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CCvsMBPT.bib
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CCvsMBPT.bib
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%% This BibTeX bibliography file was created using BibDesk.
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%% http://bibdesk.sourceforge.net/
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%% Created for Pierre-Francois Loos at 2022-10-11 13:30:09 +0200
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%% Created for Pierre-Francois Loos at 2022-10-11 16:13:34 +0200
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@article{Handy_1989,
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abstract = {A size-consistent set of equations for electron correlation which are limited to double substitutions, based on Brueckner orbitals, is discussed. Called BD theory, it is shown that at fifth order of perturbation theory, BD incorporates more terms than CCSD and QCISD. The simplicity of the equations leads to an elegant gradient theory. Preliminary applications are reported.},
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author = {Nicholas C. Handy and John A. Pople and Martin Head-Gordon and Krishnan Raghavachari and Gary W. Trucks},
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date-added = {2022-10-11 16:13:12 +0200},
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date-modified = {2022-10-11 16:13:31 +0200},
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doi = {https://doi.org/10.1016/0009-2614(89)85013-4},
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journal = {Chem. Phys. Lett.},
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number = {2},
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pages = {185-192},
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title = {Size-consistent Brueckner theory limited to double substitutions},
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volume = {164},
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year = {1989},
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bdsk-url-1 = {https://www.sciencedirect.com/science/article/pii/0009261489850134},
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bdsk-url-2 = {https://doi.org/10.1016/0009-2614(89)85013-4}}
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@article{Musial_2003a,
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author = {Musia{\l},Monika and Kucharski,Stanis{\l}aw A. and Bartlett,Rodney J.},
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date-added = {2022-10-11 16:05:07 +0200},
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date-modified = {2022-10-11 16:05:20 +0200},
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doi = {10.1063/1.1527013},
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journal = {J. Chem. Phys.},
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number = {3},
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pages = {1128-1136},
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title = {Equation-of-motion coupled cluster method with full inclusion of the connected triple excitations for ionized states: IP-EOM-CCSDT},
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volume = {118},
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year = {2003},
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bdsk-url-1 = {https://doi.org/10.1063/1.1527013}}
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@article{Musial_2003b,
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author = {Musia{\l},Monika and Bartlett,Rodney J.},
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date-added = {2022-10-11 16:00:48 +0200},
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date-modified = {2022-10-11 16:01:02 +0200},
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doi = {10.1063/1.1584657},
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journal = {J. Chem. Phys.},
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number = {4},
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pages = {1901-1908},
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title = {Equation-of-motion coupled cluster method with full inclusion of connected triple excitations for electron-attached states: EA-EOM-CCSDT},
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volume = {119},
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year = {2003},
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bdsk-url-1 = {https://doi.org/10.1063/1.1584657}}
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@article{Nooijen_1995,
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author = {Nooijen,Marcel and Bartlett,Rodney J.},
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date-added = {2022-10-11 15:59:36 +0200},
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date-modified = {2022-10-11 15:59:51 +0200},
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doi = {10.1063/1.468592},
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journal = {J. Chem. Phys.},
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number = {9},
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pages = {3629-3647},
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title = {Equation of motion coupled cluster method for electron attachment},
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volume = {102},
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year = {1995},
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bdsk-url-1 = {https://doi.org/10.1063/1.468592}}
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@article{Stanton_1994,
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author = {Stanton,John F. and Gauss,J{\"u}rgen},
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date-added = {2022-10-11 15:56:56 +0200},
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date-modified = {2022-10-11 15:57:11 +0200},
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doi = {10.1063/1.468022},
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journal = {J. Chem. Phys.},
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number = {10},
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pages = {8938-8944},
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title = {Analytic energy derivatives for ionized states described by the equation‐of‐motion coupled cluster method},
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volume = {101},
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year = {1994},
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bdsk-url-1 = {https://doi.org/10.1063/1.468022}}
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@article{Caylak_2021,
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author = {{\c C}aylak, Onur and Baumeier, Bj{\"o}rn},
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date-added = {2022-10-11 13:29:42 +0200},
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14
CCvsMBPT.tex
14
CCvsMBPT.tex
@ -171,7 +171,7 @@ Because RPA corresponds to a resummation of all ring diagrams, it is adequate in
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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 ``charged'' excitations providing a faithful description of the fundamental gap, while the remaining excitonic effect (\ie, the stabilization provided by the attraction of the excited electron and its hole left behind) is caught via BSE, hence producing overall accurate ``neutral'' excitations.
