diff --git a/SlideToulouse/Ne.png b/SlideToulouse/Ne.png new file mode 100644 index 0000000..7f35bba Binary files /dev/null and b/SlideToulouse/Ne.png differ diff --git a/SlideToulouse/SlideToulouse.bib b/SlideToulouse/SlideToulouse.bib new file mode 100644 index 0000000..21dc5d5 --- /dev/null +++ b/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} +} \ No newline at end of file diff --git a/SlideToulouse/The-energy-corrections-for-HF-at-stretched-geometry-in-the-cc-pVDZ-basis.png b/SlideToulouse/The-energy-corrections-for-HF-at-stretched-geometry-in-the-cc-pVDZ-basis.png new file mode 100644 index 0000000..7121761 Binary files /dev/null and b/SlideToulouse/The-energy-corrections-for-HF-at-stretched-geometry-in-the-cc-pVDZ-basis.png differ diff --git a/SlideToulouse/avoidedcrossing.pdf b/SlideToulouse/avoidedcrossing.pdf new file mode 100644 index 0000000..769d850 Binary files /dev/null and b/SlideToulouse/avoidedcrossing.pdf differ diff --git a/SlideToulouse/exemplesingu.pdf b/SlideToulouse/exemplesingu.pdf new file mode 100644 index 0000000..1e16199 Binary files /dev/null and b/SlideToulouse/exemplesingu.pdf differ diff --git a/SlideToulouse/figure-fig14.png b/SlideToulouse/figure-fig14.png new file mode 100644 index 0000000..c37b9f1 Binary files /dev/null and b/SlideToulouse/figure-fig14.png differ diff --git a/SlideToulouse/gill1986.png b/SlideToulouse/gill1986.png new file mode 100644 index 0000000..2517d14 Binary files /dev/null and b/SlideToulouse/gill1986.png differ diff --git a/SlideToulouse/main.tex b/SlideToulouse/main.tex new file mode 100644 index 0000000..e8aa2d5 --- /dev/null +++ b/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> 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} \ No newline at end of file diff --git a/SlideToulouse/possingu.pdf b/SlideToulouse/possingu.pdf new file mode 100644 index 0000000..2e35eeb Binary files /dev/null and b/SlideToulouse/possingu.pdf differ diff --git a/SlideToulouse/riemannsheet.png b/SlideToulouse/riemannsheet.png new file mode 100644 index 0000000..d156f9a Binary files /dev/null and b/SlideToulouse/riemannsheet.png differ