corrections

RapportStage/Rapport.bib

@@ -1,13 +1,5 @@
-%% This BibTeX bibliography file was created using BibDesk.
-%% http://bibdesk.sourceforge.net/
-
-%% Created for Pierre-Francois Loos at 2020-07-28 10:22:06 +0200 
-
-
 %% Saved with string encoding Unicode (UTF-8) 
 
-
-
 @book{BenderBook,
 	Author = {C. M. Berder and S. A. Orszag},
 	Date-Added = {2020-07-28 09:59:40 +0200},
@@ -826,3 +818,78 @@
 	Volume = {79},
 	Year = {2009},
 	Bdsk-Url-1 = {https://doi.org/10.1103/PhysRevA.79.062517}}
+
+@book{Ushveridze_1994,
+	address = {Bristol [England]; Philadelphia},
+	title = {Quasi-exactly solvable models in quantum mechanics},
+	isbn = {978-0-7503-0266-1},
+	language = {English},
+	publisher = {Institute of Physics Pub.},
+	author = {Ushveridze, Alexander G},
+	year = {1994},
+	note = {OCLC: 28421899}
+}
+
+@article{Lipkin_1965,
+	title = {Validity of many-body approximation methods for a solvable model: ({I}). {Exact} solutions and perturbation theory},
+	volume = {62},
+	doi = {10.1016/0029-5582(65)90862-X},
+	language = {en},
+	number = {2},
+	journal = {Nuclear Physics},
+	author = {Lipkin, H. J. and Meshkov, N. and Glick, A. J.},
+	month = feb,
+	year = {1965},
+	pages = {188--198},
+}
+
+
+@article{Wigner_1934,
+	title = {On the {Interaction} of {Electrons} in {Metals}},
+	volume = {46},
+	doi = {10.1103/PhysRev.46.1002},
+	number = {11},
+	journal = {Physical Review},
+	author = {Wigner, E.},
+	month = dec,
+	year = {1934},
+	pages = {1002--1011},
+}
+
+
+@article{Thompson_2005,
+	title = {A comparison of {Hartree}–{Fock} and exact diagonalization solutions for a model two-electron system},
+	volume = {122},
+	issn = {0021-9606},
+	doi = {10.1063/1.1869978},
+	number = {12},
+	journal = {The Journal of Chemical Physics},
+	author = {Thompson, David C. and Alavi, Ali},
+	month = mar,
+	year = {2005},
+	pages = {124107},
+}
+
+@article{Seidl_2007,
+	title = {Adiabatic connection in density-functional theory: {Two} electrons on the surface of a sphere},
+	volume = {75},
+	doi = {10.1103/PhysRevA.75.062506},
+	number = {6},
+	journal = {Physical Review A},
+	author = {Seidl, Michael},
+	month = jun,
+	year = {2007},
+	pages = {062506},
+}
+
+@article{Loos_2009b,
+	title = {Two {Electrons} on a {Hypersphere}: {A} {Quasiexactly} {Solvable} {Model}},
+	volume = {103},
+	doi = {10.1103/PhysRevLett.103.123008},
+	number = {12},
+	journal = {Physical Review Letters},
+	author = {Loos, Pierre-François and Gill, Peter M. W.},
+	month = sep,
+	year = {2009},
+	pages = {123008},
+}

RapportStage/Rapport.tex

@@ -492,14 +492,14 @@ One could then potentially claim that the RMP series exhibits deceptive converge
 \end{table}
 