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Practically, $GW$ produces accurate \textit{``charged''} excitations providing a faithful description of the fundamental gap, while the remaining excitonic effect (\ie, the stabilization provided by the attraction of the excited electron and its hole left behind) is caught via BSE, hence producing overall accurate \textit{``neutral''} excitations.
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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}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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@ -283,7 +283,7 @@ and matches the rCCD correlation energy
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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) and RPAx exist when the EOM space is restricted to the 1h1p configurations and only the two-body terms are dressed by rCCD correlation.
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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.
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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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@ -421,10 +421,10 @@ As in the case of RPAx (see Sec.~\ref{sec:RPAx}), several variants of the BSE co
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Following Berkelbach's analysis, \cite{Berkelbach_2018} one can extend the connection to excited states: the EOM treatment (restricted to 1h1p) of the approximate similarity-transformed rCCD Hamiltonian [see Eq.~\eqref{eq:rCCD-BSE}] provides the same excitation energies as the conventional linear-response equations \eqref{eq:BSE}.
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However, there is a significant difference with RPAx as the BSE involves $GW$ quasiparticle energies [see Eq.~\eqref{eq:A_BSE}], where some of the correlation has been already dressed, while the RPAx equations only involves (undressed) one-electron orbital energies, as shown in Eq.~\eqref{eq:EOM-rCCD}.
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In other words, in the spirit of the Brueckner version of CCD, the $GW$ post-treatment renormalizes the bare one-electron energies and, consequently, incorporates mosaic diagrams, \cite{Scuseria_2008,Scuseria_2013} a process named Brueckner-like dressing in Ref.~\onlinecite{Berkelbach_2018}.
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In other words, in the spirit of the Brueckner version of CCD, \cite{Handy_1989} the $GW$ post-treatment renormalizes the bare one-electron energies and, consequently, incorporates mosaic diagrams, \cite{Scuseria_2008,Scuseria_2013} a process named Brueckner-like dressing in Ref.~\onlinecite{Berkelbach_2018}.
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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 CCSD level, for example, this is achieved by performing IP-EOM-CCSD (up to 2h1p) and EA-EOM-CCSD (up to 2p1h) calculations prior to the EOM-CC treatment, which can then be reduced to the 1h1p sector thanks to this partial decoupling.
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At the CCSD level, for example, this is achieved by performing IP-EOM-CCSD \cite{Stanton_1994,Musial_2003a} (up to 2h1p) and EA-EOM-CCSD \cite{Nooijen_1995,Musial_2003b} (up to 2p1h) calculations prior to the EOM-CC treatment, which can then be reduced to the 1h1p sector thanks to this partial decoupling.
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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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@ -437,7 +437,7 @@ The dynamical version of BSE [where the BSE kernel is explicitly treated as freq
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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.
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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 the second-order Green's function (or second Born) \cite{Stefanucci_2013,Ortiz_2013,Phillips_2014,Rusakov_2014,Hirata_2015} or $T$-matrix.\cite{Romaniello_2012,Zhang_2017,Li_2021b,Loos_2022}
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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} 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 $G_0W_0$), the quasiparticle energies are obtained via a one-shot procedure (with or without linearization).
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@ -499,7 +499,7 @@ and the corresponding coupling blocks read
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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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Let us suppose that we are looking for the $N$ ``principal'' (\ie, quasiparticle) solutions of the eigensystem \eqref{eq:GWlin}.
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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 of size $N \times N$.
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Assuming the existence of $\bX{}{-1}$ and introducing $\bT{}{\text{2h1p}} = \bY{}{\text{2h1p}} \cdot \bX{}{-1}$ and $\bT{}{\text{2p1h}} = \bY{}{\text{2p1h}} \cdot \bX{}{-1}$, we have
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\begin{equation}
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@ -560,7 +560,7 @@ Substituting Eq.~\eqref{eq:R} into Eqs.~\eqref{eq:T1R} and \eqref{eq:T2R}, one g
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\end{split}
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\end{align}
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\end{subequations}
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In the CC language, in order to determine the 2h1p and 2p1h amplitudes, $t_{ija,p}^{\text{2h1p}}$ and $t_{iab,p}^{\text{2p1h}} $, one must solve the following residual equations
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In the CC language, in order to determine the 2h1p and 2p1h amplitudes, $t_{ija,p}^{\text{2h1p}}$ and $t_{iab,p}^{\text{2p1h}} $, one must solve the following coupled residual equations
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\begin{subequations}
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\begin{align}
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\label{eq:r_2h1p}
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