minor edits

Manuscript/EPAWTFT.tex

@@ -147,7 +147,7 @@ Each of these points is illustrated using the Hubbard dimer at half filling, whi
 
 % SPIKE THE READER
 Perturbation theory isn't usually considered in the complex plane.
-Normally it is applied using real numbers as one of very few available tools for 
+Normally, it is applied using real numbers as one of very few available tools for 
 describing realistic quantum systems.
 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.%
@@ -439,7 +439,7 @@ setting $\lambda = 1$ then yields approximate solutions to Eq.~\eqref{eq:SchrEq}
 
 % MATHEMATICAL REPRESENTATION
 Mathematically, Eq.~\eqref{eq:E_expansion} corresponds to a Taylor series expansion of the exact energy
-around the reference system $\lambda = 0$.
+around the reference system $\lambda~=~0$.
 The energy of the target ``physical'' system is recovered at the point $\lambda = 1$.
 However, like all series expansions, Eq.~\eqref{eq:E_expansion} has a radius of convergence $\rc$. 
 When $\rc < 1$, the Rayleigh--Schr\"{o}dinger expansion will diverge
@@ -576,15 +576,15 @@ In the Hubbard dimer, the HF energy can be parametrised using two rotation angle
 \label{eq:EHF}
 E_\text{HF}(\ta, \tb) = -t\, \qty( \sin \ta + \sin \tb ) + \frac{U}{2} \qty( 1 + \cos \ta \cos \tb ),
 \end{equation}
-where we have introduced \titou{occupied $\psi_1^{\sigma}$} and \titou{unoccupied $\psi_2^{\sigma}$} molecular orbitals for 
+where we have introduced \titou{occupied $\phi_1^{\sigma}$} and \titou{unoccupied $\phi_2^{\sigma}$} molecular orbitals for 
 the spin-$\sigma$ electrons as
 \begin{subequations}
 \begin{align}
 	\label{eq:psi1}
-    \titou{\psi_1^{\sigma}} & = \hphantom{-} \cos(\frac{\ts}{2}) \Lsi + \sin(\frac{\ts}{2}) \Rsi,
+    \titou{\phi_1^{\sigma}} & = \hphantom{-} \cos(\frac{\ts}{2}) \Lsi + \sin(\frac{\ts}{2}) \Rsi,
 	\\
 	\label{eq:psi2}
-	\titou{\psi_2^{\sigma}} & = - \sin(\frac{\ts}{2}) \Lsi + \cos(\frac{\ts}{2}) \Rsi
+	\titou{\phi_2^{\sigma}} & = - \sin(\frac{\ts}{2}) \Lsi + \cos(\frac{\ts}{2}) \Rsi
 \end{align}
 \label{eq:RHF_orbs}
 \end{subequations}
@@ -596,9 +596,9 @@ In the weak correlation regime $0 \le U \le 2t$, the angles which minimise the H
 \end{equation}
 giving the molecular orbitals
 \begin{align}
-	\titou{\psi_{1,\text{RHF}}^{\sigma}} & = \frac{\Lsi + \Rsi}{\sqrt{2}},
+	\titou{\phi_{1,\text{RHF}}^{\sigma}} & = \frac{\Lsi + \Rsi}{\sqrt{2}},
 	&
-	\titou{\psi_{2,\text{RHF}}^{\sigma}} & = \frac{\Lsi - \Rsi}{\sqrt{2}},
+	\titou{\phi_{2,\text{RHF}}^{\sigma}} & = \frac{\Lsi - \Rsi}{\sqrt{2}},
 \end{align}
 and the ground-state RHF energy (Fig.~\ref{fig:HF_real})
 \begin{equation}
@@ -1351,7 +1351,7 @@ idealised destination for the ionised electrons with a non-zero (yet small) hopp
 Taking the value $t=0.1$ (Fig.~\ref{subfig:rmp_cp}: dashed lines), the critical point becomes  
 an avoided crossing with a complex-conjugate pair of EPs close to the real axis (Fig.~\ref{subfig:rmp_cp_surf}).
 \titou{In contrast to the exact critical point with $t=0$, the ground-state energy remains
-smooth through this avoided crossing.}
+smooth through this avoided crossing, with a more gradual drop in the atomic site density.}
 In the limit $t \to 0$, these EPs approach the real axis (Fig.~\ref{subfig:rmp_ep_to_cp}) \titou{and the 
 avoided crossing becomes a gradient discontinuity},
 mirroring Sergeev's discussion on finite basis
@@ -1438,8 +1438,8 @@ electron density on the left and right sites, defined for each spin as
 where $\rho_{\mathcal{L}}^{\sigma}$ ($\rho_{\mathcal{R}}^{\sigma}$) is the spin-$\sigma$ electron density
 on the left (right) site. 
 This density difference is shown for the UMP ground-state at $U = 5 t$ in Fig.~\ref{fig:ump_dens} (solid lines).
-Here, the transfer of the high-spin electron from the right site to the left site can be seen as $\lambda$ passes through 1
-(and similarly for the low-spin electron).}
+Here, the transfer of the spin-up electron from the right site to the left site can be seen as $\lambda$ passes through 1
+(and similarly for the spin-down electron).}
 
 % SHARPNESS AND QPT
 The ``sharpness'' of the avoided crossing is controlled by the correlation strength $U/t$.

Response_Letter/Response_Letter.tex

@@ -192,7 +192,7 @@ These points correspond to the intersection of two surfaces rather than an avoid
 leading to the sudden change in the eigenstates.
 It is possible to encircle the CP in the complex plane, but unlike an EP, this will leave the eigenstates unchanged.
 We have included a new plot of the ``atomic'' site  electron density as a function of $\lambda$ to illustrate
-the sudden autoionisation process at the critical point.
+the sudden autoionisation process at the RMP critical point.
 We have also improved our discussion regarding these features using more explicit references to Figs.~7 and 8.
 Finally, we have endeavoured to illustrate the UMP critical point by considering the difference in electron densities on the left and right sites, as shown in an additional figure. 
 
@@ -202,8 +202,8 @@ Finally, we have endeavoured to illustrate the UMP critical point by considering
 	Presumably, the exact function is known to some order and/or inside of some radius of convergence. 
 	The approximant is fitted in this region to match the exact function and then only the approximant is used beyond the original radius of convergence or at higher order.
 \end{formal} 
-\noindent	{As mentioned in the manuscript, the Pad\'e coefficients are determined by solving a set of linear equations that relate these coefficients with the low-order terms in the Taylor series.
-	This is the only knowledge required to compute these.
+\noindent	{As mentioned in the manuscript, the Pad\'e (or quadratic) coefficients are determined by solving a set of linear equations that relate these coefficients with the low-order terms in the Taylor series.
+	This is the only knowledge required to compute these approximants.
 	A minor modification has been performed to clarify this.}
 
 \begin{formal}