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\end{eqnarray}  The natural orbitals $\phi_i(\mathbf{r})$ and their occupation numbers $n_i$ are the eigenfunctions and eigenvalues of the 1-RDM, respectively.  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, from the total energy  \begin{equation}\label{energy} \begin{eqnarray}\label{energy}  E=\sum_{i=1}^\infty\int d\mathbf{r} n_{i}\phi^{*}_{i}(\mathbf{r})\left(-\frac{\nabla^2}{2}\right) \phi_{i}(\mathbf{r})+\sum_{i=1}^\infty \int d\mathbf{r} V_{\mathrm{ext}}(\mathbf{r})n_{i}|\phi_{i}(\mathbf{r})|^{2}\nonumber\\  +\frac{1}{2}\sum_{i,j=1}^\infty n_{i} n_{j}\int d\mathbf{r} d\mathbf{r'} \frac{|\phi_{i}(\mathbf{r})|^{2} |\phi_{j}(\mathbf{r})|^{2}}{|\mathbf{r}-\mathbf{r'}|} + E_{xc}\left[\{n_{j}\},\{\phi_{j}\}\right]  \end{equation} \end{eqnarray}  the part that needs to be approximated $E_{xc}[\gamma]$ comes only from the interaction term (contrary to KS-DFT), as the interacting kinetic energy can be explicitely expressed in terms of $\gamma$. In practice, the minimization of the energy is not performed with respect to the 1-RDM but with respect to $n_{i}$ and $\phi_{i}$, separately.  For closed-shell systems the necessary and sufficient conditions for the 1-RDM to be $N$-representable, i.e.\ to correspond to a $N$-electron wavefunction is that $ 0 \leq n_{i} \leq 2$ and   \begin{eqnarray}