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% methods
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%
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% energies
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\newcommand { \Ec } { E_ \text { c} }
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% orbital energies
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% Matrix elements
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\newcommand { \MO } [2]{ \phi _ { #1} ^ { #2} }
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% Matrices
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% addresses
\newcommand { \LCPQ } { Laboratoire de Chimie et Physique Quantiques (UMR 5626), Universit\' e de Toulouse, CNRS, UPS, France}
\newcommand { \LPT } { Laboratoire de Physique Th\' eorique, Universit\' e de Toulouse, CNRS, UPS, France}
\newcommand { \ETSF } { European Theoretical Spectroscopy Facility (ETSF)}
\newcommand { \NEEL } { Universit\' e Grenoble Alpes, CNRS, Institut N\' EEL, F-38042 Grenoble, France}
\begin { document}
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\title { Connections between many-body perturbation and coupled-cluster theories}
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\author { Ra\' ul \surname { Quintero-Monsebaiz} }
\affiliation { \LCPQ }
\author { Enzo \surname { Monino} }
\affiliation { \LCPQ }
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\author { Antoine \surname { Marie} }
\affiliation { \LCPQ }
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\author { Pierre-Fran\c { c} ois \surname { Loos} }
\email { loos@irsamc.ups-tlse.fr}
\affiliation { \LCPQ }
\begin { abstract}
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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.
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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.
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The present work allows to easily transfer key developments and general knowledge gathered in CC theory to many-body perturbation theory.
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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.
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%\bigskip
%\begin{center}
% \boxed{\includegraphics[width=0.5\linewidth]{TOC}}
%\end{center}
%\bigskip
\end { abstract}
\maketitle
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\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.
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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.
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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 captures effectively long-range correlation effects (such as dispersion). \cite { Gell-Mann_ 1957,Nozieres_ 1958}
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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}
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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.
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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 $ .
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Practically, $ GW $ produces accurate \textit { ``charged''} excitations providing a faithful description of the fundamental gap via the computation of the RPA polarizability obtained by a resummation of all time-dependent ring diagrams.
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 overall accurate \textit { ``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,McKeon_ 2022}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\section { Connection between RPA and CC}
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\label { sec:RPAx}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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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}
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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, solidifying in the process the numerical evidences provided by Freeman many years before. \cite { Freeman_ 1977}
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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
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\begin { equation}
\label { eq:RPA}
\begin { pmatrix}
\bA { } { } & \bB { } { } \\
-\bB { } { } & -\bA { } { } \\
\end { pmatrix}
\cdot
\begin { pmatrix}
\bX { } { } \\
\bY { } { } \\
\end { pmatrix}
=
\begin { pmatrix}
\bX { } { } \\
\bY { } { } \\
\end { pmatrix}
\cdot
\bOm { } { }
\end { equation}
from which one gets, by introducing $ \bT { } { } = \bY { } { } \cdot \bX { } { - 1 } $ ,
\begin { equation}
\begin { pmatrix}
\bA { } { } & \bB { } { } \\
-\bB { } { } & -\bA { } { } \\
\end { pmatrix}
\cdot
\begin { pmatrix}
\bI \\
\bT { } { } \\
\end { pmatrix}
=
\begin { pmatrix}
\bI \\
\bT { } { } \\
\end { pmatrix}
\cdot
\bR { } { }
\end { equation}
where $ \bR { } { } = \bX { } { } \cdot \bOm { } { } \cdot \bX { } { - 1 } $ , or equivalently, the two following equations
\begin { subequations}
\begin { align}
\label { eq:RPA_ 1}
\bA { } { } + \bB { } { } \cdot \bT { } { } & = \bR { } { }
\\
\label { eq:RPA_ 2}
-\bB { } { } - \bA { } { } \cdot \bT { } { } & = \bT { } { } \cdot \bR { } { }
\end { align}
\end { subequations}
Substituting Eq.~\eqref { eq:RPA_ 1} into Eq.~\eqref { eq:RPA_ 2} yields the following Riccati equation
\begin { equation}
\bB { } { } + \bA { } { } \cdot \bT { } { } + \bT { } { } \cdot \bA { } { } + \bT { } { } \cdot \bB { } { } \cdot \bT { } { } = \bO
\end { equation}
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that matches the rCCD amplitude (or residual) equations
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\begin { multline}
\label { eq:rCCD}
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r_ { ij} ^ { ab}
= \dbERI { ij} { ab}
+ \Delta _ { ijab} t_ { ij} ^ { ab}
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+ \sum _ { kc} \dbERI { ic} { ak} t_ { kj} ^ { cb}
\\
+ \sum _ { kc} \dbERI { kb} { cj} t_ { ik} ^ { ac}
+ \sum _ { klcd} \dbERI { kl} { cd} t_ { ik} ^ { ac} t_ { lj} ^ { db} = 0
\end { multline}
knowing that
\begin { subequations}
\begin { align}
\label { eq:A_ RPAx}
A_ { ia,jb} & = (\e { a} { } - \e { i} { } ) \delta _ { ij} \delta _ { ab} + \dbERI { ib} { aj}
\\
\label { eq:B_ RPAx}
B_ { ia,jb} & = \dbERI { ij} { ab}
\end { align}
\end { subequations}
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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
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\begin { equation}
\label { eq:ERI}
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\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
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\end { equation}
are two-electron repulsion integrals, while
\begin { equation}
\dbERI { pq} { rs} = \braket { pq} { rs} - \braket { pq} { sr}
\end { equation}
are their anti-symmetrized versions.
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.
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In the following, $ O $ and $ V $ are the number of occupied and virtual spinorbitals, respectively, and $ N = O + V $ is the total number.
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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}
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\begin { equation}
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\label { eq:EcRPAx}
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\Ec ^ \text { RPAx} = \frac { 1} { 4} \Tr (\bOm { } { } - \bA { } { } )
\end { equation}
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and matches the rCCD correlation energy
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\begin { equation}
\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} .
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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}
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This simple and elegant proof was subsequently extended to excitation energies by Berkelbach, \cite { Berkelbach_ 2018} who showed that similitudes between equation-of-motion (EOM) rCCD (EOM-rCCD) \cite { Stanton_ 1993} and RPAx exist when the EOM space is restricted to the 1h1p configurations and only the two-body terms are dressed by rCCD correlation (see also Ref.~\onlinecite { Rishi_ 2020} ).
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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}
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\begin { equation}
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\mel * { \Psi _ { i} ^ { a} } { \bHN } { \Psi _ { j} ^ { b} } = \cF _ { ab} \delta _ { ij} - \cF _ { ij} \delta _ { ab} + \cW _ { jabi}
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\end { equation}
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where $ \bHN = e ^ { - \hT } \hH _ { N } e ^ { \hT } - \ECC $ is the (shifted) similarity-transformed normal-ordered Hamiltonian, $ \Psi _ { i } ^ { a } $ are singly-excited determinants, the one-body terms are
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\begin { subequations}
\begin { align}
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\label { eq:cFab}
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\cF _ { ab} & = \e { a} { } \delta _ { ab} - \frac { 1} { 2} \sum _ { klc} \dbERI { kl} { bc} t_ { kl} ^ { ac}
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\\
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\label { eq:cFij}
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\cF _ { ij} & = \e { i} { } \delta _ { ij} + \frac { 1} { 2} \sum _ { kcd} \dbERI { ik} { cd} t_ { jk} ^ { cd}
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\end { align}
\end { subequations}
and the two-body term is
\begin { equation}
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\label { eq:cWibaj}
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\cW _ { ibaj} = \dbERI { ib} { aj} + \sum _ { kc} \dbERI { ik} { ac} t_ { kj} ^ { cb}
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\end { equation}
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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
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\begin { equation}
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\label { eq:EOM-rCCD}
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\begin { split}
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\mel * { \Psi _ { i} ^ { a} } { \bHN } { \Psi _ { j} ^ { b} }
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& = (\e { a} { } - \e { i} { } ) \delta _ { ij} \delta _ { ab} + \dbERI { ib} { aj} + \sum _ { kc} \dbERI { ik} { ac} t_ { kj} ^ { cb}
\\
& = (\bA { } { } + \bB { } { } \cdot \bT { } { } )_ { ia,jb}
\end { split}
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\end { equation}
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which exactly matches Eq.~\eqref { eq:RPA_ 1} .
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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}
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Equation \eqref { eq:EOM-rCCD} can be more systematically derived through the formulation of $ \Lambda $ equations based on a rCCD effective Hamiltonian, as proposed by Rishi \textit { et al.} \cite { Rishi_ 2020}
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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.
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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}
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(See also Ref.~\onlinecite { Berkelbach_ 2018} for the extension to excitation energies for the pp and hh channels.)
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\section { Connection between BSE and CC}
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\label { sec:BSE}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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Within the usual static approximation of BSE, one must solve a very similar linear eigenvalue problem
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\begin { equation}
\label { eq:BSE}
\begin { pmatrix}
\bA { } { \BSE } & \bB { } { \BSE } \\
-\bB { } { \BSE } & -\bA { } { \BSE } \\
\end { pmatrix}
\cdot
\begin { pmatrix}
\bX { } { \BSE } \\
\bY { } { \BSE } \\
\end { pmatrix}
=
\begin { pmatrix}
\bX { } { \BSE } \\
\bY { } { \BSE } \\
\end { pmatrix}
\cdot
\bOm { } { \BSE }
\end { equation}
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where the matrix elements read
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\begin { subequations}
\begin { align}
\label { eq:A_ BSE}
A_ { ia,jb} ^ { \BSE } & = \delta _ { ij} \delta _ { ab} (\e { a} { \GW } - \e { i} { \GW } ) + \dbERI { ib} { aj} - W_ { ij,ba} ^ \text { stat}
\\
\label { eq:B_ BSE}
B_ { ia,jb} ^ { \BSE } & = \dbERI { ij} { ab} - W_ { ib,ja} ^ \text { stat}
\end { align}
\end { subequations}
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The quasiparticle energies $ \e { p } { \GW } $ are computed at the $ GW $ level (see below) and
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\begin { multline}
\label { eq:W}
W_ { pq,rs} ^ \text { c} (\omega ) = \sum _ { m} \sERI { pq} { m} \sERI { rs} { m}
\\
\times \qty [ \frac{1}{\omega - \Om{m}{\dRPA} + \ii \eta} - \frac{1}{\omega + \Om{m}{\dRPA} - \ii \eta} ]
\end { multline}
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are the elements of the correlation part of the dynamically-screened Coulomb potential which is set to its static limit \ie , $ W _ { pq,rs } ^ \text { stat } = W _ { pq,rs } ^ \text { c } ( \omega = 0 ) $ .
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In Eq.~\eqref { eq:W} , $ \eta $ is a positive infinitesimal, the screened two-electron integrals are
\begin { equation}
\label { eq:sERI}
\sERI { pq} { m} = \sum _ { ia} \ERI { pi} { qa} \qty ( \bX { m} { \dRPA } + \bY { m} { \dRPA } )_ { ia}
\end { equation}
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and $ \Om { m } { \dRPA } $ is the $ m $ th (positive) eigenvalue and $ \bX { m } { \dRPA } + \bY { m } { \dRPA } $ is constructed from the corresponding eigenvectors of the direct (\ie , without exchange) RPA (dRPA) problem defined as
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\begin { equation}
\label { eq:dRPA}
\begin { pmatrix}
\bA { } { \dRPA } & \bB { } { \dRPA } \\
-\bB { } { \dRPA } & -\bA { } { \dRPA } \\
\end { pmatrix}
\cdot
\begin { pmatrix}
\bX { } { \dRPA } \\
\bY { } { \dRPA } \\
\end { pmatrix}
=
\begin { pmatrix}
\bX { } { \dRPA } \\
\bY { } { \dRPA } \\
\end { pmatrix}
\cdot
\bOm { } { \dRPA }
\end { equation}
with
\begin { subequations}
\begin { align}
\label { eq:A_ dRPA}
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A_ { ia,jb} ^ { \dRPA } & = \delta _ { ij} \delta _ { ab} (\e { a} { \GW } - \e { i} { \GW } ) + \ERI { ib} { aj}
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\\
\label { eq:B_ dRPA}
B_ { ia,jb} ^ { \dRPA } & = \ERI { ij} { ab}
\end { align}
\end { subequations}
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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
\begin { equation}
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\Ec ^ \text { BSE} = \frac { 1} { 4} \Tr (\bOm { } { \BSE } - \bA { } { \BSE } ) = \frac { 1} { 4} \sum _ { ijab} \wERI { ij} { ab} \tilde { t} _ { ij} ^ { ab}
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\end { equation}
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can be equivalently 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 ,
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\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}
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with $ \Delta _ { ijab } ^ { \GW } = \e { a } { \GW } + \e { b } { \GW } - \e { i } { \GW } - \e { j } { \GW } $ .
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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.
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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 first introduced 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.
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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}
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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 ,
\begin { equation}
\label { eq:EOM-rCCD-BSE}
\mel * { \Psi _ { i} ^ { a} } { \tHN } { \Psi _ { j} ^ { b} }
= (\e { a} { \GW } - \e { i} { \GW } ) \delta _ { ij} \delta _ { ab} + \wERI { ib} { aj} + \sum _ { kc} \wERI { ik} { ac} \tilde { t} _ { kj} ^ { cb}
\end { equation}
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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} .
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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} .
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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} .}
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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.
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At the CC with singles and doubles (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.
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(An extended version of STEOM-CC has been proposed where the EOM treatment is pushed up to 2h2p. \cite { Nooijen_ 2000} )
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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).
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Then, a static BSE calculation is performed in the 1h1p sector with a two-body term dressed with correlation stemming from $ GW $ .
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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}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\section { Connection between $ GW $ and CC}
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\label { sec:GW}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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Because $ GW $ is able to capture key correlation effects as illustrated above, it is therefore interesting to investigate if it is also possible to recast the $ GW $ equations as a set of CC-like equations that can be solved iteratively using the CC machinery.
Connections between approximate IP/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.
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Note also that the procedure described below can be applied to other approximate self-energies such as second-order Green's function (or second Born) \cite { Stefanucci_ 2013,Ortiz_ 2013,Phillips_ 2014,Rusakov_ 2014,Hirata_ 2015,Hirata_ 2017} or $ T $ -matrix.\cite { Romaniello_ 2012,Zhang_ 2017,Li_ 2021b,Loos_ 2022}
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Quite unfortunately, there are several ways of computing $ GW $ quasiparticle energies. \cite { Loos_ 2018b}
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Within the perturbative $ GW $ scheme (commonly known as $ G _ 0 W _ 0 $ ), the quasiparticle energies are obtained via a one-shot procedure (with or without linearization).
