saving work on Pade

Manuscript/EPAWTFT.bbl

@@ -6,7 +6,7 @@
 %Control: page (0) single
 %Control: year (1) truncated
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-\begin{thebibliography}{131}%
+\begin{thebibliography}{132}%
 \makeatletter
 \providecommand \@ifxundefined [1]{%
  \@ifx{#1\undefined}
@@ -1277,6 +1277,16 @@
   {journal} {\bibinfo  {journal} {J. Phys. A: Math. Theor.}\ }\textbf {\bibinfo
   {volume} {40}},\ \bibinfo {pages} {581} (\bibinfo {year} {2007})}\BibitemShut
   {NoStop}%
+\bibitem [{\citenamefont {Surj{\'a}n}\ and\ \citenamefont
+  {Szabados}(2000)}]{Surjan_2000}%
+  \BibitemOpen
+  \bibfield  {author} {\bibinfo {author} {\bibfnamefont {P.~R.}\ \bibnamefont
+  {Surj{\'a}n}}\ and\ \bibinfo {author} {\bibfnamefont {{\'A}.}~\bibnamefont
+  {Szabados}},\ }\href {\doibase 10.1063/1.481006} {\bibfield  {journal}
+  {\bibinfo  {journal} {The Journal of Chemical Physics}\ }\textbf {\bibinfo
+  {volume} {112}},\ \bibinfo {pages} {4438} (\bibinfo {year} {2000})},\ \Eprint
+  {http://arxiv.org/abs/https://doi.org/10.1063/1.481006}
+  {https://doi.org/10.1063/1.481006} \BibitemShut {NoStop}%
 \bibitem [{\citenamefont {Surj{\'a}n}\ \emph {et~al.}(2018)\citenamefont
   {Surj{\'a}n}, \citenamefont {Mih{\'a}lka},\ and\ \citenamefont
   {Szabados}}]{Surjan_2018}%

Manuscript/EPAWTFT.bib

@@ -1,146 +1,178 @@
 %% This BibTeX bibliography file was created using BibDesk.
 %% http://bibdesk.sourceforge.net/
 
-%% Created for Pierre-Francois Loos at 2020-11-24 09:46:03 +0100 
+%% Created for Pierre-Francois Loos at 2020-11-25 08:29:13 +0100 
 
 
 %% Saved with string encoding Unicode (UTF-8) 
 