 In the unrestricted framework the singlet ground state wave function is allowed to mix with triplet wave functions, leading to the so-called spin contamination issue. Gill \textit{et al.}~highlighted the link between slow convergence of the UMP series and spin contamination, as shown in Table \ref{tab:SpinContamination} for \ce{H2} in a minimal basis \cite{Gill_1988}. 
-Handy and coworkers reported the same behavior of the series (oscillatory and \titou{slowly monotonically convergent}) in stretched \ce{H2O} and \ce{NH2} systems \cite{Handy_1985}. Lepetit \textit{et al.}~analyzed the difference between the MP and EN partitioning for the UHF reference \cite{Lepetit_1988}. They concluded that the slow convergence is due to the coupling of the singly- and doubly-excited configurations. 
+Handy and coworkers reported the same behavior of the series (oscillatory and slowly monotonically convergent) in stretched \ce{H2O} and \ce{NH2} systems \cite{Handy_1985}. Lepetit \textit{et al.}~analyzed the difference between the MP and EN partitioning for the UHF reference \cite{Lepetit_1988}. They concluded that the slow convergence is due to the coupling of the singly- and doubly-excited configurations. 
 %Moreover, the MP and EN numerators in Eqs.~\eqref{eq:EMP2} and \eqref{eq:EEN2} are the same and they vanish when the bond length $r$ goes to infinity. Yet the MP denominators tends towards a constant when $r \to \infty$ so the terms vanish, whereas the EN denominators tends to zero which improves the convergence but can also make the series diverge.
 Cremer and He analyzed 29 atomic and molecular systems at the FCI level \cite{Cremer_1996} and grouped them in two classes: i) the class A systems where one observes a monotonic convergence to the FCI energy, and ii) the class B for which convergence is erratic after initial oscillations. Their system set contains stretched molecules as well as molecules at their equilibrium geometry for various basis sets. They highlighted that \cite{Cremer_1996}
 \begin{quote}
 	\textit{``Class A systems are characterized by electronic structures with well-separated electron pairs while class B systems are characterized by electronic structures with electron clustering in one or more regions.''}
 \end{quote}
-\antoine{They proved that using different extrapolation formulas of the first terms of the MP series for class A and class B systems improves the precision of the results compared to the formula used without classes. The mean deviation from FCI correlation energies is $0.3$ millihartree with the adapted formula whereas with the formula that do not distinguish the system it is 12 millihartree.} Even if there were still shaded areas and that this classification was incomplete, this work showed that understanding the origin of the different modes of convergence would lead to a more rationalized use of the MP perturbation theory and to more accurate correlation energies.
+\antoine{As one can only compute the first terms of the MP$l$ series, a smart way to get more accurate results is to use extrapolation formula of this series. They proved that using specific extrapolation formulas of the MP series for class A and class B systems improves the precision of the results compared to the formula used without classes. The mean deviation from FCI correlation energies is $0.3$ millihartree with the adapted formula whereas with the formula that do not distinguish the systems it is 12 millihartree.} Even if there were still shaded areas and that this classification was incomplete, this work showed that understanding the origin of the different modes of convergence would lead to a more rationalized use of the MP perturbation theory and to more accurate correlation energies.
-%>>>>>>> 4aba1a6e837a50f3a21d610948564733096cf38e
+
 
 \subsection{Cases of divergence}
 
@@ -549,29 +549,29 @@ To understand the convergence properties of the perturbation series at $\lambda=
     ].
 \end{equation}
 
-The major difference between those two terms is that the repulsive mean field is localized around the nuclei whereas the interelectronic interaction persist away from the nuclei. If $\lambda$ becomes more and more negative the mean field becomes more and more repulsive so there exists a critical (negative) value of $\lambda$, $\lambda_\text{c}$, for which the Coulombic field created by the nuclei cannot bind the electrons anymore because of the $\lambda$-independent nature of the the electron-nucleus attraction. For $\lambda = \lambda_c$, the electrons dissociate from the nuclei and form a bound cluster which is infinitely separated from the nuclei. According to Baker \cite{Baker_1971}, this value is a critical point of the system and, by analogy with thermodynamics, the energy $E(\lambda)$ exhibits a singularity at $\lambda_\text{c}$. At this point the system undergo a phase transition \titou{and a symmetry breaking}. \titou{Beyond $\lambda_c$ there is a continuum of eigenstates with electrons dissociated from the nucleus.}
+The major difference between those two terms is that the repulsive mean field is localized around the nuclei whereas the interelectronic interaction persist away from the nuclei. If $\lambda$ becomes more and more negative the mean field becomes more and more repulsive so there exists a critical (negative) value of $\lambda$, $\lambda_\text{c}$, for which the Coulombic field created by the nuclei cannot bind the electrons anymore because of the $\lambda$-independent nature of the the electron-nucleus attraction. For $\lambda = \lambda_c$, the electrons dissociate from the nuclei and form a bound cluster which is infinitely separated from the nuclei. According to Baker \cite{Baker_1971}, this value is a critical point of the system and, by analogy with thermodynamics, the energy $E(\lambda)$ exhibits a singularity at $\lambda_\text{c}$. At this point the system undergo a phase transition and a symmetry breaking. 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, i.e., the Hamiltonian in the complete Hilbert space, this is the exact energy which exhibits this singularity on the negative real axis. However, in a finite basis set which does not span the complete Hilbert space, one can prove that, for a Hermitian Hamiltonian, the singularities of $E(\lambda)$ occurs in complex conjugate pairs with non-zero imaginary parts. Sergeev and Goodson proved \cite{Sergeev_2005}, as predicted by Stillinger \cite{Stillinger_2000}, that in a finite basis set the critical point on the real axis is modeled by a cluster of sharp avoided crossings with diffuse functions, equivalently by a cluster of $\beta$ singularities in the negative half plane. This explains that Olsen \textit{et al.}, because they used a simple $2\times2$ model, only observed the first singularity of this cluster of singularities causing the divergence.
 
