small modifs in singularity structure

Manuscript/EPAWTFT.tex

@@ -949,10 +949,10 @@ One of main take-home messages of Olsen's study is that the primary critical poi
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 \subsection{The singularity structure}
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-In the 2000's, Sergeev and Goodson \cite{Sergeev_2005, Sergeev_2006} analyzed 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. 
+In the 2000's, Sergeev and Goodson \cite{Sergeev_2005, Sergeev_2006} analyzed 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}
 They regrouped singularities in two classes: i) $\alpha$ singularities which have ``large'' imaginary parts, and ii) $\beta$ singularities which have very small imaginary parts. 
-Singularities of type $\alpha$ 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 of Stillinger. \cite{Stillinger_2000}
+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}
 
 To understand the convergence properties of the perturbation series at $\lambda=1$, one must look at the whole complex plane, in particular, for negative (\ie, real) values of $\lambda$. 
 If $\lambda$ is negative, the Coulomb interaction becomes attractive but the mean field (which has been computed at $\lambda = 1$) remains repulsive as it is proportional to $(1-\lambda)$:
@@ -970,7 +970,7 @@ If $\lambda$ is negative, the Coulomb interaction becomes attractive but the mea
 \end{multline}
 
 The major difference between these two terms is that the repulsive mean field is localised around the nuclei whereas the interelectronic interaction persist away from the nuclei. 
-If $\lambda$ becomes more and more negative the mean field becomes more and more repulsive so there exists a critical (negative) value of $\lambda$, $\lambda_\text{c}$, for which the Coulombic field created by the nuclei cannot bind the electrons anymore because of the $\lambda$-independent nature of the electron-nucleus attraction. 
+If $\lambda$ becomes more and more negative the mean field becomes more and more repulsive so there exists a critical (negative) value $\lambda_\text{c}$, for which the Coulombic field created by the nuclei cannot bind the electrons anymore because of the $\lambda$-independent nature of the electron-nucleus attraction. 
 For $\lambda = \lambda_c$, the electrons dissociate from the nuclei and form a bound cluster which is infinitely separated from the nuclei. 
 According to Baker, \cite{Baker_1971} this value is a critical point of the system and, by analogy with thermodynamics, the energy $E(\lambda)$ exhibits a singularity at $\lambda_\text{c}$. 
 At this point the system undergo a phase transition and a symmetry breaking. 
@@ -979,12 +979,12 @@ Beyond $\lambda_c$ there is a continuum of eigenstates thanks to which the elect
 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 $2\times2$ model, only observed the first singularity of this cluster of singularities causing the divergence. \cite{Olsen_2000}
+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}
 
-Finally, it was shown that $\beta$ singularities are very sensitive to changes in the basis set but not to the stretching of the system. 
+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 geometry and stretched geometry. 
+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.  
 
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