diff --git a/Manuscript/EPAWTFT.log b/Manuscript/EPAWTFT.log index cfa2391..4a9e36d 100644 --- a/Manuscript/EPAWTFT.log +++ b/Manuscript/EPAWTFT.log @@ -1,4 +1,4 @@ -This is pdfTeX, Version 3.14159265-2.6-1.40.20 (TeX Live 2019/Arch Linux) (preloaded format=pdflatex 2020.5.9) 7 JUL 2020 16:30 +This is pdfTeX, Version 3.14159265-2.6-1.40.20 (TeX Live 2019/Arch Linux) (preloaded format=pdflatex 2020.5.9) 15 JUL 2020 11:47 entering extended mode restricted \write18 enabled. %&-line parsing enabled. diff --git a/Manuscript/EPAWTFT.pdf b/Manuscript/EPAWTFT.pdf index c297842..fe1d61a 100644 Binary files a/Manuscript/EPAWTFT.pdf and b/Manuscript/EPAWTFT.pdf differ diff --git a/RapportStage/Rapport.aux b/RapportStage/Rapport.aux index 2bd4589..0da0795 100644 --- a/RapportStage/Rapport.aux +++ b/RapportStage/Rapport.aux @@ -50,7 +50,7 @@ \citation{Goodson_2011} \citation{Moller_1934} \@writefile{lof}{\contentsline {figure}{\numberline {1}{\ignorespaces \centering A generic EP with the square root branch point topology. A loop around the EP interconvert the states.}}{4}{figure.1}\protected@file@percent } -\newlabel{fig:my_label}{{1}{4}{\centering A generic EP with the square root branch point topology. A loop around the EP interconvert the states}{figure.1}{}} +\newlabel{fig:TopologyEP}{{1}{4}{\centering A generic EP with the square root branch point topology. A loop around the EP interconvert the states}{figure.1}{}} \citation{Gill_1986} \citation{Gill_1988} \citation{Handy_1985} @@ -74,9 +74,9 @@ \citation{Christiansen_1996} \citation{Olsen_2000} \@writefile{lof}{\contentsline {figure}{\numberline {2}{\ignorespaces \centering Barriers to homolytic fission of \ce {He2^2+} at MPn/STO-3G level ($n = 1$--$20$)\cite {Gill_1986}.}}{6}{figure.2}\protected@file@percent } -\newlabel{fig:my_label}{{2}{6}{\centering Barriers to homolytic fission of \ce {He2^2+} at MPn/STO-3G level ($n = 1$--$20$)\cite {Gill_1986}}{figure.2}{}} +\newlabel{fig:RUMP_Gill}{{2}{6}{\centering Barriers to homolytic fission of \ce {He2^2+} at MPn/STO-3G level ($n = 1$--$20$)\cite {Gill_1986}}{figure.2}{}} \@writefile{lot}{\contentsline {table}{\numberline {1}{\ignorespaces \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 \cite {Gill_1988}.}}{6}{table.1}\protected@file@percent } -\newlabel{tab:my_label}{{1}{6}{\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 \cite {Gill_1988}}{table.1}{}} +\newlabel{tab:SpinContamination}{{1}{6}{\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 \cite {Gill_1988}}{table.1}{}} \@writefile{toc}{\contentsline {subsection}{\numberline {3.2}Cases of divergence}{6}{subsection.3.2}\protected@file@percent } \citation{Cremer_1996} \citation{Sergeev_2005} @@ -103,14 +103,25 @@ \citation{Macek_2019} \citation{Cejnar_2009} \citation{Sachdev_2011} +\newlabel{eq:HamiltonianStillinger}{{11}{8}{The singularity structure}{equation.3.11}{}} \citation{Cejnar_2005} \citation{Cejnar_2007} \citation{Stransky_2018} -\citation{SzaboBook} \@writefile{toc}{\contentsline {subsection}{\numberline {3.4}The physics of quantum phase transition}{9}{subsection.3.4}\protected@file@percent } \@writefile{toc}{\contentsline {section}{\numberline {4}The spherium model}{9}{section.4}\protected@file@percent } +\citation{Coulson_1949} +\citation{SzaboBook} \@writefile{toc}{\contentsline {subsection}{\numberline {4.1}Weak correlation regime}{10}{subsection.4.1}\protected@file@percent } -\@writefile{toc}{\contentsline {section}{\numberline {5}To do list}{10}{section.5}\protected@file@percent } +\@writefile{toc}{\contentsline {subsubsection}{\numberline {4.1.1}Restricted and unrestricted equation for the spherium model}{10}{subsubsection.4.1.1}\protected@file@percent } +\newlabel{eq:EUHF}{{15}{10}{Restricted and unrestricted equation for the spherium model}{equation.4.15}{}} +\newlabel{eq: