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CCvsMBPT.bib
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CCvsMBPT.bib
@ -1,13 +1,124 @@
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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 11:00:19 +0200
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%% Created for Pierre-Francois Loos at 2022-10-11 16:13:34 +0200
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%% Saved with string encoding Unicode (UTF-8)
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@article{Handy_1989,
|
||||
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.},
|
||||
author = {Nicholas C. Handy and John A. Pople and Martin Head-Gordon and Krishnan Raghavachari and Gary W. Trucks},
|
||||
date-added = {2022-10-11 16:13:12 +0200},
|
||||
date-modified = {2022-10-11 16:13:31 +0200},
|
||||
doi = {https://doi.org/10.1016/0009-2614(89)85013-4},
|
||||
journal = {Chem. Phys. Lett.},
|
||||
number = {2},
|
||||
pages = {185-192},
|
||||
title = {Size-consistent Brueckner theory limited to double substitutions},
|
||||
volume = {164},
|
||||
year = {1989},
|
||||
bdsk-url-1 = {https://www.sciencedirect.com/science/article/pii/0009261489850134},
|
||||
bdsk-url-2 = {https://doi.org/10.1016/0009-2614(89)85013-4}}
|
||||
|
||||
@article{Musial_2003a,
|
||||
author = {Musia{\l},Monika and Kucharski,Stanis{\l}aw A. and Bartlett,Rodney J.},
|
||||
date-added = {2022-10-11 16:05:07 +0200},
|
||||
date-modified = {2022-10-11 16:05:20 +0200},
|
||||
doi = {10.1063/1.1527013},
|
||||
journal = {J. Chem. Phys.},
|
||||
number = {3},
|
||||
pages = {1128-1136},
|
||||
title = {Equation-of-motion coupled cluster method with full inclusion of the connected triple excitations for ionized states: IP-EOM-CCSDT},
|
||||
volume = {118},
|
||||
year = {2003},
|
||||
bdsk-url-1 = {https://doi.org/10.1063/1.1527013}}
|
||||
|
||||
@article{Musial_2003b,
|
||||
author = {Musia{\l},Monika and Bartlett,Rodney J.},
|
||||
date-added = {2022-10-11 16:00:48 +0200},
|
||||
date-modified = {2022-10-11 16:01:02 +0200},
|
||||
doi = {10.1063/1.1584657},
|
||||
journal = {J. Chem. Phys.},
|
||||
number = {4},
|
||||
pages = {1901-1908},
|
||||
title = {Equation-of-motion coupled cluster method with full inclusion of connected triple excitations for electron-attached states: EA-EOM-CCSDT},
|
||||
volume = {119},
|
||||
year = {2003},
|
||||
bdsk-url-1 = {https://doi.org/10.1063/1.1584657}}
|
||||
|
||||
@article{Nooijen_1995,
|
||||
author = {Nooijen,Marcel and Bartlett,Rodney J.},
|
||||
date-added = {2022-10-11 15:59:36 +0200},
|
||||
date-modified = {2022-10-11 15:59:51 +0200},
|
||||
doi = {10.1063/1.468592},
|
||||
journal = {J. Chem. Phys.},
|
||||
number = {9},
|
||||
pages = {3629-3647},
|
||||
title = {Equation of motion coupled cluster method for electron attachment},
|
||||
volume = {102},
|
||||
year = {1995},
|
||||
bdsk-url-1 = {https://doi.org/10.1063/1.468592}}
|
||||
|
||||
@article{Stanton_1994,
|
||||
author = {Stanton,John F. and Gauss,J{\"u}rgen},
|
||||
date-added = {2022-10-11 15:56:56 +0200},
|
||||
date-modified = {2022-10-11 15:57:11 +0200},
|
||||
doi = {10.1063/1.468022},
|
||||
journal = {J. Chem. Phys.},
|
||||
number = {10},
|
||||
pages = {8938-8944},
|
||||
title = {Analytic energy derivatives for ionized states described by the equation‐of‐motion coupled cluster method},
|
||||
volume = {101},
|
||||
year = {1994},
|
||||
bdsk-url-1 = {https://doi.org/10.1063/1.468022}}
|
||||
|
||||
@article{Caylak_2021,
|
||||
author = {{\c C}aylak, Onur and Baumeier, Bj{\"o}rn},
|
||||
date-added = {2022-10-11 13:29:42 +0200},
|
||||
date-modified = {2022-10-11 13:30:08 +0200},
|
||||
doi = {10.1021/acs.jctc.0c01099},
|
||||
journal = {J. Chem. Theory Comput.},
|
||||
number = {2},
|
||||
pages = {879-888},
|
||||
title = {Excited-State Geometry Optimization of Small Molecules with Many-Body Green's Functions Theory},
|
||||
volume = {17},
|
||||
year = {2021},
|
||||
bdsk-url-1 = {https://doi.org/10.1021/acs.jctc.0c01099}}
|
||||
|
||||
@article{IsmailBeigi_2003,
|
||||
author = {Ismail-Beigi, Sohrab and Louie, Steven G.},
|
||||
date-added = {2022-10-11 13:28:50 +0200},
|
||||
date-modified = {2022-10-11 13:29:10 +0200},
|
||||
doi = {10.1103/PhysRevLett.90.076401},
|
||||
issue = {7},
|
||||
journal = {Phys. Rev. Lett.},
|
||||
month = {Feb},
|
||||
numpages = {4},
|
||||
pages = {076401},
|
||||
publisher = {American Physical Society},
|
||||
title = {Excited-State Forces within a First-Principles Green's Function Formalism},
|
||||
url = {https://link.aps.org/doi/10.1103/PhysRevLett.90.076401},
|
||||
volume = {90},
|
||||
year = {2003},
|
||||
bdsk-url-1 = {https://link.aps.org/doi/10.1103/PhysRevLett.90.076401},
|
||||
bdsk-url-2 = {https://doi.org/10.1103/PhysRevLett.90.076401}}
|
||||
|
||||
@article{Knysh_2022,
|
||||
author = {Knysh,Iryna and Duchemin,Ivan and Blase,X. and Jacquemin,Denis M.},
|
||||
date-added = {2022-10-11 13:26:25 +0200},
|
||||
