Local density approximation

A density-functional approximation using the energy per particle of a uniform electron gas at the local density. Its exchange-energy-per-electron tail decays exponentially with an exponentially decaying density.

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.

A gradient-corrected exchange functional fitted to atomic exchange energies and designed to recover a conventional exchange-energy-per-electron tail $-1/(2r)$ for finite systems. This differs from the exchange potential tail.