From f2c9484baea44c7913876aab1c62b6f0437be4f9 Mon Sep 17 00:00:00 2001 From: Pierre-Francois Loos Date: Tue, 11 Oct 2022 11:00:07 +0200 Subject: [PATCH] almost ok with GW section --- CCvsMBPT.bib | 42 ++++++++++++++++++++++------------ CCvsMBPT.tex | 64 ++++++++++++++++++++-------------------------------- 2 files changed, 52 insertions(+), 54 deletions(-) diff --git a/CCvsMBPT.bib b/CCvsMBPT.bib index 909b20f..fe3d760 100644 --- a/CCvsMBPT.bib +++ b/CCvsMBPT.bib @@ -1,13 +1,40 @@ %% This BibTeX bibliography file was created using BibDesk. %% http://bibdesk.sourceforge.net/ -%% Created for Pierre-Francois Loos at 2022-10-10 16:49:45 +0200 +%% Created for Pierre-Francois Loos at 2022-10-11 10:48:48 +0200 %% Saved with string encoding Unicode (UTF-8) +@article{Loos_2022, + author = {Loos,Pierre-Fran{\c c}ois and Romaniello,Pina}, + date-added = {2022-10-11 10:48:31 +0200}, + date-modified = {2022-10-11 10:48:31 +0200}, + doi = {10.1063/5.0088364}, + journal = {J. Chem. Phys.}, + number = {16}, + pages = {164101}, + title = {Static and dynamic Bethe--Salpeter equations in the T-matrix approximation}, + volume = {156}, + year = {2022}, + bdsk-url-1 = {https://doi.org/10.1063/5.0088364}} + +@article{Chen_2017, + abstract = { Random-phase approximation (RPA) methods are rapidly emerging as cost-effective validation tools for semilocal density functional computations. We present the theoretical background of RPA in an intuitive rather than formal fashion, focusing on the physical picture of screening and simple diagrammatic analysis. A new decomposition of the RPA correlation energy into plasmonic modes leads to an appealing visualization of electron correlation in terms of charge density fluctuations. Recent developments in the areas of beyond-RPA methods, RPA correlation potentials, and efficient algorithms for RPA energy and property calculations are reviewed. The ability of RPA to approximately capture static correlation in molecules is quantified by an analysis of RPA natural occupation numbers. We illustrate the use of RPA methods in applications to small-gap systems such as open-shell d- and f-element compounds, radicals, and weakly bound complexes, where semilocal density functional results exhibit strong functional dependence. }, + author = {Chen, Guo P. and Voora, Vamsee K. and Agee, Matthew M. and Balasubramani, Sree Ganesh and Furche, Filipp}, + date-added = {2022-10-11 09:40:36 +0200}, + date-modified = {2022-10-11 09:41:28 +0200}, + doi = {10.1146/annurev-physchem-040215-112308}, + journal = {Ann. Rev. Phys. Chem.}, + number = {1}, + pages = {421-445}, + title = {Random-Phase Approximation Methods}, + volume = {68}, + year = {2017}, + bdsk-url-1 = {https://doi.org/10.1146/annurev-physchem-040215-112308}} + @article{Nooijen_2000, author = {Nooijen,Marcel and Lotrich,Victor}, date-added = {2022-10-10 16:49:19 +0200}, @@ -313,19 +340,6 @@ year = {2008}, bdsk-url-1 = {https://doi.org/10.1063/1.3043729}} -@article{Loos_2022, - author = {Loos,Pierre-Fran{\c c}ois and Romaniello,Pina}, - date-added = {2022-04-27 14:39:38 +0200}, - date-modified = {2022-04-27 14:39:53 +0200}, - doi = {10.1063/5.0088364}, - journal = {J. Chem. Phys.}, - number = {16}, - pages = {164101}, - title = {Static and dynamic Bethe--Salpeter equations in the T-matrix approximation}, - volume = {156}, - year = {2022}, - bdsk-url-1 = {https://doi.org/10.1063/5.0088364}} - @article{Pokhilko_2021a, author = {Pokhilko,Pavel and Zgid,Dominika}, date-added = {2022-04-24 15:40:03 +0200}, diff --git a/CCvsMBPT.tex b/CCvsMBPT.tex index 087b13f..5b45c09 100644 --- a/CCvsMBPT.tex +++ b/CCvsMBPT.tex @@ -148,11 +148,11 @@ \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 combined with the $GW$ approximation from many-body perturbation theory and coupled-cluster (CC) theory at the ground- and excited-state levels. +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 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. 