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% addresses
\newcommand{\LCPQ}{Laboratoire de Chimie et Physique Quantiques (UMR 5626), Universit\'e de Toulouse, CNRS, UPS, France}

\begin{document}	

\title{A similarity renormalization group approach to Green's function methods}
%\title{Tackling The Intruder-State Problem In Many-Body Perturbation Theory: A Similarity Renormalization Group Approach/Perspective}

\author{Antoine \surname{Marie}}
	\email{amarie@irsamc.ups-tlse.fr}
	\affiliation{\LCPQ}

\author{Pierre-Fran\c{c}ois \surname{Loos}}
	\email{loos@irsamc.ups-tlse.fr}
	\affiliation{\LCPQ}

\begin{abstract}
The family of Green's function methods based on the $GW$ approximation has gained popularity in the electronic structure theory thanks to its accuracy in weakly correlated systems and its affordability.
Despite this, self-consistent versions still pose challenges in terms of convergence.
A recent study \href{https://doi.org/10.1063/5.0089317}{[J. Chem. Phys. 156, 231101 (2022)]} has linked these convergence issues to the intruder-state problem.
In this work, a perturbative analysis of the similarity renormalization group (SRG) approach is performed on Green's function methods.
The resulting SRG-based regularized self-energy significantly accelerates the convergence of self-consistent $GW$ methods. 
Furthermore, it enables us to derive, from first principles, the expression of a new naturally Hermitian form of the static self-energy that can be employed in quasiparticle self-consistent $GW$ (qs$GW$) calculations.
%\bigskip
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\maketitle

%=================================================================%
\section{Introduction}
\label{sec:intro}
% =================================================================%

One-body Green's functions provide a natural and elegant way to access the charged excitation energies of a physical system. \cite{CsanakBook,FetterBook,Martin_2016,Golze_2019}
The non-linear Hedin equations consist of a closed set of equations leading to the exact interacting one-body Green's function and, therefore, to a wealth of properties such as the total energy, density, ionization potentials, electron affinities, as well as spectral functions, without the explicit knowledge of the wave functions associated with the neutral and charged states of the system. \cite{Hedin_1965}
Unfortunately, solving exactly Hedin's equations is usually out of reach and one must resort to approximations.
In particular, the $GW$ approximation, \cite{Hedin_1965,Aryasetiawan_1998,Onida_2002,Reining_2017,Golze_2019,Bruneval_2021} which has been first introduced in the context of solids \cite{Strinati_1980,Strinati_1982a,Strinati_1982b,Hybertsen_1985,Hybertsen_1986,Godby_1986,Godby_1987,Godby_1987a,Godby_1988,Blase_1995} and is now widely applied to molecular systems, \cite{Rohlfing_1999a,Horst_1999,Puschnig_2002,Tiago_2003,Rocca_2010,Boulanger_2014,Jacquemin_2015a,Bruneval_2015,Jacquemin_2015b,Hirose_2015,Jacquemin_2017a,Jacquemin_2017b,Rangel_2017,Krause_2017,Gui_2018,Blase_2018,Liu_2020,Li_2017,Li_2019,Li_2020,Li_2021,Blase_2020,Holzer_2018a,Holzer_2018b,Loos_2020e,Loos_2021,McKeon_2022} yields accurate charged excitation energies for weakly correlated systems \cite{Hung_2017,vanSetten_2015,vanSetten_2018,Caruso_2016,Korbel_2014,Bruneval_2021} at a relatively low computational cost. \cite{Foerster_2011,Liu_2016,Wilhelm_2018,Forster_2021,Duchemin_2019,Duchemin_2020,Duchemin_2021}

The $GW$ method approximates the self-energy $\Sigma$ which relates the exact interacting Green's function $G$ to a non-interacting reference version $G_0$ through a Dyson equation of the form
\begin{equation}
  \label{eq:dyson}
  G(1,2) = G_0(1,2) + \int d(34) G_0(1,3)\Sigma(3,4) G(4,2),
\end{equation}
where $1 = (\bx_1, t_1)$ is a composite coordinate gathering spin-space and time variables.
The self-energy encapsulates all the Hartree-exchange-correlation effects which are not taken into account in the reference system.
Approximating $\Sigma$ as the first-order term of its perturbative expansion with respect to the screened Coulomb potential $W$ yields the so-called $GW$ approximation \cite{Hedin_1965,Martin_2016}
\begin{equation}
  \label{eq:gw_selfenergy}
  \Sigma^{\GW}(1,2) = \ii G(1,2) W(1,2).
\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.

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}
Many-body perturbation theory can also be used to access optical excitation energies through the Bethe-Salpeter equation. \cite{Salpeter_1951,Strinati_1988,Blase_2018,Blase_2020} However, the accuracy is not yet satisfying for triplet excited states, where instabilities often occur. \cite{Bruneval_2015,Jacquemin_2017a,Jacquemin_2017b,Holzer_2018a}
Therefore, even if $GW$ offers a good trade-off between accuracy and computational cost, some situations might require higher precision.
Unfortunately, defining a systematic way to go beyond $GW$ via the inclusion of vertex corrections has been demonstrated to be a tricky task. \cite{Baym_1961,Baym_1962,DeDominicis_1964a,DeDominicis_1964b,Bickers_1989a,Bickers_1989b,Bickers_1991,Hedin_1999,Bickers_2004,Shirley_1996,DelSol_1994,Schindlmayr_1998,Morris_2007,Shishkin_2007b,Romaniello_2009a,Romaniello_2012,Gruneis_2014,Hung_2017,Maggio_2017b,Mejuto-Zaera_2022}
For example, Lewis and Berkelbach have shown that naive vertex corrections can even worsen the quasiparticle energies with respect to $GW$. \cite{Lewis_2019}
We refer the reader to the recent review by Golze and co-workers (see Ref.~\onlinecite{Golze_2019}) for an extensive list of current challenges in many-body perturbation theory. 

