From cf76131ece911c832508b30304b9dc6ce6bc2f43 Mon Sep 17 00:00:00 2001 From: Pierre-Francois Loos Date: Fri, 27 Nov 2020 22:13:53 +0100 Subject: [PATCH] review Hugh changes --- Manuscript/EPAWTFT.bbl | 11 ++++++++++- Manuscript/EPAWTFT.tex | 20 ++++++-------------- 2 files changed, 16 insertions(+), 15 deletions(-) diff --git a/Manuscript/EPAWTFT.bbl b/Manuscript/EPAWTFT.bbl index 87da465..5742914 100644 --- a/Manuscript/EPAWTFT.bbl +++ b/Manuscript/EPAWTFT.bbl @@ -6,7 +6,7 @@ %Control: page (0) single %Control: year (1) truncated %Control: production of eprint (0) enabled -\begin{thebibliography}{131}% +\begin{thebibliography}{132}% \makeatletter \providecommand \@ifxundefined [1]{% \@ifx{#1\undefined} @@ -1188,6 +1188,15 @@ {journal} {\bibinfo {journal} {J. Phys. A: Math. Theor.}\ }\textbf {\bibinfo {volume} {40}},\ \bibinfo {pages} {581} (\bibinfo {year} {2007})}\BibitemShut {NoStop}% +\bibitem [{\citenamefont {Kais}\ \emph {et~al.}(2006)\citenamefont {Kais}, + \citenamefont {Wenger},\ and\ \citenamefont {Wei}}]{Kais_2006}% + \BibitemOpen + \bibfield {author} {\bibinfo {author} {\bibfnamefont {S.}~\bibnamefont + {Kais}}, \bibinfo {author} {\bibfnamefont {C.}~\bibnamefont {Wenger}}, \ and\ + \bibinfo {author} {\bibfnamefont {Q.}~\bibnamefont {Wei}},\ }\href {\doibase + https://doi.org/10.1016/j.cplett.2006.03.035} {\bibfield {journal} {\bibinfo + {journal} {Chem. Phys. Lett.}\ }\textbf {\bibinfo {volume} {423}},\ \bibinfo + {pages} {45 } (\bibinfo {year} {2006})}\BibitemShut {NoStop}% \bibitem [{\citenamefont {Goodson}(2019)}]{Goodson_2019}% \BibitemOpen \bibfield {author} {\bibinfo {author} {\bibfnamefont {D.~Z.}\ \bibnamefont diff --git a/Manuscript/EPAWTFT.tex b/Manuscript/EPAWTFT.tex index f08a64b..4e45cc8 100644 --- a/Manuscript/EPAWTFT.tex +++ b/Manuscript/EPAWTFT.tex @@ -1142,12 +1142,11 @@ regardless of whether this point is inside or outside the complex unit circle. \ %======================================= % STILLINGER INTRODUCES THE CRITICAL POINT -\hugh{% In the early 2000's, Stillinger reconsidered the mathematical origin behind the divergent series with odd-even sign alternation.\cite{Stillinger_2000} This type of convergence behaviour corresponds to Cremer and He's class B systems with closely spaced electron pairs and includes \ce{Ne}, \ce{HF}, \ce{F-}, and \ce{H2O}.\cite{Cremer_1996} -Stillinger proposed that the divergence of these series occurs arise from a dominant singularity +Stillinger proposed that the divergence of these series arise from a dominant singularity on the negative real $\lambda$ axis, corresponding to a multielectron autoionisation threshold.\cite{Stillinger_2000} To understand Stillinger's argument, consider the paramterised MP Hamiltonian in the form \begin{multline} @@ -1171,12 +1170,10 @@ This autoionisation effect is closely related to the critial point for electron atoms (see Ref.~\onlinecite{Baker_1971}). Furthermore, a similar set of critical points exists along the positive real axis, corresponding to single-electron ionisation processes.\cite{Sergeev_2005} -} % CLASSIFICATIONS BY GOODSOON AND SERGEEV -\hugh{% To further develop the link between the critical point and types of MP convergence, Sergeev and Goodson investigated -the relationship with the location of the dominant singularity that controls the radius of convergence.\cite{Goodson_2004}. +the relationship with the location of the dominant singularity that controls the radius of convergence.\cite{Goodson_2004} They demonstrated that the dominant singularity in class A corresponds to a dominant EP with a positive real component, with the magnitude of the imaginary component controlling the oscillations in the signs of the MP term.