deletions | additions       diff --git a/RDMFT1.tex b/RDMFT1.tex  index db8a141..dc347e8 100644  --- a/RDMFT1.tex  +++ b/RDMFT1.tex 
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Within Reduced Density Matrix Functional Theory (RDMFT) the total energy of a system is  given as a functional of the one body reduced density matrix (1-RDM) \begin{eqnarray}  \gamma(\mathbf{r},\mathbf{r'})=N\int\cdots\int d\mathbf{r_2}...d\mathbf{r_N} \Psi^*(\mathbf{r'},\mathbf{r_2}...\mathbf{r_N})\Psi(\mathbf{r},\mathbf{r_2}...\mathbf{r_N})  \end{eqnarray} 
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In RDMFT the total energy is given by   \begin{eqnarray}  E=\sum_{i=1}^\infty\int d\mathbf{r} n_{i}\phi^{*}_{i}(\mathbf{r})\left(-\frac{\nabla^2}{2}\right) \phi_{i}(\mathbf{r})+\sum_{i=1}^\infty \int d\mathbf{r} V_{\mathrm{ext}}(\mathbf{r})n_{i}|\phi_{i}(\mathbf{r})|^{2}\nonumber\\  +\frac{1}{2}\sum_{i,j=1}^\infty n_{i} n_{j}\int d\mathbf{r} d\mathbf{r'} \frac{|\phi_{i}(\mathbf{r})|^{2} |\phi_{j}(\mathbf{r})|^{2}}{|\mathbf{r}-\mathbf{r'}|} + E_{xc}\left[\{n_{j}\},\{\phi_{j}\}\right]\label{eqenergy} E_{xc}\left[\{n_{j}\},\{\phi_{j}\}\right].\label{eqenergy}  \end{eqnarray}  As in the case of DFT, the exact functional of RDMFT is unknown. The part that needs to be approximated $E_{xc}[\gamma]$ comes only from the interaction term (contrary to KS-DFT), as the interacting kinetic energy can be explicitely expressed in terms of $\gamma$.   Different approximate functionals are employed and minimized with respect to the 1-RDM in order to find the ground state energy. A common approximation for $E_{xc}$ is the M\"uller functional, which has the following form 
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\begin{eqnarray}  F_{ki}=\theta(i-k)(\lambda_{ki}-\lambda^{*}_{ik})+\theta(k-i)(\lambda^{*}_{ik}-\lambda_{ki})\label{eqF}  \end{eqnarray}  where $\theta$ is the unit-step Heavside function. This matrix is diagonal at the extremum and hence the matrix $\mathbf{F}$ and $\gamma$ can be brought simultaneously to a diagonal form at the solution. Thus, the $\{\phi_i\}$ which are the solutions of Eq. (\ref{eqlambda}) can be found by diagonalization of $\mathbf{F}$ in an iterative manner. As the matrix $\mathbf{F}$, as defined in \ref{eqF} is zero in the diagonal, in every step the diagonal of the previous step is used and for the first step the matrix $(\lambda_{ki}+\lambda_{ik}^{*})/2$ is diagonalized (see \cite{Piris} for details) details).\par  As one needs an initial guess for the natural orbitals