Many-body perturbation theory also suffers from the infamous intruder-state problem,\cite{Andersson_1994,Andersson_1995a,Roos_1995,Forsberg_1997,Olsen_2000,Choe_2001} where they manifest themselves as solutions of the quasiparticle equation with non-negligible spectral weights.
In some cases, this transfer of spectral weight makes it difficult to distinguish between a quasiparticle and a satellite.
These multiple solutions hinder the convergence of partially self-consistent schemes, \cite{Veril_2018,Forster_2021,Monino_2022} such as quasiparticle self-consistent $GW$ \cite{Faleev_2004,vanSchilfgaarde_2006,Kotani_2007,Ke_2011,Kaplan_2016} (qs$GW$) and eigenvalue-only self-consistent $GW$ \cite{Shishkin_2007a,Blase_2011b,Marom_2012,Kaplan_2016,Wilhelm_2016} (ev$GW$). 
The simpler one-shot $G_0W_0$ scheme \cite{Strinati_1980,Hybertsen_1985a,Hybertsen_1986,Godby_1988,Linden_1988,Northrup_1991,Blase_1994,Rohlfing_1995,Shishkin_2007a} is also impacted by these intruder states, leading to discontinuities in a variety of physical quantities including charged and neutral excitation energies, correlation and total energies.\cite{Loos_2018b,Veril_2018,Loos_2020e,Berger_2021,DiSabatino_2021,Monino_2022,Scott_2023}
These convergence problems and discontinuities can even happen in the weakly correlated regime where the $GW$ approximation is supposed to be valid.

In a recent study, Monino and Loos showed that the 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, such 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 $GW$.
In particular, we focus here on the possibility of curing the qs$GW$ convergence problems using the SRG.

The SRG formalism 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.
\cite{Bogner_2007,Tsukiyama_2011,Tsukiyama_2012,Hergert_2013,Hergert_2016,Frosini_2022a,Frosini_2022b,Frosini_2022c} 
See Ref.~\onlinecite{Hergert_2016a} for a recent review in this field.

The SRG transformation aims at decoupling an internal (or reference) space from an external space while incorporating information about their coupling in the reference space.
This process often results in the appearance of intruder states. \cite{Evangelista_2014b,ChenyangLi_2019a}
However, SRG is particularly well-suited to avoid them because the decoupling of each external configuration is inversely proportional to its energy difference with the reference space.
By definition, intruder states have energies that are close to the reference energy, and therefore are the last to be decoupled.
By stopping the SRG transformation once all external configurations except the intruder states have been decoupled,
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 partially self-consistent $GW$ calculations.
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. \cite{Liebsch_1981,Bickers_1989a,Bickers_1991,Katsnelson_1999,Katsnelson_2002,Zhukov_2005,vonFriesen_2010,Romaniello_2012,Gukelberger_2015,Muller_2019,Friedrich_2019,Biswas_2021,Zhang_2017,Li_2021b,Loos_2022}

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}.
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{The $GW$ approximation}
\label{sec:gw}
%%%%%%%%%%%%%%%%%%%%%% 

The central equation of many-body perturbation theory based on Hedin's equations is the so-called quasiparticle equation which, within the $GW$ approximation, reads
\begin{equation}
  \label{eq:quasipart_eq}
  \qty[ \bF + \bSig(\omega = \epsilon_p) ] \psi_p(\bx) = \epsilon_p \psi_p(\bx),
\end{equation}
where $\bF$ is the Fock matrix in the orbital basis \cite{SzaboBook} and $\bSig(\omega)$ is (the correlation part of) the $GW$ self-energy. 
Both are $K \times K$ matrices with $K$ the number of one-electron orbitals.
Throughout the manuscript, the indices $p,q,r,s$ are general orbitals while $i,j,k,l$ and $a,b,c,d$ refers to occupied and virtual orbitals, respectively.
The indices $\mu$ and $\nu$ are composite indices, \eg $\nu=(ia)$, referring to neutral (single) excitations.

The self-energy can be physically understood as a correction to the Hartree-Fock (HF) problem (represented by $\bF$) accounting for dynamical screening effects.
Similarly to the HF case, Eq.~\eqref{eq:quasipart_eq} has to be solved self-consistently.
\titou{Note that $\bSig(\omega)$ is dynamical which implies that it depends on both the one-electron orbitals $\psi_p(\bx)$ and their corresponding energies $\epsilon_p$, while $\bF$ depends only on the orbitals.}
\PFL{I still don't like it.}

The matrix elements of $\bSig(\omega)$ have the following closed-form expression \cite{Hedin_1999,Tiago_2006,Bruneval_2012,vanSetten_2013,Bruneval_2016}
\begin{equation}
  \label{eq:GW_selfenergy}
  \Sigma_{pq}(\omega) 
  = \sum_{i\nu} \frac{W_{p,i\nu} W_{q,i\nu}}{\omega - \epsilon_i + \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_{p,q\nu} = \sum_{ia}\eri{pi}{qa}\qty(\bX+\bY)_{ia,\nu},
\end{equation}
with $\bX$ and $\bY$ the components of the eigenvectors of the direct (\ie without exchange) RPA problem defined as 
\begin{equation}
  \label{eq:full_dRPA}
  \mqty( \bA & \bB \\ -\bB & -\bA ) \mqty( \bX \\ \bY ) =  \mqty( \bX \\ \bY ) \mqty( \boldsymbol{\Omega} & \bO \\ \bO & -\boldsymbol{\Omega} ),
\end{equation}
with
\begin{subequations}
\begin{align}
  A_{ia,jb} & = (\epsilon_i - \epsilon_a) \delta_{ij}\delta_{ab} + \eri{ib}{aj},
  \\
  B_{ia,jb} & = \eri{ij}{ab},
\end{align}
\end{subequations}
and
\begin{equation}
	\braket{pq}{rs} = \iint \frac{\SO{p}(\bx_1) \SO{q}(\bx_2)\SO{r}(\bx_1) \SO{s}(\bx_2)  }{\abs{\br_1 - \br_2}} d\bx_1 d\bx_2
\end{equation}
are bare two-electron integrals in the spin-orbital basis.

