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Pierre-Francois Loos 2020-07-31 15:11:49 +02:00
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RapportStage/Rapport.tex
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@ -707,7 +707,7 @@ In addition, we can also consider the symmetry-broken solutions beyond their res
\subsection{Evolution of the radius of convergence} \subsection{Evolution of the radius of convergence}
In this subsection, we investigate how the partitioning of $\hH(\lambda)$ influence the radius of convergence of the perturbation series. Let us remind the reader that the radius of convergence is equal to the distance of the closest singularity to the origin of $E(\lambda)$. Hence, we have to determine the locations of the EPs to obtain information on the convergence properties of the perturbative series. To find them we solve simultaneously the following equations \antoine{\cite{Cejnar_2007}}: In this subsection, we investigate how the partitioning of $\hH(\lambda)$ influence the radius of convergence of the perturbation series. Let us remind the reader that the radius of convergence is equal to the distance of the closest singularity to the origin of $E(\lambda)$. Hence, we have to determine the locations of the EPs to obtain information on the convergence properties of the perturbative series. To find them we solve simultaneously the following equations \cite{Cejnar_2007}:
\begin{subequations} \begin{subequations}
\begin{align} \begin{align}
\label{eq:PolChar} \label{eq:PolChar}
@ -748,7 +748,9 @@ To simplify the problem, it is convenient to only consider basis functions of a
where $P_\ell$ are Legendre polynomials. where $P_\ell$ are Legendre polynomials.
Then, using this basis set we can compare the different partitioning of Sec.~\ref{sec:AlterPart}. Figure \ref{fig:RadiusPartitioning} shows the evolution of the radius of convergence $R_{\text{CV}}$ as a function of $R$ for the MP, the EN, the WC and the SC partitioning in a minimal basis (i.e., consisting of $P_0$ and $P_1$) of size $K = 2$, and in the same basis augmented with $P_2$ ($K = 3$). We see that, for the SC partitioning, $R_{\text{CV}}$ increases with $R$ whereas it is decreasing for the three others partitioning. This result is expected because the MP, EN, and WC partitioning use a weakly correlated reference so $\hH^{(0)}$ is a good approximation for small $R$. On the contrary, the SC partitioning consider naturally a strongly correlated reference so the SC series converges far better when the electron are strongly correlated, i.e., when $R$ is large in the spherium model. Then, using this basis set we can compare the different partitioning of Sec.~\ref{sec:AlterPart}. Figure \ref{fig:RadiusPartitioning} shows the evolution of the radius of convergence $R_{\text{CV}}$ as a function of $R$ for the MP, the EN, the WC and the SC partitioning in a minimal basis (i.e., consisting of $P_0$ and $P_1$) of size $K = 2$, and in the same basis augmented with $P_2$ ($K = 3$). We see that, for the SC partitioning, $R_{\text{CV}}$ increases with $R$ whereas it is decreasing for the three others partitioning. This result is expected because the MP, EN, and WC partitioning use a weakly correlated reference so $\hH^{(0)}$ is a good approximation for small $R$. On the contrary, the SC partitioning consider naturally a strongly correlated reference so the SC series converges far better when the electron are strongly correlated, i.e., when $R$ is large in the spherium model.
Interestingly, the radius of convergence associated with the SC partitioning is greater than one for a greater range of radii for $K = 2$ than $K = 3$. \antoine{In the complete basis the basis the radius of convergence of the SC partitioning is greater than one only for very large value of R.} Interestingly, the radius of convergence associated with the SC partitioning is greater than one for a greater range of radii for $K = 2$ than $K = 3$.
In the complete basis limit, the radius of convergence of the SC partitioning is greater than unity only for very large value of $R$.
\begin{figure} \begin{figure}
\centering \centering
\includegraphics[width=0.49\textwidth]{PartitioningRCV2.pdf} \includegraphics[width=0.49\textwidth]{PartitioningRCV2.pdf}
@ -758,7 +760,7 @@ Interestingly, the radius of convergence associated with the SC partitioning is
\end{figure} \end{figure}
The MP partitioning is always better than WC in Fig.~\ref{fig:RadiusPartitioning}. In the WC partitioning the powers of $R$ (the zeroth-order scales as $R^{-2}$ while the perturbation scales as $R^{-1}$) are well-separated so each term of the series has a well-defined power of $R$. This is not the case for the MP series. The MP partitioning is always better than WC in Fig.~\ref{fig:RadiusPartitioning}. In the WC partitioning the powers of $R$ (the zeroth-order scales as $R^{-2}$ while the perturbation scales as $R^{-1}$) are well-separated so each term of the series has a well-defined power of $R$. This is not the case for the MP series.
