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Antoine Marie 2020-07-13 10:14:49 +02:00
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commit daa5fb701b
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@ -0,0 +1,12 @@
\BOOKMARK [1][-]{section.1}{Generalities}{}% 1
\BOOKMARK [2][-]{subsection.1.1}{Objectives}{section.1}% 2
\BOOKMARK [2][-]{subsection.1.2}{Variables}{section.1}% 3
\BOOKMARK [2][-]{subsection.1.3}{``Classification'' of the bibliography}{section.1}% 4
\BOOKMARK [1][-]{section.2}{Results}{}% 5
\BOOKMARK [2][-]{subsection.2.1}{Spin contamination}{section.2}% 6
\BOOKMARK [2][-]{subsection.2.2}{Two state model}{section.2}% 7
\BOOKMARK [2][-]{subsection.2.3}{Singularity structure}{section.2}% 8
\BOOKMARK [3][-]{subsubsection.2.3.1}{Class }{subsection.2.3}% 9
\BOOKMARK [3][-]{subsubsection.2.3.2}{Class }{subsection.2.3}% 10
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\BOOKMARK [1][-]{section.3}{Open questions}{}% 12

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RapportStage/Rapport.aux
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@ -47,6 +47,7 @@
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@ -76,7 +77,13 @@
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\bibcite{Gill_1986}{29}
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27
RapportStage/Rapport.bbl
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@ -154,6 +154,11 @@ Jeppe Olsen, Poul Jørgensen, Trygve Helgaker, and Ove Christiansen.
two-state model.
\newblock {\em J. Chem. Phys.}, 112(22):9736--9748.
\bibitem{Goodson_2011}
David~Z. Goodson.
\newblock Resummation methods.
\newblock 2(5):743--761.
\bibitem{Moller_1934}
Chr. Møller and M.~S. Plesset.
\newblock Note on an approximation treatment for many-electron systems.
@ -205,4 +210,26 @@ Ove Christiansen, Jeppe Olsen, Poul Jørgensen, Henrik Koch, and Per-Åke
atom — a test case.
\newblock {\em Chemical Physics Letters}, 261(3):369--378.
\bibitem{Sergeev_2005}
Alexey~V. Sergeev, David~Z. Goodson, Steven~E. Wheeler, and Wesley~D. Allen.
\newblock On the nature of the møller-plesset critical point.
\newblock {\em J. Chem. Phys.}, 123(6):064105.
\bibitem{Sergeev_2006}
Alexey~V. Sergeev and David~Z. Goodson.
\newblock Singularities of møller-plesset energy functions.
\newblock {\em J. Chem. Phys.}, 124(9):094111.
\bibitem{Stillinger_2000}
Frank~H. Stillinger.
\newblock Mo/ller-plesset convergence issues in computational quantum
chemistry.
\newblock {\em J. Chem. Phys.}, 112(22):9711--9715.
\bibitem{Goodson_2004}
David~Z. Goodson and Alexey~V. Sergeev.
\newblock Singularity structure of møller-plesset perturbation theory.
\newblock In {\em Advances in Quantum Chemistry}, volume~47, pages 193--208.
Academic Press.
\end{thebibliography}

4
RapportStage/Rapport.bib
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@ -143,7 +143,7 @@
}
@article{Goodson_2012,
@article{Goodson_2011,
Title = {Resummation methods},
Volume = {2},
doi = {10.1002/wcms.92},
@ -151,7 +151,7 @@
number = {5},
journalTitle = {{WIREs} Computational Molecular Science},
author = {Goodson, David Z.},
Date = {2012},
Date = {2011},
}
@article{Katz_1962,

70
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@ -97,7 +97,7 @@ hyperfigures=false]
\vspace{0.5cm}
\parbox{15cm}{
\textbf{Keywords} : \it Complex chemistry, Perturbation theory, Spherium, Exceptional points, Symmetry breaking
\textbf{Keywords} : \it Quantum chemistry, Perturbation theory, Spherium, Exceptional points, Symmetry breaking
} %fin de la commande \parbox des mots clefs
\vspace{0.5cm}
@ -178,9 +178,9 @@ The energy can then be written as a power series of $\lambda$
\begin{equation}
E(\lambda) = \sum_{k=0}^\infty \lambda^k E^{(k)}
\end{equation}
where $\lambda$ is a coupling parameter set equal to 1 at the end of the calculation. However it is not guaranteed that the series $E(\lambda)$ has a radius of convergence $\abs{\lambda_0} < 1$. It means that the series is divergent for the physical system at $\lambda=1$. One can prove that $\abs{\lambda_0}$ can be obtained by extending $\lambda$ in the complex plane and looking for the singularities of $E(\lambda)$. The radius of convergence of the expansion of a function is equal to the distance of the closest singularities in the complex plane to the origin. These singularities are nothing but EPs at $\lambda_0$ and $\lambda_0^*$. Indeed, for a Hermitian Hamiltonian the singularities occur in complex conjugate pairs with non-zero imaginary parts.
