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CCvsMBPT.bib
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%% This BibTeX bibliography file was created using BibDesk. %% This BibTeX bibliography file was created using BibDesk.
%% http://bibdesk.sourceforge.net/ %% http://bibdesk.sourceforge.net/
%% Created for Pierre-Francois Loos at 2022-10-05 14:52:00 +0200 %% Created for Pierre-Francois Loos at 2022-10-10 16:49:45 +0200
%% Saved with string encoding Unicode (UTF-8) %% Saved with string encoding Unicode (UTF-8)
@article{Nooijen_2000,
author = {Nooijen,Marcel and Lotrich,Victor},
date-added = {2022-10-10 16:49:19 +0200},
date-modified = {2022-10-10 16:49:45 +0200},
doi = {10.1063/1.481828},
journal = {J. Chem. Phys.},
number = {2},
pages = {494-507},
title = {Extended similarity transformed equation-of-motion coupled cluster theory (extended-STEOM-CC): Applications to doubly excited states and transition metal compounds},
volume = {113},
year = {2000},
bdsk-url-1 = {https://doi.org/10.1063/1.481828}}
@article{Nooijen_1997c,
author = {Nooijen,Marcel and Bartlett,Rodney J.},
date-added = {2022-10-10 16:37:36 +0200},
date-modified = {2022-10-10 16:38:50 +0200},
doi = {10.1063/1.473635},
journal = {J. Chem. Phys.},
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title = {Similarity transformed equation-of-motion coupled-cluster study of ionized, electron attached, and excited states of free base porphin},
volume = {106},
year = {1997},
bdsk-url-1 = {https://doi.org/10.1063/1.473635}}
@article{Nooijen_1997b,
author = {Nooijen,Marcel and Bartlett,Rodney J.},
date-added = {2022-10-10 16:37:17 +0200},
date-modified = {2022-10-10 16:38:55 +0200},
doi = {10.1063/1.474922},
journal = {J. Chem. Phys.},
number = {17},
pages = {6812-6830},
title = {Similarity transformed equation-of-motion coupled-cluster theory: Details, examples, and comparisons},
volume = {107},
year = {1997},
bdsk-url-1 = {https://doi.org/10.1063/1.474922}}
@article{Nooijen_1997a,
author = {Nooijen,Marcel and Bartlett,Rodney J.},
date-added = {2022-10-10 16:36:53 +0200},
date-modified = {2022-10-10 16:38:27 +0200},
doi = {10.1063/1.474000},
journal = {J. Chem. Phys.},
number = {15},
pages = {6441-6448},
title = {A new method for excited states: Similarity transformed equation-of-motion coupled-cluster theory},
volume = {106},
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bdsk-url-1 = {https://doi.org/10.1063/1.474000}}
@book{Shavitt_2009,
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date-added = {2022-10-10 10:46:31 +0200},
date-modified = {2022-10-10 10:46:31 +0200},
doi = {10.1017/CBO9780511596834},
file = {/home/antoinem/Zotero/storage/HCDGARAQ/Shavitt and Bartlett - 2009 - Many-Body Methods in Chemistry and Physics MBPT a.pdf;/home/antoinem/Zotero/storage/3B8MK5GF/D12027E4DAF75CE8214671D842C6B80C.html},
isbn = {978-0-521-81832-2},
publisher = {{Cambridge University Press}},
series = {Cambridge {{Molecular Science}}},
title = {Many-{{Body Methods}} in {{Chemistry}} and {{Physics}}: {{MBPT}} and {{Coupled}}-{{Cluster Theory}}},
year = {2009},
bdsk-url-1 = {https://doi.org/10.1017/CBO9780511596834}}
@article{Musial_2007,
author = {Musia{\l}, Monika and Bartlett, Rodney J.},
date-added = {2022-10-10 10:46:26 +0200},
date-modified = {2022-10-10 10:46:26 +0200},
doi = {10.1063/1.2747245},
file = {/home/antoinem/Zotero/storage/TT5MN29Q/Musia{\l} and Bartlett - 2007 - Addition by subtraction in coupled cluster theory..pdf;/home/antoinem/Zotero/storage/R8DKBIVQ/1.html},
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title = {Addition by Subtraction in Coupled Cluster Theory. {{II}}. {{Equation}}-of-Motion Coupled Cluster Method for Excited, Ionized, and Electron-Attached States Based on the {{nCC}} Ground State Wave Function},
