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\begin{eqnarray}  \langle \phi_{i} | \phi_{j}\rangle = \delta_{ij}. \label{eqorth}  \end{eqnarray}  The bounds on the occupation numbers are automatically satisfied by setting $n_{i}=2sin^2 2\pi\vartheta_i$ $n_{i}=2\sin^2(2\pi\vartheta_i)$  and varying $\vartheta_{i}$ without constraints. The condition (\ref{eqsumocc}) is taken into account by using a Lagrange multiplyer $\mu$ and Lagrange multiplyers $\lambda_{ik}$ are used for the orthonormality constraints (\ref{eqorth}). Then, one can define the following functional \begin{eqnarray}  \Omega(N,\{\vartheta\} ,\{\phi_i(\mathbf{r})\})= E - \mu (\sum_i 2sin^2( 2\pi\vartheta_i)-N)-\sum_{ik} \lambda_{ik}(\langle\phi_k|\phi_i\rangle-\delta_{ki})  \end{eqnarray}