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\end{align}  Note that within the RDMFT implementation in octopus only closed-shell systems are treated at the momment. Minimization of the energy functional of Eq. \eqref{eqn:energy} is performed under the $N$-representability constraints and the orthonormality requierements of the natural orbitals,   \begin{eqnarray}  \langle \phi_{i} | \phi_{j}\rangle = \delta_{ij}\label{orthonorm} \delta_{ij}  \end{eqnarray}  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. The bounds on the occupation numbers are automatically satisfied by setting $n_{i}$=2sin$^2 2\pi\theta_i$ and varying $\theta_{i}$ without constraints. The occupation numbers summing up to the number of electrons is taken into account by using a Lagrange multiplyer $\mu$.