Add files via upload
This commit is contained in:
AntoineMarie2
GitHub
parent
68cd9b3891
commit
2474175f57
@ -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}
|
||||
|
||||
\pause[1]
|
||||
\begin{frame}{The Möller-Plesset theory}
|
||||
|
||||
\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 $V$: 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}
|
||||
|
||||
\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}}}
|
||||
\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).}
|
||||
\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}
|
||||
\centering \small{$\mqty(\alpha & \delta \\ \delta & \beta) =$}
|
||||
\begin{beamerboxesrounded}[scheme=foncé]{A 2x2 matrix\textsuperscript{a}}
|
||||
|
||||
\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}
|
||||
|
||||
\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}
|
||||
\begin{frame}{Why is there a class $\beta$ singularity ?}
|
||||
|
||||
\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}
|
||||
Loading…
Reference in New Issue
Block a user