Let us represent the exact exchange-correlation (xc) energy as a sum of an exchange term (x) and a correlation term (c):
$$
\tag{1}
E_{xc} = E_x + E_c~ .
$$
For a uniform electron gas (UEG), we do have an exact expression for the exchange term:
$$
\tag{2}
E_{x} = - \frac{3}{4}\left( \frac{3}{\pi} \right)^{1/3}\int\rho(\mathbf{r})^{4/3}\ \mathrm{d}\mathbf{r}\ ,
$$
but the correlation term is only known in the unphysical limit of infinitely strong or infinitely weak correlation, and for the spin-unpolarized case. In the case of infinitely strong correlation and no spin polarization, we have:
$$
\tag{3}
E_{c} = A\ln(r_{s}) + B + C\ln(r_{s})r_{s} + Dr_{s},
$$
in terms of the Wigner-Seitz radius, which I will give here for a 3D gas with a number density of $n$:
$$
\tag{4}
r_s = \left(\frac{3}{4\pi n}\right)^{1/3}.
$$
The expressions for $A,B,C$ and $D$ are not simple. For example, here is $C$ (from a 1956 paper by Gell-Mann and Brueckner) to second order in the logarithmic divergene and 4th order in perturbation theory:
$$
\tag{5}
C=\frac{2}{\pi^2}\left(1-\ln 2 \right)\left(\ \ln \left(\frac{256}{9\pi^{4}} \right)^{1/3} - \frac{1}{2} + \frac{\int R(u)^2\ln R(u)\textrm{d}u}{\int R(u)^2\textrm{d}u } \right) + \delta,\\
\delta = \frac{3}{8\pi^5}\int\!\!\!\!\int\!\!\!\!\int \frac{\textrm{d}q\textrm{d}^3p_1\textrm{d}^3p_2}{q^2 + \textbf{q}\cdot \left( \textbf{p}_1 + \textbf{p}_2\right)} + \frac{6}{\pi^3}\int\!\!\!\! \int_0^1 \frac{R(u)^2}{q}\textrm{d}q \textrm{d}u.
$$
The integrals area obtained by numerically, so they are not exact, but they can be made exact to within floating-point error if desired.
For a spin-polarized UEG, further approximations are made, and for a UEG that is not in the infinitely strong (or infinitely weak) correlation limit, we again have to rely on approximations, such as the following: VWN (Vosko-Wilk-Nusair, 1980), PZ81 (Perdew-Zunger, 1981), CP (Cole-Perdew, 1982), PW92 (Perdew-Wang 1992), Chachiyo (2016).
I will now summarize the current limitations on "exact" xc-energy functionals for a uniform electron gas (UEG). The gas has to be:
- spin-unpolarized
- in the limit of infinitely strong correlation, or infinitely weak correlation
Even in these cases, the expressions are not entirely exact because:
- they involve truncating an expansion for a logarithmic divergence (to second order, in the above example, but I suppose if one really wanted to, they could keep going until the sum of all remaining terms are deemed to not matter at machine precision),
- they involve truncating an perturbation theory expansion (to fourth order, in the above example, but machine precision can probably be achieved as in the above point),
- they involve numerical integrations, which also can in principle be done to within machine precision.
Conclusion: In theory there exists some exact functional for the UEG, but until now we only know what it is for the special case just described, and even in this special case there are series that have to be truncated to get these formulas, and even after that, there's integrals that need to be done numerically.
