this is for holding javascript data
Iris Theophilou edited RDMFT1.tex
over 9 years ago
Commit id: 4ae2e2371de23bcc705d48a72c41c287e561e733
deletions | additions
diff --git a/RDMFT1.tex b/RDMFT1.tex
index aaffe8f..638056a 100644
--- a/RDMFT1.tex
+++ b/RDMFT1.tex
...
\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$^2 2\pi\theta_i$ 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 $E(\{\theta_i\})-\mu
(\sum_{i}(2sin^2(2\pi\theta_i)-N)$. (\sum_{i}(2sin(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
$\sum_{i}(2sin^2(2\pi\theta_i)-N=0$ $\sum_{i}(2sin(2\pi\theta_i)-N=0$ is satisfied . The implementation of the natural orbital minimization follows a method by Piris and Ugalde (\cite{Piris}).
\begin{eqnarray}
\lambda_{ki}=h_{ki}n_{i}+\int d\mathbf{r} \frac{\delta V_{ee}}{\delta\phi_i(\mathbf{r})}
\end{eqnarray}