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\title [Title] { Perturbation theories in the complex plane}
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\author [] { Antoine \textsc { Marie} }
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\institute { Supervised by Pierre-François \textsc { LOOS} }
\begin { document}
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\begin { frame} [plain]
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\date { 30th June 2020}
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\titlepage
\end { frame}
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\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}
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\section { \textsc { Strange behaviors of the MP series} }
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\begin { frame} { The Møller-Plesset perturbation theory}
\pause [1]
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\begin { beamerboxesrounded} [scheme=foncé]{ \centering Partitioning of the Hamiltonian}
\begin { equation}
H = H_ 0 + \lambda V
\end { equation}
\end { beamerboxesrounded}
\begin { itemize}
\centering
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\item $ H _ 0 $ : Unperturbed Hamiltonian
\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}
\begin { equation}
F = T + J + K
\end { equation}
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\end { beamerboxesrounded}
\begin { itemize}
\centering
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\item $ T $ : Kinetic energy operator
\item $ J $ : Coulomb operator
\item $ K $ : Exchange operator
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\end { itemize}
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\pause [3]
\begin { beamerboxesrounded} [scheme=foncé]{ }
\centering
Full Configuration Interaction gives us access to high order terms of the perturbation series !
\end { beamerboxesrounded}
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\end { frame}
\begin { frame} { Deceptive or slow convergences}
\begin { figure}
\centering
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\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).}
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\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
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$ r $ & UHF & UMP2 & UMP3 & UMP4 & $ \expval { S ^ 2 } $ \\
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\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}
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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}
\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}
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But the Taylor expansion of this function does not converge for $ x \geq 1 $ ...
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\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 ^ { i 3 \pi / 4 } , e ^ { - i 3 \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*}
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H(\lambda ) = H_ 0 + \lambda V
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\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}
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\item Partitioning of the Hamiltonian: Møller-Plesset, Epstein-Nesbet,...
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\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 }
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\begin { beamerboxesrounded} [scheme=foncé]{ A 2x2 matrix}
\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}
\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}
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\begin { frame} { Two-state model}
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\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}
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For $ \lambda < 0 $ :
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\begin { equation*}
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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]
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\end { equation*}
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\footnote { stillinger, sergeev, baker}
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\end { frame}
\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}
\vspace { 0.5cm}
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\pause [2]
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\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}
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\pause [3]
\centering \Large { How is this connected???}
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\end { frame}
\begin { frame} { Singularities $ \alpha $ and $ \beta $ }
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\pause [1]
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\begin { beamerboxesrounded} [scheme=foncé]{ \centering Observation}
We can separate singularities in two parts.
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\end { beamerboxesrounded}
\pause [2]
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\begin { beamerboxesrounded} [scheme=foncé]{ \centering Singularity $ \alpha $ }
\begin { itemize}
\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}
\end { beamerboxesrounded}
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\pause [3]
\begin { beamerboxesrounded} [scheme=foncé]{ \centering Singularity $ \beta $ }
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\begin { itemize}
\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}
\end { beamerboxesrounded}
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\footnote { sergeev}
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\end { frame}
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\begin { frame} { Modeling the critical point}
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\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}
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\end { frame}
\section { The spherium model}
\begin { frame} { Spherium: a theoretical playground}
\begin { beamerboxesrounded} [scheme=foncé]{ \centering Two electrons on a sphere Hamiltonian}
\begin { equation*}
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H=-\frac { 1} { 2} (\grad _ 1^ 2 + \grad _ 2^ 2) + \frac { 1} { r_ { 12} }
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\end { equation*}
\end { beamerboxesrounded}
\vspace { 0.5cm}
\begin { columns}
\column { 0.48\textwidth }
\centering { Small $ R $ }
\vspace { 0.5cm}
\begin { itemize}
\item $ E _ { kin } >> 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}
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\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}
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\end { frame}
\begin { frame} { Conclusion}
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\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}
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\end { frame}
\end { document}