where $\lambda$ is a coupling parameter set equal to 1 at the end of the calculation. However it is not guaranteed that the series $E(\lambda)$ has a radius of convergence $\abs{\lambda_0} < 1$. It means that the series is divergent for the physical system at $\lambda=1$. One can prove that $\abs{\lambda_0}$ can be obtained by extending $\lambda$ in the complex plane and looking for the singularities of $E(\lambda)$. This is due to the following theorem \cite{Goodson_2011}: The Taylor series about a point $z_0$ of a function over the complex $z$ plane will converge at a value $z_1$ if the function is non-singular at all values of $z$ in the circular region centered at $z_0$ with radius $\abs{z_1 − z_0}$. If the function has a singular point $z_s$ such that $\abs{z_s − z_0} < \abs{z_1 − z_0}$, then the series will diverge when evaluated at $z_1$. This theorem means that the radius of convergence of the perturbation series is equal to distance to the origin of the closest singularity of $E(\lambda)$.
The discovery of a partitioning of the Hamiltonian that allowed chemists to get access to a part of the correlation energy using perturbation theory has been a major step in the development of post-Hartree-Fock methods. This case of the Rayleigh-Schrödinger perturbation theory is called the M{\o}ller-Plesset perturbation theory \cite{Moller_1934}. In the MPPT the unperturbed Hamiltonian is the sum of the $n$ mono-electronic Fock operators which are the sum of the one-electron core Hamiltonian $h(i)$, the Coulomb $J_j(i)$ and Exchange $K_j(i)$ operators.
The discovery of a partitioning of the Hamiltonian that allowed chemists to recover a part of the correlation energy (i.e. the difference between the exact energy and the Hartree-Fock energy) using perturbation theory has been a major step in the development of post-Hartree-Fock methods. This case of the Rayleigh-Schrödinger perturbation theory is called the M{\o}ller-Plesset perturbation theory \cite{Moller_1934}. In the MPPT the unperturbed Hamiltonian is the sum of the $n$ mono-electronic Fock operators which are the sum of the one-electron core Hamiltonian $h(i)$, the Coulomb $J_j(i)$ and Exchange $K_j(i)$ operators.
\begin{equation}
H_0= \sum\limits_{i=1}^{n} f(i)
@ -190,13 +190,13 @@ H_0= \sum\limits_{i=1}^{n} f(i)
f(i) = h(i) + \sum\limits_{j=1,j \neq i}^{n} \left[J_j(i) - K_j(i)\right]
\end{equation}
In Hartree-Fock theory you approximate the exact wave function as a Slater-determinant (which is an anti-symmetric combination of mono-electronic orbitals) and those wave functions are eigenvectors of the Fock operators. In the perturbation theory the energy is a power series of $\lambda$ and the physical energy is obtained by taking $\lambda$ equal to 1. We will refer to there energy up to the n-th order as the MPn energy. The MP1 energy is the Hartree-Fock energy and the correlation energy is getting recovered from the MP2 level.
In Hartree-Fock theory the exact wave function is approximated as a Slater-determinant (which is an anti-symmetric combination of mono-electronic orbitals) and those wave functions are eigenvectors of the Fock operators. In the perturbation theory the energy is a power series of $\lambda$ and the physical energy is obtained by taking $\lambda$ equal to 1. We will refer to there energy up to the n-th order as the MPn energy. The MP0 energy overestimate the energy by double counting the electron-electron interaction, the MP1 correct this effect and the MP1 energy is equal to the Hartree-Fock energy. The MP2 energy start to recover a part of the correlation energy.
