saving work in CC part

CCvsMBPT.tex

@@ -38,6 +38,7 @@
 \newcommand{\NO}[1]{\{#1\}}
 \newcommand{\HF}{\text{HF}}
 \newcommand{\GW}{GW}
+\newcommand{\GOWO}{G_0W_0}
 \newcommand{\dRPA}{\text{dRPA}}
 \newcommand{\RPAx}{\text{RPAx}}
 \newcommand{\BSE}{\text{BSE}}
@@ -83,13 +84,14 @@
 
 
 % Matrix elements
-\newcommand{\MO}[1]{\phi_{#1}}
+\newcommand{\MO}[2]{\phi_{#1}^{#2}}
-\newcommand{\SO}[1]{\psi_{#1}}
+\newcommand{\SO}[2]{\psi_{#1}^{#2}}
 \newcommand{\ERI}[2]{\braket{#1}{#2}}
 \newcommand{\sERI}[2]{(#1|#2)}
 \newcommand{\dbERI}[2]{\mel{#1}{}{#2}}
 \newcommand{\wERI}[2]{\widetilde{W}_{#1 #2}}
 \newcommand{\SigC}[1]{\Sigma^\text{c}_{#1}}
+\newcommand{\Sig}[2]{\Sigma_{#1}^{#2}}
 
 % Matrices
 \newcommand{\bO}{\boldsymbol{0}}
@@ -111,6 +113,7 @@
 \newcommand{\bZ}[2]{\boldsymbol{Z}_{#1}^{#2}}
 \newcommand{\be}[2]{\boldsymbol{\epsilon}_{#1}^{#2}}
 \newcommand{\bSig}[2]{\boldsymbol{\Sigma}_{#1}^{#2}}
+\newcommand{\bSigC}[1]{\boldsymbol{\Sigma}^\text{c}_{#1}}
 
 % orbitals, gaps, etc
 \newcommand{\IP}{I}
@@ -146,7 +149,8 @@
 
 \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 combined with the $GW$ 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.
+In particular, we show how to recast the $GW$ and Bethe-Salpeter equations as non-linear CC-like equations.
+Similitudes between BSE@$GW$ and the similarity-transformed equation-of-motion CC method introduced by Nooijen are also 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 $GW$ and BSE frameworks.
 %\bigskip
@@ -171,7 +175,7 @@ Effectively, $GW$ produces accurate ``charged'' excitations and provides a faith
 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}
 
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-\section{Connections between RPA and CC}
+\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,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.
@@ -258,10 +262,10 @@ where
 \begin{equation}
 	\Delta_{ijab} = \e{a}{} + \e{b}{} - \e{i}{} - \e{j}{}
 \end{equation}
-We assume real quantities throughout this paper, $\e{p}{}$ is the one-electron energy associated with the Hartree-Fock spinorbital $\SO{p}(\bx)$ and 
+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
+	\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}
@@ -307,6 +311,7 @@ and the two-body term is
 \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}
+\label{eq:EOM-rCCD}
 \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}
@@ -322,10 +327,10 @@ This has been also extended to the pp and hh sectors by Peng \textit{et al.} \ci
 (See also Ref.~\onlinecite{Berkelbach_2018} for the extension to excitation energies for the pp and hh channels.)
 
