revised manuscript and letter

CCvsMBPT.bib

@@ -1,13 +1,113 @@
 %% This BibTeX bibliography file was created using BibDesk.
-%% http://bibdesk.sourceforge.net/
+%% https://bibdesk.sourceforge.io/
 
-%% Created for Pierre-Francois Loos at 2022-11-16 16:20:18 +0100 
+%% Created for Pierre-Francois Loos at 2022-11-16 22:07:19 +0100 
 
 
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+@article{Damour_2021,
+	author = {Damour, Yann and V{\'{e}}ril, Micka{\"{e}}l and Kossoski, F{\'{a}}bris and Caffarel, Michel and Jacquemin, Denis and Scemama, Anthony and Loos, Pierre-Fran{\c{c}}ois},
+	date-added = {2022-11-16 22:06:07 +0100},
+	date-modified = {2022-11-16 22:06:07 +0100},
+	doi = {10.1063/5.0065314},
+	issn = {0021-9606},
+	journal = {J. Chem. Phys.},
+	number = {13},
+	pages = {134104},
+	publisher = {AIP Publishing, LLC},
+	title = {{Accurate full configuration interaction correlation energy estimates for five- and six-membered rings}},
+	url = {https://doi.org/10.1063/5.0065314},
+	volume = {155},
+	year = {2021},
+	bdsk-url-1 = {https://doi.org/10.1063/5.0065314}}
+
+@article{Kossoski_2021,
+	author = {Kossoski, F{\'a}bris and Marie, Antoine and Scemama, Anthony and Caffarel, Michel and Loos, Pierre-Fran{\c c}ois},
+	date-added = {2022-11-16 22:06:07 +0100},
+	date-modified = {2022-11-16 22:06:07 +0100},
+	doi = {10.1021/acs.jctc.1c00348},
+	journal = {J. Chem. Theory Comput.},
+	pages = {4756},
+	title = {Excited States from State-Specific Orbital-Optimized Pair Coupled Cluster},
+	volume = {17},
+	year = {2021},
+	bdsk-url-1 = {https://doi.org/10.1021/acs.jctc.1c00348}}
+
+@article{Kossoski_2022,
+	author = {Kossoski, F{\'a}bris and Damour, Yann and Loos, Pierre-Fran{\c c}ois},
+	date-added = {2022-11-16 22:06:07 +0100},
+	date-modified = {2022-11-16 22:06:07 +0100},
+	doi = {10.1021/acs.jpclett.2c00730},
+	eprint = {https://doi.org/10.1021/acs.jpclett.2c00730},
+	journal = {J. Phys. Chem. Lett.},
+	number = {19},
+	pages = {4342-4349},
+	title = {Hierarchy Configuration Interaction: Combining Seniority Number and Excitation Degree},
+	url = {https://doi.org/10.1021/acs.jpclett.2c00730},
+	volume = {13},
+	year = {2022},
+	bdsk-url-1 = {https://doi.org/10.1021/acs.jpclett.2c00730}}
+
+@article{Marie_2021b,
+	author = {Marie,Antoine and Kossoski,F{\'a}bris and Loos,Pierre-Fran{\c c}ois},
+	date-added = {2022-11-16 22:06:07 +0100},
+	date-modified = {2022-11-16 22:06:43 +0100},
+	doi = {10.1063/5.0060698},
+	journal = {J. Chem. Phys.},
+	number = {10},
+	pages = {104105},
+	title = {Variational coupled cluster for ground and excited states},
+	volume = {155},
+	year = {2021},
+	bdsk-url-1 = {https://doi.org/10.1063/5.0060698}}
+
+@article{Lee_2019,
+	author = {Lee, Joonho and Small, David W. and {Head-Gordon}, Martin},
+	date-added = {2022-11-16 22:05:49 +0100},
+	date-modified = {2022-11-16 22:05:49 +0100},
+	doi = {10.1063/1.5128795},
+	file = {/home/antoinem/Zotero/storage/2F6799N9/Lee et al. - 2019 - Excited states via coupled cluster theory without .pdf},
+	journal = {J. Chem. Phys.},
+	pages = {214103},
+	publisher = {{American Institute of Physics}},
+	title = {Excited States via Coupled Cluster Theory without Equation-of-Motion Methods: {{Seeking}} Higher Roots with Application to Doubly Excited States and Double Core Hole States},
+	volume = {151},
+	year = {2019},
+	bdsk-url-1 = {https://doi.org/10.1063/1.5128795}}
+
+@article{Mayhall_2010,
+	author = {Mayhall, Nicholas J. and Raghavachari, Krishnan},
+	date-added = {2022-11-16 22:05:38 +0100},
+	date-modified = {2022-11-16 22:05:38 +0100},
+	doi = {10.1021/ct100321k},
+	file = {/home/antoinem/Zotero/storage/YDLY5ZWI/Mayhall and Raghavachari - 2010 - Multiple Solutions to the Single-Reference CCSD Eq.pdf;/home/antoinem/Zotero/storage/2YP25YXD/ct100321k.html},
+	journal = {J. Chem. Theory Comput.},
+	pages = {2714--2720},
+	publisher = {{American Chemical Society}},
+	title = {Multiple {{Solutions}} to the {{Single}}-{{Reference CCSD Equations}} for {{NiH}}},
+	volume = {6},
+	year = {2010},
+	bdsk-url-1 = {https://doi.org/10.1021/ct100321k}}
+
+@incollection{Piecuch_2000,
+	author = {Piecuch, Piotr and Kowalski, Karol},
+	booktitle = {Computational {{Chemistry}}: {{Reviews}} of {{Current Trends}}},
+	date-added = {2022-11-16 22:05:31 +0100},
+	date-modified = {2022-11-16 22:05:31 +0100},
+	doi = {10.1142/9789812792501_0001},
+	file = {/home/antoinem/Zotero/storage/6TRIL3LE/9789812792501_0001.html},
+	isbn = {978-981-02-4371-5},
+	pages = {1--104},
+	publisher = {{WORLD SCIENTIFIC}},
+	series = {Computational {{Chemistry}}: {{Reviews}} of {{Current Trends}}},
+	title = {In {{Search}} of the {{Relationship}} between {{Multiple Solutions Characterizing Coupled}}-{{Cluster Theories}}},
+	volume = {Volume 5},
+	year = {2000},
+	bdsk-url-1 = {https://doi.org/10.1142/9789812792501_0001}}
+
 @inbook{Bartlett_1986,
 	abstract = {A diagrammatic derivation of the coupled-cluster (CC) linear response equations for gradients is presented. MBPT approximations emerge as low-order iterations of the CC equations. In CC theory a knowledge of the change in cluster amplitudes with displacement is required, which would not be necessary if the coefficients were variationally optimum, as in the CI approach. However, it is shown that the CC linear response equations can be put in a form where there is no more difficulty in evaluating CC gradients than in the variational CI procedure. This offers a powerful approach for identifying critical points on energy surfaces and in evaluating other properties than the energy.},
 	address = {Dordrecht},
@@ -609,18 +709,18 @@
 	year = 1999,
 	bdsk-url-1 = {https://doi.org/10.1088/0953-8984/11/42/201}}
 