The diagonal matrix $\boldsymbol{\Omega}$ contains the positive eigenvalues of the RPA problen 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), which is discussed in Appendix \ref{sec:nonTDA}, one sets $\bB = \bO$ in Eq.~\eqref{eq:full_dRPA} which reduces to a Hermitian eigenvalue problem of the form $\bA \bX = \bX \bOm$.

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 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.
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).
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.
\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).
However, in some cases, Eq.~\eqref{eq:G0W0} can have two (or more) solutions with similar weights, hence the quasiparticle is not well-defined.
These additional solutions with large weights are the previously mentioned intruder states. \cite{Monino_2022}

One obvious drawback of the one-shot scheme mentioned above is its starting point dependence.
Indeed, in Eq.~\eqref{eq:G0W0} we choose to rely on HF orbital energies but this is arbitrary and one could have chosen Kohn-Sham energies (and orbitals) instead.
As commonly done, one can even ``tune'' the starting point to obtain the best possible one-shot $GW$ quasiparticle energies. \cite{Korzdorfer_2012,Marom_2012,Bruneval_2013,Gallandi_2015,Caruso_2016, Gallandi_2016}
Alternatively, one may solve this set of quasiparticle equations self-consistently leading to the ev$GW$ scheme.
\ant{To do so the quasiparticle energies are used to define a new RPA problem leading to updated two-electron screened integrals.
 Then the diagonal elements of the self-energy are updated as well and Eq.~\eqref{eq:G0W0} is solved again to obtain new quasiparticle energies.}
This procedure is then iterated until convergence on the quasiparticle energies is reached.
However, if one of the quasiparticle equations does not have a well-defined quasiparticle solution, reaching self-consistency can be challenging, if not impossible. 
Even at convergence, the starting point dependence is not totally removed as the quasiparticle energies still depend on the initial set of orbitals. \cite{Marom_2012}

In order to update both the orbitals and their corresponding energies, one must consider the off-diagonal elements in $\bSig(\omega)$.
To avoid solving the non-Hermitian and dynamic quasiparticle equation defined in Eq.~\eqref{eq:quasipart_eq}, one can resort to the qs$GW$ scheme in which $\bSig(\omega)$ is replaced by a static approximation $\bSig^{\qs}$.
Then, the qs$GW$ equations are solved via a standard self-consistent field procedure similar to the HF algorithm where $\bF$ is replaced by $\bF + \bSig^{\qs}$.
Various choices for $\bSig^\qs$ are possible but the most popular is the following Hermitian approximation
\begin{equation}
  \label{eq:sym_qsgw}
  \Sigma_{pq}^\qs = \frac{1}{2}\Re[\Sigma_{pq}(\epsilon_p) + \Sigma_{pq}(\epsilon_q) ].
\end{equation}
which was first 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 functions by Ismail-Beigi. \cite{Ismail-Beigi_2017}
The corresponding matrix elements are
\begin{equation}
  \label{eq:sym_qsGW}
  \Sigma_{pq}^{\text{qs}}(\eta) = \frac{1}{2} \sum_{r\nu}  \qty( \frac{\Delta_{pr\nu}}{\Delta_{pr\nu}^2 + \titou{\eta^2}} +\frac{\Delta_{qr\nu}}{\Delta_{qr\nu}^2 + \titou{\eta^2}}  ) W_ {p,r\nu} W_{q,r\nu}.
\end{equation}
with $\Delta_{pr\nu} = \epsilon_p - \epsilon_r - \sgn(\epsilon_r-\epsilon_F)\Omega_\nu$ (where $\epsilon_F$ is the energy of the Fermi level).
One of the main results of the present manuscript is the derivation, from first principles, of an alternative static Hermitian form for the $GW$ self-energy. 

Once again, in cases where multiple solutions have large spectral weights, self-consistency can be difficult to reach at the qs$GW$ level.
Multiple solutions of Eq.~\eqref{eq:G0W0} arise due to the $\omega$ dependence of th e self-energy.
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 qs$GW$ self-consistent scheme inevitably oscillates between solutions with large spectral weights. \cite{Forster_2021}

The satellites causing convergence problems are the above-mentioned intruder states.
One can deal with them by introducing \textit{ad hoc} regularizers.
\titou{The $\ii \eta$ term that is usually added in the denominators of the self-energy} [see Eq.~\eqref{eq:GW_selfenergy}] is similar to the usual imaginary-shift regularizer employed in various other theories affected by the intruder-state problem. \cite{Surjan_1996,Forsberg_1997,Monino_2022,Battaglia_2022}
Several other regularizers are possible \cite{Stuck_2013,Rostam_2017,Lee_2018a,Evangelista_2014b,Shee_2021} and in particular, it was shown in Ref.~\onlinecite{Monino_2022} that a regularizer inspired by the SRG had some advantages over the imaginary shift.
Nonetheless, 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 central aim of the present work.

%%%%%%%%%%%%%%%%%%%%%%
\section{The similarity renormalization group}
\label{sec:srg}
%%%%%%%%%%%%%%%%%%%%%%

The SRG method aims at continuously transforming a general Hamiltonian matrix to its diagonal form, or more often, to a block-diagonal form.
Hence, the first step is to decompose this Hamiltonian matrix 
\begin{equation}
  \bH = \bH^\text{d} + \bH^\text{od}.
\end{equation}
into an off-diagonal part, $\bH^\text{od}$, that we aim at removing and the remaining diagonal part, $\bH^\text{d}$.

