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60
Manuscript/EPAWTFT.bbl
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@ -453,6 +453,25 @@
{\bibfield {journal} {\bibinfo {journal} {J. Chem. Phys.}\ }\textbf
{\bibinfo {volume} {150}},\ \bibinfo {pages} {031101} (\bibinfo {year}
{2019})}\BibitemShut {NoStop}%
\bibitem [{\citenamefont {Moiseyev}(1998)}]{Moiseyev_1998}%
\BibitemOpen
\bibfield {author} {\bibinfo {author} {\bibfnamefont {N.}~\bibnamefont
{Moiseyev}},\ }\href@noop {} {\bibfield {journal} {\bibinfo {journal}
{Phys. Rep.}\ }\textbf {\bibinfo {volume} {302}},\ \bibinfo {pages} {211}
(\bibinfo {year} {1998})}\BibitemShut {NoStop}%
\bibitem [{\citenamefont {Riss}\ and\ \citenamefont {Meyer}(1993)}]{Riss_1993}%
\BibitemOpen
\bibfield {author} {\bibinfo {author} {\bibfnamefont {U.~V.}\ \bibnamefont
{Riss}}\ and\ \bibinfo {author} {\bibfnamefont {H.-D.}\ \bibnamefont
{Meyer}},\ }\href@noop {} {\bibfield {journal} {\bibinfo {journal} {J.
Phys. B}\ }\textbf {\bibinfo {volume} {26}},\ \bibinfo {pages} {4503}
(\bibinfo {year} {1993})}\BibitemShut {NoStop}%
\bibitem [{\citenamefont {Ernzerhof}(2006)}]{Ernzerhof_2006}%
\BibitemOpen
\bibfield {author} {\bibinfo {author} {\bibfnamefont {M.}~\bibnamefont
{Ernzerhof}},\ }\href {\doibase 10.1063/1.2348880} {\bibfield {journal}
{\bibinfo {journal} {J. Chem. Phys.}\ }\textbf {\bibinfo {volume} {125}},\
\bibinfo {pages} {124104} (\bibinfo {year} {2006})}\BibitemShut {NoStop}%
\bibitem [{\citenamefont {Carrascal}\ \emph {et~al.}(2015)\citenamefont
{Carrascal}, \citenamefont {Ferrer}, \citenamefont {Smith},\ and\
\citenamefont {Burke}}]{Carrascal_2015}%
@ -481,25 +500,6 @@
{Wigner}},\ }\href {\doibase 10.1103/PhysRev.46.1002} {\bibfield {journal}
{\bibinfo {journal} {Phys. Rev.}\ }\textbf {\bibinfo {volume} {46}},\
\bibinfo {pages} {1002} (\bibinfo {year} {1934})}\BibitemShut {NoStop}%
\bibitem [{\citenamefont {Moiseyev}(1998)}]{Moiseyev_1998}%
\BibitemOpen
\bibfield {author} {\bibinfo {author} {\bibfnamefont {N.}~\bibnamefont
{Moiseyev}},\ }\href@noop {} {\bibfield {journal} {\bibinfo {journal}
{Phys. Rep.}\ }\textbf {\bibinfo {volume} {302}},\ \bibinfo {pages} {211}
(\bibinfo {year} {1998})}\BibitemShut {NoStop}%
\bibitem [{\citenamefont {Riss}\ and\ \citenamefont {Meyer}(1993)}]{Riss_1993}%
\BibitemOpen
\bibfield {author} {\bibinfo {author} {\bibfnamefont {U.~V.}\ \bibnamefont
{Riss}}\ and\ \bibinfo {author} {\bibfnamefont {H.-D.}\ \bibnamefont
{Meyer}},\ }\href@noop {} {\bibfield {journal} {\bibinfo {journal} {J.
