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Pierre-Francois Loos 2020-12-01 17:09:05 +01:00
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@ -972,15 +972,11 @@ exact correlation energy $\Delta E$ using terms up to MP6\cite{Cremer_1996}
&= \Emp^{(2)} + \Emp^{(3)} + \qty(\Emp^{(4)} + \Emp^{(5)}) \exp(\Emp^{(6)} / \Emp^{(5)}). &= \Emp^{(2)} + \Emp^{(3)} + \qty(\Emp^{(4)} + \Emp^{(5)}) \exp(\Emp^{(6)} / \Emp^{(5)}).
\end{align} \end{align}
\end{subequations} \end{subequations}
%As one can only compute the first terms of the MP series, a smart way of getting more accurate results is to use extrapolation formula, \ie, estimating the limit of the series with only few terms.
%Cremer and He 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 resorting to classes. \cite{Cremer_1996}
These class-specific formulas reduced the mean absolute error from the FCI correlation energy by a These class-specific formulas reduced the mean absolute error from the FCI correlation energy by a
factor of four compared to previous class-independent extrapolations, factor of four compared to previous class-independent extrapolations,
highlighting how one can leverage a deeper understanding of MP convergence to improve estimates of highlighting how one can leverage a deeper understanding of MP convergence to improve estimates of
the correlation energy at lower computational costs. the correlation energy at lower computational costs.
In Section~\ref{sec:Resummation}, we consider more advanced extrapolation routines that take account of EPs in the complex $\lambda$-plane. In Sec.~\ref{sec:Resummation}, we consider more advanced extrapolation routines that take account of EPs in the complex $\lambda$-plane.
%The mean absolute deviation taking the FCI correlation energies as reference is $0.3$ millihartree with the class-specific formula whereas the deviation increases to 12 millihartree using the general formula.
%Even if there were still shaded areas in their analysis and that their classification was incomplete, the work of Ref.~\onlinecite{Cremer_1996} clearly evidenced that understanding the origin of the different modes of convergence could potentially lead to a more rationalised use of MP perturbation theory and, hence, to more accurate correlation energy estimates.
In the late 90's, Olsen \etal\ discovered an even more concerning behaviour of the MP series. \cite{Olsen_1996} In the late 90's, Olsen \etal\ discovered an even more concerning behaviour of the MP series. \cite{Olsen_1996}
They showed that the series could be divergent even in systems that were considered to be well understood, They showed that the series could be divergent even in systems that were considered to be well understood,
@ -1007,7 +1003,7 @@ Using their observations in Ref.~\onlinecite{Olsen_1996}, Olsen and collaborator
a simple method that performs a scan of the real axis to detect the avoided crossing responsible a simple method that performs a scan of the real axis to detect the avoided crossing responsible
for the dominant singularities in the complex plane. \cite{Olsen_2000} for the dominant singularities in the complex plane. \cite{Olsen_2000}
By modelling this avoided crossing using a two-state Hamiltonian, one can obtain an approximation for By modelling this avoided crossing using a two-state Hamiltonian, one can obtain an approximation for
the dominant singularities as the EPs of the $2\times2$ matrix the dominant singularities as the EPs of the two-state matrix
\begin{equation} \begin{equation}
\label{eq:Olsen_2x2} \label{eq:Olsen_2x2}
\underbrace{\mqty(\alpha & \delta \\ \delta & \beta )}_{\bH} \underbrace{\mqty(\alpha & \delta \\ \delta & \beta )}_{\bH}
@ -1027,10 +1023,8 @@ These intruder-state effects are analogous to the EP that dictates the convergen
the RMP series for the Hubbard dimer (Fig.~\ref{fig:RMP}). the RMP series for the Hubbard dimer (Fig.~\ref{fig:RMP}).
Furthermore, the authors demonstrated that the divergence for \ce{Ne} is due to a back-door intruder state Furthermore, the authors demonstrated that the divergence for \ce{Ne} is due to a back-door intruder state
that arise when the ground state undergoes sharp avoided crossings with highly diffuse excited states. that arise when the ground state undergoes sharp avoided crossings with highly diffuse excited states.
%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.
This divergence is related to a more fundamental critical point in the MP energy surface that we will This divergence is related to a more fundamental critical point in the MP energy surface that we will
discuss in Section~\ref{sec:MP_critical_point}. discuss in Sec.~\ref{sec:MP_critical_point}.
Finally, Ref.~\onlinecite{Olsen_1996} proved that the extrapolation formulas of Cremer and He \cite{Cremer_1996} Finally, Ref.~\onlinecite{Olsen_1996} proved that the extrapolation formulas of Cremer and He \cite{Cremer_1996}
are not mathematically motivated when considering the complex singularities causing the divergence, and therefore are not mathematically motivated when considering the complex singularities causing the divergence, and therefore
@ -1053,7 +1047,6 @@ according to a so-called ``archetype'' that defines the overall ``shape'' of the
For Hermitian Hamiltonians, these archetypes can be subdivided into five classes For Hermitian Hamiltonians, these archetypes can be subdivided into five classes
(zigzag, interspersed zigzag, triadic, ripples, and geometric), (zigzag, interspersed zigzag, triadic, ripples, and geometric),
while two additional archetypes (zigzag-geometric and convex-geometric) are observed in non-Hermitian Hamiltonians. while two additional archetypes (zigzag-geometric and convex-geometric) are observed in non-Hermitian Hamiltonians.
%Other features characterising the convergence behaviour of a perturbation method are its rate of convergence, its length of recurring period, and its sign pattern.
The geometric archetype appears to be the most common for MP expansions,\cite{Olsen_2019} but the The geometric archetype appears to be the most common for MP expansions,\cite{Olsen_2019} but the
ripples archetype corresponds to some of the early examples of MP convergence. \cite{Handy_1985,Lepetit_1988,Leininger_2000} ripples archetype corresponds to some of the early examples of MP convergence. \cite{Handy_1985,Lepetit_1988,Leininger_2000}