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Stuff on quadratic approximant

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Manuscript/EPAWTFT.tex
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@ -109,6 +109,9 @@
\newcommand{\lc}{\lambda_{\text{c}}}
\newcommand{\lep}{\lambda_{\text{EP}}}
% Some energies
\newcommand{\Emp}{E_{\text{MP}}}
% Blackboard bold
\newcommand{\bbR}{\mathbb{R}}
\newcommand{\bbC}{\mathbb{C}}
@ -820,7 +823,7 @@ gradient discontinuities or spurious minima.
%==========================================%
\subsection{Effect of Spin-Contamination in the Hubbard Dimer}
\subsection{Spin-Contamination in the Hubbard Dimer}
%==========================================%
%%% FIG 2 %%%
@ -847,7 +850,7 @@ gradient discontinuities or spurious minima.
The behaviour of the RMP and UMP series observed in \ce{H2} can also be illustrated by considering
the analytic Hubbard dimer with a complex-valued perturbation strength.
In this system, the stretching of a chemical bond is directly mirrored by an increase in the electron correlation $U/t$.
In this system, the stretching of the \ce{H\bond{-}H} bond is directly mirrored by an increase in the electron correlation $U/t$.
Using the ground-state RHF reference orbitals leads to the parametrised RMP Hamiltonian
\begin{widetext}
\begin{equation}
@ -943,10 +946,10 @@ Using the ground-state UHF reference orbitals in the Hubbard dimer yields the pa
\end{pmatrix}.
\end{equation}
\end{widetext}
While there is a closed-form expression for the ground-state energy, it is cumbersome and we eschew reporting it.
While a closed-form expression for the ground-state energy exists, it is cumbersome and we eschew reporting it.
Instead, the radius of convergence of the UMP series can be obtained numerically as a function of $U/t$, as shown
in Fig.~\ref{fig:RadConv}.
These numerical values reveal that the UMP ground-state series has $\rc > 1$ for all $U/t$ and must always converge.
These numerical values reveal that the UMP ground-state series has $\rc > 1$ for all $U/t$ and always converges.
However, in the strong correlation limit (large $U$), this radius of convergence tends to unity, indicating that
the corresponding UMP series becomes increasingly slow.
Furthermore, the doubly-excited state using the ground-state UHF orbitals has $\rc < 1$ for almost any value
@ -960,7 +963,7 @@ These Riemann surfaces are illustrated for $U = 3t$ and $7t$ alongside the pertu
in Fig.~\ref{subfig:UMP_cvg}.
At $U = 3t$, the RMP series is convergent, while RMP becomes divergent for $U=7t$.
The ground-state UMP expansion is convergent in both cases, although the rate of convergence is significantly slower
for larger $U/t$ as the radius of convergence becomes increasingly close to 1 (Fig.~\ref{fig:RadConv}).
for larger $U/t$ as the radius of convergence becomes increasingly close to one (Fig.~\ref{fig:RadConv}).
% EFFECT OF SYMMETRY BREAKING
As the UHF orbitals break the molecular symmetry, new coupling terms emerge between the electronic states that
@ -978,8 +981,8 @@ sacrificed convergence of the excited-state series so that the ground-state conv
Since the UHF ground state already provides a good approximation to the exact energy, the ground-state sheet of
the UMP energy is relatively flat and the corresponding EP in the Hubbard dimer always lies outside the unit cylinder.
\titou{The slow convergence observed in \ce{H2}\cite{Gill_1988} can then be seen as this EP
moves closer to one at larger $U/t$ values.}
\hugh{The slow convergence observed in stretched \ce{H2}\cite{Gill_1988} can then be seen as this EP
moves increasingly close to the unit cylinder at large $U/t$ and $\rc$ approaches one (from above).}
Furthermore, the majority of the UMP expansion in this regime is concerned with removing spin-contamination from the wave
function rather than improving the energy.
It is well-known that the spin-projection needed to remove spin-contamination can require non-linear combinations
@ -1007,75 +1010,132 @@ very slowly as the perturbation order is increased.
%==========================================%
% CREMER AND HE
Cremer and He analysed 29 atomic and molecular systems at the FCI level \cite{Cremer_1996} and grouped them in two classes: i) the \textit{class A} systems where one observes a monotonic convergence \titou{of the RMP series?} to the FCI energy, and ii) the \textit{class B} systems 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}
\textit{``Class A systems are characterised 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.''}
Moreover, they analysed the contribution of the triple (T) excitations to the MP4, MP5 and MP6 energies next to the single, double and quadruple (SDQ) excitations contribution.
They showed that class A systems have very little contribution from the triple excitations and that most of the correlation energy is due to pair correlation.
On the other hand, class B systems have an important contribution from the triple excitations which alternates in sign resulting in an oscillation of the total correlation energy.
