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taking antoine comemnts into account and modifying the conclusion

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CCvsMBPT.bib
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@ -1,13 +1,26 @@
%% This BibTeX bibliography file was created using BibDesk.
%% http://bibdesk.sourceforge.net/
%% https://bibdesk.sourceforge.io/
%% Created for Pierre-Francois Loos at 2022-10-11 16:13:34 +0200
%% Created for Pierre-Francois Loos at 2022-10-11 21:51:03 +0200
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@article{McKeon_2022,
author = {McKeon,Caroline A. and Hamed,Samia M. and Bruneval,Fabien and Neaton,Jeffrey B.},
date-added = {2022-10-11 21:50:49 +0200},
date-modified = {2022-10-11 21:51:03 +0200},
doi = {10.1063/5.0097582},
journal = {J. Chem. Phys.},
number = {7},
pages = {074103},
title = {An optimally tuned range-separated hybrid starting point for ab initio GW plus Bethe--Salpeter equation calculations of molecules},
volume = {157},
year = {2022},
bdsk-url-1 = {https://doi.org/10.1063/5.0097582}}
@article{Handy_1989,
abstract = {A size-consistent set of equations for electron correlation which are limited to double substitutions, based on Brueckner orbitals, is discussed. Called BD theory, it is shown that at fifth order of perturbation theory, BD incorporates more terms than CCSD and QCISD. The simplicity of the equations leads to an elegant gradient theory. Preliminary applications are reported.},
author = {Nicholas C. Handy and John A. Pople and Martin Head-Gordon and Krishnan Raghavachari and Gary W. Trucks},

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CCvsMBPT.tex
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@ -168,13 +168,14 @@ In particular, it may provide a path for the computation of ground- and excited-
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
The random-phase approximation (RPA), introduced by Bohm and Pines \cite{Bohm_1951,Pines_1952,Bohm_1953} in the context of the uniform electron gas, \cite{Loos_2016} is a quasibosonic approximation where one treats fermion products as bosons.
In the particle-hole (ph) channel, which is quite popular in the electronic structure community, \cite{Ren_2012,Chen_2017} particle-hole fermionic excitations and deexcitations are assumed to be bosons.
Because ph-RPA corresponds to a resummation of all ring diagrams, it is adequate in the high-density (or weakly correlated) regime and catch effectively long-range correlation effects (such as dispersion). \cite{Gell-Mann_1957,Nozieres_1958} \ant{Maybe explain more this last sentence, I feel like it's a bit fast.}
Because ph-RPA takes into account dynamical screening by summing up to infinity the (time-independent) ring diagrams, it is adequate in the high-density (or weakly correlated) regime and catch effectively long-range correlation effects (such as dispersion). \cite{Gell-Mann_1957,Nozieres_1958}
Another important feature of ph-RPA compared to finite-order perturbation theory is that it does not present any divergence for small-gap or metallic systems. \cite{Gell-Mann_1957}
Roughly speaking, the Bethe-Salpeter equation (BSE) formalism \cite{Salpeter_1951,Strinati_1988,Blase_2018,Blase_2020} of many-body perturbation theory \cite{Martin_2016} can be seen as a cheap and efficient way of introducing correlation in order to go \textit{beyond} RPA physics.
In the ph channel, BSE is commonly performed on top of a $GW$ calculation \cite{Hedin_1965,Aryasetiawan_1998,Onida_2002,Reining_2017,Golze_2019,Bruneval_2021} from which one extracts the quasiparticle energies as well as the dynamically-screened Coulomb potential $W$.
Practically, $GW$ produces accurate \textit{``charged''} excitations providing a faithful description of the fundamental gap, while the remaining excitonic effect (\ie, the stabilization provided by the attraction of the excited electron and its hole left behind) is caught via BSE, hence producing overall accurate \textit{``neutral''} excitations.
