conclusion
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\begin{thebibliography}{154}%
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\begin{thebibliography}{159}%
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\makeatletter
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\providecommand \@ifxundefined [1]{%
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@ -1364,6 +1364,15 @@
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{journal} {\bibinfo {journal} {Theor. Chem. Acc.}\ }\textbf {\bibinfo
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{volume} {137}},\ \bibinfo {pages} {149} (\bibinfo {year}
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{2018})}\BibitemShut {NoStop}%
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\bibitem [{\citenamefont {Cohen}\ and\ \citenamefont
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{Mori-S\'anchez}(2016)}]{Cohen_2016}%
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{Cohen}}\ and\ \bibinfo {author} {\bibfnamefont {P.}~\bibnamefont
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{Mori-S\'anchez}},\ }\href {\doibase 10.1103/PhysRevA.93.042511} {\bibfield
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{journal} {\bibinfo {journal} {Phys. Rev. A}\ }\textbf {\bibinfo {volume}
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{93}},\ \bibinfo {pages} {042511} (\bibinfo {year} {2016})}\BibitemShut
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{NoStop}%
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\bibitem [{\citenamefont {Romaniello}\ \emph {et~al.}(2009)\citenamefont
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{Romaniello}, \citenamefont {Guyot},\ and\ \citenamefont
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{Reining}}]{Romaniello_2009}%
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@ -1408,6 +1417,16 @@
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{journal} {\bibinfo {journal} {Phys. Rev. B}\ }\textbf {\bibinfo {volume}
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{96}},\ \bibinfo {pages} {045124} (\bibinfo {year} {2017})}\BibitemShut
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{NoStop}%
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\bibitem [{\citenamefont {Olevano}\ \emph {et~al.}(2019)\citenamefont
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{Olevano}, \citenamefont {Toulouse},\ and\ \citenamefont
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{Schuck}}]{Olevano_2019}%
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\BibitemOpen
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\bibfield {author} {\bibinfo {author} {\bibfnamefont {V.}~\bibnamefont
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{Olevano}}, \bibinfo {author} {\bibfnamefont {J.}~\bibnamefont {Toulouse}}, \
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and\ \bibinfo {author} {\bibfnamefont {P.}~\bibnamefont {Schuck}},\ }\href
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{\doibase 10.1063/1.5080330} {\bibfield {journal} {\bibinfo {journal} {J.
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Chem. Phys.}\ }\textbf {\bibinfo {volume} {150}},\ \bibinfo {pages} {084112}
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(\bibinfo {year} {2019})}\BibitemShut {NoStop}%
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\bibitem [{\citenamefont {Deur}\ \emph {et~al.}(2017)\citenamefont {Deur},
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\citenamefont {Mazouin},\ and\ \citenamefont {Fromager}}]{Deur_2017}%
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\BibitemOpen
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@ -1474,4 +1493,33 @@
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{\bibfnamefont {B.}~\bibnamefont {Kirtman}}, \ and\ \bibinfo {editor}
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{\bibfnamefont {Y.}~\bibnamefont {Ozaki}}}\ (\bibinfo {publisher} {Springer,
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Singapore})\BibitemShut {NoStop}%
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\bibitem [{\citenamefont {Stein}\ \emph {et~al.}(2014)\citenamefont {Stein},
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\citenamefont {Henderson},\ and\ \citenamefont {Scuseria}}]{Stein_2014}%
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\BibitemOpen
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\bibfield {author} {\bibinfo {author} {\bibfnamefont {T.}~\bibnamefont
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{Stein}}, \bibinfo {author} {\bibfnamefont {T.~M.}\ \bibnamefont
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{Henderson}}, \ and\ \bibinfo {author} {\bibfnamefont {G.~E.}\ \bibnamefont
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{Scuseria}},\ }\href {\doibase 10.1063/1.4880819} {\bibfield {journal}
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{\bibinfo {journal} {J. Chem. Phys.}\ }\textbf {\bibinfo {volume} {140}},\
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\bibinfo {pages} {214113} (\bibinfo {year} {2014})}\BibitemShut {NoStop}%
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\bibitem [{\citenamefont {Henderson}\ \emph {et~al.}(2015)\citenamefont
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{Henderson}, \citenamefont {Bulik},\ and\ \citenamefont
