add figure 1

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  \\