Done with IIIC

Manuscript/EPAWTFT.tex

@@ -780,42 +780,6 @@ Hamiltonian.\cite{Tsuchimochi_2014,Tsuchimochi_2019}
 These methods yield more accurate spin-pure energies without 
 gradient discontinuities or spurious minima.
 
-%When one relies on MP perturbation theory (and more generally on any perturbative partitioning), it would be reasonable to ask for a systematic improvement of the energy with respect to the perturbative order, \ie, one would expect that the more terms of the perturbative series one can compute, the closer the result from the exact energy.
-%In other words, one would like a monotonic convergence of the MP series. Assuming this, the only limiting process to get the exact correlation energy (in a finite basis set) would be our ability to compute the terms of this perturbation series.
-%Unfortunately this is not as easy as one might think because i) the terms of the perturbative series become rapidly computationally cumbersome, and ii) erratic behaviour of the perturbative coefficients are not uncommon. 
-%For example, in the late 80's, Gill and Radom reported deceptive and slow convergences in stretched systems \cite{Gill_1986,Gill_1988} (see also Refs.~\onlinecite{Handy_1985,Lepetit_1988}). 
-%In Ref.~\onlinecite{Gill_1986}, the authors showed that the RMP series is convergent, yet oscillatory which is far from being convenient if one is only able to compute the first few terms of the expansion.
-%For example, in the case of the barrier to homolytic fission of \ce{He2^2+} in a minimal basis set, RMP5 is worse than RMP4 (see Fig.~2 in Ref.~\onlinecite{Gill_1986}).
-%On the other hand, the UMP series is monotonically convergent after UMP5 but very slowly . 
-%Thus, one cannot practically use it for systems where only the first terms are computationally accessible.
-
-%When a bond is stretched, in most cases the exact wave function becomes more and more of multi-reference nature. 
-%Yet, the HF wave function is restricted to be a single Slater determinant.
-%It is then inappropriate to model (even qualitatively) stretched systems. 
-%Nevertheless, as explained in Sec.~\ref{sec:HF} and illustrated in Sec.~\ref{sec:HF_hubbard}, the HF wave function can undergo symmetry breaking to lower its energy by sacrificing one of the symmetry of the exact wave function in the process (see also the pedagogical example of \ce{H2} in Ref.~\onlinecite{SzaboBook}). 
-%One could then potentially claim that the RMP series exhibits deceptive convergence properties as the RHF Slater determinant is a poor approximation of the exact wave function for stretched system. 
-%However, as illustrated above, even in the unrestricted formalism which clearly represents a better description of a stretched system, the UMP series does not have the smooth and rapidly convergent behaviour that one would wish for. 
-%The reasons behind this ambiguous behaviour are further explained below.
-
-%In the unrestricted framework the singlet ground state wave function is allowed to mix with triplet wave functions, leading to the so-called spin contamination issue. Gill \textit{et al.}~highlighted the link between slow convergence of the UMP series and spin contamination for \ce{H2} in a minimal basis. \cite{Gill_1988}
-%Handy and coworkers reported the same behaviour of the UMP series (oscillatory and slowly monotonically convergent) in stretched \ce{H2O} and \ce{NH2}. \cite{Handy_1985} Lepetit \textit{et al.}~analysed the difference between the MP and EN partitioning for the UHF reference. \cite{Lepetit_1988} 
-%They concluded that the slow convergence of the UMP series is due to i) the fact that the MP denominator (see Eq.~\ref{eq:EMP2}) tends to a constant instead of vanishing, and ii) the lack of the singly-excited configurations (which only appears at fourth order) that strongly couple to the doubly-excited configurations.
-%We believe that this divergent behaviour might therefore be attributed to the need for the single excitations to focus on correcting the structure of the reference orbitals rather than capturing the correlation energy. 
-
-%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.
-
-
 %==========================================%
 \subsection{Spin-Contamination in the Hubbard Dimer}
 \label{sec:spin_cont}
@@ -874,10 +838,6 @@ The Taylor expansion of the RMP energy can then be evaluated to obtain the $k$th
 \begin{equation}
 	E_\text{RMP}^{(k)} = U \delta_{0,k} - \frac{1}{2} \frac{U^k}{(4t)^{k-1}} \mqty( 1/2 \\ k/2).
 \end{equation}
-%with 
-%\begin{equation}
-%	E_{\text{MP}n}(\lambda) = \sum_{k=0}^n E_\text{MP}^{(k)} \lambda^k.
-%\end{equation}
  
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 % RADIUS OF CONVERGENCE PLOTS
@@ -901,9 +861,6 @@ For the divergent case, the $\lep$ lies inside this cylinder of convergence, whi
 outside this cylinder.
 In both cases, the EP connects the ground state with the doubly-excited state, and thus the convergence behaviour
 for the two states using the ground-state RHF orbitals is identical.
-% HGAB: This cannot be relevant here as the single-excitations don't couple to either ground or excited state.
-%The convergent and divergent series start to differ at fourth order, corresponding to the lowest-order contribution
-%%from the single excitations.\cite{Lepetit_1988}
 
 %%% FIG 3 %%%
 \begin{figure*}
@@ -984,22 +941,6 @@ It is well-known that the spin-projection needed to remove spin-contamination ca
 of highly-excited determinants,\cite{Lowdin_1955c} and thus it is not surprising that this process proceeds 
 very slowly as the perturbation order is increased.
 
-
-%The convergence of the UMP as a function of the ratio $U/t$ is shown in Fig.~\ref{subfig:UMP_cvg} for two specific values: the first ($U = 3t$) is well within the RMP convergence region, while the second ($U = 7t$) falls outside.
-%Note that in the case of UMP, there are now two pairs of EPs as the open-shell singlet now couples strongly with both the ground and doubly-excited states.
-%This has the clear tendency to move away from the origin the EP dictating the convergence of the ground-state energy, while deteriorating the convergence properties of the excited-state energy.
-%For $U = 3t$ (see Fig.~\ref{subfig:UMP_3}), the ground-state energy is remarkably flat since the UHF energy is already a pretty good estimate of the exact energy thanks to the symmetry-breaking process.
-%Most of the UMP expansion is actually correcting the spin-contamination in the wave function.
-%For $U = 7t$ (see Fig.~\ref{subfig:UMP_7}), we are well towards the strong correlation regime, where we see that the UMP series is slowly convergent while RMP diverges.
-%We see a single EP on the ground-state surface which falls just outside (maybe on?) the radius of convergence. 
-%An EP this close to the radius of convergence gives an increasingly slow convergence of the UMP series, but it will converge eventually as observed in Fig.~\ref{subfig:UMP_cvg}.
-%On the other hand, there is an EP on the excited energy surface that is well within the radius of convergence.
-%We can therefore say that the use of a symmetry-broken UHF wave function can retain a convergent ground-state perturbation series
-%at the expense of a divergent excited-state perturbation series. (Note: the orbitals are not optimised for excited-state here).
-%In contrast, the RMP expansion was always convergent for the open-shell excited state (which was a single CSF) while
-%the radius of convergence for the doubly-excited state was identical to the ground-state as this was the only exceptional point.
-
-
 %==========================================%
 \subsection{Classifying Types of Convergence} % Behaviour} % Further insights from a two-state model}
 %==========================================%
@@ -1011,17 +952,13 @@ Cremer and He introduced an efficient MP6 approach and used it to analyse the RM
 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. 
 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.
 
 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}