BSE@$GW$ has been shown to be highly successful to compute low-lying excited states of various natures (charge transfer, Rydberg, valence, etc) in molecular systems with a very attractive accuracy/cost ratio.\cite{Rohlfing_1999a,Horst_1999,Puschnig_2002,Tiago_2003,Rocca_2010,Boulanger_2014,Jacquemin_2015a,Bruneval_2015,Jacquemin_2015b,Hirose_2015,Jacquemin_2017a,Jacquemin_2017b,Rangel_2017,Krause_2017,Gui_2018,Blase_2018,Liu_2020,Blase_2020,Holzer_2018a,Holzer_2018b,Loos_2020e,Loos_2021}
\ant{In this paragraph we say that BSE goes beyond RPA and then we say that BSE is done on top of GW without expliciting the link between GW and RPA. I guess it's obvious for GW people but maybe just saying that the GW polarizability is obtained by summing the time dependent ring diagrams would make the connection a bit clearer.}
Practically, $GW$ produces accurate \textit{``charged''} excitations providing a faithful description of the fundamental gap via the computation of the RPA polarizability obtained by a resummation of all time-dependent ring diagrams.
The remaining excitonic effect (\ie, the stabilization provided by the attraction of the excited electron and its hole left behind) is caught via BSE, hence producing overall accurate \textit{``neutral''} excitations.
BSE@$GW$ has been shown to be highly successful to compute low-lying excited states of various natures (charge transfer, Rydberg, valence, etc) in molecular systems with a very attractive accuracy/cost ratio.\cite{Rohlfing_1999a,Horst_1999,Puschnig_2002,Tiago_2003,Rocca_2010,Boulanger_2014,Jacquemin_2015a,Bruneval_2015,Jacquemin_2015b,Hirose_2015,Jacquemin_2017a,Jacquemin_2017b,Rangel_2017,Krause_2017,Gui_2018,Blase_2018,Liu_2020,Blase_2020,Holzer_2018a,Holzer_2018b,Loos_2020e,Loos_2021,McKeon_2022}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Connection between RPA and CC}
@ -310,7 +311,7 @@ Neglecting the effect of $\hT_2$ on the one-body terms [see Eqs.~\eqref{eq:cFab}
\begin{equation}
\label{eq:EOM-rCCD}
\begin{split}
\mel*{ \Psi_{i}^{a} }{ \tHN }{ \Psi_{j}^{b} }
\mel*{ \Psi_{i}^{a} }{ \bHN }{ \Psi_{j}^{b} }
& = (\e{a}{} - \e{i}{}) \delta_{ij} \delta_{ab} + \dbERI{ib}{aj} + \sum_{kc} \dbERI{ik}{ac} t_{kj}^{cb}
\\
& = (\bA{}{} + \bB{}{} \cdot \bT{}{})_{ia,jb}
@ -422,8 +423,15 @@ with $\Delta_{ijab}^{\GW} = \e{a}{\GW} + \e{b}{\GW} - \e{i}{\GW} - \e{j}{\GW}$.
Similarly to the diagonalization of the eigensystem \eqref{eq:BSE}, these approximate CCD amplitude equations can be solved with $\order*{N^6}$ cost via the definition of appropriate intermediates.
As in the case of RPAx (see Sec.~\ref{sec:RPAx}), several variants of the BSE correlation energy do exist, either based on the plasmon formula \cite{,Li_2020,Li_2021} or the ACFDT. \cite{Maggio_2016,Holzer_2018b,Loos_2020e}
Following Berkelbach's analysis, \cite{Berkelbach_2018} one can extend the connection to excited states: the EOM treatment (restricted to 1h1p) of the approximate similarity-transformed rCCD Hamiltonian [see Eq.~\eqref{eq:rCCD-BSE}] provides the same excitation energies as the conventional linear-response equations \eqref{eq:BSE}. \ant{I'm not sure that we can call it an EOM treatment of an approximate similarity transformed rCCD Hamiltonian. Maybe we can say that we can obtain an analog of the approximate-EOM equation Eq.~(\ref{eq:EOM-rCCD}) by using the amplitudes defined by [see Eq.~\eqref{eq:rCCD-BSE}] as well as replacing $A$ and $B$ by their BSE counterparts and this equation gives the same excitations as the linear response equations}.