Phys. B}\ }\textbf {\bibinfo {volume} {26}},\ \bibinfo {pages} {4503}
(\bibinfo {year} {1993})}\BibitemShut {NoStop}%
\bibitem [{\citenamefont {Ernzerhof}(2006)}]{Ernzerhof_2006}%
\BibitemOpen
\bibfield {author} {\bibinfo {author} {\bibfnamefont {M.}~\bibnamefont
{Ernzerhof}},\ }\href {\doibase 10.1063/1.2348880} {\bibfield {journal}
{\bibinfo {journal} {J. Chem. Phys.}\ }\textbf {\bibinfo {volume} {125}},\
\bibinfo {pages} {124104} (\bibinfo {year} {2006})}\BibitemShut {NoStop}%
\bibitem [{\citenamefont {Taut}(1993)}]{Taut_1993}%
\BibitemOpen
\bibfield {author} {\bibinfo {author} {\bibfnamefont {M.}~\bibnamefont
@ -754,17 +754,6 @@
}\href {\doibase 10.1021/jp952815d} {\bibfield {journal} {\bibinfo
{journal} {J. Phys. Chem.}\ }\textbf {\bibinfo {volume} {100}},\ \bibinfo
{pages} {6173} (\bibinfo {year} {1996})}\BibitemShut {NoStop}%
\bibitem [{\citenamefont {Surj{\'a}n}\ \emph {et~al.}(2018)\citenamefont
{Surj{\'a}n}, \citenamefont {Mih{\'a}lka},\ and\ \citenamefont
{Szabados}}]{Surjan_2018}%
\BibitemOpen
\bibfield {author} {\bibinfo {author} {\bibfnamefont {P.~R.}\ \bibnamefont
{Surj{\'a}n}}, \bibinfo {author} {\bibfnamefont {Z.~{\'E}.}\ \bibnamefont
{Mih{\'a}lka}}, \ and\ \bibinfo {author} {\bibfnamefont {{\'A}.}~\bibnamefont
{Szabados}},\ }\href {\doibase 10.1007/s00214-018-2372-3} {\bibfield
{journal} {\bibinfo {journal} {Theor. Chem. Acc.}\ }\textbf {\bibinfo
{volume} {137}},\ \bibinfo {pages} {149} (\bibinfo {year}
{2018})}\BibitemShut {NoStop}%
\bibitem [{\citenamefont {Christiansen}\ \emph {et~al.}(1996)\citenamefont
{Christiansen}, \citenamefont {Olsen}, \citenamefont {J{\o}rgensen},
\citenamefont {Koch},\ and\ \citenamefont {Malmqvist}}]{Christiansen_1996}%
@ -1053,4 +1042,15 @@
{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}\ \emph {et~al.}(2018)\citenamefont
{Surj{\'a}n}, \citenamefont {Mih{\'a}lka},\ and\ \citenamefont
{Szabados}}]{Surjan_2018}%
\BibitemOpen
\bibfield {author} {\bibinfo {author} {\bibfnamefont {P.~R.}\ \bibnamefont
{Surj{\'a}n}}, \bibinfo {author} {\bibfnamefont {Z.~{\'E}.}\ \bibnamefont
{Mih{\'a}lka}}, \ and\ \bibinfo {author} {\bibfnamefont {{\'A}.}~\bibnamefont
{Szabados}},\ }\href {\doibase 10.1007/s00214-018-2372-3} {\bibfield
{journal} {\bibinfo {journal} {Theor. Chem. Acc.}\ }\textbf {\bibinfo
{volume} {137}},\ \bibinfo {pages} {149} (\bibinfo {year}
{2018})}\BibitemShut {NoStop}%
\end{thebibliography}%

191
Manuscript/EPAWTFT.tex
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@ -146,15 +146,13 @@ In particular, we highlight the seminal work of several research groups on the c
\maketitle
\tableofcontents
%%%%%%%%%%%%%%%%%%%%%%%
\section{Introduction}
\section{Background}
\label{sec:intro}
%%%%%%%%%%%%%%%%%%%%%%%
%======================%
\subsection{Background}
%=======================%
Due to the ubiquitous influence of processes involving electronic states in physics, chemistry, and biology, their faithful description from first principles has been one of the grand challenges faced by theoretical chemists since the dawn of computational chemistry.
Accurately predicting ground- and excited-state energies (hence excitation energies) is particularly valuable in this context, and it has concentrated most of the efforts within the community.
An armada of theoretical and computational methods have been developed to this end, each of them being plagued by its own flaws.
