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\citation{Gill_1988}
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@ -86,8 +87,8 @@
\citation{Goodson_2004}
\citation{Olsen_2000}
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@ -216,8 +216,6 @@ Unfortunately this is not true in generic cases and rapidly some strange behavio
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.
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.
\begin{table}[h!]
\centering
\begin{tabular}{c c c c c c c}
@ -234,9 +232,9 @@ In the unrestricted framework the ground state singlet wave function is allowed
\label{tab:my_label}
\end{table}
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}. Cremer and He performed the same analysis with 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.
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.
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}.
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}.
\subsection{Cases of divergence}
@ -295,15 +293,13 @@ Finally, it was shown that $\beta$ singularities are very sensitive to the basis
\subsection{The physics of quantum phase transition}
In the previous section, we seen that a reasoning on 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 singularity $\beta$. It is now well-known that this phenomenon is a specific case of a more general phenomenon. Indeed, theoretical physicist proved that EPs are connected to quantum phase transitions. In quantum mechanics, the Hamiltonian is almost always dependent of a parameter, in some cases the variation of a parameter can lead to abrupt changes at a critical point. Those quantum phase transitions exist both for ground and excited states. A ground-state quantum phase transition is characterized by the successive derivative of the ground-state energy with respect to a non-thermal control parameter. 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 n-th order if the n-th derivative is discontinuous. A quantum phase transition can also be identify by the discontinuity of an appropriate order parameter (or one of its derivative).
In the previous section, we saw that a reasoning on 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 singularity $\beta$. It is now well-known that this phenomenon is a specific case of a more general phenomenon. Indeed, theoretical physicists proved that EPs are connected to quantum phase transitions (citation du stransky 2018 sauf lee). In quantum mechanics, the Hamiltonian is almost always dependent of a parameter, in some cases the variation of a parameter can lead to abrupt changes at a critical point. Those quantum phase transitions exist both for ground and excited states (stransky 2017, cejnar 2006 caprio 2008). A ground-state quantum phase transition is characterized by the successive derivative of the ground-state energy with respect to a non-thermal control parameter (cejnar 2009). 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 n-th order if the n-th derivative is discontinuous. A quantum phase transition can also be identify by the discontinuity of an appropriate order parameter (or one of its derivative).
The presence of an EP close to the real axis is characteristic of a sharp avoided crossings. Yet at such an avoided crossings eigenstates change abruptly. Although it is now well understood that EPs are closely related to quantum phase transitions, the link between the type of QPT (ground state or excited state, first or superior order) and EPs still need to be clarify. One of the major challenge in order to do this reside in our 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. Cejnar et al. developped a method based on a Coulomb analogy giving access to the density of EP close to the real axis. More recently Stransky and co-workers proved
The presence of an EP close to the real axis is characteristic of a sharp avoided crossings. Yet at such an avoided crossings eigenstates change abruptly. Although it is now well understood that EPs are closely related to quantum phase transitions, the link between the type of QPT (ground state or excited state, first or superior order) and EPs still need to be clarify. One of the major challenge in order to do this reside in our 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. Cejnar et al. developped a method based on a Coulomb analogy giving access to the density of EP close to the real axis (citation cejnar 2005 2007). More recently Stransky and co-workers proved that the distribution of EPs is not the same around a QPT of first or second order (cite stransky 2018). Moreover, that when the dimension of the system increases they tends towards the real axis in a different manner, meaning respectively exponentially and algebraically.
Ajouter la biblio
Singularity $\beta$ and quantum phase transition ?
Coulson-Fisher point second QPT
Singularity $\beta$ + général
It seems like our understanding of the physics of spatial and/or spin symmetry breaking in the Hartree-Fock theory can be enlightened by quantum phase transition theory. Indeed, the second derivative of the energy is discontinuous at the Coulson-Fischer point which mean that the system undergo a second order quantum phase transition. The $\beta$ singularities introduced by Sergeev to describe the EPs modeling the formation of a bound cluster of electrons are actually a more general class of singularities.
Singularity $\alpha$ and quantum phase transition ?
%============================================================%
\section{The spherium model}

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\contentsline {subsection}{\numberline {3.2}Cases of divergence}{6}{subsection.3.2}%
\contentsline {subsection}{\numberline {3.3}The singularity structure}{7}{subsection.3.3}%
\contentsline {subsection}{\numberline {3.4}The physics of quantum phase transition}{8}{subsection.3.4}%
\contentsline {section}{\numberline {4}The spherium model}{8}{section.4}%
\contentsline {section}{\numberline {4}The spherium model}{9}{section.4}%
\contentsline {subsection}{\numberline {4.1}Weak correlation regime}{9}{subsection.4.1}%