This observation on the contribution to the MP$n$ energy corroborates the electronic structure discussed above.
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}
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.
\hugh{As computational implementations of higher-order MP terms improved, the systematic investigation
of convergence behaviour in a broader class of molecules became possible.}
Cremer and He \hugh{introduced an efficient MP6 approach and used it to analyse the RMP convergence of}
29 atomic and molecular systems with respect to the FCI energy.\cite{Cremer_1996}
They established two general classes: ``class A'' systems that exhibit monotonic convergence;
and ``class B'' systems 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.
\hugh{By analysing the different cluster contributions to the MP energy terms, they proposed that
class A systems generally include well-separated and weakly correlated electron pairs, while class B systems
are characterised by dense electron clustering in one or more spatial regions.}\cite{Cremer_1996}
%\textit{``Class A systems are characterised 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.''}
%Moreover, they analysed the contribution of the triple (T) excitations to the MP4, MP5 and MP6 energies next to the single, double and quadruple (SDQ) excitations contribution.
In class A systems, they showed that the majority of the correlation energy arises from pair correlation,
with little contribution from triple excitations.
On the other hand, triple excitations have an important contribution in class B systems, including providing
orbital relaxation, and these contributions lead to oscillations of the total correlation energy.
%This observation on the contribution to the MP$n$ energy corroborates the electronic structure discussed above.
\hugh{%
Using these classifications, Cremer and He then introduced simple extrapolation formulas for estimating the
exact correlation energy $\Delta E$ using terms up to MP6\cite{Cremer_1996}
\begin{align}
\Delta E_{\text{A}}
&= \Emp^{(2)} + \Emp^{(3)} + \Emp^{(4)}
+ \frac{\Emp^{(5)}}{1 - (\Emp^{(6)} / \Emp^{(5)})},
\\[5pt]
\Delta E_{\text{B}}
&= \Emp^{(2)} + \Emp^{(3)} + \qty(\Emp^{(4)} + \Emp^{(5)}) \exp(\Emp^{(6)} / \Emp^{(5)}).
\end{align}
%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
factor of four compared to previous class-independent extrapolations,
highlighting how one can leverage a deeper understanding of MP convergence to improve estimates of
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.
}
%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 \textit{et al.}~discovered an even more preoccupying behaviour of the MP series. \cite{Olsen_1996}
They showed that the series could be divergent even in systems that they considered as well understood like \ce{Ne} and the \ce{HF} molecule. \cite{Olsen_1996, Christiansen_1996}
Cremer and He had already studied these two systems and classified them as \textit{class B} systems.
However, the analysis of Olsen and coworkers was performed in larger basis sets containing diffuse functions.
In these basis sets, they found that the series become divergent at (very) high order.
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,
such as \ce{Ne} or the \ce{HF} molecule. \cite{Olsen_1996, Christiansen_1996}
Cremer and He had already studied these two systems and classified them as \textit{class B} systems.\cite{Cremer_1996}
However, Olsen and co-workers performed their analysis in larger basis sets containing diffuse functions,
finding that the correspoding MP series becomes divergent at (very) high order.
The discovery of this divergent behaviour is particularly worrying as large basis sets \trashHB{(as close to
the complete basis set as possible)} are required to get meaningful and accurate energies.\cite{Loos_2019d,Giner_2019}
Furthermore, diffuse functions are particularly important for anions and/or Rydberg excited states, where the wave function
is inherently more diffuse than the ground state.\cite{Loos_2018a,Loos_2020a}
The discovery of this divergent behaviour is worrying as in order to get meaningful and accurate energies, calculations must be performed in large basis sets (as close as possible from the complete basis set limit). \cite{Loos_2019d,Giner_2019}
Including diffuse functions is particularly important in the case of anions and/or Rydberg excited states where the wave function is much more diffuse than the ground-state one. \cite{Loos_2018a,Loos_2020a}
As a consequence, they investigated further the causes of these divergences as well as the reasons of the different types of convergence.
To do so, they analysed the relation between the dominant singularity (\ie, the closest singularity to the origin) and the convergence behaviour of the series. \cite{Olsen_2000} Their analysis is based on Darboux's theorem: \cite{Goodson_2011}
Olsen \etal\ investigated the causes of these divergences and the different types of convergence by
analysing the relation between the dominant singularity (\ie, the closest singularity to the origin)
and the convergence behaviour of the series.\cite{Olsen_2000}
Their analysis is based on Darboux's theorem: \cite{Goodson_2011}
\begin{quote}
\textit{``In the limit of large order, the series coefficients become equivalent to the Taylor series coefficients of the singularity closest to the origin. ''}
\end{quote}
Following the result of this theorem, the convergence patterns of the MP series can be explained by looking at the dominant singularity.
A singularity in the unit circle is designated as an intruder state, more precisely as a front-door (respectively back-door) intruder state if the real part of the singularity is positive (respectively negative).