@ -184,8 +182,17 @@ In contrast, encircling Hermitian degeneracies at conical intersections only int
More dramatically, whilst eigenvectors remain orthogonal at conical intersections, at non-Hermitian EPs the eigenvectors themselves become equivalent, resulting in a \textit{self-orthogonal} state. \cite{MoiseyevBook}
More importantly here, although EPs usually lie off the real axis, these singular points are intimately related to the convergence properties of perturbative methods and avoided crossing on the real axis are indicative of singularities in the complex plane. \cite{BenderBook,Olsen_1996,Olsen_2000,Olsen_2019,Mihalka_2017a,Mihalka_2017b,Mihalka_2019}
\titou{The use of non-Hermitian Hamiltonians in quantum chemistry has a long history; these Hamiltonians have been used extensively as a method for describing metastable resonance phenomena. \cite{MoiseyevBook}
Through a complex-scaling of the electronic or atomic coordinates,\cite{Moiseyev_1998} or by introducing a complex absorbing potential,\cite{Riss_1993,Ernzerhof_2006,Benda_2018} outgoing resonance states are transformed into square-integrable wave functions that allow the energy and lifetime of the resonance to be computed.
We refer the interested reader to the excellent book of Moiseyev for a general overview. \cite{MoiseyevBook}}
%%%%%%%%%%%%%%%%%%%%%%%
\section{Exceptional points in electronic structure}
\label{sec:EPs}
%%%%%%%%%%%%%%%%%%%%%%%
%===================================%
\subsection{Illustrative Example}
\subsection{Exceptional points in the Hubbard dimer}
\label{sec:example}
%===================================%
@ -275,14 +282,6 @@ such that $E_{\pm}(2\pi) = E_{\mp}(0)$ and $E_{\pm}(4\pi) = E_{\pm}(0)$.
As a result, completely encircling an EP leads to the interconversion of the two interacting states, while a second complete rotation returns the two states to their original energies.
Additionally, the wave functions pick up a geometric phase in the process, and four complete loops are required to recover their starting forms.\cite{MoiseyevBook}
\titou{The use of non-Hermitian Hamiltonians in quantum chemistry has a long history; these Hamiltonians have been used extensively as a method for describing metastable resonance phenomena. \cite{MoiseyevBook}
Through a complex-scaling of the electronic or atomic coordinates,\cite{Moiseyev_1998} or by introducing a complex absorbing potential,\cite{Riss_1993,Ernzerhof_2006,Benda_2018} outgoing resonance states are transformed into square-integrable wave functions that allow the energy and lifetime of the resonance to be computed.
We refer the interested reader to the excellent book of Moiseyev for a general overview. \cite{MoiseyevBook}}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Perturbation theory}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%============================================================%
\subsection{Rayleigh-Schr\"odinger perturbation theory}
%============================================================%
@ -397,10 +396,9 @@ These degeneracies can be conical intersections between two states with differen
for real values of $\lambda$,\cite{Yarkony_1996} or EPs between two states with the
same symmetry for complex values of $\lambda$.
%============================================================%
\subsection{The Hartree-Fock Hamiltonian}
%============================================================%
%===========================================%
\subsection{Hartree-Fock theory}
%===========================================%
% SUMMARY OF HF
In the Hartree-Fock (HF) approximation, the many-electron wave function is approximated as a single Slater determinant $\Psi^{\text{HF}}(\vb{x}_1,\ldots,\vb{x}_N)$, where $\vb{x} = (\sigma,\vb{r})$ is a composite vector gathering spin and spatial coordinates.
@ -460,14 +458,9 @@ such as the dissociation of the hydrogen dimer.\cite{Coulson_1949}
However, by allowing different orbitals for different spins, the UHF is no longer required to be an eigenfunction of
the total spin $\hat{\mathcal{S}}^2$ operator, leading to ``spin-contamination'' in the wave function.
%
%The spatial part of the RHF wave function is then
%\begin{equation}\label{eq:RHF_WF}
% \Psi_{\text{RHF}}(\theta_1,\theta_2) = Y_0(\theta_1) Y_0(\theta_2),
%\end{equation}
%where $\theta_i$ is the polar angle of the $i$th electron and $Y_{\ell}(\theta)$ is a zonal spherical harmonic.
%Because $Y_0(\theta) = 1/\sqrt{4\pi}$, it is clear that the RHF wave function yields a uniform one-electron density.