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{Scuseria}}]{Henderson_2015}%
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\BibitemOpen
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\bibfield {author} {\bibinfo {author} {\bibfnamefont {T.~M.}\ \bibnamefont
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{Henderson}}, \bibinfo {author} {\bibfnamefont {I.~W.}\ \bibnamefont
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{Bulik}}, \ and\ \bibinfo {author} {\bibfnamefont {G.~E.}\ \bibnamefont
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{Scuseria}},\ }\href {\doibase 10.1063/1.4921986} {\bibfield {journal}
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{\bibinfo {journal} {J. Chem. Phys.}\ }\textbf {\bibinfo {volume} {142}},\
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\bibinfo {pages} {214116} (\bibinfo {year} {2015})}\BibitemShut {NoStop}%
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\bibitem [{\citenamefont {Shepherd}\ \emph {et~al.}(2016)\citenamefont
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{Shepherd}, \citenamefont {Henderson},\ and\ \citenamefont
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{Scuseria}}]{Shepherd_2016}%
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\BibitemOpen
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\bibfield {author} {\bibinfo {author} {\bibfnamefont {J.~J.}\ \bibnamefont
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{Shepherd}}, \bibinfo {author} {\bibfnamefont {T.~M.}\ \bibnamefont
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{Henderson}}, \ and\ \bibinfo {author} {\bibfnamefont {G.~E.}\ \bibnamefont
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{Scuseria}},\ }\href {\doibase 10.1063/1.4942770} {\bibfield {journal}
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{\bibinfo {journal} {J. Chem. Phys.}\ }\textbf {\bibinfo {volume} {144}},\
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\bibinfo {pages} {094112} (\bibinfo {year} {2016})}\BibitemShut {NoStop}%
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@ -1,13 +1,75 @@
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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 2020-12-04 09:17:19 +0100
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%% Created for Pierre-Francois Loos at 2020-12-04 09:51:06 +0100
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%% Saved with string encoding Unicode (UTF-8)
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@article{Shepherd_2016,
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author = {Shepherd,James J. and Henderson,Thomas M. and Scuseria,Gustavo E.},
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date-added = {2020-12-04 09:50:38 +0100},
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date-modified = {2020-12-04 09:50:55 +0100},
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doi = {10.1063/1.4942770},
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journal = {J. Chem. Phys.},
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pages = {094112},
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title = {Using full configuration interaction quantum Monte Carlo in a seniority zero space to investigate the correlation energy equivalence of pair coupled cluster doubles and doubly occupied configuration interaction},
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volume = {144},
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year = {2016},
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Bdsk-Url-1 = {https://doi.org/10.1063/1.4942770}}
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@article{Henderson_2015,
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author = {Henderson,Thomas M. and Bulik,Ireneusz W. and Scuseria,Gustavo E.},
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date-added = {2020-12-04 09:47:58 +0100},
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date-modified = {2020-12-04 09:48:17 +0100},
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doi = {10.1063/1.4921986},
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journal = {J. Chem. Phys.},
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pages = {214116},
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title = {Pair extended coupled cluster doubles},
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volume = {142},
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year = {2015},
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Bdsk-Url-1 = {https://doi.org/10.1063/1.4921986}}
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@article{Olevano_2019,
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author = {Olevano,Valerio and Toulouse,Julien and Schuck,Peter},
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date-added = {2020-12-04 09:46:46 +0100},
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date-modified = {2020-12-04 09:46:46 +0100},
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doi = {10.1063/1.5080330},
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journal = {J. Chem. Phys.},
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number = {8},
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pages = {084112},
