saving work on Pade
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\begin{thebibliography}{132}%
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\begin{thebibliography}{138}%
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\makeatletter
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\providecommand \@ifxundefined [1]{%
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\@ifx{#1\undefined}
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@ -1277,16 +1277,63 @@
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{journal} {\bibinfo {journal} {J. Phys. A: Math. Theor.}\ }\textbf {\bibinfo
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{volume} {40}},\ \bibinfo {pages} {581} (\bibinfo {year} {2007})}\BibitemShut
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{NoStop}%
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\bibitem [{\citenamefont {{Baker Jr.}}\ and\ \citenamefont
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{Graves-Morris}(1996)}]{BakerBook}%
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\BibitemOpen
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\bibfield {author} {\bibinfo {author} {\bibfnamefont {G.~A.}\ \bibnamefont
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{{Baker Jr.}}}\ and\ \bibinfo {author} {\bibfnamefont {P.}~\bibnamefont
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{Graves-Morris}},\ }\href@noop {} {\emph {\bibinfo {title} {Pad\'e
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Approximants}}}\ (\bibinfo {publisher} {Cambridge University Press},\
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\bibinfo {year} {1996})\BibitemShut {NoStop}%
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\bibitem [{\citenamefont {Pad\'e}(1892)}]{Pade_1892}%
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\BibitemOpen
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\bibfield {author} {\bibinfo {author} {\bibfnamefont {H.}~\bibnamefont
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{Pad\'e}},\ }in\ \href@noop {} {\emph {\bibinfo {booktitle} {Annales
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scientifiques de l'{\'E}.N.S.}}},\ Vol.~\bibinfo {volume} {9},\ \bibinfo
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{editor} {edited by\ \bibinfo {editor} {\bibnamefont {Gauthier-Villars}}}\
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(\bibinfo {publisher} {{\'E}ditions scientifiques et m{\'e}dicales
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Elsevier},\ \bibinfo {year} {1892})\ pp.\ \bibinfo {pages}
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{3--93}\BibitemShut {NoStop}%
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\bibitem [{\citenamefont {Loos}(2013)}]{Loos_2013}%
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\BibitemOpen
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\bibfield {author} {\bibinfo {author} {\bibfnamefont {P.-F.}\ \bibnamefont
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{Loos}},\ }\href {\doibase 10.1063/1.4790613} {\bibfield {journal} {\bibinfo
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{journal} {J. Chem. Phys.}\ }\textbf {\bibinfo {volume} {138}},\ \bibinfo
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{pages} {064108} (\bibinfo {year} {2013})}\BibitemShut {NoStop}%
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\bibitem [{\citenamefont {Gluzman}(2020)}]{Gluzman_2020}%
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\BibitemOpen
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\bibfield {author} {\bibinfo {author} {\bibfnamefont {S.}~\bibnamefont
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{Gluzman}},\ }\href {\doibase 10.3390/sym12101600} {\bibfield {journal}
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{\bibinfo {journal} {Symmetry}\ }\textbf {\bibinfo {volume} {12}},\ \bibinfo
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{pages} {1600} (\bibinfo {year} {2020})}\BibitemShut {NoStop}%
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\bibitem [{\citenamefont {Mayer}\ and\ \citenamefont
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{Tong}(1985)}]{Mayer_1985}%
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\BibitemOpen
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\bibfield {author} {\bibinfo {author} {\bibfnamefont {I.~L.}\ \bibnamefont
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{Mayer}}\ and\ \bibinfo {author} {\bibfnamefont {B.~Y.}\ \bibnamefont
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{Tong}},\ }\href {\doibase 10.1088/0022-3719/18/17/008} {\bibfield {journal}
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{\bibinfo {journal} {J. Phys. C.: Solid State Phys.}\ }\textbf {\bibinfo
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{volume} {18}},\ \bibinfo {pages} {3297} (\bibinfo {year}
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{1985})}\BibitemShut {NoStop}%
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\bibitem [{\citenamefont {Goodson}(2019)}]{Goodson_2019}%
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\BibitemOpen
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\bibfield {author} {\bibinfo {author} {\bibfnamefont {D.~Z.}\ \bibnamefont
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{Goodson}},\ }in\ \href {\doibase
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https://doi.org/10.1016/B978-0-12-813651-5.00009-7} {\emph {\bibinfo
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{booktitle} {Mathematical Physics in Theoretical Chemistry}}},\ \bibinfo
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{series and number} {Developments in Physical {\&} Theoretical Chemistry},\
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\bibinfo {editor} {edited by\ \bibinfo {editor} {\bibfnamefont
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{S.}~\bibnamefont {Blinder}}\ and\ \bibinfo {editor} {\bibfnamefont
