large changes

Manuscript/SRGGW.tex

@@ -1,5 +1,5 @@
 \documentclass[aip,jcp,reprint,noshowkeys,superscriptaddress]{revtex4-1}
-\usepackage{graphicx,dcolumn,bm,xcolor,microtype,multirow,amscd,amsmath,amssymb,amsfonts,physics,longtable,wrapfig,bbold,siunitx,xspace}
+\usepackage{graphicx,dcolumn,bm,xcolor,microtype,multirow,amscd,amsmath,amssymb,amsfonts,physics,longtable,wrapfig,bbold,siunitx,xspace,ulem}
 \usepackage[version=4]{mhchem}
 
 \usepackage[utf8]{inputenc}
@@ -49,9 +49,10 @@
 \lstset{style=mystyle}
 
 \newcommand{\titou}[1]{\textcolor{red}{#1}}
-\newcommand{\trashPFL}[1]{\textcolor{\red}{\sout{#1}}}
+\newcommand{\trashPFL}[1]{\textcolor{red}{\sout{#1}}}
 \newcommand{\PFL}[1]{\titou{(\underline{\bf PFL}: #1)}}
 \newcommand{\ant}[1]{\textcolor{teal}{#1}}
+\newcommand{\trashant}[1]{\textcolor{teal}{\sout{#1}}}
 \newcommand{\ANT}[1]{\ant{(\underline{\bf ANT}: #1)}}
 
 % addresses
@@ -105,8 +106,8 @@ Approximating $\Sigma$ as the first-order term of its perturbative expansion wit
 \end{equation}
 Diagrammatically, $GW$ corresponds to a resummation of the (time-dependent) direct ring diagrams via the computation of the random-phase approximation (RPA) polarizability \cite{Ren_2012,Chen_2017} and is thus particularly well suited for weak correlation.
 
-\titou{Alternatively, one can choose to define $\Sigma$ as the $n$th-order expansion in terms of the bare Coulomb interaction $v$ leading to the GF($n$) class of approximations. \cite{SzaboBook,Ortiz_2013,Hirata_2015,Hirata_2017} 
+\trashant{Alternatively, one can choose to define $\Sigma$ as the $n$th-order expansion in terms of the bare Coulomb interaction $v$ leading to the GF($n$) class of approximations. \cite{SzaboBook,Ortiz_2013,Hirata_2015,Hirata_2017} 
-The GF(2) approximation \cite{Casida_1989,Casida_1991,Phillips_2014,Phillips_2015,Rusakov_2014,Rusakov_2016,Backhouse_2021,Backhouse_2020b,Backhouse_2020a,Pokhilko_2021a,Pokhilko_2021b,Pokhilko_2022} is also known as the second Born approximation in condensed matter physics. \cite{Stefanucci_2013}}
+The GF(2) approximation \cite{Casida_1989,Casida_1991,Phillips_2014,Phillips_2015,Rusakov_2014,Rusakov_2016,Backhouse_2021,Backhouse_2020b,Backhouse_2020a,Pokhilko_2021a,Pokhilko_2021b,Pokhilko_2022} is also known as the second Born approximation in condensed matter physics. \cite{Stefanucci_2013} }
 
 Despite a wide range of successes, many-body perturbation theory has well-documented limitations. \cite{Kozik_2014,Stan_2015,Rossi_2015,Tarantino_2017,Schaefer_2013,Schaefer_2016,Gunnarsson_2017,vanSetten_2015,Maggio_2017a,Duchemin_2020}
 For example, modeling core electron spectroscopy requires core ionization energies which have been proven to be challenging for routine $GW$ calculations. \cite{Golze_2018,Golze_2020,Li_2022}
@@ -123,7 +124,7 @@ These discontinuities have been traced back to a transfer of spectral weight bet
 In addition, systems, where the quasiparticle equation admits two solutions with similar spectral weights, are known to be particularly difficult to converge for partially self-consistent $GW$ schemes. \cite{Veril_2018,Forster_2021,Monino_2022}
 
