemical bonds, surfaces and molecular tails. The improvement is still approximate: adding density gradients does not automatically recover every long-distance interaction or the exact exchange potential. The Becke 1988 exchange functional is an example of a gradient-dependent exchange model.

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.