\cite { Strinati_ 1980,Hybertsen_ 1985a,Hybertsen_ 1986,Godby_ 1988,Linden_ 1988,Northrup_ 1991,Blase_ 1994,Rohlfing_ 1995,Shishkin_ 2007}
Partial self-consistency can be attained via the \textit { ``eigenvalue''} self-consistent $ GW $ (ev$ GW $ ) \cite { Hybertsen_ 1986,Shishkin_ 2007,Blase_ 2011,Faber_ 2011,Rangel_ 2016,Gui_ 2018} or the quasiparticle self-consistent $ GW $ (qs$ GW $ ) \cite { Faleev_ 2004,vanSchilfgaarde_ 2006,Kotani_ 2007,Ke_ 2011,Kaplan_ 2016} schemes.
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In the most general setting, the quasiparticle energies and their corresponding orbitals are obtained by diagonalizing the so-called non-linear and frequency-dependent quasiparticle equation
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\begin { equation}
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\label { eq:GW}
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\qty [ \be{}{} + \bSig{}{\GW}\qty(\omega = \e{p}{\GW}) ] \SO { p} { \GW } = \e { p} { \GW } \SO { p} { \GW }
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\end { equation}
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\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
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\begin { equation}
\begin { split}
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\Sig { pq} { \GW } (\omega )
& = \sum _ { im} \frac { \sERI { pi} { m} \sERI { qi} { m} } { \omega - \e { i} { \GW } + \Om { m} { \dRPA } - \ii \eta }
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\\
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& + \sum _ { am} \frac { \sERI { pa} { m} \sERI { qa} { m} } { \omega - \e { a} { \GW } - \Om { m} { \dRPA } + \ii \eta }
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\end { split}
\end { equation}
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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.
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As shown by Bintrim and Berkelbach, \cite { Bintrim_ 2021} the quasiparticle equation \eqref { eq:GW} can be recast as a larger set of linear and frequency-independent equations (that still needs to be solved self-consistently), which reads in the Tamm-Dancoff approximation
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\begin { equation}
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\label { eq:GWlin}
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\begin { pmatrix}
\be { } { } & \bV { } { \text { 2h1p} } & \bV { } { \text { 2p1h} } \\
\T { (\bV { } { \text { 2h1p} } )} & \bC { } { \text { 2h1p} } & \bO \\
\T { (\bV { } { \text { 2p1h} } )} & \bO & \bC { } { \text { 2p1h} } \\
\end { pmatrix}
\cdot
\begin { pmatrix}
\bX { } { } \\
\bY { } { \text { 2h1p} } \\
\bY { } { \text { 2p1h} } \\
\end { pmatrix}
=
\begin { pmatrix}
\bX { } { } \\
\bY { } { \text { 2h1p} } \\
\bY { } { \text { 2p1h} } \\
\end { pmatrix}
\cdot
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\be { } { \GW }
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\end { equation}
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where $ \be { } { \GW } $ is a diagonal matrix collecting the quasiparticle energies, the 2h1p and 2p1h matrix elements are
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\begin { subequations}
\begin { align}
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C^ \text { 2h1p} _ { ija,klc} & = \qty [ \qty( \e{i}{\GW} + \e{j}{\GW} - \e{a}{\GW}) \delta_{jl} \delta_{ac} - \ERI{jc}{al} ] \delta _ { ik}
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\\
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C^ \text { 2p1h} _ { iab,kcd} & = \qty [ \qty( \e{a}{\GW} + \e{b}{\GW} - \e{i}{\GW}) \delta_{ik} \delta_{ac} + \ERI{ak}{ic} ] \delta _ { bd}
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\end { align}
\end { subequations}
and the corresponding coupling blocks read
\begin { align}
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V^ \text { 2h1p} _ { p,klc} & = \ERI { pc} { kl}
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&
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V^ \text { 2p1h} _ { p,kcd} & = \ERI { pk} { dc}
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\end { align}
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Going beyond the Tamm-Dancoff approximation is possible, but more cumbersome. \cite { Bintrim_ 2021}
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Note that, contrary to the IP/EA-EOM-CC equations, $ GW $ does couple the IP and EA sectors due to the lack of exponential parametrization of the wave function. \cite { Nooijen_ 1995,Rishi_ 2020}
However, it allows to generate higher-order diagrams. \cite { Lange_ 2018,Schirmer_ 2018}
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Let us suppose that we are looking for the $ N $ \textit { ``principal''} (\ie , quasiparticle) solutions of the eigensystem \eqref { eq:GWlin} .
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Therefore, $ \bX { } { } $ and $ \be { } { \GW } $ are square matrices of size $ N \times N $ .
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Assuming the existence of $ \bX { } { - 1 } $ and introducing $ \bT { } { \text { 2 h 1 p } } = \bY { } { \text { 2 h 1 p } } \cdot \bX { } { - 1 } $ and $ \bT { } { \text { 2 p 1 h } } = \bY { } { \text { 2 p 1 h } } \cdot \bX { } { - 1 } $ , we have
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\begin { equation}
\begin { pmatrix}
\be { } { } & \bV { } { \text { 2h1p} } & \bV { } { \text { 2p1h} } \\
\T { (\bV { } { \text { 2h1p} } )} & \bC { } { \text { 2h1p} } & \bO \\
\T { (\bV { } { \text { 2p1h} } )} & \bO & \bC { } { \text { 2p1h} } \\
\end { pmatrix}
\cdot
\begin { pmatrix}
\bI \\
\bT { } { \text { 2h1p} } \\
\bT { } { \text { 2p1h} } \\
\end { pmatrix}
=
\begin { pmatrix}
\bI \\
\bT { } { \text { 2h1p} } \\
\bT { } { \text { 2p1h} } \\
\end { pmatrix}
\cdot
\bR { } { }
\end { equation}
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with $ \bR { } { } = \bX { } { } \cdot \be { } { \GW } \cdot \bX { } { - 1 } $ , which yields the three following equations
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\begin { subequations}
\begin { align}
\be { } { } + \bV { } { \text { 2h1p} } \cdot \bT { } { \text { 2h1p} } + \bV { } { \text { 2p1h} } \cdot \bT { } { \text { 2p1h} } & = \bR { } { }
\label { eq:R}
\\
\T { (\bV { } { \text { 2h1p} } )} + \bC { } { \text { 2h1p} } \cdot \bT { } { \text { 2h1p} } & = \bT { } { \text { 2h1p} } \cdot \bR { } { }
\label { eq:T1R}
\\
\T { (\bV { } { \text { 2p1h} } )} + \bC { } { \text { 2p1h} } \cdot \bT { } { \text { 2p1h} } & = \bT { } { \text { 2p1h} } \cdot \bR { } { }
\label { eq:T2R}
\end { align}
\end { subequations}
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Substituting Eq.~\eqref { eq:R} into Eqs.~\eqref { eq:T1R} and \eqref { eq:T2R} , one gets two coupled Riccati equations
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\begin { subequations}
\begin { align}
\begin { split}
\T { (\bV { } { \text { 2h1p} } )}
+ \bC { } { \text { 2h1p} } \cdot \bT { } { \text { 2h1p} }
- \bT { } { \text { 2h1p} } \cdot \be { } { }
- \bT { } { \text { 2h1p} } \cdot \bV { } { \text { 2h1p} } \cdot \bT { } { \text { 2h1p} }
\\
- \bT { } { \text { 2h1p} } \cdot \bV { } { \text { 2p1h} } \cdot \bT { } { \text { 2p1h} }
= \bO
\end { split}
\\
\begin { split}
\T { (\bV { } { \text { 2p1h} } )}
+ \bC { } { \text { 2p1h} } \cdot \bT { } { \text { 2p1h} }
- \bT { } { \text { 2p1h} } \cdot \be { } { }
- \bT { } { \text { 2p1h} } \cdot \bV { } { \text { 2h1p} } \cdot \bT { } { \text { 2h1p} }
\\
- \bT { } { \text { 2p1h} } \cdot \bV { } { \text { 2p1h} } \cdot \bT { } { \text { 2p1h} }
= \bO
\end { split}
\end { align}
\end { subequations}
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that can be converted to the following CC-like residual equations
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\begin { subequations}
\begin { align}
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\label { eq:r_ 2h1p}
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\begin { split}
r_ { ija,p} ^ { \text { 2h1p} }
& = \ERI { pa} { ij}
+ \Delta _ { ija,p} ^ { \text { 2h1p} } t_ { ija,p} ^ { \text { 2h1p} }
- \sum _ { kc} \ERI { jc} { ak} t_ { ikc,p} ^ { \text { 2h1p} }
\\
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& - \sum _ { klcq} \ERI { qc} { kl} t_ { ija,q} ^ { \text { 2h1p} } t_ { klc,p} ^ { \text { 2h1p} }
- \sum _ { kcdq} \ERI { qk} { dc} t_ { ija,q} ^ { \text { 2h1p} } t_ { kcd,p} ^ { \text { 2p1h} }
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= 0
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\end { split}
\\
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\label { eq:r_ 2p1h}
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\begin { split}
r_ { iab,p} ^ { \text { 2p1h} }
& = \ERI { pi} { ba}
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+ \Delta _ { iab,p} ^ { \text { 2p1h} } t_ { iab,p} ^ { \text { 2p1h} }
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+ \sum _ { kc} \ERI { ak} { ic} t_ { kcb,p} ^ { \text { 2p1h} }
\\
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& - \sum _ { klcq} \ERI { qc} { kl} t_ { iab,q} ^ { \text { 2p1h} } t_ { klc,p} ^ { \text { 2h1p} }
- \sum _ { kcdq} \ERI { qk} { dc} t_ { iab,q} ^ { \text { 2p1h} } t_ { kcd,p} ^ { \text { 2p1h} }
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= 0
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\end { split}
\end { align}
\end { subequations}
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with $ \Delta _ { ija,p } ^ { \text { 2 h 1 p } } = \e { i } { \GW } + \e { j } { \GW } - \e { a } { \GW } - \e { p } { } $ and $ \Delta _ { iab,p } ^ { \text { 2 p 1 h } } = \e { a } { \GW } + \e { b } { \GW } - \e { i } { \GW } - \e { p } { } $ .
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To determine the 2h1p and 2p1h amplitudes, $ t _ { ija,p } ^ { \text { 2 h 1 p } } $ and $ t _ { iab,p } ^ { \text { 2 p 1 h } } $ , one can then rely on the usual quasi-Newton iterative procedure to solve these quadratic equations by updating the amplitudes via
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\begin { subequations}
\begin { align}
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\label { eq:t_ 2h1p_ update}
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t_ { ija,p} ^ { \text { 2h1p} } & \leftarrow t_ { ija,p} ^ { \text { 2h1p} } - \qty ( \Delta _ { ija,p} ^ { \text { 2h1p} } )^ { -1} r_ { ija,p} ^ { \text { 2h1p} }
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\\
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\label { eq:t_ 2p1h_ update}
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t_ { iab,p} ^ { \text { 2p1h} } & \leftarrow t_ { iab,p} ^ { \text { 2p1h} } - \qty ( \Delta _ { iab,p} ^ { \text { 2p1h} } )^ { -1} r_ { iab,p} ^ { \text { 2p1h} }
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\end { align}
\end { subequations}
The quasiparticle energies $ \e { p } { GW } $ are thus provided by the eigenvalues of $ \be { } { } + \bSig { } { \GW } $ , where
\begin { equation}
\bSig { } { \GW } = \bV { } { \text { 2h1p} } \cdot \bT { } { \text { 2h1p} } + \bV { } { \text { 2p1h} } \cdot \bT { } { \text { 2p1h} }
\end { equation}
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\alert { Due to the non-linear nature of these equations, the iterative procedure proposed in Eqs.~\eqref { eq:t_ 2h1p_ update} and \eqref { eq:t_ 2p1h_ update} can potentially converge to satellite solutions.
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This is also the case at the CC level when one relies on more elaborated algorithms to converge the amplitude equations to higher-energy solutions. \cite { Piecuch_ 2000,Mayhall_ 2010,Lee_ 2019,Kossoski_ 2021,Marie_ 2021b} }
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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.
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Cholesky decomposition, density fitting, and other related techniques may 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}
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The $ G _ 0 W _ 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.
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%%%%%%%%%%%%%%%%%%%%%
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\section { Conclusion}
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%%%%%%%%%%%%%%%%%%%%%
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Here, we have unveiled exact similarities between CC and many-body perturbation theory at the ground- and excited-state levels.
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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.
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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.
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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.
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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.
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However, several challenges lie ahead as one must derive, for example, the $ \Lambda $ equations associated with $ GW $ \cite { Bartlett_ 1986,Rishi_ 2020} and the response of the static screening with respect to the external perturbation at the BSE level.
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The present connections between CC and $ GW $ could also provide new directions for the development of multireference $ GW $ methods \cite { Brouder_ 2009,Linner_ 2019} in order to treat strongly correlated systems. \cite { Lyakh_ 2012,Evangelista_ 2018}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\section*{Supplementary Material}
%\label{sec:supmat}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%
\acknowledgments {
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PFL thanks Xavier Blase, Pina Romaniello, and Francesco Evangelista for useful discussions.
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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).}
%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\section * { Data availability statement}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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Data sharing is not applicable to this article as no new data were created or analyzed in this study.