%
%================================================================%
\subsection{The Hartree-Fock approximation in the Hubbard dimer}
%================================================================%
%%% FIG 2 (?) %%%
% HF energies as a function of U/t
@ -612,8 +605,14 @@ role of \textit{quasi}-EPs in determining the behaviour of the HF approximation.
%It perfectly illustrates the deeper topology of electronic states revealed using a complex-scaled electron-electron interaction.
%Through the introduction of non-Hermiticity, we have provided a more general framework in which the complex and diverse characteristics of multiple solutions can be explored and understood.}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{M{\o}ller-Plesset perturbation theory}
\label{sec:MP}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%=====================================================%
\subsection{M{\o}ller-Plesset perturbation theory}
\subsection{Basics}
%=====================================================%
In electronic structure, the HF Hamiltonian \eqref{eq:HFHamiltonian} is often used as the zeroth-order Hamiltonian
@ -622,14 +621,11 @@ This approach can recover a large proportion of the electron correlation energy,
and provides the foundation for numerous post-HF approximations.
With the MP partitioning, the parametrised perturbation Hamiltonian becomes
\begin{multline}\label{eq:MPHamiltonian}
\hH(\lambda) =
\sum_{i}^{N} \Bigg(
-\frac{\grad_i^2}{2}
- \sum_{A}^{M} \frac{Z_A}{\abs{\vb{r}_i-\vb{R}_A}}
\hH(\lambda) =
\sum_{i}^{N} \qty[ - \frac{\grad_i^2}{2} - \sum_{A}^{M} \frac{Z_A}{\abs{\vb{r}_i-\vb{R}_A}} ]
\\
+ (1-\lambda) v^{\text{HF}}(\vb{x}_i)
+ \lambda\sum_{i<j}^{N}\frac{1}{\abs{\vb{r}_i-\vb{r}_j}}
\Bigg).
+ (1-\lambda) \sum_{i}^{N} v^{\text{HF}}(\vb{x}_i)
+ \lambda\sum_{i<j}^{N}\frac{1}{\abs{\vb{r}_i-\vb{r}_j}}.
\end{multline}
Any set of orbitals can be used to define the HF Hamiltonian, although usually either the RHF or UHF orbitals are chosen to
define the RMP or UMP series respectively.
@ -646,7 +642,7 @@ The second-order MP2 energy is given by
E_{\text{MP2}} = \frac{1}{4} \sum_{ij} \sum_{ab} \frac{\abs{\mel{ij}{}{ab}}^2}{\epsilon_i + \epsilon_j - \epsilon_a - \epsilon_b},
\end{equation}
where $\mel{pq}{}{rs} = \braket{pq}{rs} - \braket{pq}{sr}$ are the anti-symmetrised two-electron integrals
in the molecular spin-obital basis\cite{Gill_1994}
in the molecular spin-orbital basis\cite{Gill_1994}
\begin{equation}
\braket{pq}{rs}
= \iint \dd\vb{x}_1\dd\vb{x}_2
@ -675,6 +671,47 @@ unrestricted reference orbitals.
%For small systems, one can access the whole terms of the series using full configuration interaction (FCI).
%If the Hamiltonian $H(\lambda)$ is diagonalized in the FCI space, one gets the exact energies (in this finite Hilbert space) and the Taylor expansion with respect to $\lambda$ allows to access the MP perturbation series at any order.
Obviously, although practically convenient for electronic structure calculations, the MP partitioning is not the only possibility, and alternative partitioning have been proposed in the literature:
i) the Epstein-Nesbet (EN) partitioning which consists in taking the diagonal elements of $\hH$ as the zeroth-order Hamiltonian. \cite{Nesbet_1955,Epstein_1926}
Hence, the off-diagonal elements of $\hH$ are the perturbation operator,
ii) the weak correlation partitioning in which the one-electron part is consider as the unperturbed Hamiltonian $\hH^{(0)}$ and the two-electron part is the perturbation operator $\hV$, and
iii) the strong coupling partitioning where the two operators are inverted compared to the weak correlation partitioning. \cite{Seidl_2018}
%=====================================================%
\subsection{Behavior of the M{\o}ller-Plesset series in molecular systems}
%=====================================================%
When one relies on MP perturbation theory (and more generally on any perturbative partitioning), it would be reasonable to ask for a systematic improvement of the energy with respect to the perturbative order, \ie, one would expect that the more terms of the perturbative series one can compute, the closer the result from the exact energy.