-Finally, it was shown that $\beta$ singularities are very sensitive to changes in the basis set but not to the stretching of the system. On the contrary $\alpha$ singularities are relatively insensitive to the basis sets but very sensitive to bond stretching. \titou{According to Goodson \cite{Goodson_2004}, the singularity structure from molecules stretched from the equilibrium geometry is difficult, this is consistent with the observation of Olsen and coworkers \cite{Olsen_2000} on the \ce{HF} molecule at equilibrium geometry and stretched geometry.} To the best our knowledge, the effect of bond stretching on singularities, its link with spin contamination and symmetry breaking of the wave function has not been as well understood as the ionization phenomenon and its link with diffuse functions.  In this work, we shall 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 was shown that $\beta$ singularities are very sensitive to changes in the basis set but not to the stretching of the system. On the contrary $\alpha$ singularities are relatively insensitive to the basis sets but very sensitive to bond stretching. \antoine{According to Goodson \cite{Goodson_2004}, the singularity structure from molecules stretched from the equilibrium geometry is difficult because there is more than one significant singularity. This is consistent with the observation of Olsen and coworkers \cite{Olsen_2000} on the \ce{HF} molecule at equilibrium geometry and stretched geometry.} To the best our knowledge, the effect of bond stretching on singularities, its link with spin contamination and symmetry breaking of the wave function has not been as well understood as the ionization phenomenon and its link with diffuse functions.  In this work, we shall 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}
 
 In the previous section, we saw that a careful analysis of the structure of the Hamiltonian allows us to predict the existence of a critical point. In a finite basis set this critical point is model by a cluster of $\beta$ singularities. It is now well known that this phenomenon is a special case of a more general phenomenon. Indeed, theoretical physicists proved that EPs close to the real axis are connected to \textit{quantum phase transitions} (QPTs) \cite{Heiss_1988, Heiss_2002, Cejnar_2005, Cejnar_2007, Cejnar_2009, Borisov_2015, Sindelka_2017}. In quantum mechanics, the Hamiltonian is almost always dependent of, at least, one parameter. In some cases the variation of a parameter can lead to abrupt changes at a critical point. These QPTs exist both for ground and excited states as shown by Cejnar and coworkers \cite{Cejnar_2009, Sachdev_2011, Cejnar_2015, Cejnar_2016, Caprio_2008, Macek_2019}. A ground-state QPT is characterized by the derivatives of the ground-state energy with respect to a non-thermal control parameter \cite{Cejnar_2009, Sachdev_2011}. The transition is called discontinuous and of first order if the first derivative is discontinuous at the critical parameter value. Otherwise, it is called continuous and of $m$th order if the $m$th derivative is discontinuous. A QPT can also be identify by the discontinuity of an appropriate order parameter (or one of its derivatives). 
 
-The presence of an EP close to the real axis is characteristic of a sharp avoided crossing. Yet, at such an avoided crossing, eigenstates change abruptly. Although it is now well understood that EPs are closely related to QPTs, the link between the type of QPT (ground state or excited state, first or higher order) and EPs still need to be clarified. One of the major obstacles that one faces in order to achieve this resides in the ability to compute the distribution of EPs. The numerical assignment of an EP to two energies on the real axis is very difficult in large dimensions. Hence, the design of specific methods are required to get information on the location of EPs. Following this idea, Cejnar \textit{et al.}~developed a method based on a Coulomb analogy giving access to the density of EP close to the real axis \cite{Cejnar_2005, Cejnar_2007}. More recently Stransky and coworkers proved that the distribution of EPs is characteristic on the order of the QPT \cite{Stransky_2018}. \titou{In particular, they showed that when the dimensionality of the system increases, first- and second-order QTP behave differently, and the position of these singularities converge towards the real axis at different rates (exponentially and algebraically for the first and second order, respectively).}
+The presence of an EP close to the real axis is characteristic of a sharp avoided crossing. Yet, at such an avoided crossing, eigenstates change abruptly. Although it is now well understood that EPs are closely related to QPTs, the link between the type of QPT (ground state or excited state, first or higher order) and EPs still need to be clarified. One of the major obstacles that one faces in order to achieve this resides in the ability to compute the distribution of EPs. The numerical assignment of an EP to two energies on the real axis is very difficult in large dimensions. Hence, the design of specific methods are required to get information on the location of EPs. Following this idea, Cejnar \textit{et al.}~developed a method based on a Coulomb analogy giving access to the density of EP close to the real axis \cite{Cejnar_2005, Cejnar_2007}. More recently Stransky and coworkers proved that the distribution of EPs is characteristic on the order of the QPT \cite{Stransky_2018}. \antoine{In the thermodynamic limit some of the EPs converge towards a critical point $\lambda$\textsuperscript{c} on the real axis. They showed that within the LMG model \cite{Lipkin_1965} (which can be for example interacting fermions on two-energy levels or two interacting bosonic species) EPs associated to first- and second-order QPT behave differently when the number of particles increases. The position of these singularities converge towards the critical point and the real axis at different rates (exponentially and algebraically for the first and second order, respectively).}
 