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%%%%%%%%%
%\appendix
%%%%%%%%%
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{ 1999} )} \BibitemShut { NoStop} %
\bibitem [{ \citenamefont { van~der Horst} \ \emph { et~al.} (1999)\citenamefont
{ van~der Horst} , \citenamefont { Bobbert} , \citenamefont { Michels} ,
\citenamefont { Brocks} ,\ and\ \citenamefont { Kelly} } ]{ Horst_ 1999} %
\BibitemOpen
\bibfield { author} { \bibinfo { author} { \bibfnamefont { J.-W.} \ \bibnamefont
{ van~der Horst} } , \bibinfo { author} { \bibfnamefont { P.~A.} \ \bibnamefont
{ Bobbert} } , \bibinfo { author} { \bibfnamefont { M.~A.~J.} \ \bibnamefont
{ Michels} } , \bibinfo { author} { \bibfnamefont { G.} ~\bibnamefont { Brocks} } , \
and\ \bibinfo { author} { \bibfnamefont { P.~J.} \ \bibnamefont { Kelly} } ,\ } \href
{ \doibase 10.1103/PhysRevLett.83.4413} { \bibfield { journal} { \bibinfo
{ journal} { Phys. Rev. Lett.} \ } \textbf { \bibinfo { volume} { 83} } ,\ \bibinfo
{ pages} { 4413} (\bibinfo { year} { 1999} )} \BibitemShut { NoStop} %
\bibitem [{ \citenamefont { Puschnig} \ and\ \citenamefont
{ Ambrosch-Draxl} (2002)} ]{ Puschnig_ 2002} %
\BibitemOpen
\bibfield { author} { \bibinfo { author} { \bibfnamefont { P.} ~\bibnamefont
{ Puschnig} } \ and\ \bibinfo { author} { \bibfnamefont { C.} ~\bibnamefont
{ Ambrosch-Draxl} } ,\ } \href { \doibase 10.1103/PhysRevLett.89.056405}
{ \bibfield { journal} { \bibinfo { journal} { Phys. Rev. Lett.} \ } \textbf
{ \bibinfo { volume} { 89} } ,\ \bibinfo { pages} { 056405} (\bibinfo { year}
{ 2002} )} \BibitemShut { NoStop} %
\bibitem [{ \citenamefont { Tiago} , \citenamefont { Northrup} ,\ and\ \citenamefont
{ Louie} (2003)} ]{ Tiago_ 2003} %
\BibitemOpen
\bibfield { author} { \bibinfo { author} { \bibfnamefont { M.~L.} \ \bibnamefont
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\ and\ \bibinfo { author} { \bibfnamefont { S.~G.} \ \bibnamefont { Louie} } ,\
} \href { \doibase 10.1103/PhysRevB.67.115212} { \bibfield { journal} { \bibinfo
{ journal} { Phys. Rev. B} \ } \textbf { \bibinfo { volume} { 67} } ,\ \bibinfo
{ pages} { 115212} (\bibinfo { year} { 2003} )} \BibitemShut { NoStop} %
\bibitem [{ \citenamefont { Rocca} , \citenamefont { Lu} ,\ and\ \citenamefont
{ Galli} (2010)} ]{ Rocca_ 2010} %
\BibitemOpen
\bibfield { author} { \bibinfo { author} { \bibfnamefont { D.} ~\bibnamefont
{ Rocca} } , \bibinfo { author} { \bibfnamefont { D.} ~\bibnamefont { Lu} } , \ and\
\bibinfo { author} { \bibfnamefont { G.} ~\bibnamefont { Galli} } ,\ } \href
{ \doibase 10.1063/1.3494540} { \bibfield { journal} { \bibinfo { journal} { J.
Chem. Phys.} \ } \textbf { \bibinfo { volume} { 133} } ,\ \bibinfo { pages} { 164109}
(\bibinfo { year} { 2010} )} \BibitemShut { NoStop} %
\bibitem [{ \citenamefont { Boulanger} \ \emph { et~al.} (2014)\citenamefont
{ Boulanger} , \citenamefont { Jacquemin} , \citenamefont { Duchemin} ,\ and\
\citenamefont { Blase} } ]{ Boulanger_ 2014} %
\BibitemOpen
\bibfield { author} { \bibinfo { author} { \bibfnamefont { P.} ~\bibnamefont
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\ and\ \bibinfo { author} { \bibfnamefont { X.} ~\bibnamefont { Blase} } ,\ } \href
{ \doibase 10.1021/ct401101u} { \bibfield { journal} { \bibinfo { journal} { J.
Chem. Theory Comput.} \ } \textbf { \bibinfo { volume} { 10} } ,\ \bibinfo { pages}
{ 1212} (\bibinfo { year} { 2014} )} \BibitemShut { NoStop} %
\bibitem [{ \citenamefont { Jacquemin} , \citenamefont { Duchemin} ,\ and\
\citenamefont { Blase} (2015{ \natexlab { a} } )} ]{ Jacquemin_ 2015a} %
\BibitemOpen
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\ and\ \bibinfo { author} { \bibfnamefont { X.} ~\bibnamefont { Blase} } ,\ } \href
{ \doibase 10.1021/acs.jctc.5b00304} { \bibfield { journal} { \bibinfo
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\bibinfo { pages} { 3290} (\bibinfo { year} { 2015} { \natexlab { a} } )} \BibitemShut
{ NoStop} %
\bibitem [{ \citenamefont { Bruneval} , \citenamefont { Hamed} ,\ and\ \citenamefont
{ Neaton} (2015)} ]{ Bruneval_ 2015} %
\BibitemOpen
\bibfield { author} { \bibinfo { author} { \bibfnamefont { F.} ~\bibnamefont
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\ and\ \bibinfo { author} { \bibfnamefont { J.~B.} \ \bibnamefont { Neaton} } ,\
} \href { \doibase 10.1063/1.4922489} { \bibfield { journal} { \bibinfo
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{ pages} { 244101} (\bibinfo { year} { 2015} )} \BibitemShut { NoStop} %
\bibitem [{ \citenamefont { Jacquemin} , \citenamefont { Duchemin} ,\ and\
\citenamefont { Blase} (2015{ \natexlab { b} } )} ]{ Jacquemin_ 2015b} %
\BibitemOpen
\bibfield { author} { \bibinfo { author} { \bibfnamefont { D.} ~\bibnamefont
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\ and\ \bibinfo { author} { \bibfnamefont { X.} ~\bibnamefont { Blase} } ,\ } \href
{ \doibase 10.1021/acs.jctc.5b00619} { \bibfield { journal} { \bibinfo
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\bibinfo { pages} { 5340} (\bibinfo { year} { 2015} { \natexlab { b} } )} \BibitemShut
{ NoStop} %
\bibitem [{ \citenamefont { Hirose} , \citenamefont { Noguchi} ,\ and\ \citenamefont
{ Sugino} (2015)} ]{ Hirose_ 2015} %
\BibitemOpen
\bibfield { author} { \bibinfo { author} { \bibfnamefont { D.} ~\bibnamefont
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and\ \bibinfo { author} { \bibfnamefont { O.} ~\bibnamefont { Sugino} } ,\ } \href
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{ pages} { 205111} (\bibinfo { year} { 2015} )} \BibitemShut { NoStop} %
\bibitem [{ \citenamefont { Jacquemin} , \citenamefont { Duchemin} ,\ and\
\citenamefont { Blase} (2017)} ]{ Jacquemin_ 2017a} %
\BibitemOpen
\bibfield { author} { \bibinfo { author} { \bibfnamefont { D.} ~\bibnamefont
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\ and\ \bibinfo { author} { \bibfnamefont { X.} ~\bibnamefont { Blase} } ,\ } \href
{ \doibase 10.1021/acs.jpclett.7b00381} { \bibfield { journal} { \bibinfo
{ journal} { J. Phys. Chem. Lett.} \ } \textbf { \bibinfo { volume} { 8} } ,\ \bibinfo
{ pages} { 1524} (\bibinfo { year} { 2017} )} \BibitemShut { NoStop} %
\bibitem [{ \citenamefont { Jacquemin} \ \emph { et~al.} (2017)\citenamefont
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\BibitemOpen
\bibfield { author} { \bibinfo { author} { \bibfnamefont { D.} ~\bibnamefont
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\bibinfo { author} { \bibfnamefont { A.} ~\bibnamefont { Blondel} } , \ and\
\bibinfo { author} { \bibfnamefont { X.} ~\bibnamefont { Blase} } ,\ } \href
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\bibinfo { pages} { 767} (\bibinfo { year} { 2017} )} \BibitemShut { NoStop} %
\bibitem [{ \citenamefont { Rangel} \ \emph { et~al.} (2017)\citenamefont { Rangel} ,
\citenamefont { Hamed} , \citenamefont { Bruneval} ,\ and\ \citenamefont
{ Neaton} } ]{ Rangel_ 2017} %
\BibitemOpen
\bibfield { author} { \bibinfo { author} { \bibfnamefont { T.} ~\bibnamefont
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\bibinfo { author} { \bibfnamefont { J.~B.} \ \bibnamefont { Neaton} } ,\ } \href
{ \doibase 10.1063/1.4983126} { \bibfield { journal} { \bibinfo { journal} { J.
Chem. Phys.} \ } \textbf { \bibinfo { volume} { 146} } ,\ \bibinfo { pages} { 194108}
(\bibinfo { year} { 2017} )} \BibitemShut { NoStop} %
\bibitem [{ \citenamefont { Krause} \ and\ \citenamefont
{ Klopper} (2017)} ]{ Krause_ 2017} %
\BibitemOpen
\bibfield { author} { \bibinfo { author} { \bibfnamefont { K.} ~\bibnamefont
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{ \bibinfo { journal} { J. Comput. Chem.} \ } \textbf { \bibinfo { volume} { 38} } ,\
\bibinfo { pages} { 383} (\bibinfo { year} { 2017} )} \BibitemShut { NoStop} %
\bibitem [{ \citenamefont { Gui} , \citenamefont { Holzer} ,\ and\ \citenamefont
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\BibitemOpen
\bibfield { author} { \bibinfo { author} { \bibfnamefont { X.} ~\bibnamefont
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\bibinfo { author} { \bibfnamefont { W.} ~\bibnamefont { Klopper} } ,\ } \href
{ \doibase 10.1021/acs.jctc.8b00014} { \bibfield { journal} { \bibinfo
{ journal} { J. Chem. Theory Comput.} \ } \textbf { \bibinfo { volume} { 14} } ,\
\bibinfo { pages} { 2127} (\bibinfo { year} { 2018} )} \BibitemShut { NoStop} %
\bibitem [{ \citenamefont { Liu} \ \emph { et~al.} (2020)\citenamefont { Liu} ,
\citenamefont { Kloppenburg} , \citenamefont { Yao} , \citenamefont { Ren} ,
\citenamefont { Appel} , \citenamefont { Kanai} ,\ and\ \citenamefont
{ Blum} } ]{ Liu_ 2020} %
\BibitemOpen
\bibfield { author} { \bibinfo { author} { \bibfnamefont { C.} ~\bibnamefont
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\bibinfo { author} { \bibfnamefont { Y.} ~\bibnamefont { Yao} } , \bibinfo { author}
{ \bibfnamefont { X.} ~\bibnamefont { Ren} } , \bibinfo { author} { \bibfnamefont
{ H.} ~\bibnamefont { Appel} } , \bibinfo { author} { \bibfnamefont
{ Y.} ~\bibnamefont { Kanai} } , \ and\ \bibinfo { author} { \bibfnamefont
{ V.} ~\bibnamefont { Blum} } ,\ } \href { \doibase 10.1063/1.5123290} { \bibfield
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{ 152} } ,\ \bibinfo { pages} { 044105} (\bibinfo { year} { 2020} )} \BibitemShut
{ NoStop} %
\bibitem [{ \citenamefont { Holzer} \ and\ \citenamefont
{ Klopper} (2018)} ]{ Holzer_ 2018a} %
\BibitemOpen
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{ \bibinfo { journal} { J. Chem. Phys.} \ } \textbf { \bibinfo { volume} { 149} } ,\
\bibinfo { pages} { 101101} (\bibinfo { year} { 2018} )} \BibitemShut { NoStop} %
\bibitem [{ \citenamefont { Holzer} \ \emph { et~al.} (2018)\citenamefont { Holzer} ,
\citenamefont { Gui} , \citenamefont { Harding} , \citenamefont { Kresse} ,
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\BibitemOpen
\bibfield { author} { \bibinfo { author} { \bibfnamefont { C.} ~\bibnamefont
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\bibinfo { author} { \bibfnamefont { M.~E.} \ \bibnamefont { Harding} } , \bibinfo
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{ \bibfnamefont { W.} ~\bibnamefont { Klopper} } ,\ } \href { \doibase
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(\bibinfo { year} { 2018} )} \BibitemShut { NoStop} %
\bibitem [{ \citenamefont { Loos} \ \emph { et~al.} (2020)\citenamefont { Loos} ,
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{ Jacquemin} ,\ and\ \citenamefont { Blase} } ]{ Loos_ 2020e} %
\BibitemOpen
\bibfield { author} { \bibinfo { author} { \bibfnamefont { P.-F.} \ \bibnamefont
{ Loos} } , \bibinfo { author} { \bibfnamefont { A.} ~\bibnamefont { Scemama} } ,
\bibinfo { author} { \bibfnamefont { I.} ~\bibnamefont { Duchemin} } , \bibinfo
{ author} { \bibfnamefont { D.} ~\bibnamefont { Jacquemin} } , \ and\ \bibinfo
{ author} { \bibfnamefont { X.} ~\bibnamefont { Blase} } ,\ } \href { \doibase
10.1021/acs.jpclett.0c00460} { \bibfield { journal} { \bibinfo { journal} { J.
Phys. Chem. Lett.} \ } \textbf { \bibinfo { volume} { 11} } ,\ \bibinfo { pages}
{ 3536} (\bibinfo { year} { 2020} )} \BibitemShut { NoStop} %
\bibitem [{ \citenamefont { Loos} \ \emph { et~al.} (2021)\citenamefont { Loos} ,
\citenamefont { Comin} , \citenamefont { Blase} ,\ and\ \citenamefont
{ Jacquemin} } ]{ Loos_ 2021} %
\BibitemOpen
\bibfield { author} { \bibinfo { author} { \bibfnamefont { P.-F.} \ \bibnamefont
{ Loos} } , \bibinfo { author} { \bibfnamefont { M.} ~\bibnamefont { Comin} } ,
\bibinfo { author} { \bibfnamefont { X.} ~\bibnamefont { Blase} } , \ and\ \bibinfo
{ author} { \bibfnamefont { D.} ~\bibnamefont { Jacquemin} } ,\ } \href { \doibase
10.1021/acs.jctc.1c00226} { \bibfield { journal} { \bibinfo { journal} { J.
Chem. Theory Comput.} \ } \textbf { \bibinfo { volume} { 17} } ,\ \bibinfo { pages}
{ 3666} (\bibinfo { year} { 2021} )} \BibitemShut { NoStop} %
\bibitem [{ \citenamefont { McKeon} \ \emph { et~al.} (2022)\citenamefont { McKeon} ,
\citenamefont { Hamed} , \citenamefont { Bruneval} ,\ and\ \citenamefont
{ Neaton} } ]{ McKeon_ 2022} %
\BibitemOpen
\bibfield { author} { \bibinfo { author} { \bibfnamefont { C.~A.} \ \bibnamefont
{ McKeon} } , \bibinfo { author} { \bibfnamefont { S.~M.} \ \bibnamefont { Hamed} } ,
\bibinfo { author} { \bibfnamefont { F.} ~\bibnamefont { Bruneval} } , \ and\
\bibinfo { author} { \bibfnamefont { J.~B.} \ \bibnamefont { Neaton} } ,\ } \href
{ \doibase 10.1063/5.0097582} { \bibfield { journal} { \bibinfo { journal} { J.
