482 lines
13 KiB
TeX
482 lines
13 KiB
TeX
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\beamerboxesdeclarecolorscheme{foncé}{DarkSeaGreen4}{Ivory2}
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\title[Title]{Perturbation theories in the complex plane}
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\author[]{Antoine \textsc{Marie}}
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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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\begin{document}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\begin{frame}[plain]
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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 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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\begin{equation}
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H = H_0 + \lambda V
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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 $H_0$: 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 = \sum\limits_{i=1}^{n} f(i) \hspace{0.3cm} ; \hspace{0.3cm} f(i) = h(i) + \sum\limits_{i=1}^{n/2} \left[2J_j(i) - K_j(i)\right]
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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 $f(i)$: Fock operator
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\item $h(i)$: One electron Hamiltonian
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\item $J_j(i)$: Coulomb operator
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\item $K_j(i)$: 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 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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\begin{frame}{Deceptive or slow convergences\footcite{gill_deceptive_1986}}
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\begin{figure}
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\centering
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\includegraphics[width=0.45\textwidth]{gill1986.png}
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\caption{\centering Barriers to homolytic fission of \ce{He2^2+} at MPn/STO-3G level ($n = 1$--$20$).}
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\label{fig:my_label}
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\end{figure}
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\end{frame}
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\begin{frame}{Multi-reference and spin contamination\footcite{gill_why_1988}}
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\begin{table}
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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 & $\expval{S^2}$ \\
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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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2.00 & 0.0\% & 01.0\% & 01.8\% & 02.6\% & 0.95\\
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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 $\expval{S^2}$ 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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\end{frame}
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\begin{frame}{Divergent cases}
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\begin{figure}
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\centering
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\includegraphics[width=0.6\textwidth]{The-energy-corrections-for-HF-at-stretched-geometry-in-the-cc-pVDZ-basis.png}
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\caption{The energy corrections for HF at stretched geometry in the cc-pVDZ basis. \footcite{olsen_divergence_2000}}
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\label{fig:my_label}
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\end{figure}
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\end{frame}
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\section{The complex plane}
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\begin{frame}{A simple example}
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\begin{columns}
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\column{0.48\textwidth}
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\begin{beamerboxesrounded}[scheme=foncé]{An example function}
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\begin{equation*}
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\frac{1}{1 + x^4}
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\end{equation*}
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\end{beamerboxesrounded}
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\vspace{1cm}
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\begin{itemize}
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\item Smooth for $x \in \mathbb{R}$
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\item Infinitely differentiable in $\mathbb{R}$
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\end{itemize}
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\column{0.48\textwidth}
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\begin{figure}
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\centering
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\includegraphics[width=0.6\textwidth]{exemplesingu.pdf}
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\caption{Plot of $1/(1+x^4)$}
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\label{fig:my_label}
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\end{figure}
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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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\vspace{0.3cm}
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\centering Why ?
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\end{frame}
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\begin{frame}{And if we look in the complex plane?}
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\begin{columns}
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\column{0.48\textwidth}
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\centering The function has 4 singularities in the complex plane!
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\vspace{1cm}
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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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\centering
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\includegraphics[width=0.6\textwidth]{possingu.pdf}
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\caption{\centering Singularities of the function $1/(1+x^4)$}
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\label{fig:my_label}
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\end{figure}
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\end{columns}
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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}.
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\end{frame}
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\begin{frame}{Extending chemistry in the complex plane}
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\begin{beamerboxesrounded}[scheme=foncé]{\centering $\lambda$ a complex variable}
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\begin{equation*}
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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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\begin{columns}
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\column{0.48\textwidth}
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\begin{itemize}
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\item $n$ Riemann sheets
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\vspace{0.3cm}
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\item Exceptional points interconnecting the sheets
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\vspace{0.3cm}
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\item No ordering property in the complex plane
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\vspace{0.3cm}
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\item An avoided crossing on the real axis corresponds to two exceptionnal points in the complex plane.
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\end{itemize}
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\column{0.48\textwidth}
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\begin{figure}
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\centering
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\includegraphics[width=0.7\textwidth]{riemannsheet.png}
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\label{fig:my_label}
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\end{figure}
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\end{columns}
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\end{frame}
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\section{Classifying the singularity}
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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: Møller-Plesset, Epstein-Nesbet, \ldots
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\item Zeroth-order reference: weak or strong correlation.
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\item Finite or complete basis set.
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\item Localized or delocalized basis functions.
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\end{itemize}
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\end{frame}
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\begin{frame}{A two-state model\footcite{olsen_divergence_2000}}
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\begin{columns}
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\column{0.48\textwidth}
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\begin{figure}
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\centering
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\includegraphics[width=0.8\textwidth]{avoidedcrossing.pdf}
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\caption{Example of an avoided crossing.}
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\label{fig:my_label}
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\end{figure}
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\column{0.48\textwidth}
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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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\vspace{0.15cm}
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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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\end{columns}
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\end{frame}
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\begin{frame}{Two-state model\footcite{olsen_divergence_2000}}
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\begin{figure}
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\centering
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\includegraphics[width=0.6\textwidth]{figure-fig14.png}
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\caption{\centering The energy corrections for HF at stretched geometry in the aug'-cc-pVDZ basis with the two-state model.}
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\label{fig:my_label}
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\end{figure}
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\end{frame}
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\begin{frame}{The Møller-Plesset Hamiltonian}
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\begin{equation}
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H(\lambda)=H_0 + \lambda (H_\text{phys} - H_0)
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\end{equation}
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\begin{equation}
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H_\text{phys}=\sum\limits_{j=1}^{n}\left[ -\frac{1}{2}\grad_j^2 - \sum\limits_{k=1}^{N} \frac{Z_k}{|\vb{r}_j-\vb{R}_k|}+\sum\limits_{j<l}^{n}\frac{1}{|\vb{r}_j-\vb{r}_l|}\right]
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\end{equation}
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\begin{equation}
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H_0=\sum\limits_{j=1}^{n}\left[ -\frac{1}{2}\grad_j^2 - \sum\limits_{k=1}^{N} \frac{Z_k}{|\vb{r}_j-\vb{R}_k|}+V_j^{(scf)}\right]
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\end{equation}
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\begin{equation*}
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H(\lambda)=\sum\limits_{j=1}^{n}\left[-\frac{1}{2}\grad_j^2 - \sum\limits_{k=1}^{N} \frac{Z_k}{|\vb{r}_j-\vb{R}_k|} + (1-\lambda)V_j^{(scf)}+\lambda\sum\limits_{j<l}^{n}\frac{1}{|\vb{r}_j-\vb{r}_l|} \right]
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\end{equation*}
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\end{frame}
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\begin{frame}{Existence of a critical point\footcite{stillinger_mollerplesset_2000}}
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For $\lambda<0$:
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\begin{equation*}
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H(\lambda)=\sum\limits_{j=1}^{n}\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}^{n}\frac{1}{|\vb{r}_j-\vb{r}_l|}}_{\textcolor{blue}{Attractive}} \right]
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\end{equation*}
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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 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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\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 $\beta$ \footcite{sergeev_singularities_2006}}
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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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\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 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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\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 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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\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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\section{Conclusion}
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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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\vspace{0.5cm}
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\pause[2]
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\begin{beamerboxesrounded}[scheme=foncé]{\centering Spherium: a theoretical playground}
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We will use the spherium model (two opposite-spin electrons restricted to remain on a surface of a sphere of radius $R$) to investigate the effects of symmetry breaking on singularities.
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\end{beamerboxesrounded}
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\end{frame}
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\end{document} |