+
+
+@article{Surjan_2000,
+	author = {Surj{\'a}n,P. R. and Szabados,{\'A}.},
+	date-added = {2020-11-25 08:29:04 +0100},
+	date-modified = {2020-11-25 08:29:13 +0100},
+	doi = {10.1063/1.481006},
+	eprint = {https://doi.org/10.1063/1.481006},
+	journal = {The Journal of Chemical Physics},
+	number = {10},
+	pages = {4438-4446},
+	title = {Optimized partitioning in perturbation theory: Comparison to related approaches},
+	url = {https://doi.org/10.1063/1.481006},
+	volume = {112},
+	year = {2000},
+	Bdsk-Url-1 = {https://doi.org/10.1063/1.481006}}
+
 @article{Tsuchimochi_2019,
-    author ={Takashi Tsuchimochi and Seiichiro L. Ten-no},
+	author = {Takashi Tsuchimochi and Seiichiro L. Ten-no},
-    journal={J. Chem. Theory Comput.},
+	doi = {10.1021/acs.jctc.9b00897},
-    volume ={15},
+	journal = {J. Chem. Theory Comput.},
-    pages  ={6688},
+	pages = {6688},
-    title  ={Second-order perturbation theory with spin-symmetry-projected Hartree--Fock},
+	title = {Second-order perturbation theory with spin-symmetry-projected Hartree--Fock},
-    year   ={2019},
+	volume = {15},
-    doi    ={10.1021/acs.jctc.9b00897},
+	year = {2019},
-}
+	Bdsk-Url-1 = {https://doi.org/10.1021/acs.jctc.9b00897}}
+
 @article{Tsuchimochi_2014,
-    author ={Takashi Tsuchimochi and Troy {Van Voorhis}},
+	author = {Takashi Tsuchimochi and Troy {Van Voorhis}},
-    journal={J. Chem. Phys.},
+	doi = {10.1063/1.4898804},
-    volume ={141},
+	journal = {J. Chem. Phys.},
-    pages  ={164117},
+	pages = {164117},
-    title  ={Extended {M\oller--Plesset} perturbation theory for dynamical and static correlations},
+	title = {Extended {M\oller--Plesset} perturbation theory for dynamical and static correlations},
-    year   ={2014},
+	volume = {141},
-    doi    ={10.1063/1.4898804},
+	year = {2014},
-}
+	Bdsk-Url-1 = {https://doi.org/10.1063/1.4898804}}
+
 @article{Knowles_1988a,
-    author ={Peter J. Knowles and Nicholas C. Handy},
+	author = {Peter J. Knowles and Nicholas C. Handy},
-    journal={J. Phys. Chem.},
+	doi = {10.1021/j100322a018},
-    volume ={92},
+	journal = {J. Phys. Chem.},
-    pages  ={3097},
+	pages = {3097},
-    title  ={Convergence of projected unrestricted {Hartree--Fock M\oller--Plesset} series},
+	title = {Convergence of projected unrestricted {Hartree--Fock M\oller--Plesset} series},
-    year   ={1988},
+	volume = {92},
-    doi    ={10.1021/j100322a018},
+	year = {1988},
-}
+	Bdsk-Url-1 = {https://doi.org/10.1021/j100322a018}}
+
 @article{Knowles_1988b,
-    author ={Peter J. Knowles and Nicholas C. Handy},
+	author = {Peter J. Knowles and Nicholas C. Handy},
-    journal={J. Chem. Phys.},
+	doi = {10.1063/1.454397},
-    volume ={88},
+	journal = {J. Chem. Phys.},
-    pages  ={6991},
+	pages = {6991},
-    title  ={Projected unrestricted {M\oller--Plesset} second-order energies},
+	title = {Projected unrestricted {M\oller--Plesset} second-order energies},
-    year   ={1988},
+	volume = {88},
-    doi    ={10.1063/1.454397},
+	year = {1988},
-}
+	Bdsk-Url-1 = {https://doi.org/10.1063/1.454397}}
+
 @article{Schlegel_1986,
-    author ={H. Bernhard Schlegel},
+	author = {H. Bernhard Schlegel},
-    journal={J. Chem. Phys.},
+	doi = {10.1063/1.450026},
-    volume ={84},
+	journal = {J. Chem. Phys.},
-    pages  ={4530},
+	pages = {4530},
-    title  ={Potential energy curves using unrestricted {M\oller--Plesset} perturbation theory with spin annihilation},
+	title = {Potential energy curves using unrestricted {M\oller--Plesset} perturbation theory with spin annihilation},
-    year   ={1986},
+	volume = {84},
-    doi    ={10.1063/1.450026},
+	year = {1986},
-}
+	Bdsk-Url-1 = {https://doi.org/10.1063/1.450026}}
+
 @article{Schlegel_1988,
-    author ={H. Bernhard Schlegel},
+	author = {H. Bernhard Schlegel},
-    journal={J. Phys. Chem.},