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{J.}~\bibnamefont {House}}}\ (\bibinfo {publisher} {Elsevier},\ \bibinfo
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{year} {2019})\ pp.\ \bibinfo {pages} {295 -- 325}\BibitemShut {NoStop}%
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\bibitem [{\citenamefont {Surj{\'a}n}\ and\ \citenamefont
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{Szabados}(2000)}]{Surjan_2000}%
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\BibitemOpen
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\bibfield {author} {\bibinfo {author} {\bibfnamefont {P.~R.}\ \bibnamefont
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{Surj{\'a}n}}\ and\ \bibinfo {author} {\bibfnamefont {{\'A}.}~\bibnamefont
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{Szabados}},\ }\href {\doibase 10.1063/1.481006} {\bibfield {journal}
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{\bibinfo {journal} {The Journal of Chemical Physics}\ }\textbf {\bibinfo
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{volume} {112}},\ \bibinfo {pages} {4438} (\bibinfo {year} {2000})},\ \Eprint
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{http://arxiv.org/abs/https://doi.org/10.1063/1.481006}
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{https://doi.org/10.1063/1.481006} \BibitemShut {NoStop}%
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{\bibinfo {journal} {J. Chem. Phys.}\ }\textbf {\bibinfo {volume} {112}},\
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\bibinfo {pages} {4438} (\bibinfo {year} {2000})}\BibitemShut {NoStop}%
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\bibitem [{\citenamefont {Surj{\'a}n}\ \emph {et~al.}(2018)\citenamefont
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{Surj{\'a}n}, \citenamefont {Mih{\'a}lka},\ and\ \citenamefont
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{Szabados}}]{Surjan_2018}%
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@ -1,24 +1,97 @@
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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 2020-11-25 08:29:13 +0100
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%% Created for Pierre-Francois Loos at 2020-11-25 09:35:42 +0100
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%% Saved with string encoding Unicode (UTF-8)
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@article{Loos_2013,
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author = {Loos, Pierre-Fran{\c c}ois},
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date-added = {2020-11-25 09:34:55 +0100},
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date-modified = {2020-11-25 09:35:07 +0100},
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doi = {10.1063/1.4790613},
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journal = {J. Chem. Phys.},
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pages = {064108},
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title = {High-Density Correlation Energy Expansion of the One-Dimensional Uniform Electron Gas},
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volume = {138},
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year = {2013},
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Bdsk-Url-1 = {https://dx.doi.org/10.1063/1.4790613}}
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@article{Gluzman_2020,
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author = {S. Gluzman},
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date-added = {2020-11-25 09:32:54 +0100},
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date-modified = {2020-11-25 09:33:54 +0100},
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doi = {10.3390/sym12101600},
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journal = {Symmetry},
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pages = {1600},
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title = {Pad\'e and Post-Pad\'e Approximations for Critical Phenomena},
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volume = {12},
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year = {2020}}
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@incollection{Goodson_2019,
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abstract = {The Schr{\"o}dinger equation for an atom or molecule includes parameters, such as bond lengths or nuclear charges, and the resulting energy eigenvalue can be treated as a function with the parameter values as continuous variables. Analysis of singular points of this function, at nonphysical parameter values, can explain and predict the success or failure of quantum chemical calculation methods. An introduction to the theory of singularities in functions of a complex variable is presented and examples of applications to quantum chemistry are described, including the calculation of molecular potential energy curves, the theoretical description of ionization, and the summation of perturbation theories.},
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author = {David Z. Goodson},
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booktitle = {Mathematical Physics in Theoretical Chemistry},
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date-added = {2020-11-25 09:13:38 +0100},
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date-modified = {2020-11-25 09:14:27 +0100},
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doi = {https://doi.org/10.1016/B978-0-12-813651-5.00009-7},
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editor = {S.M. Blinder and J.E. House},
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isbn = {978-0-12-813651-5},
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keywords = {Singularities, Avoided crossings, Quadratic approximants, Molecular potential energy curves, Ionization, Finite-size scaling, Perturbation theory, Series summation},
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pages = {295 - 325},