 In a recent study, Monino and Loos showed that these discontinuities could be removed by the introduction of a regularizer inspired by the similarity renormalization group (SRG) in the quasiparticle equation. \cite{Monino_2022}
-Encouraged by the recent successes of regularization schemes in many-body quantum chemistry methods, as in single- and multi-reference perturbation theory, \cite{Lee_2018a,Shee_2021,Evangelista_2014b,ChenyangLi_2019a,Battaglia_2022} the present work investigates the application of the SRG formalism to many-body perturbation theory in its $GW$ \titou{and GF(2) variants}.
+Encouraged by the recent successes of regularization schemes in many-body quantum chemistry methods, as in single- and multi-reference perturbation theory, \cite{Lee_2018a,Shee_2021,Evangelista_2014b,ChenyangLi_2019a,Battaglia_2022} the present work investigates the application of the SRG formalism to many-body perturbation theory in its $GW$.
 The SRG has been developed independently by Wegner \cite{Wegner_1994} and Glazek and Wilson \cite{Glazek_1993,Glazek_1994} in the context of condensed matter systems and light-front quantum field theories, respectively.
 This formalism has been introduced in quantum chemistry by White \cite{White_2002} before being explored in more detail by Evangelista and coworkers in the context of multi-reference electron correlation theories. \cite{Evangelista_2014b,ChenyangLi_2015, ChenyangLi_2016,ChenyangLi_2017,ChenyangLi_2018,ChenyangLi_2019a,Zhang_2019,ChenyangLi_2021,Wang_2021,Wang_2023}
 The SRG has also been successful in the context of nuclear structure theory, where it was first developed as a mature computational tool thanks to the work of several research groups.
@@ -138,8 +139,8 @@ By stopping the SRG transformation once all external configurations except the i
 correlation effects between the internal and external spaces can be incorporated (or folded) without the presence of intruder states.
 
 The goal of this manuscript is to determine if the SRG formalism can effectively address the issue of intruder states in many-body perturbation theory, as it has in other areas of electronic and nuclear structure theory.
-This open question will lead us to an intruder-state-free static approximation of the self-energy derived from first-principles that can be employed in \titou{qs$GW$} calculations.
+This open question will lead us to an intruder-state-free static approximation of the self-energy derived from first-principles that can be employed in \ant{partially self-consistent $GW$} calculations.
-\titou{Here, we focus on the $GW$ approximation but the subsequent derivations can be straightforwardly applied to other approximations such as GF(2) or $T$-matrix.}
+\ant{Note that throughout the manuscript we focus on the $GW$ approximation but the subsequent derivations can be straightforwardly applied to other approximations such as GF(2) \cite{Casida_1989,Casida_1991,SzaboBook,Stefanucci_2013,Ortiz_2013,Phillips_2014,Phillips_2015,Rusakov_2014,Rusakov_2016,Hirata_2015,Hirata_2017,Backhouse_2021,Backhouse_2020b,Backhouse_2020a,Pokhilko_2021a,Pokhilko_2021b,Pokhilko_2022}  or $T$-matrix.}
 
 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}.
@@ -171,14 +172,14 @@ Similarly to the HF case, Eq.~\eqref{eq:quasipart_eq} has to be solved self-cons
 The matrix elements of $\bSig(\omega)$ have the following analytic expression \cite{Hedin_1999,Tiago_2006,Bruneval_2012,vanSetten_2013,Bruneval_2016}
 \begin{equation}
   \label{eq:GW_selfenergy}
-  \Sigma_{pq}(\omega) 
+  \Sigma_{pq}(\omega; \eta) 
-  = \sum_{i\nu} \frac{W_{pi\nu} W_{qi\nu}}{\omega - \epsilon_i + \Omega_{\nu} - \ii \eta} 
+  = \sum_{i\nu} \frac{W_{p,i\nu} W_{q,i\nu}}{\omega - \epsilon_i + \Omega_{\nu} - \ii \eta} 
-  + \sum_{a\nu} \frac{W_{pa\nu}W_{qa\nu}}{\omega - \epsilon_a - \Omega_{\nu} + \ii \eta},
+  + \sum_{a\nu} \frac{W_{p,a\nu}W_{q,a\nu}}{\omega - \epsilon_a - \Omega_{\nu} + \ii \eta},
 \end{equation}
 where $\eta$ is a positive infinitesimal and the screened two-electron integrals are
 \begin{equation}
   \label{eq:GW_sERI}
-  W_{pq\nu} = \sum_{ia}\eri{pi}{qa}\qty( \bX_{\nu}+\bY_{\nu})_{ia},
+  W_{p,q\nu} = \sum_{ia}\eri{pi}{qa}\qty( \bX_{\nu}+\bY_{\nu})_{ia},
 \end{equation}
 where $\bX$ and $\bY$ are the components of the eigenvectors of the particle-hole direct RPA problem defined as 
 \begin{equation}
@@ -200,7 +201,7 @@ The indices $\mu$ and $\nu$ are composite indices, \eg $\nu=(ia)$, referring to
 Because of the frequency dependence, fully solving the quasi-particle equation is a rather complicated task. 
 Hence, several approximate schemes have been developed to bypass self-consistency.
 The most popular one is the one-shot (perturbative) scheme, known as $G_0W_0$, where the self-consistency is completely abandoned, and the off-diagonal elements of Eq.~\eqref{eq:quasipart_eq} are neglected.
-\titou{Assuming a HF starting point,} this results in $K$ quasi-particle equations that read
+Assuming a HF starting point, this results in $K$ quasi-particle equations that read
 \begin{equation}
   \label{eq:G0W0}
   \epsilon_p^{\HF} + \Sigma_{pp}(\omega) - \omega = 0,
@@ -218,13 +219,10 @@ These additional solutions with large weights are the previously mentioned intru
 