This transformation can be performed continuously via a unitary matrix $\bU(s)$, as follows:
\begin{equation}
  \label{eq:SRG_Ham}
  \bH(s) = \bU(s) \, \bH \, \bU^\dag(s),
\end{equation}
where $s$ is the so-called flow parameter that controls the extent of the decoupling.
By definition, we have $\bH(s=0) = \bH$ [or $\bU(s=0) = \bI$] and $\bH^\text{od}(s=\infty) = \bO$.

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}
  \dv{\bH(s)}{s} = \comm{\boldsymbol{\eta}(s)}{\bH(s)},
\end{equation}
where $\boldsymbol{\eta}(s)$, the flow generator, is defined as
\begin{equation}
\boldsymbol{\eta}(s) = \dv{\bU(s)}{s}	\bU^\dag(s)	= - \boldsymbol{\eta}^\dag(s).
\end{equation}

To solve the flow equation at a lower cost than the one associated with the diagonalization of the initial Hamiltonian, one must introduce an approximate form for $\boldsymbol{\eta}(s)$.
In this work, we consider Wegner's canonical generator \cite{Wegner_1994}
\begin{equation}
  \boldsymbol{\eta}^\text{W}(s) = \comm{\bH^\text{d}(s)}{\bH(s)} = \comm{\bH^\text{d}(s)}{\bH^\text{od}(s)},
\end{equation}
which satisfied the following condition \cite{Kehrein_2006}
\begin{equation}
  \label{eq:derivative_trace}
  \dv{s}\text{Tr}\left[ \bH^\text{od}(s)^2 \right] \leq 0.
\end{equation}
This implies that the matrix elements of the off-diagonal part decrease in a monotonic way throughout the transformation.
Moreover, the coupling coefficients associated with the highest-energy determinants are removed first as we shall evidence in the perturbative analysis below.
The main drawback of this generator is that it generates a stiff set of ODE which is therefore difficult to solve numerically. \cite{Hergert_2016a}
However, here we will not tackle the full SRG problem but only consider analytical low-order perturbative expressions so we will not be affected by this problem. \cite{Evangelista_2014,Hergert_2016}

Let us now perform the perturbative analysis of the SRG equations.
For $s=0$, the initial problem is
\begin{equation}
  \bH(0) = \bH^\text{d}(0) + \lambda \bH^\text{od}(0),
\end{equation}
where $\lambda$ is the usual perturbation parameter and the off-diagonal part of the Hamiltonian has been defined as the perturbation.
For finite values of $s$, we have the following perturbation expansion of the Hamiltonian
\begin{equation}
  \label{eq:perturbation_expansionH}
  \bH(s) = \bH^{(0)}(s) + \lambda ~ \bH^{(1)}(s) + \lambda^2 \bH^{(2)}(s) + \cdots.
\end{equation}
Hence, the generator $\boldsymbol{\eta}(s)$ admits a similar perturbation expansion.
Then, as performed in Sec.~\ref{sec:srggw}, one can collect order by order the terms in Eq.~\eqref{eq:flowEquation} and solve analytically the low-order differential equations.

%%%%%%%%%%%%%%%%%%%%%%
\section{Regularized $GW$ approximation}
\label{sec:srggw}
%%%%%%%%%%%%%%%%%%%%%% 

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.
A way around this problem is to transform Eq.~\eqref{eq:quasipart_eq} to its upfolded version which elegantly highlights the coupling terms: \cite{Bintrim_2021,Tolle_2022}
\begin{equation}
  \label{eq:GWlin}
  \begin{pmatrix}
    \bF				&	\bW^{\text{2h1p}}		&	\bW^{\text{2p1h}}	\\
    (\bW^{\text{2h1p}})^\dag	&	\bC^{\text{2h1p}}		&	\bO					\\
    (\bW^{\text{2p1h}})^\dag	&	\bO				&	\bC^{\text{2p1h}}	\\
  \end{pmatrix},
%  \begin{pmatrix}
%    \bZ^{\text{1h/1p}}	\\
%    \bZ^{\text{2h1p}}	\\
%    \bZ^{\text{2p1h}}	\\
%  \end{pmatrix}
%  =
%  \begin{pmatrix}
%    \bZ^{\text{1h/1p}}	\\
%    \bZ^{\text{2h1p}}	\\
%    \bZ^{\text{2p1h}}	\\
%  \end{pmatrix}
%  \boldsymbol{\epsilon},
\end{equation}
%where $\boldsymbol{\epsilon}$ is a diagonal matrix collecting the quasiparticle and satellite energies, 
where the 2h1p and 2p1h matrix elements are
\begin{subequations}
  \begin{align}
    C^\text{2h1p}_{i\nu,j\mu} & =  \left(\epsilon_i - \Omega_\nu\right)\delta_{ij}\delta_{\nu\mu},
    \\
    C^\text{2p1h}_{a\nu,b\mu} & = \left(\epsilon_a + \Omega_\nu\right)\delta_{ab}\delta_{\nu\mu},
  \end{align}
\end{subequations}
and the corresponding coupling blocks read [see Eq.~(\ref{eq:GW_sERI})]
\begin{align}
  W^\text{2h1p}_{p,i\nu} & = W_{p,i\nu},
  &
    W^\text{2p1h}_{p,a\nu} & = W_{p,a\nu}.
\end{align}
The usual $GW$ non-linear equation can be obtained by applying L\"owdin partitioning technique \cite{Lowdin_1963} to Eq.~\eqref{eq:GWlin} yielding \cite{Bintrim_2021} 
\begin{equation}
\begin{split}
  \bSig(\omega) 
  & = \bW^{\hhp} \qty(\omega \bI - \bC^{\hhp})^{-1}  (\bW^{\hhp})^\dag
  \\
  & + \bW^{\pph} \qty(\omega \bI - \bC^{\pph})^{-1}  (\bW^{\pph})^\dag, 
\end{split}
\end{equation}
which can be further developed to recover exactly Eq.~\eqref{eq:GW_selfenergy}.

