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adding bunch of refs and other stuff

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Manuscript/EPAWTFT.bbl
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@ -530,12 +530,12 @@
{journal} {\bibinfo {journal} {Phys. Rev. Lett.}\ }\textbf {\bibinfo
{volume} {108}},\ \bibinfo {pages} {083002} (\bibinfo {year}
{2012})}\BibitemShut {NoStop}%
\bibitem [{\citenamefont {Goodson}(2012)}]{Goodson_2012}%
\bibitem [{\citenamefont {Goodson}(2011)}]{Goodson_2011}%
\BibitemOpen
\bibfield {author} {\bibinfo {author} {\bibfnamefont {D.~Z.}\ \bibnamefont
{Goodson}},\ }\href {\doibase 10.1002/wcms.92} {\bibfield {journal}
{\bibinfo {journal} {{WIREs} Comput. Mol. Sci.}\ }\textbf {\bibinfo {volume}
{2}},\ \bibinfo {pages} {743} (\bibinfo {year} {2012})}\BibitemShut {NoStop}%
{2}},\ \bibinfo {pages} {743} (\bibinfo {year} {2011})}\BibitemShut {NoStop}%
\bibitem [{\citenamefont {Cejnar}\ \emph {et~al.}(2007)\citenamefont {Cejnar},
\citenamefont {Heinze},\ and\ \citenamefont {Macek}}]{Cejnar_2007}%
\BibitemOpen

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Manuscript/EPAWTFT.bib
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%% This BibTeX bibliography file was created using BibDesk.
%% http://bibdesk.sourceforge.net/
%% Created for Pierre-Francois Loos at 2020-11-20 09:33:35 +0100
%% Created for Pierre-Francois Loos at 2020-11-20 09:53:53 +0100
%% Saved with string encoding Unicode (UTF-8)
@ -762,16 +762,16 @@
year = {2004},
Bdsk-Url-1 = {https://doi.org/10.1016/S0065-3276(04)47011-7}}
@article{Goodson_2012,
@article{Goodson_2011,
author = {Goodson, David Z.},
date-modified = {2020-08-22 22:13:28 +0200},
date-modified = {2020-11-20 09:53:53 +0100},
doi = {10.1002/wcms.92},
journal = {{WIREs} Comput. Mol. Sci.},
number = {5},
pages = {743--761},
title = {Resummation methods},
volume = {2},
year = {2012},
year = {2011},
Bdsk-Url-1 = {https://doi.org/10.1002/wcms.92}}
@article{Katz_1962,

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Manuscript/EPAWTFT.tex
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@ -349,7 +349,7 @@ of the reference and perturbation Hamiltonians in Eq.~\eqref{eq:SchrEq-PT}.\cite
% LAMBDA IN THE COMPLEX PLANE
From complex-analysis, \cite{BenderBook} the radius of convergence for the energy can be obtained by looking for the
singularities of $E(\lambda)$ in the complex $\lambda$ plane.
This property arises from the following theorem: \cite{Goodson_2012}
This property arises from the following theorem: \cite{Goodson_2011}
\begin{quote}
\it
``The Taylor series about a point $z_0$ of a function over the complex $z$ plane will converge at a value $z_1$
@ -372,7 +372,7 @@ that complex singularities are essential to fully understand the series converge
The radius of convergence for the perturbation series is therefore dictated by the magnitude $\abs{\lambda_0}$ of the
singularity in $E(\lambda)$ that is closest to the origin.
Like the exact system in Section~\ref{sec:example}, the perturbation energy $E(\lambda)$ represents
Like the exact system in Sec.~\ref{sec:example}, the perturbation energy $E(\lambda)$ represents
a ``one-to-many'' function with the output elements representing an approximation to both the ground and excited states.
The most common singularities on $E(\lambda)$ therefore correspond to non-analytic EPs in the complex
$\lambda$ plane where two states become degenerate.
@ -881,7 +881,7 @@ Even if there were still shaded areas in their analysis and that their classific
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}
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 taking again as an example the water molecule in a stretched geometry.
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.
However, the choice of the functional form of the fit remains a subtle task.
This technique was first generalised by using complex scaling parameters and applying analytic continuation by solving the Laplace equation, \cite{Surjan_2018} and then further improved thanks to Cauchy's integral formula \cite{Mihalka_2019}
@ -898,15 +898,17 @@ The authors illustrate this protocol on the dissociation curve of \ce{LiH} and t
\subsection{Insights from a two-state model}
%==========================================%
In the late 90's, Olsen \textit{et al.}~discovered an even more preoccupying behavior of the MP series. \cite{Olsen_1996} They showed that the series could be divergent even in systems that they considered as well understood like \ce{Ne} and \ce{HF}. \cite{Olsen_1996, Christiansen_1996} Cremer and He had already studied these two systems and classified them as \textit{class B} systems. However, the analysis of Olsen and coworkers was performed in larger basis sets containing diffuse functions. In these basis sets, they found that the series become divergent at (very) high order.
In the late 90's, Olsen \textit{et al.}~discovered an even more preoccupying behavior of the MP series. \cite{Olsen_1996}
They showed that the series could be divergent even in systems that they considered as well understood like \ce{Ne} and the \ce{HF} molecule. \cite{Olsen_1996, Christiansen_1996} Cremer and He had already studied these two systems and classified them as \textit{class B} systems. However, the analysis of Olsen and coworkers was performed in larger basis sets containing diffuse functions.
In these basis sets, they found that the series become divergent at (very) high order.
The discovery of this divergent behaviour is worrying as in order to get meaningful and accurate energies, calculations must be performed in large basis sets (as close as possible from the complete basis set limit).
Including diffuse functions is particularly important in the case of anions and/or Rydberg excited states where the wave function is much more diffuse than the ground-state one. As a consequence, they investigated further the causes of these divergences as well as the reasons of the different types of convergence.
To do so, they analysed the relation between the dominant singularity (\ie, the closest singularity to the origin) and the convergence behaviour of the series. \cite{Olsen_2000} Their analysis is based on Darboux's theorem:
To do so, they analysed the relation between the dominant singularity (\ie, the closest singularity to the origin) and the convergence behaviour of the series. \cite{Olsen_2000} Their analysis is based on Darboux's theorem: \cite{Goodson_2011}
\begin{quote}
\textit{``In the limit of large order, the series coefficients become equivalent to the Taylor series coefficients of the singularity closest to the origin. Following the result of this theorem, the convergence patterns of the MP series can be explained by looking at the dominant singularity.''}
\textit{``In the limit of large order, the series coefficients become equivalent to the Taylor series coefficients of the singularity closest to the origin. ''}
\end{quote}
Following the result of this theorem, the convergence patterns of the MP series can be explained by looking at the dominant singularity.
A singularity in the unit circle is designated as an intruder state, more precisely as a front-door (respectively back-door) intruder state if the real part of the singularity is positive (respectively negative).
Their method consists in performing a scan of the real axis to detect the avoided crossing responsible for the pair of dominant singularities.
Then, by modelling this avoided crossing via a two-state Hamiltonian one can get an approximation of the dominant conjugate pair of singularities by finding the EPs of the following $2\times2$ matrix