\cite{Goodson_2000a,Goodson_2000b} @@ -1187,10 +1184,8 @@ HF potential $v^{\text{HF}}$ is relatively localised and the autoionization is f $\lambda$ values closer to the origin. With these insights, they regrouped the systems into new classes: i) $\alpha$ singularities which have ``large'' imaginary parts, and ii) $\beta$ singularities which have very small imaginary parts.\cite{Goodson_2004,Sergeev_2006} -} % RELATIONSHIP TO BASIS SET SIZE -\hugh{% The existence of the MP critical point can also explain why the divergence observed by Olsen \etal\ in the \ce{Ne} atom and \ce{HF} molecule occured when diffuse basis functions were included.\cite{Olsen_1996} Clearly diffuse basis functions are required for the electrons to dissociate from the nuclei, and indeed using @@ -1200,10 +1195,9 @@ by a cluster of sharp avoided crossings between the ground state and high-lying Alternatively, Sergeev \etal\ demonstrated that the inclusion of a ``ghost'' atom also allows the formation of the critical point as the electrons form a bound cluster occuping the ghost atom orbitals.\cite{Sergeev_2005} This effect explains the origin of the divergence in the \ce{HF} molecule as the \ce{F} valence electrons jump to \ce{H} -for a sufficiently negative $\lambda$.\cite{Sergeev_2005} +for a sufficiently negative $\lambda$ value.\cite{Sergeev_2005} Furthermore, the two-state model of Olsen \etal{}\cite{Olsen_2000} was simply too minimal to understand the complexity of divergences caused by the MP critical point. -} % BASIS SET DEPENDENCE (INCLUDE?) %Finally, it was shown that $\beta$ singularities are very sensitive to changes in the basis set but not to the bond stretching. @@ -1223,21 +1217,19 @@ divergences caused by the MP critical point. %This explains that Olsen \textit{et al.}, because they used a simple two-state model, only observed the first singularity of this cluster of singularities causing the divergence. \cite{Olsen_2000} % RELATIONSHIP TO QUANTUM PHASE TRANSITION -\hugh{% When a Hamiltonian is parametrised by a variable such as $\lambda$, the existence of abrupt changes in the eigenstates as a function of $\lambda$ indicate the presence of a zero-temperature quantum phase transition (QPT).% \cite{Heiss_1988,Heiss_2002,Borisov_2015,Sindelka_2017,CarrBook,Vojta_2003,SachdevBook,GilmoreBook} -As an avoided crossing becomes increasingly sharp, the corresponding EPs move increasingly close to the real axis. +As an avoided crossing becomes increasingly sharp, the corresponding EPs move increasingly close to the real axis, eventually forming a critical point. The existence of an EP \emph{on} the real axis is therefore diagnostic of a QPT.\cite{Cejnar_2005, Cejnar_2007a} Since the MP critical point corresponds to a singularity on the real $\lambda$ axis, it can immediately be recognised a QPT within the perturbation theory approximation. However, a conventional QPT can only occur in the thermodynamic limit, which here is analogous to the complete -basis set limit.\cite{Kais_2006} +basis set limit in our case.\cite{Kais_2006} The MP critical point $\beta$ singularity in a finite basis must therefore be modelled by pairs of EPs -that tend towards the real axis, exactly as described by Sergeev \etal.\cite{Sergeev_2005} +that tend towards the real axis, exactly as described by Sergeev \etal\cite{Sergeev_2005} In contrast, $\alpha$ singularities correspond to large avoided crossings that are indicative of low-lying excited states which share the symmetry of the ground state,\cite{Goodson_2004} and are thus not manifestations of a QPT. -} %======================================= \subsection{Critical Point in the Hubbard Dimer}