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% coordinates
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% methods
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% 
\newcommand{\Ne}{N}
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% Matrix elements
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% Matrices
\newcommand{\bO}{\boldsymbol{0}}
\newcommand{\bI}{\boldsymbol{1}}
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% addresses
\newcommand{\LCPQ}{Laboratoire de Chimie et Physique Quantiques (UMR 5626), Universit\'e de Toulouse, CNRS, UPS, France}
\newcommand{\LPT}{Laboratoire de Physique Th\'eorique, Universit\'e de Toulouse, CNRS, UPS, France}
\newcommand{\ETSF}{European Theoretical Spectroscopy Facility (ETSF)}
\newcommand{\NEEL}{Universit\'e Grenoble Alpes, CNRS, Institut N\'EEL, F-38042 Grenoble, France}

\begin{document}	

\title{Connections between many-body perturbation and coupled-cluster theories}

\author{Ra\'ul \surname{Quintero-Monsebaiz}}
	\affiliation{\LCPQ}
\author{Enzo \surname{Monino}}
	\affiliation{\LCPQ}
\author{Pierre-Fran\c{c}ois \surname{Loos}}
	\email{loos@irsamc.ups-tlse.fr}
	\affiliation{\LCPQ}

\begin{abstract}
Here, we build on the works of Scuseria \textit{et al.} [\href{http://dx.doi.org/10.1063/1.3043729}{J.~Chem.~Phys.~\textbf{129}, 231101 (2008)}] and Berkelbach [\href{https://doi.org/10.1063/1.5032314}{J.~Chem.~Phys.~\textbf{149}, 041103 (2018)}] to show intimate connections between the Bethe-Salpeter equation (BSE) formalism and $GW$ approximation from many-body perturbation theory and coupled-cluster (CC) theory at the ground- and excited-state levels.
Similitudes between BSE@$GW$ and the similarity-transformed equation-of-motion CC method introduced by Nooijen are put forward.
The present work allows to easily transfer key developments and general knowledge gathered in CC theory to many-body perturbation theory.
In particular, it provides a clear path for the computation of ground- and excited-state properties (such as nuclear gradients) within the BSE framework.
%\bigskip
%\begin{center}
%	\boxed{\includegraphics[width=0.5\linewidth]{TOC}}
%\end{center}
%\bigskip
\end{abstract}

\maketitle

%%%%%%%%%%%%%%%%%%%%%%
%\section{RPA Physics and Beyond}
%%%%%%%%%%%%%%%%%%%%%%
The random-phase approximation (RPA), introduced by Bohm and Pines 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, particle-hole fermionic excitations and deexcitations are assumed to be bosons. 
Because 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} 

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 effective way of introducing correlation 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 $W$. 
This scheme 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}


%When one is looking for high accuracy, the workhorse of molecular electronic structure is certainly coupled cluster theory.
%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.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\section{Bethe-Salpeter equation}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
In a landmark paper, Scuseria \textit{et al.} \cite{Scuseria_2008} have proven that rCCD is equivalent to RPAx for the computation of the correlation energy.
Assuming the existence of $\bX{}{-1}$ (which can be proven as long as the RPAx problem is stable), this proof can be quickly summarised starting from the ubiquitous RPAx eigensystem
\begin{equation}
	\label{eq:RPA}
	\begin{pmatrix}
		\bA{}{}		&	\bB{}{}	\\
		-\bB{}{}	&	-\bA{}{}	\\
	\end{pmatrix}
	\cdot
	\begin{pmatrix}
		\bX{}{}	\\
		\bY{}{}	\\
	\end{pmatrix}
	=
	\begin{pmatrix}
		\bX{}{}	\\
		\bY{}{}	\\
	\end{pmatrix}
	\cdot
	\bOm{}{}
\end{equation}
from which one gets, by introducing $\bT{}{} = \bY{}{} \cdot \bX{}{-1}$,
\begin{equation}
	\begin{pmatrix}
		\bA{}{}		&	\bB{}{}	\\
		-\bB{}{}	&	-\bA{}{}	\\
	\end{pmatrix}
	\cdot
	\begin{pmatrix}
		\bI	\\
		\bT{}{}	\\
	\end{pmatrix}
	=
	\begin{pmatrix}
		\bI	\\
		\bT{}{}	\\
	\end{pmatrix}
	\cdot
	\bR{}{}
\end{equation}
where $\bR{}{} = \bX{}{} \cdot \bOm{}{} \cdot \bX{}{-1}$, or equivalently, the two following equations
\begin{subequations}
\begin{align}
	\label{eq:RPA_1}
	\bA{}{}	+ \bB{}{} \cdot \bT{}{} & = \bR{}{}
	\\
	\label{eq:RPA_2}
	-\bB{}{} - \bA{}{} \cdot \bT{}{} & = \bT{}{} \cdot \bR{}{}
\end{align}
\end{subequations}

