further corrections

Manuscript/seniority.tex

@@ -224,21 +224,15 @@ From the PECs, we have also extracted the vibrational frequencies and equilibriu
 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.
+\fk{It is worth mentioning that the determinant-driven framework of {\QP} allows the inclusion of any arbitrary set of determinants.}
 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 (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 ground state energy to be converged when the second-order perturbation correction \fk{from the truncated Hilbert space} (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.
-We have to mention this somewhere.
-Also, it is worth mentioning that one uses Davidson's iterative algorithm to seek for the ground-state energy which means that the computation and storage cost us $\order*{\Ndet^2}$ and $\order*{\Ndet}$, respectively.
-This shows that the determinant-driven algorithm is definitely not optimal.
+\fk{The ground-state CI energy is obtained with the Davidson's iterative algorithm \cite{Davidson_1975} [Titou, please add the ref.],
+which in the present implementation of {\QP} means that the computation and storage cost us $\order*{\Ndet^{3/2}}$ and $\order*{\Ndet}$, respectively.
+This shows that the determinant-driven algorithm is not optimal in general.
 However, the selected nature of the CIPSI algorithm means that the actual number of determinants is quite small and therefore calculations are technically feasable.}
 
 The CI calculations were performed with both canonical HF orbitals and optimized orbitals.
@@ -250,6 +244,8 @@ 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).
+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.
 
 It is worth mentioning that obtaining smooth PECs for the orbital optimized calculations proved to be far from trivial.
 First, the orbital optimization started from the HF orbitals of each geometry.
@@ -361,7 +357,7 @@ Up to this point, all results and discussions have been based on CI calculations
 \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).}
+Thus, we decided to further assess the role of orbital optimization (occupied-virtual rotations included) for each class of CI methods.}
 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.
@@ -419,26 +415,23 @@ Nevertheless, double (ethylene) and even triple (\ce{N2}) bond breaking still ap
 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 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)
+We evaluated the performance of hCI against excitation-based CI and seniority-based CI,
+by comparing PECs and derived quantities 
 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.
 The comparison to seniority-based CI is less trivial.
 DOCI (the first level of seniority-based CI) often provides even lower NPEs for a similar $\Ndet$, but it falls short in describing the other properties investigated here.
-If higher accuracy is desired, than the convergence is faster with hCI (and also excitation-based CI) than seniority-based CI, at least for HF orbitals.
+In addition, if higher accuracy is desired, convergence was found to be faster with hCI (and also excitation-based CI) than seniority-based CI, at least for HF orbitals.
 Finally, the exponential scaling of seniority-based CI in practice precludes this approach for larger systems and basis sets,
 while the favorable polynomial scaling and encouraging performance of hCI is an alternative.
  
 We found surprisingly good results for the first level of hCI (hCI1) and the orbital optimized version of CIS (oo-CIS), two methods with very favorable computational scaling.
 In particular, oo-CIS correctly describes single bond breaking.
 We hope to report on generalizations to excited states in the future.
-
-%For the challenging cases of \ce{H4} and \ce{H8}, hCI and excitation-based CI perform similarly.
-An important conclusion is that orbital optimization at the CI level is not necessarily a recommended strategy, 
+In contrast, an important conclusion is that orbital optimization at higher CI levels is not necessarily a recommended strategy, 
 given the overall modest improvement in convergence when compared to results with canonical HF orbitals.
 One should bear in mind that optimizing the orbitals is always accompanied with well-known challenges (several solutions, convergence issues, etc)
 and may imply a significant computational burden (associated with the calculations of the orbital gradient and Hessian, and the many iterations that are often required),

cp_to_manuscript.sh

@@ -0,0 +1,22 @@
+#!/bin/bash
+
+path='/home/fabris/ongoing_projects/seniority/Manuscript'
+
+#molecules=( HF F2 ethylene N2 H4 H8 )
+#molecules=( ethylene )
+molecules=( H8 )
+
+for mol in "${molecules[@]}"
+do
+
+#echo  "${mol}_cc-pvdz/plot_pes.pdf"
+
+cp ${mol}_cc-pvdz/plot_pes.pdf      $path/${mol}_pes.pdf
+cp ${mol}_cc-pvdz/plot_error.pdf    $path/${mol}_pes_error.pdf
+cp ${mol}_cc-pvdz/plot_stat.pdf     $path/${mol}_npe.pdf
+cp ${mol}_cc-pvdz/plot_distance.pdf $path/${mol}_distance.pdf
+cp ${mol}_cc-pvdz/freq.pdf          $path/${mol}_freq.pdf
+cp ${mol}_cc-pvdz/force.pdf         $path/${mol}_force.pdf
+cp ${mol}_cc-pvdz/xe.pdf            $path/${mol}_xe.pdf
+
+done