ajout des slides

SlideToulouse/SlideToulouse.bib

@@ -0,0 +1,104 @@
+
+@article{gill_why_1988,
+	title = {Why does unrestricted Mo/ller–Plesset perturbation theory converge so slowly for spin‐contaminated wave functions?},
+	volume = {89},
+	issn = {0021-9606},
+	url = {https://aip.scitation.org/doi/abs/10.1063/1.455312},
+	doi = {10.1063/1.455312},
+	pages = {7307--7314},
+	number = {12},
+	journaltitle = {The Journal of Chemical Physics},
+	shortjournal = {J. Chem. Phys.},
+	author = {Gill, Peter M. W. and Pople, John A. and Radom, Leo and Nobes, Ross H.},
+	urldate = {2020-06-10},
+	date = {1988-12-15},
+	note = {Publisher: American Institute of Physics},
+	file = {Snapshot:/home/amarie/Zotero/storage/4SDW37YN/1.html:text/html}
+}
+
+@article{sergeev_nature_2005,
+	title = {On the nature of the Møller-Plesset critical point},
+	volume = {123},
+	issn = {0021-9606},
+	url = {https://aip.scitation.org/doi/10.1063/1.1991854},
+	doi = {10.1063/1.1991854},
+	pages = {064105},
+	number = {6},
+	journaltitle = {The Journal of Chemical Physics},
+	shortjournal = {J. Chem. Phys.},
+	author = {Sergeev, Alexey V. and Goodson, David Z. and Wheeler, Steven E. and Allen, Wesley D.},
+	urldate = {2020-06-10},
+	date = {2005-08-08},
+	note = {Publisher: American Institute of Physics},
+	file = {Snapshot:/home/amarie/Zotero/storage/5N2L8RQ6/1.html:text/html}
+}
+
+@article{sergeev_singularities_2006,
+	title = {Singularities of Møller-Plesset energy functions},
+	volume = {124},
+	issn = {0021-9606},
+	url = {https://aip.scitation.org/doi/10.1063/1.2173989},
+	doi = {10.1063/1.2173989},
+	pages = {094111},
+	number = {9},
+	journaltitle = {The Journal of Chemical Physics},
+	shortjournal = {J. Chem. Phys.},
+	author = {Sergeev, Alexey V. and Goodson, David Z.},
+	urldate = {2020-06-10},
+	date = {2006-03-07},
+	note = {Publisher: American Institute of Physics},
+	file = {Snapshot:/home/amarie/Zotero/storage/IP28R6TR/1.html:text/html}
+}
+
+@article{olsen_divergence_2000,
+	title = {Divergence in Møller–Plesset theory: A simple explanation based on a two-state model},
+	volume = {112},
+	issn = {0021-9606},
+	url = {https://aip.scitation.org/doi/10.1063/1.481611},
+	doi = {10.1063/1.481611},
+	shorttitle = {Divergence in Møller–Plesset theory},
+	pages = {9736--9748},
+	number = {22},
+	journaltitle = {The Journal of Chemical Physics},
+	shortjournal = {J. Chem. Phys.},
+	author = {Olsen, Jeppe and Jørgensen, Poul and Helgaker, Trygve and Christiansen, Ove},
+	urldate = {2020-06-10},
+	date = {2000-05-31},
+	note = {Publisher: American Institute of Physics},
+	file = {Snapshot:/home/amarie/Zotero/storage/NNNBDR3R/1.html:text/html}
+}
+
+@article{loos_ground_2009,
+	title = {Ground state of two electrons on a sphere},
+	volume = {79},
+	url = {https://link.aps.org/doi/10.1103/PhysRevA.79.062517},
+	doi = {10.1103/PhysRevA.79.062517},
+	abstract = {We have performed a comprehensive study of the singlet ground state of two electrons on the surface of a sphere of radius R. We have used electronic structure models ranging from restricted and unrestricted Hartree-Fock theories to explicitly correlated treatments, the last of which leads to near-exact wave functions and energies for any value of R. Møller-Plesset energy corrections (up to fifth-order) are also considered, as well as the asymptotic solution in the large-R regime.},
