reviewing hugh stuff, good as usual

Manuscript/EPAWTFT.tex

@@ -27,7 +27,7 @@
 %\newcommand{\latin}[1]{\textit{#1}}
 \newcommand{\ie}{\latin{i.e.}}
 \newcommand{\eg}{\latin{e.g.}}
-\newcommand{\etal}{\textit{et.\ al}}
+\newcommand{\etal}{\textit{et al.}}
 
 \newcommand{\mc}{\multicolumn}
 \newcommand{\fnm}{\footnotemark}
@@ -711,7 +711,6 @@ to the convergence properties of the MP expansion.
 %=====================================================%
 
 % GENERAL DESIRE FOR WELL-BEHAVED CONVERGENCE AND LOW-ORDER TERMS
-\hugh{%
  Among the most desirable properties of any electronic structure technique is the existence of 
 a systematic route to increasingly accurate energies. 
 In the context of MP theory, one would like a monotonic convergence of the perturbation
@@ -721,10 +720,8 @@ terms in the series becomes the only barrier to computing the exact correlation
 Unfortunately, the computational scaling of each term in the MP series increases with the perturbation
 order, and practical calculations must rely on fast convergence
 to obtain high-accuracy results using only the lowest order terms.
-}
 
 % INITIAL POSITIVITY AROUND THE CONVERGENCE PROPERTIES AND EARLY WORK SCOPE
-\hugh{%
 MP theory was first introduced to quantum chemistry through the pioneering
 works of Bartlett \etal\ in the context of many-body perturbation theory,\cite{Bartlett_1975}
 and Pople and co-workers in the context of determinantal expansions.\cite{Pople_1976,Pople_1978}
@@ -734,12 +731,10 @@ However, it was quickly realised that the MP series often demonstrated very slow
 or erratic convergence, with the UMP series showing particularly slow convergence.\cite{Laidig_1985,Knowles_1985,Handy_1985}
 For example, RMP5 is worse than RMP4 for predicting the homolytic barrier fission of \ce{He2^2+} using a minimal basis set, 
 while the UMP series monotonically converges but becomes increasingly slow beyond UMP5.\cite{Gill_1986}
-The first examples of divergent MP series were observed in the heavy-atom \ce{N2} and \ce{F2} 
+The first examples of divergent MP series were observed in the \ce{N2} and \ce{F2} 
 diatomics, where low-order RMP and UMP expansions give qualitatively wrong binding curves.\cite{Laidig_1987} 
-}
 
 % SLOW UMP CONVERGENCE AND SPIN CONTAMINATION
-\hugh{%
 The divergence of RMP expansions for stretched bonds can be easily understood from two perspectives.\cite{Gill_1988a}
 Firstly, the exact wave function becomes increasingly multi-configurational as the bond is stretched, and the 
 HF wave function no longer provides a qualitatively correct reference for the perturbation expansion.
@@ -749,16 +744,14 @@ expansion Eq.~\eqref{eq:EMP2}.
 In contrast, the origin of slow UMP convergence is less obvious as the reference UHF energy remains
 qualitatively correct at large bond lengths and the orbital degeneracy is avoided.
 Furthermore, this slow convergence can also be observed in molecules with a UHF ground state at the equilibrium
-geometry (\eg, \ce{CN^{-}}), suggesting a more fundamental link with spin-contamination 
+geometry (\eg, \ce{CN-}), suggesting a more fundamental link with spin-contamination 
 in the reference wave function.\cite{Nobes_1987}
-}
 
-\hugh{%
-Using the UHF framework allows} the singlet ground state wave function to mix with triplet wave functions, 
-leading to \hugh{spin contamination where the wave function is no longer an eigenfunction of the $\Hat{\cS}^2$ operator.}
-\hugh{The link between slow UMP convergence and this spin-contamination was first systematically investigated} 
+Using the UHF framework allows the singlet ground state wave function to mix with triplet wave functions, 
+leading to spin contamination where the wave function is no longer an eigenfunction of the $\Hat{\cS}^2$ operator.
+The link between slow UMP convergence and this spin-contamination was first systematically investigated
 by Gill \etal\ using the minimal basis \ce{H2} model.\cite{Gill_1988}
-\hugh{In this work, the authors compared the UMP series with the exact RHF- and UHF-based FCI expansions
+In this work, the authors compared the UMP series with the exact RHF- and UHF-based FCI expansions
 and identified that the slow UMP convergence arises from its failure to correctly predict the amplitude of the
 low-lying double excitation.
 This erroneous description of the double excitation amplitude has the same origin as the spin-contamination in the reference
@@ -773,14 +766,12 @@ appear at fourth-order.\cite{Lepetit_1988}
 Drawing these ideas together, we believe that slow UMP convergence occurs because the single excitations must focus on removing
 spin-contamination from the reference wave function, limiting their ability to fine-tune the amplitudes of the higher 
 excitations that capture the correlation energy.
-}
 