Substituting Eq.~\eqref{eq:RPA_1} into Eq.~\eqref{eq:RPA_2} yields the following Riccati equation
\begin{equation}
	\bB{}{} + \bA{}{} \cdot \bT{}{} + \bT{}{} \cdot \bA{}{} + \bT{}{} \cdot \bB{}{} \cdot \bT{}{} = \bO
\end{equation}
that matches the well-known rCCD amplitude equations
\begin{multline}
\label{eq:rCCD}
	\dbERI{ij}{ab} 
	+ \qty( \e{a}{} + \e{b}{} - \e{i}{} - \e{j}{} ) t_{ij}^{ab}
	+ \sum_{kc} \dbERI{ic}{ak} t_{kj}^{cb} 
	\\
	+ \sum_{kc} \dbERI{kb}{cj} t_{ik}^{ac} 
	+ \sum_{klcd} \dbERI{kl}{cd} t_{ik}^{ac} t_{lj}^{db} = 0
\end{multline}
knowing that
\begin{subequations}
\begin{align}
	\label{eq:A_RPAx}
	A_{ia,jb} & = (\e{a}{} - \e{i}{}) \delta_{ij} \delta_{ab} + \dbERI{ib}{aj}
	\\ 
	\label{eq:B_RPAx}
	B_{ia,jb} & = \dbERI{ij}{ab}
\end{align}
\end{subequations}
We assume real quantities throughout this paper, $\e{p}{}$ is the one-electron energy associated with the Hartree-Fock spinorbital $\SO{p}(\bx)$ and 
\begin{equation}
	\label{eq:ERI}
	\braket{pq}{rs} = \iint \SO{p}(\bx_1) \SO{q}(\bx_2) \frac{1}{\abs{\br_1 - \br_2}} \SO{r}(\bx_1) \SO{s}(\bx_2)  d\bx_1 d\bx_2
\end{equation}
are two-electron repulsion integrals, while 
\begin{equation}
	\dbERI{pq}{rs} = \braket{pq}{rs} - \braket{pq}{sr}
\end{equation}
are their anti-symmetrized versions.
The composite variable $\bx$ gathers spin and spatial ($\br$) variables.
The indices $i$, $j$, $k$, and $l$ are occupied (hole) orbitals; $a$, $b$, $c$, and $d$ are unoccupied (particle) orbitals; $p$, $q$, $r$, and $s$ indicate arbitrary orbitals; and $m$ labels single excitations or deexcitations.
In the following, $O$ and $V$ are the number of occupied and virtual spinorbitals, respectively, and $N = O + V$ is the total number.


There are various ways of computing the RPAx correlation energy, \cite{Angyan_2011} but the usual plasmon (or trace) formula yields
\begin{equation}
\label{eq:EcRPAx}
	\Ec^\text{RPAx} = \frac{1}{4} \Tr(\bOm{}{} - \bA{}{})
\end{equation}
and matches the rCCD correlation energy
\begin{equation}
	\Ec^\text{rCCD} = \frac{1}{4} \sum_{ijab} \dbERI{ij}{ab} t_{ij}^{ab} = \frac{1}{4} \Tr(\bB{}{} \cdot \bT{}{})
\end{equation}
because $\Tr(\bOm{}{} - \bA{}{}) = \Tr(\bR{}{} - \bA{}{}) = \Tr(\bB{}{} \cdot \bT{}{})$, as evidenced by Eq.~\eqref{eq:RPA_1}.
Note that the same expression as Eq.~\eqref{eq:EcRPAx} can be derived from the adiabatic connection fluctuation dissipation theorem.\cite{Angyan_2011}

