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\usepackage[english]{babel}
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\usepackage[T1]{fontenc}
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\usepackage[utf8]{inputenc}
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\usepackage{xcolor}
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\usepackage{siunitx}
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\usepackage{graphicx}
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\usepackage{physics}
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\usepackage{multimedia}
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\usepackage{subfigure}
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\usepackage{xcolor}
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\usepackage[absolute,overlay]{textpos}
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\usepackage{ragged2e}
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\usepackage{amssymb}
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\usepackage[version=4]{mhchem}
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\renewcommand{\thefootnote}{\alph{footnote}}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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@ -61,9 +61,8 @@
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\beamerboxesdeclarecolorscheme{clair}{Coral4}{Ivory2}
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\beamerboxesdeclarecolorscheme{foncé}{DarkSeaGreen4}{Ivory2}
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\title[Title]{Perturbative theories in the complex plane}
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\title[Title]{Perturbation theories in the complex plane}
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\author[]{Antoine \textsc{Marie}}
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\date{30 Juin 2020}
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\setbeamersize{text margin left=5mm}
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\setbeamersize{text margin right=5mm}
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\institute{Supervised by Pierre-François \textsc{LOOS}}
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@ -71,21 +70,50 @@
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\begin{document}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\begin{frame}[plain]
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\date{24 Avril 2020}
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\date{30th June 2020}
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\titlepage
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\end{frame}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\begin{frame}{Why do we use perturbation theories in computational chemistry?}
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\pause[1]
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The Hartree-Fock theory is \textcolor{Green4}{computationally cheap} and can be applied even to \textcolor{Green4}{large systems}.
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But this method is missing the \textcolor{red}{correlation energy}...
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\vspace{0.5cm}
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\pause[2]
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$\rightarrow$ We need methods to get this correlation energy!
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\vspace{0.5cm}
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\pause[3]
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\begin{beamerboxesrounded}[scheme=foncé]{\centering A general method}
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In physics perturbation theory is often a good way to improve the obtained results with an approximated Hamiltonian.
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\end{beamerboxesrounded}
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\end{frame}
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\section{\textsc{Strange behaviors of the MP series}}
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\begin{frame}{The Möller-Plesset theory}
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\begin{frame}{The Møller-Plesset perturbation theory}
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\pause[1]
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\begin{beamerboxesrounded}[scheme=foncé]{\centering Partitioning of the Hamiltonian}
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@ -97,23 +125,33 @@
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\begin{itemize}
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\centering
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\item $H_0$ : Unperturbed Hamiltonian
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\item $V$ : Perturbation operator
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\item $H_0$: Unperturbed Hamiltonian
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\item $V$: Perturbation operator
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\end{itemize}
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\pause[2]
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\begin{beamerboxesrounded}[scheme=foncé]{\centering The Fock operator}
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\begin{equation}
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F = T + J + K
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\end{equation}
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\end{beamerboxesrounded}
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\begin{itemize}
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\centering
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\item $T$ : Kinetic energy operator
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\item $J$ : Coulomb operator
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\item $K$ : Exchange operator
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\item $T$: Kinetic energy operator
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\item $J$: Coulomb operator
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\item $K$: Exchange operator
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\end{itemize}
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\pause[3]
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\begin{beamerboxesrounded}[scheme=foncé]{}
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\centering
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Full Configuration Interaction gives us access to high order terms of the perturbation series !
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\end{beamerboxesrounded}
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\end{frame}
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@ -121,12 +159,11 @@
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\begin{figure}
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\centering
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\includegraphics[width=0.5\textwidth]{gill1986.png}
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\caption{\centering Barriers to homolytic fission of \ce{He2^2+} using minimal basis set MPn theory (n~=~1-20).}
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\includegraphics[width=0.4\textwidth]{gill1986.png}
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\caption{\centering Barriers to homolytic fission of \ce{He2^2+} using STO-3G basis set MPn theory (n~=~1-20).}
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\label{fig:my_label}
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\end{figure}
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\footnotetext{\tiny{Gill et al.~Deceptive convergence in Møller-Plesset perturbation energies, \textit{Chemical Physics Letter}, 1986}}
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\end{frame}
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@ -135,7 +172,7 @@
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\centering
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\begin{tabular}{c c c c c c c}
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\hline
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$r$ & UHF & UMP2 & UMP3 & UMP4 & $<S^2>$ \\
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$r$ & UHF & UMP2 & UMP3 & UMP4 & $\expval{S^2}$ \\
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\hline
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0.75 & 0.0\% & 63.8\% & 87.4\% & 95.9\% & 0.00\\
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1.35 & 0.0\% & 15.2\% & 26.1\% & 34.9\% & 0.49\\
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@ -143,7 +180,7 @@
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2.50 & 0.0\% & 00.1\% & 00.3\% & 00.4\% & 0.99\\
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\hline
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\end{tabular}
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\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.}
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\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 STO-3G basis set.}
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\label{tab:my_label}
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\end{table}
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@ -199,7 +236,7 @@
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\end{columns}
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But the Taylor expansion of this function does not converge for $x\geq1$ ...
