minor corrections in results

Manuscript/seniority.tex

@@ -1,8 +1,8 @@
 \documentclass[aip,jcp,reprint,noshowkeys,superscriptaddress]{revtex4-1}
 \usepackage{graphicx,dcolumn,bm,xcolor,microtype,multirow,amscd,amsmath,amssymb,amsfonts,physics,wrapfig,txfonts,siunitx}
 \usepackage[version=4]{mhchem}
-\usepackage{natbib}
-\bibliographystyle{achemso}
+%\usepackage{natbib}
+%\bibliographystyle{achemso}
 
 \newcommand{\fk}[1]{\textcolor{blue}{#1}}
 
@@ -106,7 +106,7 @@ Importantly, the number of determinants $\Ndet$ (which is the key parameter gove
 %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.
+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}.
 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}
@@ -267,14 +267,14 @@ We recall that saddle point solutions were purposely avoided in our orbital opti
 %\subsection{Non-parallelity errors and dissociation energies}
 %\subsection{Non-parallelity errors}
 
-In Fig.~\ref{fig:plot_stat} we present the NPEs for the six systems studied, and for the three classes of CI methods,
+In Fig.~\ref{fig:plot_stat}, we present the NPEs for the six systems studied, and for the three classes of CI methods,
 as functions of $\Ndet$.
-The corresponding PECs and the energy differences with respect to the FCI results can be found in the \SupInf.
-The main result contained in Fig.~\ref{fig:plot_stat} concerns the overall faster convergence of the hCI methods when compared to excitation-based and seniority-based CI methods.
+The corresponding PECs and the energy differences with respect to FCI can be found in the \SupInf.
+The main result contained in Fig.~\ref{fig:plot_stat} concerns the overall faster convergence of hCI when compared to excitation-based and seniority-based CI.
 This is observed for single bond breaking (\ce{HF} and \ce{F2}) as well as the more challenging double (ethylene), triple (\ce{N2}), and quadruple (\ce{H4}) bond breaking.
 For \ce{H8}, hCI and excitation-based CI perform similarly.
-The convergence with respect to $\Ndet$ is slower in the latter, more challenging cases, irrespective of the class of CI methods, as would be expected.
-But more importantly, the superiority of the hCI methods appears to be highlighted in the multiple bond break systems (compare ethylene and \ce{N2} with \ce{HF} and \ce{F2} in Fig.~\ref{fig:plot_stat}).
+The convergence with respect to $\Ndet$ is slower in the latter, more challenging cases, irrespective of the class of CI methods, as expected.
+But more importantly, the superiority of hCI appears to be highlighted in the multiple bond break systems (compare ethylene and \ce{N2} with \ce{HF} and \ce{F2} in Fig.~\ref{fig:plot_stat}).
 
 %%% FIG 2 %%%
 \begin{figure}[h!]
@@ -292,18 +292,18 @@ This result demonstrates the importance of higher-order excitations with low sen
 which are accounted for in hCI but not in excitation-based CI (for a given scaling with $\Ndet$).
  
 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 $\Ndet$ (hCI2.5 and CISDT).
-However, convergence is clearly slower for the next levels (sCI2 and sCI4), whereas excitation-based CI and specially hCI methods converge faster.
+tends to offer a rather low NPE when compared to the other CI methods with a similar $\Ndet$ scaling (hCI2.5 and CISDT).
+However, convergence is clearly slower for the next levels (sCI2 and sCI4), whereas excitation-based CI and specially hCI converge faster.
 Furthermore, seniority-based CI becomes less attractive for larger basis set in view of its exponential scaling.
 This can be seen in Fig.~Sx of the \SupInf, which shows that augmenting the basis set leads to a much steeper increase of $\Ndet$ for seniority-based CI.
 
