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\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}}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
@ -61,9 +61,8 @@
\beamerboxesdeclarecolorscheme{clair}{Coral4}{Ivory2}
\beamerboxesdeclarecolorscheme{foncé}{DarkSeaGreen4}{Ivory2}
\title[Title]{Perturbative theories in the complex plane}
\title[Title]{Perturbation 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}}
@ -71,21 +70,50 @@
\begin{document}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}[plain]
\date{24 Avril 2020}
\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 Möller-Plesset theory}
\begin{frame}{The Møller-Plesset perturbation theory}
\pause[1]
\begin{beamerboxesrounded}[scheme=foncé]{\centering Partitioning of the Hamiltonian}
@ -97,23 +125,33 @@
\begin{itemize}
\centering
\item $H_0$ : Unperturbed Hamiltonian
\item $V$ : Perturbation operator
\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
\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}
@ -121,12 +159,11 @@
\begin{figure}
\centering
\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).}
\includegraphics[width=0.4\textwidth]{gill1986.png}
\caption{\centering Barriers to homolytic fission of \ce{He2^2+} using STO-3G 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}
@ -135,7 +172,7 @@
\centering
\begin{tabular}{c c c c c c c}
\hline
$r$ & UHF & UMP2 & UMP3 & UMP4 & $<S^2>$ \\
$r$ & 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\\
@ -143,7 +180,7 @@
2.50 & 0.0\% & 00.1\% & 00.3\% & 00.4\% & 0.99\\
\hline
\end{tabular}
\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.}
\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.}
\label{tab:my_label}
\end{table}
@ -199,7 +236,7 @@
\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 ?
@ -217,8 +254,6 @@ 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}
@ -239,7 +274,7 @@ 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 = H_0 + \lambda V
H(\lambda) = H_0 + \lambda V
\end{equation*}
\end{beamerboxesrounded}
@ -273,7 +308,7 @@ 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: Møller-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.
@ -296,27 +331,12 @@ The \textcolor{red}{radius of convergence} of the Taylor expansion of a function
\column{0.48\textwidth}
\begin{beamerboxesrounded}[scheme=foncé]{A 2x2 matrix\textsuperscript{a}}
\begin{beamerboxesrounded}[scheme=foncé]{A 2x2 matrix}
\centering \small{$\mqty(\alpha & \delta \\ \delta & \beta) =$}
$
\vspace{0.15cm}
\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}}
$
\small{$\mqty(\alpha + \alpha_s & 0 \\ 0 & \beta + \beta_s ) + \mqty(- \alpha_s & \delta \\ \delta & - \beta_s)$}
\end{beamerboxesrounded}
\vspace{1cm}
@ -326,7 +346,7 @@ $
\end{frame}
\begin{frame}{Two state model}
\begin{frame}{Two-state model}
\begin{figure}
\centering
@ -341,57 +361,91 @@ $
\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}\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]
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 value below $z_{crit}$.
$E(z)$ 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}
\centering \Large{How is this connected ???}
\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}
\end{beamerboxesrounded}
\pause[2]
\begin{beamerboxesrounded}[scheme=foncé]{\centering Singularity $\alpha$}
\begin{itemize}
\item Large avoided crossing
\item Interaction with a low lying doubly excited states
\item Non-zero imaginary part
\item Interaction with a low lying doubly excited states
\end{itemize}
\end{beamerboxesrounded}
\begin{beamerboxesrounded}[scheme=foncé]{\centering Singularity $\alpha$}
\pause[3]
\begin{beamerboxesrounded}[scheme=foncé]{\centering Singularity $\beta$}
\begin{itemize}
\item Sharp avoided crossing
\item Interaction with a diffuse function
\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}
@ -401,7 +455,7 @@ We can separate singularities in two parts.
\begin{beamerboxesrounded}[scheme=foncé]{\centering Two electrons on a sphere Hamiltonian}
\begin{equation*}
H=-\frac{1}{2}(\nabla_1^2 + \nabla_2^2) + \frac{1}{r_{12}}
H=-\frac{1}{2}(\grad_1^2 + \grad_2^2) + \frac{1}{r_{12}}
\end{equation*}
\end{beamerboxesrounded}
\vspace{0.5cm}
@ -436,11 +490,52 @@ We can separate singularities in two parts.
\end{frame}
\begin{frame}{Why is there a class $\beta$ singularity ?}
\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}