starting working on response letter
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%Control: page (0) single
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\begin{thebibliography}{179}%
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\begin{thebibliography}{183}%
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\makeatletter
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\providecommand \@ifxundefined [1]{%
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\@ifx{#1\undefined}
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@ -234,6 +234,15 @@
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{journal} {\bibinfo {journal} {J. Chem. Phys.}\ }\textbf {\bibinfo {volume}
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{89}},\ \bibinfo {pages} {998} (\bibinfo {year} {1988})}\BibitemShut
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{NoStop}%
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\bibitem [{\citenamefont {Malrieu}\ and\ \citenamefont
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{Angeli}(2013)}]{Malrieu_2003}%
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\BibitemOpen
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\bibfield {author} {\bibinfo {author} {\bibfnamefont {J.-P.}\ \bibnamefont
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{Malrieu}}\ and\ \bibinfo {author} {\bibfnamefont {C.}~\bibnamefont
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{Angeli}},\ }\href {\doibase 10.1080/00268976.2013.788745} {\bibfield
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{journal} {\bibinfo {journal} {Mol. Phys.}\ }\textbf {\bibinfo {volume}
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{111}},\ \bibinfo {pages} {1092} (\bibinfo {year} {2013})}\BibitemShut
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{NoStop}%
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\bibitem [{\citenamefont {Bender}(2019)}]{BenderPTBook}%
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\BibitemOpen
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\bibfield {author} {\bibinfo {author} {\bibfnamefont {C.~M.}\ \bibnamefont
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@ -896,6 +905,20 @@
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}\href {\doibase 10.1063/1.481764} {\bibfield {journal} {\bibinfo {journal}
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{J. Chem. Phys.}\ }\textbf {\bibinfo {volume} {112}},\ \bibinfo {pages}
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{9213} (\bibinfo {year} {2000})}\BibitemShut {NoStop}%
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\bibitem [{\citenamefont {Surj{\'a}n}\ and\ \citenamefont
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{Szabados}(2004)}]{Surjan_2004}%
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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}},\ }\enquote {\bibinfo {title} {Appendix to ``studies in
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perturbation theory'': The problem of partitioning},}\ in\ \href {\doibase
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10.1007/978-94-017-0448-9_8} {\emph {\bibinfo {booktitle} {Fundamental World
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of Quantum Chemistry: A Tribute to the Memory of Per-Olov L{\"o}wdin Volume
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III}}},\ \bibinfo {editor} {edited by\ \bibinfo {editor} {\bibfnamefont
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{E.~J.}\ \bibnamefont {Br{\"a}ndas}}\ and\ \bibinfo {editor} {\bibfnamefont
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{E.~S.}\ \bibnamefont {Kryachko}}}\ (\bibinfo {publisher} {Springer
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Netherlands},\ \bibinfo {address} {Dordrecht},\ \bibinfo {year} {2004})\ pp.\
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\bibinfo {pages} {129--185}\BibitemShut {NoStop}%
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\bibitem [{\citenamefont {Nesbet}\ and\ \citenamefont
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{Hartree}(1955)}]{Nesbet_1955}%
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\BibitemOpen
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@ -1361,6 +1384,15 @@
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{Shanks}},\ }\href {\doibase https://doi.org/10.1002/sapm19553411} {\bibfield
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{journal} {\bibinfo {journal} {J. Math. Phys.}\ }\textbf {\bibinfo {volume}
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{34}},\ \bibinfo {pages} {1} (\bibinfo {year} {1955})}\BibitemShut {NoStop}%
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\bibitem [{\citenamefont {Szabados}\ and\ \citenamefont
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{Surjan}(1999)}]{Szabados_1999}%
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\BibitemOpen
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\bibfield {author} {\bibinfo {author} {\bibfnamefont {A.}~\bibnamefont
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{Szabados}}\ and\ \bibinfo {author} {\bibfnamefont {P.}~\bibnamefont
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{Surjan}},\ }\href {\doibase https://doi.org/10.1016/S0009-2614(99)00647-8}
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{\bibfield {journal} {\bibinfo {journal} {Chem. Phys. Lett.}\ }\textbf
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{\bibinfo {volume} {308}},\ \bibinfo {pages} {303 } (\bibinfo {year}
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{1999})}\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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@ -1369,6 +1401,15 @@
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{Szabados}},\ }\href {\doibase 10.1063/1.481006} {\bibfield {journal}
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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 {Szabados}\ and\ \citenamefont
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{Surj{\'a}n}(2003)}]{Szabados_2003}%
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\BibitemOpen
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\bibfield {author} {\bibinfo {author} {\bibfnamefont {{\'A}.}~\bibnamefont
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{Szabados}}\ and\ \bibinfo {author} {\bibfnamefont {P.~R.}\ \bibnamefont
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{Surj{\'a}n}},\ }\href {\doibase https://doi.org/10.1002/qua.10502}
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{\bibfield {journal} {\bibinfo {journal} {Int. J. Quantum Chem.}\ }\textbf
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{\bibinfo {volume} {92}},\ \bibinfo {pages} {160} (\bibinfo {year}
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{2003})}\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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@ -1,7 +1,7 @@
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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 2020-12-14 09:50:05 +0100
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%% Created for Pierre-Francois Loos at 2021-01-29 20:57:57 +0100
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%% Saved with string encoding Unicode (UTF-8)
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@ -165,7 +165,7 @@ The popularity of MP theory stems from its black-box nature, size-extensivity, a
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making it easily applied in a broad range of molecular research.\cite{HelgakerBook}
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However, it is now widely recognised that the series of MP approximations (defined for a given perturbation
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order $n$ as MP$n$) can show erratic, slow, or divergent behaviour that limit its systematic improvability.%
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\cite{Laidig_1985,Knowles_1985,Handy_1985,Gill_1986,Laidig_1987,Nobes_1987,Gill_1988,Gill_1988a,Lepetit_1988}
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\cite{Laidig_1985,Knowles_1985,Handy_1985,Gill_1986,Laidig_1987,Nobes_1987,Gill_1988,Gill_1988a,Lepetit_1988,Malrieu_2003}
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As a result, practical applications typically employ only the lowest-order MP2 approach, while
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the successive MP3, MP4, and MP5 (and higher order) terms are generally not considered to offer enough improvement
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to justify their increased cost.