-It is worth mentioning the surprisingly good performance of the hCI1 and hCI1.5 methods.
-For \ce{HF}, \ce{F2}, and ethylene, they presented lower NPEs than the much more expensive CISDT method, being slightly higher in the case of \ce{N2}.
-For the same systems we also see the NPEs increase from hCI1.5 to hCI2, and decreasing to lower values only at the hCI3 level. 
+It is worth mentioning the surprisingly good performance of hCI1 and hCI1.5.
+For \ce{HF}, \ce{F2}, and ethylene, they yield lower NPEs than the much more expensive CISDT method, and only slightly higher in the case of \ce{N2}.
+For the same systems, we also see the NPEs increase from hCI1.5 to hCI2, and decreasing to lower values only at the hCI3 level. 
 (Even than, it is important to remember that the hCI2 results remain overall superior to their excitation-based counterparts.)
 Both findings are not observed for \ce{H4} and \ce{H8}.
 It seems that both the relative worsening of hCI2 and the success of hCI1 and hCI1.5
-become less apparent as progressively more bonds are being broken (compare for instance \ce{F2}, \ce{N2}, and \ce{H8} in Fig.~\ref{fig:plot_stat}).
+become less apparent as progressively more bonds are being broken (compare, for instance, \ce{F2}, \ce{N2}, and \ce{H8} in Fig.~\ref{fig:plot_stat}).
 This reflects the fact that higher-order excitations are needed to properly describe multiple bond breaking,
 and also hints at some cancelation of errors in low-order hCI methods for single bond breaking.
 
@@ -319,7 +319,7 @@ as functions of $\Ndet$, for the three classes of CI methods.
 For the equilibrium geometries, hCI performs slightly better overall than excitation-based CI.
 A more significant advantage of hCI can be seen for the vibrational frequencies.
 For both observables, hCI and excitation-based CI largely outperform seniority-based CI.
-Similarly to what we observed for the NPEs, the convergence of hCI was also found to be non-monotonic in some cases.
+Similarly to what we have observed for the NPEs, the convergence of hCI is also found to be non-monotonic in some cases.
 This oscillatory behavior is particularly evident for \ce{F2}, also noticeable for \ce{HF}, becoming less apparent for ethylene, virtually absent for \ce{N2},
 and showing up again for \ce{H4} and \ce{H8}.
 Interestingly, equilibrium geometries and vibrational frequencies of \ce{HF} and \ce{F2} (single bond breaking), 
@@ -343,10 +343,10 @@ are rather accurate when evaluated at the hCI1.5 level, bearing in mind its rela
 \end{figure}
 %%% %%% %%%
 
-For the \ce{HF} molecule we have also evaluated how the convergence is affected by increasing the basis sets, going from cc-pVDZ to cc-pVTZ and cc-pVQZ (see Fig.~Sx and Fig.~Sy in the \SupInf).
+For the \ce{HF} molecule we have also evaluated how the convergence is affected by increasing the size of the basis set, going from cc-pVDZ to cc-pVTZ and cc-pVQZ (see Fig.~Sx and Fig.~Sy in the \SupInf).
 While a larger $\Ndet$ is required to achieve the same level of convergence, as expected,
 the convergence profiles remain very similar for all basis sets.
-Vibrational frequency and equilibrium geometry present less oscillations for the hCI methods.
+Vibrational frequency and equilibrium geometry present less oscillations for hCI.
 We thus believe that the main findings discussed here for the other systems would be equally basis set independent.
 
 %\subsection{Orbital optimized configuration interaction}
@@ -354,12 +354,12 @@ We thus believe that the main findings discussed here for the other systems woul
 \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 optimizing the orbitals at each given CI calculation.
-Due to the significantly higher computational cost and numerical difficulties for optimizing the orbitals at higher levels of CI,
+Now we discuss the role of further orbital optimization for each given CI calculation.
+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.
 
