ok with intro

Manuscript/seniority.tex

@@ -57,19 +57,10 @@
 
 % Abstract
 \begin{abstract}
-%Aiming at recovering both static and dynamic correlation,
 We propose a novel partitioning of the Hilbert space, hierarchy configuration interaction (hCI), 
 where the excitation degree (with respect to a given reference determinant) and the seniority number (\ie, the number of unpaired electrons) are combined in a single hierarchy parameter.
 The key appealing feature of hCI is that each hierarchy level accounts for all classes of determinants whose number share the same scaling with system size.
-%number of electrons and basis functions.
-%In this way, it accounts for low-seniority high-excitation determinants lacking in excitation-based CI, while keeping the same computational scaling with system size.
 By surveying the dissociation of multiple molecular systems, we found that the overall performance of hCI usually exceeds or, at least, parallels that of excitation-based CI.
-%By surveying the dissociation of multiple molecular systems, we examined how fast hCI and their excitation-based and seniority-based parents converge as we step up towards the exact full CI limit.
-%The overall performance of hCI usually exceeds or at least parallels that of excitation-based CI.
-%For small systems and basis sets, doubly-occupied CI (the first level of seniority-based CI) often remains the best option, but becomes impractical for larger systems or basis sets, and for higher accuracy.
-%However, for larger systems or basis sets, and for higher accuracy, seniority-based CI becomes impractical.
-%However, some of its interesting features, particularly the small non-parallelity errors, are partially recovered with hCI, at only a polynomial cost.
-%We have further explored the role of optimizing the orbitals at several levels of CI.
 For higher orders of hCI and excitation-based CI, 
 the additional computational burden related to orbital optimization usually do not compensate the marginal improvements compared with results obtained with Hartree-Fock orbitals.
 The exception is orbital-optimized CI with single excitations, a minimally correlated model displaying the qualitatively correct description of single bond breaking,
@@ -98,21 +89,20 @@ The question is then how to construct an effective and computationally tractable
 that quickly recover the correlation energy, understood as the energy difference between the FCI and the mean-field Hartree-Fock (HF) solutions.
 
 Excitation-based CI is surely the most well-known and popular class of CI methods.
-In this context, one accounts for all determinants generated by exciting up to $e$ electrons from a given reference, which is usually the HF solution, but does not have to.
+In this context, one accounts for all determinants generated by exciting up to $e$ electrons from a given reference, which is usually the HF determinant, but does not have to.
 In this way, the excitation degree $e$ defines the following sequence of models:
 CI with single excitations (CIS), CI with single and double excitations (CISD), CI with single, double, and triple excitations (CISDT), and so on.
 Excitation-based CI manages to quickly recover weak (dynamic) correlation effects, but struggles in strong (static) correlation regimes.
-It also famously lacks size-consistency which explains issues at dissociations.
+It also famously lacks size-consistency which explains issues, for example, when dissociating chemical bonds.
 Importantly, the number of determinants $\Ndet$ (which is the key parameter governing the computational cost, as discussed later) scales polynomially with the number of basis functions $\Nbas$ as $\Nbas^{2e}$.
-%This means that the contribution of higher excitations become progressively smaller.
  
 Alternatively, seniority-based CI methods (sCI) have been proposed in both nuclear \cite{Ring_1980} and electronic \cite{Bytautas_2011} structure calculations.
 In short, the seniority number $s$ is the number of unpaired electrons in a given determinant.
 By truncating at the seniority zero ($s = 0$) sector (sCI0), one obtains the well-known doubly-occupied CI (DOCI) method, \cite{Bytautas_2011,Allen_1962,Smith_1965,Veillard_1967}
 which has been shown to be particularly effective at catching static correlation,
 while higher sectors tend to contribute progressively less. \cite{Bytautas_2011,Bytautas_2015,Alcoba_2014b,Alcoba_2014}
-In addition, sCI0 is size-consistent, a property that is not shared by higher orders of seniority-based CI.
+\titou{In addition, sCI0 is size-consistent, a property that is not shared by higher orders of seniority-based CI.}
-However, already at the sCI0 level, $\Ndet$ scales exponentially with $\Nbas$, since excitations of all excitation degrees are included.
+However, already at the sCI0 level, $\Ndet$ scales exponentially with $\Nbas$, since excitations of all degrees are included.
 Therefore, despite the encouraging successes of seniority-based CI methods, their unfavorable computational scaling restricts applications to very small systems. \cite{Shepherd_2016}
 Besides CI, other methods that exploit the concept of seniority number have been pursued. \cite{Limacher_2013,Limacher_2014,Tecmer_2014,Boguslawski_2014a,Boguslawski_2015,Boguslawski_2014b,Boguslawski_2014c,Johnson_2017,Fecteau_2020,Johnson_2020,Henderson_2014,Stein_2014,Henderson_2015,Chen_2015,Bytautas_2018}
 
