almost OK with letter

Manuscript/ufGW.bib

@@ -1,13 +1,52 @@
 %% This BibTeX bibliography file was created using BibDesk.
-%% http://bibdesk.sourceforge.net/
+%% https://bibdesk.sourceforge.io/
 
-%% Created for Pierre-Francois Loos at 2022-04-21 13:21:26 +0200 
+%% Created for Pierre-Francois Loos at 2022-04-24 15:40:25 +0200 
 
 
 %% Saved with string encoding Unicode (UTF-8) 
 
 
 
+@article{Pokhilko_2021a,
+	author = {Pokhilko,Pavel and Zgid,Dominika},
+	date-added = {2022-04-24 15:40:03 +0200},
+	date-modified = {2022-04-24 15:40:15 +0200},
+	doi = {10.1063/5.0055191},
+	journal = {J. Chem. Phys.},
+	number = {2},
+	pages = {024101},
+	title = {Interpretation of multiple solutions in fully iterative GF2 and GW schemes using local analysis of two-particle density matrices},
+	volume = {155},
+	year = {2021},
+	bdsk-url-1 = {https://doi.org/10.1063/5.0055191}}
+
+@article{Pokhilko_2021b,
+	author = {Pokhilko,Pavel and Iskakov,Sergei and Yeh,Chia-Nan and Zgid,Dominika},
+	date-added = {2022-04-24 15:38:35 +0200},
+	date-modified = {2022-04-24 15:39:02 +0200},
+	doi = {10.1063/5.0054661},
+	journal = {J. Chem. Phys.},
+	number = {2},
+	pages = {024119},
+	title = {Evaluation of two-particle properties within finite-temperature self-consistent one-particle Green's function methods: Theory and application to GW and GF2},
+	volume = {155},
+	year = {2021},
+	bdsk-url-1 = {https://doi.org/10.1063/5.0054661}}
+
+@article{Pokhilko_2022,
+	author = {Pokhilko,Pavel and Yeh,Chia-Nan and Zgid,Dominika},
+	date-added = {2022-04-24 15:35:30 +0200},
+	date-modified = {2022-04-24 15:35:48 +0200},
+	doi = {10.1063/5.0082586},
+	journal = {J. Chem. Phys.},
+	number = {9},
+	pages = {094101},
+	title = {Iterative subspace algorithms for finite-temperature solution of Dyson equation},
+	volume = {156},
+	year = {2022},
+	bdsk-url-1 = {https://doi.org/10.1063/5.0082586}}
+
 @article{Hedin_1999,
 	abstract = {The GW approximation (GWA) extends the well-known Hartree-Fock approximation (HFA) for the self-energy (exchange potential), by replacing the bare Coulomb potential v by the dynamically screened potential W, e.g. Vex = iGv is replaced by GW = iGW. Here G is the one-electron Green's function. The GWA like the HFA is self-consistent, which allows for solutions beyond perturbation theory, like say spin-density waves. In a first approximation, iGW is a sum of a statically screened exchange potential plus a Coulomb hole (equal to the electrostatic energy associated with the charge pushed away around a given electron). The Coulomb hole part is larger in magnitude, but the two parts give comparable contributions to the dispersion of the quasi-particle energy. The GWA can be said to describe an electronic polaron (an electron surrounded by an electronic polarization cloud), which has great similarities to the ordinary polaron (an electron surrounded by a cloud of phonons). The dynamical screening adds new crucial features beyond the HFA. With the GWA not only bandstructures but also spectral functions can be calculated, as well as charge densities, momentum distributions, and total energies. We will discuss the ideas behind the GWA, and generalizations which are necessary to improve on the rather poor GWA satellite structures in the spectral functions. We will further extend the GWA approach to fully describe spectroscopies like photoemission, x-ray absorption, and electron scattering. Finally we will comment on the relation between the GWA and theories for strongly correlated electronic systems. In collecting the material for this review, a number of new results and perspectives became apparent, which have not been published elsewhere.},
 	author = {Lars Hedin},

