Hugh added some spice to the opening paragraph

Manuscript/EPAWTFT.tex

@@ -165,38 +165,53 @@ Each of these points is further illustrated with the ubiquitous Hubbard dimer at
 \label{sec:intro}
 %%%%%%%%%%%%%%%%%%%%%%%
 
+% SPIKE THE READER
+\hugh{Perturbation theory isn't usually considered in the complex plane.
+Normally it is applied using real numbers as one of very few availabe tools for 
+describing realistic quantum systems where exact solutions of the Schr\"odinger equation are impossible.\cite{Dirac_1929}
+In particular, time-independent Rayleigh--Schr\"odinger perturbation theory\cite{RayleighBook,Schrodinger_1926} 
+has emerged as an instrument of choice among the vast array of methods developed for this purpose.%
+\cite{SzaboBook,JensenBook,CramerBook,HelgakerBook,ParrBook,FetterBook,ReiningBook}
+However, the properties of perturbation theory in the complex plane
+are essential for understanding the quality of perturbative approximations on the real axis.
+}
+
 % Good old Schroedinger
-The electronic Schr\"odinger equation,
-\begin{equation}
-	\hH \Psi = E \Psi,
-\end{equation}
-is the starting point for a fundamental understanding of the behaviour of electrons and, thence, of chemical structure, bonding and reactivity. 
-However, as famously stated by Dirac: \cite{Dirac_1929}
-\begin{quote}
-	\textit{``The underlying physical laws necessary for the mathematical theory of a large part of physics and the whole of chemistry are thus completely known, and the difficulty is only that the exact application of these laws leads to equations much too complicated to be soluble. 
-	It therefore becomes desirable that approximate practical methods of applying quantum mechanics should be developed, which can lead to an explanation of the main features of complex atomic systems without too much computation.''}
-\end{quote}
-Indeed, as anticipated by Dirac, accurately predicting the energetics of electronic states from first principles has been one of the grand challenges faced by theoretical chemists and physicists since the dawn of quantum mechanics.
+%The electronic Schr\"odinger equation,
+%\begin{equation}
+%	\hH \Psi = E \Psi,
+%\end{equation}
+%is the starting point for a fundamental understanding of the behaviour of electrons and, thence, of chemical structure, bonding and reactivity. 
+%However, as famously stated by Dirac: \cite{Dirac_1929}
+%\begin{quote}
+%	\textit{``The underlying physical laws necessary for the mathematical theory of a large part of physics and the whole of chemistry are thus completely known, and the difficulty is only that the exact application of these laws leads to equations much too complicated to be soluble. 
+%	It therefore becomes desirable that approximate practical methods of applying quantum mechanics should be developed, which can lead to an explanation of the main features of complex atomic systems without too much computation.''}
+%\end{quote}
+%Indeed, as anticipated by Dirac, accurately predicting the energetics of electronic states from first principles has been one of the grand challenges faced by theoretical chemists and physicists since the dawn of quantum mechanics.
 
 % RSPT
-Together with the variational principle, perturbation theory is one of the very few essential tool for describing realistic quantum systems for which it is impossible to find the exact solution of the Schr\"odinger equation. 
-In particular, time-independent Rayleigh--Schr\"odinger perturbation theory \cite{RayleighBook,Schrodinger_1926} has cemented itself as an instrument of choice in the armada of theoretical and computational methods that have been developed for this purpose. \cite{SzaboBook,JensenBook,CramerBook,HelgakerBook,ParrBook,FetterBook,ReiningBook}
+%Together with the variational principle, perturbation theory is one of the very few essential tool for describing realistic quantum systems for which it is impossible to find the exact solution of the Schr\"odinger equation. 
+%In particular, time-independent Rayleigh--Schr\"odinger perturbation theory \cite{RayleighBook,Schrodinger_1926} has cemented itself as an instrument of choice in the armada of theoretical and computational methods that have been developed for this purpose. \cite{SzaboBook,JensenBook,CramerBook,HelgakerBook,ParrBook,FetterBook,ReiningBook}
 