 % SPIN-PROJECTION SCHEMES
-\hugh{%
 A number of spin-projected extensions have been derived to reduce spin-contamination in the wave function
 and overcome the slow UMP convergence.
-Early versions of these theories, introduced by Schlegel\cite{Schlegel_1986, Schlegel_1988} or 
-Knowles and Handy\cite{Knowles_1988a,Knowles_1988b}, exploited the ``projection-after-variation'' philosophy,
+Early versions of these theories, introduced by Schlegel \cite{Schlegel_1986, Schlegel_1988} or 
+Knowles and Handy,\cite{Knowles_1988a,Knowles_1988b} exploited the ``projection-after-variation'' philosophy,
 where the spin-projection is applied directly to the UMP expansion.
 These methods succeeded in accelerating the convergence of the projected MP series and were 
 considered as highly effective methods for capturing the electron correlation at low computational cost.\cite{Knowles_1988b}
@@ -791,7 +782,6 @@ More recent formulations of spin-projected perturbations theories have considere
 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.
@@ -855,11 +845,10 @@ gradient discontinuities or spurious minima.
 	\label{fig:RMP}}
 \end{figure*}
 
-\hugh{The behaviour of the RMP and UMP series observed in \ce{H2} can also be illustrated by considering
+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 $U/t$.
-}
-Using the ground-state RHF reference orbitals leads to the \hugh{parametrised} RMP Hamiltonian
+Using the ground-state RHF reference orbitals leads to the parametrised RMP Hamiltonian
 \begin{widetext}
 \begin{equation}
 \label{eq:H_RMP}
@@ -940,8 +929,8 @@ for the two states using the ground-state RHF orbitals is identical.
 \end{figure*}
 
 The behaviour of the UMP series is more subtle than the RMP series as the spin-contamination in the wave function
-\hugh{introduces additional coupling between the singly- and doubly-excited configurations.}
-Using the ground-state UHF reference orbitals in the Hubbard dimer yields the \hugh{parametrised} UMP Hamiltonian
+introduces additional coupling between the singly- and doubly-excited configurations.
+Using the ground-state UHF reference orbitals in the Hubbard dimer yields the parametrised UMP Hamiltonian
 \begin{widetext}
 \begin{equation}
 \label{eq:H_UMP}
@@ -980,24 +969,22 @@ For example, while the RMP energy shows only one EP between the ground state and
 the doubly-excited state (Fig.~\ref{fig:RMP}), the UMP energy has two EPs: one connecting the ground state with the
 singly-excited open-shell singlet, and the other connecting this single excitation to the 
 doubly-excited second excitation (Fig.~\ref{fig:UMP}).
-\hugh{%
 This new ground-state EP always appears outside the unit cylinder and guarantees convergence of the ground-state energy.
-However, the excited-state EP is moved within} the unit cylinder and causes the 
+However, the excited-state EP is moved within the unit cylinder and causes the 
 convergence of the excited-state UMP series to deteriorate.
-\hugh{Our interpretation of this effect is that the symmetry-broken orbital optimisation has redistributed the strong 
+Our interpretation of this effect is that the symmetry-broken orbital optimisation has redistributed the strong 
 coupling between the ground- and doubly-excited states into weaker couplings between all states, and has thus
-sacrificed convergence of the excited-state series so that the ground-state convergence can be maximised.}
+sacrificed convergence of the excited-state series so that the ground-state convergence can be maximised.
 
 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.
-\hugh{The slow convergence observed in \ce{H2}\cite{Gill_1988} can then be seen as this EP 
+\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.}
 Furthermore, the majority of the UMP expansion in this regime is concerned with removing spin-contamination from the wave 
-function \hugh{rather than improving the energy.
+function rather than improving the energy.
 It is well-known that the spin-projection needed to remove spin-contamination can require non-linear combinations
 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.
@@ -1020,7 +1007,7 @@ very slowly as the perturbation order is increased.
 %==========================================%
 
 % CREMER AND HE
-\hugh{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. 
+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.''}
@@ -1031,7 +1018,7 @@ This observation on the contribution to the MP$n$ energy corroborates the electr
 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.}
+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}