 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 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}
+Therefore, one can \ant{tune} 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 set of quasi-particle equations self-consistently leading to the eigenvalue-only self-consistent scheme (ev$GW$). \cite{Shishkin_2007a,Blase_2011b,Marom_2012,Kaplan_2016,Wilhelm_2016}
 The solutions $\epsilon_p$  are used to build Eq.~\eqref{eq:G0W0} instead of the HF ones and then these equation are solved for $\omega$ again.
 This procedure is iterated until convergence for $\epsilon_p$ is reached.
-\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.}
-\ANT{Is it better now ?}
 However, if one of the quasi-particle equations does not have a well-defined quasi-particle solution, 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 initial molecular orbitals. \cite{Marom_2012}
 
@@ -313,13 +311,25 @@ Therefore, these blocks will be the target of the SRG transformation but before
 \label{sec:srg}
 %%%%%%%%%%%%%%%%%%%%%%
 
-The similarity renormalization group method aims at continuously transforming a Hamiltonian to a diagonal form, or more often to a block-diagonal form.
+The SRG method aims at continuously transforming a Hamiltonian to a diagonal form, or more often to a block-diagonal form.
-Therefore, the transformed Hamiltonian
+\ant{Hence, the first step is to decompose the Hamiltonian into an off-diagonal part which will be suppressed and the remaining diagonal part
+\begin{equation}
+  \bH = \underbrace{\bH^\text{d}}_{\text{diagonal}} + \underbrace{\bH^\text{od}}_{\text{off-diagonal}}.
+\end{equation}
+The SRG transformed Hamiltonian
 \begin{equation}
   \label{eq:SRG_Ham}
-  \bH(s) = \bU(s) \, \bH \, \bU^\dag(s)
+  \bH(s) = \bU(s) \, \bH \, \bU^\dag(s),
 \end{equation}
-depends on a flow parameter $s$, such that $\bH(s=0)$ is the initial untransformed Hamiltonian and $\bH(s=\infty)$ is the (block-)diagonal Hamiltonian.
+and the unitary matrix $\bU(s)$ both depend on a flow parameter $s$, such that $\bH(s=0)$ is the initial untransformed Hamiltonian and $\bU(s=0)$ is the identity.
+By definition, we also have the following condition on the off-diagonal part
+\begin{equation}
+  \bH^\text{od}(s=\infty) = \boldsymbol{0}.
+\end{equation}
+Therefore, the flow parameter controls the extent of the decoupling.
+}
+
+
 An evolution equation for $\bH(s)$ can be easily obtained by differentiating Eq.~\eqref{eq:SRG_Ham} with respect to $s$, yielding the flow equation
 \begin{equation}
   \label{eq:flowEquation}
@@ -330,16 +340,7 @@ where $\boldsymbol{\eta}(s)$, the flow generator, is defined as
 \boldsymbol{\eta}(s) = \dv{\bU(s)}{s}	\bU^\dag(s)	= - \boldsymbol{\eta}^\dag(s).
 \end{equation}
 To solve this equation at a lower cost than the one of diagonalizing the initial Hamiltonian, one must introduce an approximate form for $\boldsymbol{\eta}(s)$.
-\titou{Before defining such an approximation, we need to define what are the blocks to suppress to obtain a block-diagonal Hamiltonian.
+
-The Hamiltonian is separated into two parts as
-\begin{equation}
-  \bH(s) = \underbrace{\bH^\text{d}(s)}_{\text{diagonal}} + \underbrace{\bH^\text{od}(s)}_{\text{off-diagonal}},
-\end{equation}
-and, by definition, we have the following condition on $\bH^\text{od}$
-\begin{equation}
-  \bH^\text{od}(s=\infty) = \boldsymbol{0}.
-\end{equation}}
-\PFL{Move this part at the start of the section.}
 