+	pages = {062517},
+	number = {6},
+	journaltitle = {Physical Review A},
+	shortjournal = {Phys. Rev. A},
+	author = {Loos, Pierre-François and Gill, Peter M. W.},
+	urldate = {2020-06-11},
+	date = {2009-06-30},
+	note = {Publisher: American Physical Society},
+	file = {APS Snapshot:/home/amarie/Zotero/storage/WWCNWCPS/PhysRevA.79.html:text/html;Submitted Version:/home/amarie/Zotero/storage/5DIQ69YK/Loos and Gill - 2009 - Ground state of two electrons on a sphere.pdf:application/pdf}
+}
+
+@article{gill_deceptive_1986,
+	title = {Deceptive convergence in møller-plesset perturbation energies},
+	volume = {132},
+	issn = {0009-2614},
+	url = {http://www.sciencedirect.com/science/article/pii/0009261486806868},
+	doi = {10.1016/0009-2614(86)80686-8},
+	abstract = {Meller-Plesset perturbation calculations ({MPn}) up to fiftieth order, within both the restricted ({RHF}) and unrestricted Hartree-Fock ({UHF}) frameworks, have been used to examine the He2+2 ground-state potential curve. The bond lengths of the equilibrium and transition structures have been optimized at all orders of perturbation theory. It is found that {RMP} n describes the homolytic dissociation better than {UMPn} for all n {\textgreater} 2. This unexpected behaviour may be attributed to spin contamination in the {UHF} wavefunction. The {UMPn} barriers deceptively appear convergent for small n and the results may be indicative of dangers inherent generally in using the {UMP} approach with significantly spin-contaminated wavefunctions.},
+	pages = {16--22},
+	number = {1},
+	journaltitle = {Chemical Physics Letters},
+	shortjournal = {Chemical Physics Letters},
+	author = {Gill, Peter M. W. and Radom, Leo},
+	urldate = {2020-06-28},
+	date = {1986-11-28},
+	langid = {english},
+	file = {ScienceDirect Snapshot:/home/amarie/Zotero/storage/YV2LVWML/0009261486806868.html:text/html;Submitted Version:/home/amarie/Zotero/storage/U8VEPSSU/Gill and Radom - 1986 - Deceptive convergence in møller-plesset perturbati.pdf:application/pdf}
+}

SlideToulouse/main.tex

@@ -0,0 +1,542 @@
+\documentclass[xcolor=x11names,compress]{beamer}
+
+%% General document %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\usepackage{graphicx}
+\usepackage{tikz}
+\usetikzlibrary{decorations.fractals}
+\usepackage{mathpazo}
+\usepackage[english]{babel}
+\usepackage[T1]{fontenc}
+\usepackage[utf8]{inputenc}
+\usepackage{xcolor}
+\usepackage{siunitx}
+\usepackage{graphicx}
+\usepackage{physics}
+\usepackage{multimedia}
+\usepackage{subfigure}
+\usepackage[absolute,overlay]{textpos}
+\usepackage{ragged2e}
+\usepackage{amssymb}
+\usepackage[version=4]{mhchem}
+
+
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+
+%% Beamer Layout %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\useoutertheme[subsection=false,shadow]{miniframes}
+\useinnertheme{default}
+
+
+\setbeamerfont{title like}{shape=\scshape}
+\setbeamerfont{frametitle}{shape=\scshape}