+	doi = {10.1021/j100322a014},
-    volume ={91},
+	journal = {J. Phys. Chem.},
-    pages  ={3075},
+	pages = {3075},
-    title  ={{M\oller--Plesset} perturbation theory with spin projection},
+	title = {{M\oller--Plesset} perturbation theory with spin projection},
-    year   ={1988},
+	volume = {91},
-    doi    ={10.1021/j100322a014},
+	year = {1988},
-}
+	Bdsk-Url-1 = {https://doi.org/10.1021/j100322a014}}
+
 @article{Gill_1988a,
-    author ={P. M. W. Gill and M. W. Wong and R. H. Nobes and L. Radom},
+	author = {P. M. W. Gill and M. W. Wong and R. H. Nobes and L. Radom},
-    journal={Chem. Phys. Lett.},
+	doi = {10.1016/0009-2614(88)80328-2},
-    volume ={148},
+	journal = {Chem. Phys. Lett.},
-    pages  ={541},
+	pages = {541},
-    title  ={How well can {RMP4} theory treat homolytic fragmentations?},
+	title = {How well can {RMP4} theory treat homolytic fragmentations?},
-    year   ={1988},
+	volume = {148},
-    doi    ={10.1016/0009-2614(88)80328-2},
+	year = {1988},
-} 
+	Bdsk-Url-1 = {https://doi.org/10.1016/0009-2614(88)80328-2}}
+
 @article{Nobes_1987,
-    author ={R. H. Nobes and J. A. Pople and L. Radom and N. C. Handy and P. J. Knowles},
+	author = {R. H. Nobes and J. A. Pople and L. Radom and N. C. Handy and P. J. Knowles},
-    journal={Chem. Phys. Lett.},
+	doi = {10.1016/0009-2614(87)80545-6},
-    volume ={138},
+	journal = {Chem. Phys. Lett.},
-    pages  ={481},
+	pages = {481},
-    title  ={Slow convergence of the {M\oller--Plesset} perturbation series: the dissociation energy of hydrogen cyanide and the electron affinity of the cyano radical},
+	title = {Slow convergence of the {M\oller--Plesset} perturbation series: the dissociation energy of hydrogen cyanide and the electron affinity of the cyano radical},
-    year   ={1987},
+	volume = {138},
-    doi    ={10.1016/0009-2614(87)80545-6},
+	year = {1987},
-}
+	Bdsk-Url-1 = {https://doi.org/10.1016/0009-2614(87)80545-6}}
+
 @article{Laidig_1987,
-    author ={William D. Laidig and Paul Saxe and Rodney J. Bartlett},
+	author = {William D. Laidig and Paul Saxe and Rodney J. Bartlett},
-    journal={J. Chem. Phys.},
+	doi = {10.1063/1.452291},
-    volume ={86},
+	journal = {J. Chem. Phys.},
-    pages  ={887},
+	pages = {887},
-    title  ={The description of \ce{N2} and \ce{F2} potential energy surfaces using multireference coupled cluster theory},
+	title = {The description of \ce{N2} and \ce{F2} potential energy surfaces using multireference coupled cluster theory},
-    year   ={1987},
+	volume = {86},
-    doi    ={10.1063/1.452291},
+	year = {1987},
-}
+	Bdsk-Url-1 = {https://doi.org/10.1063/1.452291}}
+
 @article{Bartlett_1975,
-    author ={R. J. Bartlett and D. M. Silver},
+	author = {R. J. Bartlett and D. M. Silver},
-    journal={J. Chem. Phys.},
+	doi = {10.1063/1.430878},
-    volume ={62},
+	journal = {J. Chem. Phys.},
-    pages  ={3258},
+	pages = {3258},
-    title  ={Many-body perturbation theory applied to electron pair correlation energies. I. Closed-shell first-row diatomic hydrides},
+	title = {Many-body perturbation theory applied to electron pair correlation energies. I. Closed-shell first-row diatomic hydrides},
-    year   ={1975},
+	volume = {62},
-    doi    ={10.1063/1.430878},
+	year = {1975},
-}
+	Bdsk-Url-1 = {https://doi.org/10.1063/1.430878}}
+
 @article{Krishnan_1980,
-    author ={R. Krishnan and M. J. Frisch and J. A. Pople},
+	author = {R. Krishnan and M. J. Frisch and J. A. Pople},
-    journal={J. Chem. Phys.},
+	doi = {10.1063/1.439657},
-    volume ={72},
+	journal = {J. Chem. Phys.},
-    pages  ={4244},
+	pages = {4244},
-    title  ={Contribution of triple substitutions to the electron correlation energy in fourth order perturbation theory},
+	title = {Contribution of triple substitutions to the electron correlation energy in fourth order perturbation theory},