 In this work, we consider Wegner's canonical generator which is defined as \cite{Wegner_1994}
 \begin{equation}
@@ -444,15 +445,15 @@ The last equation can be solved by introducing $\bU$ the matrix that diagonalize
 where $\bW^{(0)}(s)= \bV^{(0)}(s) \bU$.
 The matrix elements of $\bU$ and $\bD^{(0)}$ are
 \begin{align}
-  U_{(p,v),(q,w)} &= \delta_{pq} \bX_{v,w} \\
+  U_{p\nu,q\mu} &= \delta_{pq} \bX_{\nu\mu} \\
-  D_{(p,v),(q,w)}^{(0)} &= \left(\epsilon_p + \text{sign}(\epsilon_p-\epsilon_\text{F})\Omega_v\right)\delta_{pq}\delta_{vw}
+  D_{p\nu,q\mu}^{(0)} &= \left(\epsilon_p + \text{sign}(\epsilon_p-\epsilon_\text{F})\Omega_\nu\right)\delta_{pq}\delta_{\nu\mu}
 \end{align}
 where $\epsilon_\text{F}$ is the Fermi level.
 Note that the matrix $\bU$ is also used in the downfolding process of Eq.~\eqref{eq:GWlin}. \cite{Bintrim_2021}
 
 Thanks to the diagonal structure of $\bF^{(0)}$ and $\bD^{(0)}$, Eq.~\eqref{eq:eqdiffW0} can be easily solved and give 
 \begin{equation}
-  W_{p,(q,v)}^{(0)}(s) = W_{p,(q,v)}^{(0)}(0) e^{- (F_{pp}^{(0)} - D_{(q,v),(q,v)}^{(0)})^2 s}
+  W_{p,q\nu}^{(0)}(s) = W_{p,q\nu}^{(0)}(0) e^{- (F_{pp}^{(0)} - D_{q\nu,q\nu}^{(0)})^2 s}
 \end{equation}
 Due to the initial conditions  $\bV^{(0)}(0) = \bO$, we have $\bW^{(0)}(s)=\bO$ and therefore $\bV^{(0)}(s)=\bO=\bV^{(0)}(0) $.
 Therefore, the zeroth order Hamiltonian is
@@ -477,11 +478,11 @@ Once again the two first equations are easily solved
 \end{align}
 and the first order coupling elements are given by (up to a multiplication by $\bU^{-1}$)
 \begin{align}
-  W_{p,(q,v)}^{(1)}(s) &= W_{p,(q,v)}^{(1)}(0) e^{- (F_{pp}^{(0)} - D_{(q,v),(q,v)}^{(0)})^2 s} \\
+  W_{p,q\nu}^{(1)}(s) &= W_{p,q\nu}^{(1)}(0) e^{- (F_{pp}^{(0)} - D_{q\nu,q\nu}^{(0)})^2 s} \\
-  &= W_{p,(q,v)}^{(1)}(0) e^{- (\epsilon_p - \epsilon_q - \text{sign}(\epsilon_q-\epsilon_F)\Omega_v)^2 s} \notag
+  &= W_{p,q\nu}^{(1)}(0) e^{- (\epsilon_p - \epsilon_q - \text{sign}(\epsilon_q-\epsilon_F)\Omega_\nu)^2 s} \notag
 \end{align}
-At $s=0$ the elements  $W_{p,(q,v)}^{(1)}(0)$ are equal to the two-electron screened integrals defined in Eq.~\eqref{eq:GW_sERI} while for $s\to\infty$ they go to zero.
+At $s=0$ the elements  $W_{p,q\nu}^{(1)}(0)$ are equal to the two-electron screened integrals defined in Eq.~\eqref{eq:GW_sERI} while for $s\to\infty$ they go to zero.
-Therefore, $W_{p,(q,v)}^{(1)}(s)$ are renormalized two-electrons screened integrals.
+Therefore, $W_{p,q\nu}^{(1)}(s)$ are renormalized two-electrons screened integrals.
 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}).
 