 % Moller-Plesset
-The workhorse of time-independent perturbation theory is most certainly M\o{}ller--Plesset (MP) perturbation 
-theory \cite{Moller_1934} which remains one of the most popular methods for computing the electron correlation energy, 
-an old yet important concept, first introduced by Wigner \cite{Wigner_1934} and later defined by L\"owdin. \cite{Lowdin_1958} 
-In this approach, the exact electronic energy is estimated by constructing a perturbative correction on top
+%\hugh{Accurately predicting the electronic energy is the primary focus of electronic structure theory, in 
+%principle providing a fundamental understanding of chemical structure, bonding, and reactivity.
+\hugh{In electronic structure theory,} the workhorse of time-independent perturbation theory is M\o{}ller--Plesset (MP) %perturbation 
+theory,\cite{Moller_1934} which remains one of the most popular methods for computing the electron 
+correlation energy.\cite{Wigner_1934,Lowdin_1958}
+%\trashHB{an old yet important concept, first introduced by Wigner \cite{Wigner_1934} and later defined by L\"owdin. \cite{Lowdin_1958}}
+This approach estimates the exact electronic energy by constructing a perturbative correction on top
 of a mean-field Hartree--Fock (HF) approximation.\cite{SzaboBook}
-The popularity of MP theory stems from its black-box nature and relatively low computational scaling, 
-making it easily applied in a broad range of molecular research.
-However, it is now widely understood that the series of MP approximations (defined for a given perturbation
+The popularity of MP theory stems from its black-box nature, \hugh{size-extensivity,} and relatively low computational scaling, 
+making it easily applied in a broad range of molecular research.\cite{HelgakerBook}
+However, it is now widely recognised that the series of MP approximations (defined for a given perturbation
 order $n$ as MP$n$) can show erratic, slow, or divergent behaviour that limit its systematic improvability.%
 \cite{Laidig_1985,Knowles_1985,Handy_1985,Gill_1986,Laidig_1987,Nobes_1987,Gill_1988,Gill_1988a,Lepetit_1988} 
 As a result, practical applications typically employ only the lowest-order MP2 approach, while 
-the successive MP3, MP4, and MP5 (and higher) terms are generally not considered to offer enough improvement
+the successive MP3, MP4, and MP5 (and higher \hugh{order}) terms are generally not considered to offer enough improvement
 to justify their increased cost.
-Turning the MP approximations into a convergent and systematically improvable series largely remains an open challenge.
+Turning the MP approximations into a convergent and 
+systematically improvable series largely remains an open challenge.
 
 % COMPLEX PLANE
 Our conventional view of electronic structure theory is centred around the Hermitian notion of quantised energy levels,
@@ -207,7 +222,7 @@ However, an entirely different perspective on quantisation can be found by analy
 quantum mechanics into the complex domain.
 In this inherently non-Hermitian framework, the energy levels emerge as individual \textit{sheets} of a complex
 multi-valued function and can be connected as one continuous \textit{Riemann surface}.\cite{BenderPTBook}
-This connection is possible because orderability of real numbers is lost when energies are extended to the
+This connection is possible because the orderability of real numbers is lost when energies are extended to the
 complex domain.
 As a result, our quantised view of conventional quantum mechanics only arises from
 restricting our domain to Hermitian approximations.
@@ -254,12 +269,16 @@ In doing so, we will demonstrate how understanding the MP energy in the complex
 be harnessed to significantly improve estimates of the exact energy using only the lowest-order terms
 in the MP series.
 
-The present review is organised as follows.
-In Sec.~\ref{sec:EPs}, we introduce key concepts, such as Rayleigh-Schr\"odinger perturbation theory and the mean-field HF approximation, and discuss their analytic continuation into the complex plane.
-Section \ref{sec:MP} deals with MP perturbation theory and we report an exhaustive historical overview of the various research activities that have been performed on the physics of singularities.
-We discuss several resummation techniques (such as Pad\'e and quadratic approximants) in Sec.~\ref{sec:Resummation}.
-Finally, we draw our conclusions in Sec.~\ref{sec:ccl}.
-For most of the concepts presented in this review, we report illustrative and pedagogical examples based on the ubiquitous Hubbard dimer, showing the amazing versatility of this simple yet powerful model system.
+\trashHB{The present review is organised as follows.}
+In Sec.~\ref{sec:EPs}, we introduce the key concepts such as Rayleigh-Schr\"odinger perturbation theory and the mean-field HF approximation, and discuss their \hugh{non-Hermitian} analytic continuation into the complex plane.
+Section \ref{sec:MP} presents MP perturbation theory and we report a comprehensive historical overview of the research that
+has been performed on the physics of MP singularities.
+In Sec.~\ref{sec:Resummation}, we discuss several resummation techniques for improving the accuracy
+of low-order MP approximations, including Pad\'e and quadratic approximants.
+Finally, we draw our conclusions in Sec.~\ref{sec:ccl} \hugh{and highlight our perspective on directions for 
+future research}.
+Throughout this review, we present illustrative and pedagogical examples based on the ubiquitous 
+Hubbard dimer, \hugh{reinforcing} the amazing versatility of this powerful simplistic model.
 
 %%%%%%%%%%%%%%%%%%%%%%%
 \section{Exceptional Points in Electronic Structure}
@@ -1776,7 +1795,7 @@ the cost of larger denominators is an overall slower rate of convergence.
     \includegraphics[width=\linewidth]{fig12}
     \caption{%
         Comparison of the scaled RMP10 Taylor expansion with the exact RMP energy as a function
-        of $\lambda$ for the symmetric Hubbard dimer at $U/t = 4$. 
+        of $\lambda$ for the symmetric Hubbard dimer at $U/t = 4.5$. 
         The two functions correspond closely within the radius of convergence.
     }
     \label{fig:rmp_anal_cont}