The local density approximation

The exchange-correlation functional $E_{xc}$ is known almost exactly for the uniform electron gas, and so one possible way to obtain an approximation, presumably good in systems in which the electron density does not vary too strongly, is to use the local density approximation (LDA). Here one considers contributions to $E_{xc}$ from each element of volume $d^3{\bf r}$, given by the exchange-correlation energy per particle of the uniform electron gas with density $\rho({\bf r})$, $\epsilon_{xc}[\rho({\bf r})]$. The total XC energy is then:

$$E_{xc}^{LDA} = \int_V \epsilon_{xc}[\rho({\bf r})]\rho({\bf r}) d^3 {\bf r},$$ (8.42)

This was the original proposal of Kohn and Sham, which turned out to be reasonably accurate also for systems with highly non-uniform densities, such as atoms and molecules.