This simple and elegant proof was subsequently extended to excitation energies by Berkerbach, \cite{Berkelbach_2018} who showed that similitudes between EE-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}
	\mel*{ \Psi_{i}^{a} }{ \bHN }{ \Psi_{j}^{b} } = \cF_{ab} \delta_{ij} - \cF_{ij} \delta_{ab} + \cW_{jabi}
\end{equation}
where $\bHN = e^{-\hT} \hH e^{\hT} - E_\text{CCD}$ is the normal-ordered similarity-transformed Hamiltonian, $\Psi_{i}^{a}$ are singly-excited determinants, the one-body terms are
\begin{subequations}
\begin{align}
\label{eq:cFab}
	\cF_{ab} & = \e{a}{} \delta_{ab} - \frac{1}{2} \sum_{klc} \ERI{kl}{bc} t_{kl}^{ac}
	\\
\label{eq:cFij}
	\cF_{ij} & = \e{i}{} \delta_{ij} + \frac{1}{2} \sum_{kcd} \ERI{ik}{cd} t_{jk}^{cd}
\end{align}
\end{subequations}
and the two-body term is
\begin{equation}
\label{eq:cWibaj}
	\cW_{ibaj} = \ERI{ib}{aj} + \sum_{kc} \ERI{ik}{ac} t_{kj}^{cb}
\end{equation}
Neglecting the effect of $\hT_2$ on the one-body terms [see Eqs.~\eqref{eq:cFab} and \eqref{eq:cFij}] and relying on the rCCD amplitudes in the two-body terms, Eq.~\eqref{eq:cWibaj}, yields 
\begin{equation}
\begin{split}
	\mel*{ \Psi_{i}^{a} }{ \tHN }{ \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}
\end{split}
\end{equation}
which exactly matches Eq.~\eqref{eq:RPA_1}.
Although the excitation energies of this approximate EE-EOM-rCCD scheme are equal to the RPAx ones, it has been shown that the transition amplitudes (or residues) are distinct and only agrees at the lowest order in the Coulomb interaction. \cite{Emrich_1981,Berkelbach_2018}

This does not hold only for RPAx as it is actually quite general and can be applied to any ph problem with the structure of Eq.~\eqref{eq:RPA}, such as TD-DFT, BSE, and many others.
This has been also extended to the pp and hh sectors by Peng \textit{et al.} \cite{Peng_2013} and Scuseria \textit{et al.} \cite{Scuseria_2013} (see also Ref.~\onlinecite{Berkelbach_2018} for the extension to excitation energies for the pp and hh channels).