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@ -447,6 +447,7 @@ As a result, the radius of convergence for a function is equal to the distance f
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in the complex plane, referred to as the ``dominant'' singularity.
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This singularity may represent a pole of the function, or a branch point (\eg, square-root or logarithmic)
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in a multi-valued function.
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\titou{T2: define here critical point.}
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For example, the simple function
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\begin{equation} \label{eq:DivExample}
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@ -551,18 +552,22 @@ the total spin operator $\hat{\mathcal{S}}^2$, leading to ``spin-contamination''
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In the Hubbard dimer, the HF energy can be parametrised using two rotation angles $\ta$ and $\tb$ as
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\begin{equation}
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\label{eq:EHF}
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E_\text{HF}(\ta, \tb) = -t\, \qty( \sin \ta + \sin \tb ) + \frac{U}{2} \qty( 1 + \cos \ta \cos \tb ),
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\end{equation}
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where we have introduced bonding $\mathcal{B}^{\sigma}$ and antibonding $\mathcal{A}^{\sigma}$ molecular orbitals for
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where we have introduced \titou{occupied $\psi_1^{\sigma}$} and \titou{unoccupied $\psi_2^{\sigma}$} molecular orbitals for
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the spin-$\sigma$ electrons as
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\begin{subequations}
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\begin{align}
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\mathcal{B}^{\sigma} & = \hphantom{-} \cos(\frac{\ts}{2}) \Lsi + \sin(\frac{\ts}{2}) \Rsi,
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\label{eq:psi1}
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\titou{\psi_1^{\sigma}} & = \hphantom{-} \cos(\frac{\ts}{2}) \Lsi + \sin(\frac{\ts}{2}) \Rsi,
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\\
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\mathcal{A}^{\sigma} & = - \sin(\frac{\ts}{2}) \Lsi + \cos(\frac{\ts}{2}) \Rsi
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\label{eq:psi2}
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\titou{\psi_2^{\sigma}} & = - \sin(\frac{\ts}{2}) \Lsi + \cos(\frac{\ts}{2}) \Rsi
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\end{align}
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\label{eq:RHF_orbs}
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\end{subequations}
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\titou{Equations \eqref{eq:EHF}, \eqref{eq:psi1}, and \eqref{eq:psi2} are valid for both RHF and UHF.}
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In the weak correlation regime $0 \le U \le 2t$, the angles which minimise the HF energy,
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\ie, $\pdv*{E_\text{HF}}{\ts} = 0$, are
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\begin{equation}
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@ -570,9 +575,9 @@ In the weak correlation regime $0 \le U \le 2t$, the angles which minimise the H
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\end{equation}
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giving the symmetry-pure molecular orbitals
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\begin{align}
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\mathcal{B}_\text{RHF}^{\sigma} & = \frac{\Lsi + \Rsi}{\sqrt{2}},
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\titou{\psi_{1,\text{RHF}}^{\sigma}} & = \frac{\Lsi + \Rsi}{\sqrt{2}},
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&
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\mathcal{A}_\text{RHF}^{\sigma} & = \frac{\Lsi - \Rsi}{\sqrt{2}},
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\titou{\psi_{2,\text{RHF}}^{\sigma}} & = \frac{\Lsi - \Rsi}{\sqrt{2}},
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\end{align}
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and the ground-state RHF energy (Fig.~\ref{fig:HF_real})
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\begin{equation}
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@ -733,12 +738,12 @@ be computed to understand the convergence of the MP$n$ series.\cite{Handy_1985}
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systematically improvable theory.