Equations \eqref{eq:GWlin} and \eqref{eq:quasipart_eq} yield exactly the quasiparticle and satellite energies but one is linear and the other is not.
The price to pay for this linearity is that the size of the matrix in the former is $\order{K^3}$ while it is only $\order{K}$ in the latter.
We refer to Ref.~\onlinecite{Bintrim_2021} for a detailed discussion of the up/downfolding processes of the $GW$ equations (see also Refs.~\onlinecite{Tolle_2022,Scott_2023}).

As can be readily seen in Eq.~\eqref{eq:GWlin}, the blocks $\bW^\text{2h1p}$ and $\bW^\text{2p1h}$ are coupling the 1h and 1p configuration to the 2h1p and 2p1h configurations.
Therefore, it is natural to define, within the SRG formalism, the diagonal and off-diagonal parts of the $GW$ effective Hamiltonian as
\begin{subequations}
\begin{align}
  \label{eq:diag_and_offdiag}
  \bH^\text{d}(s) &= 
  \begin{pmatrix}
    \bF	&	\bO	                	&     \bO		\\
    \bO	&	\bC^{\text{2h1p}}		&	\bO			\\
    \bO	&	\bO				&	\bC^{\text{2p1h}}	\\
  \end{pmatrix},
  \\
    \bH^\text{od}(s)  &= 
  \begin{pmatrix}
    \bO			                &     \bW^{\text{2h1p}} &     \bW^{\text{2p1h}} \\
    (\bW^{\text{2h1p}})^\dag	&	\bO	   	        &	\bO                    	\\
    (\bW^{\text{2p1h}})^\dag	&	\bO		        &	\bO	                \\
  \end{pmatrix},
\end{align}
\end{subequations}
where we omit the $s$ dependence of the matrices for the sake of brevity.
Then, the aim is to solve, order by order, the flow equation \eqref{eq:flowEquation} knowing that the initial conditions are
\begin{subequations}
\begin{align}
  \bHd{0}(0) & =   \mqty( \bF & \bO \\ \bO & \bC ),
  & 
  \bHod{0}(0) & = \bO,
  \\
  \bHd{1}(0) & = \bO, 
  &
  \bHod{1}(0) & = \mqty( \bO & \bW \\ \bW^{\dagger} & \bO ),
\end{align}
\end{subequations}
where the supermatrices 
\begin{subequations}
\begin{align}
  \bC & = \mqty( \bC^{\text{2h1p}}  &  \bO  \\  \bO  &  \bC^{\text{2p1h}}  )
  \\
  \bW & = \mqty(  \bW^{\text{2h1p}} &  \bW^{\text{2p1h}} )
\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:GWlin} before applying the downfolding process to obtain a renormalized version of the quasiparticle equation.
In particular, we focus here on the second-order renormalized quasiparticle equation.

%///////////////////////////%
\subsection{Zeroth-order matrix elements}
% ///////////////////////////%

The choice of Wegner's generator in the flow equation [see Eq.~\eqref{eq:flowEquation}] implies that the off-diagonal correction is of order $\order*{\lambda}$ while the correction to the diagonal block is at least $\order*{\lambda^2}$.
Therefore, the zeroth-order Hamiltonian is independent of $s$ and we have
\begin{equation}
  \bH^{(0)}(s) = \bH^{(0)}(0).
\end{equation}

%///////////////////////////%
\subsection{First-order matrix elements}
%///////////////////////////%

Knowing that $\bHod{0}(s)=\bO$, the first-order flow equation is
\begin{equation}
  \dv{\bH^{(1)}}{s} = \comm{\comm{\bHd{0}}{\bHod{1}}}{\bHd{0}},
\end{equation}
which gives the following system of equations
\begin{align}
  \label{eq:F0_C0}
  \dv{\bF^{(0)}}{s}&=\bO,   &  \dv{\bC^{(0)}}{s}&=\bO,
\end{align}
and
\begin{multline}
  \label{eq:W1}
  \dv{\bW^{(1)}}{s}
   = 2 \bF^{(0)}\bW^{(1)}\bC^{(0)} 
   \\
    - (\bF^{(0)})^2\bW^{(1)} - \bW^{(1)}(\bC^{(0)})^2.
\end{multline}
%\begin{subequations}
%  \begin{align}
%    \label{eq:W1}
%    \begin{split}
%      \dv{\bW^{(1),\dagger}}{s}
%      & = 2 \bC^{(0)}\bW^{(1),\dagger}\bF^{(0)} 
%      \\
%      & - \bW^{(1),\dagger}(\bF^{(0)})^2 - (\bC^{(0)})^2\bW^{(1),\dagger},
%    \end{split}
%    \\
%    \begin{split}
%     \label{eq:W1dag}
%     \dv{\bW^{(1)}}{s}
%      & = 2 \bF^{(0)}\bW^{(1)}\bC^{(0)} 
%      \\
%      & - (\bF^{(0)})^2\bW^{(1)} - \bW^{(1)}(s)(\bC^{(0)})^2.
%    \end{split}
%  \end{align}
%\end{subequations}
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 \titou{diagonal structure of $\bF^{(0)}$} 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.
It is worth noting the close similarity of the first-order elements with the ones derived by Evangelista in Ref.~\onlinecite{Evangelista_2014b} \titou{in a different context} following a second quantization formalism (see also Ref.~\onlinecite{Hergert_2016}).

