first complete draft; still some cleanup to do

CCvsMBPT.bib

@@ -1,7 +1,7 @@
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-%% Created for Pierre-Francois Loos at 2022-10-11 10:48:48 +0200 
+%% Created for Pierre-Francois Loos at 2022-10-11 11:00:19 +0200 
 
 
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CCvsMBPT.tex

@@ -178,9 +178,7 @@ BSE@$GW$ has been shown to be highly successful to compute low-lying excited sta
 \section{Connection between RPA and CC}
 \label{sec:RPAx}
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-Interestingly, RPA has strong connections with coupled-cluster (CC) theory, \cite{Freeman_1977,Scuseria_2008,Jansen_2010,Scuseria_2013,Peng_2013,Berkelbach_2018,Rishi_2020} the workhorse of molecular electronic structure when one is looking for high accuracy. \cite{Cizek_1966,Paldus_1972,Crawford_2000,Piecuch_2002,Bartlett_2007,Shavitt_2009} %At low orders, \eg, coupled-cluster with singles and doubles, CC can treat weakly correlated systems, while when one can afford to ramp up the excitation degree to triplets or quadruples, even strongly correlated systems can be treated, hence eschewing to resort to multireference methods.
-%Link between BSE and STEOM-CC.
-%A route towards the obtention of BSE gradients for ground and excited states.
+Interestingly, RPA has strong connections with coupled-cluster (CC) theory, \cite{Freeman_1977,Scuseria_2008,Jansen_2010,Scuseria_2013,Peng_2013,Berkelbach_2018,Rishi_2020} the workhorse of molecular electronic structure when one is looking for high accuracy. \cite{Cizek_1966,Paldus_1972,Crawford_2000,Piecuch_2002,Bartlett_2007,Shavitt_2009} 
 
 In a landmark paper, Scuseria \textit{et al.} \cite{Scuseria_2008} have proven that ring CC with doubles (rCCD) is equivalent to RPA with exchange (RPAx) for the computation of the correlation energy, which solidifies the numerical evidences provided by Freeman many years before. \cite{Freeman_1977}
 Assuming the existence of $\bX{}{-1}$ (which can be proven as long as the RPAx problem is stable \cite{Scuseria_2008}), this proof can be quickly summarized starting from the RPAx linear eigensystem
@@ -285,7 +283,7 @@ and matches the rCCD correlation energy
 because $\Tr(\bOm{}{} - \bA{}{}) = \Tr(\bR{}{} - \bA{}{}) = \Tr(\bB{}{} \cdot \bT{}{})$, as evidenced by Eq.~\eqref{eq:RPA_1}.
 Note that, in the case of RPAx, the same expression as Eq.~\eqref{eq:EcRPAx} can be derived from the adiabatic connection fluctuation dissipation theorem \cite{Furche_2005} (ACFDT) when exchange is included in the interaction kernel. \cite{Angyan_2011}
 
-This simple and elegant proof was subsequently extended to excitation energies by Berkerbach, \cite{Berkelbach_2018} who showed that similitudes between equation-of-motion (EOM) rCCD (EOM-rCCD) and RPAx exist when the EOM space is restricted to the 1h1p configurations and only the two-body terms are dressed by rCCD correlation.
+This simple and elegant proof was subsequently extended to excitation energies by Berkelbach, \cite{Berkelbach_2018} who showed that similitudes between equation-of-motion (EOM) rCCD (EOM-rCCD) and RPAx exist when the EOM space is restricted to the 1h1p configurations and only the two-body terms are dressed by rCCD correlation.
 
 To be more specific, restricting ourselves to CCD, \ie, $\hT = \hT_2$, the elements of the 1h1p block of the EOM Hamiltonian read \cite{Stanton_1993}
 \begin{equation}
@@ -439,6 +437,7 @@ The dynamical version of BSE [where the BSE kernel is explicitly treated as freq
 
 Because $GW$ is able to capture key correlation effects as illustrated above, it is therefore interesting to investigate if it is also possible to recast the $GW$ equations as a set of CC-like equations that can be solved iteratively using the CC machinery.
 Connections between approximate IP/EA-EOM-CC schemes and the $GW$ approximation have been already studied in details by Lange and Berkelbach, \cite{Lange_2018} but we believe that the present work proposes a different perspective on this particular subject as we derive genuine CC equations that do not decouple the 2h1p and 2p1h sectors.
+Note also that the procedure described below can be applied to other approximate self-energies such as the second-order Green's function (or second Born) \cite{Stefanucci_2013,Ortiz_2013,Phillips_2014,Rusakov_2014,Hirata_2015} or $T$-matrix.\cite{Romaniello_2012,Zhang_2017,Li_2021b,Loos_2022}
 
 Quite unfortunately, there are several ways of computing $GW$ quasiparticle energies.
 Within the perturbative $GW$ scheme (commonly known as $G_0W_0$), the quasiparticle energies are obtained via a one-shot procedure (with or without linearization).
@@ -603,10 +602,8 @@ 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}
-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 the judicious intermediates.
-\titou{Discuss cost and gradients? $\Lambda$ equations for $GW$?}
+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.
 The $G_0W_0$ quasiparticle energies can be easily obtained via the procedure described in Ref.~\onlinecite{Monino_2022} by solving the previous equation for each value of $p$ separately.
-It can be applied to other approximate self-energies such as the second-order Green's function (or second Born) \cite{Stefanucci_2013,Ortiz_2013,Phillips_2014,Rusakov_2014,Hirata_2015} or $T$-matrix.\cite{Romaniello_2012,Zhang_2017,Li_2021b,Loos_2022}
 
 %======================
 %\subsection{EE-EOM-CCD}
@@ -722,9 +719,11 @@ It can be applied to other approximate self-energies such as the second-order Gr
 %%%%%%%%%%%%%%%%%%%%%
 \section{Conclusion}
 %%%%%%%%%%%%%%%%%%%%%
-The conventional and CC-based versions of the BSE and GW schemes have been implemented in the electronic structure package QuAcK \cite{QuAcK} which is freely available at \url{https://github.com/pfloos/QuAcK}, with which we have numerically check the present equivalences between many-body perturbation and CC theories.
-%A route towards the obtention of BSE gradients for ground and excited states.
-%and provides a clear path for the computation of ground- and excited-state properties (such as nuclear gradients) within the BSE framework
+Here, we have unveiled exact similarities between the BSE formalism and the $GW$ approximation from many-body perturbation theory and CC theory at the ground- and excited-state levels.
+The conventional and CC-based versions of the BSE and $GW$ schemes that we have described in the present work have been implemented in the electronic structure package QuAcK \cite{QuAcK} (available at \url{https://github.com/pfloos/QuAcK}) with which we have numerically checked these exact equivalences.
+Similitudes between BSE@$GW$ and the STEOM-CC method have been put forward.
+We hope that the present work may provide a path for the computation of ground- and excited-state properties (such as nuclear gradients) within the $GW$ and BSE frameworks.
+
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 %\section*{Supplementary Material}
 %\label{sec:supmat}
@@ -738,7 +737,7 @@ This project has received funding from the European Research Council (ERC) under
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 \section*{Data availability statement}
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-The data that supports the findings of this study are available within the article.% and its supplementary material.
+Data sharing is not applicable to this article as no new data were created or analyzed in this study.
 
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 %\appendix