commit b4 tricky part

Manuscript/SRGGW.tex

@@ -336,8 +336,8 @@ Then, as performed in Sec.~\ref{sec:srggw}, one can collect order by order the t
 
 By applying the SRG to $GW$, our aim is to gradually remove the coupling between the quasiparticle and the satellites resulting in a renormalized quasiparticle equation.
 However, to do so, one must identify the coupling terms in Eq.~\eqref{eq:quasipart_eq}, which is not straightforward.
-\ant{A way around this problem is to transform Eq.~\eqref{eq:quasipart_eq} to an equivalent upfolded form which elegantly highlights the coupling terms.
-Indeed, the $GW$ quasiparticle equation is equivalent to the diagonalization of the following matrix \cite{Bintrim_2021,Tolle_2022} }
+A way around this problem is to transform Eq.~\eqref{eq:quasipart_eq} to an equivalent upfolded form which elegantly highlights the coupling terms.
+Indeed, the $GW$ quasiparticle equation is equivalent to the diagonalization of the following matrix \cite{Bintrim_2021,Tolle_2022}
 \begin{equation}
   \label{eq:GWlin}
   \begin{pmatrix}
@@ -431,7 +431,7 @@ where the supermatrices
 \end{align}
 \end{subequations}
 collect the 2h1p and 2p1h channels.
-Once the closed-form expressions of the low-order perturbative expansions are known, they can be inserted in Eq.~\eqref{eq:downfolded_sigma}\trashant{\eqref{eq:GWlin} before applying the downfolding process to obtain} to define a renormalized version of the quasiparticle equation.
+Once the closed-form expressions of the low-order perturbative expansions are known, they can be inserted in Eq.~\eqref{eq:downfolded_sigma} to define a renormalized version of the quasiparticle equation.
 In particular, we focus here on the second-order renormalized quasiparticle equation.
 
 %///////////////////////////%
@@ -488,13 +488,13 @@ Equation \eqref{eq:F0_C0} implies
 \begin{align}
   \bF^{(1)}(s) &= \bF^{(1)}(0) = \bO, & \bC^{(1)}(s) &= \bC^{(1)}(0) = \bO,
 \end{align}
-and, thanks to the \ant{diagonal structure of $\bF^{(0)}$ (which is a consequence of the HF starting point)} and $\bC^{(0)}$, the differential equation for the coupling block in Eq.~\eqref{eq:W1} is easily solved and yields
+and, thanks to the diagonal structure of $\bF^{(0)}$ (which is a consequence of the HF starting point) and $\bC^{(0)}$, the differential equation for the coupling block in Eq.~\eqref{eq:W1} is easily solved and yields
 \begin{equation}
   W_{p,q\nu}^{(1)}(s) = W_{p,q\nu}^{(1)}(0) e^{- (F_{pp}^{(0)} - C_{q\nu,q\nu}^{(0)})^2 s}
 \end{equation}
 At $s=0$ the elements  $W_{p,q\nu}^{(1)}(0)$ are equal to the screened two-electron integrals defined in Eq.~\eqref{eq:GW_sERI}, while for $s\to\infty$, they tend to zero.
 Therefore, $W_{p,q\nu}^{(1)}(s)$ are genuine renormalized two-electron screened integrals.
-\ant{It is worth noting the close similarity of the first-order elements with the ones derived by Evangelista in Ref.~\onlinecite{Evangelista_2014b} for the usual electronic Hamiltonian in the context of wave-function theory within a second quantization formalism (see also Ref.~\onlinecite{Hergert_2016}).}
+It is worth noting the close similarity of the first-order elements with the ones derived by Evangelista in Ref.~\onlinecite{Evangelista_2014b} in the context of single- and multi-reference perturbation theory (see also Ref.~\onlinecite{Hergert_2016}).
 
 %///////////////////////////%
 \subsection{Second-order matrix elements}
@@ -532,7 +532,7 @@ At $s=0$, the second-order correction vanishes while, for $s\to\infty$, it tends
   F_{pq}^{(2)}(\infty) =  \sum_{r\nu}  \frac{\Delta_{pr\nu}+ \Delta_{qr\nu}}{\Delta_{pr\nu}^2 + \Delta_{qr\nu}^2} W_{p,r\nu} W_{q,r\nu}.
 \end{equation}
 Note that, in the limit $s\to\infty$, the dynamic part of the self-energy [see Eq.~\eqref{eq:srg_sigma}] tends to zero.
-\ant{Therefore, the SRG flow continuously kills the dynamic part $\widetilde{\bSig}(\omega; s)$ while creating a static correction $\widetilde{\bF}(s)$.}
+\titou{Therefore, the SRG flow continuously kills the dynamic part $\widetilde{\bSig}(\omega; s)$ while creating a static correction $\widetilde{\bF}(s)$.}
 This transformation is done gradually starting from the states that have the largest denominators in Eq.~\eqref{eq:static_F2}.
 
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