 %///////////////////////////%
@@ -507,28 +508,26 @@ Collecting every second-order terms in the flow equation and performing the bloc
   \dv{\bF^{(2)}}{s}  = \bF^{(0)}\bV^{(1)}\bV^{(1),\dagger} + \bV^{(1)}\bV^{(1),\dagger}\bF^{(0)} - 2 \bV^{(1)}\bC^{(0)}\bV^{(1),\dagger} .
 \end{equation}
 This can be solved by simple integration along with the initial condition $\bF^{(2)}=\bO$ to give
-\begin{align}
+\begin{multline}
-  &F_{pq}^{(2)}(s) =  \\
+  F_{pq}^{(2)}(s) =  \sum_{r\mu} \frac{\Delta_{pr\mu}+ \Delta_{qr\mu}}{\Delta_{pr\mu}^2 + \Delta_{qr\mu}^2} W_{p,r\mu} W^{\dagger}_{r\mu,q} \times \\
-  &\sum_{r,v} \frac{\Delta_{prv}+ \Delta_{qrv}}{\Delta_{prv}^2 + \Delta_{qrv}^2} W_{p,(r,v)} W^{\dagger}_{(r,v),q}\left(1 - e^{-(\Delta_{prv}^2 + \Delta_{qrv}^2) s}\right), \notag
+  \left(1 - e^{-(\Delta_{pr\mu}^2 + \Delta_{qr\mu}^2) s}\right),
-\end{align}
+\end{multline}
 with $\Delta_{prv} = \epsilon_p - \epsilon_r - \text{sign}(\epsilon_r-\epsilon_F)\Omega_v$.
 
 At $s=0$, this second-order correction is null while for $s\to\infty$ it tends towards the following static limit
 \begin{equation}
   \label{eq:static_F2}
-  F_{pq}^{(2)}(\infty) =  \sum_{r,v}  \frac{\Delta_{prv}+ \Delta_{qrv}}{\Delta_{prv}^2 + \Delta_{qrv}^2} W_{p,(r,v)} W^{\dagger}_{(r,v),q}.
+  F_{pq}^{(2)}(\infty) =  \sum_{r,v}  \frac{\Delta_{prv}+ \Delta_{qrv}}{\Delta_{prv}^2 + \Delta_{qrv}^2} W_{p,r\mu} W^{\dagger}_{r\mu,q}.
 \end{equation}
 Note that in the $s\to\infty$ limit the dynamic part of the self-energy [see Eq.~\eqref{eq:srg_sigma}] tends to zero.
 Therefore, the SRG flow transforms the dynamic part of $\bSig(\omega)$ into a static correction.
 This transformation is done gradually starting from the states that have the largest denominators in Eq.~\eqref{eq:static_F2}.
-
+y
-\ANT{Change notation here and use multline}
 Interestingly, the static limit, \ie $s\to\infty$ limit, of Eq.~\eqref{eq:GW_renorm} defines an alternative qs$GW$ approximation to the one defined by Eq.~\eqref{eq:sym_qsgw} which matrix elements read as
-\begin{align}
+\begin{equation}
   \label{eq:sym_qsGW}
-  &\Sigma_{pq}^{\text{qs}GW}(\eta) = \\
+    \Sigma_{pq}^{\text{qs}}(\eta) = \frac{1}{2} \sum_{r\mu}  \left( \frac{\Delta_{pr\mu}}{\Delta_{pr\mu}^2 + \eta^2} +\frac{\Delta_{qr\mu}}{\Delta_{qr\mu}^2 + \eta^2}  \right) W_{p,r\mu} W^{\dagger}_{r\mu,q}.
-  &\frac{1}{2} \sum_{r,v}  \left( \frac{\Delta_{prv}}{\Delta_{prv}^2 + \eta^2} +\frac{\Delta_{qrv}}{\Delta_{qrv}^2 + \eta^2}  \right) W_{p,(r,v)} W^{\dagger}_{(r,v),q}. \notag 
+\end{equation}
-\end{align}
 Yet, both approximation are closely related as they share the same diagonal terms when $\eta=0$.
 Also, note that the SRG static approximation is naturally Hermitian as opposed to the symmetrized case where it is enforced by symmetrization.
 