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bdsk-url-1 = {https://doi.org/10.1063/1.2747245}}
@article{Bartlett_2007,
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date-added = {2022-10-10 10:46:26 +0200},
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bdsk-url-1 = {https://doi.org/10.1103/RevModPhys.79.291}}
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booktitle = {Reviews in {{Computational Chemistry}}},
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file = {/home/antoinem/Zotero/storage/SS7HANWJ/9780470125915.html},
isbn = {978-0-470-12591-5},
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publisher = {{John Wiley \& Sons, Ltd}},
title = {An {{Introduction}} to {{Coupled Cluster Theory}} for {{Computational Chemists}}},
year = {2000},
bdsk-url-1 = {https://doi.org/10.1002/9780470125915.ch2}}
@article{Paldus_1972,
author = {Paldus, J. and \ifmmode \check{C}\else \v{C}\fi{}\'{\i}\ifmmode \check{z}\else \v{z}\fi{}ek, J. and Shavitt, I.},
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doi = {10.1103/PhysRevA.5.50},
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month = {Jan},
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pages = {50--67},
publisher = {American Physical Society},
title = {Correlation Problems in Atomic and Molecular Systems. IV. Extended Coupled-Pair Many-Electron Theory and Its Application to the B${\mathrm{H}}_{3}$ Molecule},
url = {https://link.aps.org/doi/10.1103/PhysRevA.5.50},
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@article{Cizek_1966,
author = {{\v C}{\'\i}{\v z}ek, Ji{\v r}{\'\i}},
date-added = {2022-10-10 10:46:12 +0200},
date-modified = {2022-10-10 10:46:12 +0200},
doi = {10.1063/1.1727484},
file = {/home/antoinem/Zotero/storage/PR39PXU8/1.html},
journal = {J. Chem. Phys.},
pages = {4256--4266},
publisher = {{American Institute of Physics}},
title = {On the {{Correlation Problem}} in {{Atomic}} and {{Molecular Systems}}. {{Calculation}} of {{Wavefunction Components}} in {{Ursell}}-{{Type Expansion Using Quantum}}-{{Field Theoretical Methods}}},
volume = {45},
year = {1966},
bdsk-url-1 = {https://doi.org/10.1063/1.1727484}}
@article{Bohm_1953,
author = {Bohm, David and Pines, David},
date-added = {2022-10-10 10:36:26 +0200},
date-modified = {2022-10-10 10:36:34 +0200},
doi = {10.1103/PhysRev.92.609},
issue = {3},
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month = {Nov},
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pages = {609--625},
publisher = {American Physical Society},
title = {A Collective Description of Electron Interactions: III. Coulomb Interactions in a Degenerate Electron Gas},
url = {https://link.aps.org/doi/10.1103/PhysRev.92.609},
volume = {92},
year = {1953},
bdsk-url-1 = {https://link.aps.org/doi/10.1103/PhysRev.92.609},
bdsk-url-2 = {https://doi.org/10.1103/PhysRev.92.609}}
@article{Pines_1952,
author = {Pines, David and Bohm, David},
date-added = {2022-10-10 10:36:03 +0200},
date-modified = {2022-10-10 10:36:11 +0200},
doi = {10.1103/PhysRev.85.338},
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url = {https://link.aps.org/doi/10.1103/PhysRev.85.338},
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bdsk-url-1 = {https://link.aps.org/doi/10.1103/PhysRev.85.338},
bdsk-url-2 = {https://doi.org/10.1103/PhysRev.85.338}}
@article{Bohm_1951,
author = {Bohm, David and Pines, David},
date-added = {2022-10-10 10:35:34 +0200},
date-modified = {2022-10-10 10:35:41 +0200},
doi = {10.1103/PhysRev.82.625},
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bdsk-url-1 = {https://link.aps.org/doi/10.1103/PhysRev.82.625},
bdsk-url-2 = {https://doi.org/10.1103/PhysRev.82.625}}
@article{Rishi_2020, @article{Rishi_2020,
