ok with Sec II

Manuscript/SRGGW.tex

@@ -212,24 +212,24 @@ and where
 are bare two-electron integrals in the spin-orbital basis.
 
 The diagonal matrix $\boldsymbol{\Omega}$ contains the positive eigenvalues of the RPA problem defined in Eq.~\eqref{eq:full_dRPA} and its elements $\Omega_\nu$ appear in Eq.~\eqref{eq:GW_selfenergy}.
-In the Tamm-Dancoff approximation (TDA), one sets $\bB = \bO$ in Eq.~\eqref{eq:full_dRPA} which reduces to a Hermitian eigenvalue problem of the form $\bA \bX = \bX \bOm$ (hence $\bY=0$).
+%In the Tamm-Dancoff approximation (TDA), one sets $\bB = \bO$ in Eq.~\eqref{eq:full_dRPA} which reduces to a Hermitian eigenvalue problem of the form $\bA \bX = \bX \bOm$ (hence $\bY=0$).
 
 As mentioned above, because of the frequency dependence of the self-energy, solving exactly the quasiparticle equation \eqref{eq:quasipart_eq} is a rather complicated task. 
 Hence, several approximate schemes have been developed to bypass full self-consistency.
-The most popular strategy is the one-shot (perturbative) $G_0W_0$ scheme, where the self-consistency is completely abandoned, and the off-diagonal elements of Eq.~\eqref{eq:quasipart_eq} are neglected.
+The most popular strategy is the one-shot (perturbative) $GW$ scheme, $G_0W_0$, where the self-consistency is completely abandoned, and the off-diagonal elements of Eq.~\eqref{eq:quasipart_eq} are neglected.
 Assuming a HF starting point, this results in $K$ quasiparticle equations that read
 \begin{equation}
   \label{eq:G0W0}
   \epsilon_p^{\HF} + \Sigma_{pp}(\omega) - \omega = 0,
 \end{equation}
 where $\Sigma_{pp}(\omega)$ are the diagonal elements of $\bSig$ and $\epsilon_p^{\HF}$ are the HF orbital energies.
-The previous equations are non-linear with respect to $\omega$ and therefore have multiple solutions $\epsilon_{p,s}$ for a given $p$ (where the index $s$ is numbering solutions).
+The previous equations are non-linear with respect to $\omega$ and therefore have multiple solutions $\epsilon_{p,z}$ for a given $p$ (where the index $z$ is numbering solutions).
 These solutions can be characterized by their spectral weight given by the renormalization factor
 \begin{equation}
   \label{eq:renorm_factor}
-  0 \leq Z_{p,s} = \qty[ 1 - \eval{\pdv{\Sigma_{pp}(\omega)}{\omega}}_{\omega=\epsilon_{p,s}} ]^{-1} \leq 1.
+  0 \leq Z_{p,z} = \qty[ 1 - \eval{\pdv{\Sigma_{pp}(\omega)}{\omega}}_{\omega=\epsilon_{p,z}} ]^{-1} \leq 1.
 \end{equation}
-The solution with the largest weight is referred to as the quasiparticle while the others are known as satellites (or shake-up transitions).
+The solution with the largest weight $Z_p \equiv Z_{p,z=0}$ is referred to as the quasiparticle while the others are known as satellites (or shake-up transitions).
 However, in some cases, Eq.~\eqref{eq:G0W0} can have two (or more) solutions with similar weights, hence the quasiparticle is not well-defined.
 
 One obvious drawback of the one-shot scheme mentioned above is its starting point dependence.
@@ -264,7 +264,7 @@ Multiple solutions of Eq.~\eqref{eq:G0W0} arise due to the $\omega$ dependence o
 Therefore, by suppressing this dependence, the static approximation relies on the fact that there is well-defined quasiparticle solutions.
 If it is not the case, the self-consistent qs$GW$ scheme inevitably oscillates between solutions with large spectral weights. \cite{Forster_2021}
 
-The satellites causing convergence problems are the above-mentioned intruder states. \cite{Monino_2022}
+The satellites causing convergence issues are the above-mentioned intruder states. \cite{Monino_2022}
 One can deal with them by introducing \textit{ad hoc} regularizers.
 For example, the $\ii\eta$ term in the denominators of Eq.~\eqref{eq:GW_selfenergy}, sometimes referred to as a broadening parameter linked to the width of the quasiparticle peak, is similar to the usual imaginary-shift regularizer employed in various other theories plagued by the intruder-state problem. \cite{Surjan_1996,Forsberg_1997,Monino_2022,Battaglia_2022}.