%///////////////////////////%
\subsection{Second-order matrix elements}
% ///////////////////////////%

The second-order renormalized quasiparticle equation is given by
\begin{equation}
  \label{eq:GW_renorm}
%  \qty[ \widetilde{\bF}(s) + \widetilde{\bSig}(\omega; s) ] \bX = \omega \bX,
  \qty[ \widetilde{\bF}(s) + \widetilde{\bSig}(\omega = \epsilon_p; s) ] \psi_p(\bx) = \epsilon_p \psi_p(\bx),
\end{equation}
with
\begin{subequations}
\begin{align}
  \widetilde{\bF}(s) &= \bF^{(0)}+\bF^{(2)}(s),\\
  \label{eq:srg_sigma}
  \widetilde{\bSig}(\omega; s) &= \bV^{(1)}(s) \left(\omega \bI - \bC^{(0)}\right)^{-1}  (\bV^{(1)}(s))^{\dagger}.
\end{align}
\end{subequations}
As can be readily seen above, $\bF^{(2)}$ is the only second-order block of the effective Hamiltonian contributing to the second-order SRG quasiparticle equation.
Collecting every second-order term in the flow equation and performing the block matrix products results in the following differential equation
\begin{multline}
  \label{eq:diffeqF2}
  \dv{\bF^{(2)}}{s}  = \bF^{(0)}\bW^{(1)}\bW^{(1),\dagger} + \bW^{(1)}\bW^{(1),\dagger}\bF^{(0)} \\ - 2 \bW^{(1)}\bC^{(0)}\bW^{(1),\dagger},
\end{multline}
which can be solved by simple integration along with the initial condition $\bF^{(2)}=\bO$ to give
\begin{multline}
  F_{pq}^{(2)}(s) =  \sum_{r\nu} \frac{\Delta_{pr\nu}+ \Delta_{qr\nu}}{\Delta_{pr\nu}^2 + \Delta_{qr\nu}^2} W_{p,r\nu} W^{\dagger}_{r\nu,q}  \\
  \times \qty[1 - e^{-(\Delta_{pr\nu}^2 + \Delta_{qr\nu}^2) s}].
\end{multline}

At $s=0$, the second-order correction vanishes while, for $s\to\infty$, it tends towards the following static limit
\begin{equation}
  \label{eq:static_F2}
  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.
\titou{Therefore, the SRG flow continuously transforms the dynamic part $\widetilde{\bSig}(\omega; s)$ into 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}.

%%% FIG 1 %%% 
\begin{figure*}
  \centering
  \includegraphics[width=\linewidth]{fig1.pdf}
  \caption{
    Functional form of the qs$GW$ self-energy (left) for $\eta = 1$ and the SRG-$GW$ self-energy (right) for $s = 1/(2\eta^2)$.
    \label{fig:fig1}}
\end{figure*}
%%% %%% %%% %%%

%///////////////////////////%
\subsection{Alternative form of the static self-energy}
% ///////////////////////////%

Because the $s\to\infty$ limit of Eq.~(\ref{eq:GW_renorm}) is purely static, it can be seen as a qs$GW$ calculation with an alternative static approximation than the usual one of Eq.~(\ref{eq:sym_qsgw}).
Unfortunately, as we shall discuss further in Sec.~\ref{sec:results}, in the $s\to\infty$ limit self-consistently solving the renormalized quasi-particle equation is once again quite difficult, if not impossible.
However, one can define a more flexible new static self-energy, which will be referred to as SRG-qs$GW$ in the following, by discarding the dynamic part in Eq.~(\ref{eq:GW_renorm}).
This yields a $s$-dependent static self-energy which matrix elements read
\begin{multline}
  \label{eq:SRG_qsGW}
  \Sigma_{pq}^{\text{SRG}}(s) = \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}  \\ \times \qty[(1 - e^{-(\Delta_{pr\nu}^2 + \Delta_{qr\nu}^2) s} ],
\end{multline}
Note that the SRG static approximation is naturally Hermitian as opposed to the usual case [see Eq.~(\ref{eq:sym_qsGW})] where it is enforced by symmetrization.
Another important difference is that the SRG regularizer is energy dependent while the imaginary shift is the same for every state.
Yet, these approximations are closely related because for $\eta=0$ and $s\to\infty$ they share the same diagonal terms.

It is well-known that in traditional qs$GW$ calculation increasing the imaginary shift $\ii\eta$ to ensure convergence in difficult cases is most often unavoidable.
Similarly, in SRG-qs$GW$ one might need to decrease the value of $s$ to ensure convergence.
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, one should use a value of $s$ large enough to include almost every state but small enough to avoid intruder states.

It is instructive to plot both regularizing functions, this is done in Fig.~\ant{???}.
The surfaces correspond to a value of the regularizing parameter value of 1.
The SRG surface is much smoother than its qs counterpart.
In fact the SRG regularization has less work to do because for $\eta=0$ there is a single singularity at $x=y=0$.
On the other hand the function $f_{\text{qs}}(x,y;0)$ is singular on two entire axis, $x=0$ and $y=0$.
The smoothness of the SRG surface might induce a smoother convergence of SRG-qs$GW$ compared to qs$GW$.
The convergence properties and the accuracy of both static approximations will be quantitatively gauged in the results section.

To conclude this section, the case of discontinuities will be briefly discussed.
Indeed, it has been previously mentioned that intruder states are responsible for both the poor convergence of qs$GW$ and discontinuities in physical quantities at the $\GOWO$ level.
So is it possible to use the SRG machinery developed above to remove discontinuities?
Not directly because discontinuities are due to intruder states in the dynamic part while we have seen just above that a finite value of $s$ is well-designed to avoid the intruder states in the static part.
However, doing a change of variable such that
\begin{align}
  e^{-s} &= 1-e^{-t} & 1 - e^{-s} &= e^{-t} 
\end{align}
will reverse the situation and now a finite value of $t$ will be well-designed to avoid discontinuities in the renormalized dynamic part.
The dynamic part after the change of variable is actually closely related to the SRG-inspired regularizer introduced by Monino and Loos in Ref.~\onlinecite{Monino_2022}.

