cover letter done

Cover_Letter/CoverLetter.tex

@@ -15,10 +15,16 @@
 Please find enclosed our manuscript entitled \textit{``Unphysical Discontinuities, Intruder States and Regularization in GW Methods''}, which we would like you to consider as a Communication in the \textit{Journal of Chemical Physics}.
 This contribution has never been submitted in total nor in parts to any other journal, and has been seen and approved by both authors.
 
+Originally developed in the framework of nuclear physics and popularized in condensed-matter physics, one of the new emerging method in the computational chemistry landscape is the $GW$ approximation of many-body perturbation theory which allows to compute accurate charged excitation (i.e., ionization potentials, electron affinities and fundamental gaps) in solids and molecules.
 
-Because of the important new insights provided by this work and its potential impact in quantum chemistry and condensed matter physics, we expect it to be of interest to a wide audience within the chemistry and physics communities.
+In the present Communication, thanks to an upfolding process of the non-linear $GW$ equation, we provide convincing and clear physical and mathematical explanations behind the appearance of multiple solutions and unphysical discontinuities in key experimentally measurable properties computed within the $GW$ approximation, an issue that hamper the applicability of Green's function methods in the molecular electronic structure community and beyond.
+More precisely, we evidence that intruder states are the main cause behind these issues and that this downfall of $GW$ is a key signature of strong correlation. 
+A simple and efficient regularization procedure inspired by the similarity renormalization group is proposed to remove these discontinuities. 
+As a byproduct, this regularization of the self-energy significantly speeds up the convergence of (partially) self-consistent $GW$ methods.
+
+Because of the importance of the new insights and developments provided by this work and their potential impact in quantum chemistry and condensed matter physics, we expect it to be of interest to a wide audience within the chemistry and physics communities.
 We suggest Timothy Berkelbach, Francesco Evangelista, George Booth, Fabien Bruneval, Patrick Rinke, and Lucia Reining as potential referees.
-We look forward to hearing from you soon.
+We look forward to hearing from you.
 
 \closing{Sincerely, the authors.}
 

Manuscript/ufGW.tex

@@ -163,7 +163,7 @@ However, none of these solutions is completely satisfying as a static approximat
 In the present article, via an upfolding process of the non-linear $GW$ equation, \cite{Bintrim_2021a} we provide further physical insights into the origin of these discontinuities by highlighting, in particular, the role of intruder states.
 Inspired by regularized electronic structure theories, \cite{Lee_2018a,Evangelista_2014b} these new insights allow us to propose a cheap and efficient regularization scheme in order to avoid these issues and speed up convergence of partially self-consistent $GW$ calculations.
 
-Here, we consider the one-shot {\GOWO} \cite{Strinati_1980,Hybertsen_1985a,Hybertsen_1986,Godby_1988,Linden_1988,Northrup_1991,Blase_1994,Rohlfing_1995,Shishkin_2007} for the sake of simplicity but the same analysis can be performed in the case of (partially) self-consistent schemes such as ev$GW$, \cite{Hybertsen_1986,Shishkin_2007,Blase_2011,Faber_2011,Rangel_2016} where one updates only the quasiparticle energies, and qs$GW$, \cite{Gui_2018,Faleev_2004,vanSchilfgaarde_2006,Kotani_2007,Ke_2011,Kaplan_2016} where both quasiparticle energies and orbitals are updated at each iteration. 
+Here, for the sake of simplicity, we consider the one-shot {\GOWO} \cite{Strinati_1980,Hybertsen_1985a,Hybertsen_1986,Godby_1988,Linden_1988,Northrup_1991,Blase_1994,Rohlfing_1995,Shishkin_2007} but the same analysis can be performed in the case of (partially) self-consistent schemes such as ev$GW$ \cite{Hybertsen_1986,Shishkin_2007,Blase_2011,Faber_2011,Rangel_2016} (where one updates only the quasiparticle energies) and qs$GW$ \cite{Gui_2018,Faleev_2004,vanSchilfgaarde_2006,Kotani_2007,Ke_2011,Kaplan_2016} (where both quasiparticle energies and orbitals are updated at each iteration). 
 Moreover, we consider a Hartree-Fock (HF) starting point but it can be straightforwardly extended to a Kohn-Sham starting point.
 Throughout this article, $p$ and $q$ are general (spatial) orbitals, $i$, $j$, $k$, and $l$ denotes occupied orbitals, $a$, $b$, $c$, and $d$ are vacant orbitals, while $m$ labels single excitations $i \to a$.
 Atomic units are used throughout.