-It seems like our understanding of the physics of spatial and/or spin symmetry breaking in HF theory can be enlightened by QPT theory. Indeed, the second derivative of the energy is discontinuous at the \titou{Coulson-Fischer point (not defined)} which means that the system undergo a second-order QPT. Moreover, the $\beta$ singularities introduced by Sergeev and coworkers to describe the EPs modeling the formation of a bound cluster of electrons are actually a more general class of singularities. The EPs close to the real axis (the so-called $\beta$ singularities) are connected to QPT because they result from a sharp avoided crossings at which the eigenstates change quickly. However, the $\alpha$ singularities arise from large avoided crossings. Thus, they cannot be connected to QPT. The avoided crossings generating $\alpha$ singularities generally involve the ground state and low-lying doubly-excited states. Those excited states have a non-negligible contribution to the exact FCI solution because they have the same spatial and spin symmetry as the ground state. We believe that $\alpha$ singularities are connected to states with non-negligible contribution in the CI expansion thus to the dynamical part of the correlation energy, while $\beta$ singularities are linked to symmetry breaking and phase transitions of the wave function, i.e., to the multi-reference nature of the wave function thus to the static part of the correlation energy.
+It seems like our understanding of the physics of spatial and/or spin symmetry breaking in HF theory can be enlightened by QPT theory. \antoine{Indeed, the second derivative of the HF ground-state energy is discontinuous at the point of spin symmetry-breaking which means that the system undergo a second-order QPT.} Moreover, the $\beta$ singularities introduced by Sergeev and coworkers to describe the EPs modeling the formation of a bound cluster of electrons are actually a more general class of singularities. The EPs close to the real axis (the so-called $\beta$ singularities) are connected to QPT because they result from a sharp avoided crossings at which the eigenstates change quickly. However, the $\alpha$ singularities arise from large avoided crossings. Thus, they cannot be connected to QPT. The avoided crossings generating $\alpha$ singularities generally involve the ground state and low-lying doubly-excited states. Those excited states have a non-negligible contribution to the exact FCI solution because they have the same spatial and spin symmetry as the ground state. We believe that $\alpha$ singularities are connected to states with non-negligible contribution in the CI expansion thus to the dynamical part of the correlation energy, while $\beta$ singularities are linked to symmetry breaking and phase transitions of the wave function, i.e., to the multi-reference nature of the wave function thus to the static part of the correlation energy.
 
 
 %============================================================%
 \section{The spherium model}\label{sec:spherium}
 %============================================================%
 
