diff --git a/Manuscript/EPAWTFT.tex b/Manuscript/EPAWTFT.tex index b39db11..8723ec9 100644 --- a/Manuscript/EPAWTFT.tex +++ b/Manuscript/EPAWTFT.tex @@ -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.''} diff --git a/hugh_notes.txt b/hugh_notes.txt index 1c1de71..e42750f 100644 --- a/hugh_notes.txt +++ b/hugh_notes.txt @@ -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. + + + + + +