-At a given CI level, orbital optimization will lead to lower energies than with HF orbitals. 
+Of course, at a given CI level, orbital optimization will lead to lower energies than with HF orbitals. 
 However, even though the energy is lowered (thus improved) at each geometry, such improvement may vary largely along the PEC, which may or may not decrease the NPE.
 More often than not, the NPEs do decrease upon orbital optimization, though not always.
 For example, compared with their non-optimized counterparts, oo-hCI1 and oo-hCI1.5 provide somewhat larger NPEs for \ce{HF} and \ce{F2},
@@ -373,10 +373,10 @@ Based on the present oo-CI results, hCI still has the upper hand when compared w
 
 Orbital optimization usually reduces the NPE for seniority-based CI (in this case we only considered oo-DOCI) as well.
 The gain is specially noticeable for \ce{H4} and \ce{H8} (where the orbitals become symmetry-broken \cite{}),
-and much less so for \ce{HF}, ethylene, and \ce{N2} (where the orbitals remain symmetry-preserving).
+and much less so for \ce{HF}, ethylene, and \ce{N2} (where the orbitals remain symmetry-preserved).
 This is in line with what has been observed before for \ce{N2}. \cite{Bytautas_2011}
 For \ce{F2}, we found that orbital optimization actually increases the NPE (though by a small amount),
-due to the larger energy lowering at the Franck-Condon region than at dissociation.
+due to the larger energy lowering in the Franck-Condon region than at dissociation.
 These results suggest that, when bond breaking involves one site, orbital optimization at the DOCI level does not have such an important role,
 at least in the sense of decreasing the NPE.
 
@@ -386,12 +386,12 @@ The large oscillations observed in the hCI convergence with HF orbitals (for \ce
 
 We come back to the surprisingly good performance of oo-CIS, which is interesting due to its low computational cost.
 The PECs are compared with those of HF and FCI in Fig.~Sx of the \SupInf.
-At this level, the orbital rotations provide an optimized reference (different from the HF solution), from which only single excitations are performed.
-Since the reference is not the HF one, Brillouin's theorem no longer holds, and single excitations actually connect with the reference.
+At this level, the orbital rotations provide an optimized reference (different from the HF determinant), from which only single excitations are performed.
+Since the reference is not the HF determinant, Brillouin's theorem no longer holds, and single excitations actually connect with the reference.
 Thus, with only single excitations (and a reference that is optimized in the presence of these excitations), one obtains a minimally correlated model.
 Surprisingly, oo-CIS recovers a non-negligible fraction (15\%-40\%) of the correlation energy around the equilibrium geometries.
 For all systems, significantly more correlation energy (25\%-65\% of the total) is recovered at dissociation.
-In fact, the larger account of correlation at dissociation implies in the relatively small NPEs encountered at the oo-CIS level.
+In fact, the larger account of correlation at dissociation is responsible of the relatively small NPEs encountered at the oo-CIS level.
 We also found that the NPE drops more significantly (with respect to the HF one) for the single bond breaking cases (\ce{HF} and \ce{F2}), 
 followed by the double (ethylene) and triple (\ce{N2}) bond breaking, then \ce{H4}, and finally \ce{H8}.
 
@@ -401,7 +401,7 @@ The reference has a decreasing weight in the CI expansion as the bond is stretch
 However, that is the reference one needs to achieve the correct open-shell character of the fragments when the single excitations of oo-CIS are accounted for.
 Indeed, the most important single excitations promote the electron from the negative to the positive fragment, resulting in two singly open-shell radicals.
 This is enough to obtain the qualitatively correct description of single bond breaking, hence the relatively low NPEs observed for \ce{HF} and \ce{F2}.
-In contrast, the oo-CIS method can only explicitly account for one unpaired electron at each fragment, such that multiple bond breaking become insufficiently described.
+In contrast, the oo-CIS method can only explicitly account for one unpaired electron on each fragment, such that multiple bond breaking become insufficiently described.
 Nevertheless, double (ethylene) and even triple (\ce{N2}) bond breaking still appear to be reasonably well-described at the oo-CIS level.
 
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