At the BSE level, and within the static approximation, one must solve a very similar linear eigenvalue problem
\begin{equation}
	\label{eq:BSE}
	\begin{pmatrix}
		\bA{}{\BSE}		&	\bB{}{\BSE}	\\
		-\bB{}{\BSE}	&	-\bA{}{\BSE}	\\
	\end{pmatrix}
	\cdot
	\begin{pmatrix}
		\bX{}{\BSE}	\\
		\bY{}{\BSE}	\\
	\end{pmatrix}
	=
	\begin{pmatrix}
		\bX{}{\BSE}	\\
		\bY{}{\BSE}	\\
	\end{pmatrix}
	\cdot
	\bOm{}{\BSE}
\end{equation}
where
\begin{subequations}
\begin{align}
	\label{eq:A_BSE}
	A_{ia,jb}^{\BSE} & = \delta_{ij} \delta_{ab} (\e{a}{\GW} - \e{i}{\GW}) + \dbERI{ib}{aj} - W_{ij,ba}^\text{stat}
	\\ 
	\label{eq:B_BSE}
	B_{ia,jb}^{\BSE} & = \dbERI{ij}{ab} - W_{ib,ja}^\text{stat}
\end{align}
\end{subequations}
and 
\begin{multline}
	\label{eq:W}
	W_{pq,rs}^\text{c}(\omega) = \sum_{m} \sERI{pq}{m} \sERI{rs}{m} 
	\\
	\times \qty[ \frac{1}{\omega - \Om{m}{\dRPA} + \ii \eta} - \frac{1}{\omega + \Om{m}{\dRPA} - \ii \eta} ]
\end{multline}
are the elements of the correlation part of the dynamically-screened Coulomb potential which is set to its static limit \ie, $W_{pq,rs}^\text{stat} = W_{pq,rs}^\text{c}(\omega = 0)$.
In Eq.~\eqref{eq:W}, $\eta$ is a positive infinitesimal, the screened two-electron integrals are
\begin{equation}
	\label{eq:sERI}
	\sERI{pq}{m} = \sum_{ia} \ERI{pi}{qa} \qty( \bX{m}{\dRPA} + \bY{m}{\dRPA} )_{ia}
\end{equation}
and $\Om{m}{\dRPA}$ is the $m$th (positive) eigenvalues and $\bX{m}{\dRPA} + \bY{m}{\dRPA}$ is constructed from the corresponding eigenvectors of the direct (\ie, without exchange) RPA (dRPA) problem defined as
\begin{equation}
	\label{eq:dRPA}
	\begin{pmatrix}
		\bA{}{\dRPA}		&	\bB{}{\dRPA}	\\
		-\bB{}{\dRPA}	&	-\bA{}{\dRPA}	\\
	\end{pmatrix}
	\cdot
	\begin{pmatrix}
		\bX{}{\dRPA}	\\
		\bY{}{\dRPA}	\\
	\end{pmatrix}
	=
	\begin{pmatrix}
		\bX{}{\dRPA}	\\
		\bY{}{\dRPA}	\\
	\end{pmatrix}
	\cdot
	\bOm{}{\dRPA}
\end{equation}
with
\begin{subequations}
\begin{align}
	\label{eq:A_dRPA}
	A_{ia,jb}^{\dRPA} & = \delta_{ij} \delta_{ab} (\e{a}{} - \e{i}{}) + \ERI{ib}{aj}
	\\ 
	\label{eq:B_dRPA}
	B_{ia,jb}^{\dRPA} & = \ERI{ij}{ab}
\end{align}
\end{subequations}
As readily seen in Eqs.~\eqref{eq:A_RPAx}, \eqref{eq:B_RPAx}, \eqref{eq:A_BSE} and \eqref{eq:B_BSE}, the only difference between RPAx and BSE lies in the definition of the matrix elements, where one includes, via the screening of the electron-electron interaction [see Eq.~\eqref{eq:W}], correlation effects at the BSE level.

Note also that, within BSE, the orbital energies in Eqs.~\eqref{eq:A_BSE} and \eqref{eq:B_BSE} are usually computed at the $GW$ level to be consistent with the introduction of $W$ in the BSE kernel.
For example, at the one-shot $GW$ level, \cite{Strinati_1980,Hybertsen_1985a,Hybertsen_1986,Godby_1988,Linden_1988,Northrup_1991,Blase_1994,Rohlfing_1995,Shishkin_2007} we have
\begin{equation}
	\label{eq:G0W0}
	\e{p}{\GW} = \e{p}{\HF} + Z_p \SigC{p}(\omega = \e{p}{\HF})
\end{equation}
where the elements of the correlation part of the dynamical self-energy are
\begin{equation}
\begin{split}
	\SigC{pq}(\omega) 
	& = \sum_{im} \frac{\sERI{pi}{m} \sERI{qi}{m}}{\omega - \e{i}{\HF} + \Om{m}{\dRPA} - \ii \eta}
	\\
	& + \sum_{am} \frac{\sERI{pa}{m} \sERI{qa}{m}}{\omega - \e{a}{\HF} - \Om{m}{\dRPA} + \ii \eta}
\end{split}
\end{equation}
and the renormalization factor is
\begin{equation}
	Z_p = \qty[ 1 - \eval{\pdv{\SigC{pp}(\omega)}{\omega}}_{\omega = \e{p}{\HF}} ]^{-1}
\end{equation}


Therefore, following a similar procedure, one can show that the BSE correlation energy obtained using the trace formula 
\begin{equation}
	\Ec^\text{BSE} = \frac{1}{4} \Tr(\bOm{}{\BSE} - \bA{}{\BSE})
\end{equation}
can be obtained via a set of rCCD equations, where one substitutes in Eq.~\eqref{eq:rCCD} all the two-electron integrals $\dbERI{pq}{rs}$ by $\dbERI{pq}{rs} - W_{ps,qr}^\text{stat}$.
Similarly to the diagonalization of the eigensystem \eqref{eq:BSE}, these rCCD-based equations can be solved in $\order*{N^6}$ cost, and provides a clear path for the computation of ground- and excited-state properties (such as nuclear gradients) within the BSE framework.

