From 4fff7c9fd2adab865c0fee03b5e358b02cc163aa Mon Sep 17 00:00:00 2001
From: Hugh Burton <hugh_burton@hotmail.com>
Date: Mon, 30 Nov 2020 21:34:39 +0000
Subject: [PATCH] minor edits

---
 Manuscript/EPAWTFT.tex | 14 +++++++-------
 1 file changed, 7 insertions(+), 7 deletions(-)

diff --git a/Manuscript/EPAWTFT.tex b/Manuscript/EPAWTFT.tex
index 9845ad9..1878208 100644
--- a/Manuscript/EPAWTFT.tex
+++ b/Manuscript/EPAWTFT.tex
@@ -1295,11 +1295,11 @@ orbitals [see Eq.~\eqref{eq:RHF_orbs}] with $\theta_{\alpha}^{\text{RHF}} = \the
 %\begin{equation}
 %   E_\text{HF}(0, 0) = \frac{1}{2} (2 U - 4 \epsilon).
 %\end{equation}
-With this representation, the parametrised RMP Hamiltonian becomes
+With this representation, the parametrised \hugh{atomic} RMP Hamiltonian becomes
 \begin{widetext}
 \begin{equation}
 \label{eq:H_RMP}
-\bH_\text{RMP}\qty(\lambda) = 
+\hugh{\bH_\text{atom}\qty(\lambda)} = 
 \begin{pmatrix}
     2(U-\epsilon) - \lambda U &	-\lambda t				 &	-\lambda t	            &	0	        \\
     -\lambda t				  &	(U-\epsilon) - \lambda U &	0		                &	-\lambda t	\\
@@ -1337,7 +1337,8 @@ In contrast, smaller $\epsilon$ gives a weaker attraction to the atomic site,
 representing strong screening of the nuclear attraction by core and valence electrons, 
 and again a less negative $\lambda$ is required for ionisation to occur.
 Both of these factors are common in atoms on the right-hand side of the periodic table, \eg\ \ce{F},
-\ce{O}, \ce{Ne}, and thus molecules containing these atoms are often class $\beta$ systems with
+\ce{O}, \ce{Ne}.
+Molecules containing these atoms are therefore often class $\beta$ systems with
 a divergent RMP series due to the MP critical point. \cite{Goodson_2004,Sergeev_2006} 
 
 % EXACT VERSUS APPROXIMATE
@@ -1384,10 +1385,9 @@ For $\lambda>1$, the HF potential becomes an attractive component in Stillinger'
 Hamiltonian displayed in Eq.~\eqref{eq:HamiltonianStillinger}, while the explicit electron-electron interaction
 becomes increasingly repulsive.
 Critical points along the positive real $\lambda$ axis for closed-shell molecules then represent 
-points where the two-electron repulsion overcomes the attractive HF potential and an electron
-are successively expelled from the molecule.\cite{Sergeev_2006}
+points where the two-electron repulsion overcomes the attractive HF potential and a single electron dissociates from the molecule.\cite{Sergeev_2006}
 
-Symmetry-breaking in the UMP reference creates different HF potentials for the spin-up and spin-down electrons.
+In contrast, symmetry-breaking in the UMP reference creates different HF potentials for the spin-up and spin-down electrons.
 Consider the reference UHF solution where the spin-up and spin-down electrons are localised on the left and
 right sites respectively.  
 The spin-up HF potential will then be a repulsive interaction from the spin-down electron 
@@ -1413,7 +1413,7 @@ occurs exactly at $\lambda = 1$.
 In this limit, the ground-state EPs approach the real axis (Fig.~\ref{subfig:ump_ep_to_cp}) and the avoided 
 crossing creates a gradient discontinuity in the ground-state energy (dashed lines in Fig.~\ref{subfig:ump_cp}).
 We therefore find that, in the strong correlation limit, the symmetry-broken ground-state EP becomes
-a new type of MP critical and represents a QPT in the perturbation approximation. 
+a new type of MP critical point and represents a QPT in the perturbation approximation. 
 Furthermore, this argument explains why the dominant UMP singularity lies so close, but always outside, the 
 radius of convergence (see Fig.~\ref{fig:RadConv}).