corrections

HF_cc-pvtz/pes_s4.dat

@@ -32,4 +32,11 @@
 2.6         -100.10849486
 2.7         -100.10698502
 2.8         -100.10581861
-2.9         -100.10497295
+2.9         -100.10497292
+3.0         -100.10435365
+3.5         -100.10307608
+4.0         -100.10299962
+4.5         -100.10323678
+5.0         -100.10356103
+5.5         -100.10388771
+6.0         -100.10418171

Manuscript/TOC/TOC.tex

@@ -10,8 +10,8 @@
 %\end{tcolorbox}
 \smartdiagramset{
 circular distance=3.75cm,
-font=\large,
-text width=3.2cm,
+font=\Large,
+text width=4.2cm,
 set color list={red!40,blue!40,green!40}}
 \smartdiagram[circular diagram:clockwise]{
 	Excitation degree $e$\\

Manuscript/seniority.tex

@@ -59,8 +59,8 @@
 \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 \titou{in order to recover both static and dynamic correlations at a similar rate?}
-The key appealing feature of hCI is that each hierarchy level accounts \titou{for all classes of determinants that share the same scaling with system size.}
+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.
@@ -95,28 +95,27 @@ by expanding the wave function as a linear combination of Slater determinants (o
 At the full CI (FCI) level, the complete Hilbert space is spanned in the wave function expansion, leading to the exact solution for a given one-electron basis set.
 Except for very small systems, \cite{Knowles_1984,Knowles_1989} the FCI limit is unattainable, and in practice the expansion of the CI wave function must be truncated.
 The question is then how to construct an effective and computationally tractable hierarchy of truncated CI methods
-that quickly recover the correlation energy, understood as the energy difference between the FCI and the mean-field \titou{restricted?} Hartree-Fock (HF) solutions.
+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 \titou{closed-shell?} reference, which is usually the \titou{restricted?} 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 solution, 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.
-\titou{It also famously lacks size-consistency which explains issues at dissociations?}
-Importantly, the number of determinants $\Ndet$ (which is the key parameter governing the computational cost) scales polynomially with the number of \titou{basis functions} $\Nbas$ as $\Nbas^{2e}$.
+It also famously lacks size-consistency which explains issues at dissociations.
+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 \titou{and takes only even values for a closed-shell system}.
+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.
 However, already at the sCI0 level, $\Ndet$ scales exponentially with $\Nbas$, since excitations of all excitation degrees are included.
-\titou{Is it therefore size-consistent?}
 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}
 
-
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 %\section{Hierarchy configuration interaction}
 %\label{sec:hCI}
@@ -133,7 +132,7 @@ It combines both the excitation degree $e$ and the seniority number $s$ into one
 	h = \frac{e+s/2}{2},
 \end{equation}
 which assumes half-integer values.
-% open-shell
+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.
 
 %%% FIG 1 %%%
@@ -180,13 +179,13 @@ In the latter case, the scaling of $\Ndet$ would be dominated by the rightmost b
 Bytautas \textit{et al.}\cite{Bytautas_2015} explored a different hybrid scheme combining determinants having a maximum seniority number and those from a complete active space.
 In comparison to previous approaches, our hybrid hCI scheme has two key advantages.
 First, it is defined by a single parameter that unifies excitation degree and seniority number [see Eq.~\eqref{eq:h}].
-Second and most importantly, each next level includes all classes of determinants \titou{sharing the same scaling with system size}, as discussed before, thus preserving the polynomial cost of the method.
+Second and most importantly, each next level includes all classes of determinants whose number share the same scaling with system size, as discussed before, thus preserving the polynomial cost of the method.
 
