small corrections in GW part

Manuscript/SRGGW.tex

@@ -134,17 +134,17 @@ The goal of this manuscript is to determine if the SRG formalism can effectively
 
 The manuscript is organized as follows.
 We begin by reviewing the $GW$ approximation in Sec.~\ref{sec:gw} and then briefly review the SRG formalism in Sec.~\ref{sec:srg}.
-Section~\ref{sec:theoretical} is concluded by a perturbative analysis of SRG applied to $GW$ (see Sec.~\ref{sec:srggw}).
+A perturbative analysis of SRG applied to $GW$ is presented in Sec.~\ref{sec:srggw}.
 The computational details are provided in Sec.~\ref{sec:comp_det} before turning to the results section.
 This section starts by
 
 %=================================================================%
-\section{Theoretical background}
- \label{sec:theoretical}
+%\section{Theoretical background}
+% \label{sec:theoretical}
 %=================================================================%
 
 %%%%%%%%%%%%%%%%%%%%%%
-\subsection{Many-body perturbation theory in the GW approximation}
+\section{The $GW$ approximation}
 \label{sec:gw}
 %%%%%%%%%%%%%%%%%%%%%% 
 
@@ -183,22 +183,25 @@ In fact, these cases are related to the discontinuities and convergence problems
 
 One obvious flaw of the one-shot scheme mentioned above is its starting point dependence.
 Indeed, in Eq.~\eqref{eq:G0W0} we chose to use the HF orbital energies but this is arbitrary and one could have chosen Kohn-Sham orbitals for example.
-Therefore, one could try to optimize the starting point to obtain the best one-shot energies possible. \cite{Korzdorfer_2012,Marom_2012,Bruneval_2013,Gallandi_2015,Caruso_2016, Gallandi_2016}
-Alternatively, one could solve this equation self-consistently leading to the eigenvalue-only self-consistent scheme. \cite{Shishkin_2007,Blase_2011b,Marom_2012,Kaplan_2016,Wilhelm_2016}
+Therefore, one can \titou{optimize} the starting point to obtain the best one-shot energies possible, which is commonly done. \cite{Korzdorfer_2012,Marom_2012,Bruneval_2013,Gallandi_2015,Caruso_2016, Gallandi_2016}
+\PFL{Maybe it is worth mentioning here that is is a fairly heuristic approach that is obviously system dependent?}
+Alternatively, one could solve this equation self-consistently leading to the eigenvalue-only self-consistent scheme (ev$GW$). \cite{Shishkin_2007,Blase_2011b,Marom_2012,Kaplan_2016,Wilhelm_2016}
 To do so the energy of the quasi-particle solution of the previous iteration is used to build Eq.~\eqref{eq:G0W0} and then this equation is solved for $\omega$ again until convergence is reached.
-However, if the quasi-particle solution is not well-defined, self-consistency can be quite difficult, if not impossible, to reach. 
-Even if self-consistency has been reached, the starting point dependence has not been totally removed because the results still depend on the starting molecular orbitals. \cite{Marom_2012}
+\PFL{This is not quite right. It is probably going to be easier to explain when you're going to introduce the explicit expressions of these quantities.}
+However, if the quasi-particle solution is not well-defined, reaching self-consistency can be quite difficult, if not impossible. 
+Even at convergence, the starting point dependence is not totally removed as the results still depend on the starting orbitals. \cite{Marom_2012}
 
-To update both the energies and the molecular orbitals, one needs to take into account the off-diagonal elements in Eq.~\eqref{eq:quasipart_eq}.
+To update both energies and orbitals, one must take into account the off-diagonal elements in Eq.~\eqref{eq:quasipart_eq}.
 To take into account the effect of off-diagonal elements without fully solving the quasi-particle equation, one can resort to the quasi-particle self-consistent (qs) scheme in which $\bSig(\omega)$ is replaced by a static approximation $\bSig^{\qs}$.
 The algorithm to solve the qs problem is totally analog to the HF case with $\bF$ replaced by $\bF + \bSig^{\qs}$.
 Various choices for $\bSig^\qs$ are possible but the most popular one is the following Hermitian approximation
 \begin{equation}
   \label{eq:sym_qsgw}
-  \Sigma_{pq}^\qs = \frac{1}{2}\Re\left(\Sigma_{pq}(\epsilon_p) + \Sigma_{pq}(\epsilon_q) \right).
+  \Sigma_{pq}^\qs = \frac{1}{2}\Re[\Sigma_{pq}(\epsilon_p) + \Sigma_{pq}(\epsilon_q) ].
 \end{equation}
-This form has first been introduced by Faleev and co-workers \cite{Faleev_2004,vanSchilfgaarde_2006,Kotani_2007} before being derived as the effective Hamiltonian that minimizes the length of the gradient of the Klein functional for non-interacting Green's function. \cite{Ismail-Beigi_2017}
-One of the main results of this manuscript is the derivation from first principles of an alternative static Hermitian form, this will be done in the next section.
+This form has first been introduced by Faleev and co-workers \cite{Faleev_2004,vanSchilfgaarde_2006,Kotani_2007} before being derived as the \titou{form of the} effective Hamiltonian that minimizes the length of the gradient of the Klein functional for non-interacting Green's function. \cite{Ismail-Beigi_2017}
+One of the main results of this manuscript is the derivation from first principles of an alternative static Hermitian form. 
+This will be done in the next section.
 
