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Iris Theophilou edited RDMFT1.tex
over 9 years ago
Commit id: 082bc06d9253eb9decb7552240c46f79721bf9f7
deletions | additions
diff --git a/RDMFT1.tex b/RDMFT1.tex
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\begin{eqnarray}
\lambda_{ki}=h_{ki}n_i+\int d\mathbf{r} \frac{\delta V_{ee}}{\delta \phi_i^{*}(\mathbf{r})}\phi_k^{*}(\mathbf{r}).
\end{eqnarray}
At the extremum, the matrix of the Lagrange multiplyers must be Hermitian, i.e.
$\lambda_{ki}=\lambda_{ik}^{*}$ \begin{eqnarray}\label{lambdaeq}
\lambda_{ki}-\lambda_{ik}^{*}=0
\end{eqnarray}
Then one can define the off diagonal elements of a Hermitian matrix $\mathbf{F}$ as:
\begin{eqnarray}
F_{ki}=\theta(i-k)(\lambda_{ki}-\lambda^{*}_{ik})+\theta(k-i)(\lambda^{*}_{ik}-\lambda_{ki})
\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. form at the solution.