RHFWF}{{16}{10}{Restricted and unrestricted equation for the spherium model}{equation.4.16}{}} +\newlabel{eq:ERHF}{{17}{10}{Restricted and unrestricted equation for the spherium model}{equation.4.17}{}} +\@writefile{toc}{\contentsline {subsubsection}{\numberline {4.1.2}The minimal basis example}{10}{subsubsection.4.1.2}\protected@file@percent } +\@writefile{toc}{\contentsline {subsubsection}{\numberline {4.1.3}Symmetry-broken solutions}{11}{subsubsection.4.1.3}\protected@file@percent } +\@writefile{toc}{\contentsline {subsection}{\numberline {4.2}Strongly correlated regime}{11}{subsection.4.2}\protected@file@percent } +\@writefile{toc}{\contentsline {section}{\numberline {5}Perturbation theory and exceptional points}{11}{section.5}\protected@file@percent } +\@writefile{toc}{\contentsline {section}{\numberline {6}To do list}{11}{section.6}\protected@file@percent } +\@writefile{toc}{\contentsline {section}{\numberline {7}Conclusion}{11}{section.7}\protected@file@percent } \bibstyle{unsrt} \bibdata{Rapport} \bibcite{Bittner_2012}{1} @@ -168,3 +179,5 @@ \bibcite{Caprio_2008}{53} \bibcite{Macek_2019}{54} \bibcite{Stransky_2018}{55} +\bibcite{Coulson_1949}{56} +\@writefile{toc}{\contentsline {section}{\numberline {A}ERHF and EUHF}{15}{appendix.A}\protected@file@percent } diff --git a/RapportStage/Rapport.bbl b/RapportStage/Rapport.bbl index 61d242d..d5f1c71 100644 --- a/RapportStage/Rapport.bbl +++ b/RapportStage/Rapport.bbl @@ -311,4 +311,10 @@ Pavel Stránský, Martin Dvořák, and Pavel Cejnar. transitions. \newblock 97(1):012112. +\bibitem{Coulson_1949} +C.~A. Coulson and I.~Fischer. +\newblock {XXXIV}. notes on the molecular orbital treatment of the hydrogen + molecule. +\newblock 40(303):386--393. + \end{thebibliography} diff --git a/RapportStage/Rapport.bib b/RapportStage/Rapport.bib index 7bc90aa..d9b89b4 100644 --- a/RapportStage/Rapport.bib +++ b/RapportStage/Rapport.bib @@ -739,4 +739,15 @@ shortjournal = {Phys. Scr.}, author = {Cejnar, Pavel and Stránský, Pavel}, date = {2016-07}, +} + +@article{Coulson_1949, + title = {{XXXIV}. Notes on the molecular orbital treatment of the hydrogen molecule}, + volume = {40}, + doi = {10.1080/14786444908521726}, + pages = {386--393}, + number = {303}, + journaltitle = {The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science}, + author = {Coulson, C. 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PDF statistics: - 421 PDF objects out of 1000 (max. 8388607) - 360 compressed objects within 4 object streams - 97 named destinations out of 1000 (max. 500000) - 131 words of extra memory for PDF output out of 10000 (max. 10000000) + 486 PDF objects out of 1000 (max. 8388607) + 422 compressed objects within 5 object streams + 115 named destinations out of 1000 (max. 500000) + 187 words of extra memory for PDF output out of 10000 (max. 10000000) diff --git a/RapportStage/Rapport.out b/RapportStage/Rapport.out index f84c5cf..58d5a8c 100644 --- a/RapportStage/Rapport.out +++ b/RapportStage/Rapport.out @@ -7,4 +7,11 @@ \BOOKMARK [2][-]{subsection.3.4}{The physics of quantum phase transition}{section.3}% 7 \BOOKMARK [1][-]{section.4}{The spherium model}{}% 8 \BOOKMARK [2][-]{subsection.4.1}{Weak correlation regime}{section.4}% 9 -\BOOKMARK [1][-]{section.5}{To do list}{}% 10 +\BOOKMARK [3][-]{subsubsection.4.1.1}{Restricted and unrestricted equation for the spherium model}{subsection.4.1}% 10 +\BOOKMARK [3][-]{subsubsection.4.1.2}{The minimal basis example}{subsection.4.1}% 11 +\BOOKMARK [3][-]{subsubsection.4.1.3}{Symmetry-broken solutions}{subsection.4.1}% 12 +\BOOKMARK [2][-]{subsection.4.2}{Strongly correlated regime}{section.4}% 13 +\BOOKMARK [1][-]{section.5}{Perturbation theory and exceptional points}{}% 14 +\BOOKMARK [1][-]{section.6}{To do list}{}% 15 +\BOOKMARK [1][-]{section.7}{Conclusion}{}% 16 +\BOOKMARK [1][-]{appendix.A}{ERHF and EUHF}{}% 17 diff --git a/RapportStage/Rapport.pdf b/RapportStage/Rapport.pdf index 3868c04..626b500 100644 Binary files a/RapportStage/Rapport.pdf and b/RapportStage/Rapport.pdf differ diff --git a/RapportStage/Rapport.synctex.gz b/RapportStage/Rapport.synctex.gz index 1579c24..3b7008a 100644 Binary files a/RapportStage/Rapport.synctex.gz and b/RapportStage/Rapport.synctex.gz differ diff --git a/RapportStage/Rapport.tex b/RapportStage/Rapport.tex index 37f91f8..432649c 100644 --- a/RapportStage/Rapport.tex +++ b/RapportStage/Rapport.tex @@ -48,16 +48,6 @@ hyperfigures=false] \fancyhead[R]{\scriptsize \textsc{Antoine \textsc{MARIE}}} \fancyfoot[C]{ \thepage} -% commande d'annulation du correcteur typographique du package [francais]{babel} qui force l'espace avant ':' (parfois utile pour la bibliographie) -\makeatletter -\@ifpackageloaded{babel}% - {\newcommand{\nospace}[1]{{\NoAutoSpaceBeforeFDP{}#1}}}% % !! double {{}} pour cantonner l'effet à l'argument #1 !! - {\newcommand{\nospace}[1]{#1}} -\makeatother - -% commande de déplacement d'un objet -\newcommand{\drawat}[3]{\makebox[0pt][l]{\raisebox{#2}{\hspace*{#1}#3}}} - \newcommand{\bH}{\mathbf{H}} \newcommand{\bV}{\mathbf{V}} @@ -128,6 +118,8 @@ Laboratoire de Chimie et Physique Quantiques \newpage +\setlength{\parindent}{17pt} + \section*{Acknowledgments} \tableofcontents @@ -155,7 +147,7 @@ Exceptional points (EPs) \cite{Heiss_1990, Heiss_1999, Heiss_2012, Heiss_2016} a CIs are ubiquitous in non-adiabatic processes and play a key role in photo-chemical mechanisms. In the case of auto-ionizing resonances, EPs have a role in deactivation processes similar to CIs in the decay of bound excited states. Although Hermitian and non-Hermitian Hamiltonians are closely related, the behavior of their eigenvalues near degeneracies is starkly different. -For example, encircling non-Hermitian degeneracies at EPs leads to an interconversion of states, and two loops around the EP are necessary to recover the initial energy (see Figure 1 for a graphical example). +For example, encircling non-Hermitian degeneracies at EPs leads to an interconversion of states, and two loops around the EP are necessary to recover the initial energy (see \autoref{fig:TopologyEP} for a graphical example). Additionally, the wave function picks up a geometric phase (also known as Berry phase \cite{Berry_1984}) and four loops are required to recover the initial wave function. In contrast, encircling Hermitian degeneracies at CIs only introduces a geometric phase while leaving the states unchanged. More dramatically, whilst eigenvectors remain orthogonal at CIs, at non-Hermitian EPs the eigenvectors themselves become equivalent, resulting in a \textit{self-orthogonal} state. \cite{MoiseyevBook} @@ -165,7 +157,7 @@ More importantly here, although EPs usually lie off the real axis, these singula \centering \includegraphics[width=0.7\textwidth]{TopologyEP.pdf} \caption{\centering A generic EP with the square root branch point topology. A loop around the EP interconvert the states.} - \label{fig:my_label} + \label{fig:TopologyEP} \end{figure} %============================================================% @@ -212,16 +204,16 @@ But as mentioned before \textit{a priori} there are no reasons that this power s \subsection{Behavior of the M{\o}ller-Plesset series} When we use M{\o}ller-Plesset perturbation theory it would be very convenient that each time a higher order term is computed the result obtained is closer to exact energy. In other words, that the M{\o]ller-Plesset series would be monotonically convergent. Assuming this, the only limiting process to get the exact correlation energy in a finite basis set is our ability to compute the terms of the perturbation