some spell-checking

Manuscript/EPAWTFT.tex

@@ -1137,7 +1137,7 @@ This analysis highlights the importance of the primary critical point in control
 regardless of whether this point is inside or outside the complex unit circle. \cite{Handy_1985,Olsen_2000}
 
 %=======================================
-\subsection{The M{\o}ller--Plesset Critical Point}
+\subsection{M{\o}ller--Plesset Critical Point}
 \label{sec:MP_critical_point}
 %=======================================
 
@@ -1146,9 +1146,9 @@ In the early 2000's, Stillinger reconsidered the mathematical origin behind the
 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 arise from a dominant singularity
+Stillinger proposed that these series diverge due to 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
+To understand Stillinger's argument, consider the parametrised MP Hamiltonian in the form
 \begin{multline}
 \label{eq:HamiltonianStillinger}
     \hH(\lambda) = 
@@ -1164,7 +1164,7 @@ The mean-field potential $v^{\text{HF}}$ essentially represents a negatively cha
 controlled by the extent of the HF orbitals, usually located close to the nuclei.
 When $\lambda$ is negative, the mean-field potential becomes increasingly repulsive, while the explicit two-electron 
 Coulomb interaction becomes attractive.
-There is therefore a critical value $\lc < 0$ where it becomes energetically favourable for the electrons 
+There is therefore a negative critical value $\lc$ where it becomes energetically favourable for the electrons 
 to dissociate and form a bound cluster at an infinite separation from the nuclei.\cite{Stillinger_2000}
 This autoionisation effect is closely related to the critial point for electron binding in two-electron 
 atoms (see Ref.~\onlinecite{Baker_1971}).
@@ -1174,9 +1174,9 @@ processes.\cite{Sergeev_2005}
 % CLASSIFICATIONS BY GOODSOON AND SERGEEV
 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}
-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}
+They demonstrated that the dominant singularity in class A systems corresponds to a dominant EP with a positive real component, 
+with the magnitude of the imaginary component controlling the oscillations in the signs of successive MP 
+terms.\cite{Goodson_2000a,Goodson_2000b}
 In contrast, class B systems correspond to a dominant singularity on the negative real $\lambda$ axis representing
 the MP critical point.
 The divergence of class B systems, which contain closely spaced electrons (\eg, \ce{F-}), can then be understood as the 
@@ -1187,15 +1187,15 @@ and ii) $\beta$ singularities which have very small imaginary parts.\cite{Goodso
 
 % RELATIONSHIP TO BASIS SET SIZE
 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}
+and \ce{HF} molecule occurred 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
 only compact basis functions causes the critical point to disappear.
 While a finite basis can only predict complex-conjugate branch point singularities, the critical point is modelled
 by a cluster of sharp avoided crossings between the ground state and high-lying excited states.\cite{Sergeev_2005}
 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 $\lambda$ value.\cite{Sergeev_2005}
+allows the formation of the critical point as the electrons form a bound cluster occupying the ghost atom orbitals.\cite{Sergeev_2005}
+This effect explains the origin of the divergence in the \ce{HF} molecule as the fluorine valence electrons jump to hydrogen at
+ 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.
 
@@ -1220,12 +1220,12 @@ divergences caused by the MP critical point.
 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, eventually forming a critical point.
+Meanwhile, 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.
+recognised as 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 in our case.\cite{Kais_2006}
+basis set limit.\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}
 In contrast, $\alpha$ singularities correspond to large avoided crossings that are indicative of low-lying excited