Chem. Phys.} \ } \textbf { \bibinfo { volume} { 157} } ,\ \bibinfo { pages} { 074103}
(\bibinfo { year} { 2022} )} \BibitemShut { NoStop} %
\bibitem [{ \citenamefont { Freeman} (1977)} ]{ Freeman_ 1977} %
\BibitemOpen
\bibfield { author} { \bibinfo { author} { \bibfnamefont { D.~L.} \ \bibnamefont
{ Freeman} } ,\ } \href { \doibase 10.1103/PhysRevB.15.5512} { \bibfield { journal}
{ \bibinfo { journal} { Phys. Rev. B} \ } \textbf { \bibinfo { volume} { 15} } ,\
\bibinfo { pages} { 5512} (\bibinfo { year} { 1977} )} \BibitemShut { NoStop} %
\bibitem [{ \citenamefont { Scuseria} , \citenamefont { Henderson} ,\ and\
\citenamefont { Sorensen} (2008)} ]{ Scuseria_ 2008} %
\BibitemOpen
\bibfield { author} { \bibinfo { author} { \bibfnamefont { G.~E.} \ \bibnamefont
{ Scuseria} } , \bibinfo { author} { \bibfnamefont { T.~M.} \ \bibnamefont
{ Henderson} } , \ and\ \bibinfo { author} { \bibfnamefont { D.~C.} \ \bibnamefont
{ Sorensen} } ,\ } \href { \doibase 10.1063/1.3043729} { \bibfield { journal}
{ \bibinfo { journal} { J. Chem. Phys.} \ } \textbf { \bibinfo { volume} { 129} } ,\
\bibinfo { pages} { 231101} (\bibinfo { year} { 2008} )} \BibitemShut { NoStop} %
\bibitem [{ \citenamefont { Jansen} , \citenamefont { Liu} ,\ and\ \citenamefont
{ { \' A} ngy{ \' a} n} (2010)} ]{ Jansen_ 2010} %
\BibitemOpen
\bibfield { author} { \bibinfo { author} { \bibfnamefont { G.} ~\bibnamefont
{ Jansen} } , \bibinfo { author} { \bibfnamefont { R.-F.} \ \bibnamefont { Liu} } , \
and\ \bibinfo { author} { \bibfnamefont { J.~G.} \ \bibnamefont
{ { \' A} ngy{ \' a} n} } ,\ } \href { \doibase 10.1063/1.3481575} { \bibfield { journal}
{ \bibinfo { journal} { J. Chem. Phys.} \ } \textbf { \bibinfo { volume} { 133} } ,\
\bibinfo { pages} { 154106} (\bibinfo { year} { 2010} )} \BibitemShut { NoStop} %
\bibitem [{ \citenamefont { Scuseria} , \citenamefont { Henderson} ,\ and\
\citenamefont { Bulik} (2013)} ]{ Scuseria_ 2013} %
\BibitemOpen
\bibfield { author} { \bibinfo { author} { \bibfnamefont { G.~E.} \ \bibnamefont
{ Scuseria} } , \bibinfo { author} { \bibfnamefont { T.~M.} \ \bibnamefont
{ Henderson} } , \ and\ \bibinfo { author} { \bibfnamefont { I.~W.} \ \bibnamefont
{ Bulik} } ,\ } \href { \doibase 10.1063/1.4820557} { \bibfield { journal}
{ \bibinfo { journal} { J. Chem. Phys.} \ } \textbf { \bibinfo { volume} { 139} } ,\
\bibinfo { pages} { 104113} (\bibinfo { year} { 2013} )} \BibitemShut { NoStop} %
\bibitem [{ \citenamefont { Peng} \ \emph { et~al.} (2013)\citenamefont { Peng} ,
\citenamefont { Steinmann} , \citenamefont { { van Aggelen} } ,\ and\ \citenamefont
{ Yang} } ]{ Peng_ 2013} %
\BibitemOpen
\bibfield { author} { \bibinfo { author} { \bibfnamefont { D.} ~\bibnamefont
{ Peng} } , \bibinfo { author} { \bibfnamefont { S.~N.} \ \bibnamefont { Steinmann} } ,
\bibinfo { author} { \bibfnamefont { H.} ~\bibnamefont { { van Aggelen} } } , \ and\
\bibinfo { author} { \bibfnamefont { W.} ~\bibnamefont { Yang} } ,\ } \href { \doibase
10.1063/1.4820556} { \bibfield { journal} { \bibinfo { journal} { J. Chem.
Phys.} \ } \textbf { \bibinfo { volume} { 139} } ,\ \bibinfo { pages} { 104112}
(\bibinfo { year} { 2013} )} \BibitemShut { NoStop} %
\bibitem [{ \citenamefont { Berkelbach} (2018)} ]{ Berkelbach_ 2018} %
\BibitemOpen
\bibfield { author} { \bibinfo { author} { \bibfnamefont { T.~C.} \ \bibnamefont
{ Berkelbach} } ,\ } \href { \doibase 10.1063/1.5032314} { \bibfield { journal}
{ \bibinfo { journal} { J. Chem. Phys.} \ } \textbf { \bibinfo { volume} { 149} } ,\
\bibinfo { pages} { 041103} (\bibinfo { year} { 2018} )} \BibitemShut { NoStop} %
\bibitem [{ \citenamefont { Rishi} , \citenamefont { Perera} ,\ and\ \citenamefont
{ Bartlett} (2020)} ]{ Rishi_ 2020} %
\BibitemOpen
\bibfield { author} { \bibinfo { author} { \bibfnamefont { V.} ~\bibnamefont
{ Rishi} } , \bibinfo { author} { \bibfnamefont { A.} ~\bibnamefont { Perera} } , \
and\ \bibinfo { author} { \bibfnamefont { R.~J.} \ \bibnamefont { Bartlett} } ,\
} \href { \doibase 10.1063/5.0023862} { \bibfield { journal} { \bibinfo
{ journal} { J. Chem. Phys.} \ } \textbf { \bibinfo { volume} { 153} } ,\ \bibinfo
{ pages} { 234101} (\bibinfo { year} { 2020} )} \BibitemShut { NoStop} %
\bibitem [{ \citenamefont { { \v C} { \' \i } { \v z} ek} (1966)} ]{ Cizek_ 1966} %
\BibitemOpen
\bibfield { author} { \bibinfo { author} { \bibfnamefont { J.} ~\bibnamefont { { \v
C} { \' \i } { \v z} ek} } ,\ } \href { \doibase 10.1063/1.1727484} { \bibfield
{ journal} { \bibinfo { journal} { J. Chem. Phys.} \ } \textbf { \bibinfo { volume}
{ 45} } ,\ \bibinfo { pages} { 4256} (\bibinfo { year} { 1966} )} \BibitemShut
{ NoStop} %
\bibitem [{ \citenamefont { Paldus} , \citenamefont { \ifmmode \check { C} \else
\v { C} \fi { } \' { \i } \ifmmode ~\check { z} \else \v { z} \fi { } ek} ,\ and\ \citenamefont
{ Shavitt} (1972)} ]{ Paldus_ 1972} %
\BibitemOpen
\bibfield { author} { \bibinfo { author} { \bibfnamefont { J.} ~\bibnamefont
{ Paldus} } , \bibinfo { author} { \bibfnamefont { J.} ~\bibnamefont { \ifmmode
\check { C} \else \v { C} \fi { } \' { \i } \ifmmode ~\check { z} \else \v { z} \fi { } ek} } , \ and\
\bibinfo { author} { \bibfnamefont { I.} ~\bibnamefont { Shavitt} } ,\ } \href
{ \doibase 10.1103/PhysRevA.5.50} { \bibfield { journal} { \bibinfo { journal}
{ Phys. Rev. A} \ } \textbf { \bibinfo { volume} { 5} } ,\ \bibinfo { pages} { 50}
(\bibinfo { year} { 1972} )} \BibitemShut { NoStop} %
\bibitem [{ \citenamefont { Crawford} \ and\ \citenamefont
{ Schaefer} (2000)} ]{ Crawford_ 2000} %
\BibitemOpen
\bibfield { author} { \bibinfo { author} { \bibfnamefont { T.~D.} \ \bibnamefont
{ Crawford} } \ and\ \bibinfo { author} { \bibfnamefont { H.~F.} \ \bibnamefont
{ Schaefer} } ,\ } in\ \href { \doibase 10.1002/9780470125915.ch2} { \emph
{ \bibinfo { booktitle} { Reviews in { { Computational Chemistry} } } } } \ (\bibinfo
{ publisher} { { John Wiley \& Sons, Ltd} } ,\ \bibinfo { year} { 2000} )\ pp.\
\bibinfo { pages} { 33--136} \BibitemShut { NoStop} %
\bibitem [{ \citenamefont { Piecuch} \ \emph { et~al.} (2002)\citenamefont
{ Piecuch} , \citenamefont { Kowalski} , \citenamefont { Pimienta} ,\ and\
\citenamefont { Mcguire} } ]{ Piecuch_ 2002} %
\BibitemOpen
\bibfield { author} { \bibinfo { author} { \bibfnamefont { P.} ~\bibnamefont
{ Piecuch} } , \bibinfo { author} { \bibfnamefont { K.} ~\bibnamefont { Kowalski} } ,
\bibinfo { author} { \bibfnamefont { I.~S.~O.} \ \bibnamefont { Pimienta} } , \ and\
\bibinfo { author} { \bibfnamefont { M.~J.} \ \bibnamefont { Mcguire} } ,\ } \href
{ \doibase 10.1080/0144235021000053811} { \bibfield { journal} { \bibinfo
{ journal} { Int. Rev. Phys. Chem.} \ } \textbf { \bibinfo { volume} { 21} } ,\
\bibinfo { pages} { 527} (\bibinfo { year} { 2002} )} \BibitemShut { NoStop} %
\bibitem [{ \citenamefont { Bartlett} \ and\ \citenamefont
{ Musia{ \l } } (2007)} ]{ Bartlett_ 2007} %
\BibitemOpen
\bibfield { author} { \bibinfo { author} { \bibfnamefont { R.~J.} \ \bibnamefont
{ Bartlett} } \ and\ \bibinfo { author} { \bibfnamefont { M.} ~\bibnamefont
{ Musia{ \l } } } ,\ } \href { \doibase 10.1103/RevModPhys.79.291} { \bibfield
{ journal} { \bibinfo { journal} { Rev. Mod. Phys.} \ } \textbf { \bibinfo { volume}
{ 79} } ,\ \bibinfo { pages} { 291} (\bibinfo { year} { 2007} )} \BibitemShut
{ NoStop} %
\bibitem [{ \citenamefont { Shavitt} \ and\ \citenamefont
{ Bartlett} (2009)} ]{ Shavitt_ 2009} %
\BibitemOpen
\bibfield { author} { \bibinfo { author} { \bibfnamefont { I.} ~\bibnamefont
{ Shavitt} } \ and\ \bibinfo { author} { \bibfnamefont { R.~J.} \ \bibnamefont
{ Bartlett} } ,\ } \href { \doibase 10.1017/CBO9780511596834} { \emph { \bibinfo
{ title} { Many-{ { Body Methods} } in { { Chemistry} } and { { Physics} } : { { MBPT} } and
{ { Coupled} } -{ { Cluster Theory} } } } } ,\ Cambridge { { Molecular Science} } \
(\bibinfo { publisher} { { Cambridge University Press} } ,\ \bibinfo { address}
{ { Cambridge} } ,\ \bibinfo { year} { 2009} )\BibitemShut { NoStop} %
\bibitem [{ \citenamefont { Furche} (2008)} ]{ Furche_ 2008} %
\BibitemOpen
\bibfield { author} { \bibinfo { author} { \bibfnamefont { F.} ~\bibnamefont
{ Furche} } ,\ } \href { \doibase 10.1063/1.2977789} { \bibfield { journal}
{ \bibinfo { journal} { J. Chem. Phys.} \ } \textbf { \bibinfo { volume} { 129} } ,\
\bibinfo { pages} { 114105} (\bibinfo { year} { 2008} )} \BibitemShut { NoStop} %
\bibitem [{ \citenamefont { Angyan} \ \emph { et~al.} (2011)\citenamefont { Angyan} ,
\citenamefont { Liu} , \citenamefont { Toulouse} ,\ and\ \citenamefont
{ Jansen} } ]{ Angyan_ 2011} %
\BibitemOpen
\bibfield { author} { \bibinfo { author} { \bibfnamefont { J.~G.} \ \bibnamefont
{ Angyan} } , \bibinfo { author} { \bibfnamefont { R.-F.} \ \bibnamefont { Liu} } ,
\bibinfo { author} { \bibfnamefont { J.} ~\bibnamefont { Toulouse} } , \ and\
\bibinfo { author} { \bibfnamefont { G.} ~\bibnamefont { Jansen} } ,\ } \href
{ \doibase 10.1021/ct200501r} { \bibfield { journal} { \bibinfo { journal} { J.