Manuscript/ufGW.tex

@@ -153,9 +153,9 @@ Electron correlation is then explicitly incorporated into one-body quantities vi
 In recent studies, \cite{Loos_2018b,Veril_2018,Loos_2020e,Berger_2021,DiSabatino_2021} we discovered that one can observe (unphysical) irregularities and/or discontinuities in the energy surfaces of several key quantities (ionization potential, electron affinity, fundamental and optical gaps, total and correlation energies, as well as excitation energies) even in the weakly-correlated regime. 
 These issues were discovered in Ref.~\onlinecite{Loos_2018b} while studying a model two-electron system \cite{Seidl_2007,Loos_2009a,Loos_2009c} and they were further investigated in Ref.~\onlinecite{Veril_2018}, where we provided additional evidences and explanations of these undesirable features in real molecular systems.
 In particular, we showed that each branch of the self-energy $\Sigma$ is associated with a distinct quasiparticle solution, and that each switch between solutions implies a significant discontinuity in the quasiparticle energy due to the transfer of weight between two solutions of the quasiparticle equation. \cite{Veril_2018}
-Multiple solution issues in $GW$ appears frequently, \cite{vanSetten_2015,Maggio_2017,Duchemin_2020} especially for orbitals that are energetically far from the Fermi level, such as in core ionized states. \cite{Golze_2018,Golze_2020}
+Multiple solution issues in $GW$ appears frequently \cite{vanSetten_2015,Maggio_2017,Duchemin_2020} \alert{(even at finite temperature \cite{Pokhilko_2021a,Pokhilko_2021b})}, especially for orbitals that are energetically far from the Fermi level, such as in core ionized states. \cite{Golze_2018,Golze_2020} and finite-temperature scheme. 
 
-In addition to obvious irregularities in potential energy surfaces that hampers the accurate determination of properties such as equilibrium bond lengths and harmonic vibrational frequencies, \cite{Loos_2020e,Berger_2021} one direct consequence of these discontinuities is the difficulty to converge (partially) self-consistent $GW$ calculations as the self-consistent procedure jumps erratically from one solution to the other even if convergence accelerator techniques such as DIIS are employed. \cite{Pulay_1980,Pulay_1982,Veril_2018}
+In addition to obvious irregularities in potential energy surfaces that hampers the accurate determination of properties such as equilibrium bond lengths and harmonic vibrational frequencies, \cite{Loos_2020e,Berger_2021} one direct consequence of these discontinuities is the difficulty to converge (partially) self-consistent $GW$ calculations as the self-consistent procedure jumps erratically from one solution to the other \alert{even if convergence accelerator techniques such as DIIS \cite{Pulay_1980,Pulay_1982,Veril_2018} or more elaborate schemes  \cite{Pokhilko_2022} are employed.}
 Note in passing that the present issues do not only appear in $GW$ as the $T$-matrix \cite{Romaniello_2012,Zhang_2017,Li_2021b,Loos_2022} and second-order Green's function (or second Born) formalisms \cite{SzaboBook,Casida_1989,Casida_1991,Stefanucci_2013,Ortiz_2013,Phillips_2014,Phillips_2015,Rusakov_2014,Rusakov_2016,Hirata_2015,Hirata_2017} exhibit the same drawbacks.
 