 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, \titou{which has the same computational scaling}.
-Of course, in practice an integer-$h$ hCI method has more determinants than its excitation-based counterpart \titou{(despite the same scaling)},
+because if one can afford for, say, a CISDT calculation, then one could probably afford a hCI3 calculation, due to the same scaling of $\Ndet$.
+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).
 Therefore, here we decided to discuss the results in terms of $\Ndet$, rather than the formal scaling of $\Ndet$ as a function of $\Nbas$,
@@ -216,7 +215,8 @@ The NPE is defined as the maximum minus the minimum differences between the PECs
 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.
 From the PECs, we have also extracted the vibrational frequencies and equilibrium geometries (details can be found in the \SupInf).
- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 %\section{Computational details}
 %\label{sec:compdet}
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
@@ -225,13 +225,14 @@ The hCI method was implemented in {\QP} via a straightforward adaptation of the
 \textit{configuration interaction using a perturbative selection made iteratively} (CIPSI) algorithm, \cite{Huron_1973,Giner_2013,Giner_2015,Garniron_2018}
 by allowing only for determinants having a given maximum hierarchy $h$ to be selected.
 The excitation-based CI, seniority-based CI, and FCI calculations presented here were also performed with the CIPSI algorithm implemented in {\QP}. \cite{Garniron_2019}
-In practice, we consider, for a given CI level, the CI energy to be converged when the second-order perturbation correction \titou{(which approximately measures the error between the selective and complete calculations)} lies below \SI{0.01}{\milli\hartree}. \cite{Garniron_2018}
+In practice, we consider, for a given CI level, the CI energy to be converged when the second-order perturbation correction (which approximately measures the error between the selective and complete calculations) lies below \SI{0.01}{\milli\hartree}. \cite{Garniron_2018}
 These selected versions of CI require considerably fewer determinants than the formal number of determinants (understood as all those that belong to a given CI level, regardless of their weight or symmetry) of their complete counterparts.
 Nevertheless, we decided to present the results as functions of the formal number of determinants, 
 which are not related to the particular algorithmic choices of the CIPSI calculations.
 All CI calculations were performed for the cc-pVDZ basis set and with frozen core orbitals.
 For the \ce{HF} molecule we have also tested basis set effects, by considered the cc-pVTZ and cc-pVQZ basis sets.
 
+\fk{I'll work on this.}
 \titou{T2: I think it might be worth mentioning that the determinant-driven framework of {\QP} allows to include any arbitrary set of determinants.
 This would also justify why we are focusing on the number of determinants instead of the actual scaling of the method.
 I think this is a important point because the CISD Hilbert space has a size proportional to $N^4$ but the cost associated with solving the CISD equations scales as $N^6$... Actually, it follows the same rules as CC: CISD scales as $N^6$, CISDT as $N^8$, CISDTQ as $N^{10}$, etc.
@@ -244,8 +245,9 @@ The CI calculations were performed with both canonical HF orbitals and optimized
 In the latter case, the energy is obtained variationally in the CI space and in the orbital parameter space, hence defining orbital-optimized CI (oo-CI) methods.
 We employed the algorithm described elsewhere \cite{Damour_2021} and also implemented in {\QP} for optimizing the orbitals within a CI wave function.
 In order to avoid converging to a saddle point solution, we employed a similar strategy as recently described in Ref.~\onlinecite{Elayan_2022}.
-Namely, whenever the eigenvalue of the orbital rotation Hessian is negative and the corresponding gradient component $g_i$ lies below a given threshold $g_0$ \titou{(typically equal to ?)},
+Namely, whenever the eigenvalue of the orbital rotation Hessian is negative and the corresponding gradient component $g_i$ lies below a given threshold $g_0$,
 then this gradient component is replaced by $g_0 \abs{g_i}/g_i$.
+\fk{Here we took $g_0 = $ \SI{1}{\micro\hartree}, and considered the orbitals to be converged when the maximum orbital rotation gradient lies below \SI{0.1}{\milli\hartree}.}
 While we cannot ensure that the obtained solutions are global minima in the orbital parameter space, we verified that in all stationary solutions surveyed here 
 correspond to real minima (rather than maxima or saddle points).
 