Chem. Theory Comput.} \ } \textbf { \bibinfo { volume} { 7} } ,\ \bibinfo { pages}
{ 3116} (\bibinfo { year} { 2011} )} \BibitemShut { NoStop} %
\bibitem [{ \citenamefont { Sawada} (1957)} ]{ Sawada_ 1957a} %
\BibitemOpen
\bibfield { author} { \bibinfo { author} { \bibfnamefont { K.} ~\bibnamefont
{ Sawada} } ,\ } \href { \doibase 10.1103/PhysRev.106.372} { \bibfield { journal}
{ \bibinfo { journal} { Phys. Rev.} \ } \textbf { \bibinfo { volume} { 106} } ,\
\bibinfo { pages} { 372} (\bibinfo { year} { 1957} )} \BibitemShut { NoStop} %
\bibitem [{ \citenamefont { Rowe} (1968)} ]{ Rowe_ 1968} %
\BibitemOpen
\bibfield { author} { \bibinfo { author} { \bibfnamefont { D.~J.} \ \bibnamefont
{ Rowe} } ,\ } \href { \doibase 10.1103/PhysRev.175.1283} { \bibfield { journal}
{ \bibinfo { journal} { Phys. Rev.} \ } \textbf { \bibinfo { volume} { 175} } ,\
\bibinfo { pages} { 1283} (\bibinfo { year} { 1968} )} \BibitemShut { NoStop} %
\bibitem [{ \citenamefont { Ring} \ and\ \citenamefont
{ Schuck} (2004)} ]{ Schuck_ book} %
\BibitemOpen
\bibfield { author} { \bibinfo { author} { \bibfnamefont { P.} ~\bibnamefont
{ Ring} } \ and\ \bibinfo { author} { \bibfnamefont { P.} ~\bibnamefont { Schuck} } ,\
} \href @noop { } { \emph { \bibinfo { title} { The Nuclear Many-Body Problem} } } \
(\bibinfo { publisher} { Springer} ,\ \bibinfo { year} { 2004} )\BibitemShut
{ NoStop} %
\bibitem [{ Note1()} ]{ Note1} %
\BibitemOpen
\bibinfo { note} { The factor $ 1 / 4 $ in Eq.~\protect \textup { \hbox
{ \mathsurround \z @ \protect \normalfont (\ignorespaces \ref
{ eq:EcRPAx} \unskip \@ @italiccorr )} } is sometimes replaced by a factor $ 1 / 2 $ ,
which corresponds to a different choice for the interaction kernel. See
Ref.~\protect \rev @citealpnum { Angyan_ 2011} for more details} \BibitemShut
{ NoStop} %
\bibitem [{ \citenamefont { Furche} \ and\ \citenamefont { { Van
Voorhis} } (2005)} ]{ Furche_ 2005} %
\BibitemOpen
\bibfield { author} { \bibinfo { author} { \bibfnamefont { F.} ~\bibnamefont
{ Furche} } \ and\ \bibinfo { author} { \bibfnamefont { T.} ~\bibnamefont { { Van
Voorhis} } } ,\ } \href { \doibase 10.1063/1.1884112} { \bibfield { journal}
{ \bibinfo { journal} { J. Chem. Phys.} \ } \textbf { \bibinfo { volume} { 122} } ,\
\bibinfo { pages} { 164106} (\bibinfo { year} { 2005} )} \BibitemShut { NoStop} %
\bibitem [{ \citenamefont { Stanton} \ and\ \citenamefont
{ Bartlett} (1993)} ]{ Stanton_ 1993} %
\BibitemOpen
\bibfield { author} { \bibinfo { author} { \bibfnamefont { J.~F.} \ \bibnamefont
{ Stanton} } \ and\ \bibinfo { author} { \bibfnamefont { R.~J.} \ \bibnamefont
{ Bartlett} } ,\ } \href { \doibase 10.1063/1.464746} { \bibfield { journal}
{ \bibinfo { journal} { J. Chem. Phys.} \ } \textbf { \bibinfo { volume} { 98} } ,\
\bibinfo { pages} { 7029} (\bibinfo { year} { 1993} )} \BibitemShut { NoStop} %
\bibitem [{ \citenamefont { Emrich} (1981)} ]{ Emrich_ 1981} %
\BibitemOpen
\bibfield { author} { \bibinfo { author} { \bibfnamefont { K.} ~\bibnamefont
{ Emrich} } ,\ } \href { \doibase https://doi.org/10.1016/0375-9474(81)90179-2}
{ \bibfield { journal} { \bibinfo { journal} { Nuc. Phys. A} \ } \textbf { \bibinfo
{ volume} { 351} } ,\ \bibinfo { pages} { 379} (\bibinfo { year}
{ 1981} )} \BibitemShut { NoStop} %
\bibitem [{ \citenamefont { Runge} \ and\ \citenamefont
{ Gross} (1984)} ]{ Runge_ 1984} %
\BibitemOpen
\bibfield { author} { \bibinfo { author} { \bibfnamefont { E.} ~\bibnamefont
{ Runge} } \ and\ \bibinfo { author} { \bibfnamefont { E.~K.~U.} \ \bibnamefont
{ Gross} } ,\ } \href { \doibase 10.1103/PhysRevLett.52.997} { \bibfield { journal}
{ \bibinfo { journal} { Phys. Rev. Lett.} \ } \textbf { \bibinfo { volume} { 52} } ,\
\bibinfo { pages} { 997} (\bibinfo { year} { 1984} )} \BibitemShut { NoStop} %
\bibitem [{ \citenamefont { Casida} (1995)} ]{ Casida_ 1995} %
\BibitemOpen
\bibfield { author} { \bibinfo { author} { \bibfnamefont { M.~E.} \ \bibnamefont
{ Casida} } ,\ } \enquote { \bibinfo { title} { Time-dependent density functional
response theory for molecules} ,} \ \ (\bibinfo { publisher} { World Scientific,
Singapore} ,\ \bibinfo { year} { 1995} )\ pp.\ \bibinfo { pages}
{ 155--192} \BibitemShut { NoStop} %
\bibitem [{ Note2()} ]{ Note2} %
\BibitemOpen
\bibinfo { note} { \protect \leavevmode { \color { black} To the best of our
knowledge, the trace (or plasmon) formula has been first introduced 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.~\protect \rev @citealpnum { Li_ 2020} for more
details} .} \BibitemShut { Stop} %
\bibitem [{ \citenamefont { Li} \ \emph { et~al.} (2020)\citenamefont { Li} ,
\citenamefont { Duchemin} , \citenamefont { Blase} ,\ and\ \citenamefont
{ Olevano} } ]{ Li_ 2020} %
\BibitemOpen
\bibfield { author} { \bibinfo { author} { \bibfnamefont { J.} ~\bibnamefont
{ Li} } , \bibinfo { author} { \bibfnamefont { I.} ~\bibnamefont { Duchemin} } ,
\bibinfo { author} { \bibfnamefont { X.} ~\bibnamefont { Blase} } , \ and\ \bibinfo
{ author} { \bibfnamefont { V.} ~\bibnamefont { Olevano} } ,\ } \href { \doibase
10.21468/SciPostPhys.8.2.020} { \bibfield { journal} { \bibinfo { journal}
{ SciPost Phys.} \ } \textbf { \bibinfo { volume} { 8} } ,\ \bibinfo { pages} { 20}
(\bibinfo { year} { 2020} )} \BibitemShut { NoStop} %
\bibitem [{ \citenamefont { Li} \ and\ \citenamefont { Olevano} (2021)} ]{ Li_ 2021} %
\BibitemOpen
\bibfield { author} { \bibinfo { author} { \bibfnamefont { J.} ~\bibnamefont
{ Li} } \ and\ \bibinfo { author} { \bibfnamefont { V.} ~\bibnamefont { Olevano} } ,\
} \href { \doibase 10.1103/PhysRevA.103.012809} { \bibfield { journal} { \bibinfo
{ journal} { Phys. Rev. A} \ } \textbf { \bibinfo { volume} { 103} } ,\ \bibinfo
{ pages} { 012809} (\bibinfo { year} { 2021} )} \BibitemShut { NoStop} %
\bibitem [{ \citenamefont { Di~Sabatino} , \citenamefont { Loos} ,\ and\
\citenamefont { Romaniello} (2021)} ]{ DiSabatino_ 2021} %
\BibitemOpen
\bibfield { author} { \bibinfo { author} { \bibfnamefont { S.} ~\bibnamefont
{ Di~Sabatino} } , \bibinfo { author} { \bibfnamefont { P.-F.} \ \bibnamefont
{ Loos} } , \ and\ \bibinfo { author} { \bibfnamefont { P.} ~\bibnamefont
{ Romaniello} } ,\ } \href { \doibase 10.3389/fchem.2021.751054} { \bibfield
{ journal} { \bibinfo { journal} { Front. Chem.} \ } \textbf { \bibinfo { volume}
{ 9} } ,\ \bibinfo { pages} { 751054} (\bibinfo { year} { 2021} )} \BibitemShut
{ NoStop} %
\bibitem [{ \citenamefont { Maggio} \ and\ \citenamefont
{ Kresse} (2016)} ]{ Maggio_ 2016} %
\BibitemOpen
\bibfield { author} { \bibinfo { author} { \bibfnamefont { E.} ~\bibnamefont
{ Maggio} } \ and\ \bibinfo { author} { \bibfnamefont { G.} ~\bibnamefont
{ Kresse} } ,\ } \href { \doibase 10.1103/PhysRevB.93.235113} { \bibfield
{ journal} { \bibinfo { journal} { Phys. Rev. B} \ } \textbf { \bibinfo { volume}
{ 93} } ,\ \bibinfo { pages} { 235113} (\bibinfo { year} { 2016} )} \BibitemShut
{ NoStop} %
\bibitem [{ \citenamefont { Berger} , \citenamefont { Loos} ,\ and\ \citenamefont
{ Romaniello} (2020)} ]{ Berger_ 2021} %
\BibitemOpen
\bibfield { author} { \bibinfo { author} { \bibfnamefont { J.~A.} \ \bibnamefont
{ Berger} } , \bibinfo { author} { \bibfnamefont { P.-F.} \ \bibnamefont { Loos} } , \
and\ \bibinfo { author} { \bibfnamefont { P.} ~\bibnamefont { Romaniello} } ,\
} \href { \doibase 10.1021/acs.jctc.0c00896} { \bibfield { journal} { \bibinfo
{ journal} { J. Chem. Theory Comput.} \ } \textbf { \bibinfo { volume} { 17} } ,\
\bibinfo { pages} { 191} (\bibinfo { year} { 2020} )} \BibitemShut { NoStop} %
\bibitem [{ \citenamefont { Handy} \ \emph { et~al.} (1989)\citenamefont { Handy} ,
\citenamefont { Pople} , \citenamefont { Head-Gordon} , \citenamefont
{ Raghavachari} ,\ and\ \citenamefont { Trucks} } ]{ Handy_ 1989} %
\BibitemOpen
\bibfield { author} { \bibinfo { author} { \bibfnamefont { N.~C.} \ \bibnamefont
{ Handy} } , \bibinfo { author} { \bibfnamefont { J.~A.} \ \bibnamefont { Pople} } ,
\bibinfo { author} { \bibfnamefont { M.} ~\bibnamefont { Head-Gordon} } , \bibinfo
{ author} { \bibfnamefont { K.} ~\bibnamefont { Raghavachari} } , \ and\ \bibinfo
{ author} { \bibfnamefont { G.~W.} \ \bibnamefont { Trucks} } ,\ } \href { \doibase
https://doi.org/10.1016/0009-2614(89)85013-4} { \bibfield { journal} { \bibinfo
{ journal} { Chem. Phys. Lett.} \ } \textbf { \bibinfo { volume} { 164} } ,\ \bibinfo
{ pages} { 185} (\bibinfo { year} { 1989} )} \BibitemShut { NoStop} %
\bibitem [{ \citenamefont { Lange} \ and\ \citenamefont
{ Berkelbach} (2018)} ]{ Lange_ 2018} %
\BibitemOpen
\bibfield { author} { \bibinfo { author} { \bibfnamefont { M.~F.} \ \bibnamefont
{ Lange} } \ and\ \bibinfo { author} { \bibfnamefont { T.~C.} \ \bibnamefont
{ Berkelbach} } ,\ } \href { \doibase 10.1021/acs.jctc.8b00455} { \bibfield
{ journal} { \bibinfo { journal} { J. Chem. Theory. Comput.} \ } \textbf { \bibinfo
{ volume} { 14} } ,\ \bibinfo { pages} { 4224} (\bibinfo { year}
{ 2018} )} \BibitemShut { NoStop} %
\bibitem [{ \citenamefont { Nooijen} \ and\ \citenamefont
{ Bartlett} (1997{ \natexlab { a} } )} ]{ Nooijen_ 1997c} %
\BibitemOpen
\bibfield { author} { \bibinfo { author} { \bibfnamefont { M.} ~\bibnamefont
{ Nooijen} } \ and\ \bibinfo { author} { \bibfnamefont { R.~J.} \ \bibnamefont
{ Bartlett} } ,\ } \href { \doibase 10.1063/1.473635} { \bibfield { journal}
{ \bibinfo { journal} { J. Chem. Phys.} \ } \textbf { \bibinfo { volume} { 106} } ,\
\bibinfo { pages} { 6449} (\bibinfo { year} { 1997} { \natexlab { a} } )} \BibitemShut
{ NoStop} %
\bibitem [{ \citenamefont { Nooijen} \ and\ \citenamefont
{ Bartlett} (1997{ \natexlab { b} } )} ]{ Nooijen_ 1997b} %
\BibitemOpen
\bibfield { author} { \bibinfo { author} { \bibfnamefont { M.} ~\bibnamefont
{ Nooijen} } \ and\ \bibinfo { author} { \bibfnamefont { R.~J.} \ \bibnamefont
{ Bartlett} } ,\ } \href { \doibase 10.1063/1.474922} { \bibfield { journal}
{ \bibinfo { journal} { J. Chem. Phys.} \ } \textbf { \bibinfo { volume} { 107} } ,\
\bibinfo { pages} { 6812} (\bibinfo { year} { 1997} { \natexlab { b} } )} \BibitemShut
{ NoStop} %
\bibitem [{ \citenamefont { Nooijen} \ and\ \citenamefont
{ Bartlett} (1997{ \natexlab { c} } )} ]{ Nooijen_ 1997a} %
\BibitemOpen
\bibfield { author} { \bibinfo { author} { \bibfnamefont { M.} ~\bibnamefont
{ Nooijen} } \ and\ \bibinfo { author} { \bibfnamefont { R.~J.} \ \bibnamefont
{ Bartlett} } ,\ } \href { \doibase 10.1063/1.474000} { \bibfield { journal}
{ \bibinfo { journal} { J. Chem. Phys.} \ } \textbf { \bibinfo { volume} { 106} } ,\
\bibinfo { pages} { 6441} (\bibinfo { year} { 1997} { \natexlab { c} } )} \BibitemShut
{ NoStop} %
\bibitem [{ \citenamefont { Stanton} \ and\ \citenamefont
{ Gauss} (1994)} ]{ Stanton_ 1994} %
\BibitemOpen
\bibfield { author} { \bibinfo { author} { \bibfnamefont { J.~F.} \ \bibnamefont
{ Stanton} } \ and\ \bibinfo { author} { \bibfnamefont { J.} ~\bibnamefont
{ Gauss} } ,\ } \href { \doibase 10.1063/1.468022} { \bibfield { journal} { \bibinfo
{ journal} { J. Chem. Phys.} \ } \textbf { \bibinfo { volume} { 101} } ,\ \bibinfo
{ pages} { 8938} (\bibinfo { year} { 1994} )} \BibitemShut { NoStop} %
\bibitem [{ \citenamefont { Musia{ \l } } , \citenamefont { Kucharski} ,\ and\
\citenamefont { Bartlett} (2003)} ]{ Musial_ 2003a} %
\BibitemOpen
\bibfield { author} { \bibinfo { author} { \bibfnamefont { M.} ~\bibnamefont
{ Musia{ \l } } } , \bibinfo { author} { \bibfnamefont { S.~A.} \ \bibnamefont
{ Kucharski} } , \ and\ \bibinfo { author} { \bibfnamefont { R.~J.} \ \bibnamefont