@@ -536,11 +535,11 @@ However, as will be discussed in more detail in the results section, the converg
 This is similar to the symmetric case when the imaginary shift $\ii \eta$ is set to zero.
 Indeed, in qs$GW$ calculation using the symmetrized static form, increasing $\eta$ to ensure convergence in difficult cases is most often unavoidable.
 Therefore, we will define the SRG-qs$GW$ static effective Hamiltonian as
-\begin{align}
+
+\begin{multline}
   \label{eq:SRG_qsGW}
-  &\Sigma_{pq}^{\text{SRG}}(s) =  \\
+  \Sigma_{pq}^{\text{SRG}}(s) = \sum_{r\mu} \frac{\Delta_{pr\mu}+ \Delta_{qr\mu}}{\Delta_{pr\mu}^2 + \Delta_{qr\mu}^2} W_{p,r\mu} W_{q,r\mu} \times \\ \left(1 - e^{-(\Delta_{pr\mu}^2 + \Delta_{qr\mu}^2) s}\right)  
-  &\sum_{r,v} \frac{\Delta_{prv}+ \Delta_{qrv}}{\Delta_{prv}^2 + \Delta_{qrv}^2} W_{p,(r,v)} W_{q,(r,v)}\left(1 - e^{-(\Delta_{prv}^2 + \Delta_{qrv}^2) s}\right) \notag 
+\end{multline}
-\end{align}
 which depends on one regularising parameter $s$ analogously to $\eta$ in the usual case.
 The fact that the $s\to\infty$ static limit does not always converge when used in a qs$GW$ calculation could have been predicted because in this limit even the intruder states have been included in $\tilde{\bF}$.
 Therefore, we should use a value of $s$ large enough to include almost every states but small enough to avoid intruder states.
@@ -561,12 +560,18 @@ In fact, the dynamic part after the change of variable is closely related to the
 \label{sec:comp_det}
 %=================================================================%
 
-\ANT{Frozen-core}
+% Reference comp det
+The geometries have been optimized without frozen core approximation at the CC3 level in the aug-cc-pVTZ basis set using the CFOUR program.
+The reference CCSD(T) ionization potential (IP) energies have been obtained using default parameters of Gaussian 16.
+This means that the cations used an unrestricted HF reference while the neutral ground-state energies have been obtained in a restricted HF formalism.
+
+% GW comp det
 The two qs$GW$ variants considered in this work have been implemented in an in-house program.
 The $GW$ implementation closely follows the one of mol$GW$. \cite{Bruneval_2016}
-The geometries have been optimized at the CC3 level in the aug-cc-pVTZ basis set without frozen core using the CFOUR program.
+All $GW$ calculations were performed without the frozen-core approximation.
-The reference CCSD(T) IP energies have been obtained using default parameters of Gaussian 16.
+The DIIS space size and the maximum of iterations were set to 5 and 64, respectively.
-This means that the cations used an unrestricted HF reference while the neutral ground-state energies have been obtained in a restricted formalism.
+In practice, one could (and should) achieve convergence in some cases by adjusting these parameters or by using an alternative mixing scheme.
+However, in order to perform a black-box comparison of the methods these parameters have been fixed.
 
 %=================================================================%
 \section{Results}
@@ -575,56 +580,68 @@ This means that the cations used an unrestricted HF reference while the neutral
 
 The results section is divided in two parts.
 The first step will be to analyse in depth the behavior of the two static self-energy approximations in a few test cases.
-Then the accuracy of the IP yielded by the Sym and SRG schemes will be statistically gauged over a test set of molecules.
+Then the accuracy of the IP yielded by the traditional and SRG schemes will be statistically gauged over a test set of molecules.
 
