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The variation of $\Omega$ is done in two steps: for a fixed set of orbitals the energy functional is minimized with respect to occupation numbers and accordingly for a fixed set of occupations the energy functional is minimized with respect to variations of the orbitals until overall convergence of the absolute value of the energy is achieved. As a starting point we use results from a Hartree-Fock calculation and first optimize the occupation numbers. Since the correct $\mu$ is not known it is determined via bisection: for every $\mu$ the objective functional is minimized with respect to $\vartheta_i$ until the condition \ref{eqsumocc} is satisfied.\par  Due to the dependence on the occupation numbers, the natural orbital minimization does not lead to an eigenvalue equation like in Hartree-Fock. The implementation of the natural orbital minimization follows the method by Piris and Ugalde (\cite{Piris}). As one can show for fixed occupation numbers one obtains  \begin{eqnarray}  \lambda_{ji} = n_i\langle\phi_j|-\frac(\nabla^2}{2}+v_{ext}|\phi_i\langle\\ n_i\langle\phi_j|-\frac{\nabla^2}{2}+v_{ext}|\phi_i\langle\\  +\int d\mathbf{r} \frac{\delta E_{Hxc}}{\delta \phi_i^{*}(\mathbf{r})}\phi_k^{*}(\mathbf{r}).  \end{eqnarray}  At the extremum, the matrix of the Lagrange multiplyers must be Hermitian, i.e.