From 2a428b01c3f1df1cf04df06fd4d09d2412d98152 Mon Sep 17 00:00:00 2001
From: Pierre-Francois Loos <pierrefrancois.loos@gmail.com>
Date: Thu, 23 Jul 2020 17:06:35 +0200
Subject: [PATCH] Sec 3.3

---
 RapportStage/Rapport.tex | 14 ++++++++------
 1 file changed, 8 insertions(+), 6 deletions(-)

diff --git a/RapportStage/Rapport.tex b/RapportStage/Rapport.tex
index 7b92fad..3173c52 100644
--- a/RapportStage/Rapport.tex
+++ b/RapportStage/Rapport.tex
@@ -461,19 +461,21 @@ Moreover they proved that the extrapolation formulas of Cremer and He \cite{Crem
 
 \subsection{The singularity structure}
 
-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 his co-workers were trying to find the dominant singularity causing the divergence. They regrouped singularities in two classes: the $\alpha$ singularities which have unit order imaginary parts and the $\beta$ singularities which have very small imaginary parts. The singularities $\alpha$ are related to large avoided crossing between the ground state and a low-lying excited states. Whereas the singularities $\beta$ 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 using a previous work of Stillinger \cite{Stillinger_2000}. 
+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. 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 using a previous work of Stillinger \cite{Stillinger_2000}. 
 
-To understand the convergence properties at $\lambda=1$ of the perturbation series one need to look at the whole complex plane. In particular for real negative value of $\lambda$. If $\lambda$ is negative, the Coulomb interaction becomes attractive but the mean field stays repulsive as it is proportional to $(1-\lambda)$.
+To understand the convergence properties of the perturbation series at $\lambda=1$, one must look at the whole complex plane, in particular, for negative (i.e., 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 $(1-\lambda)$:
 
 \begin{equation}\label{eq:HamiltonianStillinger}
-    \bH(\lambda)=\sum\limits_{i=1}^{n}\left[ \underbrace{-\frac{1}{2}\grad_i^2 - \sum\limits_{k=1}^{N} \frac{Z_k}{|\vb{r}_i-\vb{R}_k|}}_{\text{Independent of }\lambda} + \overbrace{(1-\lambda)V_i^{\text{HF}}}^{\textcolor{red}{Repulsive}}+\underbrace{\lambda\sum\limits_{i<j}^{n}\frac{1}{|\vb{r}_i-\vb{r}_j|}}_{\textcolor{blue}{Attractive}}  \right]
+    \bH(\lambda)=\sum\limits_{i=1}^{n}\left[ \underbrace{-\frac{1}{2}\grad_i^2 - \sum\limits_{k=1}^{N} \frac{Z_k}{|\vb{r}_i-\vb{R}_k|}}_{\text{Independent of $\lambda$}} + \overbrace{(1-\lambda)V_i^{\text{HF}}}^{\textcolor{red}{\text{Repulsive}}}+\underbrace{\lambda\sum\limits_{i<j}^{n}\frac{1}{|\vb{r}_i-\vb{r}_j|}}_{\textcolor{blue}{\text{Attractive}}}  \right]
 \end{equation}
 
-The major difference between those two terms is that the repulsive mean field is localized around nucleus whereas the electrons interactions persist away from nuclei. If $\lambda$ becomes more and more negative the mean field becomes more and more repulsive so the nuclei can not bind the electrons anymore because the electron-nucleus attraction is not scaled with $\lambda$. There is a real negative value $\lambda_c$ where the electrons form a bound cluster and goes to infinity. According to Baker this value is a critical point of the system and by analogy with thermodynamics the energy $E(\lambda)$ exhibits a singularity at $\lambda_c$ \cite{Baker_1971}. At this point the system undergo a phase transition and a symmetry breaking. Beyond $\lambda_c$ there is a continuum of eigenstates with electrons dissociated from the nucleus.
+The major difference between those two terms is that the repulsive mean field is localized 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 the electron-nucleus attraction. For $\lambda = \lambda_c$, the electrons form a bound cluster that dissociates from the nuclei \titou{by going towards $\lambda = -\infty$.} 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 $\lambda_c$. At this point the system undergo a phase transition \titou{and a symmetry breaking}. \titou{Beyond $\lambda_c$ there is a continuum of eigenstates with electrons dissociated from the nucleus.}
 
-This reasoning is done on the exact Hamiltonian and energy, this is the exact energy which exhibits this singularity on the negative real axis. But in a finite basis set, 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, that in a finite basis set the critical point on the real axis is modeled by a cluster of sharp avoided crossings with diffuse functions, equivalently by a cluster of $\beta$ singularities in the negative half plane. They explain that Olsen et al., because they used a $2\times2$ model, only observed the first singularity of this cluster of singularities causing the divergence.
+This reasoning is done on the exact Hamiltonian and energy, this is the exact energy which exhibits this singularity on the negative real axis. 
+\titou{The Hamiltonian can be exact in a finite basis. I wonder if it would not be more judicious to talk about complete/infinite Hilbert space and finite Hilbert space.}
+However, in a finite basis set, 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{} that in a finite basis set the critical point on the real axis is modeled by a cluster of sharp avoided crossings with diffuse functions, equivalently by a cluster of $\beta$ singularities in the negative half plane. They explain that Olsen et al., \cite{} because they used a $2\times2$ model, only observed the first singularity of this cluster of singularities causing the divergence.
 
-Finally, it was shown that $\beta$ singularities are very sensitive to changes of the basis set but not to the stretching of the system. On the contrary $\alpha$ singularities are relatively insensitive to the basis sets but very sensitive to bond stretching. According to Goodson the singularity structure from molecules stretched from the equilibrium geometry is difficult \cite{Goodson_2004}, this is consistent with the observation of Olsen and co-workers on the \ce{HF} molecule at equilibrium geometry and stretched geometry \cite{Olsen_2000}. To our knowledge the effect of bond stretching on singularities, its link with spin contamination and symmetry breaking of the wave function hasn't been as well understood as the ionization effect and its link with diffuse function. In this work we try to improve our understanding of the effect of symmetry breaking on the singularities of $E(\lambda)$ and we hope that it will lead to a deeper understanding of perturbation theory.
+Finally, it was shown that $\beta$ singularities are very sensitive to changes of the basis set but not to the stretching of the system. 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 from molecules stretched from the equilibrium geometry is difficult, this is consistent with the observation of Olsen and co-workers  \cite{Olsen_2000} on the \ce{HF} molecule at equilibrium geometry and stretched geometry. To 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 effect and its link with diffuse function. In this work we shall try to improve our understanding of the effect of symmetry breaking on the singularities of $E(\lambda)$ and we hope that it will lead to a deeper understanding of perturbation theory.
 
 \subsection{The physics of quantum phase transition}