{ Bartlett} } ,\ } \href { \doibase 10.1063/1.1527013} { \bibfield { journal}
{ \bibinfo { journal} { J. Chem. Phys.} \ } \textbf { \bibinfo { volume} { 118} } ,\
\bibinfo { pages} { 1128} (\bibinfo { year} { 2003} )} \BibitemShut { NoStop} %
\bibitem [{ \citenamefont { Nooijen} \ and\ \citenamefont
{ Bartlett} (1995)} ]{ Nooijen_ 1995} %
\BibitemOpen
\bibfield { author} { \bibinfo { author} { \bibfnamefont { M.} ~\bibnamefont
{ Nooijen} } \ and\ \bibinfo { author} { \bibfnamefont { R.~J.} \ \bibnamefont
{ Bartlett} } ,\ } \href { \doibase 10.1063/1.468592} { \bibfield { journal}
{ \bibinfo { journal} { J. Chem. Phys.} \ } \textbf { \bibinfo { volume} { 102} } ,\
\bibinfo { pages} { 3629} (\bibinfo { year} { 1995} )} \BibitemShut { NoStop} %
\bibitem [{ \citenamefont { Musia{ \l } } \ and\ \citenamefont
{ Bartlett} (2003)} ]{ Musial_ 2003b} %
\BibitemOpen
\bibfield { author} { \bibinfo { author} { \bibfnamefont { M.} ~\bibnamefont
{ Musia{ \l } } } \ and\ \bibinfo { author} { \bibfnamefont { R.~J.} \ \bibnamefont
{ Bartlett} } ,\ } \href { \doibase 10.1063/1.1584657} { \bibfield { journal}
{ \bibinfo { journal} { J. Chem. Phys.} \ } \textbf { \bibinfo { volume} { 119} } ,\
\bibinfo { pages} { 1901} (\bibinfo { year} { 2003} )} \BibitemShut { NoStop} %
\bibitem [{ \citenamefont { Nooijen} \ and\ \citenamefont
{ Lotrich} (2000)} ]{ Nooijen_ 2000} %
\BibitemOpen
\bibfield { author} { \bibinfo { author} { \bibfnamefont { M.} ~\bibnamefont
{ Nooijen} } \ and\ \bibinfo { author} { \bibfnamefont { V.} ~\bibnamefont
{ Lotrich} } ,\ } \href { \doibase 10.1063/1.481828} { \bibfield { journal}
{ \bibinfo { journal} { J. Chem. Phys.} \ } \textbf { \bibinfo { volume} { 113} } ,\
\bibinfo { pages} { 494} (\bibinfo { year} { 2000} )} \BibitemShut { NoStop} %
\bibitem [{ \citenamefont { Monino} \ and\ \citenamefont
{ Loos} (2022)} ]{ Monino_ 2022} %
\BibitemOpen
\bibfield { author} { \bibinfo { author} { \bibfnamefont { E.} ~\bibnamefont
{ Monino} } \ and\ \bibinfo { author} { \bibfnamefont { P.-F.} \ \bibnamefont
{ Loos} } ,\ } \href { \doibase 10.1063/5.0089317} { \bibfield { journal} { \bibinfo
{ journal} { J. Chem. Phys.} \ } \textbf { \bibinfo { volume} { 156} } ,\ \bibinfo
{ pages} { 231101} (\bibinfo { year} { 2022} )} \BibitemShut { NoStop} %
\bibitem [{ \citenamefont { Strinati} , \citenamefont { Mattausch} ,\ and\
\citenamefont { Hanke} (1980)} ]{ Strinati_ 1980} %
\BibitemOpen
\bibfield { author} { \bibinfo { author} { \bibfnamefont { G.} ~\bibnamefont
{ Strinati} } , \bibinfo { author} { \bibfnamefont { H.~J.} \ \bibnamefont
{ Mattausch} } , \ and\ \bibinfo { author} { \bibfnamefont { W.} ~\bibnamefont
{ Hanke} } ,\ } \href { \doibase 10.1103/PhysRevLett.45.290} { \bibfield { journal}
{ \bibinfo { journal} { Phys. Rev. Lett.} \ } \textbf { \bibinfo { volume} { 45} } ,\
\bibinfo { pages} { 290} (\bibinfo { year} { 1980} )} \BibitemShut { NoStop} %
\bibitem [{ \citenamefont { Strinati} (1982)} ]{ Strinati_ 1982} %
\BibitemOpen
\bibfield { author} { \bibinfo { author} { \bibfnamefont { G.} ~\bibnamefont
{ Strinati} } ,\ } \href { \doibase 10.1103/PhysRevLett.49.1519} { \bibfield
{ journal} { \bibinfo { journal} { Phys. Rev. Lett.} \ } \textbf { \bibinfo
{ volume} { 49} } ,\ \bibinfo { pages} { 1519} (\bibinfo { year}
{ 1982} )} \BibitemShut { NoStop} %
\bibitem [{ \citenamefont { Strinati} (1984)} ]{ Strinati_ 1984} %
\BibitemOpen
\bibfield { author} { \bibinfo { author} { \bibfnamefont { G.} ~\bibnamefont
{ Strinati} } ,\ } \href { \doibase 10.1103/PhysRevB.29.5718} { \bibfield
{ journal} { \bibinfo { journal} { Phys. Rev. B} \ } \textbf { \bibinfo { volume}
{ 29} } ,\ \bibinfo { pages} { 5718} (\bibinfo { year} { 1984} )} \BibitemShut
{ NoStop} %
\bibitem [{ \citenamefont { Rohlfing} \ and\ \citenamefont
{ Louie} (2000)} ]{ Rohlfing_ 2000} %
\BibitemOpen
\bibfield { author} { \bibinfo { author} { \bibfnamefont { M.} ~\bibnamefont
{ Rohlfing} } \ and\ \bibinfo { author} { \bibfnamefont { S.~G.} \ \bibnamefont
{ Louie} } ,\ } \href { \doibase 10.1103/PhysRevB.62.4927} { \bibfield { journal}
{ \bibinfo { journal} { Phys. Rev. B} \ } \textbf { \bibinfo { volume} { 62} } ,\
\bibinfo { pages} { 4927} (\bibinfo { year} { 2000} )} \BibitemShut { NoStop} %
\bibitem [{ \citenamefont { Romaniello} \ \emph { et~al.} (2009)\citenamefont
{ Romaniello} , \citenamefont { Sangalli} , \citenamefont { Berger} , \citenamefont
{ Sottile} , \citenamefont { Molinari} , \citenamefont { Reining} ,\ and\
\citenamefont { Onida} } ]{ Romaniello_ 2009b} %
\BibitemOpen
\bibfield { author} { \bibinfo { author} { \bibfnamefont { P.} ~\bibnamefont
{ Romaniello} } , \bibinfo { author} { \bibfnamefont { D.} ~\bibnamefont
{ Sangalli} } , \bibinfo { author} { \bibfnamefont { J.~A.} \ \bibnamefont
{ Berger} } , \bibinfo { author} { \bibfnamefont { F.} ~\bibnamefont { Sottile} } ,
\bibinfo { author} { \bibfnamefont { L.~G.} \ \bibnamefont { Molinari} } , \bibinfo
{ author} { \bibfnamefont { L.} ~\bibnamefont { Reining} } , \ and\ \bibinfo
{ author} { \bibfnamefont { G.} ~\bibnamefont { Onida} } ,\ } \href { \doibase
10.1063/1.3065669} { \bibfield { journal} { \bibinfo { journal} { J. Chem.
Phys.} \ } \textbf { \bibinfo { volume} { 130} } ,\ \bibinfo { pages} { 044108}
(\bibinfo { year} { 2009} )} \BibitemShut { NoStop} %
\bibitem [{ \citenamefont { Loos} \ and\ \citenamefont
{ Blase} (2020)} ]{ Loos_ 2020h} %
\BibitemOpen
\bibfield { author} { \bibinfo { author} { \bibfnamefont { P.-F.} \ \bibnamefont
{ Loos} } \ and\ \bibinfo { author} { \bibfnamefont { X.} ~\bibnamefont { Blase} } ,\
} \href { \doibase 10.1063/5.0023168} { \bibfield { journal} { \bibinfo
{ journal} { J. Chem. Phys.} \ } \textbf { \bibinfo { volume} { 153} } ,\ \bibinfo
{ pages} { 114120} (\bibinfo { year} { 2020} )} \BibitemShut { NoStop} %
\bibitem [{ \citenamefont { Authier} \ and\ \citenamefont
{ Loos} (2020)} ]{ Authier_ 2020} %
\BibitemOpen
\bibfield { author} { \bibinfo { author} { \bibfnamefont { J.} ~\bibnamefont
{ Authier} } \ and\ \bibinfo { author} { \bibfnamefont { P.-F.} \ \bibnamefont
{ Loos} } ,\ } \href { \doibase 10.1063/5.0028040} { \bibfield { journal} { \bibinfo
{ journal} { J. Chem. Phys.} \ } \textbf { \bibinfo { volume} { 153} } ,\ \bibinfo
{ pages} { 184105} (\bibinfo { year} { 2020} )} \BibitemShut { NoStop} %
\bibitem [{ \citenamefont { Monino} \ and\ \citenamefont
{ Loos} (2021)} ]{ Monino_ 2021} %
\BibitemOpen
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{ \bibinfo { journal} { J. Chem. Theory Comput.} \ } \textbf { \bibinfo { volume}
{ 17} } ,\ \bibinfo { pages} { 2852} (\bibinfo { year} { 2021} )} \BibitemShut
{ NoStop} %
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{ Berkelbach} (2022)} ]{ Bintrim_ 2022} %
\BibitemOpen
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{ \bibinfo { journal} { J. Chem. Phys.} \ } \textbf { \bibinfo { volume} { 156} } ,\
\bibinfo { pages} { 044114} (\bibinfo { year} { 2022} )} \BibitemShut { NoStop} %
\bibitem [{ \citenamefont { Stefanucci} \ and\ \citenamefont { van
Leeuwen} (2013)} ]{ Stefanucci_ 2013} %
\BibitemOpen
\bibfield { author} { \bibinfo { author} { \bibfnamefont { G.} ~\bibnamefont
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Leeuwen} } ,\ } \href @noop { } { \emph { \bibinfo { title} { Nonequilibrium Many-Body
Theory of Quantum Systems: A Modern Introduction} } } \ (\bibinfo { publisher}
{ { Cambridge University Press} } ,\ \bibinfo { address} { Cambridge} ,\ \bibinfo
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\bibitem [{ \citenamefont { Ortiz} (2013)} ]{ Ortiz_ 2013} %
\BibitemOpen
\bibfield { author} { \bibinfo { author} { \bibfnamefont { J.~V.} \ \bibnamefont
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{ \bibinfo { volume} { 3} } ,\ \bibinfo { pages} { 123} (\bibinfo { year}
{ 2013} )} \BibitemShut { NoStop} %
\bibitem [{ \citenamefont { Phillips} \ and\ \citenamefont
{ Zgid} (2014)} ]{ Phillips_ 2014} %
\BibitemOpen
\bibfield { author} { \bibinfo { author} { \bibfnamefont { J.~J.} \ \bibnamefont
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\bibitem [{ \citenamefont { Rusakov} , \citenamefont { Phillips} ,\ and\
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\BibitemOpen
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{ pages} { 194105} (\bibinfo { year} { 2014} )} \BibitemShut { NoStop} %
\bibitem [{ \citenamefont { Hirata} \ \emph { et~al.} (2015)\citenamefont { Hirata} ,
\citenamefont { Hermes} , \citenamefont { Simons} ,\ and\ \citenamefont
{ Ortiz} } ]{ Hirata_ 2015} %
\BibitemOpen
\bibfield { author} { \bibinfo { author} { \bibfnamefont { S.} ~\bibnamefont
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\bibinfo { author} { \bibfnamefont { J.} ~\bibnamefont { Simons} } , \ and\ \bibinfo
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10.1021/acs.jctc.5b00005} { \bibfield { journal} { \bibinfo { journal} { J.
Chem. Theory Comput.} \ } \textbf { \bibinfo { volume} { 11} } ,\ \bibinfo { pages}
{ 1595} (\bibinfo { year} { 2015} )} \BibitemShut { NoStop} %
\bibitem [{ \citenamefont { Hirata} \ \emph { et~al.} (2017)\citenamefont { Hirata} ,
\citenamefont { Doran} , \citenamefont { Knowles} ,\ and\ \citenamefont
{ Ortiz} } ]{ Hirata_ 2017} %
\BibitemOpen
\bibfield { author} { \bibinfo { author} { \bibfnamefont { S.} ~\bibnamefont
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\bibinfo { author} { \bibfnamefont { P.~J.} \ \bibnamefont { Knowles} } , \ and\
\bibinfo { author} { \bibfnamefont { J.~V.} \ \bibnamefont { Ortiz} } ,\ } \href
{ \doibase 10.1063/1.4994837} { \bibfield { journal} { \bibinfo { journal} { J.
Chem. Phys.} \ } \textbf { \bibinfo { volume} { 147} } ,\ \bibinfo { pages} { 044108}
(\bibinfo { year} { 2017} )} \BibitemShut { NoStop} %
\bibitem [{ \citenamefont { Romaniello} , \citenamefont { Bechstedt} ,\ and\
\citenamefont { Reining} (2012)} ]{ Romaniello_ 2012} %
\BibitemOpen
\bibfield { author} { \bibinfo { author} { \bibfnamefont { P.} ~\bibnamefont
{ Romaniello} } , \bibinfo { author} { \bibfnamefont { F.} ~\bibnamefont
{ Bechstedt} } , \ and\ \bibinfo { author} { \bibfnamefont { L.} ~\bibnamefont
{ Reining} } ,\ } \href { \doibase 10.1103/PhysRevB.85.155131} { \bibfield
{ journal} { \bibinfo { journal} { Phys. Rev. B} \ } \textbf { \bibinfo { volume}
{ 85} } ,\ \bibinfo { pages} { 155131} (\bibinfo { year} { 2012} )} \BibitemShut
{ NoStop} %
\bibitem [{ \citenamefont { Zhang} , \citenamefont { Su} ,\ and\ \citenamefont
{ Yang} (2017)} ]{ Zhang_ 2017} %
\BibitemOpen
\bibfield { author} { \bibinfo { author} { \bibfnamefont { D.} ~\bibnamefont
{ Zhang} } , \bibinfo { author} { \bibfnamefont { N.~Q.} \ \bibnamefont { Su} } , \
and\ \bibinfo { author} { \bibfnamefont { W.} ~\bibnamefont { Yang} } ,\ } \href
{ \doibase 10.1021/acs.jpclett.7b01275} { \bibfield { journal} { \bibinfo
{ journal} { J. Phys. Chem. Lett.} \ } \textbf { \bibinfo { volume} { 8} } ,\ \bibinfo
{ pages} { 3223} (\bibinfo { year} { 2017} )} \BibitemShut { NoStop} %
\bibitem [{ \citenamefont { Li} , \citenamefont { Chen} ,\ and\ \citenamefont
{ Yang} (2021)} ]{ Li_ 2021b} %
\BibitemOpen
\bibfield { author} { \bibinfo { author} { \bibfnamefont { J.} ~\bibnamefont
{ Li} } , \bibinfo { author} { \bibfnamefont { Z.} ~\bibnamefont { Chen} } , \ and\
\bibinfo { author} { \bibfnamefont { W.} ~\bibnamefont { Yang} } ,\ } \href { \doibase
10.1021/acs.jpclett.1c01723} { \bibfield { journal} { \bibinfo { journal} { J.
