# Can DFT be 'exact' in the limit of the uniform electron gas?

I was wondering if DFT (or specifically, LDA ?) can be exact in the limit of the homogeneous electron gas? In that case, shouldn't the self-interaction error perfectly cancel out? I realise that such a material might not exist, but in theory, is this true?

I am aware that nothing is perfectly exact without accounting for relativity, quantum electrodynamics, quantum gravity, etc. So by exact I just mean to say that the energy corresponding to this specific Hamiltonian is exact (meaning no self-interaction error):

• +1. Of course there is no way to account for quantum gravity (there is no consistent theory for it yet), and even special relativity cannot be treated exactly with LDA (except for the case of a single electron). Quantum electrodynamics also has to be treated numerically, which has errors. What is it that you mean by exact? Jun 16 '20 at 19:06
• Nike, what I meant by 'exact' is when the self-interaction error is perfectly canceled out. When people say that the 'exact' form of DFT is not yet known (even in the ground state), I am thinking whether there are in fact, cases where it can be exact - case in point, the limit of the homogeneous electron gas, same as in the question. Jun 16 '20 at 19:09
• Before several people start giving answers about all the ways in which LDA is not exact for the UEG, I have given a specific Hamiltonian to consider. Now we can talk about exact within the approximations already inherent in this Hamiltonian. We can further improve this, but I wanted to do this early before too many people show up talking about all the different approximations that aren't covered by DFT. Jun 16 '20 at 19:16
• There is an LDA xc-functional for the infinity density limit, in the spin un-polarized case. If the UEG has a physically realistic density though, I think approximations still have to be made, such as in the following LDAs: VWN, PZ81, CP, PW92, Chachiyo2016. Jun 16 '20 at 19:31

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.

• I'm happy to help you. My friend Tyberius will be adding an answer that explains some of the points which I might have missed. Jun 17 '20 at 22:32
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