-@article{Marie_2021,
+@article{Marie_2021a,
 	abstract = {We explore the non-Hermitian extension of quantum chemistry in the complex plane and its link with perturbation theory. We observe that the physics of a quantum system is intimately connected to the position of complex-valued energy singularities, known as exceptional points. After presenting the fundamental concepts of non-Hermitian quantum chemistry in the complex plane, including the mean-field Hartree--Fock approximation and Rayleigh--Schr{\"o}dinger perturbation theory, we provide a historical overview of the various research activities that have been performed on the physics of singularities. In particular, we highlight seminal work on the convergence behaviour of perturbative series obtained within M{\o}ller--Plesset perturbation theory, and its links with quantum phase transitions. We also discuss several resummation techniques (such as Pad{\'e} and quadratic approximants) that can improve the overall accuracy of the M{\o}ller--Plesset perturbative series in both convergent and divergent cases. Each of these points is illustrated using the Hubbard dimer at half filling, which proves to be a versatile model for understanding the subtlety of analytically-continued perturbation theory in the complex plane.},
 	author = {Antoine Marie and Hugh G A Burton and Pierre-Fran{\c{c}}ois Loos},
 	date-added = {2022-02-24 14:47:18 +0100},
-	date-modified = {2022-02-24 14:47:49 +0100},
+	date-modified = {2022-11-16 22:06:38 +0100},
 	doi = {10.1088/1361-648x/abe795},
 	journal = {J. Phys. Condens. Matter},
 	number = {28},
 	pages = {283001},
 	title = {Perturbation theory in the complex plane: exceptional points and where to find them},
 	volume = {33},
-	year = 2021,
+	year = {2021},
 	bdsk-url-1 = {https://doi.org/10.1088/1361-648x/abe795}}
 
