minor corrections again

CCvsMBPT.tex

@@ -290,9 +290,9 @@ This simple and elegant proof was subsequently extended to excitation energies b
 
 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}
+	\mel*{ \Psi_{i}^{a} }{ \bHN}{ \Psi_{j}^{b} } = \cF_{ab} \delta_{ij} - \cF_{ij} \delta_{ab} + \cW_{jabi}
 \end{equation}
-where \textcolor{blue}{ $\bHN = e^{-\hT} \hH_{N} e^{\hT} $} is the normal-ordered similarity-transformed Hamiltonian, $\Psi_{i}^{a}$ are singly-excited determinants, the one-body terms are
+where $\bHN = e^{-\hT} \hH_{N} e^{\hT} - \ECC $ is the (shifted) similarity-transformed Hamiltonian, $\Psi_{i}^{a}$ are singly-excited determinants, the one-body terms are
 \begin{subequations}
 \begin{align}
 \label{eq:cFab}
@@ -319,6 +319,7 @@ Neglecting the effect of $\hT_2$ on the one-body terms [see Eqs.~\eqref{eq:cFab}
 \end{equation}
 which exactly matches Eq.~\eqref{eq:RPA_1}.
 Although the excitation energies of this approximate 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}
+Equation \eqref{eq:EOM-rCCD} can be more systematically derived through the formulation of $\Lambda$ equations based on a rCCD effective Hamiltonian, as proposed by Rishi \textit{et al.} \cite{Rishi_2020}
 
 As we shall see below, the connection between a ph eigensystem with the structure of Eq.~\eqref{eq:RPA} and a set of CC-like amplitude equations does not hold only for RPAx as it is actually quite general and can be applied to most ph problems, such as time-dependent density-functional theory (TD-DFT), \cite{Runge_1984,Casida_1995} BSE, and many others.
 This analysis has also been extended to the pp and hh sectors independently by Peng \textit{et al.} \cite{Peng_2013} and Scuseria \textit{et al.} \cite{Scuseria_2013} 
@@ -429,7 +430,8 @@ Indeed, one can obtain an analog of the 1h1p block of the approximate EOM-rCCD H
 	\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}.
+This equation provides the same excitation energies as the conventional linear-response equations \eqref{eq:BSE}, and the corresponding $\Lambda$ equations based on the BSE effective Hamiltonian $\tHN$ can be derived following Ref.~\onlinecite{Rishi_2020}.
+
 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}.
 
@@ -503,11 +505,13 @@ where $\be{}{\GW}$ is a diagonal matrix collecting the quasiparticle energies, t
 \end{subequations}
 and the corresponding coupling blocks read
 \begin{align}
-	V^\text{2h1p}_{p,klc} & =\textcolor{blue}{\sqrt{2}}\ERI{pc}{kl}
+	V^\text{2h1p}_{p,klc} & = \ERI{pc}{kl}
 	&
-	V^\text{2p1h}_{p,kcd} & =\textcolor{blue}{\sqrt{2}}  \ERI{pk}{dc}
+	V^\text{2p1h}_{p,kcd} & = \ERI{pk}{dc}
 \end{align}
 Going beyond the Tamm-Dancoff approximation is possible, but more cumbersome. \cite{Bintrim_2021}
+Note that, contrary to the IP/EA-EOM-CC equations, $GW$ does couple the IP and EA sectors due to the lack of exponential parametrization of the wave function. \cite{Nooijen_1995,Rishi_2020}
+However, it allows to generate higher-order diagrams. \cite{Lange_2018,Schirmer_2018}
 
 Let us suppose that we are looking for the $N$ \textit{``principal''} (\ie, quasiparticle) solutions of the eigensystem \eqref{eq:GWlin}. 
 Therefore, $\bX{}{}$ and $\be{}{\GW}$ are square matrices of size $N \times N$.
@@ -625,7 +629,7 @@ The conventional and CC-based versions of the BSE and $GW$ schemes that we have
 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 consistent approach 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.
-However, several challenges lie ahead as one must derive, for example, the associated $\Lambda$ equations \cite{Bartlett_1986,Rishi_2020} and the response of the static screening with respect to the external perturbation at the BSE level.
+However, several challenges lie ahead as one must derive, for example, the $\Lambda$ equations associated with $GW$\cite{Bartlett_1986,Rishi_2020} and the response of the static screening with respect to the external perturbation at the BSE level.
 The present connections between CC and $GW$ could also provide new directions for the development of multireference $GW$ methods \cite{Brouder_2009,Linner_2019} in order to treat strongly correlated systems. \cite{Lyakh_2012,Evangelista_2018}
 
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