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8
CCvsMBPT.bib
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@ -1,7 +1,7 @@
%% This BibTeX bibliography file was created using BibDesk. %% This BibTeX bibliography file was created using BibDesk.
%% https://bibdesk.sourceforge.io/ %% http://bibdesk.sourceforge.net/
%% Created for Pierre-Francois Loos at 2022-10-12 13:42:59 +0200 %% Created for Pierre-Francois Loos at 2022-11-16 16:20:18 +0100
%% Saved with string encoding Unicode (UTF-8) %% Saved with string encoding Unicode (UTF-8)
@ -6382,10 +6382,10 @@
year = {2014}, year = {2014},
bdsk-url-1 = {https://doi.org/10.1063/1.4871875}} bdsk-url-1 = {https://doi.org/10.1063/1.4871875}}
@book{Schuck_Book, @book{Schuck_book,
author = {P. Ring and P. Schuck}, author = {P. Ring and P. Schuck},
date-added = {2020-01-04 20:15:01 +0100}, date-added = {2020-01-04 20:15:01 +0100},
date-modified = {2020-01-04 20:16:49 +0100}, date-modified = {2022-11-16 16:17:45 +0100},
publisher = {Springer}, publisher = {Springer},
title = {The Nuclear Many-Body Problem}, title = {The Nuclear Many-Body Problem},
year = {2004}} year = {2004}}

13
CCvsMBPT.tex
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@ -274,7 +274,7 @@ 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. 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. 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{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} There are various ways of computing the RPAx correlation energy, \cite{Furche_2008,Jansen_2010,Angyan_2011} but the usual plasmon (or trace) formula \cite{Sawada_1957a,Rowe_1968,Schuck_book} 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} \begin{equation}
\label{eq:EcRPAx} \label{eq:EcRPAx}
\Ec^\text{RPAx} = \frac{1}{4} \Tr(\bOm{}{} - \bA{}{}) \Ec^\text{RPAx} = \frac{1}{4} \Tr(\bOm{}{} - \bA{}{})
@ -421,7 +421,9 @@ can be equivalently obtained via a set of rCCD-like amplitude equations, where o
\end{multline} \end{multline}
with $\Delta_{ijab}^{\GW} = \e{a}{\GW} + \e{b}{\GW} - \e{i}{\GW} - \e{j}{\GW}$. 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. 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,DiSabatino_2021} or the ACFDT. \cite{Maggio_2016,Holzer_2018b,Loos_2020e,Berger_2021,DiSabatino_2021} As in the case of RPAx (see Sec.~\ref{sec:RPAx}), several variants of the BSE correlation energy do exist, \footnote{\alert{To the best of our knowledge, the trace (or plasmon) formula has been introduced first by Sawada \cite{Sawada_1957a} to calculate the correlation energy of the uniform electron gas as an alternative to the Gell-Mann-Brueckner formulation \cite{Gell-Mann_1957} where one integrates along the adiabatic connection path.
More precisely, the trace formula can be justified via the introduction of a quadratic Hamiltonian made of boson transition operators (quasiboson approximation). See Ref.~\onlinecite{Li_2020} for more details}.} either based on the plasmon formula \cite{Li_2020,Li_2021,DiSabatino_2021} or the ACFDT. \cite{Maggio_2016,Holzer_2018b,Loos_2020e,Berger_2021,DiSabatino_2021}
Following Berkelbach's analysis, \cite{Berkelbach_2018} one can extend the connection to excited states. 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, 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,
@ -433,7 +435,7 @@ Indeed, one can obtain an analog of the 1h1p block of the approximate EOM-rCCD H
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}. 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}. 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 $GW$ pre-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}. In other words, in the spirit of the Brueckner version of CCD, \cite{Handy_1989} \alert{the $GW$ pre-treatment renormalizes the bare one-electron energies and, consequently, incorporates mosaic \cite{Scuseria_2008,Scuseria_2013} as well as additional diagrams, \cite{Lange_2018} 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. 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 \cite{Stanton_1994,Musial_2003a} (up to 2h1p) and EA-EOM-CCSD \cite{Nooijen_1995,Musial_2003b} (up to 2p1h) calculations prior to the EOM-CC treatment, which can then be reduced to the 1h1p sector thanks to this partial decoupling. At the CCSD level, for example, this is achieved by performing IP-EOM-CCSD \cite{Stanton_1994,Musial_2003a} (up to 2h1p) and EA-EOM-CCSD \cite{Nooijen_1995,Musial_2003b} (up to 2p1h) calculations prior to the EOM-CC treatment, which can then be reduced to the 1h1p sector thanks to this partial decoupling.
@ -441,7 +443,6 @@ At the CCSD level, for example, this is achieved by performing IP-EOM-CCSD \cite
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). 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 $GW$. 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 is explicitly treated as frequency-dependent in Eq.~\eqref{eq:BSE}] takes partially into account the 2h2p configurations. \cite{Strinati_1980,Strinati_1982,Strinati_1984,Strinati_1988,Rohlfing_2000,Romaniello_2009b,Loos_2020h,Authier_2020,Monino_2021,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_1980,Strinati_1982,Strinati_1984,Strinati_1988,Rohlfing_2000,Romaniello_2009b,Loos_2020h,Authier_2020,Monino_2021,Bintrim_2022}
\titou{More equations and/or figures?}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Connection between $GW$ and CC} \section{Connection between $GW$ and CC}
@ -462,7 +463,8 @@ In the most general setting, the quasiparticle energies and their corresponding
\label{eq:GW} \label{eq:GW}
\qty[ \be{}{} + \bSig{}{\GW}\qty(\omega = \e{p}{\GW}) ] \SO{p}{\GW} = \e{p}{\GW} \SO{p}{\GW} \qty[ \be{}{} + \bSig{}{\GW}\qty(\omega = \e{p}{\GW}) ] \SO{p}{\GW} = \e{p}{\GW} \SO{p}{\GW}
\end{equation} \end{equation}
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 \alert{which gives also access to the satellite solutions}.