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-\section{Connections between BSE and CC}
+\section{Connection between BSE and CC}
 \label{sec:BSE}
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-At the BSE level, and within the static approximation, one must solve a very similar linear eigenvalue problem
+Within the usual static approximation of BSE, one must solve a very similar linear eigenvalue problem
 \begin{equation}
 	\label{eq:BSE}
 	\begin{pmatrix}
@@ -392,7 +397,7 @@ with
 \begin{subequations}
 \begin{align}
 	\label{eq:A_dRPA}
-	A_{ia,jb}^{\dRPA} & = \delta_{ij} \delta_{ab} (\e{a}{} - \e{i}{}) + \ERI{ib}{aj}
+	A_{ia,jb}^{\dRPA} & = \delta_{ij} \delta_{ab} (\e{a}{\GW} - \e{i}{\GW}) + \ERI{ib}{aj}
 	\\ 
 	\label{eq:B_dRPA}
 	B_{ia,jb}^{\dRPA} & = \ERI{ij}{ab}
@@ -402,9 +407,9 @@ with
 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 presence of the $GW$ quasiparticle energies in the one-body terms and the screening of the electron-electron interaction [see Eq.~\eqref{eq:W}] in the two-body terms, correlation effects at the BSE level.
 Therefore, following the derivation detailed in Sec.~\ref{sec:RPAx}, 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})
+	\Ec^\text{BSE} = \frac{1}{4} \Tr(\bOm{}{\BSE} - \bA{}{\BSE}) = \frac{1}{4} \sum_{ijab} \wERI{ij}{ab} \tilde{t}_{ij}^{ab} 
 \end{equation}
-can be obtained via a set of rCCD-like amplitude equations, where one substitutes in Eq.~\eqref{eq:rCCD} the HF orbital energies by the $GW$ quasiparticle energies and all the antisymmetrized two-electron integrals $\dbERI{pq}{rs}$ by $\wERI{pq}{rs} = \dbERI{pq}{rs} - W_{ps,qr}^\text{stat}$, \ie,
+can be equivalently obtained via a set of rCCD-like amplitude equations $\tilde{t}_{ij}^{ab}$, where one substitutes in Eq.~\eqref{eq:rCCD} the HF orbital energies by the $GW$ quasiparticle energies and all the antisymmetrized two-electron integrals $\dbERI{pq}{rs}$ by $\wERI{pq}{rs} = \dbERI{pq}{rs} - W_{ps,qr}^\text{stat}$, \ie,
 \begin{multline}
 \label{eq:rCCD-BSE}
 	\Tilde{r}_{ij}^{ab}
@@ -422,47 +427,51 @@ with
 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 \eqref{eq:rCCD-BSE}] provides the same excitation energies as the conventional linear -response equations \eqref{eq:BSE}.
+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}.
-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.
+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}.
-In other words, in the spirit of the Brueckner version of CCD, the $GW$ quasiparticles renormalize the bare one-electron energies and can be seen as incorporating mosaic terms. \cite{Scuseria_2008,Scuseria_2013,Berkelbach_2018}
+In other words, in the spirit of the Brueckner version of CCD, the $GW$ quasiparticles renormalize the bare one-electron energies and, consequently, incorporate mosaic diagrams. \cite{Scuseria_2008,Scuseria_2013,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.
 At the CCSD level, for example, this is achieved by performing IP-EOM-CCSD (up to 2h1p) and EA-EOM-CCSD (up to 2p1h) calculations prior to the EOM-CC treatment, which can then be reduced to the 1h1p sector thanks to this partial decoupling.
 (An extended version of STEOM-CC has been proposed where the EOM treatment is pushed up to 2h2p. \cite{Nooijen_2000})
 Following the same philosophy, in the BSE@$GW$ method, one performs first a $GW$ calculation (which corresponds to an approximate and simultaneous treatment of the IP and EA sectors up to 2h1p and 2p1h \cite{Lange_2018,Monino_2022}) in order to renormalize the one-electron energies (see Sec.~\ref{sec:GW} for more details).
 Then, a static BSE calculation is performed in the 1h1p sector with a two-body term dressed with correlation stemming from $GW$.
-The dynamical version of BSE \cite{Strinati_1988,Rohlfing_2000,Romaniello_2009b,Loos_2020h,Authier_2020,Bintrim_2022} (where the BSE kernel \eqref{eq:W} is explicitly treated as frequency-dependent) takes partially into account the 2h2p configurations. 
+The dynamical version of BSE [where the BSE kernel given by Eq.~\eqref{eq:W} is explicitly treated as frequency-dependent] takes partially into account the 2h2p configurations. \cite{Strinati_1988,Rohlfing_2000,Romaniello_2009b,Loos_2020h,Authier_2020,Bintrim_2022} 
 
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-\section{Connections between $GW$ and CC}
+\section{Connection between $GW$ and CC}
 \label{sec:GW}
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 
-Because $GW$ is able to capture key correlation effects as illustrated above, it is therefore interesting to investigate if it possible also the recast the $GW$ equations as a set of CC-like equations.
+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- 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 that do not decouple the 2h1p and 2p1h sectors.
+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.
 