+\setbeamerfont{framesubtitle}{size=\normalsize}
+\setbeamerfont{caption}{size=\scriptsize}
+\setbeamercolor*{lower separation line head}{bg=DeepSkyBlue4} 
+\setbeamercolor*{normal text}{fg=black,bg=white} 
+\setbeamercolor*{alerted text}{fg=red} 
+\setbeamercolor*{example text}{fg=black} 
+\setbeamercolor*{structure}{fg=black} 
+
+ 
+\setbeamercolor*{palette tertiary}{fg=black,bg=black!10} 
+\setbeamercolor*{palette quaternary}{fg=black,bg=black!10} 
+\setbeamercolor{caption name}{fg=DeepSkyBlue4}
+\setbeamercolor{title}{fg=DeepSkyBlue4}
+\setbeamercolor{itemize item}{fg=DeepSkyBlue4}
+\setbeamercolor{frametitle}{fg=DeepSkyBlue4}
+\renewcommand{\(}{\begin{columns}}
+\renewcommand{\)}{\end{columns}}
+\newcommand{\<}[1]{\begin{column}{#1}}
+\renewcommand{\>}{\end{column}}
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+
+\setbeamertemplate{navigation symbols}{} 
+\setbeamertemplate{footline}[frame number]
+\setbeamertemplate{caption}[numbered]
+\setbeamertemplate{section in toc}[ball]
+\setbeamertemplate{itemize items}[circle]
+\beamerboxesdeclarecolorscheme{clair}{Coral4}{Ivory2}
+\beamerboxesdeclarecolorscheme{foncé}{DarkSeaGreen4}{Ivory2}
+
+\title[Title]{Perturbation theories in the complex plane}
+\author[]{Antoine \textsc{Marie}}
+  \setbeamersize{text margin left=5mm}
+  \setbeamersize{text margin right=5mm}
+\institute{Supervised by Pierre-François \textsc{LOOS}} 
+
+\begin{document}
+
+
+
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+\begin{frame}[plain]
+
+\date{30th June 2020}
+\titlepage
+\end{frame}
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+\begin{frame}{Why do we use perturbation theories in computational chemistry?}
+
+\pause[1]
+
+The Hartree-Fock theory is \textcolor{Green4}{computationally cheap} and can be applied even to \textcolor{Green4}{large systems}.
+
+But this method is missing the \textcolor{red}{correlation energy}...
+
+\vspace{0.5cm}
+
+\pause[2]
+
+$\rightarrow$ We need methods to get this correlation energy! 
+
+\vspace{0.5cm}
+
+\pause[3]
+
+\begin{beamerboxesrounded}[scheme=foncé]{\centering A general method}
+In physics perturbation theory is often a good way to improve the obtained results with an approximated Hamiltonian.
+\end{beamerboxesrounded}
+
+    
+\end{frame}
+
+\section{\textsc{Strange behaviors of the MP series}}
+
+\begin{frame}{The Møller-Plesset perturbation theory}
+
+\pause[1]
+
+\begin{beamerboxesrounded}[scheme=foncé]{\centering Partitioning of the Hamiltonian}
+
+\begin{equation}
+   H = H_0 + \lambda V
+\end{equation}
+
+\end{beamerboxesrounded}
+
+\begin{itemize}
+\centering
+    \item $H_0$: Unperturbed Hamiltonian
+    \item $V$: Perturbation operator
+\end{itemize}
+
+\pause[2]
+
+\begin{beamerboxesrounded}[scheme=foncé]{\centering The Fock operator}
+
+\begin{equation}
+   F = T + J + K
+\end{equation}
+
+\end{beamerboxesrounded}
+
+\begin{itemize}
+\centering
+    \item $T$: Kinetic energy operator
+    \item $J$: Coulomb operator
+    \item $K$: Exchange operator
+\end{itemize}
+
+\pause[3]
+
+\begin{beamerboxesrounded}[scheme=foncé]{}
+\centering
+Full Configuration Interaction gives us access to high order terms of the perturbation series !
