It is used in several areas of physics and chemistry and its success in predicting the electronic properties of atoms, molecules and materials is unprecedented. Among the many functionals that have been developed over the years [@MarHea-17], the *Local Density Approximation* (LDA) is the standard and simplest scheme [@HohKoh-64; @KohSha-65; @DreGro-90; @ParYan-94; @PerKur-03]. It is not as accurate as its successors involving gradient corrections, but it is considered as *“the mother of all approximations”* [@PerSch-01] and it is still one of the methods of choice in solid state physics.

In the orbital-free formulation of Density Functional Theory [@Levy-79; @Lieb-83b], the Local Density Approximation consists in replacing the full ground state energy by a local functional as follows: $$F_{\rm LL}(\rho)\approx\frac{1}{2}\int_{{\mathbb{R}}^3}\int_{{\mathbb{R}}^3}\frac{\rho(x)\rho(y)}{|x-y|}\,dx\,dy+\int_{{\mathbb{R}}^3}e_{\rm UEG}\big(\rho(x)\big)\,dx.
 \label{eq:LDA}$$ Here $\rho$ is the given one-particle density of the system and $F_{\rm LL}(\rho)$ is the *Levy-Lieb functional* [@Levy-79; @Lieb-83b], the main object of interest in DFT. This is the lowest possible Schrödinger energy of all quantum states having the prescribed density $\rho$. The first term on the right side is called the *direct* or *Hartree term*. It is the classical electrostatic interaction energy of the density $\rho$ and it is the only nonlocal term in the LDA. The second 