 Quite unfortunately, there are several ways of computing $GW$ quasiparticle energies.
-For example, at the one-shot $GW$ level, or $G_0W_0$, \cite{Strinati_1980,Hybertsen_1985a,Hybertsen_1986,Godby_1988,Linden_1988,Northrup_1991,Blase_1994,Rohlfing_1995,Shishkin_2007} we have
+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).
+\cite{Strinati_1980,Hybertsen_1985a,Hybertsen_1986,Godby_1988,Linden_1988,Northrup_1991,Blase_1994,Rohlfing_1995,Shishkin_2007}
+Partial self-consistency can be attained via the \textit{``eigenvalue''} self-consistent $GW$ (ev$GW$)  \cite{Hybertsen_1986,Shishkin_2007,Blase_2011,Faber_2011,Rangel_2016,Gui_2018} or the quasiparticle self-consistent $GW$ (qs$GW$) \cite{Faleev_2004,vanSchilfgaarde_2006,Kotani_2007,Ke_2011,Kaplan_2016} schemes.
+
+In the most general setting, the quasiparticle energies and their corresponding orbitals are obtained by diagonalizing the so-called non-linear and frequency-dependent quasiparticle equation 
 \begin{equation}
-	\label{eq:G0W0}
+	\label{eq:GW}
-	\e{p}{\GW} = \e{p}{} + \SigC{pp}(\omega = \e{p}{\GW})
+	\qty[ \be{}{} + \bSig{}{\GW}\qty(\e{p}{\GW}) ] \SO{p}{\GW} = \e{p}{\GW} \SO{p}{\GW}
 \end{equation}
-where the elements of the correlation part of the dynamical self-energy are
+where $\be{}{}$ is a diagonal matrix gathering the HF orbital energies and the elements of the correlation part of the dynamical (and non-hermitian) $GW$ self-energy are
 \begin{equation}
 \begin{split}
-	\SigC{pq}(\omega) 
+	\Sig{pq}{\GW}(\omega) 
-	& = \sum_{im} \frac{\sERI{pi}{m} \sERI{qi}{m}}{\omega - \e{i}{} + \Om{m}{\dRPA} - \ii \eta}
+	& = \sum_{im} \frac{\sERI{pi}{m} \sERI{qi}{m}}{\omega - \e{i}{\GW} + \Om{m}{\dRPA} - \ii \eta}
 	\\
-	& + \sum_{am} \frac{\sERI{pa}{m} \sERI{qa}{m}}{\omega - \e{a}{} - \Om{m}{\dRPA} + \ii \eta}
+	& + \sum_{am} \frac{\sERI{pa}{m} \sERI{qa}{m}}{\omega - \e{a}{\GW} - \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}
+Because both the left- and right-hand sides of Eq.~\eqref{eq:GW} depend on $\e{p}{\GW}$, this equation has to be solved iteratively via a self-consistent process.
 
-
+As shown by Bintrim and Berkelbach, \cite{Bintrim_2021} the frequency-dependent $GW$ equations can be recast as a larger set of frequency-independent equations, which reads in the Tamm-Dancoff 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}
 \label{eq:GWlin}
 	\begin{pmatrix}
@@ -483,14 +492,14 @@ As shown by Bintrim and Berkelbach, \cite{Bintrim_2021} the non-linear $GW$ equa
 		\bY{}{\text{2p1h}}	\\
 	\end{pmatrix}
 	\cdot
-	\bom{}{}
+	\be{}{\GW}
 \end{equation}
-where $\be{}{}$ is a diagonal matrix gathered the orbital energies, the 2h1p and 2p1h matrix elements are
+where $\be{}{\GW}$ is a diagonal matrix gathering the quasiparticle 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{2h1p}_{ija,klc} & = \qty[ \qty( \e{i}{\GW} + \e{j}{\GW} - \e{a}{\GW}) \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} 
+	C^\text{2p1h}_{iab,kcd} & = \qty[ \qty( \e{a}{\GW} + \e{b}{\GW} - \e{i}{\GW}) \delta_{ik} \delta_{ac} + \ERI{ak}{ic} ] \delta_{bd} 
 \end{align}
 \end{subequations}
 and the corresponding coupling blocks read
@@ -502,8 +511,8 @@ and the corresponding coupling blocks read
 Going beyond the Tamm-Dancoff approximation is possible, but more cumbersome. \cite{Bintrim_2021}
 
