The local density approximation - LDA

In the local density approximation (LDA), the value of $E_{xc}[n({\bf r})]$ is approximated by the exchange-correlation energy of an electron in an homogeneous electron gas of the same density $n({\bf r})$, i.e.

$$E_{xc}^{LDA}[n({\bf r})] = \int \epsilon_{xc}(n({\bf r}))n({\bf r})d{\bf r} \;\;\; .$$

The most accurate data for $\epsilon_{xc}(n({\bf r}))$ is from Quantum Monte Carlo calculations. The LDA is often surprisingly accurate and for systems with slowly varying charge densities generally gives very good results. The failings of the LDA are now well established: it has a tendency to favour more homogeneous systems and over-binds molecules and solids. In weakly bonded systems these errors are exaggerated and bond lengths are too short. In good systems where the LDA works well, often those mostly consisting of $sp$ bonds, geometries are good and bond lengths and angles are accurate to within a few percent. Quantities such as the dielectric and piezoelectric constant are approximately 10% too large.

The principle advantage of LDA-DFT over methods such as Hartree-Fock is that where the LDA works well (correlation effects are well accounted for) many experimentally relevant physical properties can be determined to a useful level of accuracy. Difficulties arise where it is not clear whether the LDA is applicable. For example, although the LDA performs well in bulk group-IV semiconductors it is not immediately clear how well it performs at surfaces of these materials.