The Uniform Electron Gas

There is no systematic way to find or improve a density functional. The most appealing way forward is to find the exact solution for a model system, and then assume that the system of interest behaves similarly to the model. The first density functionals were due to Thomas [78], Fermi [79,80,81] and Dirac [82], all of which used the uniform electron gas as their model.

The uniform electron gas is defined as a large number of electrons N in a cube of volume V, throughout which there is a uniform spread of positive charge sufficient to make the system neutral. The uniform gas is then defined as the limit $N\to\infty$, $V\to\infty$, with the density $\rho=N/V$ remaining finite. Although it does bear some resemblance to electrons in metals, its widespread use is due to its simplicity -- it is completely defined by one variable, the electron density $\rho$.

Using the uniform electron gas, an expression for the kinetic energy (the Thomas-Fermi kinetic functional) can be derived [83]
$$\begin{align}T^{TF27}[\rho_{\sigma}] = \frac{3}{10}(6\pi^{2})^{2/3}\int\rho_{\sigma}^{5/3}({\bf r})\,d{\bf r}, \end{align}$$
where $\sigma$ can take the values of $\alpha$ or $\beta$. When applied to atoms and molecules the Thomas-Fermi functional yields kinetic energies that are about 10% too small.

Similarly, an expression for the exchange energy of the uniform electron gas can be calculated (the Dirac exchange functional) [83]
$$\begin{align}E_{x}^{D30}[\rho_{\sigma}] = -\frac{3}{2}\left(\frac{3}{4\pi}\right)^{1/3}\int\rho_{\sigma}^{4/3}({\bf r})\,d{\bf r}. \end{align}$$
The Dirac functional also gives exchange energies that are roughly 10% smaller than those from HF theory [84]. More worrying is that the spurious self-interaction of electrons is not exactly canceled.

A closed shell functional for the correlation energy of the uniform electron was determined by Vosko, Wilk and Nusair [85], who combined analytic information about the high and low density limits with the quantum Monte-Carlo simulation results of Ceperly and Alder [86]. The VWN functional usually overestimates the correlation energy of atoms and molecules by approximately a factor of two [87]. The uniform electron gas is obviously a better reference system for exchange energies than it is for correlation energies.