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a/Notes/Notes.tex b/Notes/Notes.tex index 379cb4b..f833fe5 100644 --- a/Notes/Notes.tex +++ b/Notes/Notes.tex @@ -840,6 +840,14 @@ Finally, we discuss the renormalized correlation self-energy introduced in this In this case the situation is reversed, \ie the divergent denominators will be the last removed when $\Lambda$ is increased. Therefore the renormalized self-energy seems not to be the good strategy to remove discontinuities. However, it defines SRG-PT2 approximations to the quasiparticle energies which have the same pros as the SRG-MP2 discussed above. + Schematically, the determinants included in $\Tilde{F}$ and $\Tilde{\Sigma}$ are like in the following figure where blue means that the determinant is included. + + \begin{figure} + \centering + \includegraphics[width=0.9\linewidth]{Figures/renormalizedF.pdf} + \caption{Determinants included at a finite value of $s$ according to the diagonal denominators of $\Tilde{F}$} + \label{fig:fig_1} + \end{figure} %=================================================================% \section{An alternative partitioning designed for discontinuities} @@ -873,7 +881,7 @@ The idea to obtain this is to start from the full Hamiltonian and use a perturbe \\ \bV{}{}(s) & = \bV{}{'(0)}(s) + \lambda' \bV{}{'(1)}(s) + \lambda'^2 \bV{}{'(2)}(s) + \cdots \end{align} - We can use the expansion in terms of $\lambda$ and transform them to $\lambda^'$ and then identify with the expressions above, for example for $\bF{}{}$ + We can use the expansion in terms of $\lambda$ and transform them to $\lambda'$ and then identify with the expressions above, for example for $\bF{}{}$ \begin{align} \bF{}{}(s) & = \bF{}{(0)}(s) + (1 - \lambda') \bF{}{(1)}(s) + (1 - \lambda')^2 \bF{}{(2)}(s) + \cdots \\ &= \qty( \bF{}{(0)}(s) + \bF{}{(1)}(s) + \bF{}{(2)}(s) + \cdots) \notag \\