Following Berkelbach's analysis, \cite{Berkelbach_2018} one can extend the connection to excitation energies.
However, there is a significant difference 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
It is therefore interesting to investigate if it possible also the recast the $GW$ equations as a set of CC-like equations.
Connections between approximate IP- and 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 and we do not decouple the 2h1p and 2p1h sectors.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\section{$GW$ approximation}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
As shown by Bintrim and Berkelbach, \cite{Bintrim_2021} the non-linear $GW$ equations can be recast as a set of linear equations, which reads in the Tamm-Dancoff approximation
\begin{equation}
	\begin{pmatrix}
		\be{}{}						&	\bV{}{\text{2h1p}}		&	\bV{}{\text{2p1h}}	\\
		\T{(\bV{}{\text{2h1p}})}	&	\bC{}{\text{2h1p}}		&	\bO					\\
		\T{(\bV{}{\text{2p1h}})}	&	\bO						&	\bC{}{\text{2p1h}}	\\
	\end{pmatrix}
	\cdot
	\begin{pmatrix}
		\bX{}{}	\\
		\bY{}{\text{2h1p}}	\\
		\bY{}{\text{2p1h}}	\\
	\end{pmatrix}
	=
	\begin{pmatrix}
		\bX{}{}	\\
		\bY{}{\text{2h1p}}	\\
		\bY{}{\text{2p1h}}	\\
	\end{pmatrix}
	\cdot
	\bOm{}{}
\end{equation}
where $\be{}{}$ is a diagonal matrix gathered the orbital energies, the 2h1p and 2p1h matrix elements are
\begin{subequations}
\begin{align}
	C^\text{2h1p}_{ija,klc} & = \qty[ \qty( \e{i}{} + \e{j}{} - \e{a}{}) \delta_{jl} \delta_{ac} - \ERI{jc}{al} ] \delta_{ik} 
	\\
	C^\text{2p1h}_{iab,kcd} & = \qty[ \qty( \e{a}{} + \e{b}{} - \e{i}{}) \delta_{ik} \delta_{ac} + \ERI{ak}{ic} ] \delta_{bd} 
\end{align}
\end{subequations}
and the corresponding coupling blocks read
\begin{align}
	V^\text{2h1p}_{p,klc} & = \ERI{pc}{kl}
	&
	V^\text{2p1h}_{p,kcd} & = \ERI{pk}{dc}
\end{align}
Going beyond the Tamm-Dancoff approximation is possible, but more cumbersome. \cite{Bintrim_2021}

Introducing $\bT{}{\text{2h1p}} = \bY{}{\text{2h1p}} \cdot \bX{}{-1}$ and $\bT{}{\text{2p1h}} = \bY{}{\text{2p1h}} \cdot \bX{}{-1}$, we have
\begin{equation}
	\begin{pmatrix}
		\be{}{}						&	\bV{}{\text{2h1p}}		&	\bV{}{\text{2p1h}}	\\
		\T{(\bV{}{\text{2h1p}})}	&	\bC{}{\text{2h1p}}		&	\bO					\\
		\T{(\bV{}{\text{2p1h}})}	&	\bO						&	\bC{}{\text{2p1h}}	\\
	\end{pmatrix}
	\cdot
	\begin{pmatrix}
		\bI	\\
		\bT{}{\text{2h1p}}	\\
		\bT{}{\text{2p1h}}	\\
	\end{pmatrix}
	=
	\begin{pmatrix}
		\bI	\\
		\bT{}{\text{2h1p}}	\\
		\bT{}{\text{2p1h}}	\\
	\end{pmatrix}
	\cdot
	\bR{}{}
\end{equation}
with $\bR{}{} = \bX{}{} \cdot \bOm{}{} \cdot \bX{}{-1}$.
The matrix $\bX{}{-1}$ can be proven to exist when the system is stable.