Phys. Chem. Lett.} \ } \textbf { \bibinfo { volume} { 12} } ,\ \bibinfo { pages}
{ 6203} (\bibinfo { year} { 2021} )} \BibitemShut { NoStop} %
\bibitem [{ \citenamefont { Loos} \ and\ \citenamefont
{ Romaniello} (2022)} ]{ Loos_ 2022} %
\BibitemOpen
\bibfield { author} { \bibinfo { author} { \bibfnamefont { P.-F.} \ \bibnamefont
{ Loos} } \ and\ \bibinfo { author} { \bibfnamefont { P.} ~\bibnamefont
{ Romaniello} } ,\ } \href { \doibase 10.1063/5.0088364} { \bibfield { journal}
{ \bibinfo { journal} { J. Chem. Phys.} \ } \textbf { \bibinfo { volume} { 156} } ,\
\bibinfo { pages} { 164101} (\bibinfo { year} { 2022} )} \BibitemShut { NoStop} %
\bibitem [{ \citenamefont { Loos} , \citenamefont { Romaniello} ,\ and\
\citenamefont { Berger} (2018)} ]{ Loos_ 2018b} %
\BibitemOpen
\bibfield { author} { \bibinfo { author} { \bibfnamefont { P.~F.} \ \bibnamefont
{ Loos} } , \bibinfo { author} { \bibfnamefont { P.} ~\bibnamefont { Romaniello} } , \
and\ \bibinfo { author} { \bibfnamefont { J.~A.} \ \bibnamefont { Berger} } ,\
} \href { \doibase 10.1021/acs.jctc.8b00260} { \bibfield { journal} { \bibinfo
{ journal} { J. Chem. Theory Comput.} \ } \textbf { \bibinfo { volume} { 14} } ,\
\bibinfo { pages} { 3071} (\bibinfo { year} { 2018} )} \BibitemShut { NoStop} %
\bibitem [{ \citenamefont { Hybertsen} \ and\ \citenamefont
{ Louie} (1985)} ]{ Hybertsen_ 1985a} %
\BibitemOpen
\bibfield { author} { \bibinfo { author} { \bibfnamefont { M.~S.} \ \bibnamefont
{ Hybertsen} } \ and\ \bibinfo { author} { \bibfnamefont { S.~G.} \ \bibnamefont
{ Louie} } ,\ } \href { \doibase 10.1103/PhysRevLett.55.1418} { \bibfield
{ journal} { \bibinfo { journal} { Phys. Rev. Lett.} \ } \textbf { \bibinfo
{ volume} { 55} } ,\ \bibinfo { pages} { 1418} (\bibinfo { year}
{ 1985} )} \BibitemShut { NoStop} %
\bibitem [{ \citenamefont { Hybertsen} \ and\ \citenamefont
{ Louie} (1986)} ]{ Hybertsen_ 1986} %
\BibitemOpen
\bibfield { author} { \bibinfo { author} { \bibfnamefont { M.~S.} \ \bibnamefont
{ Hybertsen} } \ and\ \bibinfo { author} { \bibfnamefont { S.~G.} \ \bibnamefont
{ Louie} } ,\ } \href { \doibase 10.1103/PhysRevB.34.5390} { \bibfield { journal}
{ \bibinfo { journal} { Phys. Rev. B} \ } \textbf { \bibinfo { volume} { 34} } ,\
\bibinfo { pages} { 5390} (\bibinfo { year} { 1986} )} \BibitemShut { NoStop} %
\bibitem [{ \citenamefont { Godby} , \citenamefont { Schl\" uter} ,\ and\
\citenamefont { Sham} (1988)} ]{ Godby_ 1988} %
\BibitemOpen
\bibfield { author} { \bibinfo { author} { \bibfnamefont { R.~W.} \ \bibnamefont
{ Godby} } , \bibinfo { author} { \bibfnamefont { M.} ~\bibnamefont { Schl\" uter} } , \
and\ \bibinfo { author} { \bibfnamefont { L.~J.} \ \bibnamefont { Sham} } ,\ } \href
{ \doibase 10.1103/PhysRevB.37.10159} { \bibfield { journal} { \bibinfo
{ journal} { Phys. Rev. B} \ } \textbf { \bibinfo { volume} { 37} } ,\ \bibinfo
{ pages} { 10159} (\bibinfo { year} { 1988} )} \BibitemShut { NoStop} %
\bibitem [{ \citenamefont { von~der Linden} \ and\ \citenamefont
{ Horsch} (1988)} ]{ Linden_ 1988} %
\BibitemOpen
\bibfield { author} { \bibinfo { author} { \bibfnamefont { W.} ~\bibnamefont
{ von~der Linden} } \ and\ \bibinfo { author} { \bibfnamefont { P.} ~\bibnamefont
{ Horsch} } ,\ } \href { \doibase 10.1103/PhysRevB.37.8351} { \bibfield { journal}
{ \bibinfo { journal} { Phys. Rev. B} \ } \textbf { \bibinfo { volume} { 37} } ,\
\bibinfo { pages} { 8351} (\bibinfo { year} { 1988} )} \BibitemShut { NoStop} %
\bibitem [{ \citenamefont { Northrup} , \citenamefont { Hybertsen} ,\ and\
\citenamefont { Louie} (1991)} ]{ Northrup_ 1991} %
\BibitemOpen
\bibfield { author} { \bibinfo { author} { \bibfnamefont { J.~E.} \ \bibnamefont
{ Northrup} } , \bibinfo { author} { \bibfnamefont { M.~S.} \ \bibnamefont
{ Hybertsen} } , \ and\ \bibinfo { author} { \bibfnamefont { S.~G.} \ \bibnamefont
{ Louie} } ,\ } \href { \doibase 10.1103/PhysRevLett.66.500} { \bibfield { journal}
{ \bibinfo { journal} { Phys. Rev. Lett.} \ } \textbf { \bibinfo { volume} { 66} } ,\
\bibinfo { pages} { 500} (\bibinfo { year} { 1991} )} \BibitemShut { NoStop} %
\bibitem [{ \citenamefont { Blase} , \citenamefont { Zhu} ,\ and\ \citenamefont
{ Louie} (1994)} ]{ Blase_ 1994} %
\BibitemOpen
\bibfield { author} { \bibinfo { author} { \bibfnamefont { X.} ~\bibnamefont
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\bibinfo { author} { \bibfnamefont { S.~G.} \ \bibnamefont { Louie} } ,\ } \href
{ \doibase 10.1103/PhysRevB.49.4973} { \bibfield { journal} { \bibinfo
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{ pages} { 4973} (\bibinfo { year} { 1994} )} \BibitemShut { NoStop} %
\bibitem [{ \citenamefont { Rohlfing} , \citenamefont { Kr{ \" u} ger} ,\ and\
\citenamefont { Pollmann} (1995)} ]{ Rohlfing_ 1995} %
\BibitemOpen
\bibfield { author} { \bibinfo { author} { \bibfnamefont { M.} ~\bibnamefont
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{ Pollmann} } ,\ } \href { \doibase 10.1103/PhysRevB.52.1905} { \bibfield
{ journal} { \bibinfo { journal} { Phys. Rev. B} \ } \textbf { \bibinfo { volume}
{ 52} } ,\ \bibinfo { pages} { 1905} (\bibinfo { year} { 1995} )} \BibitemShut
{ NoStop} %
\bibitem [{ \citenamefont { Shishkin} \ and\ \citenamefont
{ Kresse} (2007)} ]{ Shishkin_ 2007} %
\BibitemOpen
\bibfield { author} { \bibinfo { author} { \bibfnamefont { M.} ~\bibnamefont
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{ 75} } ,\ \bibinfo { pages} { 235102} (\bibinfo { year} { 2007} )} \BibitemShut
{ NoStop} %
\bibitem [{ \citenamefont { Blase} \ and\ \citenamefont
{ Attaccalite} (2011)} ]{ Blase_ 2011} %
\BibitemOpen
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{ \bibinfo { journal} { Appl. Phys. Lett.} \ } \textbf { \bibinfo { volume} { 99} } ,\
\bibinfo { pages} { 171909} (\bibinfo { year} { 2011} )} \BibitemShut { NoStop} %
\bibitem [{ \citenamefont { Faber} \ \emph
{ et~al.} (2011{ \natexlab { a} } )\citenamefont { Faber} , \citenamefont
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\citenamefont { Blase} } ]{ Faber_ 2011} %
\BibitemOpen
\bibfield { author} { \bibinfo { author} { \bibfnamefont { C.} ~\bibnamefont
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\bibitem [{ \citenamefont { Rangel} \ \emph { et~al.} (2016)\citenamefont { Rangel} ,
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{ Neaton} } ]{ Rangel_ 2016} %
\BibitemOpen
\bibfield { author} { \bibinfo { author} { \bibfnamefont { T.} ~\bibnamefont
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\bibitem [{ \citenamefont { Faleev} , \citenamefont { { van Schilfgaarde} } ,\ and\
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\BibitemOpen
\bibfield { author} { \bibinfo { author} { \bibfnamefont { S.~V.} \ \bibnamefont
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{ 2004} )} \BibitemShut { NoStop} %
\bibitem [{ \citenamefont { { van Schilfgaarde} } , \citenamefont { Kotani} ,\ and\
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\BibitemOpen
\bibfield { author} { \bibinfo { author} { \bibfnamefont { M.} ~\bibnamefont { { van
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{ Kotani} } , \ and\ \bibinfo { author} { \bibfnamefont { S.} ~\bibnamefont
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{ 2006} )} \BibitemShut { NoStop} %
\bibitem [{ \citenamefont { Kotani} , \citenamefont { { van Schilfgaarde} } ,\ and\
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\BibitemOpen
\bibfield { author} { \bibinfo { author} { \bibfnamefont { T.} ~\bibnamefont
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\bibnamefont { Faleev} } ,\ } \href { \doibase 10.1103/PhysRevB.76.165106}
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{ 2007} )} \BibitemShut { NoStop} %
\bibitem [{ \citenamefont { Ke} (2011)} ]{ Ke_ 2011} %
\BibitemOpen
\bibfield { author} { \bibinfo { author} { \bibfnamefont { S.-H.} \ \bibnamefont
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{ \bibinfo { journal} { Phys. Rev. B} \ } \textbf { \bibinfo { volume} { 84} } ,\
\bibinfo { pages} { 205415} (\bibinfo { year} { 2011} )} \BibitemShut { NoStop} %
\bibitem [{ \citenamefont { Kaplan} \ \emph { et~al.} (2016)\citenamefont { Kaplan} ,
\citenamefont { Harding} , \citenamefont { Seiler} , \citenamefont { Weigend} ,
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\BibitemOpen
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{ 2528} (\bibinfo { year} { 2016} )} \BibitemShut { NoStop} %
\bibitem [{ \citenamefont { Bintrim} \ and\ \citenamefont
{ Berkelbach} (2021)} ]{ Bintrim_ 2021} %
\BibitemOpen
\bibfield { author} { \bibinfo { author} { \bibfnamefont { S.~J.} \ \bibnamefont
{ Bintrim} } \ and\ \bibinfo { author} { \bibfnamefont { T.~C.} \ \bibnamefont
{ Berkelbach} } ,\ } \href { \doibase 10.1063/5.0035141} { \bibfield { journal}
{ \bibinfo { journal} { J. Chem. Phys.} \ } \textbf { \bibinfo { volume} { 154} } ,\
\bibinfo { pages} { 041101} (\bibinfo { year} { 2021} )} \BibitemShut { NoStop} %
\bibitem [{ \citenamefont { Schirmer} (2018)} ]{ Schirmer_ 2018} %
\BibitemOpen
\bibfield { author} { \bibinfo { author} { \bibfnamefont { J.} ~\bibnamefont
{ Schirmer} } ,\ } \href @noop { } { \emph { \bibinfo { title} { Many-Body Methods for
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\bibinfo { year} { 2018} )\BibitemShut { NoStop} %
\bibitem [{ \citenamefont { Piecuch} \ and\ \citenamefont
{ Kowalski} (2000)} ]{ Piecuch_ 2000} %
\BibitemOpen
\bibfield { author} { \bibinfo { author} { \bibfnamefont { P.} ~\bibnamefont
{ Piecuch} } \ and\ \bibinfo { author} { \bibfnamefont { K.} ~\bibnamefont
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{ \bibinfo { booktitle} { Computational { { Chemistry} } : { { Reviews} } of { { Current
Trends} } } } } ,\ \bibinfo { series} { Computational { { Chemistry} } : { { Reviews} } of
{ { Current Trends} } } , Vol.\ \bibinfo { volume} { Volume 5} \ (\bibinfo
{ publisher} { { WORLD SCIENTIFIC} } ,\ \bibinfo { year} { 2000} )\ pp.\ \bibinfo
{ pages} { 1--104} \BibitemShut { NoStop} %
\bibitem [{ \citenamefont { Mayhall} \ and\ \citenamefont
{ Raghavachari} (2010)} ]{ Mayhall_ 2010} %
\BibitemOpen
\bibfield { author} { \bibinfo { author} { \bibfnamefont { N.~J.} \ \bibnamefont
{ Mayhall} } \ and\ \bibinfo { author} { \bibfnamefont { K.} ~\bibnamefont
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{ \bibinfo { journal} { J. Chem. Theory Comput.} \ } \textbf { \bibinfo { volume}
{ 6} } ,\ \bibinfo { pages} { 2714} (\bibinfo { year} { 2010} )} \BibitemShut
{ NoStop} %
\bibitem [{ \citenamefont { Lee} , \citenamefont { Small} ,\ and\ \citenamefont
{ { Head-Gordon} } (2019)} ]{ Lee_ 2019} %
\BibitemOpen
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{ journal} { J. Chem. Phys.} \ } \textbf { \bibinfo { volume} { 151} } ,\ \bibinfo
{ pages} { 214103} (\bibinfo { year} { 2019} )} \BibitemShut { NoStop} %
\bibitem [{ \citenamefont { Kossoski} \ \emph { et~al.} (2021)\citenamefont
{ Kossoski} , \citenamefont { Marie} , \citenamefont { Scemama} , \citenamefont
{ Caffarel} ,\ and\ \citenamefont { Loos} } ]{ Kossoski_ 2021} %
\BibitemOpen
\bibfield { author} { \bibinfo { author} { \bibfnamefont { F.} ~\bibnamefont
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\bibinfo { author} { \bibfnamefont { A.} ~\bibnamefont { Scemama} } , \bibinfo
{ author} { \bibfnamefont { M.} ~\bibnamefont { Caffarel} } , \ and\ \bibinfo
{ author} { \bibfnamefont { P.-F.} \ \bibnamefont { Loos} } ,\ } \href { \doibase
10.1021/acs.jctc.1c00348} { \bibfield { journal} { \bibinfo { journal} { J.
