ok with Sec III

Manuscript/SRGGW.tex

@@ -488,9 +488,13 @@ and
 %  \end{align}
 %\end{subequations}
 Equation \eqref{eq:F0_C0} implies
+\begin{subequations}
 \begin{align}
-  \bF^{(1)}(s) &= \bF^{(1)}(0) = \bO, & \bC^{(1)}(s) &= \bC^{(1)}(0) = \bO,
+  \bF^{(1)}(s) &= \bF^{(1)}(0) = \bO, &
+  \\
+   \bC^{(1)}(s) &= \bC^{(1)}(0) = \bO,
 \end{align}
+\end{subequations}
 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_{pq}^{\nu(1)}(s) = W_{pq}^{\nu} e^{-(\Delta_{pq}^{\nu})^2 s}
@@ -499,7 +503,7 @@ At $s=0$, $W_{pq}^{\nu(1)}(s)$ reduces to the screened two-electron integrals de
 \begin{equation}
 	\lim_{s\to\infty} W_{pq}^{\nu(1)}(s) = 0.
 \end{equation}
-Therefore, $W_{pq}^{\nu(1)}(s)$ are genuine renormalized two-electron screened integrals.
+Therefore, $W_{pq}^{\nu(1)}(s)$ is a genuine renormalized two-electron screened integral.
 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}).
 
 %///////////////////////////%
@@ -620,7 +624,7 @@ The convergence properties and the accuracy of both static approximations are qu
 To conclude this section, we briefly discussed the case of discontinuities mentioned in Sec.~\ref{sec:intro}.
 Indeed, it has been previously mentioned that intruder states are responsible for both the poor convergence of qs$GW$ and discontinuities in physical quantities. \cite{Loos_2018b,Veril_2018,Loos_2020e,Berger_2021,DiSabatino_2021,Monino_2022,Scott_2023}
 Is it then possible to rely on the SRG machinery to remove discontinuities?
-Not directly because discontinuities are due to intruder states in the dynamic part of the quasiparticle equation, while, as we have seen just above, a finite $s$ values are suitable to avoid intruder states in its static part.
+Not directly because discontinuities are due to intruder states in the dynamic part of the quasiparticle equation, while, as we have seen just above, a finite value of $s$ is suitable to avoid intruder states in its static part.
 However, performing the following bijective transformation
 \titou{\begin{align}
   e^{-s} &= 1-e^{-t}, & 1 - e^{-s} &= e^{-t},