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publisher = {Elsevier},
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series = {Developments in Physical {\&} Theoretical Chemistry},
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title = {Chapter 9 - Singularity analysis in quantum chemistry},
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year = {2019},
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Bdsk-Url-1 = {http://www.sciencedirect.com/science/article/pii/B9780128136515000097},
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Bdsk-Url-2 = {https://doi.org/10.1016/B978-0-12-813651-5.00009-7}}
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@article{Mayer_1985,
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abstract = {The quadratic Pade method-a new method for calculating the local density of states in various physical systems-is introduced and discussed. The method is based upon the use of Hermite-Pade polynomials and it makes the calculation of densities of states a straightforward and relatively simple matter. Its advantages over other methods with similar generality and complexity are outlined and numerical results for various systems, which illustrate the virtues of the new method, are presented and discussed.},
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author = {I L Mayer and B Y Tong},
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date-added = {2020-11-25 09:01:38 +0100},
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date-modified = {2020-11-25 09:03:36 +0100},
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doi = {10.1088/0022-3719/18/17/008},
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journal = {J. Phys. C.: Solid State Phys.},
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pages = {3297--3318},
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title = {The quadratic Pade approximant method and its application for calculating densities of states},
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volume = {18},
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year = 1985,
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Bdsk-Url-1 = {https://doi.org/10.1088%2F0022-3719%2F18%2F17%2F008},
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Bdsk-Url-2 = {https://doi.org/10.1088/0022-3719/18/17/008}}
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@book{BakerBook,
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author = {G. A. {Baker Jr.} and P. Graves-Morris},
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date-added = {2020-11-25 08:58:42 +0100},
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date-modified = {2020-11-25 08:59:33 +0100},
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publisher = {Cambridge University Press},
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title = {Pad\'e Approximants},
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year = {1996}}
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@incollection{Pade_1892,
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author = {H. Pad\'e},
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booktitle = {Annales scientifiques de l'{\'E}.N.S.},
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date-added = {2020-11-25 08:48:36 +0100},
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date-modified = {2020-11-25 08:56:44 +0100},
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editor = {Gauthier-Villars},
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pages = {3--93},
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publisher = {{\'E}ditions scientifiques et m{\'e}dicales Elsevier},
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title = {Sur la repr{\'e}sentation approch{\'e}e d'une fonction par des fractions rationnelles},
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volume = {9},
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year = {1892}}
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@article{Surjan_2000,
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author = {Surj{\'a}n,P. R. and Szabados,{\'A}.},
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date-added = {2020-11-25 08:29:04 +0100},
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date-modified = {2020-11-25 08:29:13 +0100},
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date-modified = {2020-11-25 09:15:28 +0100},
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doi = {10.1063/1.481006},
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eprint = {https://doi.org/10.1063/1.481006},
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journal = {The Journal of Chemical Physics},
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journal = {J. Chem. Phys.},
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number = {10},
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pages = {4438-4446},
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title = {Optimized partitioning in perturbation theory: Comparison to related approaches},
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url = {https://doi.org/10.1063/1.481006},
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volume = {112},
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year = {2000},
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Bdsk-Url-1 = {https://doi.org/10.1063/1.481006}}
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@ -1161,27 +1161,31 @@ Those excited states have a non-negligible contribution to the exact FCI solutio
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We believe that $\alpha$ singularities are connected to states with non-negligible contribution in the CI expansion thus to the dynamical part of the correlation energy, while $\beta$ singularities are linked to symmetry breaking and phase transitions of the wave function, \ie, to the multi-reference nature of the wave function thus to the static part of the correlation energy.
