the classical case where the kinetic energy is neglected, the limit of $E_{\rm cl}(\rho_N)/N$ was found in our previous work [@LewLieSei-18], but without a quantitative estimate on the remainder. We can give an estimate similar to  in the classical case, with a lower power of ${\varepsilon}$ in front of the gradient term. This is just a slight adaptation of the proof in [@LewLieSei-18], which is much easier than the quantum case. The argument is explained for completeness in Appendix \[app:classical\].

In the classical case the limit for $E_{\rm cl}(\rho_N)/N$ was later extended to Riesz interactions $|x|^{-s}$ and other dimensions $d{\geqslant}1$ in [@CotPet-19]. Although our result  in the quantum case can probably be extended to other Riesz interactions by using ideas from [@Fefferman-85; @Hugues-85; @Gregg-89; @CotPet-19], we only consider here the physically relevant 3D Coulomb case for shortness.

We expect an inequality similar to  for the (mixed) canonical version of $E(\rho)$ where $\int_{{\mathbb{R}}^3}\rho=N\in\mathbb{N}$ and $\Gamma_n=0$ for $n\neq N$. However our proof does not adapt in an obvious way to this case.

\[rmk:exchange\_correlation\] In physics and chemistry, the LDA is usually expressed in terms of the *exchange-correlation energy*. In the grand-canonical setting it is defined by $$E_{\rm xc}(\rho)=E(\rho)-T(\rho),$$ with $T(\rho)$ the lowest 