\begin{equation}
E_{\text{MP\textsubscript{n}}}= \sum_{k=0}^n E^{(k)}
\end{equation}
But \textit{a priori} there is no reason that this power series is always convergent for $\lambda$=1 when n goes to infinity. In fact, it is known that when the Hartree-Fock wave function is a bad approximation of the exact wave function, for example for multi-reference states, the M{\o}ller-Plesset will give bad results\cite{Gill_1986, Gill_1988, Handy_1985, Lepetit_1988}. A smart way to investigate the convergence properties of the MP series is to transform the coupling parameter $\lambda$ into a complex variable. By doing this the Hamiltonian and the energy become functions of this variable. So by searching the singularities of the function $E(\lambda)$ we can get information on the convergence properties of the MPPT. Those singularities of the energy are exactly the exceptional points connecting the electronic states mentioned in the introduction. The direct computation of the terms of the series is relatively easy up to the 4th order and the 5th and 6th order can be obtained at high cost. But to understand deeply the behavior of the MP series and how it is connected to the singularities, we need to have access to high order terms of the series. For small systems we can have access to the whole series using Full Configuration Interaction. If you diagonalize the Hamiltonian $H(\lambda)$ in the FCI basis you get the exact energies (in this finite basis set) and expanding in $\lambda$ allow you to get the M{\o}ller-Plesset perturbation series at every order.
But as mentioned before \textit{a priori} there are no reasons that this power series is always convergent for $\lambda$=1 when n goes to infinity. In fact, it is known that when the Hartree-Fock wave function is a bad approximation of the exact wave function, for example for multi-reference states, the M{\o}ller-Plesset will give bad results\cite{Gill_1986, Gill_1988, Handy_1985, Lepetit_1988}. A smart way to investigate the convergence properties of the MP series is to transform the coupling parameter $\lambda$ into a complex variable. By doing this the Hamiltonian and the energy become functions of this variable. By doing this the energy become a multivalued function on $n$ Riemann sheets. As mentioned above by searching the singularities of the function $E(\lambda)$ we can get information on the convergence properties of the MPPT. Those singularities of the energy are exactly the exceptional points connecting the electronic states mentioned in the introduction. The direct computation of the terms of the series is relatively easy up to the 4th order and the 5th and 6th order can be obtained at high cost. But to understand deeply the behavior of the MP series and how it is connected to the singularities, we need to have access to high order terms of the series. For small systems we can have access to the whole series using Full Configuration Interaction. If you diagonalize the Hamiltonian $H(\lambda)$ in the FCI basis set you get the exact energies (in this finite basis set) and expanding in $\lambda$ allow you to get the M{\o}ller-Plesset perturbation series at every order.
%============================================================%
\section{Historical overview}
@ -251,19 +251,25 @@ But Olsen et al. have discovered even more preoccupying behavior of the MP serie
\label{fig:my_label}
\end{figure}
The discovery of those divergent behavior was really worrying because to get more and more accurate results theoretical chemists need to work in large basis sets. So they investigated the causes of those divergences and in the same time the reasons of the different types of convergence. In order to do this they analyzed the relation between the dominant singularity (i.e. the closest singularity to the origin) and the convergence behavior of the series \cite{Olsen_2000}. A singularity in the unit circle is designated as an intruder state, more precisely as a front-door (respectively back-door) intruder state if the real part of the singularity is positive (respectively negative). The method used is to do a scan of the real axis to identify the avoided crossing responsible for the pair of dominant singularity. Then by modeling this avoided crossing by a two-state Hamiltonian one can get an approximation of the dominant conjugate pair of singularity by finding the EPs of the 2x2 Hamiltonian. The diagonal matrix is the unperturbed Hamiltonian and the other matrix is the perturbation part of the Hamiltonian.