 Let us suppose that we are looking for the $N$ ``principal'' (\ie, quasiparticle) solutions of the eigensystem \eqref{eq:GWlin}. 
-Therefore, $\bX{}{}$ and $\bom{}{}$ are of size $N \times N$, and we can safely define $\bX{}{-1}$.
+Therefore, $\bX{}{}$ and $\be{}{\GW}$ are of size $N \times N$.
-Introducing $\bT{}{\text{2h1p}} = \bY{}{\text{2h1p}} \cdot \bX{}{-1}$ and $\bT{}{\text{2p1h}} = \bY{}{\text{2p1h}} \cdot \bX{}{-1}$, we have
+Assuming the existence of $\bX{}{-1}$ and 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}}	\\
@@ -525,9 +534,7 @@ Introducing $\bT{}{\text{2h1p}} = \bY{}{\text{2h1p}} \cdot \bX{}{-1}$ and $\bT{}
 	\cdot
 	\bR{}{}
 \end{equation}
-with $\bR{}{} = \bX{}{} \cdot \bom{}{} \cdot \bX{}{-1}$.
+with $\bR{}{} = \bX{}{} \cdot \be{}{\GW} \cdot \bX{}{-1}$, which yields the three following equations
-
-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{}{}
@@ -540,7 +547,7 @@ This yields the three following equations
 	\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 
+Substituting Eq.~\eqref{eq:R} into Eqs.~\eqref{eq:T1R} and \eqref{eq:T2R}, one gets the two coupled Riccati equations 
 \begin{subequations}
 \begin{align}
 	\begin{split}
@@ -564,7 +571,7 @@ Substituting Eq.~\eqref{eq:R} into Eqs.~\eqref{eq:T1R} and \eqref{eq:T2R} yields
 	\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 residual equations
+In the CC language, in order to determine the 2h1p and 2p1h amplitudes, $t_{ija,p}^{\text{2h1p}}$ and $t_{iab,p}^{\text{2p1h}} $, one must solve the following residual equations
 \begin{subequations}
 \begin{align}
 	\begin{split}
@@ -575,6 +582,7 @@ In the CC language, in order to determine the 2h1p and 2p1h amplitudes, $t_{ija,
 		\\
 		& - \sum_{klcq} \ERI{qc}{kl} t_{ija,q}^{\text{2h1p}} t_{klc,p}^{\text{2h1p}} 
 		- \sum_{kcdq} \ERI{qk}{dc} t_{ija,q}^{\text{2h1p}} t_{kcd,p}^{\text{2p1h}} 
+		= 0
 	\end{split}
 	\\
 	\begin{split}
@@ -585,6 +593,7 @@ In the CC language, in order to determine the 2h1p and 2p1h amplitudes, $t_{ija,
 		\\
 		& - \sum_{klcq} \ERI{qc}{kl} t_{iab,q}^{\text{2p1h}} t_{klc,p}^{\text{2h1p}} 
 		- \sum_{kcdq} \ERI{qk}{dc} t_{iab,q}^{\text{2p1h}} t_{kcd,p}^{\text{2p1h}} 
+		= 0
 	\end{split}
 \end{align}
 \end{subequations}
@@ -596,7 +605,7 @@ with
 	\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
+One can then employed the usual quasi-Newton iterative procedure to solve these quadratic 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_{ija,p}^{\text{2h1p}}}
@@ -610,8 +619,9 @@ The quasiparticle energies $\e{p}{GW}$ are thus provided by the eigenvalues of $
 	\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.
+\titou{Discuss cost and gradients? $\Lambda$ equations for $GW$?}
 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.
-
+\titou{It can be applied to other approximate self-energy such as GF2 and $T$-matrix.}
 
 %======================
 %\subsection{EE-EOM-CCD}
@@ -727,7 +737,7 @@ The $G_0W_0$ quasiparticle energies can be easily obtained via the procedure des
 %%%%%%%%%%%%%%%%%%%%%
 \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 proof of equivalence.
+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.
 %Link between BSE and STEOM-CC.
 %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