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little paragraph on Olsen 2019

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Pierre-Francois Loos 2020-11-12 14:33:30 +01:00
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\citation{Olsen_2000} \citation{Olsen_2000}
\citation{Goodson_2004} \citation{Goodson_2004}
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\citation{Cejnar_2009,Sachdev_2011,Cejnar_2015,Cejnar_2016,Caprio_2008,Macek_2019} \citation{Cejnar_2009,Sachdev_2011,Cejnar_2015,Cejnar_2016,Caprio_2008,Macek_2019}
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\citation{Stransky_2018} \citation{Stransky_2018}
\citation{Lipkin_1965} \citation{Lipkin_1965}
@ -160,14 +160,14 @@
\bibcite{Baker_1971}{{56}{1971}{{{Baker}}}{{}}} \bibcite{Baker_1971}{{56}{1971}{{{Baker}}}{{}}}
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\bibcite{Sindelka_2017}{{63}{2017}{{{\v S}indelka\ \emph {et~al.}}}{{{\v S}indelka, Santos,\ and\ Moiseyev}}} \bibcite{Sindelka_2017}{{63}{2017}{{{\v S}indelka\ \emph {et~al.}}}{{{\v S}indelka, Santos,\ and\ Moiseyev}}}
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\bibcite{Cejnar_2016}{{66}{2016}{{Cejnar\ and\ Str{\'a}nsk{\'y}}}{{}}} \bibcite{Cejnar_2016}{{66}{2016}{{Cejnar\ and\ Str{\'a}nsk{\'y}}}{{}}}
\bibcite{Caprio_2008}{{67}{2008}{{Caprio\ \emph {et~al.}}}{{Caprio, Cejnar,\ and\ Iachello}}} \bibcite{Caprio_2008}{{67}{2008}{{Caprio\ \emph {et~al.}}}{{Caprio, Cejnar,\ and\ Iachello}}}

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@ -367,7 +367,10 @@ A singularity in the unit circle is designated as an intruder state, more precis
\underbrace{\mqty(\alpha & \delta \\ \delta & \beta)}_{\bH} = \underbrace{\mqty(\alpha + \alpha_s & 0 \\ 0 & \beta + \beta_s )}_{\bH^{(0)}} + \underbrace{\mqty(- \alpha_s & \delta \\ \delta & - \beta_s)}_{\bV}, \underbrace{\mqty(\alpha & \delta \\ \delta & \beta)}_{\bH} = \underbrace{\mqty(\alpha + \alpha_s & 0 \\ 0 & \beta + \beta_s )}_{\bH^{(0)}} + \underbrace{\mqty(- \alpha_s & \delta \\ \delta & - \beta_s)}_{\bV},
\end{equation} \end{equation}
where the diagonal matrix is the unperturbed Hamiltonian matrix $\bH^{(0)}$ and the second matrix in the right-hand-side $\bV$ is the perturbation. where the diagonal matrix is the unperturbed Hamiltonian matrix $\bH^{(0)}$ and the second matrix in the right-hand-side $\bV$ is the perturbation.
See also Ref.~\onlinecite{Olsen_2019} where the present model is generalized to a non-symmetric (i.e, non-Hermitian) Hamiltonian.
In Ref.~\onlinecite{Olsen_2019}, the simple $2 \times 2$ model proposed by Olsen \textit{et al.} is generalized to a non-symmetric (i.e, non-Hermitian) Hamiltonian and various choice of perturbation (not only the MP partioning).
Interestingly, they showed that the convergence pattern of a given perturbation method can be characterized by its archetype which defines the overall ``shape'' of the energy convergence and can be subdivided in five classes for Hermitian Hamiltonians (zigzag, interspersed zigzag, triadic, ripples, and geometric), while two additional archetypes (zigzag-geometric and convex-geometric) are observed in non-Hermitian Hamiltonians.
Other features characterizing the convergence behavior of a perturbation method are its rate of convergence, its length of recurring period, and its sign pattern.
They first studied molecules with low-lying doubly-excited states of the same spatial and spin symmetry. They first studied molecules with low-lying doubly-excited states of the same spatial and spin symmetry.
The exact wave function has a non-negligible contribution from the doubly-excited states, so these low-lying excited states were good candidates for being intruder states. \titou{For \ce{CH_2} in a large basis set, the series is convergent up to the 50th order. They showed that the dominant singularity lies outside the unit circle but close to it causing the slow convergence.} The exact wave function has a non-negligible contribution from the doubly-excited states, so these low-lying excited states were good candidates for being intruder states. \titou{For \ce{CH_2} in a large basis set, the series is convergent up to the 50th order. They showed that the dominant singularity lies outside the unit circle but close to it causing the slow convergence.}
@ -375,7 +378,9 @@ The exact wave function has a non-negligible contribution from the doubly-excite
Then they demonstrated that the divergence for \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. Then they demonstrated that the divergence for \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 concluded that the divergence of the series was due to the interaction with a highly diffuse excited state. %They concluded that the divergence of the series was due to the interaction with a highly diffuse excited state.
Moreover they proved that the extrapolation formulas of Cremer and He \cite{Cremer_1996} cannot be used for all systems, and that these formulas were not mathematically motivated when looking 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 results in alternated terms up to 10th order. For higher orders, the series is monotonically convergent. This surprising behavior is due to the fact that two pairs of singularities are approximately at the same distance from the origin. Moreover they proved that the extrapolation formulas of Cremer and He \cite{Cremer_1996} cannot be used for all systems, and that these formulas were not mathematically motivated when looking 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 results in alternated terms up to 10th order.
For higher orders, the series is monotonically convergent. This surprising behavior is due to the fact that two pairs of singularities are approximately at the same distance from the origin.
%======================================= %=======================================
\subsection{The singularity structure} \subsection{The singularity structure}