Done with IIIE

Manuscript/EPAWTFT.tex

@@ -1096,7 +1096,7 @@ processes.\cite{Sergeev_2005}
 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 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 
+where the magnitude of the imaginary component controls 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.
@@ -1117,26 +1117,9 @@ Alternatively, Sergeev \etal\ demonstrated that the inclusion of a ``ghost'' ato
 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 
+Furthermore, the two-state model of Olsen and collaborators \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. 
-%On the contrary, $\alpha$ singularities are relatively insensitive to the basis sets but very sensitive to bond stretching. 
-%According to Goodson, \cite{Goodson_2004} the singularity structure of stretched molecules is difficult because there is more than one significant singularity. 
-%This is consistent with the observation of Olsen and coworkers \cite{Olsen_2000} on the \ce{HF} molecule at equilibrium and stretched geometries. 
-%To the best of our knowledge, the effect of bond stretching on singularities, its link with spin contamination and symmetry breaking of the wave function has not been as well understood as the ionization phenomenon and its link with diffuse functions.  
-
-%In the 2000's, Sergeev and Goodson \cite{Sergeev_2005, Sergeev_2006} analysed this problem from a more mathematical point of view by looking at the whole singularity structure where Olsen and collaborators were trying to find the dominant singularity causing the divergence. \cite{Olsen_1996,Olsen_2000,Olsen_2019}
-%Singularities of $\alpha$-type are related to large avoided crossing between the ground and low-lying excited states, whereas $\beta$ singularities come from a sharp avoided crossing between the ground state and a highly diffuse state. 
-%They succeeded to explain the divergence of the series caused by $\beta$ singularities following previous work by Stillinger. \cite{Stillinger_2000}
-
-
-%This reasoning is done on the exact Hamiltonian and energy, \ie, the Hamiltonian in the complete Hilbert space, this is the exact energy which exhibits this singularity on the negative real axis. 
-%However, in a finite basis set which does not span the complete Hilbert space, one can prove that, for a Hermitian Hamiltonian, the singularities of $E(\lambda)$ occurs in complex conjugate pairs with non-zero imaginary parts. 
-%Sergeev and Goodson proved, \cite{Sergeev_2005} as predicted by Stillinger, \cite{Stillinger_2000} that in a finite basis set the critical point on the real axis is modelled by a cluster of sharp avoided crossings with diffuse functions, equivalently by a cluster of $\beta$ singularities in the negative half plane. 
-%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
 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).%