This yields the three following equations
\begin{subequations}
\begin{align}
	\be{}{} + \bV{}{\text{2h1p}} \cdot \bT{}{\text{2h1p}} + \bV{}{\text{2p1h}} \cdot \bT{}{\text{2p1h}} & = \bR{}{}
	\label{eq:R}
	\\
	\T{(\bV{}{\text{2h1p}})} + \bC{}{\text{2h1p}} \cdot \bT{}{\text{2h1p}} & = \bT{}{\text{2h1p}} \cdot \bR{}{}
	\label{eq:T1R}
	\\
	\T{(\bV{}{\text{2p1h}})} + \bC{}{\text{2p1h}} \cdot \bT{}{\text{2p1h}} & = \bT{}{\text{2p1h}} \cdot \bR{}{}
	\label{eq:T2R}
\end{align}
\end{subequations}
Substituting Eq.~\eqref{eq:R} into Eqs.~\eqref{eq:T1R} and \eqref{eq:T2R} yields the two following coupled Riccati equations 
\begin{subequations}
\begin{align}
	\begin{split}
		\T{(\bV{}{\text{2h1p}})} 
		+ \bC{}{\text{2h1p}} \cdot \bT{}{\text{2h1p}} 
		- \bT{}{\text{2h1p}} \cdot \be{}{} 
		- \bT{}{\text{2h1p}} \cdot \bV{}{\text{2h1p}} \cdot \bT{}{\text{2h1p}}
		\\
		- \bT{}{\text{2h1p}} \cdot \bV{}{\text{2p1h}} \cdot \bT{}{\text{2p1h}}
		= \bO
	\end{split}
	\\
	\begin{split}
		\T{(\bV{}{\text{2p1h}})} 
		+ \bC{}{\text{2p1h}} \cdot \bT{}{\text{2p1h}} 
		- \bT{}{\text{2p1h}} \cdot \be{}{} 
		- \bT{}{\text{2p1h}} \cdot \bV{}{\text{2h1p}} \cdot \bT{}{\text{2h1p}}
		\\
		- \bT{}{\text{2p1h}} \cdot \bV{}{\text{2p1h}} \cdot \bT{}{\text{2p1h}}
		= \bO
	\end{split}
\end{align}
\end{subequations}
In the CC language, in order to determine the 2h1p and 2p1h amplitudes, $t_{ija,p}^{\text{2h1p}}$ and $t_{iab,p}^{\text{2p1h}} $, it means that one must solve the following amplitude equations
\begin{subequations}
\begin{align}
	\begin{split}
		r_{ija,p}^{\text{2h1p}}
		& = \ERI{pa}{ij}
		+ \Delta_{ija,p}^{\text{2h1p}} t_{ija,p}^{\text{2h1p}} 
		- \sum_{kc} \ERI{jc}{ak} t_{ikc,p}^{\text{2h1p}} 
		\\
		& - \sum_{klcq} t_{ija,q}^{\text{2h1p}} \ERI{qc}{kl} t_{klc,p}^{\text{2h1p}} 
		- \sum_{kcdq} t_{ija,q}^{\text{2h1p}} \ERI{qk}{dc} t_{kcd,p}^{\text{2p1h}} 
	\end{split}
	\\
	\begin{split}
		r_{iab,p}^{\text{2p1h}}
		& = \ERI{pi}{ba}
		+ \Delta_{iab,p}^{\text{2p1h}} t_{ija,p}^{\text{2p1h}} 
		+ \sum_{kc} \ERI{ak}{ic} t_{kcb,p}^{\text{2p1h}} 
		\\
		& - \sum_{klcq} t_{iab,q}^{\text{2p1h}} \ERI{qc}{kl} t_{klc,p}^{\text{2h1p}} 
		- \sum_{kcdq} t_{iab,q}^{\text{2p1h}} \ERI{qk}{dc} t_{kcd,p}^{\text{2p1h}} 
	\end{split}
\end{align}
\end{subequations}
with 
\begin{subequations}
\begin{align}
	\Delta_{ija,p}^{\text{2h1p}} & = \e{i}{} + \e{j}{} - \e{a}{} - \e{p}{}
	\\
	\Delta_{iab,p}^{\text{2p1h}} & = \e{a}{} + \e{b}{} - \e{i}{} - \e{p}{} 
\end{align}
\end{subequations}
One can then employed the usual iterative procedure to solve these non-linear equations by updating the amplitudes via
\begin{subequations}
\begin{align}
	t_{ija,p}^{\text{2h1p}} & \leftarrow t_{ija,p}^{\text{2h1p}} - \frac{r_{ija,p}^{\text{2h1p}}}{\Delta_{ij}^{ap}}
	\\
	t_{iab,p}^{\text{2h1p}} & \leftarrow t_{iab,p}^{\text{2p1h}} - \frac{r_{iab,p}^{\text{2p1h}}}{\Delta_{ab}^{ip}}
\end{align}
\end{subequations}