Interestingly, it can be proved that the $m$th order energy of the WC series can be obtained as a Taylor series of MP$m$ with respect to $R$. Interestingly, it can be proved that the $m$th order energy of the WC series can be obtained as a Taylor series of MP$m$ with respect to $R$.
It seems that the EN is better than MP for very small $R$ in the minimal basis. In fact, it is just an artefact of the minimal basis because, for $K = 3$, the MP series has a greater radius of convergence for all values of $R$ \antoine{and this is still true for $K>3$.} It seems that the EN is better than MP for very small $R$ in the minimal basis. In fact, it is just an artefact of the minimal basis because, for $K = 3$, the MP series has a greater radius of convergence for all values of $R$. It holds true for $K>3$.
Figure \ref{fig:RadiusBasis} shows that the radius of convergence is not very sensitive to the size of the basis set. The CSFs have all the same spin and spatial symmetries so we expect that the singularities obtained within this basis set will be $\alpha$ singularities. Table \ref{tab:SingAlpha} shows that the singularities considered in this case are indeed $\alpha$ singularities. This is consistent with the observation of Goodson and Sergeev \cite{Goodson_2004} who stated that $\alpha$ singularities are relatively insensitive to the basis set size. The discontinuities observed in Fig.~\ref{fig:RadiusBasis} for the MP partitioning are due to changes in dominant singularity. We can observe this change in Table \ref{tab:SingAlpha}, the value for $R=1$ and $R=2$ are respectively in the positive and negative planes. Figure \ref{fig:RadiusBasis} shows that the radius of convergence is not very sensitive to the size of the basis set. The CSFs have all the same spin and spatial symmetries so we expect that the singularities obtained within this basis set will be $\alpha$ singularities. Table \ref{tab:SingAlpha} shows that the singularities considered in this case are indeed $\alpha$ singularities. This is consistent with the observation of Goodson and Sergeev \cite{Goodson_2004} who stated that $\alpha$ singularities are relatively insensitive to the basis set size. The discontinuities observed in Fig.~\ref{fig:RadiusBasis} for the MP partitioning are due to changes in dominant singularity. We can observe this change in Table \ref{tab:SingAlpha}, the value for $R=1$ and $R=2$ are respectively in the positive and negative planes.
@ -881,7 +883,7 @@ This confirms that $\beta$ singularities can occur for other types of transition
It would be interesting to investigate the difference between the different type of symmetry breaking and how it affects the singularity structure. It would be interesting to investigate the difference between the different type of symmetry breaking and how it affects the singularity structure.
Moreover the singularity structure in the non-Hermitian case still need to be investigated. Moreover the singularity structure in the non-Hermitian case still need to be investigated.
In the holomorphic domain, some singularities lie on the real axis and it would also be interesting to look at the differences between the different symmetry breaking and their respective holomorphic domain. In the holomorphic domain, some singularities lie on the real axis and it would also be interesting to look at the differences between the different symmetry breaking and their respective holomorphic domain.
\antoine{Furthermore, in this study we have used spherical harmonics (or combination of spherical harmonics) as basis functions which have a delocalized nature. It would also be interesting to investigate the use of localized basis functions \cite{Seidl_2018} (for example gaussians) because these functions would be more adapted to describe the strongly correlated regime. More generally, to investigate the effect of the type of basis on the physics of EPs.} Furthermore, in this study we have used spherical harmonics (or combination of spherical harmonics) as basis functions which have a delocalized nature. It would also be interesting to investigate the use of localized basis functions \cite{Seidl_2018} (for example gaussians) because these functions would be more adapted to describe the strongly correlated regime. %More generally, to investigate the effect of the type of basis on the physics of EPs.
To conclude, this work shows that our understanding of the singularity structure of the energy is still incomplete but we hope that it opens new perspectives for the understanding of the physics of EPs in electronic structure theory. To conclude, this work shows that our understanding of the singularity structure of the energy is still incomplete but we hope that it opens new perspectives for the understanding of the physics of EPs in electronic structure theory.