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CCvsMBPT.bib
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@ -1,13 +1,62 @@
%% This BibTeX bibliography file was created using BibDesk.
%% https://bibdesk.sourceforge.io/
%% Created for Pierre-Francois Loos at 2022-10-12 08:56:18 +0200
%% Created for Pierre-Francois Loos at 2022-10-12 10:53:04 +0200
%% Saved with string encoding Unicode (UTF-8)
@article{Forster_2020,
author = {F{\"o}rster, Arno and Visscher, Lucas},
date-added = {2022-10-12 10:52:33 +0200},
date-modified = {2022-10-12 10:52:55 +0200},
doi = {10.1021/acs.jctc.0c00693},
journal = {J. Chem. Theory Comput.},
number = {12},
pages = {7381-7399},
title = {Low-Order Scaling G0W0 by Pair Atomic Density Fitting},
volume = {16},
year = {2020},
bdsk-url-1 = {https://doi.org/10.1021/acs.jctc.0c00693}}
@article{Linner_2019,
author = {Linn\'er, E. and Aryasetiawan, F.},
date-added = {2022-10-12 10:39:35 +0200},
date-modified = {2022-10-12 10:39:45 +0200},
doi = {10.1103/PhysRevB.100.235106},
issue = {23},
journal = {Phys. Rev. B},
month = {Dec},
numpages = {6},
pages = {235106},
publisher = {American Physical Society},
title = {Ensemble Green's function theory for interacting electrons with degenerate ground states},
url = {https://link-aps-org-s.docadis.univ-tlse3.fr/doi/10.1103/PhysRevB.100.235106},
volume = {100},
year = {2019},
bdsk-url-1 = {https://link-aps-org-s.docadis.univ-tlse3.fr/doi/10.1103/PhysRevB.100.235106},
bdsk-url-2 = {https://doi.org/10.1103/PhysRevB.100.235106}}
@article{Brouder_2009,
author = {Brouder, Christian and Panati, Gianluca and Stoltz, Gabriel},
date-added = {2022-10-12 10:38:17 +0200},
date-modified = {2022-10-12 10:38:26 +0200},
doi = {10.1103/PhysRevLett.103.230401},
issue = {23},
journal = {Phys. Rev. Lett.},
month = {Dec},
numpages = {4},
pages = {230401},
publisher = {American Physical Society},
title = {Many-Body Green Function of Degenerate Systems},
url = {https://link.aps.org/doi/10.1103/PhysRevLett.103.230401},
volume = {103},
year = {2009},
bdsk-url-1 = {https://link.aps.org/doi/10.1103/PhysRevLett.103.230401},
bdsk-url-2 = {https://doi.org/10.1103/PhysRevLett.103.230401}}
@article{Lyakh_2012,
author = {Lyakh, Dmitry I. and Musia{\l}, Monika and Lotrich, Victor F. and Bartlett, Rodney J.},
date-added = {2022-10-12 08:56:00 +0200},

126
CCvsMBPT.tex
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@ -169,7 +169,7 @@ In particular, it may provide a path for the computation of ground- and excited-
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, \cite{Ren_2012,Chen_2017} particle-hole fermionic excitations and deexcitations are assumed to be bosons.
Because ph-RPA takes into account dynamical screening by summing up to infinity the (time-independent) 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}
Another important feature of ph-RPA compared to finite-order perturbation theory is that it does not present any divergence for small-gap or metallic systems. \cite{Gell-Mann_1957}
Another important feature of ph-RPA compared to finite-order perturbation theory is that it does not exhibit divergences for small-gap or metallic systems. \cite{Gell-Mann_1957}
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 $W$.
@ -321,7 +321,7 @@ 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 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 independently by Peng \textit{et al.} \cite{Peng_2013} and Scuseria \textit{et al.} \cite{Scuseria_2013}
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}
(See also Ref.~\onlinecite{Berkelbach_2018} for the extension to excitation energies for the pp and hh channels.)
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
@ -401,7 +401,6 @@ with
B_{ia,jb}^{\dRPA} & = \ERI{ij}{ab}
\end{align}
\end{subequations}
%\ant{We maybe we can remove the above equations and just say that it's Eq.~\eqref{eq:RPA}\eqref{eq:A_RPAx}\eqref{eq:B_RPAx} without double bars. If we keep the equations, I think the GW superscript on the energies is wrong.}
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
@ -421,7 +420,7 @@ can be equivalently obtained via a set of rCCD-like amplitude equations, where o
\end{multline}
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}
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}
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,
@ -614,119 +613,9 @@ 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}
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 judicious intermediates.
Cholesky decomposition, density fitting, and other related techniques might be employed to further reduce this scaling as it is done in conventional $GW$ calculations. \cite{Bintrim_2021,Forster_2020,Forster_2021,Duchemin_2019,Duchemin_2020,Duchemin_2021}
The $G_0W_0$ quasiparticle energies can be easily obtained via the procedure described in Ref.~\onlinecite{Monino_2022} by solving the previous equations for each value of $p$ separately.
%======================
%\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}
%%%%%%%%%%%%%%%%%%%%%
@ -734,8 +623,10 @@ Here, we have unveiled exact similarities between CC and many-body perturbation
More specifically, we have shown how to recast $GW$ and BSE as non-linear CC-like equations that can be solved with the usual CC machinery at the same computational cost.
The conventional and CC-based versions of the BSE and $GW$ schemes that we have described in the present work have been implemented in the electronic structure package QuAcK \cite{QuAcK} (available at \url{https://github.com/pfloos/QuAcK}) with which we have numerically checked these exact equivalences.
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 path 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.
Thanks to the connections between CC and $GW$, it could also provide new directions for the development of multireference $GW$ methods. \cite{Lyakh_2012,Evangelista_2018}
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 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}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\section*{Supplementary Material}
@ -744,6 +635,7 @@ Thanks to the connections between CC and $GW$, it could also provide new directi
%%%%%%%%%%%%%%%%%%
\acknowledgments{
PFL thanks Xavier Blase, Pina Romaniello, and Francesco Evangelista for useful discussions.
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).}
%%%%%%%%%%%%%%%%%%