## 2 Theoretical Part  
### Uniform Electron Gas  
Let us consider a UEG with \(N\) electrons at temperature \(T\), contained in the cubic cell with volume \(V\). We assume that the system is unpolarized, and do not take into account spin interactions, so the numbers of electrons with two different spin projections are equal and constant: \(N_{u}=N_{d}=N/2\). Further, we do not taken into account any spin interactions. In order to reduce the finite-size effects, we apply the periodic boundary conditions (PBC). In addition to the Coulomb interactions in the main cell, one has to take into account the interactions with all periodic images using the Ewald summation technique [35]. As a result, the effective Hamiltonian of the UEG is as follows [36]:  
\[\hat{H}=\sum_{a=1}^{N}\frac{\hat{\mathbf{p}}_{a}^{2}}{2m}+\sum_{a=1}^{N}\sum_{ b=a+1}^{N}e^{2}\Psi(\mathbf{q}_{a},\mathbf{q}_{b})+\frac{1}{2}e^{2}N\bar{\xi}_{M}, \tag{1}\]  
where \(m\) and \(e\) are the electronic mass and charge, \(\hat{\mathbf{p}}_{a}\) and \(\mathbf{q}_{a}\) are the momentum operator and the coordinate of an \(a\)-th electron. The periodic Ewald pair potential is:  
\[\Psi(\mathbf{r},\mathbf{s})=\frac{1}{V\pi}\sum_{\mathbf{G}\
eq 0}G^{-2}\mathbf{e} ^{-\frac{n^{2}Q^{2}}{\kappa^{2}}+2\pi\mathbf{i}\mathbf{G}(\mathbf{r}-\mathbf{ s})}-\frac{\pi}{\kappa^{2}V}+\sum_{\mathbf{R}}\frac{\text{erfc}(\kappa| \mathbf{r}-\mathbf{s}+\mathbf{R}|)}{|\mathbf{r}-\mathbf{s}+\mathbf{R}|}, \tag{2}\]  
and the Madelung constant is:  
\[\bar{\xi}_{M}=\frac{1}{V\pi}\sum_{\mathbf{G}\
eq 0}G^{-2}\mathbf{e}^{-\frac{n^{2} Q^{2}}{\kappa^{2}}}-\frac{\pi}{\kappa^{2}V}+\sum_{\mathbf{R}\
eq 0}\frac{\text{ erfc}(\kappa|\mathbf{r}-\mathbf{s}|)}{|\mathbf{r}- \mathbf{s}|}. \tag{3}\]  
Here, \(\mathbf{R}=\mathbf{n}_{d}L\) and \(\mathbf{G}=\mathbf{n}_{r}/L\), with \(\mathbf{n}_{d}\) and \(\mathbf{n}_{r}\) having integer components. Note that the Ewald parameter \(\kappa\) does not affect the values and should be chosen for efficiency reasons.  
The MDF of the UEG describes the probability density of the electrons to have certain values of momentum. Due to the kinematic and dynamic identity of the electrons in the state of thermodynamic equilibrium, a single-particle MDF is of the most interest:  
\[F(\mathbf{p})=\int\cdots\int\mathrm{d}\mathbf{p}_{2}\ldots\mathrm{d}\mathbf{p }_{N}F(\mathbf{p}_{1},\ldots,\mathbf{p}_{N})\Big{|}_{\mathbf{p}=\mathbf{p}_{1 }}, \tag{4}\]  
where \(F(\mathbf{p}_{1},\ldots,\mathbf{p}_{N})\) is the \(N\)-particle MDF, and the integration is taken over all electrons except for the first one (any other electron can also be chosen).  
The PCFs \(g_{uu}(r)\), \(g_{dd}(r)\) and \(g_{ud}(r)\) describe the distribution of distances between the electrons with corresponding spin projections:  
\[\begin{split}& g_{uu}(r)\propto\int\cdots\int\mathrm{d}\mathbf{q}_{ u3}\ldots\mathrm{d}\mathbf{q}_{uN}\mathrm{d}\mathbf{q}_{d1}\ldots\mathrm{d} \mathbf{q}_{uN}g(\mathbf{q}_{1},\ldots,\mathbf{q}_{N})\Big{|}_{r=|q_{ud}-q_{d2}| },\\ & g_{dd}(r)\propto\int\cdots\int\mathrm{d}\mathbf{q}_{u1}\ldots \mathrm{d}\mathbf{q}_{uN}\mathrm{d}\mathbf{q}_{d3}\ldots\mathrm{d}\mathbf{q}_{ dN}g(\mathbf{q}_{1},\ldots,\mathbf{q}_{N})\Big{|}_{r=|q_{ud}-q_{d2}|},\\ & g_{ud}(r)\propto\int\cdots\int\mathrm{d}\mathbf{q}_{u2}\ldots \mathrm{d}\mathbf{q}_{uN}\mathrm{d}\mathbf{q}_{d2}\ldots\mathrm{d}\mathbf{q}_{ dN}g(\mathbf{q}_{1},\ldots,\mathbf{q}_{N})\Big{|}_{r=|q_{ud}-q_{d1}|},\end{split} \tag{5}\]  
where \(g(\mathbf{q}_{1},\ldots,\mathbf{q}_{N})\) is the \(N\)-particle spatial distribution of the electrons, \(\mathbf{q}_{ua}\) and \(\mathbf{q}_{da}\) specify the electrons with different spin projections, and the exact values of the normalization factors depending of \(N_{u}\), \(N_{d}\), and \(V\) are not required for our purposes.