-    year   ={1980},
+	volume = {72},
-    doi    ={10.1063/1.439657},
+	year = {1980},
-}
+	Bdsk-Url-1 = {https://doi.org/10.1063/1.439657}}
+
 @article{Pople_1978,
-    author ={J. A. Pople and R. Krishnan and H. B. Schlegel and J. S. Binkley},
+	author = {J. A. Pople and R. Krishnan and H. B. Schlegel and J. S. Binkley},
-    journal={Int. J. Quantum Chem.},
+	doi = {10.1002/qua.560140503},
-    volume ={14},
+	journal = {Int. J. Quantum Chem.},
-    pages  ={545},
+	pages = {545},
-    title  ={Electron correlation theories and their application to the study of simple reaction potential surfaces},
+	title = {Electron correlation theories and their application to the study of simple reaction potential surfaces},
-    year   ={1978},
+	volume = {14},
-    doi    ={10.1002/qua.560140503},
+	year = {1978},
-}
+	Bdsk-Url-1 = {https://doi.org/10.1002/qua.560140503}}
+
 @article{Pople_1976,
-    author ={John A. Pople and Stephen Binkley and Rolf Seeger},
+	author = {John A. Pople and Stephen Binkley and Rolf Seeger},
-    journal={Int. J. Quantum Chem. Symp.},
+	doi = {10.1002/qua.560100802},
-    volume ={10},
+	journal = {Int. J. Quantum Chem. Symp.},
-    pages  ={1},
+	pages = {1},
-    title  ={Theoretical models incorporating electron correlation},
+	title = {Theoretical models incorporating electron correlation},
-    year   ={1976},
+	volume = {10},
-    doi    ={10.1002/qua.560100802},
+	year = {1976},
-}
+	Bdsk-Url-1 = {https://doi.org/10.1002/qua.560100802}}
+
 @article{Knowles_1985,
-    author ={P. J. Knowles and K. Somasundram and N. C. Handy and K. Hirao},
+	author = {P. J. Knowles and K. Somasundram and N. C. Handy and K. Hirao},
-    journal={Chem. Phys. Lett.},
+	doi = {10.1016/0009-2614(85)85002-8},
-    volume ={113},
+	journal = {Chem. Phys. Lett.},
-    pages  ={8},
+	pages = {8},
-    title  ={The Calculation of High-Order Energies in the Many-Body Perturbation Theory Series},
+	title = {The Calculation of High-Order Energies in the Many-Body Perturbation Theory Series},
-    year   ={1985},
+	volume = {113},
-    doi    ={10.1016/0009-2614(85)85002-8},
+	year = {1985},
-}
+	Bdsk-Url-1 = {https://doi.org/10.1016/0009-2614(85)85002-8}}
+
 @article{Laidig_1985,
-    author ={William D. Laidig and George Fitzgerald and Rodney J. Bartlett},
+	author = {William D. Laidig and George Fitzgerald and Rodney J. Bartlett},
-    journal={Chem. Phys. Lett.},
+	doi = {10.1016/0009-2614(85)80934-9},
-    volume ={113},
+	journal = {Chem. Phys. Lett.},
-    pages  ={151},
+	pages = {151},
-    title  ={Is Fifth-Order MBPT Enough?},
+	title = {Is Fifth-Order MBPT Enough?},
-    year   ={1985},
+	volume = {113},
-    doi    ={10.1016/0009-2614(85)80934-9}
+	year = {1985},
-}
+	Bdsk-Url-1 = {https://doi.org/10.1016/0009-2614(85)80934-9}}
+
 @article{Hall_1951,
 	abstract = { An analysis of the `linear combination of atomic orbitals' approximation using the accurate molecular orbital equations shows that it does not lead to equations of the form usually assumed in the semi-empirical molecular orbital method. A new semi-empirical method is proposed, therefore, in terms of equivalent orbitals. The equations obtained, which do have the usual form, are applicable to a large class of molecules and do not involve the approximations that were thought necessary. In this method the ionization potentials are calculated by treating certain integrals as semi-empirical parameters. The value of these parameters is discussed in terms of the localization of equivalent orbitals and some approximate rules are suggested. As an illustration the ionization potentials of the paraffin series are considered and good agreement between the observed and calculated values is found. },
 	author = {Hall, G. G. and Lennard-Jones, John Edward},