\trashHB{Following the result of this theorem, the convergence patterns of the MP series can be explained by looking at the dominant singularity.}
\hugh{Following this theory,} a singularity in the unit circle is designated as an intruder state,
with a front-door (or back-door) intruder state if the real part of the singularity is positive (or negative).
Following their observations from Ref.~\onlinecite{Olsen_1996}, Olsen and collaborators later proposed a simple method which consists in performing a scan of the real axis to detect the avoided crossing responsible for the pair of dominant singularities. \cite{Olsen_2000}
Then, by modelling this avoided crossing via a two-state Hamiltonian one can get an approximation of the dominant conjugate pair of singularities by finding the EPs of the following $2\times2$ matrix
Using their observations in Ref.~\onlinecite{Olsen_1996}, Olsen and collaborators proposed
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}
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
\begin{equation}
\label{eq:Olsen_2x2}
\underbrace{\mqty(\alpha & \delta \\ \delta & \beta)}_{\bH} = \underbrace{\mqty(\alpha & 0 \\ 0 & \beta + \gamma )}_{\bH^{(0)}} + \underbrace{\mqty( 0 & \delta \\ \delta & - \gamma)}_{\bV},
\underbrace{\mqty(\alpha & \delta \\ \delta & \beta )}_{\bH}
= \underbrace{\mqty(\alpha + \alpha_{\text{s}} & 0 \\ 0 & \beta + \beta_{\text{s}} )}_{\bH^{(0)}}
+ \underbrace{\mqty( -\alpha_{\text{s}} & \delta \\ \delta & - \beta_{\text{s}})}_{\bV},
\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 $\bH^{(0)}$ with level shifts
$\alpha_{\text{s}}$ and $\beta_{\text{s}}$, \hugh{and $\bV$ represents the perturbation.}
They first studied an example of molecules with low-lying doubly-excited states of the same spatial and spin symmetry as the ground state. \cite{Olsen_2000}
In such a case, the exact wave function has a non-negligible contribution from the doubly-excited states, so these low-lying excited states are good candidates for being intruder states.
For \ce{CH_2} in a diffuse yet rather small basis set, the series is convergent at least up to the 50th order.
They showed that the dominant singularity lies outside the unit circle but close to it causing the slow convergence.
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.
The authors first considered molecules with low-lying doubly-excited states with the same spatial
and spin symmetry as the ground state. \cite{Olsen_2000}
In these systems, the exact wave function has a non-negligible contribution from the doubly-excited states,
and thus the low-lying excited states are likely to become intruder states.
For \ce{CH_2} in a diffuse, yet rather small basis set, the series is convergent at least up to the 50th order, and
the dominant singularity lies close (but outside) the unit circle, causing slow convergence of the series.
\hugh{These intruder-state effects are analogous to the EP that dictates the convergence behaviour of
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
that arise when the ground state undergos 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.
\hugh{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}.}
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 behaviour is due to the fact that two pairs of singularities are approximately at the same distance from the origin.
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
cannot be applied for all systems.
For example, \ce{HF} contains both back-door intruder states and low-lying doubly-excited states that
result in alternating terms up to 10th order.
The series becomes monotonically convergent at higher orders since
the two pairs of singularities are approximately the same distance from the origin.
In Ref.~\onlinecite{Olsen_2019}, the simple two-state model proposed by Olsen \textit{et al.} [see Eq.~\eqref{eq:Olsen_2x2}] is generalised to a non-symmetric Hamiltonian
More recently, this two-state model has been extended to non-symmetric Hamiltonians as\cite{Olsen_2019}
\begin{equation}
\underbrace{\mqty(\alpha & \delta_1 \\ \delta_2 & \beta)}_{\bH} = \underbrace{\mqty(\alpha & 0 \\ 0 & \beta + \gamma )}_{\bH^{(0)}} + \underbrace{\mqty( 0 & \delta_2 \\ \delta_1 & - \gamma)}_{\bV}.
\end{equation}
allowing an analysis of various choice of perturbation (not only the MP partitioning) such as coupled cluster perturbation expansions \cite{Pawlowski_2019a,Pawlowski_2019b,Pawlowski_2019c,Pawlowski_2019d,Pawlowski_2019e} and other non-Hermitian perturbation methods.
It is worth noting that only cases where $\text{sgn}(\delta_1) = - \text{sgn}(\delta_2)$ leads to new forms of perturbation expansions.
Interestingly, they showed that the convergence pattern of a given perturbation method can be characterised by its archetype which defines the overall ``shape'' of the energy convergence.
These so-called archetypes 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 characterising the convergence behaviour of a perturbation method are its rate of convergence, its length of recurring period, and its sign pattern.