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. +In particular, it may provide a path for the computation of ground- and excited-state properties (such as nuclear gradients) within the $GW$ and BSE frameworks. %\bigskip %\begin{center} % \boxed{\includegraphics[width=0.5\linewidth]{TOC}} @@ -166,24 +166,24 @@ In particular, it provides a clear path for the computation of ground- and excit \section{RPA Physics and Beyond} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 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, particle-hole fermionic excitations and deexcitations are assumed to be 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 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 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 $W$. -Effectively, $GW$ produces accurate ``charged'' excitations and provides a faithful description of the fundamental gap, while the remaining excitonic effect (\ie, the stabilization provided by the attraction of the excited electron and the hole left behind) is caught via BSE, hence producing accurate ``neutral'' excitations. +Practically, $GW$ produces accurate ``charged'' excitations and provides 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 accurate ``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} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \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. +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. 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), this proof can be quickly summarised starting from the ubiquitous RPAx eigensystem +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 \begin{equation} \label{eq:RPA} \begin{pmatrix} @@ -258,11 +258,8 @@ knowing that B_{ia,jb} & = \dbERI{ij}{ab} \end{align} \end{subequations} -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 +where $\Delta_{ijab} = \e{a}{} + \e{b}{} - \e{i}{} - \e{j}{}$. +We assume real quantities throughout this paper, $\e{p}{}$ is the one-electron energy associated with the Hartree-Fock (HF) 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 @@ -276,7 +273,7 @@ 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{Jansen_2010,Angyan_2011} but the usual plasmon (or trace) formula yields\footnote{The factor $1/4$ in Eq.~\eqref{eq:EcRPAx} is sometimes replaced by a factor $1/2$, which corresponds to a different choice for the interaction kernel. See Ref.~\onlinecite{Angyan_2011} for more details} +There are various ways of computing the RPAx correlation energy, \cite{Furche_2008,Jansen_2010,Angyan_2011} but the usual plasmon (or trace) formula yields\footnote{The factor $1/4$ in Eq.~\eqref{eq:EcRPAx} is sometimes replaced by a factor $1/2$, which corresponds to a different choice for the interaction kernel. See Ref.~\onlinecite{Angyan_2011} for more details} \begin{equation} \label{eq:EcRPAx} \Ec^\text{RPAx} = \frac{1}{4} \Tr(\bOm{}{} - \bA{}{}) @@ -286,7 +283,7 @@ and matches the rCCD correlation energy \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, in the case of RPAx, the same expression as Eq.~\eqref{eq:EcRPAx} can be derived from the adiabatic connection fluctuation dissipation theorem (ACFDT). \cite{Angyan_2011} +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. @@ -322,7 +319,7 @@ Neglecting the effect of $\hT_2$ on the one-body terms [see Eqs.~\eqref{eq:cFab} 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} -As we shall see below, the connection between a ph linear system 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 with the structure of Eq.