+\end{beamerboxesrounded}
+    
+\end{frame}
+
+\begin{frame}{Deceptive or slow convergences}
+
+\begin{figure}
+    \centering
+    \includegraphics[width=0.4\textwidth]{gill1986.png}
+    \caption{\centering Barriers to homolytic fission of \ce{He2^2+} using minimal basis set MPn theory (n~=~1-20).\footnote{\cite{gill_deceptive_1986}}}
+    \label{fig:my_label}
+\end{figure}
+
+    
+\end{frame}
+
+\begin{frame}{Multi-reference and spin contamination}
+\begin{table}
+    \centering
+    \begin{tabular}{c c c c c c c}
+\hline
+ $r$ & UHF & UMP2 & UMP3 & UMP4 & $\expval{S^2}$ \\
+\hline
+0.75 & 0.0\% & 63.8\% & 87.4\% & 95.9\% & 0.00\\
+1.35 & 0.0\% & 15.2\% & 26.1\% & 34.9\% & 0.49\\
+2.00 & 0.0\% & 01.0\% & 01.8\% & 02.6\% & 0.95\\
+2.50 & 0.0\% & 00.1\% & 00.3\% & 00.4\% & 0.99\\
+\hline
+\end{tabular}
+    \caption{\centering Percentage of electron correlation energy recovered and $\expval{S^2}$ for the \ce{H2} molecule as a function of bond length (r,\si{\angstrom}) in the minimal basis.}
+    \label{tab:my_label}
+\end{table}
+
+\footnotetext{\tiny{Gill et al. Why does unrestricted Møller–Plesset perturbation theory converge so slowly for spin-contaminated wave functions, \textit{Journal of chemical physics}, 1988}}
+    
+\end{frame}
+
+\begin{frame}{Divergent cases}
+  
+\begin{figure}
+    \centering
+    \includegraphics[width=0.6\textwidth]{The-energy-corrections-for-HF-at-stretched-geometry-in-the-cc-pVDZ-basis.png}
+    \caption{The energy corrections for HF at stretched geometry in the cc-pVDZ basis.}
+    \label{fig:my_label}
+\end{figure}
+    
+\footnotetext{\tiny{Olsen et al. Divergence in Møller–Plesset theory: A simple explanation based on a two-state model, \textit{Journal of chemical physics}, 2000}} 
+
+\end{frame}
+
+\section{The complex plane}
+
+\begin{frame}{A simple example}
+
+\begin{columns}
+
+\column{0.48\textwidth}
+
+\begin{beamerboxesrounded}[scheme=foncé]{An example function}
+
+\begin{equation*}
+   \frac{1}{1 + x^4}
+\end{equation*}
+
+\end{beamerboxesrounded}
+
+\vspace{1cm}
+
+\begin{itemize}
+    \item Smooth for $x \in \mathbb{R}$
+    
+    \item Infinitely differentiable on $\mathbb{R}$
+\end{itemize}
+
+\column{0.48\textwidth}
+
+    \begin{figure}
+    \centering
+    \includegraphics[width=0.6\textwidth]{exemplesingu.pdf}
+    \caption{Plot of $1/(1+x^4)$}
+    \label{fig:my_label}
+\end{figure}
+
+\end{columns}
+
+But the Taylor expansion of this function does not converge for $x\geq1$... 
+\vspace{0.3cm}
+\centering Why ?
+
+\end{frame}
+
+\begin{frame}{And if we look in the complex plane ?}
+
+\begin{columns}
+
+\column{0.48\textwidth}
+
+\centering The function has 4 singularities in the complex plane !
+
+\vspace{1cm}
+
+$x = e^{i\pi/4}, e^{-i\pi/4}, e^{i3\pi/4}, e^{-i3\pi/4}$
+
+\column{0.48\textwidth}
+
+\begin{figure}
+    \centering
+    \includegraphics[width=0.6\textwidth]{possingu.pdf}
+    \caption{\centering Singularities of the function $1/(1+x^4)$}
+    \label{fig:my_label}
+\end{figure}
+
+\end{columns}
+
+The \textcolor{red}{radius of convergence} of the Taylor expansion of a function is equal to the distance of the \textcolor{red}{closest singularity} to the origin in the \textcolor{red}{complex plane}.
+    
+\end{frame}
+
+\begin{frame}{Extending chemistry in the complex plane}
+
+\begin{beamerboxesrounded}[scheme=foncé]{\centering $\lambda$ a complex variable}
+
+\begin{equation*}
+   H(\lambda) = H_0 + \lambda V
+\end{equation*}
+
+\end{beamerboxesrounded}
+
+\begin{columns}
+
+\column{0.48\textwidth}
+
+\begin{itemize}
+    \item $n$ Riemann sheets
+    \vspace{0.3cm}
+    \item Exceptional points interconnecting the sheets
+    \vspace{0.3cm}
+    \item No ordering property in the complex plane
+\end{itemize}
+
+\column{0.48\textwidth}
+
+\begin{figure}
+    \centering
+    \includegraphics[width=0.7\textwidth]{riemannsheet.png}
+    \label{fig:my_label}
+\end{figure}
+
+\end{columns}
+
+\end{frame}
+
+\section{Classifying the singularity}
+
+\begin{frame}{Which features of the system localize the singularities ?}
+
+\begin{itemize}
+    \item Partitioning of the Hamiltonian: Möller-Plesset, Epstein-Nesbet,...
+    \item Zeroth order reference: weak correlation or strongly correlated electrons.
+    \item Finite or complete basis set.
+    \item Localized or delocalized basis functions.
+\end{itemize}
+    
+\end{frame}
+
+\begin{frame}{A two-state model}
+
+\begin{columns}
+
+\column{0.48\textwidth}
+
+\begin{figure}
+    \centering
+    \includegraphics[width=0.8\textwidth]{avoidedcrossing.pdf}
+    \caption{Example of an avoided crossing.}
+    \label{fig:my_label}
+\end{figure}
+
+\column{0.48\textwidth}
+
+\begin{beamerboxesrounded}[scheme=foncé]{A 2x2 matrix}
+\centering \small{$\mqty(\alpha & \delta \\ \delta & \beta)  =$} 
+
+\vspace{0.15cm}
+
+\small{$\mqty(\alpha + \alpha_s & 0 \\ 0 & \beta + \beta_s ) + \mqty(- \alpha_s & \delta \\ \delta & - \beta_s)$}
+
+\end{beamerboxesrounded}
+\vspace{1cm}
+\end{columns}
+
+\footnotetext{\tiny{Olsen et al. Divergence in Møller–Plesset theory: A simple explanation based on a two-state model, \textit{Journal of chemical physics}, 2000}} 
+    
+\end{frame}
+
+\begin{frame}{Two-state model}
+  
+\begin{figure}
+    \centering
+    \includegraphics[width=0.6\textwidth]{figure-fig14.png}
+    \caption{\centering The energy corrections for HF at stretched geometry in the aug'-cc-pVDZ basis with the two-state model.\textsuperscript{a}}
+    \label{fig:my_label}
+\end{figure}
+    
+\footnotetext{\tiny{Olsen et al. Divergence in Møller–Plesset theory: A simple explanation based on a two-state model, \textit{Journal of chemical physics}, 2000}} 
+
+\end{frame}
+
+\begin{frame}{Existence of a critical point}
+
+For $\lambda<0$:
+
+\begin{equation*}
+    H(\lambda)=\sum\limits_{j=1}^{2n}\left[ \underbrace{-\frac{1}{2}\grad_j^2 - \sum\limits_{k=1}^{N} \frac{Z_k}{|\vb{r}_j-\vb{R}_k|}}_{\text{Independant of }\lambda} + \overbrace{(1-\lambda)V_j^{(scf)}}^{\textcolor{red}{Repulsive}}+\underbrace{\lambda\sum\limits_{j<l}^{2n}\frac{1}{|\vb{r}_j-\vb{r}_l|}}_{\textcolor{blue}{Attractive}}  \right]
+\end{equation*}
+
+\footnote{stillinger, sergeev, baker}
+
+\end{frame}
+
+\begin{frame}{Critical point in a finite basis set}
+
+\pause[1]
+    
+\begin{beamerboxesrounded}[scheme=foncé]{\centering Exact energy $E(z)$}
+$E(z)$ has a critical point on the negative real axis and $E(z)$ is continue for real values below $z_{crit}$.
+\end{beamerboxesrounded}
+
+\vspace{0.5cm}
+
+\pause[2]
+
+\begin{beamerboxesrounded}[scheme=foncé]{\centering In a finite basis set}
+The singularities occur in complex conjugate pairs with non-zero imaginary parts and the energies are discrete. 
+\end{beamerboxesrounded}
+
+\vspace{0.5cm}
+
+\pause[3]
+
+\centering \Large{How is this connected???}
+    
+\end{frame}
+
+\begin{frame}{Singularities $\alpha$ and $\beta$}
+
+\pause[1]
+
+\begin{beamerboxesrounded}[scheme=foncé]{\centering Observation}
+We can separate singularities in two parts. 