author = {Rishi,Varun and Perera,Ajith and Bartlett,Rodney J.}, author = {Rishi,Varun and Perera,Ajith and Bartlett,Rodney J.},
date-added = {2022-10-05 14:51:45 +0200}, date-added = {2022-10-05 14:51:45 +0200},
@ -10985,23 +11177,6 @@
year = {1999}, year = {1999},
bdsk-url-1 = {https://doi.org/10.1016/S1386-1425(98)00261-3}} bdsk-url-1 = {https://doi.org/10.1016/S1386-1425(98)00261-3}}
@article{Nooijen_2000,
author = {Nooijen, Marcel},
date-added = {2020-01-01 21:36:51 +0100},
date-modified = {2020-01-01 21:36:52 +0100},
doi = {10.1021/jp993983z},
issn = {1089-5639, 1520-5215},
journal = {J. Phys. Chem. A},
language = {en},
month = may,
number = {19},
pages = {4553-4561},
shorttitle = {Electronic {{Excitation Spectrum}} of {\emph{s}} -{{Tetrazine}}},
title = {Electronic {{Excitation Spectrum}} of {\emph{s}} -{{Tetrazine}}: {{An Extended}}-{{STEOM}}-{{CCSD Study}}},
volume = {104},
year = {2000},
bdsk-url-1 = {https://doi.org/10.1021/jp993983z}}
@article{Noro_2000, @article{Noro_2000,
author = {Noro, Takeshi and Sekiya, Masahiro and Koga, Toshikatsu and Matsuyama, Hisashi}, author = {Noro, Takeshi and Sekiya, Masahiro and Koga, Toshikatsu and Matsuyama, Hisashi},
date-added = {2020-01-01 21:36:51 +0100}, date-added = {2020-01-01 21:36:51 +0100},

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CCvsMBPT.tex
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@ -88,6 +88,7 @@
\newcommand{\ERI}[2]{\braket{#1}{#2}} \newcommand{\ERI}[2]{\braket{#1}{#2}}
\newcommand{\sERI}[2]{(#1|#2)} \newcommand{\sERI}[2]{(#1|#2)}
\newcommand{\dbERI}[2]{\mel{#1}{}{#2}} \newcommand{\dbERI}[2]{\mel{#1}{}{#2}}
\newcommand{\wERI}[2]{\widetilde{W}_{#1 #2}}
\newcommand{\SigC}[1]{\Sigma^\text{c}_{#1}} \newcommand{\SigC}[1]{\Sigma^\text{c}_{#1}}
% Matrices % Matrices
@ -137,15 +138,17 @@
\affiliation{\LCPQ} \affiliation{\LCPQ}
\author{Enzo \surname{Monino}} \author{Enzo \surname{Monino}}
\affiliation{\LCPQ} \affiliation{\LCPQ}
\author{Antoine \surname{Marie}}
\affiliation{\LCPQ}
\author{Pierre-Fran\c{c}ois \surname{Loos}} \author{Pierre-Fran\c{c}ois \surname{Loos}}
\email{loos@irsamc.ups-tlse.fr} \email{loos@irsamc.ups-tlse.fr}
\affiliation{\LCPQ} \affiliation{\LCPQ}
\begin{abstract} \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 and $GW$ 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 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. Similitudes between BSE@$GW$ and the similarity-transformed equation-of-motion CC method introduced by Nooijen are put forward.
The present work allows to easily transfer key developments and general knowledge gathered in CC theory to many-body perturbation theory. 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 BSE framework. 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 %\bigskip
%\begin{center} %\begin{center}
% \boxed{\includegraphics[width=0.5\linewidth]{TOC}} % \boxed{\includegraphics[width=0.5\linewidth]{TOC}}
@ -155,24 +158,27 @@ In particular, it provides a clear path for the computation of ground- and excit
\maketitle \maketitle
%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\section{RPA Physics and Beyond} \section{RPA Physics and Beyond}
%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
The random-phase approximation (RPA), introduced by Bohm and Pines in the context of the uniform electron gas, \cite{Loos_2016} is a quasibosonic approximation where one treats fermion products as bosons. 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, 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} 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 effective way of introducing correlation in order to go \textit{beyond} RPA physics. 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 $W$. 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 $W$.
