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RapportStage/Rapport.tex
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@ -380,8 +380,8 @@ where $\braket{ij}{ab}$ is the two-electron integral
Additionally, we will consider two other partitioning. The electronic Hamiltonian can be separated in a one-electron part and in a two-electron part as seen previously. We can use this separation to create two other partitioning:
\begin{itemize}
\item The Weak Correlation partitioning in which the one-electron part is consider as the unperturbed Hamiltonian $\bH^{(0)}$ and the two-electron part is the perturbation operator $\bV$.
\item The Strong Coupling partitioning where the two operators are inverted compared to the weak correlation partitioning.
\item The Weak Correlation (WC) partitioning in which the one-electron part is consider as the unperturbed Hamiltonian $\bH^{(0)}$ and the two-electron part is the perturbation operator $\bV$.
\item The Strong Coupling (SC) partitioning where the two operators are inverted compared to the weak correlation partitioning.
\end{itemize}
%============================================================%
@ -504,7 +504,7 @@ The laplacian operators are the kinetic operators for each electrons and $r_{12}
\item Radius of the spherium that ultimately dictates the correlation regime.
\end{itemize}
In the restricted Hartree-Fock formalism (RHF), the wave function cannot model properly the physics of the system at large R because the spatial orbitals are restricted to be the same. If the spatial orbitals are the same a fortiori it cannot represent two electrons on opposite side of the sphere. In the unrestricted Hartree-Fock (UHF) formalism there is a critical value of R, called the Coulson-Fischer point \cite{Coulson_1949}, at which a second UHF solution appears. This solution is symmetry-broken as the two electrons tends to localize on opposite side of the sphere. The UHF wave function is defined as:
In the RHF formalism, the wave function cannot model properly the physics of the system at large R because the spatial orbitals are restricted to be the same. If the spatial orbitals are the same a fortiori it cannot represent two electrons on opposite side of the sphere. In the UHF formalism there is a critical value of R, called the Coulson-Fischer point \cite{Coulson_1949}, at which a second UHF solution appears. This solution is symmetry-broken as the two electrons tends to localize on opposite side of the sphere. The UHF wave function is defined as:
\begin{equation}\label{eq:UHF_WF}
\Psi_{\text{UHF}}(\theta_1,\theta_2)=\phi_\alpha(\theta_1)\phi_\beta(\theta_2)
@ -602,7 +602,7 @@ In addition, we can also consider the symmetry-broken solutions beyond their res
\subsection{Evolution of the radius of convergence}
In this part, we will try to investigate how some parameters of $\bH(\lambda)$ influence the radius of convergence of the perturbation series. The radius of convergence is equal to the distance of the closest singularity to the origin of $E(\lambda)$. Hence we need to determine the locations of the exceptional points to obtain information on the convergence properties. To find them we solve simultaneously the equations \eqref{eq:PolChar} and \eqref{eq:DPolChar}. The equation \eqref{eq:PolChar} is the well-known secular equation giving the energies of the system, if an energy is also solution of \eqref{eq:DPolChar} then this energy is degenerate. In this case the energies obtained are dependent of $\lambda$ so solving those equations with respect to $E$ and $\lambda$ gives the value of $\lambda$ where two energies are degenerate i.e. the exceptional points.
In this part, we will try to investigate how some parameters of $\bH(\lambda)$ influence the radius of convergence of the perturbation series. The radius of convergence is equal to the distance of the closest singularity to the origin of $E(\lambda)$. Hence we need to determine the locations of the exceptional points to obtain information on the convergence properties. To find them we solve simultaneously the equations \eqref{eq:PolChar} and \eqref{eq:DPolChar}. The equation \eqref{eq:PolChar} is the well-known secular equation giving the energies of the system, if an energy is also solution of \eqref{eq:DPolChar} then this energy is degenerate. In this case the energies obtained are dependent of $\lambda$ so solving those equations with respect to $E$ and $\lambda$ gives the value of $\lambda$ where two energies are degenerate. These degeneracies can be conical intersections between two states with different symmetry for real value of $\lambda$ or exceptional points between two states with the same symmetry for complex value of $\lambda$.
\begin{equation}\label{eq:PolChar}
\text{det}[E-\bH(\lambda)]=0
@ -638,8 +638,8 @@ To simplify the problem, it is convenient to only consider basis functions with
\end{equation}
where $P_l$ are the Legendre polynomial and $\theta$ is the interelectronic angle.
Then using this basis set we can compare the different partitioning of \autoref{sec:AlterPart}. \autoref{fig:RadiusPartitioning} shows the evolution of the radius of convergence $R_{\text{CV}}$ in function of $R$ for the MP, the EN, the Weak Correlation and the Strong Coupling partitioning in a minimal basis i.e. $P_0$ and $P_1$ and in the same basis augmented with $P_2$. We see that the radius of convergence of the strong coupling partitioning is growing with R whereas it is decreasing for the three others partitioning. This result was expected because the three decreasing partitioning use a weakly correlated reference so $\bH^{(0)}$ is a good approximation for small $R$. On the contrary, the strong coupling one uses a strongly correlated reference so this series converge better when the electron are strongly correlated i.e. when $R$ is large for the spherium model.