 %%%%%%%%%%%%%%%%%%%%%%
-\subsection{Flow parameter dependence of the SRG/Sym-qs$GW$/GF(2) schemes}
+\subsection{Flow parameter dependence of the qs$GW$ and SRG-qs$GW$ schemes}
 \label{sec:flow_param_dep}
 %%%%%%%%%%%%%%%%%%%%%% 
 
 %%% FIG 1 %%%
-\begin{figure*}
+\begin{figure}
   \centering
   \includegraphics[width=\linewidth]{fig1.pdf}
   \caption{
-    Add caption \ANT{Should we add $G_0W_0$?}
+    Principal IP of the water molecule in the aug-cc-pVTZ cartesian basis set as a function of the flow parameter $s$ for the SRG-qs$GW$ method with and without TDA.
+    Reference values (HF, qs$GW$ with and without TDA) are also reported as dashed lines.
     \label{fig:fig1}}
-\end{figure*}
+\end{figure}
 %%% %%% %%% %%%
 
-This section starts by considering the Neon atom and the water molecule in the aug-cc-pVTZ cartesian basis set in Fig.~\ref{fig:fig1}.
+This section starts by considering a prototypical molecular system, \ie the water molecule, in the aug-cc-pVTZ cartesian basis set.
-Figure~\ref{fig:fig1} shows the error of various methods for the principal IP with respect to CCSD(T).
+Figure~\ref{fig:fig1} shows the error of various methods for the principal IP with respect to (w.r.t.) the CCSD(T) reference value.
-The HF IPs (cyan lines) are overestimated, this is a consequence of the missing correlation, a result which is now well-understood. \cite{Lewis_2019} \ANT{I should maybe search for another ref as well.}
+The HF IP (dashed black line) is overestimated, this is a consequence of the missing correlation, a result which is now well-understood. \cite{Lewis_2019} \ANT{I should maybe search for another ref as well.}
-The usual Sym-qs$GW$ scheme (green lines) brings a quantitative improvement as both IP energies are now within \SI{0.5}{\electronvolt} of the reference.
+The usual qs$GW$ scheme (dashed blue line) brings a quantitative improvement as the IP is now within \SI{0.3}{\electronvolt} of the reference.
-The Neon atom is a well-behaved system and could be converged without regularisation parameter while for water $\eta$ was set to 0.01 to help convergence.
+%The Neon atom is a well-behaved system and could be converged without regularisation parameter while for water $\eta$ was set to 0.01 to help convergence.
 
-Figure~\ref{fig:fig1} also displays the SRG-qs$GW$ IPs as a function of the flow parameter (blue curves).
+Figure~\ref{fig:fig1} also displays the SRG-qs$GW$ IP as a function of the flow parameter (blue curve).
-At $s=0$, the IPs are equal to their HF counterparts as expected from the discussion of Sec.~\ref{sec:srggw}.
+At $s=0$, the IP is equal to its HF counterpart as expected from the discussion of Sec.~\ref{sec:srggw}.
-For $s\to\infty$ both IPs reach a plateau at an error that is significantly smaller than their $s=0$ starting point.
+For $s\to\infty$, the IP reaches a plateau at an error that is significantly smaller than their $s=0$ starting point.
-Even more, the values associated with these plateau are more accurate than their Sym-qs$GW$ counterparts.
+Even more, the value associated with this plateau is slightly more accurate than its qs$GW$ counterpart.
-However, the SRG-qs$GW$ error do not decrease smoothly between the HF values and their limits as for small $s$ values they are actually worst than the HF IPs.
+However, the SRG-qs$GW$ error do not decrease smoothly between the initial HF value and the $s\to\infty$ limit as for small $s$ values it is actually worst than the HF starting point.
 
-In addition, we also considered the qs$GW$ and SRG-qs$GW$ methods based on a TDA screening.
+This behavior as a function of $s$ can be \ant{approximately} streamlined by applying matrix perturbation theory on Eq.~(\ref{eq:GWlin}).
-The TDA IPs are now understimated unlike their RPA counterparts.
+Through second order, the principal IP is
-The SRG-qs$GW$ absolute error for the IPs clearly deteriorates when going from RPA to TDA.
+\begin{equation}
-On the other hand, for qs$GW$ the TDA-based IP is better than the RPA one for Neon while it is the other way around for water.
+  \label{eq:2nd_order_IP}
+  I_k(2) = - \epsilon_k - \sum_{i\nu} \frac{W_{k,i\nu}^2}{\epsilon_k - \epsilon_i + \Omega_\nu} - \sum_{a\nu} \frac{W_{k,a\nu}^2}{\epsilon_k - \epsilon_a - \Omega_\nu} + \order{3}
+\end{equation}
+where $k$ is the index of the highest molecular MO (HOMO).
+The first term is the zeroth order IP and the two following terms come from the 2h1p and 2p1h coupling, respectively.
+The denominator of the last term is always positive while the 2h1p term is negative.
+
+When $s$ is increased, the first states that will be decoupled from the HOMO will be the 2p1h ones because their energy difference with the HOMO is larger than the ones of the 2h1p block.
+Therefore, for small $s$ only the last term of Eq.~\eqref{eq:2nd_order_IP} will be partially included resulting in a positive correction to the IP.
+As soon as $s$ is large enough to decouple the 2h1p block as well the IP will start to decrease and eventually go below the $s=0$ initial value as observed in Fig.~\ref{fig:fig1}.
+
+In addition, the qs$GW$ and SRG-qs$GW$ methods based on a TDA screening are also considered in Fig.~\ref{fig:fig1}.
+The TDA IPs are now underestimated unlike their RPA counterparts.
+For both static self-energies, the TDA leads to a slight increase of the absolute error.
 This trend will be investigated in more details in the next subsection.
 