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But the Taylor expansion of this function does not converge for $x\geq1$...
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\vspace{0.3cm}
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\centering Why ?
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@ -217,8 +254,6 @@ But the Taylor expansion of this function does not converge for $x\geq1$ ...
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$x = e^{i\pi/4}, e^{-i\pi/4}, e^{i3\pi/4}, e^{-i3\pi/4}$
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\column{0.48\textwidth}
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\begin{figure}
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@ -239,7 +274,7 @@ The \textcolor{red}{radius of convergence} of the Taylor expansion of a function
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\begin{beamerboxesrounded}[scheme=foncé]{\centering $\lambda$ a complex variable}
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\begin{equation*}
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H = H_0 + \lambda V
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H(\lambda) = H_0 + \lambda V
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\end{equation*}
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\end{beamerboxesrounded}
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@ -273,7 +308,7 @@ The \textcolor{red}{radius of convergence} of the Taylor expansion of a function
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\begin{frame}{Which features of the system localize the singularities ?}
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\begin{itemize}
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\item Partitioning of the Hamiltonian: Möller-Plesset, Epstein-Nesbet, ...
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\item Partitioning of the Hamiltonian: Møller-Plesset, Epstein-Nesbet,...
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\item Zeroth order reference: weak correlation or strongly correlated electrons.
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\item Finite or complete basis set.
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\item Localized or delocalized basis functions.
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@ -296,27 +331,12 @@ The \textcolor{red}{radius of convergence} of the Taylor expansion of a function
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\column{0.48\textwidth}
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\begin{beamerboxesrounded}[scheme=foncé]{A 2x2 matrix\textsuperscript{a}}
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\begin{beamerboxesrounded}[scheme=foncé]{A 2x2 matrix}
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\centering \small{$\mqty(\alpha & \delta \\ \delta & \beta) =$}
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$
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\vspace{0.15cm}
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\small{\centering \begin{pmatrix}
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\alpha & \delta \\
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\delta & \beta
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\end{pmatrix} =
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\vspace{0.3cm}
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\begin{pmatrix}
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\alpha + \alpha_s & 0 \\
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0 & \beta + \beta_s
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\end{pmatrix} +
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\begin{pmatrix}
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- \alpha_s & \delta \\
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\delta & - \beta_s
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\end{pmatrix}}
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$
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\small{$\mqty(\alpha + \alpha_s & 0 \\ 0 & \beta + \beta_s ) + \mqty(- \alpha_s & \delta \\ \delta & - \beta_s)$}
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\end{beamerboxesrounded}
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\vspace{1cm}
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@ -326,7 +346,7 @@ $
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\end{frame}
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\begin{frame}{Two state model}
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\begin{frame}{Two-state model}
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\begin{figure}
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\centering
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@ -341,57 +361,91 @@ $
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\begin{frame}{Existence of a critical point}
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For $\lambda<0$ :
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For $\lambda<0$:
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\begin{equation*}
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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]
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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]
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\end{equation*}
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\footnote{stillinger, sergeev, baker}
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\end{frame}
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\begin{frame}{Critical point in a finite basis set}
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\pause[1]
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\begin{beamerboxesrounded}[scheme=foncé]{\centering Exact energy $E(z)$}
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$E(z)$ has a critical point on the negative real axis and $E(z)$ is continue for real value below $z_{crit}$.