%In other words, each time a higher-order term is computed, one would like to obtained an overall result closer to the exact energy.
In other words, one would like a monotonic convergence of the MP series. Assuming this, the only limiting process to get the exact correlation energy (in a finite basis set) would be our ability to compute the terms of this perturbation series.
Unfortunately this is not as easy as one might think because i) the terms of the perturbative series become rapidly computationally cumbersome, and ii) erratic behavior of the perturbative coefficients are not uncommon. For example, in the late 80's, Gill and Radom reported deceptive and slow convergences in stretched systems \cite{Gill_1986, Gill_1988} (see also Refs.~\onlinecite{Handy_1985, Lepetit_1988}).
In Ref.~\onlinecite{Gill_1986}, the authors showed that the RMP series is convergent, yet oscillatory which is far from being convenient if one is only able to compute the first few terms of the expansion (for example, in the case of \ce{He2^2+}, RMP5 is worse than RMP4).
On the other hand, the UMP series is monotonically convergent (except for the first few orders) but very slowly.
Thus, one cannot practically use it for systems where only the first terms can be computed.
When a bond is stretched, in most cases the exact wave function becomes more and more of multi-reference nature.
Yet the HF wave function is restricted to be a single Slater determinant.
It is then inappropriate to model (even qualitatively) stretched systems. Nevertheless, the HF wave function can undergo symmetry breaking to lower its energy by sacrificing one of the symmetry of the exact wave function in the process (see for example the case of \ce{H2} in Ref.~\onlinecite{SzaboBook}).
One could then potentially claim that the RMP series exhibits deceptive convergence properties as the RHF Slater determinant is a poor approximation of the exact wave function for stretched system. However, even in the unrestricted formalism which clearly represents a better description of a stretched system, the UMP series does not have the smooth and rapidly convergent behaviour that one would wish for.
In the unrestricted framework the singlet ground state wave function is allowed to mix with triplet wave functions, leading to the so-called spin contamination issue. Gill \textit{et al.}~highlighted the link between slow convergence of the UMP series and spin contamination for \ce{H2} in a minimal basis. \cite{Gill_1988}
Handy and coworkers reported the same behaviour of the series (oscillatory and slowly monotonically convergent) in stretched \ce{H2O} and \ce{NH2}. \cite{Handy_1985} Lepetit \textit{et al.}~analysed the difference between the MP and EN partitioning for the UHF reference. \cite{Lepetit_1988}
They concluded that the slow convergence is due to the coupling of the singly- and doubly-excited configurations.
Cremer and He analysed 29 atomic and molecular systems at the FCI level \cite{Cremer_1996} and grouped them in two classes: i) the \textit{class A} systems where one observes a monotonic convergence to the FCI energy, and ii) the \textit{class B} systems for which convergence is erratic after initial oscillations. Their system set contains stretched molecules as well as molecules at their equilibrium geometry for various basis sets. They highlighted that \cite{Cremer_1996}
\textit{``Class A systems are characterised by electronic structures with well-separated electron pairs while class B systems are characterized by electronic structures with electron clustering in one or more regions.''}
Moreover, they analyzed the contribution of the triple (T) excitations to the MP4, MP5 and MP6 energies next to the single, double and quadruple (SDQ) excitations contribution.
They showed that class A systems have very little contribution from the triple excitations and that most of the correlation energy is due to pair correlation. On the other hand, class B systems have an important contribution from the triple excitations which alternates in sign resulting in an oscillation of the total correlation energy.
This observation on the contribution to the MPn energy corroborates the electronic structure discussed above.
As one can only compute the first terms of the MP series, a smart way of getting more accurate results is to use extrapolation formula, \ie, estimating the limit of the series with only few terms.
Cremer and He proved that using specific extrapolation formulas of the MP series for class A and class B systems improves the precision of the results compared to the formula used without resorting to classes. The mean absolute deviation taking the FCI correlation energies as reference is $0.3$ millihartree with the class-specific formula whereas the deviation increases to 12 millihartree using the general formula.
Even if there were still shaded areas in their analysis and that their classification was incomplete, the work of Cremer and He clearly evidenced that understanding the origin of the different modes of convergence could potentially lead to a more rationalised use of MP perturbation theory and, hence, to more accurate correlation energy estimates.