The discovery of those divergent behavior was really worrying because in order to get more and more accurate results theoretical chemists need to work in large basis sets. So they investigated the causes of those divergences and in the same time the reasons of the different types of convergence. In order to do this they analyzed the relation between the dominant singularity (i.e. the closest singularity to the origin) and the convergence behavior of the series \cite{Olsen_2000}. Their analysis is based on Darboux's theorem: In the limit of large order, the series coefficients become equivalent to the Taylor series coefficients of the singularity closest to the origin. Following the result of this theorem, one can explain the convergence patterns of the MP series by looking at the dominant singularity
A singularity in the unit circle is designated as an intruder state, more precisely as a front-door (respectively back-door) intruder state if the real part of the singularity is positive (respectively negative). The method used is to do a scan of the real axis to identify the avoided crossing responsible for the pair of dominant singularity. Then by modeling this avoided crossing by a two-state Hamiltonian one can get an approximation of the dominant conjugate pair of singularity by finding the EPs of the 2x2 Hamiltonian. The diagonal matrix is the unperturbed Hamiltonian and the other matrix is the perturbation part of the Hamiltonian.
\begin{equation}
\mqty(\alpha & \delta \\ \delta & \beta) = \mqty(\alpha + \alpha_s & 0 \\ 0 & \beta + \beta_s ) + \mqty(- \alpha_s & \delta \\ \delta & - \beta_s)
\end{equation}
They have shown that those divergences were due to a backdoor intruder state i.e. a pair of singularity in the negative half plane. Their method to investigate links between singularities and those behaviors of the series is to scan energies on the real axis to list the avoided crossings. An avoided crossing on the real axis is indicative of a pair of complex conjugate singularities, their real parts are equal to the value of $\lambda$ at the minimum difference of energy of the avoided crossing. Then modeling an avoided crossing with a two level model, they get an approximation of the pair of singularities by finding the EPs of the two-state model. When the basis set is augmented with diffuse functions the ground state undergoes a shark avoided crossing with a diffuse state on the negative real axis. When there are enough diffuse function in the basis set this pair of singularities is in the unit circle causing the divergence of the series. Olsen and his coworkers conclude that the divergence of the series in those case was due to the interaction of the ground state with a highly diffuse state.
They also analyzed the convergence of the series for the \ce{CH_2} as an example molecule containing low-lying doubly excited states. In this molecule the doubly excited states are strongly interacting with the ground state because they have the same spatial and spin symmetry. Thus the single reference wave function is a poor approximation. They have shown that there is a singularity close to the unit circle resulting from an interaction with the lowest doubly excited states and that this singularity is causing the slow convergence for the series of \ce{CH_2}.
They first studied molecules with low-lying doubly excited states of the same spatial and spin symmetry because in those systems the HF wave function is a bad approximation. The exact wave function have non-negligible contribution from the doubly excited states so those low-lying excited states were good candidates for being intruder states. For \ce{CH_2} in a large basis set, the series is convergent up to the 50th order. They have shown that the dominant singularity lies outside the unit circle but close to it causing the slow convergence.
Then they have shown that the divergence for the \ce{Ne} is due to a back-door intruder state. When the basis set is augmented with diffuse functions, the ground state undergo sharp avoided crossings with highly diffuse excited states leading to a back door intruder state. They used their two-state model on this avoided crossings and the model was actually predicting the divergence of the series. They conclude that the divergence of the series was due to the interaction with a highly diffuse excited states.
Moreover they proved that the extrapolation formula of Cremer and He \cite{Cremer_1996} can't be used for all systems. Even more that those formula were not mathematically motivated when one look at the singularity causing the divergence. For example the hydrogen fluoride molecule contains both back-door intruder states and low-lying doubly excited states which result in alternated terms up to order 10 and then the series is monotonically convergent. This is due to the fact that two pairs of singularity are approximately at the same distance of the origin.
\subsection{The singularity structure}
In the 2000's Sergeev and Goodson analyzed this problem from a more mathematical point of view by looking at the whole singularity structure where Olsen and his co-workers were trying to find the singularity causing the divergence. They regroup singularities in two classes: the $\alpha$ singularities which have unit order imaginary parts and the $\beta$ singularities which have very small imaginary parts. The singularities $\alpha$ are related to large avoided crossing between the ground state and a low-lying excited states. Whereas the singularities $\beta$ come from a sharp avoided crossing between the ground state and a highly diffuse state. They used the work of Stillinger to explain the physical signification of those $\beta$ singularities.
The M{\o}ller-Plesset Hamiltonian is defined as above and by reassembling the term we get the expression (10).
In the 2000's Sergeev and Goodson \cite{Sergeev_2005, Sergeev_2006} analyzed this problem from a more mathematical point of view by looking at the whole singularity structure where Olsen and his co-workers were trying to find the dominant singularity causing the divergence. They regroup singularities in two classes: the $\alpha$ singularities which have unit order imaginary parts and the $\beta$ singularities which have very small imaginary parts. The singularities $\alpha$ are related to large avoided crossing between the ground state and a low-lying excited states. Whereas the singularities $\beta$ come from a sharp avoided crossing between the ground state and a highly diffuse state. They succeed to explain the divergence of the series caused by $\beta$ singularities using a previous work of Stillinger \cite{Stillinger_2000}.