-\ANT{Maybe we should add GF(2) because it allows us to explain the behavior of the SRG curve using perturbation theory.}
+Before going to the statistical study, the behavior of three particular molecules is investigated.
-The behavior of the SRG-qsGF2 IPS is similar to the SRG-qs$GW$ one.
+The Lithium dimer \ce{Li2} will be considered as a case where HF actually underestimate the IP.
-Add sentence about $GW$ better than GF2 when the results will be here.
+The Lithium hydrid will also be investigated because in this case the usual qs$GW$ IP is worst than the HF one.
-The decrease and then increase behavior of the IPs can be rationalised using results from perturbation theory for GF(2).
+Finally, the Beryllium oxyde will be studied as a prototypical example of a molecular system difficult to converge because of intruder states. \cite{vanSetten_2015,Veril_2018,Forster_2021}
-We refer the reader to the chapter 8 of Ref.~\onlinecite{Schirmer_2018} for more details about this analysis.
-The GF(2) IP admits the following perturbation expansion... \ANT{Remove GF2 and try matrix perturbation theory on $GW$, cf Evangelista's talk.}
 
-Because $GW$ relies on an infinite resummation of diagram such a perturbation analysis is difficult to make in this case.
+% \ANT{Maybe we should add GF(2) because it allows us to explain the behavior of the SRG curve using perturbation theory.}
-But the mechanism causing the increase/decrease of the $GW$ IPs as a function of $s$ should be closely related to the GF(2) one exposed above.
+% The behavior of the SRG-qsGF2 IPS is similar to the SRG-qs$GW$ one.
-
+% Add sentence about $GW$ better than GF2 when the results will be here.
-\begin{itemize}
+% The decrease and then increase behavior of the IPs can be rationalised using results from perturbation theory for GF(2).
-\item Li2 interesting because HF underestimate IP
+% We refer the reader to the chapter 8 of Ref.~\onlinecite{Schirmer_2018} for more details about this analysis.
-\item LiH interesting because Sym-qs$GW$ worst than HF
+% The GF(2) IP admits the following perturbation expansion... \ANT{Remove GF2 and try matrix perturbation theory on $GW$, cf Evangelista's talk.}
-\item BeO interesting because usually difficult to converge in qs$GW$ (cf Forster 2021)
+% Because $GW$ relies on an infinite resummation of diagram such a perturbation analysis is difficult to make in this case.
-\end{itemize}
+% But the mechanism causing the increase/decrease of the $GW$ IPs as a function of $s$ should be closely related to the GF(2) one exposed above.
 
 %%% FIG 2 %%% 
 \begin{figure*}
@@ -643,7 +660,7 @@ But the mechanism causing the increase/decrease of the $GW$ IPs as a function of
 
 The test set considered in this study is composed of the GW20 set of molecules introduced by Lewis and Berkelbach. \cite{Lewis_2019}
 This set is made of the 20 smallest atoms and molecules of the GW100 benchmark set.
-We also added the MgO and O3 molecules which are part of GW100 and are known to be difficult to converged for qs$GW$.
+We also added the MgO and O3 molecules which are part of GW100 and are known to be difficult to converged for qs$GW$. \cite{vanSetten_2015,Forster_2021}
 In addition, we considered the Quest 1 and 2 sets which is made of small and medium size organic molecules.