modifications in Part 1

MidTermSummary.tex

@@ -8,11 +8,10 @@
 \usepackage[top=3cm,bottom=5cm,left=3cm,right=3cm]{geometry}
 \usepackage{caption}
 
-\title{Summary sheet: half of the internship}
+\title{Internship Summary: Mid-Term Review}
+
 \author{Antoine Marie}
 
-
-
 \begin{document}
 
 \maketitle
@@ -20,28 +19,27 @@
 \section{Generalities}
 
 \subsection{Objectives}
-Our goal is to understand the physics of exceptional points and perturbation series i.e. where are EP localize in the complex plane and how do they affect the convergence properties of the perturbation series. To do this we use the spherium model as a benchmark.
+Our goal is to understand the physics of exceptional points (EPs) and perturbation series, i.e., where are EPs localize in the complex plane and how do they affect the convergence properties of the perturbation series. To do this we use the electronic ground-state of the spherium model (i.e., two opposite-spin electrons restricted to remain on a surface of a sphere of radius $R$) as a playground.
 
 \subsection{Variables}
 
-There are many variables that influence the physics of exceptional points and we need to rationalize how each variable affects the exceptional points.
-
+There are many variables that influence the physics of EPs and we must rationalize how each variable affects the location of these EPs.
 \begin{itemize}
-\item Partitioning of the Hamiltonian: Weak correlation reference (RHF or UHF, Möller-Plesset or Epstein-Nesbet), Strongly correlated reference
-\item The basis set: minimal basis or infinite basis, localized or not basis functions
-\item Radius of the spherium
+	\item Partitioning of the Hamiltonian and the actual zeroth-order reference: weak correlation reference [restricted Hartree-Fock (RHF) or unrestricted Hartree-Fock (UHF) references, M{\o}ller-Plesset (MP) or Epstein-Nesbet (EN) partitioning], or strongly correlated reference.
+	\item Basis set: minimal basis or infinite (i.e., complete) basis made of localized or delocalized basis functions
+	\item Radius of the spherium that ultimately dictates the correlation regime, i.e., weak correlation regime at small $R$ where the kinetic energy dominates, or strong correlation regime where the electron repulsion term drives the physics.
 \end{itemize}
 
-\subsection{"Classification" of the bibliography}
+\subsection{``Classification'' of the bibliography}
 
 We can categorize what have been done already in 3 groups:
 
 \begin{itemize}
-\item Gill, Handy, Nobes ... in the late 80's noticed that there are links between deceptive/erratic convergence of the Möller-Plesset pertubation series and spin contamination of the wavefunction.
+\item Gill, Handy, Nobes, \ldots in the late 80's noticed that there are links between deceptive/erratic convergence of the MP pertubation series and spin contamination of the wavefunction.
 
-\item Chaudhuri, Olsen ... in the late 90's highlighted links between singularities in the complex plane and convergence/divergence schemes of the MP series with a two-state model.
+\item Chaudhuri, Olsen, \ldots in the late 90's highlighted links between singularities in the complex plane and convergence/divergence schemes of the MP series with a two-state model.
 
-\item Sergeev, Goodson, Stillinger in the 00's studied this problem from a more mathematical point of view: $\alpha / \beta$ classification of the singularities, Pade approximant, critical point on the real axis ...
+\item Sergeev, Goodson, Stillinger in the 2000's studied this problem from a more mathematical point of view: $\alpha$/$\beta$ classification of the singularities, Pad\'e approximant, critical point on the real axis, etc.
 \end{itemize}
 
 \section{Results}