This scheme 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} 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.
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}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
Interestingly, RPA has strong connections with coupled-cluster (CC) theory, the workhorse of molecular electronic structure is when one is looking for high accuracy. \cite{Freeman_1977,Scuseria_2008,Jansen_2010,Scuseria_2013,Peng_2013,Berkelbach_2018,Rishi_2020} \section{Connections between RPA and CC}
%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. \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.
%Link between BSE and STEOM-CC. %Link between BSE and STEOM-CC.
%A route towards the obtention of BSE gradients for ground and excited states. %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 rCCD is equivalent to RPAx for the computation of the correlation energy. 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), this proof can be quickly summarised starting from the ubiquitous RPAx eigensystem
\begin{equation} \begin{equation}
\label{eq:RPA} \label{eq:RPA}
@ -227,11 +233,12 @@ Substituting Eq.~\eqref{eq:RPA_1} into Eq.~\eqref{eq:RPA_2} yields the following
\begin{equation} \begin{equation}
\bB{}{} + \bA{}{} \cdot \bT{}{} + \bT{}{} \cdot \bA{}{} + \bT{}{} \cdot \bB{}{} \cdot \bT{}{} = \bO \bB{}{} + \bA{}{} \cdot \bT{}{} + \bT{}{} \cdot \bA{}{} + \bT{}{} \cdot \bB{}{} \cdot \bT{}{} = \bO
\end{equation} \end{equation}
that matches the well-known rCCD amplitude equations that matches the well-known rCCD amplitude (or residual) equations
\begin{multline} \begin{multline}
\label{eq:rCCD} \label{eq:rCCD}
\dbERI{ij}{ab} r_{ij}^{ab}
+ \qty( \e{a}{} + \e{b}{} - \e{i}{} - \e{j}{} ) t_{ij}^{ab} = \dbERI{ij}{ab}
+ \Delta_{ijab} t_{ij}^{ab}
+ \sum_{kc} \dbERI{ic}{ak} t_{kj}^{cb} + \sum_{kc} \dbERI{ic}{ak} t_{kj}^{cb}
\\ \\
+ \sum_{kc} \dbERI{kb}{cj} t_{ik}^{ac} + \sum_{kc} \dbERI{kb}{cj} t_{ik}^{ac}
@ -247,6 +254,10 @@ knowing that
B_{ia,jb} & = \dbERI{ij}{ab} B_{ia,jb} & = \dbERI{ij}{ab}
\end{align} \end{align}
\end{subequations} \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 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} \begin{equation}
\label{eq:ERI} \label{eq:ERI}
@ -261,8 +272,7 @@ 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{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{Jansen_2010,Angyan_2011} but the usual plasmon (or trace) formula yields
\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{}{})
@ -272,9 +282,9 @@ 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{}{}) \Ec^\text{rCCD} = \frac{1}{4} \sum_{ijab} \dbERI{ij}{ab} t_{ij}^{ab} = \frac{1}{4} \Tr(\bB{}{} \cdot \bT{}{})
\end{equation} \end{equation}
because $\Tr(\bOm{}{} - \bA{}{}) = \Tr(\bR{}{} - \bA{}{}) = \Tr(\bB{}{} \cdot \bT{}{})$, as evidenced by Eq.~\eqref{eq:RPA_1}. because $\Tr(\bOm{}{} - \bA{}{}) = \Tr(\bR{}{} - \bA{}{}) = \Tr(\bB{}{} \cdot \bT{}{})$, as evidenced by Eq.~\eqref{eq:RPA_1}.
Note that the same expression as Eq.~\eqref{eq:EcRPAx} can be derived from the adiabatic connection fluctuation dissipation theorem.\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 (ACFDT). \cite{Angyan_2011}
This simple and elegant proof was subsequently extended to excitation energies by Berkerbach, \cite{Berkelbach_2018} who showed that similitudes between EE-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. 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.