series. -Unfortunately this is not true in generic cases and rapidly some strange behaviors of the series were exhibited. In the late 80's Gill et al. reported deceptive and slow convergences in stretch systems\cite{Gill_1986, Gill_1988, Handy_1985, Lepetit_1988}. In the figure below we can see that the restricted M{\o}ller-Plesset series is convergent but oscillating which is not convenient if you are only able to compute few terms (for example here RMP5 is worse than RMP4). On the other hand, the unrestricted M{\o}ller-Plesset series is monotonically converging (except for the first few orders) but very slowly so we can't use it for systems where we can only compute the first terms. +Unfortunately this is not true in generic cases and rapidly some strange behaviors of the series were exhibited. In the late 80's Gill et al. reported deceptive and slow convergences in stretch systems\cite{Gill_1986, Gill_1988, Handy_1985, Lepetit_1988}. In the \autoref{fig:RUMP_Gill} we can see that the restricted M{\o}ller-Plesset series is convergent but oscillating which is not convenient if you are only able to compute few terms (for example here RMP5 is worse than RMP4). On the other hand, the unrestricted M{\o}ller-Plesset series is monotonically converging (except for the first few orders) but very slowly so we can't use it for systems where we can only compute the first terms. \begin{figure}[h!] \centering \includegraphics[width=0.45\textwidth]{gill1986.png} \caption{\centering Barriers to homolytic fission of \ce{He2^2+} at MPn/STO-3G level ($n = 1$--$20$)\cite{Gill_1986}.} - \label{fig:my_label} + \label{fig:RUMP_Gill} \end{figure} -When a bond is stretched the exact function can undergo a symmetry breaking becoming multi-reference during this process (see for example the case of \ce{H_2} in \cite{SzaboBook}). A restricted HF Slater determinant is a poor approximation of a broken symmetry wave function but even in the unrestricted formalism, where the spatial orbitals of electrons $\alpha$ and $\beta$ are not restricted to be the same\cite{Fukutome_1981}, which allows a better description of broken symmetry system, the series doesn't give accurate results at low orders. Even with this improvement of the zeroth order wave function the series doesn't have the smooth and rapidly converging behavior wanted. +When a bond is stretched the exact function can undergo a symmetry breaking becoming multi-reference during this process (see for example the case of \ce{H_2} in \cite{SzaboBook}). A restricted HF Slater determinant is a poor approximation of a symmetry-broken wave function but even in the unrestricted formalism, where the spatial orbitals of electrons $\alpha$ and $\beta$ are not restricted to be the same\cite{Fukutome_1981}, which allows a better description of symmetry-broken system, the series doesn't give accurate results at low orders. Even with this improvement of the zeroth order wave function the series doesn't have the smooth and rapidly converging behavior wanted. \begin{table}[h!] \centering @@ -236,10 +228,10 @@ When a bond is stretched the exact function can undergo a symmetry breaking beco \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 STO-3G basis set \cite{Gill_1988}.} - \label{tab:my_label} + \label{tab:SpinContamination} \end{table} -In the unrestricted framework the ground state singlet wave function is allowed to mix with triplet states which leads to spin contamination. Gill et al. highlighted the link between the slow convergence of the unrestricted MP series and the spin contamination of the wave function as shown in the Table 1 in the example of \ce{H_2} in a minimal basis. Handy and co-workers exhibited the same behaviors of the series (oscillating and monotonically slowly) in stretched \ce{H_2O} and \ce{NH_2} systems \cite{Handy_1985}. Lepetit et al. analyzed the difference between the M{\o}ller-Plesset and Epstein-Nesbet partitioning for the unrestricted Hartree-Fock reference \cite{Lepetit_1988}. They concluded that the slow convergence is due to the coupling of the single with the double excited configuration. Moreover the MP denominators tends towards a constant so each contribution become very small when the bond is stretched. +In the unrestricted framework the ground state singlet wave function is allowed to mix with triplet states which leads to spin contamination. Gill et al. highlighted the link between the slow convergence of the unrestricted MP series and the spin contamination of the wave function as shown in the \autoref{tab:SpinContamination} in the example of \ce{H_2} in a minimal basis. Handy and co-workers exhibited the same behaviors of the series (oscillating and monotonically slowly) in stretched \ce{H_2O} and \ce{NH_2} systems \cite{Handy_1985}. Lepetit et al. analyzed the difference between the M{\o}ller-Plesset and Epstein-Nesbet partitioning for the unrestricted Hartree-Fock reference \cite{Lepetit_1988}. They concluded that the slow convergence is due to the coupling of the single with the double excited configuration. Moreover the MP denominators tends towards a constant so each contribution become very small when the bond is stretched. Cremer and He analyzed 29 FCI systems \cite{Cremer_1996} and regrouped all the systems in two classes. The class A systems which have a monotonic convergence to the FCI value and the class B which converge erratically after initial oscillations. The sample of systems contains stretched molecules and also some at equilibrium geometry, there are also some systems in various basis sets. They highlighted that systems with class A convergence have well-separated electrons pairs whereas class B systems present electrons clustering. This classification was encouraging in order to develop methods based on perturbation theory as it rationalizes the two different observed convergence modes. If it is possible to predict if a system is class A or B, then one can use extrapolation method of the first terms adapted to the class of the systems \cite{Cremer_1996}. @@ -258,7 +250,7 @@ However Olsen et al. have discovered an even more preoccupying behavior of the M The discovery of this divergent behavior was really worrying because in order to get more and more accurate results theoretical chemists need to work in large basis sets. As a consequence they investigated the causes of those divergences and in the same time the reasons of the different types of convergence. 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, the convergence patterns of the MP series can be explained 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 $22\times$ Hamiltonian. The diagonal matrix is the unperturbed Hamiltonian and the other matrix is the perturbative part of the Hamiltonian. +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 $2\times2$ Hamiltonian. The diagonal matrix is the unperturbed Hamiltonian and the other matrix is the perturbative 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) @@ -274,7 +266,7 @@ Moreover they proved that the extrapolation formula of Cremer and He \cite{Creme 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 regrouped 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 succeeded 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 (11). +The M{\o}ller-Plesset Hamiltonian is defined as below and by reassembling the term we get the expression \eqref{eq:HamiltonianStillinger}. \begin{equation} H(\lambda)=H_0 + \lambda (H_\text{phys} - H_0) @@ -290,6 +282,7 @@ H(\lambda)=H_0 + \lambda (H_\text{phys} - H_0) \begin{equation} 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