modifs in Sec I

Manuscript/SRGGW.tex

@@ -114,7 +114,7 @@ Approximating $\Sigma$ as the first-order term of its perturbative expansion wit
 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}
+For example, modeling core-electron spectroscopy requires core ionization energies which have been proven to be challenging for routine $GW$ calculations. \cite{vanSetten_2018,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}
@@ -128,7 +128,7 @@ The simpler one-shot $G_0W_0$ scheme \cite{Strinati_1980,Hybertsen_1985a,Hyberts
 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 $GW$.
+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 in $GW$-based methods.
 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.
@@ -146,13 +146,14 @@ correlation effects between the internal and external spaces can be incorporated
 
 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}
+Note that throughout the manuscript we focus on the $GW$ approximation but the subsequent derivations can be straightforwardly applied to other self-energies such as the one derived from the 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 second Born) 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} (or Bethe-Goldstone) approximations.
 
 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}.
+We begin by reviewing the $GW$ approximation in Sec.~\ref{sec:gw} and then briefly introduce 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
+Our conclusions are drawn in Sec.~\ref{sec:conclusion}.
+Unless otherwise stated, atomic units are used throughout.
 
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 %\section{Theoretical background}