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\end{eqnarray}  As the exact functional is unknown different approximate functionals are employed and minimized with respect to the 1-RDM in order to find the ground state energy. However, the part that needs to be approximated $E_{xc}(\gamma)$ comes only from the interaction term (contrary to KS-DFT),as the kinetic energy can be explicitely expressed with respect to $\gamma$. In practice the minimization of the energy is not performed with respect to the 1-RDM ($\gamma$) but with respect to its eigenvalues which are named occupation numbers n$_{i}$ and eigenfunctions, the natural orbitals $\phi_{i}$ separately.  \begin{eqnarray}  E=-\frac{1}{2}\int dx\sum n_{i}\phi^{*}_{i}(x)\nabla^{2} \phi_{i}(x)+\int dx V_{ext}(x)\sum n_{i}|\phi_{i}(x)|^{2} +\frac{1}{2} n_{i}|\phi_{i}(x)|^{2}+\nonumber\\  \frac{1}{2}  \frac{\int dx dx' \sum_{i} n_{i}|\phi_{i}(x)|^{2}\sum_{j} n_{j} |\phi_{j}(x)|^{2}}{|x-x'|} + E_{xc}(\{n_{j}\},\{\phi_{j}\}) \end{eqnarray}  The normalization of the occupation numbers that we use is to sum up to the total number of electrons $N$  \begin{eqnarray}