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\section{Exploiting complex analysis}
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\section{Resummation techniques}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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As frequently claimed by Carl Bender, the most stupid thing that one can do with a series is to sum it.
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Nonetheless, quantum chemists are basically doing exactly this on a daily basis.
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The inability of Taylor series to model properly the energy function $E(\lambda$) can be simply understood by the fact that one aims at modelling a complicated function with potentially poles and singularities by a simple polynomial of finite order.
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%==========================================%
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\subsection{Pad\'e approximant}
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%==========================================%
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According to Wikipedia, \textit{``a Pad\'e approximant is the best approximation of a function by a rational function of given order''}.
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A $[d_A/d_B]$ Pad\'e approximant is defined as \cite{Pade}
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According to Wikipedia, \textit{``a Pad\'e approximant is the best approximation of a function by a rational function of given order''}. \cite{BakerBook}
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A $[d_A/d_B]$ Pad\'e approximant is defined as \cite{Pade_1892}
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\begin{equation}
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\label{eq:PadeApp}
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E_{[d_A/d_B]}(\lambda) = \frac{A(\lambda)}{B(\lambda)} = \frac{\sum_{k=0}^{d_A} a_k \lambda^k}{\sum_{k=0}^{d_B} b_k \lambda^k}
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\end{equation}
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(with $b_0 = 1$), where the coefficients of the polynomials are determined by collecting terms according to power of $\lambda$.
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Pad\'e approximants are nice and they can model poles (the poles of a $[d_A/d_B]$ Pad\'e approximant appears at the roots of the polynomial $B(\lambda)$), but they cannot model functions with square-root branch points, and that sucks because it is not consistent with the singularity structure of a typical perturbative treatment.
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(with $b_0 = 1$), where the coefficients of the polynomials $A(\lambda)$ and $B(\lambda)$ are determined by collecting terms according to power of $\lambda$.
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Pad\'e approximants are extremely useful in many areas of physics and chemistry \cite{Loos_2013,Gluzman_2020} as they can model poles [the poles of a Pad\'e approximant appears at the roots of the polynomial $B(\lambda)$].
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However, they are unable to model functions with square-root branch points, which are ubiquitous in the singularity structure of a typical perturbative treatment.
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%==========================================%
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\subsection{Quadratic approximant}
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%==========================================%
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In a nutshell, the idea behind quadratic approximant is to model the singularity structure of the energy function $E(\lambda)$ via a generalized version of the square-root singularity expression
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In a nutshell, the idea behind quadratic approximant is to model the singularity structure of the energy function $E(\lambda)$ via a generalized version of the square-root singularity expression \cite{Mayer_1985,Goodson_2011,Goodson_2019}
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\begin{equation}
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\label{eq:QuadApp}
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E(\lambda) = \frac{1}{2 Q(\lambda)} \qty[ P(\lambda) \pm \sqrt{P^2(\lambda) - 4 Q(\lambda) R(\lambda)} ]
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@ -1195,7 +1199,7 @@ where
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&
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R(\lambda) & = \sum_{k=0}^{d_R} r_k \lambda^k
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\end{align}
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are polynomials, such that $d_P + d_Q + d_R = n - 1$ where $n$ is the highest-order series coefficient known from the Taylor expansion of $E(\lambda)$.
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are polynomials, such that $d_P + d_Q + d_R = n - 1$, where $n$ is the highest-order series coefficient known from the Taylor expansion of $E(\lambda)$.
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Recasting Eq.~\eqref{eq:QuadApp} as a second-order expression in $E(\lambda)$, \ie,
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\begin{equation}
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Q(\lambda) E^2(\lambda) - P(\lambda) E(\lambda) + R(\lambda) \sim \order*{\lambda^{n+1}}
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