From 68cd9b38915855962fecce72ecef3405c694702c Mon Sep 17 00:00:00 2001 From: AntoineMarie2 Date: Sun, 28 Jun 2020 17:42:47 +0200 Subject: [PATCH] ajout des slides --- SlideToulouse/Ne.png | Bin 0 -> 22204 bytes SlideToulouse/SlideToulouse.bib | 104 ++++ ...tretched-geometry-in-the-cc-pVDZ-basis.png | Bin 0 -> 8023 bytes SlideToulouse/avoidedcrossing.pdf | Bin 0 -> 7454 bytes SlideToulouse/exemplesingu.pdf | Bin 0 -> 11281 bytes SlideToulouse/figure-fig14.png | Bin 0 -> 7586 bytes SlideToulouse/gill1986.png | Bin 0 -> 19076 bytes SlideToulouse/main.tex | 542 ++++++++++++++++++ SlideToulouse/possingu.pdf | Bin 0 -> 5563 bytes SlideToulouse/riemannsheet.png | Bin 0 -> 84948 bytes 10 files changed, 646 insertions(+) create mode 100644 SlideToulouse/Ne.png create mode 100644 SlideToulouse/SlideToulouse.bib create mode 100644 SlideToulouse/The-energy-corrections-for-HF-at-stretched-geometry-in-the-cc-pVDZ-basis.png create mode 100644 SlideToulouse/avoidedcrossing.pdf create mode 100644 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+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + + +\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> 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. 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