In Eq.~\eqref{eq:GW}, $\be{}{}$ 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{equation}
\begin{split} \begin{split}
\Sig{pq}{\GW}(\omega) \Sig{pq}{\GW}(\omega)
@ -612,7 +614,6 @@ To determine the 2h1p and 2p1h amplitudes, $t_{ija,p}^{\text{2h1p}}$ and $t_{iab
t_{iab,p}^{\text{2p1h}} & \leftarrow t_{iab,p}^{\text{2p1h}} - \qty( \Delta_{iab,p}^{\text{2p1h}} )^{-1} r_{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{align}
\end{subequations} \end{subequations}
\titou{Maybe analyze further these equations and mention that we obtain an eigensystem, not an energy like in CC?}
The quasiparticle energies $\e{p}{GW}$ are thus provided by the eigenvalues of $\be{}{} + \bSig{}{\GW}$, where The quasiparticle energies $\e{p}{GW}$ are thus provided by the eigenvalues of $\be{}{} + \bSig{}{\GW}$, where
\begin{equation} \begin{equation}

29
Response_Letter/Response_Letter.tex
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@ -38,7 +38,8 @@ J. Chem. Phys. is a good venue for this work, and I am happy to recommend public
} }
\\ \\
\alert{ \alert{
bla bla bla We thank the reviewer for supporting the publication of the present manuscript.
Below, we address his/her suggestions.
} }
\begin{enumerate} \begin{enumerate}
@ -52,7 +53,15 @@ If so, I think a few words to make this connection would be valuable to the read
It's not obvious (to me at least) that all methods that allow a Tamm-Dancoff approximation can rigorously be related to a correlation energy expression in the form of the plasmon trace formula.} It's not obvious (to me at least) that all methods that allow a Tamm-Dancoff approximation can rigorously be related to a correlation energy expression in the form of the plasmon trace formula.}
\\ \\
\alert{ \alert{
bla bla bla Like in the case of RPA, there are many ways of defining the correlation energy at the BSE level (as mentioned in the original manuscript).
The trace formula (9) is one of them and it has been recently used to compute the potential energy surfaces of diatomics [see Refs.~(62) and (63)].
Another way is to rely on the adiabatic-connection fluctuation-dissipation theorem as done by Holzer et al.~[see Ref.~(37)] and some of the authors [see Ref.~(38)].
\\
To the best of our knowledge, the trace (or plasmon) formula has been introduced first by Sawada to calculate the correlation energy of the uniform electron gas as an alternative to the Gell-Mann \& Brueckner formula which integrates along the adiabatic connection path.
These two approaches were later found to be equivalent in the case of direct RPA for both homogeneous and inhomogeneous systems.
\\
More precisely, the trace formula can be justified via the introduction of a quadratic Hamiltonian made of boson transition operators (quasiboson approximation) as discussed in Refs.~(62).
A footnote gathering these information has been added to the revised version of the manuscript alongside additional references.
} }
\item \item
@ -61,7 +70,8 @@ The mosaic diagrams are the only ones that are captured at the CCSD level, but i
} }
\\ \\
\alert{ \alert{
bla bla bla The reviewer is right.
We have modified this statement as ``The $GW$ pre-treatment renormalizes the bare one-electron energies and, consequently, incorporates mosaic as well as additional diagrams, a process named Brueckner-like dressing in Ref.~46.''
} }
\item \item
@ -70,7 +80,7 @@ bla bla bla
} }
\\ \\
\alert{ \alert{
bla bla bla The reviewer is right again. We have mentioned this just below Eq.~(24).
} }
\item \item
@ -118,8 +128,9 @@ However, I reemphasize that in both cases, the ``CC-like'' equations that result
} }
\\ \\
\alert{ \alert{
bla bla bla We believe that the connection between $GW$ and $CC$ is more profound and this is nicely illustrated by the work of Lange and Berkelbach [see Ref.~
} This nonlinear equation has more similarities between drCCD and dRPA that Scuseria already described in Ref[44].
As it is discussed in the conclusion : If we consider that this non linear eigenvalue problem comes from MBPT there is a conection with CC at the ground and excited state level.}
\item \item
{A confusion I have in my own discussion above is related to a question I have about the final Eq.~(34) of the present work. {A confusion I have in my own discussion above is related to a question I have about the final Eq.~(34) of the present work.
@ -130,7 +141,11 @@ What am I missing here?
If the authors appreciate this confusion, then some discussion in the manuscript would be welcome.} If the authors appreciate this confusion, then some discussion in the manuscript would be welcome.}
\\ \\
\alert{ \alert{
bla bla bla This is explicitly stated before Eq.~(29) as ``Let us suppose that we are looking for the $N$ ``principal'' (i.e., quasiparticle) solutions of the eigensystem (26).''
However, we understand the confusion of the reviewer.
Let us be more precise.
The iterative algorithm proposed here [see Eqs.~(33)] could potentially converge to satellite solutions, and this is also the case at the CC level when one relies on a Newton-Raphson algorithm to converge to higher-energy solutions by modifying the updating procedure in Eqs.~(33) [see JCTC 17 (2021) 4756. ].
We discuss further this point below Eq.~(34).
} }
\end{enumerate} \end{enumerate}