date-modified = {2022-10-11 13:26:41 +0200},
|
||||
doi = {10.1063/5.0121121},
|
||||
journal = {J. Chem. Phys.},
|
||||
number = {ja},
|
||||
pages = {null},
|
||||
title = {Modelling of excited state potential energy surfaces with the Bethe−Salpeter equation formalism: The 4-(dimethylamino)benzonitrile twist},
|
||||
volume = {0},
|
||||
year = {0},
|
||||
bdsk-url-1 = {https://doi.org/10.1063/5.0121121}}
|
||||
|
||||
@article{Loos_2022,
|
||||
author = {Loos,Pierre-Fran{\c c}ois and Romaniello,Pina},
|
||||
date-added = {2022-10-11 10:48:31 +0200},
|
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|
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35
CCvsMBPT.tex
35
CCvsMBPT.tex
@ -172,7 +172,7 @@ Because \ant{ph-}RPA corresponds to a resummation of all ring diagrams, it is ad
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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 and provides a faithful description of the fundamental gap, while the remaining excitonic effect (\ie, the stabilization provided by the attraction of the excited electron and its hole left behind) is caught via BSE, hence producing 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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\ant{In this paragraph we say that BSE goes beyond RPA and then we say that BSE is done on top of GW without expliciting the link between GW and RPA. I guess it's obvious for GW people but maybe just saying that the GW polarizability is obtained by summing the time dependent ring diagrams would make the connection a bit clearer.}
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@ -182,7 +182,7 @@ BSE@$GW$ has been shown to be highly successful to compute low-lying excited sta
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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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, which solidifies the numerical evidences provided by Freeman many years before. \cite{Freeman_1977}
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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}
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\label{eq:RPA}
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@ -237,7 +237,7 @@ Substituting Eq.~\eqref{eq:RPA_1} into Eq.~\eqref{eq:RPA_2} yields the following
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\begin{equation}
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\bB{}{} + \bA{}{} \cdot \bT{}{} + \bT{}{} \cdot \bA{}{} + \bT{}{} \cdot \bB{}{} \cdot \bT{}{} = \bO
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\end{equation}
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that matches the well-known rCCD amplitude (or residual) equations
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that matches the rCCD amplitude (or residual) equations
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\begin{multline}
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\label{eq:rCCD}
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r_{ij}^{ab}
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@ -285,7 +285,7 @@ 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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@ -320,7 +320,7 @@ 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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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 has been also extended to the pp and hh sectors by Peng \textit{et al.} \cite{Peng_2013} and Scuseria \textit{et al.} \cite{Scuseria_2013}
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This has been also extended to the pp and hh sectors independently by Peng \textit{et al.} \cite{Peng_2013} and Scuseria \textit{et al.} \cite{Scuseria_2013}
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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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@ -407,7 +407,7 @@ Therefore, following the derivation detailed in Sec.~\ref{sec:RPAx}, one can sho
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\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 $\tilde{t}_{ij}^{ab}$, 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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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}
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\label{eq:rCCD-BSE}
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\Tilde{r}_{ij}^{ab}
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@ -424,14 +424,14 @@ 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}. \ant{I'm not sure that we can call it an EOM treatment of an approximate similarity transformed rCCD Hamiltonian. Maybe we can say that we can obtain an analog of the approximate-EOM equation Eq.~(\ref{eq:EOM-rCCD}) by using the amplitudes defined by [see Eq.~\eqref{eq:rCCD-BSE}] as well as replacing $A$ and $B$ by their BSE counterparts and this equation gives the same excitations as the linear response equations}.