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In fact, when the reference HF wave function is a poor approximation to the exact wave function,
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for example in multi-configurational systems, MP theory can yield highly oscillatory,
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slowly convergent, or catastrophically divergent results.\cite{Gill_1986,Gill_1988,Handy_1985,Lepetit_1988,Leininger_2000}
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slowly convergent, or catastrophically divergent results.\cite{Gill_1986,Gill_1988,Handy_1985,Lepetit_1988,Leininger_2000,Malrieu_2003}
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Furthermore, the convergence properties of the MP series can depend strongly on the choice of restricted or
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unrestricted reference orbitals.
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Although practically convenient for electronic structure calculations, the MP partitioning is not
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the only possibility and alternative partitionings have been considered including:
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the only possibility and alternative partitionings have been considered \cite{Surjan_2004} including:
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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}
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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
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iii) the strong coupling partitioning where the two operators are inverted compared to the weak correlation partitioning. \cite{Seidl_2018,Daas_2020}
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@ -1047,8 +1052,8 @@ Their analysis is based on Darboux's theorem: \cite{Goodson_2011}
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\textit{``In the limit of large order, the series coefficients become equivalent to
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the Taylor series coefficients of the singularity closest to the origin. ''}
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\end{quote}
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Following this theory, a singularity in the unit circle is designated as an intruder state,
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with a front-door (or back-door) intruder state if the real part of the singularity is positive (or negative).
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\titou{Following this theory, a singularity in the unit circle is designated as an intruder state,
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with a front-door (or back-door) intruder state if the real part of the singularity is positive (or negative).}
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Using their observations in Ref.~\onlinecite{Olsen_1996}, Olsen and collaborators proposed
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a simple method that performs a scan of the real axis to detect the avoided crossing responsible
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@ -1100,7 +1105,7 @@ For Hermitian Hamiltonians, these archetypes can be subdivided into five classes
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while two additional archetypes (zigzag-geometric and convex-geometric) are observed in non-Hermitian Hamiltonians.
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%
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The geometric archetype appears to be the most common for MP expansions,\cite{Olsen_2019} but the
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ripples archetype corresponds to some of the early examples of MP convergence. \cite{Handy_1985,Lepetit_1988,Leininger_2000}
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ripples archetype corresponds to some of the early examples of MP convergence. \cite{Handy_1985,Lepetit_1988,Leininger_2000,Malrieu_2003}
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The three remaining Hermitian archetypes seem to be rarely observed in MP perturbation theory.
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In contrast, the non-Hermitian coupled cluster perturbation theory,%
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\cite{Pawlowski_2019a,Pawlowski_2019b,Pawlowski_2019c,Pawlowski_2019d,Pawlowski_2019e} exhibits a range of archetypes
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@ -1751,7 +1756,7 @@ terms of a perturbation series, even if it diverges.
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Recently, Mih\'alka \etal\ have studied the effect of different partitionings, such as MP or EN theory, on the position of
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branch points and the convergence properties of Rayleigh--Schr\"odinger perturbation theory\cite{Mihalka_2017b} (see also
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Ref.~\onlinecite{Surjan_2000}).
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Refs.~\onlinecite{Szabados_1999,Surjan_2000,Szabados_2003}).
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Taking the equilibrium and stretched water structures as an example, they estimated the radius of convergence using quadratic
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Pad\'e approximants.
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The EN partitioning provided worse convergence properties than the MP partitioning, which is believed to be
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@ -1825,7 +1830,7 @@ We then provided a comprehensive review of the various research that has been pe
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around the physics of complex singularities in perturbation theory, with a particular focus on M{\o}ller--Plesset theory.
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Seminal contributions from various research groups have revealed highly oscillatory,
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slowly convergent, or catastrophically divergent behaviour of the restricted and/or unrestricted MP perturbation series.%
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\cite{Laidig_1985,Knowles_1985,Handy_1985,Gill_1986,Laidig_1987,Nobes_1987,Gill_1988,Gill_1988a,Lepetit_1988}
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\cite{Laidig_1985,Knowles_1985,Handy_1985,Gill_1986,Laidig_1987,Nobes_1987,Gill_1988,Gill_1988a,Lepetit_1988,Malrieu_2003}
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In particular, the spin-symmetry-broken unrestricted MP series is notorious
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for giving incredibly slow convergence.\cite{Gill_1986,Nobes_1987,Gill_1988a,Gill_1988}
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All these behaviours can be rationalised and explained by the position of exceptional points
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@ -124,7 +124,8 @@ We look forward to hearing from you.
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I raise this point because the journal is JPCM, and these terms are not common in condensed matter literature.
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Are intruder states equivalent to states creating exceptional points that ruin the series convergence?}
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\\
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\alert{The classification of front-door and back-door intruder states have been clarified.}
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\alert{The classification of front-door and back-door intruder states have been clarified.
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"Following this theory, a singularity in the unit circle is designated as an intruder state, with a front-door (or back-door) intruder state if the real part of the singularity is positive (or negative)."}
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\item
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{IIIe.\\
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