%==========================================%
\subsection{Behavior of the M{\o}ller-Plesset series in the Hubbard dimer}
%==========================================%
To illustrate the behaviour of the RMP and UMP series, we can again consider the Hubbard dimer.
Using the ground-state RHF reference orbitals leads to the RMP Hamiltonian
\begin{widetext}
@ -840,62 +877,9 @@ spin-contamination from the wave function.
\label{fig:UMP}}
\end{figure*}
Obviously, although practically convenient for electronic structure calculations, the MP partitioning is not the only possibility, and alternative partitioning have been proposed in the literature:
i) the Epstein-Nesbet (EN) partitioning which consists in taking the diagonal elements of $\hH$ as the zeroth-order Hamiltonian. \cite{Nesbet_1955,Epstein_1926}
Hence, the off-diagonal elements of $\hH$ are the perturbation operator,
ii) the weak correlation partitioning in which the one-electron part is consider as the unperturbed Hamiltonian $\hH^{(0)}$ and the two-electron part is the perturbation operator $\hV$, and
iii) the strong coupling partitioning where the two operators are inverted compared to the weak correlation partitioning. \cite{Seidl_2018}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Historical overview}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%=====================================================%
\subsection{Behavior of the M{\o}ller-Plesset series}
%=====================================================%
When one relies on MP perturbation theory (and more generally on any perturbative partitioning), it would be reasonable to ask for a systematic improvement of the energy with respect to the perturbative order, \ie, one would expect that the more terms of the perturbative series one can compute, the closer the result from the exact energy.
%In other words, each time a higher-order term is computed, one would like to obtained an overall result closer to the exact energy.
In other words, one would like a monotonic convergence of the MP series. Assuming this, the only limiting process to get the exact correlation energy (in a finite basis set) would be our ability to compute the terms of this perturbation series.
Unfortunately this is not as easy as one might think because i) the terms of the perturbative series become rapidly computationally cumbersome, and ii) erratic behavior of the perturbative coefficients are not uncommon. For example, in the late 80's, Gill and Radom reported deceptive and slow convergences in stretched systems \cite{Gill_1986, Gill_1988} (see also Refs.~\onlinecite{Handy_1985, Lepetit_1988}).
In Ref.~\onlinecite{Gill_1986}, the authors showed that the RMP series is convergent, yet oscillatory which is far from being convenient if one is only able to compute the first few terms of the expansion (for example, in the case of \ce{He2^2+}, RMP5 is worse than RMP4).
On the other hand, the UMP series is monotonically convergent (except for the first few orders) but very slowly.
Thus, one cannot practically use it for systems where only the first terms can be computed.
When a bond is stretched, in most cases the exact wave function becomes more and more of multi-reference nature.
Yet the HF wave function is restricted to be a single Slater determinant.
It is then inappropriate to model (even qualitatively) stretched systems. Nevertheless, the HF wave function can undergo symmetry breaking to lower its energy by sacrificing one of the symmetry of the exact wave function in the process (see for example the case of \ce{H2} in Ref.~\onlinecite{SzaboBook}).
One could then potentially claim that the RMP series exhibits deceptive convergence properties as the RHF Slater determinant is a poor approximation of the exact wave function for stretched system. However, even in the unrestricted formalism which clearly represents a better description of a stretched system, the UMP series does not have the smooth and rapidly convergent behaviour that one would wish for.
In the unrestricted framework the singlet ground state wave function is allowed to mix with triplet wave functions, leading to the so-called spin contamination issue. Gill \textit{et al.}~highlighted the link between slow convergence of the UMP series and spin contamination for \ce{H2} in a minimal basis. \cite{Gill_1988}
Handy and coworkers reported the same behaviour of the series (oscillatory and slowly monotonically convergent) in stretched \ce{H2O} and \ce{NH2}. \cite{Handy_1985} Lepetit \textit{et al.}~analysed the difference between the MP and EN partitioning for the UHF reference. \cite{Lepetit_1988}
They concluded that the slow convergence is due to the coupling of the singly- and doubly-excited configurations.