@@ -121,7 +111,7 @@ Besides CI, other methods that exploit the concept of seniority number have been
 %\label{sec:hCI}
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 
-At this point, we notice the current dicothomy.
+At this point, we notice the current dichotomy.
 When targeting static correlation, seniority-based CI methods tend to have a better performance than excitation-based CI, despite their higher computational cost.
 The latter class of methods, in contrast, are well-suited for recovering dynamic correlation, and only at polynomial cost with system size.
 Ideally, we aim for a method that captures most of both static and dynamic correlation, with as few determinants as possible.
@@ -133,7 +123,7 @@ It combines both the excitation degree $e$ and the seniority number $s$ into one
 \end{equation}
 which assumes half-integer values.
 Here we only consider systems with an even number of electrons, meaning that $s$ takes only even values as well.
-Figure \ref{fig:allCI} shows how the Hilbert space is populated in excitation-based CI, seniority-based CI, and our hybrid hCI methods.
+Figure \ref{fig:allCI} shows how the Hilbert space is progressively populated in excitation-based CI, seniority-based CI, and our hybrid hCI methods.
 
 %%% FIG 1 %%%
 \begin{figure*}%[h!]
@@ -184,7 +174,7 @@ Second and most importantly, each next level includes all classes of determinant
 Each level of excitation-based CI has a hCI counterpart with the same scaling of $\Ndet$ with respect to $\Nbas$.
 For example, $\Ndet = \order*{\Nbas^4}$ in both hCI2 and CISD, whereas $\Ndet = \order*{\Nbas^6}$ in hCI3 and CISDT, and so on.
 From this computational perspective, hCI can be seen as a more natural choice than the traditional excitation-based CI,
-because if one can afford for, say, a CISDT calculation, then one could probably afford a hCI3 calculation, due to the same scaling of $\Ndet$.
+because if one can afford for, say, CISDT, then one could probably afford hCI3, due to the same scaling of $\Ndet$.
 Of course, in practice an integer-$h$ hCI method has more determinants than its excitation-based counterpart (despite the same scaling of $\Ndet$),
 and thus one should first ensure whether including the lower-triangular blocks (going from CISDT to hCI3 in our example)
 is a better strategy than adding the next column (going from CISDT to CISDTQ).
@@ -196,21 +186,21 @@ In contrast, the paired double excitations of hCI1 do connect with the reference
 Therefore, while CIS based on HF orbitals does not improve with respect to the mean-field HF wave function,
 the hCI1 counterpart already represents a minimally correlated model, with the same and favorable $\Ndet = \order*{\Nbas^2}$ scaling.
 hCI also allows for half-integer values of $h$, with no equivalent in excitation-based CI.
-This gives extra flexibility in terms of choice of method.
+This gives extra flexibility in terms of methodological choice.
 For a particular application with excitation-based CI, CISD might be too inaccurate, for example, while the improved accuracy of CISDT might be too expensive.
 hCI2.5 could represent an alternative, being more accurate than hCI2 and less expensive than hCI3.
 
 Our main goal here is to assess the performance of hCI against excitation-based and seniority-based CI.
 To do so, we have evaluated how fast different observables converge to the FCI limit as a function of $\Ndet$.
-We have calculated the potential energy curves (PECs) for the dissociation of six systems, 
+In particular, we have calculated the potential energy curves (PECs) for the dissociation of six systems: 
 \ce{HF}, \ce{F2}, \ce{N2}, ethylene, \ce{H4}, and \ce{H8},
 which display a variable number of bond breaking.
-For the latter two molecules, we considered linearly arranged with equally spaced hydrogen atoms, and computed PECs along the symmetric dissociation coordinate.
+For the latter two molecules, we have considered linearly arranged with equally spaced hydrogen atoms, and computed PECs along the symmetric dissociation coordinate.
-For ethylene, we considered the \ce{C=C} double bond breaking, while freezing the remaining internal coordinates.
+For ethylene, we consider the \ce{C=C} double bond breaking, while freezing the remaining internal coordinates.
 Its equilibrium geometry was taken from Ref.~\onlinecite{Loos_2018} and is reproduced in the \SupInf.
 Due to the (multiple) bond breaking, these are challenging systems for electronic structure methods,
 being often considered when assessing novel methodologies.
-We evaluated the convergence of four observables: the non-parallelity error (NPE), the distance error, the vibrational frequencies, and the equilibrium geometries.
+More precisely, we have evaluated the convergence of four observables: the non-parallelity error (NPE), the distance error, the vibrational frequencies, and the equilibrium geometries.
 The NPE is defined as the maximum minus the minimum differences between the PECs obtained at given CI level and the exact FCI result.
 We define the distance error as the maximum plus the minimum differences between a given PEC and the FCI result.
 Thus, while the NPE probes the similarity regarding the shape of the PECs, the distance error provides a measure of how their overall magnitudes compare.