The M{\o}ller-Plesset Hamiltonian is defined as below and by reassembling the term we get the expression (10).
\begin{equation}
H(\lambda)=H_0 + \lambda (H_\text{phys} - H_0)
@ -283,9 +289,9 @@ H(\lambda)=H_0 + \lambda (H_\text{phys} - H_0)
The first two terms, the kinetic energy and the electron-nucleus attraction, form the mono-electronic core Hamiltonian which is independant of $\lambda$. The third term is the mean field repulsion of the Hartree-Fock calculation done to get $H_0$ and the last term is the Coulomb repulsion. If $\lambda$ is negative, the Coulomb interaction becomes attractive but the mean field stay repulsive as it is proportional to $(1-\lambda)$. If $\lambda$ becomes more and more negative the mean field becomes more and more repulsive so the nucleus can't bind anymore the electrons because the electron-nucleus attraction is not scaled with $\lambda$. But the repulsive mean field is localized around nucleus whereas the electrons interactions persist away from nucleus. So there is a real negative value $\lambda_c$ where the electron form a bound cluster and goes to infinity. According to Baker this value is a critical point of the system and by analogy with thermodynamics the energy $E(\lambda)$ exhibits a singularity at $\lambda_c$. Beyond $\lambda_c$ there is a continuum of eigenstates with electrons dissociated from the nucleus.
This reasoning is done on the exact Hamiltonian and energy, this is the exact energy which exhibits this singularity on the negative real axis. But in finite basis set, one can prove that for a Hermitian Hamiltonian the singularities of $E(\lambda)$ occurs in complex conjugate pair with non-zero imaginary parts. Sergeev and Goodson proved that, as predicted by Stillinger, in a finite basis set the critical point on the real axis modeled by a cluster of sharp avoided crossings with diffuse functions, equivalently by a cluster of $\beta$ singularities in the negative half plane.
This reasoning is done on the exact Hamiltonian and energy, this is the exact energy which exhibits this singularity on the negative real axis. But in finite basis set, one can prove that for a Hermitian Hamiltonian the singularities of $E(\lambda)$ occurs in complex conjugate pair with non-zero imaginary parts. Sergeev and Goodson proved that, as predicted by Stillinger, in a finite basis set the critical point on the real axis modeled by a cluster of sharp avoided crossings with diffuse functions, equivalently by a cluster of $\beta$ singularities in the negative half plane. They explain that Olsen et al., because they used a 2x2 model, only observed the first singularity of this cluster of singularities causing the divergence.
Finally, it has been shown that $\beta$ singularities are very sensitive to the basis sets but not to stretching of the system. On the contrary $\alpha$ singularities are relatively insensitive to the basis sets but very sensitive to bond stretching. According to Goodson the singularity structure from molecules stretched from the equilibrium geometry is difficult. To our knowledge the effect of bond stretching on singularities, its link with spin contamination and symmetry breaking of the wave function haven't been as well understood as the ionization effect and its link with diffuse function. In this work we try to improve our understanding of the effect of symmetry breaking on the singularities of $E(\lambda)$ and we hope that it will lead to a deeper understanding of perturbation theory.
Finally, it has been shown that $\beta$ singularities are very sensitive to the basis sets but not to stretching of the system. On the contrary $\alpha$ singularities are relatively insensitive to the basis sets but very sensitive to bond stretching. According to Goodson the singularity structure from molecules stretched from the equilibrium geometry is difficult \cite{Goodson_2004}, this is conform with the observation of Olsen and co-workers on the \ce{HF} molecule at equilibrium geometry and stretched geometry \cite{Olsen_2000}. To our knowledge the effect of bond stretching on singularities, its link with spin contamination and symmetry breaking of the wave function haven't been as well understood as the ionization effect and its link with diffuse function. In this work we try to improve our understanding of the effect of symmetry breaking on the singularities of $E(\lambda)$ and we hope that it will lead to a deeper understanding of perturbation theory.
\subsection{The physics of quantum phase transition}