Importantly, they observed that the geometric archetype is the most common for MP expansions but that the ripples archetype sometimes occurs. \cite{Handy_1985,Lepetit_1988,Leininger_2000}
The three remaining archetypes seem to be rarely observed in MP perturbation theory.
However, in the non-Hermitian setting of coupled cluster perturbation theory, \cite{Pawlowski_2019a,Pawlowski_2019b,Pawlowski_2019c,Pawlowski_2019d,Pawlowski_2019e} on can encounter interspersed zigzag, triadic, ripple, geometric, and zigzag-geometric archetypes.
One of main take-home messages of Olsen's study is that the primary critical point defines the high-order convergence, irrespective of whether this point is inside or outside the complex unit circle. \cite{Handy_1985,Olsen_2000}
This extension allows various choices of perturbation to be analysed, including coupled cluster
perturbation expansions \cite{Pawlowski_2019a,Pawlowski_2019b,Pawlowski_2019c,Pawlowski_2019d,Pawlowski_2019e}
and other non-Hermitian perturbation methods.
Note that new forms of perturbation expansions only occur when the sign of $\delta_1$ and $\delta_2$ differ.
Using these non-Hermitian two-state model, the convergence of a perturbation series can be characterised
according to a so-called ``archetype'' that defines the overall ``shape'' of the energy convergence.\cite{Olsen_2019}
For Hermitian Hamiltonians, these archetypes can be subdivided into five classes
(zigzag, interspersed zigzag, triadic, ripples, and geometric),
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
ripples archetype corresponds to some of the early examples of MP convergence. \cite{Handy_1985,Lepetit_1988,Leininger_2000}
The three remaining Hermitian archetypes seem to be rarely observed in MP perturbation theory.
In contrast, the non-Hermitian coupled cluster perturbation theory,%
\cite{Pawlowski_2019a,Pawlowski_2019b,Pawlowski_2019c,Pawlowski_2019d,Pawlowski_2019e} exhibits a range of archetypes
including the interspersed zigzag, triadic, ripple, geometric, and zigzag-geometric forms.
\hugh{This analysis highlights the importance of the primary critical point in controlling the high-order convergergence,
regardless of whether this point is inside or outside the complex unit circle. \cite{Handy_1985,Olsen_2000}}
%=======================================
\subsection{The singularity structure}
\label{sec:MP_critical_point}
%=======================================
In the 2000's, Sergeev and Goodson \cite{Sergeev_2005, Sergeev_2006} analyzed this problem from a more mathematical point of view by looking at the whole singularity structure where Olsen and collaborators were trying to find the dominant singularity causing the divergence. \cite{Olsen_1996,Olsen_2000,Olsen_2019}
They regrouped singularities in two classes: i) $\alpha$ singularities which have ``large'' imaginary parts, and ii) $\beta$ singularities which have very small imaginary parts.
@ -1162,6 +1222,7 @@ We believe that $\alpha$ singularities are connected to states with non-negligib
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Resummation Methods}
\label{sec:Resummation}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
As frequently claimed by Carl Bender, \textit{``the most stupid thing that one can do with a series is to sum it.''}

38
hugh_notes.txt
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@ -194,3 +194,41 @@ The resulting SUPT2 provided more accurate binding curves than EMP2, which the a
the SUPT2 approach correctly handles the redundancy of internal rotations in the effective active space of the
reference spin projection.
+==========================================================+
| Classifying Convergence Behaviour |
+==========================================================+
Cremer and He, JPC (1996):
--------------------------
Consider the MP6 energy as this is the next even order after MP2 and MP4 so introduces new excitations
(in this case pentuples and hextuples).
They decompose their MPn correlation into pair-pair, pair pair pair, etc terms to try and understand the
convergence behaviour:
SDQ = SS + SD + DD + DQ + QQ (singles, doubles, quadruples)
T = ST + DT + TT + TQ (terms including triple excitations)
They intend to show:
Class A) Monotonic convergence expected for systems in which the electron pairs are well-separated and weakly couple.
including eg BH, NH2, CH3, CH2 etc
Generally include well-separated electron pairs such that three-electron correlation effects are weak.
Class B) Initial oscillatory convergence with strong pair and three-electron correlation effects.
eg. Ne, F, F^-, FH
In these systems, there are closely spaced electron pairs that cluster in a small region of space.
One might imagine that this requires greater orbital relaxation, perhaps ``breating'' relaxation,
to allow the electron pairs to become separated? Or maybe that it generally introduces stronger
dynamic correlation effects? Orbital relaxation plus pair correlation comes through T1 T2 terms.
They observe both E(SDQ) and E(T) terms negative in Class A systems, but E_MP5(SDTQ) terms generally positive
in Class B. Oscillations in the T correlation terms drive the oscillatory convergence behaviour. This behaviour
does not appear to be caused by multiconfigurational effects, but may be amplified by them.
Class B has more improtant orbital relaxation effects and three-electron correlation than Class A.