possible kinetic energy . The functional $T(\rho)$ is studied in Section \[sec:a\_priori\] below, where it is proved that $$\left|T(\rho)-q^{-\frac23}c_{\rm TF}\int_{{\mathbb{R}}^3}\rho(x)^{\frac53}\,dx\right|
{\leqslant}{\varepsilon}q^{-\frac23}\int_{{\mathbb{R}}^3}\rho(x)^{\frac53}\,dx+\frac{C}{{\varepsilon}^{\frac{13}3}}\int_{{\mathbb{R}}^3}|\nabla\sqrt\rho(x)|^2\,dx
\label{eq:estim_T}$$ with $c_{\rm TF}=3^{5/3}4^{1/3}\pi^{4/3}/5$ the Thomas-Fermi constant. The lower bound was derived by Nam [@Nam-18] and we prove the missing upper bound (with a better power of ${\varepsilon}$) in Theorem \[thm:upper\_bound\_T\] below. Actually, by following our proof of Theorem \[thm:LDA\] (simply discarding the Coulomb interaction) we can also prove a lower bound on $T(\rho)$, with an error similar to the right side of  but with a smaller power of ${\varepsilon}$ in front of $|\nabla\rho^\theta|^p$. This provides the following estimate on the exchange-correlation energy $$\begin{gathered}
 \left|E_{\rm xc}(\rho)- \int_{{\mathbb{R}}^3}e_{\rm UEG}\big(\rho(x)\big)\,dx+q^{-\frac23}c_{\rm TF}\int_{{\mathbb{R}}^3}\rho(x)^{\frac53}\,dx\right|\\
 {\leqslant}{\varepsilon}\int_{{\mathbb{R}}^3}\big(\rho(x)+\rho(x)^2\big)\,dx
 +\frac{C(1+{\varepsilon})}{{\varepsilon}}\int_{{\mathbb{R}}^3}|\nabla\sqrt\rho(x)|^2\,dx\\+\frac{C}{{\varepsilon}^{4p-1}}\int_{{\mathbb{R}}^3}|\nabla\rho^\theta(x)|^p\,dx.
 \label{eq:LDA_exchange_correlation}\end{gathered}$$ For a rescaled density $\rho_N(x)=\rho(x/N^{1/3})$ we obtain the same rate of convergence $N^{11/12}$ as in .

\[rmk:LSDA\] In practice, it is often convenient to not fix the total density but, rather, the density of each spin component $$\begin{gathered}
\rho_\sigma(x)=\\
\sum_{n{\geqslant}1}n \sum_{\substack{\sigma_2,...,\sigma_n\\ \in\{1,...,q\}}}\int_{{\mathbb{R}}^{3(n-1)}}
\Gamma_n(x,\sigma,x_2,...,x_n,\sigma_n;x,\sigma,x_2,...,x_n,\sigma_n)dx_2\cdots dx_n \end{gathered}$$ for $\sigma\in\{1,...,q\}$. Similarly as in Theorem \[thm:UEG\_thermo\_limit\_quantum\] one can define the corresponding spin-polarized UEG energy $e_{\rm UEG}(\rho_1,...,\rho_q)$ of the uniform electron gas where the electrons of spin $\sigma$ are assumed to have the constant density $\rho_\sigma$. By following the arguments in this 