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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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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_1988,Rohlfing_2000,Romaniello_2009b,Loos_2020h,Authier_2020,Bintrim_2022}
|
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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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||||
\section{Connection between $GW$ and CC}
|
||||
@ -440,9 +440,9 @@ 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.
|
||||
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.
|
||||
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}
|
||||
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.
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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).
|
||||
\cite{Strinati_1980,Hybertsen_1985a,Hybertsen_1986,Godby_1988,Linden_1988,Northrup_1991,Blase_1994,Rohlfing_1995,Shishkin_2007}
|
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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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@ -502,7 +502,7 @@ 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
|
||||
\begin{equation}
|
||||
@ -563,7 +563,7 @@ Substituting Eq.~\eqref{eq:R} into Eqs.~\eqref{eq:T1R} and \eqref{eq:T2R}, one g
|
||||
\end{split}
|
||||
\end{align}
|
||||
\end{subequations}
|
||||
In the CC language, in order to determine the 2h1p and 2p1h amplitudes, $t_{ija,p}^{\text{2h1p}}$ and $t_{iab,p}^{\text{2p1h}} $, one must solve the following residual equations
|
||||
In the CC language, in order to determine the 2h1p and 2p1h amplitudes, $t_{ija,p}^{\text{2h1p}}$ and $t_{iab,p}^{\text{2p1h}} $, one must solve the following coupled residual equations
|
||||
\begin{subequations}
|
||||
\begin{align}
|
||||
\label{eq:r_2h1p}
|
||||
@ -606,7 +606,7 @@ The quasiparticle energies $\e{p}{GW}$ are thus provided by the eigenvalues of $
|
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\bSig{}{\GW} = \bV{}{\text{2h1p}} \cdot \bT{}{\text{2h1p}} + \bV{}{\text{2p1h}} \cdot \bT{}{\text{2p1h}}
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\end{equation}
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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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The $G_0W_0$ quasiparticle energies can be easily obtained via the procedure described in Ref.~\onlinecite{Monino_2022} by solving the previous equation for each value of $p$ separately.
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The $G_0W_0$ quasiparticle energies can be easily obtained via the procedure described in Ref.~\onlinecite{Monino_2022} by solving the previous equations for each value of $p$ separately.
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|
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%======================
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%\subsection{EE-EOM-CCD}
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@ -722,10 +722,11 @@ The $G_0W_0$ quasiparticle energies can be easily obtained via the procedure des
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%%%%%%%%%%%%%%%%%%%%%
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\section{Conclusion}
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%%%%%%%%%%%%%%%%%%%%%
|
||||
Here, we have unveiled exact similarities between the BSE formalism and the $GW$ approximation from many-body perturbation theory and CC theory at the ground- and excited-state levels.
|
||||
Here, we have unveiled exact similarities between CC and many-body perturbation theory at the ground- and excited-state levels.
|
||||
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 with the same computational cost.
|
||||
The conventional and CC-based versions of the BSE and $GW$ schemes that we have described in the present work have been implemented in the electronic structure package QuAcK \cite{QuAcK} (available at \url{https://github.com/pfloos/QuAcK}) with which we have numerically checked these exact equivalences.
|
||||
Similitudes between BSE@$GW$ and the STEOM-CC method have been put forward.
|
||||
We hope that the present work may provide a path for the computation of ground- and excited-state properties (such as nuclear gradients) within the $GW$ and BSE frameworks.
|
||||
Similitudes between BSE@$GW$ and STEOM-CC have been put forward.
|
||||
We hope that the present work may provide a path for the computation of ground- and excited-state properties (such as nuclear gradients) within the $GW$ \cite{Lazzeri_2008,Faber_2011b,Yin_2013,Montserrat_2016,Zhenglu_2019} and BSE \cite{IsmailBeigi_2003,Caylak_2021,Knysh_2022} frameworks, and broaden the applicability of Green's function methods in the molecular electronic structure community and beyond.
|
||||
|
||||
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
%\section*{Supplementary Material}
|
||||
|
||||
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Reference in New Issue
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