%=================================================================%
\section{Computational details}
\label{sec:comp_det}
%=================================================================%

% 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}
All $GW$ calculations were performed without the frozen-core approximation.
The DIIS space size and the maximum of iterations were set to 5 and 64, respectively.
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}
\label{sec:results}
%=================================================================%

The results section is divided into two parts.
The first step will be to analyze 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 traditional and SRG schemes will be statistically gauged over a test set of molecules.

%%%%%%%%%%%%%%%%%%%%%%
\subsection{Flow parameter dependence of the SRG-qs$GW$ scheme}
\label{sec:flow_param_dep}
%%%%%%%%%%%%%%%%%%%%%% 

%%% FIG 1 %%%
\begin{figure}
  \centering
  \includegraphics[width=\linewidth]{fig2.pdf}
  \caption{
    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:fig2}}
\end{figure}
%%% %%% %%% %%%

%%% FIG 2 %%% 
\begin{figure*}
  \centering
  \includegraphics[width=\linewidth]{fig3.pdf}
  \caption{
    Add caption
    \label{fig:fig3}}
\end{figure*}
%%% %%% %%% %%%

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 the CCSD(T) reference value.
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.}
\PFL{Check Szabo\&Ostlund, section on Koopman's theorem.}
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.

Figure~\ref{fig:fig1} also displays the SRG-qs$GW$ IP as a function of the flow parameter (blue curve).
At $s=0$, the IP is equal to its HF counterpart as expected from the discussion of Sec.~\ref{sec:srggw}.
For $s\to\infty$, the IP reaches a plateau at an error that is significantly smaller than their $s=0$ starting point.
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 initial HF value and the $s\to\infty$ limit as for small $s$ values it is actually worst than the HF starting point.

This behavior as a function of $s$ can be approximately rationalized by applying matrix perturbation theory on Eq.~(\ref{eq:GWlin}).
Through second order in the coupling block, the principal IP is
\begin{equation}
  \label{eq:2nd_order_IP}
  I_k = - \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 in the absolute error.
This trend will be investigated in more detail in the next subsection.

Now the flow parameter dependence of the SRG-qs$GW$ method will be investigated in three less well-behaved molecular systems.
The left panel of Fig.~\ref{fig:fig2} shows the results for the Lithium dimer, \ce{Li2} is an interesting case because the HP IP is actually underestimated.
On the other hand, the qs-$GW$ and SRG-qs$GW$ IPs are overestimated
Indeed, we can see that the positive increase of the SRG-qs$GW$ IP is proportionally more important than for water.
In addition, the plateau is reached for larger values of $s$ in comparison to Fig.~\ref{fig:fig1}.
Both TDA results are worse than their RPA counterparts but in this case the SRG-qs$GW_\TDA$ is more accurate than the qs$GW_\TDA$.

Now turning to the lithium hydride heterodimer, see the middle panel of Fig.~\ref{fig:fig2}.
In this case, the qs$GW$ IP is actually worse than the HF one which is already pretty accurate.
However, the SRG-qs$GW$ can improve slightly the accuracy with respect to HF.
Finally, the beryllium oxide is considered a prototypical example of a molecular system difficult to converge because of intruder states. \cite{vanSetten_2015,Veril_2018,Forster_2021}
The SRG-qs$GW$ could be converged without any problem even for large values of $s$.
Once again, a plateau is attained and the corresponding value is slightly more accurate than its qs$GW$ counterpart.

Note that for \ce{LiH} and \ce{BeO} the TDA actually improves the accuracy compared to RPA-based qs$GW$ schemes.
However, as we will see in the next subsection these are just particular molecular systems and on average the RPA polarizability performs better than the TDA one.
Also, the SRG-qs$GW_\TDA$ is better than qs$GW_\TDA$ in the three cases of Fig.~\ref{fig:fig2} but this is the other way around.
Therefore, it seems that the effect of the TDA can not be systematically predicted.



%%%%%%%%%%%%%%%%%%%%%%
\subsection{Statistical analysis}
\label{sec:SRG_vs_Sym}
%%%%%%%%%%%%%%%%%%%%%%

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. \cite{vanSetten_2015}
In addition, the MgO and O3 molecules (which are part of GW100 as well) has been added to the test set because they are known to be difficult to be plagued by intruder states. \cite{vanSetten_2015,Forster_2021}

%=================================================================%
\section{Conclusion}
\label{sec:conclusion}
%=================================================================%

Here comes the conclusion.

%%%%%%%%%%%%%%%%%%%%%%%%
\acknowledgements{
The authors thank Francesco Evangelista for inspiring discussions.
This project has received funding from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme (Grant agreement No.~863481).}
%%%%%%%%%%%%%%%%%%%%%%%%

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section*{Data availability statement}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
The data that supports the findings of this study are available within the article.% and its supplementary material.

\appendix

%%%%%%%%%%%%%%%%%%%%%%
\section{Non-TDA $GW$  and $GW_{\text{TDHF}}$ equations}
\label{sec:nonTDA}
%%%%%%%%%%%%%%%%%%%%%%

The matrix elements of the $GW$ self-energy within the TDA are the same as in Eq.~\eqref{eq:GW_selfenergy} but the screened integrals are now defined as
\begin{equation}
  \label{eq:GWnonTDA_sERI}
  W_{p,q\nu} = \sum_{ia}\eri{pi}{qa}\qty( \bX_{\nu})_{ia},
\end{equation}
where $\bX$ is the eigenvector matrix of the TDA particle-hole dRPA problem obtained by setting $\bB= \bO$ in Eq.~\eqref{eq:full_dRPA}, \ie
\begin{equation}
  \label{eq:TDA_dRPA}
  \bA \bX = \bX \boldsymbol{\Omega}.
\end{equation}