However, there is a significant difference with RPAx as the BSE involves $GW$ quasiparticle energies [see Eq.~\eqref{eq:A_BSE}], where some of the correlation has been already dressed, while the RPAx equations only involves (undressed) one-electron orbital energies, as shown in Eq.~\eqref{eq:EOM-rCCD}.
Following Berkelbach's analysis, \cite{Berkelbach_2018} one can extend the connection to excited states.
Indeed, one can obtain an analog of the 1h1p block of the approximate EOM-rCCD Hamiltonian [see Eq.\eqref{eq:EOM-rCCD}] using the amplitudes resulting from Eq.~\eqref{eq:rCCD-BSE} as well as replacing $\bA{}{}$ and $\bB{}{}$ by their BSE counterparts, \ie,
\begin{equation}
\label{eq:EOM-rCCD-BSE}
\mel*{ \Psi_{i}^{a} }{ \tHN }{ \Psi_{j}^{b} }
= (\e{a}{\GW} - \e{i}{\GW}) \delta_{ij} \delta_{ab} + \wERI{ib}{aj} + \sum_{kc} \wERI{ik}{ac} \tilde{t}_{kj}^{cb}
\end{equation}
This equation provides the same excitation energies as the conventional linear-response equations \eqref{eq:BSE}.
However, there is a significant difference with RPAx as the BSE involves $GW$ quasiparticle energies, where some of the correlation has been already dressed, while the RPAx equations only involves (undressed) one-electron orbital energies, as shown in Eq.~\eqref{eq:EOM-rCCD}.
In other words, in the spirit of the Brueckner version of CCD, \cite{Handy_1989} the $GW$ post-treatment renormalizes the bare one-electron energies and, consequently, incorporates mosaic diagrams, \cite{Scuseria_2008,Scuseria_2013} a process named Brueckner-like dressing in Ref.~\onlinecite{Berkelbach_2018}.
This observation evidences clear similitudes between BSE@$GW$ and the similarity-transformed EOM-CC (STEOM-CC) method introduced by Nooijen, \cite{Nooijen_1997c,Nooijen_1997b,Nooijen_1997a} where one performs a second similarity transformation to partially decouple the 1h determinants from the 2h1p ones in the ionization potential (IP) sector and the 1p determinants from the 1h2p ones in the electron affinity (EA) sector.
@ -723,10 +731,10 @@ The $G_0W_0$ quasiparticle energies can be easily obtained via the procedure des
\section{Conclusion}
%%%%%%%%%%%%%%%%%%%%%
Here, we have unveiled exact similarities between CC and many-body perturbation theory at the ground- and excited-state levels.
More specifically, we have shown how to recast $GW$ and BSE as non-linear CC-like equations that can be solved with the usual CC machinery with the same computational cost.
More specifically, we have shown how to recast $GW$ and BSE as non-linear CC-like equations that can be solved with the usual CC machinery at the same computational cost.
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 STEOM-CC 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 $GW$ \cite{Lazzeri_2008,Faber_2011b,Yin_2013,Montserrat_2016,Zhenglu_2019} and BSE \cite{IsmailBeigi_2003,Caylak_2021,Knysh_2022} frameworks, and broaden the applicability of Green's function methods in the molecular electronic structure community and beyond.
Similitudes between BSE@$GW$ and STEOM-CC have been also highlighted, and may explain the reliability of BSE@$GW$ for the computation of optical excitations in molecular systems.
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 $GW$ \cite{Lazzeri_2008,Faber_2011b,Yin_2013,Montserrat_2016,Zhenglu_2019} and BSE \cite{IsmailBeigi_2003,Caylak_2021,Knysh_2022} frameworks, hence broadening the applicability of these formalisms in computational photochemistry.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\section*{Supplementary Material}