 It was shown that these problems can be tamed by using a static Coulomb-hole plus screened-exchange (COHSEX) \cite{Hedin_1965,Hybertsen_1986,Hedin_1999,Bruneval_2006} self-energy \cite{Berger_2021} or by considering a fully self-consistent $GW$ scheme, \cite{Stan_2006,Stan_2009,Rostgaard_2010,Caruso_2012,Caruso_2013,Caruso_2013a,Caruso_2013b,Koval_2014,Wilhelm_2018} where one considers not only the quasiparticle solution but also the satellites at each iteration. \cite{DiSabatino_2021}

Response_Letter/Response_Letter.tex

@@ -16,7 +16,7 @@ Please find attached a revised version of the manuscript entitled
 \begin{quote}
 	\textit{``Unphysical Discontinuities, Intruder States and Regularization in $GW$ Methods''}.
 \end{quote}
-We thank the reviewers for their constructive comments.
+We thank the reviewers for their constructive comments and to support publication of the present manuscript.
 Our detailed responses to their comments can be found below.
 For convenience, changes are highlighted in red in the revised version of the manuscript. 
 
@@ -48,17 +48,18 @@ In such cases, the potential energy curves are smooth within a region where one
 The reviewer is correct; these issues are more general and do not originate solely from the vanishing denominator.
 For example, we have recently studied multiple solutions in another non-linear method, coupled-cluster theory [J. Chem. Theory Comput. 17, 4756 (2021)], where their origin stems from the non-linear nature of the equations, only.
 However, in the case of $GW$ at least, we have clearly observed that these multiple solution issues appear when denominators vanish.
-More precisely, in most $GW$ methods, because one only considers the quasiparticle solution at each iteration while discarding the other solutions (known as satellites, discontinuities are observed. 
+More precisely, in most $GW$ methods, because one only considers the quasiparticle solution at each iteration while discarding the other solutions (known as satellites), discontinuities are observed. 
 This has been studied in details in a previous study [J. Chem. Theory Comput. 14, 5220 (2018)].
-In a fully self-consistent scheme where one takes into account all these solutions at each iteration, these issues do not appear, as we have recently discussed in a recent paper [Front. Chem. (9, 751054 (2021)]. 
+In a fully self-consistent scheme where one takes into account all these solutions at each iteration, these issues do not appear, as we have recently discussed in a recent paper [Front. Chem. 9, 751054 (2021)]. 
 The references provided by the reviewer has been added in due place and this point has been clarified in the revised version of the manuscript.}
 
 {Convergence accelerators such as DIIS, KAIN, LCIS, can be successfully used and smooth PES can be produced when one is relatively close to a local minima, one should not expect that DIIS will help with convergence where multiple close lying solutions exists and when two solutions are competing, see J. Chem. Phys. 156, 094101 (2022).}
 \\
 \alert{
 Indeed, convergence accelerators such as DIIS can be used to ease convergence but they will not make these discontinuities disappear as their origin is more profond.
-This particular case is discussed in a recent paper [J. Chem. Theory Comput. 14, 5220 (2018)] where we have provided our implementation of DIIS within $GW$ methods (see Appendix).
-This point has been further stressed in the revised version of the manuscript.}
+This point was already stressed in our original manuscript.
+This particular case is further discussed in a recent paper [J. Chem. Theory Comput. 14, 5220 (2018)] where we have provided our implementation of DIIS within $GW$ methods (see Appendix).
+This point has been clearified in the revised version of the manuscript where we have also added the reference provided by Reviewer \#1.}
 
 
 {It is my understanding that the calculation on the illustrative example H2 in 6-31G basis illustrates the existence of multiple such solutions. 
@@ -68,7 +69,7 @@ Frankly, I find it infuriating that in the GW community many times it is not cle
 }
 \\
 \alert{We agree with the reviewer that, historically, the literature has been unclear on how to solve the $GW$ equations, but we would like to stress that our group has made a clear effort to provide all the working equations and details necessary to solve these equations in the different cases.
-We have added additional details and references in order to guide the readers to the relevant papers when one can find all the necessary details regarding the implementation of $GW$ methods.}
+We have added all the required details and references in order to guide the readers to the relevant papers when one can find all the necessary details regarding the implementation of $GW$ methods.}
 
 {Here, I will describe more specific issues that I noticed during the reading of the manuscript:}
 \\