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Iris Theophilou edited RDMFT1.tex
over 9 years ago
Commit id: 95df61484ffaa39de7b8723aecef63b98e0d795e
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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_i\}
,\{\phi_i(\mathbf{r})\})= ,\{\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}