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title = {A formally exact one-frequency-only Bethe-Salpeter-like equation. Similarities and differences between GW+BSE and self-consistent RPA},
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volume = {150},
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year = {2019},
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Bdsk-Url-1 = {https://doi.org/10.1063/1.5080330}}
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@article{Stein_2014,
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author = {Stein,Tamar and Henderson,Thomas M. and Scuseria,Gustavo E.},
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date-added = {2020-12-04 09:43:40 +0100},
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date-modified = {2020-12-04 09:43:58 +0100},
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doi = {10.1063/1.4880819},
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journal = {J. Chem. Phys.},
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pages = {214113},
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title = {Seniority zero pair coupled cluster doubles theory},
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volume = {140},
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year = {2014},
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Bdsk-Url-1 = {https://doi.org/10.1063/1.4880819}}
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@article{Cohen_2016,
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author = {Cohen, Aron J. and Mori-S\'anchez, Paula},
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date-added = {2020-12-04 09:41:48 +0100},
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date-modified = {2020-12-04 09:42:00 +0100},
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doi = {10.1103/PhysRevA.93.042511},
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journal = {Phys. Rev. A},
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pages = {042511},
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title = {Landscape of an exact energy functional},
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volume = {93},
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year = {2016},
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Bdsk-Url-1 = {https://link.aps.org/doi/10.1103/PhysRevA.93.042511},
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Bdsk-Url-2 = {https://doi.org/10.1103/PhysRevA.93.042511}}
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@inbook{Lowdin_1958,
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abstract = {Summary This chapter contains sections titled: Introduction Formulation of the Correlation Problem Methods for Treating Electronic Correlation Recent Developments; Concluding Remarks},
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author = {L\"owdin, Per-Olov},
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@ -1840,7 +1840,7 @@ However, it is worth mentioning that the construction of these approximants requ
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The Shanks transformation presented in Sec.~\ref{sec:Shanks} can, in some cases, alleviate this issue.
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Most of the physical concepts and mathematical tools reviewed in the present manuscript has been illustrated on the symmetric (or asymmetric in one occasion) Hubbard dimer at half-filling.
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Although extremely simple, this clearly illustrates the obvious versatility of the Hubbard model to understand the subtle notions linked to the extension of perturbation theory into the complex plane, as well as other concepts such as Kohn-Sham density-functional theory (DFT), \cite{Carrascal_2015} linear-response theory, \cite{Carrascal_2018} many-body perturbation theory, \cite{Romaniello_2009,Romaniello_2012,DiSabatino_2015,Tarantino_2017}, DFT for ensembles, \cite{Deur_2017,Deur_2018,Senjean_2018,Sagredo_2018,Fromager_2020} thermal DFT, \cite{Smith_2016,Smith_2018}, correlated methods, \cite{} and many others.
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Although extremely simple, this clearly illustrates the obvious versatility of the Hubbard model to understand the subtle notions linked to the extension of perturbation theory into the complex plane, as well as other concepts such as Kohn-Sham density-functional theory (DFT), \cite{Carrascal_2015,Cohen_2016} linear-response theory, \cite{Carrascal_2018} many-body perturbation theory, \cite{Romaniello_2009,Romaniello_2012,DiSabatino_2015,Tarantino_2017,Olevano_2019} DFT for ensembles, \cite{Deur_2017,Deur_2018,Senjean_2018,Sagredo_2018,Fromager_2020} thermal DFT, \cite{Smith_2016,Smith_2018} coupled cluster theory, \cite{Stein_2014,Henderson_2015,Shepherd_2016} and many others.
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We believe that the Hubbard dimer could then be used for further developments and comprehension around perturbation theory.
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As a concluding remark and from a broader point of view, the present work shows that our understanding of the singularity structure of the energy is still incomplete.
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