~\eqref{eq:RPA}, such as time-dependent density-functional theory (TD-DFT), \cite{Runge_1984,Casida_1995} BSE, and many others. +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 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.) @@ -420,23 +417,20 @@ can be equivalently obtained via a set of rCCD-like amplitude equations $\tilde{ + \sum_{kc} \wERI{kb}{cj} \tilde{t}_{ik}^{ac} + \sum_{klcd} \wERI{kl}{cd} \tilde{t}_{ik}^{ac} \tilde{t}_{lj}^{db} = 0 \end{multline} -with -\begin{equation} - \Delta_{ijab}^{\GW} = \e{a}{\GW} + \e{b}{\GW} - \e{i}{\GW} - \e{j}{\GW} -\end{equation} +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}. 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, consequently, incorporate mosaic diagrams. \cite{Scuseria_2008,Scuseria_2013,Berkelbach_2018} +In other words, in the spirit of the Brueckner version of CCD, 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. 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). +Following the same philosophy, in BSE@$GW$, 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 [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} +The dynamical version of BSE [where the BSE kernel is explicitly treated as frequency-dependent in Eq.~\eqref{eq:BSE}] takes partially into account the 2h2p configurations. \cite{Strinati_1988,Rohlfing_2000,Romaniello_2009b,Loos_2020h,Authier_2020,Bintrim_2022} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \section{Connection between $GW$ and CC} @@ -465,11 +459,7 @@ where $\be{}{}$ is a diagonal matrix gathering the HF orbital energies and the e & + \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. +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 procedure. 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 \begin{equation} @@ -547,7 +537,7 @@ with $\bR{}{} = \bX{}{} \cdot \be{}{\GW} \cdot \bX{}{-1}$, which yields the thre \label{eq:T2R} \end{align} \end{subequations} -Substituting Eq.~\eqref{eq:R} into Eqs.~\eqref{eq:T1R} and \eqref{eq:T2R}, one gets the two coupled Riccati equations +Substituting Eq.~\eqref{eq:R} into Eqs.~\eqref{eq:T1R} and \eqref{eq:T2R}, one gets two coupled Riccati equations \begin{subequations} \begin{align} \begin{split} @@ -574,6 +564,7 @@ Substituting Eq.~\eqref{eq:R} into Eqs.~\eqref{eq:T1R} and \eqref{eq:T2R}, one g 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} + \label{eq:r_2h1p} \begin{split} r_{ija,p}^{\text{2h1p}} & = \ERI{pa}{ij} @@ -585,6 +576,7 @@ In the CC language, in order to determine the 2h1p and 2p1h amplitudes, $t_{ija, = 0 \end{split} \\ + \label{eq:r_2p1h} \begin{split} r_{iab,p}^{\text{2p1h}} & = \ERI{pi}{ba} @@ -597,20 +589,13 @@ In the CC language, in order to determine the 2h1p and 2p1h amplitudes, $t_{ija, \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} +with $\Delta_{ija,p}^{\text{2h1p}} = \e{i}{} + \e{j}{} - \e{a}{} - \e{p}{}$ and $\Delta_{iab,p}^{\text{2p1h}} = \e{a}{} + \e{b}{} - \e{i}{} - \e{p}{}$. 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}}} + t_{ija,p}^{\text{2h1p}} & \leftarrow t_{ija,p}^{\text{2h1p}} - \qty( \Delta_{ija,p}^{\text{2h1p}} )^{-1} r_{ija,p}^{\text{2h1p}} \\ - t_{iab,p}^{\text{2h1p}} & \leftarrow t_{iab,p}^{\text{2p1h}} - \frac{r_{iab,p}^{\text{2p1h}}}{\Delta_{iab,p}^{\text{2p1h}}} + t_{iab,p}^{\text{2p1h}} & \leftarrow t_{iab,p}^{\text{2p1h}} - \qty( \Delta_{iab,p}^{\text{2p1h}} )^{-1} r_{iab,p}^{\text{2p1h}} \end{align} \end{subequations} @@ -618,10 +603,10 @@ 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} -is the correlation part of the $GW$ self-energy. +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 $GW$?} 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.} +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} @@ -738,7 +723,6 @@ 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 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 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%