 In this case, as well self-consistency can be difficult to reach in cases where multiple solutions have large spectral weights.
 Multiple solutions arise due to the $\omega$ dependence of the self-energy.
@@ -207,7 +210,7 @@ If it is not the case, the qs scheme will oscillate between the solutions with l
 
 Convergence problems arise when a shake-up configuration has an energy similar to the associated quasi-particle solution, this satellite is the so-called intruder state.
 The intruder state problem can be dealt with by introducing \textit{ad hoc} regularizers.
-The $\ii eta$ term that is usually added in the denominator of the self-energy (see below) is the usual imaginary-shift regularizer used in various other theories flawed by intruder states. \cite{Battaglia_2022} \ant{more ref...}
+The $\ii \eta$ term that is usually added in the denominator of the self-energy (see below) is the usual imaginary-shift regularizer used in various other theories flawed by intruder states. \cite{Battaglia_2022} \ant{more ref...}
 Various other regularizers are possible and in particular one of us has shown that a regularizer inspired by the SRG had some advantages, in the $GW$ case, over the imaginary shift one. \cite{Monino_2022}
 But it would be more rigorous, and more instructive, to obtain this regularizer from first principles by applying the SRG formalism to many-body perturbation theory.
 This is the aim of this work.
@@ -294,7 +297,7 @@ As can be readily seen in Eq.~\eqref{eq:GWlin}, the blocks $V^\text{2h1p}$ and $
 Therefore, these blocks will be the target of our SRG transformation but before going into more detail we will review the SRG formalism.
 
 %%%%%%%%%%%%%%%%%%%%%%
-\subsection{The similarity renormalization group}
+\section{The similarity renormalization group}
 \label{sec:srg}
 %%%%%%%%%%%%%%%%%%%%%%
 
@@ -355,7 +358,7 @@ Hence, the generator $\boldsymbol{\eta}(s)$ admits a perturbation expansion as w
 Then, one can collect order by order the terms in Eq.~\eqref{eq:flowEquation} and solve analytically the low-order differential equations.
 
 %%%%%%%%%%%%%%%%%%%%%%
-\subsection{Renormalized GW}
+\section{Regularized $GW$ approximation}
 \label{sec:srggw}
 %%%%%%%%%%%%%%%%%%%%%% 
 
@@ -403,7 +406,7 @@ Once the analytical low-order perturbative expansions are known they can be inse
 In particular, in this manuscript, the focus will be on the second-order renormalized quasi-particle equation.
 
 %///////////////////////////%
-\subsubsection{Zeroth-order matrix elements}
+\subsection{Zeroth-order matrix elements}
 %///////////////////////////%
 
 There is only one zeroth order term in the right-hand side of the flow equation
@@ -447,7 +450,7 @@ Therefore, the zeroth order Hamiltonian is
 \ie it is independent of $s$.
 
 %///////////////////////////%
-\subsubsection{First-order matrix elements}
+\subsection{First-order matrix elements}
 %///////////////////////////%
 
 Knowing that $\bHod{0}(s)=\bO$, the first order flow equation is
@@ -470,7 +473,7 @@ Therefore, $W_{p,(q,v)}^{(1)}(s)$ are renormalized two-electrons screened integr
 Note the close similarity of the first-order element expressions with the ones of Evangelista in Ref.~\onlinecite{Evangelista_2014b} obtained in a second quantization formalism (see also Ref.~\onlinecite{Hergert_2016}).
 
 %///////////////////////////%
-\subsubsection{Second-order matrix elements}
+\subsection{Second-order matrix elements}
 % ///////////////////////////%
 
 The second-order renormalized quasi-particle equation is given by