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$E(z)$ has a critical point on the negative real axis and $E(z)$ is continue for real values below $z_{crit}$.
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\end{beamerboxesrounded}
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\vspace{0.5cm}
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\pause[2]
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\begin{beamerboxesrounded}[scheme=foncé]{\centering In a finite basis set}
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The singularities occur in complex conjugate pairs with non-zero imaginary parts and the energies are discrete.
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\end{beamerboxesrounded}
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\vspace{0.5cm}
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\centering \Large{How is this connected ???}
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\pause[3]
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\centering \Large{How is this connected???}
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\end{frame}
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\begin{frame}{Singularities $\alpha$ and $\beta$}
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\pause[1]
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\begin{beamerboxesrounded}[scheme=foncé]{\centering Observation}
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We can separate singularities in two parts.
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\end{beamerboxesrounded}
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\end{beamerboxesrounded}
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\pause[2]
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\begin{beamerboxesrounded}[scheme=foncé]{\centering Singularity $\alpha$}
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\begin{itemize}
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\item Large avoided crossing
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\item Interaction with a low lying doubly excited states
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\item Non-zero imaginary part
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\item Interaction with a low lying doubly excited states
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\end{itemize}
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\end{beamerboxesrounded}
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\begin{beamerboxesrounded}[scheme=foncé]{\centering Singularity $\alpha$}
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\pause[3]
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\begin{beamerboxesrounded}[scheme=foncé]{\centering Singularity $\beta$}
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\begin{itemize}
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\item Sharp avoided crossing
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\item Interaction with a diffuse function
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\item Very small imaginary part
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\item Interaction with a diffuse function
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\end{itemize}
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\end{beamerboxesrounded}
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\footnote{sergeev}
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\end{frame}
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\begin{frame}{Modeling the critical point}
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\pause[1]
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\begin{beamerboxesrounded}[scheme=foncé]{\centering Stillinger}
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\begin{quote}
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\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."}
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\end{quote}
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\end{beamerboxesrounded}
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\vspace{0.5cm}
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\pause[2]
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\begin{beamerboxesrounded}[scheme=foncé]{\centering Sergeev et al.}
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Proof of the existence of this group of sharp avoided crossings for Ne, He and HF when the basis set contains diffuse functions.
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\end{beamerboxesrounded}
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\end{frame}
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@ -401,7 +455,7 @@ We can separate singularities in two parts.
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\begin{beamerboxesrounded}[scheme=foncé]{\centering Two electrons on a sphere Hamiltonian}
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\begin{equation*}
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H=-\frac{1}{2}(\nabla_1^2 + \nabla_2^2) + \frac{1}{r_{12}}
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H=-\frac{1}{2}(\grad_1^2 + \grad_2^2) + \frac{1}{r_{12}}
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\end{equation*}
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\end{beamerboxesrounded}
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\vspace{0.5cm}
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@ -436,11 +490,52 @@ We can separate singularities in two parts.
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\end{frame}
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\begin{frame}{Why is there a class $\beta$ singularity ?}
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\begin{frame}{Apparition of a class $\beta$ singularity}
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\pause[1]
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\begin{beamerboxesrounded}[scheme=foncé]{\centering Expectation}
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The electrons are restricted to the surface of the sphere so we should not observe singularities characteristic of ionization processes.
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\end{beamerboxesrounded}
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\vspace{0.5cm}
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\pause[2]
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\large But for some values of R... we actually observe some $\beta$ singularities!
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\centering Why?
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\vspace{0.5cm}
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\pause[3]
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\begin{beamerboxesrounded}[scheme=foncé]{\centering Symmetry breaking}
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The $\beta$ singularities observed are connected to the symmetry breaking of the wave function.
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\end{beamerboxesrounded}
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\end{frame}
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\begin{frame}{Conclusion}
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\pause[1]
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\begin{beamerboxesrounded}[scheme=foncé]{\centering Møller-Plesset perturbation theory}
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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.
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\end{beamerboxesrounded}
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\pause[2]
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\vspace{0.5cm}
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But there is an other secret application of exceptional points...
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\pause[3]
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\vspace{0.5cm}
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\begin{beamerboxesrounded}[scheme=foncé]{\centering A new way to excited states energies}
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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.
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\end{beamerboxesrounded}
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\end{frame}
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