deletions | additions       diff --git a/RDMFT1.tex b/RDMFT1.tex  index d82f438..aaffe8f 100644  --- a/RDMFT1.tex  +++ b/RDMFT1.tex 
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\begin{eqnarray}  \langle \phi_{i} | \phi_{j}\rangle = \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. This does not lead to eigenvalue equation..The bounds on the occupation numbers are automatically satisfied by setting $n_{i}=2sin^2 $n_{i}$=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 $\mu$. The minimization with respect to occupation numbers corresponds to the minimization of the objectivefunctional $E(\{\theta_i\})-\mu (\sum_{i}(2sin(2\pi\theta_i)-N)$. (\sum_{i}(2sin^2(2\pi\theta_i)-N)$.  As the correct $\mu$ is not known bisection is used. For every $\mu$, the objective functional is minimized with respect to $\theta_i$ until $\sum_{i}(2sin(2\pi\theta_i)-N=0$ $\sum_{i}(2sin^2(2\pi\theta_i)-N=0$  is satisfied . The implementation of the natural orbital minimization follows a method by Piris and Ugalde (\cite{Piris}).