The quasiparticle energies $\e{p}{GW}$ are thus provided by the eigenvalues of $\be{}{} + \bSig{}{\GW}$, where
\begin{equation}
	\bSig{}{\GW} = \bV{}{\text{2h1p}} \cdot \bT{}{\text{2h1p}} + \bV{}{\text{2p1h}} \cdot \bT{}{\text{2p1h}} 
\end{equation}
is the correlation part of the $GW$ self-energy.
The $G_0W_0$ quasiparticle energies can be easily obtained via the procedure described in Ref.~\onlinecite{Monino_2022}.

%======================
%\subsection{EE-EOM-CCD}
%======================

%Up to 2h2p, the EE-EOM-CC matrix has the simple form
%\begin{equation}
%	\bdH_\text{2h2p}^\text{EE} = 
%	\begin{pmatrix}
%		\mel*{ \Phi_{i}^{a} }{ \bHN }{ \Phi_{k}^{c} }		&	\mel*{ \Phi_{i}^{a} }{ \bHN }{ \Phi_{kl}^{cd} } 
%		\\
%		\mel*{ \Phi_{ij}^{ab} }{ \bHN }{ \Phi_{kl}^{cd} }	&	\mel*{ \Phi_{ij}^{ab} }{ \bHN }{ \Phi_{kl}^{cd} } 
%		\\
%	\end{pmatrix}
%\end{equation}
%Restricting ourselves to CCD, the elements of the 1h1p block read
%\begin{equation}
%	\mel*{ \Phi_{i}^{a} }{ \bHN }{ \Phi_{j}^{b} } = \cF_{ab} \delta_{ij} - \cF_{ij} \delta_{ab} + \cW_{jabi}
%\end{equation}
%where the one-body terms are
%\begin{subequations}
%\begin{align}
%	\cF_{ab} & = \e{a}{} \delta_{ab} - \frac{1}{2} \sum_{klc} \ERI{kl}{bc} t_{kl}^{ac}
%	\\
%	\cF_{ij} & = \e{i}{} \delta_{ij} + \frac{1}{2} \sum_{kcd} \ERI{ik}{cd} t_{jk}^{cd}
%\end{align}
%\end{subequations}
%and the two-body term is
%\begin{equation}
%	\cW_{ibaj} = \ERI{ib}{aj} + \sum_{kc} \ERI{ik}{ac} t_{kj}^{cb}
%\end{equation}
%It is interesting to note that, in the case where $t_{ij}^{ab} = 0$, the 1h1p block reduces to the well-known random-phase approximation (with exchange).