Defining an unfold version of this equation that does not require a diagonalization of the RPA problem before unfolding is a tricky task (see supplementary material of Ref.~\onlinecite{Bintrim_2021}).
However, because we will eventually downfold again the upfolded matrix, we can use the following matrix \cite{Tolle_2022}
\begin{equation}
  \label{eq:nonTDA_upfold}
    \begin{pmatrix}
      \bF				&	\bW^{\text{2h1p}}		&	\bW^{\text{2p1h}}	\\
    (\bW^{\text{2h1p}})^{\mathrm{T}}	&	\bD^{\text{2h1p}}		&	\bO					\\
    (\bW^{\text{2p1h}})^{\mathrm{T}}	&	\bO				&	\bD^{\text{2p1h}}	\\
  \end{pmatrix}
  \cdot
  \begin{pmatrix}
    \bX	\\
    \bY^{\text{2h1p}}	\\
    \bY^{\text{2p1h}}	\\
  \end{pmatrix}
  =
  \begin{pmatrix}
    \bX	\\
    \bY^{\text{2h1p}}	\\
    \bY^{\text{2p1h}}	\\
  \end{pmatrix}
  \cdot
  \boldsymbol{\epsilon},
\end{equation}
which already depends on the screened integrals and therefore require the knowledge of the eigenvectors of the dRPA problem defined in Eq.~\eqref{eq:full_dRPA}.
Within the TDA the renormalized matrix elements have the same $s$ dependence as in the RPA case but the $s=0$ screened integrals $W_{p,q\nu}$ and eigenvalues $\Omega_\nu$ are replaced by the ones of Eq.~\eqref{eq:GWnonTDA_sERI} and \eqref{eq:TDA_dRPA}, respectively.

% %%%%%%%%%%%%%%%%%%%%%%
% \section{GF(2) equations \ant{NOT SURE THAT WE KEEP IT}}
% \label{sec:GF2}
% %%%%%%%%%%%%%%%%%%%%%%

% The GF($n$) formalism is defined such that the self-energy includes every diagram up to $n$-th order of M\"oller-Plesset perturbation theory.
% The matrix elements of its second-order version read as 
% \begin{align}
%   \label{eq:GF2_selfenergy}
%   \Sigma_{pq}^{\text{GF(2)}}(\omega) 
%   &=  \frac{1}{\sqrt{2}} \sum_{ija} \frac{\aeri{pa}{ij}\aeri{qa}{ij}}{\omega + \epsilon _a -\epsilon_i -\epsilon_j - \ii \eta} \\
%   &+  \frac{1}{\sqrt{2}} \sum_{iab} \frac{\aeri{pi}{ab}\aeri{qi}{ab}}{\omega + \epsilon _i -\epsilon_a -\epsilon_b + \ii \eta} 
% \end{align}
% This self-energy can be upfolded similarly to the $GW$ case and one obtain the following ``super-matrix''
% \begin{equation}
%   \label{eq:unfolded_matrice}
%   \bH =
%   \begin{pmatrix}
%     \bF			                        &	\bV^{\text{2h1p}}		&	\bV^{\text{2p1h}}	\\
%     (\bV^{\text{2h1p}})^{\mathsf{T}}	&	\bC^{\text{2h1p}}		&	\bO				\\
%     (\bV^{\text{2p1h}})^{\mathsf{T}}	&	\bO					&	\bC^{\text{2p1h}}	\\
%   \end{pmatrix}
% \end{equation}
% The expression of the coupling blocks $\bV{}{}$ and the diagonal blocks $\bC{}{}$ is given below.
%   \begin{align}
%     \label{eq:GF2_unfolded}
%     V^\text{2h1p}_{p,ija} & = \frac{1}{\sqrt{2}}\aeri{pa}{ij}
%     \\
%     V^\text{2p1h}_{p,iab} & = \frac{1}{\sqrt{2}}\aeri{pi}{ab} 
%     \\
%     C^\text{2h1p}_{ija,klc} & = \qty( \epsilon_i + \epsilon_j - \epsilon_a) \delta_{jl} \delta_{ac} \delta_{ik} 
%     \\
%     C^\text{2p1h}_{iab,kcd} & = \qty( \epsilon_a + \epsilon_b - \epsilon_i) \delta_{ik} \delta_{ac} \delta_{bd} 
%   \end{align}
%   Note that this matrix is exactly the ADC(2) matrix for charged excitations.
%   The fact that the integrals are not screened in GF(2) manifests itself in the fact that the $\bC$ matrices are already diagonal.

%   Applying the SRG formalism to this matrix is completely analog to the derivation exposed in the main text.
%   We only give the analytical expressions of the matrix elements needed for the second-order SRG-GF(2) quasiparticle equations.

%   \begin{equation}
%     (V^\text{2h1p}_{p,ija})^{(1)}(s) = \frac{1}{\sqrt{2}}\aeri{pa}{ij} e^{- (\epsilon_p + \epsilon_a - \epsilon_i - \epsilon_j)^2 s}
%   \end{equation}
%   \begin{equation}
%    (V^\text{2h1p}_{p,iab})^{(1)}(s) = \frac{1}{\sqrt{2}}\aeri{pi}{ab} e^{- (\epsilon_p + \epsilon_i - \epsilon_a - \epsilon_b)^2 s}
%  \end{equation}

% We define $ \Delta_{pq,rs} = \epsilon_p + \epsilon_q - \epsilon_r - \epsilon_s $
 
%  \begin{align}
%   F_{pq}^{(2)}(s) &= \sum_{ria} \frac{\epsilon_{p} + \epsilon_{q} - 2 (\epsilon_r \pm \Omega_v)}{(\epsilon_p - \epsilon_r \pm \Omega_v)^2 + (\epsilon_q - \epsilon_r \pm \Omega_v)^2} W_{p,(r,v)} \notag \\
%                   &\times W^{\dagger}_{(r,v),q}\left(1 - e^{-(\epsilon_p - \epsilon_r \pm \Omega_v)^2s}   e^{-(\epsilon_q - \epsilon_r \pm \Omega_v)^2s}\right).
% \end{align}


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\bibliography{SRGGW}
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\end{document}
