The Almost Uniform Electron Gas

The electron densities of atoms and molecules are often far from uniform, so functionals based on systems which include an inhomogeneous density should perform better. In 1935 von Weizsacker [88] placed infinitesimally small ripples on the uniform electron gas and calculated the second order correction to the kinetic energy
$$\begin{align}T^{W35}[\rho_{\sigma}] = T^{TF27}[\rho_{\sigma}] + \frac{1}{8}\int\rho_{\sigma}^{5/3}x^{2}_{\sigma}\, d{\bf r} \end{align}$$
where $x({\bf r})$ is a dimensionless quantity, the reduced density gradient
$$\begin{align}x({\bf r}) = \frac{\vert\nabla \rho({\bf r})\vert}{\rho^{4/3}({\bf r})}. \end{align}$$
Unfortunately the original derivation was flawed and the above functional is too large by a factor of nine [83]. The corrected functional is a large improvement on $T^{TF27}[\rho]$, yielding kinetic energies typically within 1% of HF theory. The fourth order [89] and sixth order [90] corrections have subsequently been computed; however, the series is divergent due to the extremely large values of $x({\bf r})$ in the Rydberg regions, and it is advantageous to stop after second order.

A similar correction was made to the Dirac exchange functional by Sham [91]. Kleinman [92] later showed that the Sham derivation was too small by 10/7. The second order correction to the exchange energy is
$$\begin{align}E_{x}^{SK71}[\rho_{\sigma}] = E_{x}^{D30}[\rho_{\sigma}] - \frac{5}{(36\pi)^{5/3}}\int\rho_{\sigma}^{4/3} x^{2}_{\sigma}\, d{\bf r}. \end{align}$$
The corrected functional gives exchange energies that are typically within 3% of HF; however, it is not seen as an improvement over the Dirac functional, as the potential is unbounded in the Rydberg regions of atoms and molecules.