%================================
%\subsection{IP- and EA-EOM-CCD}
%================================
%Up to 2h1p and 2p1h, the IP-EOM-CC and EA-EOM-CC matrices, respectively, have the simple form
%\begin{subequations}
%\begin{align}
%	\bdH_\text{2h1p}^\text{IP} & = 
%	\begin{pmatrix}
%		\mel*{ \Phi_{i} }{ \bHN }{ \Phi_{k} } &	\mel{ \Phi_{i} }{ \bHN }{ \Phi_{kl}^{c} } 
%		\\
%		\mel*{ \Phi_{ij}^{a} }{ \bHN }{ \Phi_{k} } &	\mel{ \Phi_{ij}^{a} }{ \bHN }{ \Phi_{kl}^{c} } 
%		\\
%	\end{pmatrix}
%	\\
%	\bdH_\text{2p1h}^\text{EA} & = 
%	\begin{pmatrix}
%		\mel*{ \Phi^{a} }{ \bHN }{ \Phi^{c} }		&	\mel*{ \Phi^{a} }{ \bHN }{ \Phi_{k}^{cd} } 
%		\\
%		\mel*{ \Phi_{i}^{ab} }{ \bHN }{ \Phi_{k}^{cd} }	&	\mel*{ \Phi_{i}^{ab} }{ \bHN }{ \Phi_{k}^{cd} } 
%		\\
%	\end{pmatrix}
%\end{align}
%\end{subequations}
%
%\begin{align}
%	\mel*{ \Phi_{i} }{ \bHN }{ \Phi_{k} } & = \titou{??}
%	\\
%	\mel*{ \Phi_{ij}^{a} }{ \bHN }{ \Phi_{kl}^{c} } & = \titou{??}
%	\\
%	\mel*{ \Phi_{i} }{ \bHN }{ \Phi_{kl}^{c} } & = \cW_{ickl}
%	\\
%	\mel*{ \Phi_{ij}^{a} }{ \bHN }{ \Phi_{k} } & = \ERI{ij}{ka}
%\end{align}
%and
%\begin{multline}
%	\cW_{iakl} = \ERI{ia}{kl} + \sum_{me} \ERI{im}{ke} t_{lm}^{ae} 
%	\\
%	- \sum_{me} \ERI{im}{le} t_{km}^{ae} 
%	+ \frac{1}{2} \sum_{ef} \ERI{ia}{ef} t_{kl}^{ef}
%\end{multline}
%================================
%\subsection{DIP- and DEA-EOM-CCD}
%================================
%Up to 3h1p and 3p1h, the DIP-EOM-CC and DEA-EOM-CC matrices, respectively, have the simple form
%\begin{subequations}
%\begin{align}
%	\bdH_\text{3h2p}^\text{DIP} & = 
%	\begin{pmatrix}
%		\mel*{ \Phi_{ij} }{ \bHN }{ \Phi_{lm} }			&	\mel*{ \Phi_{ij} }{ \bHN }{ \Phi_{lmn}^{d} } 
%		\\
%		\mel*{ \Phi_{ijk}^{a} }{ \bHN }{ \Phi_{lm} }	&	\mel*{ \Phi_{ijk}^{a} }{ \bHN }{ \Phi_{lmn}^{d} } 
%		\\
%	\end{pmatrix}
%	\\
%	\bdH_\text{3p2h}^\text{DEA} & = 
%	\begin{pmatrix}
%		\mel*{ \Phi_{i}^{abc} }{ \bHN }{ \Phi^{de} }	&	\mel*{ \Phi_{i}^{abc} }{ \bHN }{ \Phi_{l}^{def} } 
%		\\
%		\mel*{ \Phi^{ab} }{ \bHN }{ \Phi^{de} }			&	\mel*{ \Phi^{ab} }{ \bHN }{ \Phi_{l}^{def} } 
%	\end{pmatrix}
%\end{align}
%\end{subequations}
%
%\begin{subequations}
%\begin{align}
%	\mel*{ \Phi_{ij} }{ \bHN }{ \Phi_{kl} } & = - \cF_{ij} \delta_{kl} -  \delta_{ij} \cF_{kl}+ \cW_{ijkl}
%		\\
%	\mel*{ \Phi^{ab} }{ \bHN }{ \Phi^{cd} } & = + \cF_{ab} \delta_{cd} + \delta_{ab} \cF_{cd} + \cW_{abcd}
%\end{align}
%\end{subequations}
%with
%\begin{subequations}
%\begin{align}
%	\cW_{ijkl} & = \ERI{ij}{kl} + \sum_{a<b} \ERI{ij}{ab} t_{ij}^{ab}
%	\\
%	\cW_{abcd} & = \ERI{ab}{cd} + \sum_{i<j} \ERI{ab}{ij} t_{ij}^{ab}
%\end{align}
%\end{subequations}

%%%%%%%%%%%%%%%%%%%%%
%\section{Conclusion}
%%%%%%%%%%%%%%%%%%%%%
Here comes the conclusion.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\section*{Supplementary Material}
%\label{sec:supmat}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%%%%%%%%%%%%%%%%%%
\acknowledgments{
This project has received funding from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme (Grant agreement No.~863481).}
%%%%%%%%%%%%%%%%%%

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\section*{Data availability statement}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%The data that supports the findings of this study are available within the article and its supplementary material.

%%%%%%%%%
%\appendix
%%%%%%%%%

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\bibliography{CCvsMBPT}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\end{document}