Cremer and He analysed 29 atomic and molecular systems at the FCI level \cite{Cremer_1996} and grouped them in two classes: i) the \textit{class A} systems where one observes a monotonic convergence to the FCI energy, and ii) the \textit{class B} systems for which convergence is erratic after initial oscillations. Their system set contains stretched molecules as well as molecules at their equilibrium geometry for various basis sets. They highlighted that \cite{Cremer_1996}
\textit{``Class A systems are characterised by electronic structures with well-separated electron pairs while class B systems are characterized by electronic structures with electron clustering in one or more regions.''}
Moreover, they analyzed the contribution of the triple (T) excitations to the MP4, MP5 and MP6 energies next to the single, double and quadruple (SDQ) excitations contribution.
They showed that class A systems have very little contribution from the triple excitations and that most of the correlation energy is due to pair correlation. On the other hand, class B systems have an important contribution from the triple excitations which alternates in sign resulting in an oscillation of the total correlation energy.
This observation on the contribution to the MPn energy corroborates the electronic structure discussed above.
As one can only compute the first terms of the MP series, a smart way of getting more accurate results is to use extrapolation formula, \ie, estimating the limit of the series with only few terms.
Cremer and He proved that using specific extrapolation formulas of the MP series for class A and class B systems improves the precision of the results compared to the formula used without resorting to classes. The mean absolute deviation taking the FCI correlation energies as reference is $0.3$ millihartree with the class-specific formula whereas the deviation increases to 12 millihartree using the general formula.
Even if there were still shaded areas in their analysis and that their classification was incomplete, the work of Cremer and He clearly evidenced that understanding the origin of the different modes of convergence could potentially lead to a more rationalised use of MP perturbation theory and, hence, to more accurate correlation energy estimates.
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 \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}
\begin{equation}
\label{eq:Cauchy}
\frac{1}{2\pi i} \oint_{\gamma} \frac{E(\lambda)}{\lambda - a} = E(a),
\end{equation}
which states that the value of the energy can be computed at $\lambda=a$ inside the complex contour $\gamma$ only by the knowledge of its values on the same contour.
Their method consists in refining self-consistently the values of $E(\lambda)$ computed on a contour going through the physical point at $\lambda = 1$ and encloses points of the ``trusted'' region (where the MP series is convergent). The shape of this contour is arbitrary but no singularities are allowed inside the contour to ensure $E(\lambda)$ is analytic.
When the values of $E(\lambda)$ on the so-called contour are converged, Cauchy's integrals formula \eqref{eq:Cauchy} is invoked to compute the values at $E(\lambda=1)$ which corresponds to the final estimate of the FCI energy.
The authors illustrate this protocol on the dissociation curve of \ce{LiH} and the stretched water molecule showing encouraging results. \cite{Mihalka_2019}
%==========================================%
\subsection{Insights from a two-state model}
\subsection{Further 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}
@ -1030,6 +1014,35 @@ The avoided crossings generating $\alpha$ singularities generally involve the gr
Those excited states have a non-negligible contribution to the exact FCI solution because they have (usually) the same spatial and spin symmetry as the ground state.
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.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Exploiting complex analysis}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%==========================================%
\subsection{Pad\'e approximant}
%==========================================%
%==========================================%
\subsection{Analytic continuation}
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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 \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}
\begin{equation}
\label{eq:Cauchy}
\frac{1}{2\pi i} \oint_{\gamma} \frac{E(\lambda)}{\lambda - a} = E(a),
\end{equation}
which states that the value of the energy can be computed at $\lambda=a$ inside the complex contour $\gamma$ only by the knowledge of its values on the same contour.
Their method consists in refining self-consistently the values of $E(\lambda)$ computed on a contour going through the physical point at $\lambda = 1$ and encloses points of the ``trusted'' region (where the MP series is convergent). The shape of this contour is arbitrary but no singularities are allowed inside the contour to ensure $E(\lambda)$ is analytic.
When the values of $E(\lambda)$ on the so-called contour are converged, Cauchy's integrals formula \eqref{eq:Cauchy} is invoked to compute the values at $E(\lambda=1)$ which corresponds to the final estimate of the FCI energy.
The authors illustrate this protocol on the dissociation curve of \ce{LiH} and the stretched water molecule showing encouraging results. \cite{Mihalka_2019}
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\section{Conclusion}
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