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Pierre-Francois Loos 2020-07-29 22:53:39 +02:00
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@ -821,16 +821,16 @@ For a non-Hermitian Hamiltonian the exceptional points can lie on the real axis.
In order to model accurately chemical systems, one must choose, in a ever growing zoo of methods, which computational protocol is adapted to the system of interest.
This choice can be, moreover, motivated by the type of properties that one is interested in.
That means that one must understand the strengths and weaknesses of each method, i.e., why one method might fail in some cases and work beautifully in others.
We have seen that for methods relying on perturbation theory the successes and failures are directly connected to the position of EPs in the complex plane.
We have seen that for methods relying on perturbation theory, their successes and failures are directly connected to the position of EPs in the complex plane.
Exhaustive studies have been performed on the causes of failure of MP perturbation theory.
First, it has been understood that, for chemical systems for which the HF method is a poor approximation to the exact wave function, MP perturbation theory will fail too. Such systems can be, for example, systems where the exact wave function is dominated by more than one configuration, i.e., multi-reference systems.
More preoccupying cases were reported.
First, it was understood that, for chemical systems for which the HF Slater determinant is a poor approximation to the exact wave function, MP perturbation theory fails too. Such systems can be, for example, molecules where the exact ground-state wave function is dominated by more than one configuration, i.e., multi-reference systems.
More preoccupying cases were also reported.
For instance, it has been shown that systems considered as well-understood (e.g., \ce{Ne}) can exhibit divergent behavior when the basis set is augmented with diffuse functions.
Later, these behaviors of the perturbation series have been investigated and rationalized in terms of avoided crossings and singularities in the complex plane. It has been shown that the singularities can be classified in two families.
Later, these erratic behaviors of the perturbation series were investigated and rationalized in terms of avoided crossings and singularities in the complex plane. It was shown that the singularities can be classified in two families.
The first family includes $\alpha$ singularities resulting from a large avoided crossing between the ground state and a low-lying doubly-excited states.
The $\beta$ singularities, which constitutes the second family, are artifacts generated by the incompleteness of the Hilbert space, and they are directly connected to an ionization phenomenon occurring in the complete Hilbert space.
These singularities are close to the real axis and connected with sharp avoided crossing between the ground state and a highly diffuse state.
We have seen that the $\beta$ singularities modeling the ionization phenomenon described by Sergeev and Goodson are actually part of a more general class of singularities. Indeed, those singularities close to the real axis are connected to quantum phase transition and symmetry breaking, and theoretical physics have demonstrated that the behavior of the EPs depends of the type of transitions from which the EPs result (first or higher orders, ground state or excited state transitions).
We have found that the $\beta$ singularities modeling the ionization phenomenon described by Sergeev and Goodson are actually part of a more general class of singularities. Indeed, those singularities close to the real axis are connected to quantum phase transition and symmetry breaking, and theoretical physics have demonstrated that the behavior of the EPs depends of the type of transitions from which the EPs result (first or higher orders, ground state or excited state transitions).
In this work, we have shown that $\beta$ singularities are involved in the spin symmetry breaking of the UHF wave function.
This confirms that $\beta$ singularities can occur for other types of transition and symmetry breaking than just the formation of a bound cluster of electrons.