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SlideToulouse/main.tex

@@ -8,19 +8,19 @@
 \usepackage[english]{babel}
 \usepackage[T1]{fontenc}
 \usepackage[utf8]{inputenc}
-\usepackage{xcolor}
 \usepackage{siunitx}
 \usepackage{graphicx}
 \usepackage{physics}
 \usepackage{multimedia}
 \usepackage{subfigure}
+\usepackage{xcolor}
 \usepackage[absolute,overlay]{textpos}
 \usepackage{ragged2e}
 \usepackage{amssymb}
 \usepackage[version=4]{mhchem}
 
 
-
+\renewcommand{\thefootnote}{\alph{footnote}}
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 
 
@@ -61,8 +61,9 @@
 \beamerboxesdeclarecolorscheme{clair}{Coral4}{Ivory2}
 \beamerboxesdeclarecolorscheme{foncé}{DarkSeaGreen4}{Ivory2}
 
-\title[Title]{Perturbation theories in the complex plane}
+\title[Title]{Perturbative theories in the complex plane}
 \author[]{Antoine \textsc{Marie}}
+\date{30 Juin 2020}
   \setbeamersize{text margin left=5mm}
   \setbeamersize{text margin right=5mm}
 \institute{Supervised by Pierre-François \textsc{LOOS}} 
@@ -70,50 +71,21 @@
 \begin{document}
 
 
-
-
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 
 \begin{frame}[plain]
 
-\date{30th June 2020}
+\date{24 Avril 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}
+\begin{frame}{The Möller-Plesset theory}
-
-\pause[1]
 
 \begin{beamerboxesrounded}[scheme=foncé]{\centering Partitioning of the Hamiltonian}
 
@@ -125,45 +97,36 @@ In physics perturbation theory is often a good way to improve the obtained resul
 
 \begin{itemize}
 \centering
-    \item $H_0$: Unperturbed Hamiltonian
+    \item $H_0$ : Unperturbed Hamiltonian
-    \item $V$: Perturbation operator
+    \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 $T$ : Kinetic energy operator
-    \item $J$: Coulomb operator
+    \item $J$ : Coulomb operator
-    \item $K$: Exchange 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}
+    \includegraphics[width=0.5\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}}}
+    \caption{\centering Barriers to homolytic fission of \ce{He2^2+} using minimal basis set MPn theory (n~=~1-20).}
     \label{fig:my_label}
 \end{figure}
 
+\footnotetext{\tiny{Gill et al.~Deceptive convergence in Møller-Plesset perturbation  energies, \textit{Chemical Physics Letter}, 1986}}
     
 \end{frame}
 
@@ -172,7 +135,7 @@ Full Configuration Interaction gives us access to high order terms of the pertur
     \centering
     \begin{tabular}{c c c c c c c}
 \hline
- $r$ & UHF & UMP2 & UMP3 & UMP4 & $\expval{S^2}$ \\
+ $r$ & UHF & UMP2 & UMP3 & UMP4 & $<S^2>$ \\
 \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\\
@@ -180,7 +143,7 @@ Full Configuration Interaction gives us access to high order terms of the pertur
 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.}
+    \caption{\centering Percentage of electron correlation energy recovered and $<S^2>$ for the \ce{H2} molecule as a function of bond length (r,A) in the minimal basis.}
     \label{tab:my_label}
 \end{table}
 
@@ -236,7 +199,7 @@ Full Configuration Interaction gives us access to high order terms of the pertur
 
 \end{columns}
 
-But the Taylor expansion of this function does not converge for $x\geq1$... 
+But the Taylor expansion of this function does not converge for $x\geq1$ ... 
 \vspace{0.3cm}
 \centering Why ?
 
@@ -254,6 +217,8 @@ But the Taylor expansion of this function does not converge for $x\geq1$...
 
 $x = e^{i\pi/4}, e^{-i\pi/4}, e^{i3\pi/4}, e^{-i3\pi/4}$
 
+
+
 \column{0.48\textwidth}
 
 \begin{figure}
@@ -274,7 +239,7 @@ The \textcolor{red}{radius of convergence} of the Taylor expansion of a function
 \begin{beamerboxesrounded}[scheme=foncé]{\centering $\lambda$ a complex variable}
 
 \begin{equation*}
-   H(\lambda) = H_0 + \lambda V
+   H = H_0 + \lambda V
 \end{equation*}
 
 \end{beamerboxesrounded}
@@ -308,7 +273,7 @@ The \textcolor{red}{radius of convergence} of the Taylor expansion of a function
 \begin{frame}{Which features of the system localize the singularities ?}
 
 \begin{itemize}
-    \item Partitioning of the Hamiltonian: Möller-Plesset, Epstein-Nesbet,...
+    \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.
@@ -331,12 +296,27 @@ The \textcolor{red}{radius of convergence} of the Taylor expansion of a function
 
 \column{0.48\textwidth}
 
-\begin{beamerboxesrounded}[scheme=foncé]{A 2x2 matrix}
+\begin{beamerboxesrounded}[scheme=foncé]{A 2x2 matrix\textsuperscript{a}}
-\centering \small{$\mqty(\alpha & \delta \\ \delta & \beta)  =$} 
 
-\vspace{0.15cm}
+$
 
-\small{$\mqty(\alpha + \alpha_s & 0 \\ 0 & \beta + \beta_s ) + \mqty(- \alpha_s & \delta \\ \delta & - \beta_s)$}
+\small{\centering \begin{pmatrix}
+   \alpha & \delta \\
+   \delta & \beta 
+\end{pmatrix} = 
+
+\vspace{0.3cm}
+
+\begin{pmatrix}
+
+   \alpha + \alpha_s & 0 \\
+   0 & \beta + \beta_s
+\end{pmatrix} +
+\begin{pmatrix}
+   - \alpha_s & \delta \\
+   \delta & - \beta_s
+\end{pmatrix}}
+$
 
 \end{beamerboxesrounded}
 \vspace{1cm}
@@ -346,7 +326,7 @@ The \textcolor{red}{radius of convergence} of the Taylor expansion of a function
     
 \end{frame}
 
-\begin{frame}{Two-state model}
+\begin{frame}{Two state model}
   
 \begin{figure}
     \centering
@@ -361,92 +341,58 @@ The \textcolor{red}{radius of convergence} of the Taylor expansion of a function
 
 \begin{frame}{Existence of a critical point}
 
-For $\lambda<0$:
+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]
+    H(\lambda)=\sum\limits_{j=1}^{2n}\left[ \underbrace{-\frac{1}{2}\nabla_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 $z_{crit}$.
+$E(z)$ has a critical point on the negative real axis and $E(z)$ is continue for real value 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 ???}
-
-\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
+    \item Non-zero imaginary part
 \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 
+    \item Very small imaginary part
 \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}
@@ -455,7 +401,7 @@ Proof of the existence of this group of sharp avoided crossings for Ne, He and H
 
 \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}}
+    H=-\frac{1}{2}(\nabla_1^2 + \nabla_2^2) + \frac{1}{r_{12}}
 \end{equation*} 
 \end{beamerboxesrounded}
 \vspace{0.5cm}
@@ -490,53 +436,12 @@ Proof of the existence of this group of sharp avoided crossings for Ne, He and H
 
 \end{frame}
 
-\begin{frame}{Apparition of a class $\beta$ singularity}
+\begin{frame}{Why is there 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}