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\end{eqnarray}  The bounds on the occupation numbers are automatically satisfied by setting $n_{i}=2\sin^2(2\pi\vartheta_i)$ and varying $\vartheta_{i}$ without constraints. The conditions (\ref{eqsumocc}) and (\ref{eqorth}) are taken into account via Lagrange multipliers $\mu$ and $\lambda_{ij}$, respectively. Then, one can define the following functional  \begin{eqnarray}  \Omega(N,\{\vartheta\} ,\{\phi_i(\mathbf{r})\})= E - \mu \left(\sum_{i=1}^\infty 2sin^2( 2\pi\vartheta_i)-N\right)-\sum_{i,j=1}^\infty 2\pi\vartheta_i)-N\right)-\\\nonumber  \sum_{i,j=1}^\infty  \lambda_{ji}(\langle\phi_i|\phi_j\rangle-\delta_{ij}) \end{eqnarray}  which has to be stationary with respect to variations in $\{\vartheta_i\}$, $\{\phi_i(\mathbf{r})\}$ and $\{\phi_i^{*}(\mathbf{r})\}$. In any practical calculation the infinite sums have to be truncated including only a finite number of occupation numbers and natural orbitals. However, since the occupation numbers $n_j$ decay very quickly for $j>N$ this is not problematic.  %\begin{eqnarray}