-Simple systems that are analytically solvable (or at least quasi-exactly solvable \titou{(please define)}) are of great importance in theoretical chemistry. 
+Simple systems that are analytically solvable (or at least quasi-exactly solvable \antoine{i.e. model for which it is possible to obtain a finite portion of the exact solutions of the Schrödinger equation \eqref{eq:SchrEq} \cite{Ushveridze_1994}}) are of great importance in theoretical chemistry. 
 These systems are very useful to perform benchmark studies in order to test new methods as the mathematics are easier than in realistic systems (such as molecules or solids) but retain much of the key physics. 
 To investigate the physics of EPs we consider one such system named \textit{spherium}. 
-It consists of two electrons confined to the surface of a sphere interacting through the long-range Coulomb potential. 
+It consists of two electrons confined to the surface of a sphere interacting through the long-range Coulomb potential \cite{Thompson_2005, Seidl_2007, Loos_2009b}. 
 Thus, the Hamiltonian is
 \begin{equation}
 	\hH = -\frac{\grad_1^2 + \grad_2^2}{2} + \frac{1}{r_{12}},
@@ -583,10 +583,10 @@ or
 where $\omega$ is the interelectronic angle.
 The Laplace operators are the kinetic operators for each electrons and $r_{12}^{-1} = \abs{\vb{r}_1 - \vb{r}_2}^{-1}$ is the Coulomb operator. 
 Note that, as readily seen by the definition of the interelectronic distance $r_{12}$, the electrons interact through the sphere.
-The radius of the sphere $R$ dictates the correlation regime. 
+The radius of the sphere $R$ dictates the correlation regime \cite{Loos_2009}. 
 In the weak correlation regime (i.e., small $R$), the kinetic energy (which scales as $R^{-2}$) dominates and the electrons are delocalized over the sphere.
 For large $R$ (or strong correlation regime), the electron repulsion term (which scales as $R^{-1}$) drives the physics and the electrons localize on opposite side of the sphere to minimize their Coulomb repulsion. 
-This phenomenon is sometimes referred to as a Wigner crystallization.
+This phenomenon is sometimes referred to as a Wigner crystallization \cite{Wigner_1934}.
 \titou{T2: Missing references in this part.}
 
 We will use this model in order to rationalize the effects of the parameters that may influence the physics of EPs:
@@ -628,7 +628,7 @@ with
 	V = \frac{1}{R} \sum_{\ell_1,\ell_2,\ell_3,\ell_4=0}^{\infty} \sum_{L=0}^{\infty} 
 	(-1)^{\ell_3+\ell_4} v^\alpha_{\ell_1,\ell_2,L} v^\beta_{\ell_3,\ell_4,L} 
 \end{gather}
-and
+\antoine{and $v^\sigma_{\ell_1,\ell_2,L}$ is expressed in terms of the Wigner 3j-symbols \cite{AngularBook}}
 \begin{equation}
 	v^\sigma_{\ell_1,\ell_2,L} 
 	= \sqrt{(2\ell_1+1)(2\ell_2+1)} C_{\sigma,\ell_1}C_{\sigma,\ell_2}
@@ -638,13 +638,12 @@ and
 		0 & 0 & 0
 	\end{pmatrix}^2 
 \end{equation}
-\titou{T2: please define Wigner 3j symbols.}
 \titou{STOPPED HERE.}
 
-We obtained Eq.~\eqref{eq:EUHF} for the general form of the wave function \eqref{eq:UHF_WF}, but to be associated with a physical wave function the energy needs to be stationary with respect to the coefficients. The general method is to use the Hartree-Fock self-consistent field method \cite{SzaboBook} to get the coefficients of the wave functions corresponding to physical solutions. We will work in a minimal basis, composed  of a $Y_{00}$ and a $Y_{10}$ spherical harmonic, or equivalently a s and a p\textsubscript{z} orbital, to illustrate the difference between the RHF and UHF solutions. In this basis there is a shortcut to find the stationary solutions. One can define the one-electron wave functions $\phi(\theta)$ using a mixing angle between the two basis functions. Hence we just need to minimize the energy with respect to the two mixing angles $\chi_\alpha$ and $\chi_\beta$.
+We obtained Eq.~\eqref{eq:EUHF} for the general form of the wave function \eqref{eq:UHF_WF}, but to be associated with a physical wave function the energy needs to be stationary with respect to the coefficients. The general method is to use the Hartree-Fock self-consistent field method \cite{SzaboBook} to get the coefficients of the wave functions corresponding to physical solutions. We will work in a minimal basis, composed  of a $Y_{0}$ and a $Y_{1}$ spherical harmonic, or equivalently a s and a p\textsubscript{z} orbital, to illustrate the difference between the RHF and UHF solutions. In this basis there is a shortcut to find the stationary solutions. One can define the one-electron wave functions $\phi(\theta)$ using a mixing angle between the two basis functions. Hence we just need to minimize the energy with respect to the two mixing angles $\chi_\alpha$ and $\chi_\beta$.
 
 \begin{equation}
-\phi_\alpha(\theta_1)= \cos(\chi_\alpha)\frac{Y_{00}(\theta_1)}{R} + \sin(\chi_\alpha)\frac{Y_{10}(\theta_1)}{R}
+\phi_\sigma(\theta_i)= \cos(\chi_\sigma)\frac{Y_{0}(\theta_i)}{R} + \sin(\chi_\sigma)\frac{Y_{1}(\theta_i)}{R}
 \end{equation}
  
 The minimization gives the three following solutions valid for all value of R: