minor corrections

RapportStage/Rapport.tex

@@ -660,7 +660,7 @@ This process provides the three following solutions valid for all value of $R$,
 \end{itemize}
 \titou{STOPPED HERE.}
 
-In addition, the minimization process gives also the well-known symmetry-broken UHF (sb-UHF) solution. In this case the Coulson-Fischer point associated to this solution is $R=3/2$. For $R>3/2$ the sb-UHF solution is the global minimum of the HF equations and the RHF solution presented before is a local minimum. This solution corresponds to the configuration with the electron $\alpha$ on one side of the sphere and the electron $\beta$ on the opposite side and the configuration the other way round. The electrons can be on opposite sides of the sphere because the choice of p\textsubscript{z} as a basis function induced a privileged axis on the sphere for the electrons. This solution has the energy \eqref{eq:EsbUHF} for $R>3/2$.
+In addition, the minimization process gives also the well-known symmetry-broken UHF (sb-UHF) solution. In this case the Coulson-Fischer point associated to this solution is $R=3/2$. For $R>3/2$ the sb-UHF solution is the global minimum of the HF equations and the RHF solution presented before is a local minimum. This solution corresponds to the configuration with the electron $\alpha$ in an orbital on one side of the sphere and the electron $\beta$ in a symmetric orbital on the opposite side and the configuration the other way round. The electrons can be on opposite sides of the sphere because the choice of p\textsubscript{z} as a basis function induced a privileged axis on the sphere for the electrons. This solution has the energy \eqref{eq:EsbUHF} for $R>3/2$.
 
 \begin{equation}\label{eq:EsbUHF}
 	E_{\text{sb-UHF}}=-\frac{75}{112R^3}+\frac{25}{28R^2}+\frac{59}{84R}
@@ -693,13 +693,13 @@ Exact 	& 9.783874 & 0.852781 & 0.391959 & 0.247898 & 0.139471 & 0.064525 & 0.005
     \label{fig:SpheriumNrj}
 \end{wrapfigure}
 
-There is also another symmetry-broken solution for $R>75/38$ but this one corresponds to a maximum of the HF equations. This solution is associated with another type of symmetry breaking somewhat less known. It corresponds to a configuration where both electrons are on the same side of the sphere, in the same spatial orbital. This solution is called symmetry-broken RHF (sb-RHF). At the critical value of $R$, the repulsion of the two electrons on the same side of the sphere maximizes more the energy than the kinetic energy of the p\textsubscript{z} orbitals. This symmetry breaking is associated with a charge density wave, the system oscillates between the situations with the two electrons on each side \cite{GiulianiBook}.
+There is also another symmetry-broken solution for $R>75/38$ but this one corresponds to a maximum of the HF equations. This solution is associated with another type of symmetry breaking somewhat less known. It corresponds to a configuration where both electrons are on the same side of the sphere, in the same spatial orbital. This solution is called symmetry-broken RHF (sb-RHF). At the critical value of $R$, the repulsion of the two electrons being in the same orbital on the same side of the sphere maximizes more the energy than the kinetic energy of the two electrons in the p\textsubscript{z} orbital. This configuration break the spatial symmetry of charge. Hence this symmetry breaking is associated with a charge density wave, the system oscillates between the situations with the two electrons on each side \cite{GiulianiBook}.
 
 \begin{equation}
 E_{\text{sb-RHF}}=\frac{75}{88R^3}+\frac{25}{22R^2}+\frac{91}{66R}
 \end{equation}
 
-We can also consider negative values of $R$. This corresponds to the situation where one of the electrons is replaced by a positron. There are also a sb-RHF ($R<-3/2$) and a sb-UHF ($R<-75/38$) solution for negative values of $R$ (see Fig.~\ref{fig:SpheriumNrj}) but in this case the sb-RHF solution is a minimum and the sb-UHF is a maximum. Indeed, the sb-RHF states minimize the attraction energy by placing the electron and the positron on the same side of the sphere. And the sb-UHF states maximize the energy because the two attracting particles are on opposite sides of the sphere.
+We can also consider negative values of $R$. This corresponds to the situation where one of the electrons is replaced by a positron (the anti-particle of the electron). There are also a sb-RHF ($R<-3/2$) and a sb-UHF ($R<-75/38$) solution for negative values of $R$ (see Fig.~\ref{fig:SpheriumNrj}) but in this case the sb-RHF solution is a minimum and the sb-UHF is a maximum of the HF equations. Indeed, the sb-RHF state minimizes the attraction energy by placing the electron and the positron on the same side of the sphere. And the sb-UHF state maximizes the energy because the two attracting particles are on opposite sides of the sphere.
 
 In addition, we can also consider the symmetry-broken solutions beyond their respective Coulson-Fischer points by analytically continuing their respective energies leading to the so-called holomorphic solutions \cite{Hiscock_2014, Burton_2019, Burton_2019a}. All those energies are plotted in Fig.~\ref{fig:SpheriumNrj}. The dotted curves corresponds to the holomorphic domain of the energies.