+\end{beamerboxesrounded}   
+
+\pause[2]
+
+\begin{beamerboxesrounded}[scheme=foncé]{\centering Singularity $\alpha$}
+\begin{itemize}
+    \item Large avoided crossing
+    \item Non-zero imaginary part
+    \item Interaction with a low lying doubly excited states
+\end{itemize} 
+\end{beamerboxesrounded}  
+
+\pause[3]
+
+\begin{beamerboxesrounded}[scheme=foncé]{\centering Singularity $\alpha$}
+\begin{itemize}
+    \item Sharp avoided crossing
+    \item Very small imaginary part
+    \item Interaction with a diffuse function 
+\end{itemize} 
+\end{beamerboxesrounded} 
+
+\footnote{sergeev}
+
+\end{frame}
+
+
+
+\begin{frame}{Modeling the critical point}
+
+\pause[1]
+
+\begin{beamerboxesrounded}[scheme=foncé]{\centering Stillinger}
+\begin{quote}
+    \textit{"One might expect that $E_{FCI}(z) $ would try to model a continuum at $z_c$ with a grouping of discrete but closely spaced eigenstates that undergo sharp avoided crossing with the ground states."}
+\end{quote}
+\end{beamerboxesrounded}
+
+\vspace{0.5cm}
+
+\pause[2]
+
+\begin{beamerboxesrounded}[scheme=foncé]{\centering Sergeev et al.}
+Proof of the existence of this group of sharp avoided crossings for Ne, He and HF when the basis set contains diffuse functions.
+\end{beamerboxesrounded}
+    
+\end{frame}
+
+\section{The spherium model}
+
+\begin{frame}{Spherium: a theoretical playground}
+
+\begin{beamerboxesrounded}[scheme=foncé]{\centering Two electrons on a sphere Hamiltonian}
+\begin{equation*}
+    H=-\frac{1}{2}(\grad_1^2 + \grad_2^2) + \frac{1}{r_{12}}
+\end{equation*} 
+\end{beamerboxesrounded}
+\vspace{0.5cm}
+
+\begin{columns}
+
+\column{0.48\textwidth}
+
+\centering{Small $R$}
+\vspace{0.5cm}
+
+\begin{itemize}
+    \item $E_{kin} >> E_p$
+    \item Uniform density of electrons
+    \item \textcolor{red}{Weak correlation}
+\end{itemize}
+
+\vspace{0.5cm}
+
+\column{0.48\textwidth}
+
+\centering{Large $R$}
+\vspace{0.5cm}
+
+\begin{itemize}
+    \item $E_{kin} << E_p$
+    \item Electrons on the opposite sides of the sphere
+    \item \textcolor{red}{Strong correlation}
+    \end{itemize}
+
+\end{columns}
+
+\end{frame}
+
+\begin{frame}{Apparition of a class $\beta$ singularity}
+
+\pause[1]
+
+\begin{beamerboxesrounded}[scheme=foncé]{\centering Expectation}
+The electrons are restricted to the surface of the sphere so we should not observe singularities characteristic of ionization processes.
+\end{beamerboxesrounded}
+
+\vspace{0.5cm}
+
+\pause[2]
+
+\large But for some values of R... we actually observe some $\beta$ singularities!
+\centering Why?
+
+\vspace{0.5cm}
+
+\pause[3]
+
+\begin{beamerboxesrounded}[scheme=foncé]{\centering Symmetry breaking}
+The $\beta$ singularities observed are connected to the symmetry breaking of the wave function.
+\end{beamerboxesrounded}
+
+\end{frame}
+
+\begin{frame}{Conclusion}
+
+\pause[1]
+
+\begin{beamerboxesrounded}[scheme=foncé]{\centering Møller-Plesset perturbation theory}
+By understanding how the singularities are localized in the complex plane we hope that it will gives us a deep understanding of the strengths and weaknesses of the Møller-Plesset method to get the correlation energy.
+\end{beamerboxesrounded}
+
+\pause[2]
+
+\vspace{0.5cm}
+
+But there is an other secret application of exceptional points...
+
+\pause[3]
+
+\vspace{0.5cm}
+
+\begin{beamerboxesrounded}[scheme=foncé]{\centering A new way to excited states energies}
+The exceptionnal points connect ground and excited states in the complex plane. Using those properties one can smoothly morph a ground state in an excited state. 
+\end{beamerboxesrounded}
+    
+\end{frame}
+
+\end{document}