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} 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} \begin{equation}
@ -284,16 +294,16 @@ where $\bHN = e^{-\hT} \hH e^{\hT} - E_\text{CCD}$ is the normal-ordered similar
\begin{subequations} \begin{subequations}
\begin{align} \begin{align}
\label{eq:cFab} \label{eq:cFab}
\cF_{ab} & = \e{a}{} \delta_{ab} - \frac{1}{2} \sum_{klc} \ERI{kl}{bc} t_{kl}^{ac} \cF_{ab} & = \e{a}{} \delta_{ab} - \frac{1}{2} \sum_{klc} \dbERI{kl}{bc} t_{kl}^{ac}
\\ \\
\label{eq:cFij} \label{eq:cFij}
\cF_{ij} & = \e{i}{} \delta_{ij} + \frac{1}{2} \sum_{kcd} \ERI{ik}{cd} t_{jk}^{cd} \cF_{ij} & = \e{i}{} \delta_{ij} + \frac{1}{2} \sum_{kcd} \dbERI{ik}{cd} t_{jk}^{cd}
\end{align} \end{align}
\end{subequations} \end{subequations}
and the two-body term is and the two-body term is
\begin{equation} \begin{equation}
\label{eq:cWibaj} \label{eq:cWibaj}
\cW_{ibaj} = \ERI{ib}{aj} + \sum_{kc} \ERI{ik}{ac} t_{kj}^{cb} \cW_{ibaj} = \dbERI{ib}{aj} + \sum_{kc} \dbERI{ik}{ac} t_{kj}^{cb}
\end{equation} \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 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} \begin{equation}
@ -305,16 +315,16 @@ Neglecting the effect of $\hT_2$ on the one-body terms [see Eqs.~\eqref{eq:cFab}
\end{split} \end{split}
\end{equation} \end{equation}
which exactly matches Eq.~\eqref{eq:RPA_1}. which exactly matches Eq.~\eqref{eq:RPA_1}.
Although the excitation energies of this approximate EE-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} 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.
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.)
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\section{Bethe-Salpeter equation} \section{Connections between BSE and CC}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \label{sec:BSE}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
This does not hold only for RPAx as it is actually quite general and can be applied to any ph problem with the structure of Eq.~\eqref{eq:RPA}, such as TD-DFT, 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).
At the BSE level, and within the static approximation, one must solve a very similar linear eigenvalue problem At the BSE level, and within the static approximation, one must solve a very similar linear eigenvalue problem
\begin{equation} \begin{equation}
\label{eq:BSE} \label{eq:BSE}
@ -335,7 +345,7 @@ At the BSE level, and within the static approximation, one must solve a very sim
\cdot \cdot
\bOm{}{\BSE} \bOm{}{\BSE}
\end{equation} \end{equation}
where where the matrix elements read
\begin{subequations} \begin{subequations}
\begin{align} \begin{align}
\label{eq:A_BSE} \label{eq:A_BSE}
@ -345,7 +355,7 @@ where
B_{ia,jb}^{\BSE} & = \dbERI{ij}{ab} - W_{ib,ja}^\text{stat} B_{ia,jb}^{\BSE} & = \dbERI{ij}{ab} - W_{ib,ja}^\text{stat}
\end{align} \end{align}
\end{subequations} \end{subequations}
and The quasiparticle energies $\e{p}{\GW}$ are computed at the $GW$ level (see below) and
\begin{multline} \begin{multline}
\label{eq:W} \label{eq:W}
W_{pq,rs}^\text{c}(\omega) = \sum_{m} \sERI{pq}{m} \sERI{rs}{m} W_{pq,rs}^\text{c}(\omega) = \sum_{m} \sERI{pq}{m} \sERI{rs}{m}
@ -388,45 +398,70 @@ with
B_{ia,jb}^{\dRPA} & = \ERI{ij}{ab} B_{ia,jb}^{\dRPA} & = \ERI{ij}{ab}
\end{align} \end{align}
\end{subequations} \end{subequations}
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 screening of the electron-electron interaction [see Eq.~\eqref{eq:W}], correlation effects at the BSE level.
Note also that, within BSE, the orbital energies in Eqs.~\eqref{eq:A_BSE} and \eqref{eq:B_BSE} are usually computed at the $GW$ level to be consistent with the introduction of $W$ in the BSE kernel. 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.
