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Hugh Burton 2020-11-25 11:54:57 +00:00
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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
@ -1007,18 +1010,45 @@ 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}
@ -1162,6 +1192,7 @@ 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.