@@ -292,7 +294,8 @@ For all systems (specially ethylene and \ce{N2}), hCI2 is better than CISD, two
 hCI2.5 is better than CISDT (except for \ce{H8}), despite its lower computational cost, whereas hCI3 is much better than CISDT, and comparable in accuracy with CISDTQ (again for all systems).
 Inspection of the PECs (see \SupInf) reveals that the lower NPEs observed for hCI stem mostly from the contribution of the dissociation region.
 This result demonstrates the importance of higher-order excitations with low seniority number in this strong correlation regime, 
-which are accounted for in hCI but not in excitation-based CI (for a given scaling with $\Ndet$).
+which are accounted for in hCI but not in excitation-based CI (for a given scaling of $\Ndet$).
+\fk{These determinants are responsible for alleviating the size-consistency problem when going from excitation-based CI to hCI.}
  
 Meanwhile, the first level of seniority-based CI (sCI0, which is the same as DOCI)
 tends to offer a rather low NPE when compared to the other CI methods with a similar $\Ndet$ scaling (hCI2.5 and CISDT).
@@ -354,10 +357,11 @@ We thus believe that the main findings discussed here for the other systems woul
 
 %\subsection{Orbital optimized configuration interaction}
 
-\titou{T2: Would it be a good idea to have mentioned that seniority-based schemes are not invariant with respect to orbital rotations?}
-
 Up to this point, all results and discussions have been based on CI calculations with HF orbitals.
-Now we discuss the role of further orbital optimization for each given CI calculation.
+\fk{We recall that seniority-based CI (in contrast to excitation-based CI) is not invariant with respect to orbital rotations within the occupied and virtual subspaces, \cite{Bytautas_2011}
+and for this reason it is customary to optimize the corresponding wave function by performing such rotations.
+Similarly, hCI wave functions are not invariant under orbital rotations within each subspace.
+Thus, we decided to further assess the role of orbital optimization for each class of CI methods (also including occupied-virtual rotations).}
 Due to the significantly higher computational cost and numerical difficulties associated with orbital optimization at higher CI levels,
 such calculations were typically limited up to oo-CISD (for excitation-based), oo-DOCI (for seniority-based), and oo-hCI2 (for hCI).
 The PECs and analogous results to those of Figs.~\ref{fig:plot_stat}, \ref{fig:xe}, and \ref{fig:freq} are shown in the \SupInf.
@@ -412,13 +416,14 @@ Nevertheless, double (ethylene) and even triple (\ce{N2}) bond breaking still ap
 %\label{sec:ccl}
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 
-In summary, here we have proposed a new scheme for truncating the Hilbert space in configuration interaction calculations, named hierarchy CI (hCI).
+In this Letter, we have proposed a new scheme for truncating the Hilbert space in configuration interaction calculations, named hierarchy CI (hCI).
 By merging the excitation degree and the seniority number into a single hierarchy parameter $h$, 
-the hCI method ensures that all classes of determinants sharing \titou{the same scaling with the number of electrons} are included in each level of the hierarchy.
+the hCI method ensures that all classes of determinants sharing the same scaling of $\Ndet$ with the number of basis functions are included in each level of the hierarchy.
 We evaluated the performance of hCI against the traditional excitation-based CI and seniority-based CI,
 by comparing PECs and derived quantities (non-parallelity errors, distance errors, vibrational frequencies, and equilibrium geometries)
 for six systems, ranging from single to multiple bond breaking.
 
+\fk{I'll still rearrange this somewhat.}
 Our key finding is that the overall performance of hCI either surpasses or equals that of excitation-based CI,
 in the sense of convergence with respect to $\Ndet$.
 The superiority of hCI methods is more noticeable for the non-parallelity and distance errors, but also observed to a lesser extent for the vibrational frequencies and equilibrium geometries.

plot_all/cp_to_manuscript.sh

@@ -1,6 +1,7 @@
 #!/bin/bash
 
-path='/home/fabris/ongoing_projects/seniority/Manuscript'
+#path='/home/fabris/ongoing_projects/seniority/Manuscript'
+path='/home/fabris/seniority/Manuscript'
 
 cp plot_pes.pdf          $path/
 cp plot_stat.pdf         $path/