For example, at the one-shot $GW$ level, \cite{Strinati_1980,Hybertsen_1985a,Hybertsen_1986,Godby_1988,Linden_1988,Northrup_1991,Blase_1994,Rohlfing_1995,Shishkin_2007} we have 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})
\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,
\begin{multline}
\label{eq:rCCD-BSE}
\Tilde{r}_{ij}^{ab}
= \wERI{ij}{ab}
+ \Delta_{ijab}^{\GW} \tilde{t}_{ij}^{ab}
+ \sum_{kc} \wERI{ic}{ak} \tilde{t}_{kj}^{cb}
\\
+ \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}
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}.
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.
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}
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.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Connections 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.
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.
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
\begin{equation} \begin{equation}
\label{eq:G0W0} \label{eq:G0W0}
\e{p}{\GW} = \e{p}{\HF} + Z_p \SigC{p}(\omega = \e{p}{\HF}) \e{p}{\GW} = \e{p}{} + \SigC{pp}(\omega = \e{p}{\GW})
\end{equation} \end{equation}
where the elements of the correlation part of the dynamical self-energy are where the elements of the correlation part of the dynamical self-energy are
\begin{equation} \begin{equation}
\begin{split} \begin{split}
\SigC{pq}(\omega) \SigC{pq}(\omega)
& = \sum_{im} \frac{\sERI{pi}{m} \sERI{qi}{m}}{\omega - \e{i}{\HF} + \Om{m}{\dRPA} - \ii \eta} & = \sum_{im} \frac{\sERI{pi}{m} \sERI{qi}{m}}{\omega - \e{i}{} + \Om{m}{\dRPA} - \ii \eta}
\\ \\
& + \sum_{am} \frac{\sERI{pa}{m} \sERI{qa}{m}}{\omega - \e{a}{\HF} - \Om{m}{\dRPA} + \ii \eta} & + \sum_{am} \frac{\sERI{pa}{m} \sERI{qa}{m}}{\omega - \e{a}{} - \Om{m}{\dRPA} + \ii \eta}
\end{split} \end{split}
\end{equation} \end{equation}
and the renormalization factor is %and the renormalization factor is
\begin{equation} %\begin{equation}
Z_p = \qty[ 1 - \eval{\pdv{\SigC{pp}(\omega)}{\omega}}_{\omega = \e{p}{\HF}} ]^{-1} % Z_p = \qty[ 1 - \eval{\pdv{\SigC{pp}(\omega)}{\omega}}_{\omega = \e{p}{\HF}} ]^{-1}
\end{equation} %\end{equation}
Therefore, following a similar procedure, 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})
\end{equation}
can be obtained via a set of rCCD equations, where one substitutes in Eq.~\eqref{eq:rCCD} all the two-electron integrals $\dbERI{pq}{rs}$ by $\dbERI{pq}{rs} - W_{ps,qr}^\text{stat}$.
Similarly to the diagonalization of the eigensystem \eqref{eq:BSE}, these rCCD-based equations can be solved in $\order*{N^6}$ cost, and provides a clear path for the computation of ground- and excited-state properties (such as nuclear gradients) within the BSE framework.
Following Berkelbach's analysis, \cite{Berkelbach_2018} one can extend the connection to excitation energies.
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
It is therefore interesting to investigate if it possible also the recast the $GW$ equations as a set of CC-like equations.