The MP partitioning is always better than the weak correlation in \autoref{fig:RadiusPartitioning}. In the weak correlation partitioning the powers of $R$ are well-separated so each term of the series is a different power of $R$. Whereas the MP reference is proportionnal to $R^{-1}$ and $R^{-2}$ so the MP series is not well-defined in terms of powers of $R$. Moreover it can be proved that the $n$-th order energy of the weak correlation series can be obtained as a Taylor approximation of MP$n$ respective to $R$. It seems that the EN partitioning is better than the MP one for very small R in the minimal basis. In fact, it is just an artifact of the minimal basis because in the minimal basis augmented with $P_2
Then using this basis set we can compare the different partitioning of \autoref{sec:AlterPart}. \autoref{fig:RadiusPartitioning} shows the evolution of the radius of convergence $R_{\text{CV}}$ in function of $R$ for the MP, the EN, the WC and the SC partitioning in a minimal basis i.e. $P_0$ and $P_1$ and in the same basis augmented with $P_2$. We see that the radius of convergence of the SC partitioning is growing with R whereas it is decreasing for the three others partitioning. This result was expected because the three decreasing partitioning use a weakly correlated reference so $\bH^{(0)}$ is a good approximation for small $R$. On the contrary, the strong coupling one uses a strongly correlated reference so this series converge better when the electron are strongly correlated i.e. when $R$ is large for the spherium model.
The MP partitioning is always better than the weak correlation in \autoref{fig:RadiusPartitioning}. In the weak correlation partitioning the powers of $R$ are well-separated so each term of the series is a different power of $R$. Whereas the MP reference is proportionnal to $R^{-1}$ and $R^{-2}$ so the MP series is not well-defined in terms of powers of $R$. Moreover it can be proved that the $n$-th order energy of the weak correlation series can be obtained as a Taylor approximation of MP$n$ respective to $R$. It seems that the EN partitioning is better than the MP one for very small R in the minimal basis. In fact, it is just an artifact of the minimal basis because in the minimal basis augmented with $P_2
$ (and in larger basis set) the MP series has a greater radius of convergence for all value of $R$.
\begin{figure}[h!]
@ -662,7 +662,7 @@ $ (and in larger basis set) the MP series has a greater radius of convergence fo
\begin{table}[h!]
\centering
\caption{\centering Dominant singularity in the CSF basis set ($n=8$) for various value of R. The first line is the value for the MP partitioning and the second for the WC one.}
\caption{\centering Dominant singularity in the CSF basis set ($K=8$) for various value of R. The first line is the value for the MP partitioning and the second for the WC one.}
\begin{tabular}{cccccccc}
\hline
\hline
@ -715,7 +715,7 @@ In this study we have used spherical harmonics (or combination of spherical harm
\subsection{Exceptional points in the UHF formalism}\label{sec:uhfSing}
In the RHF case there are only $\alpha$ singularities and large avoided crossings but we can see in \autoref{fig:UHFMiniBas} that in the UHF case there are sharp avoided crossings which are connected to $\beta$ singularities. For example at $R=10$ the pair of singularities connected to the avoided crossing between s\textsuperscript{2} and sp\textsubscript{z} $^{3}P$ is $0.999\pm0.014i$. And the one between sp\textsubscript{z} $^{3}P$ and p\textsubscript{z}\textsuperscript{2} is connected with the singularities $2.207\pm0.023i$. However in the spherium the electrons can't be ionized so those singularities are not the $\beta$ singularities highlighted by Sergeev and Goodson \cite{Sergeev_2005}. We can see in \autoref{fig:UHFEP} that the degeneracy between the s\textsuperscript{2} singlet and the sp\textsubscript{z} triplet at $R=3/2$. For $R>3/2$, it becomes an avoided crossing on the real axis and the degeneracies are moved in the complex plane. The wave function is spin contaminated for $R>3/2$ this why the s\textsuperscript{2} singlet energy can not cross the sp\textsubscript{z} triplet curves anymore. When $R$ increases this avoided crossing becomes sharper. As presented before $\beta$ singularities are linked to quantum phase transition so it seems that this singularity is linked to the spin symmetry breaking of the UHF wave function. The fact that a similar pair of $\beta$ singularities appears for $R<-75/62$ confirms this assumption. A second pair of $\beta$ singularities appear for $R\gtrsim 2.5$, this is probably due to an excited-state quantum phase transition but this still need to be investigated.
In the RHF case there are only $\alpha$ singularities and large avoided crossings but we can see in \autoref{fig:UHFMiniBas} that in the UHF case there are sharp avoided crossings which are known to be connected to $\beta$ singularities. For example at $R=10$ the pair of singularities connected to the avoided crossing between s\textsuperscript{2} and sp\textsubscript{z} $^{3}P$ is $0.999\pm0.014i$. And the one between sp\textsubscript{z} $^{3}P$ and p\textsubscript{z}\textsuperscript{2} is connected with the singularities $2.207\pm0.023i$. However in the spherium the electrons can't be ionized so those singularities are not the $\beta$ singularities highlighted by Sergeev and Goodson \cite{Sergeev_2005}. We can see in \autoref{fig:UHFEP} that the degeneracy between the s\textsuperscript{2} singlet and the sp\textsubscript{z} triplet at $R=3/2$. For $R>3/2$, it becomes an avoided crossing on the real axis and the degeneracies are moved in the complex plane. The wave function is spin contaminated for $R>3/2$ this why the s\textsuperscript{2} singlet energy can not cross the sp\textsubscript{z} triplet curves anymore. When $R$ increases this avoided crossing becomes sharper. As presented before $\beta$ singularities are linked to quantum phase transition so it seems that this singularity is linked to the spin symmetry breaking of the UHF wave function. The fact that a similar pair of $\beta$ singularities appears for $R<-75/62$ confirms this assumption. A second pair of $\beta$ singularities appear for $R\gtrsim 2.5$, this is probably due to an excited-state quantum phase transition but this still need to be investigated.
\begin{figure}[h!]
\centering
@ -742,6 +742,8 @@ For a non-Hermitian Hamiltonian the exceptional points can lie on the real axis.
\section{Conclusion}
\newpage
\printbibliography