 @article{Rostam_2017,

CCvsMBPT.tex

@@ -609,8 +609,10 @@ with $\Delta_{ija,p}^{\text{2h1p}} = \e{i}{} + \e{j}{} - \e{a}{} - \e{p}{}$ and
 To determine the 2h1p and 2p1h amplitudes, $t_{ija,p}^{\text{2h1p}}$ and $t_{iab,p}^{\text{2p1h}} $, one can then rely on the usual quasi-Newton iterative procedure to solve these quadratic equations by updating the amplitudes via
 \begin{subequations}
 \begin{align}
+	\label{eq:t_2h1p_update}
 	t_{ija,p}^{\text{2h1p}} & \leftarrow t_{ija,p}^{\text{2h1p}} - \qty( \Delta_{ija,p}^{\text{2h1p}} )^{-1} r_{ija,p}^{\text{2h1p}}
 	\\
+	\label{eq:t_2p1h_update}
 	t_{iab,p}^{\text{2p1h}} & \leftarrow t_{iab,p}^{\text{2p1h}} - \qty( \Delta_{iab,p}^{\text{2p1h}} )^{-1} r_{iab,p}^{\text{2p1h}}
 \end{align}
 \end{subequations}
@@ -619,6 +621,9 @@ The quasiparticle energies $\e{p}{GW}$ are thus provided by the eigenvalues of $
 \begin{equation}
 	\bSig{}{\GW} = \bV{}{\text{2h1p}} \cdot \bT{}{\text{2h1p}} + \bV{}{\text{2p1h}} \cdot \bT{}{\text{2p1h}} 
 \end{equation}
+\alert{Due to the non-linear nature of these equations, the iterative procedure proposed in Eqs.~\eqref{eq:t_2h1p_update} and \eqref{eq:t_2p1h_update} can potentially converge to satellite solutions. 
+This is also the case at the CC level when one relies on more elaborated algorithm to converge the amplitude equations to higher-energy solutions. \cite{Piecuch_2000,Mayhall_2010,Lee_2019,Kossoski_2021,Marie_2021b}}
+
 Again, similarly to the dynamical equations defined in Eq.~\eqref{eq:GW} which requires the diagonalization of the dRPA eigenproblem [see Eq.~\eqref{eq:dRPA}], the CC equations reported in Eqs.~\eqref{eq:r_2h1p} and \eqref{eq:r_2p1h} can be solved with $\order*{N^6}$ cost by defining judicious intermediates.
 Cholesky decomposition, density fitting, and other related techniques may be employed to further reduce this scaling as it is done in conventional $GW$ calculations. \cite{Bintrim_2021,Forster_2020,Forster_2021,Duchemin_2019,Duchemin_2020,Duchemin_2021}
 The $G_0W_0$ quasiparticle energies can be easily obtained via the procedure described in Ref.~\onlinecite{Monino_2022} by solving the previous equations for each value of $p$ separately.

Response_Letter/Response_Letter.tex

@@ -128,9 +128,8 @@ However, I reemphasize that in both cases, the ``CC-like'' equations that result
 }
 \\
 \alert{
-We believe that the connection between $GW$ and $CC$ is more profound and this is nicely illustrated by the work of Lange and Berkelbach [see Ref.~
+The reviewer is right in some aspects but we believe that the connection between $GW$ and CC is more profound and this is nicely illustrated by the work of Lange and Berkelbach [see Ref.~(72)], where they show diagrammatic connections between $GW$ and approximate EOM-CCD.
-This nonlinear equation has more similarities between drCCD and dRPA that Scuseria already described in Ref[44]. 
+}
-As it is discussed in the conclusion : If we consider that this non linear  eigenvalue problem comes from MBPT there is a conection with CC at the ground and excited state level.}
 
 \item 
 {A confusion I have in my own discussion above is related to a question I have about the final Eq.~(34) of the present work. 
@@ -144,7 +143,7 @@ If the authors appreciate this confusion, then some discussion in the manuscript
 This is explicitly stated before Eq.~(29) as ``Let us suppose that we are looking for the $N$ ``principal'' (i.e., quasiparticle) solutions of the eigensystem (26).''
 However, we understand the confusion of the reviewer.
 Let us be more precise.
-The iterative algorithm proposed here [see Eqs.~(33)] could potentially converge to satellite solutions, and this is also the case at the CC level when one relies on a Newton-Raphson algorithm to converge to higher-energy solutions by modifying the updating procedure in Eqs.~(33) [see JCTC 17 (2021) 4756. ]. 
+The iterative algorithm proposed here [see Eqs.~(33)] could potentially converge to satellite solutions, and this is also the case at the CC level when one relies on a Newton-Raphson algorithm to converge to higher-energy solutions by modifying the updating procedure in Eqs.~(33) [see JCTC 17 (2021) 4756 and references therein].
 We discuss further this point below Eq.~(34).
 }