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 and we do not decouple the 2h1p and 2p1h sectors.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\section{$GW$ 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 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} \begin{equation}
\label{eq:GWlin} \label{eq:GWlin}
@ -529,7 +564,7 @@ Substituting Eq.~\eqref{eq:R} into Eqs.~\eqref{eq:T1R} and \eqref{eq:T2R} yields
\end{split} \end{split}
\end{align} \end{align}
\end{subequations} \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 amplitude 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}} $, it means that one must solve the following residual equations
\begin{subequations} \begin{subequations}
\begin{align} \begin{align}
\begin{split} \begin{split}
@ -538,8 +573,8 @@ In the CC language, in order to determine the 2h1p and 2p1h amplitudes, $t_{ija,
+ \Delta_{ija,p}^{\text{2h1p}} t_{ija,p}^{\text{2h1p}} + \Delta_{ija,p}^{\text{2h1p}} t_{ija,p}^{\text{2h1p}}
- \sum_{kc} \ERI{jc}{ak} t_{ikc,p}^{\text{2h1p}} - \sum_{kc} \ERI{jc}{ak} t_{ikc,p}^{\text{2h1p}}
\\ \\
& - \sum_{klcq} t_{ija,q}^{\text{2h1p}} \ERI{qc}{kl} t_{klc,p}^{\text{2h1p}} & - \sum_{klcq} \ERI{qc}{kl} t_{ija,q}^{\text{2h1p}} t_{klc,p}^{\text{2h1p}}
- \sum_{kcdq} t_{ija,q}^{\text{2h1p}} \ERI{qk}{dc} t_{kcd,p}^{\text{2p1h}} - \sum_{kcdq} \ERI{qk}{dc} t_{ija,q}^{\text{2h1p}} t_{kcd,p}^{\text{2p1h}}
\end{split} \end{split}
\\ \\
\begin{split} \begin{split}
@ -548,8 +583,8 @@ In the CC language, in order to determine the 2h1p and 2p1h amplitudes, $t_{ija,
+ \Delta_{iab,p}^{\text{2p1h}} t_{ija,p}^{\text{2p1h}} + \Delta_{iab,p}^{\text{2p1h}} t_{ija,p}^{\text{2p1h}}
+ \sum_{kc} \ERI{ak}{ic} t_{kcb,p}^{\text{2p1h}} + \sum_{kc} \ERI{ak}{ic} t_{kcb,p}^{\text{2p1h}}
\\ \\
& - \sum_{klcq} t_{iab,q}^{\text{2p1h}} \ERI{qc}{kl} t_{klc,p}^{\text{2h1p}} & - \sum_{klcq} \ERI{qc}{kl} t_{iab,q}^{\text{2p1h}} t_{klc,p}^{\text{2h1p}}
- \sum_{kcdq} t_{iab,q}^{\text{2p1h}} \ERI{qk}{dc} t_{kcd,p}^{\text{2p1h}} - \sum_{kcdq} \ERI{qk}{dc} t_{iab,q}^{\text{2p1h}} t_{kcd,p}^{\text{2p1h}}
\end{split} \end{split}
\end{align} \end{align}
\end{subequations} \end{subequations}
@ -564,9 +599,9 @@ with
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 iterative procedure to solve these non-linear equations by updating the amplitudes via
\begin{subequations} \begin{subequations}
\begin{align} \begin{align}
t_{ija,p}^{\text{2h1p}} & \leftarrow t_{ija,p}^{\text{2h1p}} - \frac{r_{ija,p}^{\text{2h1p}}}{\Delta_{ij}^{ap}} t_{ija,p}^{\text{2h1p}} & \leftarrow t_{ija,p}^{\text{2h1p}} - \frac{r_{ija,p}^{\text{2h1p}}}{\Delta_{ija,p}^{\text{2h1p}}}
\\ \\
t_{iab,p}^{\text{2h1p}} & \leftarrow t_{iab,p}^{\text{2p1h}} - \frac{r_{iab,p}^{\text{2p1h}}}{\Delta_{ab}^{ip}} t_{iab,p}^{\text{2h1p}} & \leftarrow t_{iab,p}^{\text{2p1h}} - \frac{r_{iab,p}^{\text{2p1h}}}{\Delta_{iab,p}^{\text{2p1h}}}
\end{align} \end{align}
\end{subequations} \end{subequations}
@ -690,10 +725,12 @@ The $G_0W_0$ quasiparticle energies can be easily obtained via the procedure des
%\end{subequations} %\end{subequations}
%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%
%\section{Conclusion} \section{Conclusion}
%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%
Here comes the 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.
%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
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\section*{Supplementary Material} %\section*{Supplementary Material}
%\label{sec:supmat} %\label{sec:supmat}
@ -705,9 +742,9 @@ This project has received funding from the European Research Council (ERC) under
%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\section*{Data availability statement} \section*{Data availability statement}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%The data that supports the findings of this study are available within the article and its supplementary material. The data that supports the findings of this study are available within the article.% and its supplementary material.
%%%%%%%%% %%%%%%%%%
%\appendix %\appendix