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{ 4756} (\bibinfo { year} { 2021} )} \BibitemShut { NoStop} %
\bibitem [{ \citenamefont { Marie} , \citenamefont { Kossoski} ,\ and\ \citenamefont
{ Loos} (2021)} ]{ Marie_ 2021b} %
\BibitemOpen
\bibfield { author} { \bibinfo { author} { \bibfnamefont { A.} ~\bibnamefont
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and\ \bibinfo { author} { \bibfnamefont { P.-F.} \ \bibnamefont { Loos} } ,\ } \href
{ \doibase 10.1063/5.0060698} { \bibfield { journal} { \bibinfo { journal} { J.
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(\bibinfo { year} { 2021} )} \BibitemShut { NoStop} %
\bibitem [{ \citenamefont { F{ \" o} rster} \ and\ \citenamefont
{ Visscher} (2020)} ]{ Forster_ 2020} %
\BibitemOpen
\bibfield { author} { \bibinfo { author} { \bibfnamefont { A.} ~\bibnamefont
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{ 2020} )} \BibitemShut { NoStop} %
\bibitem [{ \citenamefont { F{ \" o} rster} \ and\ \citenamefont
{ Visscher} (2021)} ]{ Forster_ 2021} %
\BibitemOpen
\bibfield { author} { \bibinfo { author} { \bibfnamefont { A.} ~\bibnamefont
{ F{ \" o} rster} } \ and\ \bibinfo { author} { \bibfnamefont { L.} ~\bibnamefont
{ Visscher} } ,\ } \href { \doibase 10.3389/fchem.2021.736591} { \bibfield
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{ 9} } ,\ \bibinfo { pages} { 736591} (\bibinfo { year} { 2021} )} \BibitemShut
{ NoStop} %
\bibitem [{ \citenamefont { Duchemin} \ and\ \citenamefont
{ Blase} (2019)} ]{ Duchemin_ 2019} %
\BibitemOpen
\bibfield { author} { \bibinfo { author} { \bibfnamefont { I.} ~\bibnamefont
{ Duchemin} } \ and\ \bibinfo { author} { \bibfnamefont { X.} ~\bibnamefont
{ Blase} } ,\ } \href { \doibase 10.1063/1.5090605} { \bibfield { journal}
{ \bibinfo { journal} { J. Chem. Phys.} \ } \textbf { \bibinfo { volume} { 150} } ,\
\bibinfo { pages} { 174120} (\bibinfo { year} { 2019} )} \BibitemShut { NoStop} %
\bibitem [{ \citenamefont { Duchemin} \ and\ \citenamefont
{ Blase} (2020)} ]{ Duchemin_ 2020} %
\BibitemOpen
\bibfield { author} { \bibinfo { author} { \bibfnamefont { I.} ~\bibnamefont
{ Duchemin} } \ and\ \bibinfo { author} { \bibfnamefont { X.} ~\bibnamefont
{ Blase} } ,\ } \href { \doibase 10.1021/acs.jctc.9b01235} { \bibfield { journal}
{ \bibinfo { journal} { J. Chem. Theory Comput.} \ } \textbf { \bibinfo { volume}
{ 16} } ,\ \bibinfo { pages} { 1742} (\bibinfo { year} { 2020} )} \BibitemShut
{ NoStop} %
\bibitem [{ \citenamefont { Duchemin} \ and\ \citenamefont
{ Blase} (2021)} ]{ Duchemin_ 2021} %
\BibitemOpen
\bibfield { author} { \bibinfo { author} { \bibfnamefont { I.} ~\bibnamefont
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{ \bibinfo { journal} { J. Chem. Theory Comput.} \ } \textbf { \bibinfo { volume}
{ 17} } ,\ \bibinfo { pages} { 2383} (\bibinfo { year} { 2021} )} \BibitemShut
{ NoStop} %
\bibitem [{ \citenamefont { Loos} (2019)} ]{ QuAcK} %
\BibitemOpen
\bibfield { author} { \bibinfo { author} { \bibfnamefont { P.~F.} \ \bibnamefont
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{ title} { { { QuAcK: a software for emerging quantum electronic structure
methods} } } ,} \ } (\bibinfo { year} { 2019} ),\ \bibinfo { note}
{ \url { https://github.com/pfloos/QuAcK} } \BibitemShut { NoStop} %
\bibitem [{ \citenamefont { Lazzeri} \ \emph { et~al.} (2008)\citenamefont
{ Lazzeri} , \citenamefont { Attaccalite} , \citenamefont { Wirtz} ,\ and\
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\BibitemOpen
\bibfield { author} { \bibinfo { author} { \bibfnamefont { M.} ~\bibnamefont
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\ and\ \bibinfo { author} { \bibfnamefont { F.} ~\bibnamefont { Mauri} } ,\ } \href
{ \doibase 10.1103/PhysRevB.78.081406} { \bibfield { journal} { \bibinfo
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\bibitem [{ \citenamefont { Faber} \ \emph
{ et~al.} (2011{ \natexlab { b} } )\citenamefont { Faber} , \citenamefont { Janssen} ,
\citenamefont { C\^ ot\' e} , \citenamefont { Runge} ,\ and\ \citenamefont
{ Blase} } ]{ Faber_ 2011b} %
\BibitemOpen
\bibfield { author} { \bibinfo { author} { \bibfnamefont { C.} ~\bibnamefont
{ Faber} } , \bibinfo { author} { \bibfnamefont { J.~L.} \ \bibnamefont { Janssen} } ,
\bibinfo { author} { \bibfnamefont { M.} ~\bibnamefont { C\^ ot\' e} } , \bibinfo
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{ \bibfnamefont { X.} ~\bibnamefont { Blase} } ,\ } \href { \doibase
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(\bibinfo { year} { 2011} { \natexlab { b} } )} \BibitemShut { NoStop} %
\bibitem [{ \citenamefont { Yin} , \citenamefont { Kutepov} ,\ and\ \citenamefont
{ Kotliar} (2013)} ]{ Yin_ 2013} %
\BibitemOpen
\bibfield { author} { \bibinfo { author} { \bibfnamefont { Z.~P.} \ \bibnamefont
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\bibinfo { author} { \bibfnamefont { G.} ~\bibnamefont { Kotliar} } ,\ } \href
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\bibitem [{ \citenamefont { Monserrat} (2016)} ]{ Montserrat_ 2016} %
\BibitemOpen
\bibfield { author} { \bibinfo { author} { \bibfnamefont { B.} ~\bibnamefont
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{ NoStop} %
\bibitem [{ \citenamefont { Li} \ \emph { et~al.} (2019)\citenamefont { Li} ,
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and\ \citenamefont { Louie} } ]{ Zhenglu_ 2019} %
\BibitemOpen
\bibfield { author} { \bibinfo { author} { \bibfnamefont { Z.} ~\bibnamefont
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{ \bibfnamefont { S.~G.} \ \bibnamefont { Louie} } ,\ } \href { \doibase
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{ 186402} (\bibinfo { year} { 2019} )} \BibitemShut { NoStop} %
\bibitem [{ \citenamefont { Ismail-Beigi} \ and\ \citenamefont
{ Louie} (2003)} ]{ IsmailBeigi_ 2003} %
\BibitemOpen
\bibfield { author} { \bibinfo { author} { \bibfnamefont { S.} ~\bibnamefont
{ Ismail-Beigi} } \ and\ \bibinfo { author} { \bibfnamefont { S.~G.} \ \bibnamefont
{ Louie} } ,\ } \href { \doibase 10.1103/PhysRevLett.90.076401} { \bibfield
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{ 2003} )} \BibitemShut { NoStop} %
\bibitem [{ \citenamefont { { \c C} aylak} \ and\ \citenamefont
{ Baumeier} (2021)} ]{ Caylak_ 2021} %
\BibitemOpen
\bibfield { author} { \bibinfo { author} { \bibfnamefont { O.} ~\bibnamefont { { \c
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{ volume} { 17} } ,\ \bibinfo { pages} { 879} (\bibinfo { year} { 2021} )} \BibitemShut
{ NoStop} %
\bibitem [{ \citenamefont { Knysh} \ \emph { et~al.} (2022)\citenamefont { Knysh} ,
\citenamefont { Duchemin} , \citenamefont { Blase} ,\ and\ \citenamefont
{ Jacquemin} } ]{ Knysh_ 2022} %
\BibitemOpen
\bibfield { author} { \bibinfo { author} { \bibfnamefont { I.} ~\bibnamefont
{ Knysh} } , \bibinfo { author} { \bibfnamefont { I.} ~\bibnamefont { Duchemin} } ,
\bibinfo { author} { \bibfnamefont { X.} ~\bibnamefont { Blase} } , \ and\ \bibinfo
{ author} { \bibfnamefont { D.~M.} \ \bibnamefont { Jacquemin} } ,\ } \href { \doibase
10.1063/5.0121121} { \bibfield { journal} { \bibinfo { journal} { J. Chem.
Phys.} \ } \textbf { \bibinfo { volume} { 157} } ,\ \bibinfo { pages} { 194102}
(\bibinfo { year} { 2022} )} \BibitemShut { NoStop} %
\bibitem [{ \citenamefont { Bartlett} (1986)} ]{ Bartlett_ 1986} %
\BibitemOpen
\bibfield { author} { \bibinfo { author} { \bibfnamefont { R.~J.} \ \bibnamefont
{ Bartlett} } ,\ } \enquote { \bibinfo { title} { Analytical evaluation of gradients
in coupled-cluster and many-body perturbation theory} ,} \ in\ \href { \doibase
10.1007/978-94-009-4584-5_ 4} { \emph { \bibinfo { booktitle} { Geometrical
Derivatives of Energy Surfaces and Molecular Properties} } } ,\ \bibinfo
{ editor} { edited by\ \bibinfo { editor} { \bibfnamefont { P.} ~\bibnamefont
{ J{ \o } rgensen} } \ and\ \bibinfo { editor} { \bibfnamefont { J.} ~\bibnamefont
{ Simons} } } \ (\bibinfo { publisher} { Springer Netherlands} ,\ \bibinfo
{ address} { Dordrecht} ,\ \bibinfo { year} { 1986} )\ pp.\ \bibinfo { pages}
{ 35--61} \BibitemShut { NoStop} %
\bibitem [{ \citenamefont { Brouder} , \citenamefont { Panati} ,\ and\ \citenamefont
{ Stoltz} (2009)} ]{ Brouder_ 2009} %
\BibitemOpen
\bibfield { author} { \bibinfo { author} { \bibfnamefont { C.} ~\bibnamefont
{ Brouder} } , \bibinfo { author} { \bibfnamefont { G.} ~\bibnamefont { Panati} } , \
and\ \bibinfo { author} { \bibfnamefont { G.} ~\bibnamefont { Stoltz} } ,\ } \href
{ \doibase 10.1103/PhysRevLett.103.230401} { \bibfield { journal} { \bibinfo
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{ pages} { 230401} (\bibinfo { year} { 2009} )} \BibitemShut { NoStop} %
\bibitem [{ \citenamefont { Linn\' er} \ and\ \citenamefont
{ Aryasetiawan} (2019)} ]{ Linner_ 2019} %
\BibitemOpen
\bibfield { author} { \bibinfo { author} { \bibfnamefont { E.} ~\bibnamefont
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{ Aryasetiawan} } ,\ } \href { \doibase 10.1103/PhysRevB.100.235106} { \bibfield
{ journal} { \bibinfo { journal} { Phys. Rev. B} \ } \textbf { \bibinfo { volume}
{ 100} } ,\ \bibinfo { pages} { 235106} (\bibinfo { year} { 2019} )} \BibitemShut
{ NoStop} %
\bibitem [{ \citenamefont { Lyakh} \ \emph { et~al.} (2012)\citenamefont { Lyakh} ,
\citenamefont { Musia{ \l } } , \citenamefont { Lotrich} ,\ and\ \citenamefont
{ Bartlett} } ]{ Lyakh_ 2012} %
\BibitemOpen
\bibfield { author} { \bibinfo { author} { \bibfnamefont { D.~I.} \ \bibnamefont
{ Lyakh} } , \bibinfo { author} { \bibfnamefont { M.} ~\bibnamefont { Musia{ \l } } } ,
\bibinfo { author} { \bibfnamefont { V.~F.} \ \bibnamefont { Lotrich} } , \ and\
\bibinfo { author} { \bibfnamefont { R.~J.} \ \bibnamefont { Bartlett} } ,\ } \href
{ \doibase 10.1021/cr2001417} { \bibfield { journal} { \bibinfo { journal}
{ Chem. Rev.} \ } \textbf { \bibinfo { volume} { 112} } ,\ \bibinfo { pages} { 182}
(\bibinfo { year} { 2012} )} \BibitemShut { NoStop} %
\bibitem [{ \citenamefont { Evangelista} (2018)} ]{ Evangelista_ 2018} %
\BibitemOpen
\bibfield { author} { \bibinfo { author} { \bibfnamefont { F.~A.} \ \bibnamefont
{ Evangelista} } ,\ } \href { \doibase 10.1063/1.5039496} { \bibfield { journal}
{ \bibinfo { journal} { J. Chem. Phys.} \ } \textbf { \bibinfo { volume} { 149} } ,\
\bibinfo { pages} { 030901} (\bibinfo { year} { 2018} )} \BibitemShut { NoStop} %
\end { thebibliography} %
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\end { document}