Manuscript/EPAWTFT.tex

@@ -757,7 +757,7 @@ low-lying double excitation.
 This erroneous description of the double excitation amplitude has the same origin as the spin-contamination in the reference
 UHF wave function, creating the first direct link between spin-contamination and slow UMP convergence.\cite{Gill_1988}
 Lepetit \etal\ later analysed the difference between perturbation convergence using the unrestricted MP 
-and EN partitionings.\cite{Lepetit_1988}
+and EN partitionings. \cite{Lepetit_1988}
 They argued that the slow UMP convergence for stretched molecules arises from 
 (i) the fact that the MP denominator (see Eq.~\ref{eq:EMP2})
 tends to a constant value instead of vanishing, and (ii) the slow convergence of contributions from the 
@@ -1164,17 +1164,19 @@ We believe that $\alpha$ singularities are connected to states with non-negligib
 \section{Exploiting complex analysis}
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 
+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.
+
 %==========================================%
 \subsection{Pad\'e approximant}
 %==========================================%
 According to Wikipedia, \textit{``a Pad\'e approximant is the best approximation of a function by a rational function of given order''}.
-A $[d_A/d_B]$ Pad\'e approximant is defined as
+A $[d_A/d_B]$ Pad\'e approximant is defined as \cite{Pade}
 \begin{equation}
 	\label{eq:PadeApp}
-	E(\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}
+	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}
 \end{equation}
-with $b_0 = 1$.
+(with $b_0 = 1$), where the coefficients of the polynomials are determined by collecting terms according to power of $\lambda$.
-Pad\'e approximants are nice and they can model poles, but they cannot model functions with square-root branch points, and that sucks.
+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.
 
 %==========================================%
 \subsection{Quadratic approximant}
@@ -1212,7 +1214,7 @@ However, by ramping up high enough the degree of the polynomials, one is able to
 \subsection{Analytic continuation}
 %==========================================%
 
-Recently, Mih\'alka \textit{et al.} studied the partitioning effect on the convergence properties of Rayleigh-Schr\"odinger perturbation theory by considering the MP and the EN partitioning as well as an alternative partitioning. \cite{Mihalka_2017a} 
+Recently, Mih\'alka \textit{et al.} studied the partitioning effect on the convergence properties of Rayleigh-Schr\"odinger perturbation theory by considering the MP and the EN partitioning as well as an alternative partitioning \cite{Mihalka_2017a} (see also Ref.~\onlinecite{Surjan_2000}).
 Taking as an example (in particular) the water molecule at equilibrium and at stretched geometries, they could estimate the radius of convergence via a quadratic Pad\'e approximant and convert divergent perturbation expansions to convergent ones in some cases thanks to a judicious choice of the level shift parameter.
 In a subsequent study by the same group, \cite{Mihalka_2017b} they use analytic continuation techniques to resum divergent MP series \cite{Goodson_2011} taking again as an example the water molecule in a stretched geometry.
 In a nutshell, their idea consists in calculating the energy of the system for several values of $\lambda$ for which the MP series is rapidly convergent (\ie, for $\lambda < r_c$), and to extrapolate the final energy to the physical system at $\lambda = 1$ via a polynomial- or Pad\'e-based fit.