+Although, even in such a simple case, no exact form of the correlation part of $\epsilon_{\text{xc}}(n)$ is known, highly accurate numerical estimates using Monte Carlo methods \cite{ceperley80} and corresponding paramterizations exist \cite{perdew81}.
+Obviously exact for the homogeneous electron gas, the LDA was {\em a priori} expected to be useful only for densities varying slowly on scales of the local Fermi or TF wavelength.
+Nevertheless, LDA turned out to be extremely successful in describing some properties of highly inhomogeneous systems accurately within a few percent.
+Although LDA is known to overestimate the cohesive energy in solids by \unit[10-20]{\%}, the lattice parameters are typically determined with an astonishing accuracy of about \unit[1]{\%}.
+
+More accurate approximations of the exchange-correlation functional are realized by the introduction of gradient corrections with respect to the density \cite{kohn65}.
+Respective considerations are based on the concept of an exchange-correlation hole density describing the depletion of the electron density in the vicinity of an electron.
+The averaged hole density can be used to give a formally exact expression for $E_{\text{xc}}[n(vec{r})]$ and an equivalent expression \cite{kohn96,kohn98}, which makes use of the electron density distribution $n(~\vec{r})$ at positions $~\vec{r}$ near $\vec{r}$, yielding the form
+\begin{equation}
+E_{\text{xc}}[n(vec{r})]=\int\epsilon_{\text{xc}}(\vec{r};[n(~\vec{r})])n(\vec{r}) d\vec{r}
+\end{equation}
+for the exchange-correlation energy, where $\epsilon_{\text{xc}}(\vec{r};[n(~\vec{r})])$ becomes a nearsighted functional of $n(~\vec{r})$.
+Expressing $n(~\vec{r})$ in a Taylor series, $\epsilon_{\text{xc}}$ can be thought of as a function of coefficients, which correspond to the respective terms of the expansion.
+Neglecting all terms of order $\mathcal{O}(\nabla n(\vec{r})$ results in the functional equal to LDA, which requires the function of variable $n$.
+Including the next element of the Taylor series introduces the gradient correction to the functional, which requires the function of variables $n$ and $|\nabla n|$.
+This is called the generalized gradient approximation (GGA), which expresses the exchange-correlation energy density as a function of the local density and the local gradient of the density.
+\begin{equation}
+E^{\text{GGA}}_{\text{xc}}[n(vec{r})]=\int\epsilon_{\text{xc}}(n(\vec{r}),|\nabla n(\vec{r})|)n(\vec{r}) d\vec{r}
+\text{ .}
+\end{equation}
+These functionals constitute the simplest extensions of LDA for inhomogeneous systems.
+At modest computational costs gradient-corrected functionals very often yield much better results than the LDA with respect to cohesive and binding energies.