revision done

Manuscript/Cover_Letter/Cover_Letter.tex

@@ -18,9 +18,9 @@ This contribution has never been submitted in total nor in parts to any other jo
 Several quantum chemistry methodologies, including single- and multi-reference configuration interaction (CI) and coupled cluster, 
 rely on an educated truncation of the full Hilbert space, while trying to keep most of its relevant physics.
 From the birth of quantum chemistry and for many decades, most methods have been developed under an \textit{excitation}-based framework,
-which are usually successful in weak-correlation regimes, and, importantly, present favourable (polynomial) computational scaling.
+which are usually successful in weak correlation regimes, and, importantly, present favourable (polynomial) computational scaling.
 Over the past ten years, there has been a growing interest in \textit{seniority}-based methods,
-which are more suitable for strong-correlation regimes, even though they display a much steeper (exponential) computational scaling.
+which are more suitable for strong correlation regimes, even though they display a much steeper (exponential) computational scaling.
 
 Aiming at recovering the best of both worlds, here we present a novel partitioning of the Hilbert space, \textit{hierarchy} configuration interaction (hCI).
 By benchmarking hCI against its excitation- and seniority-based parents and numerically exact results, for a series of challenging molecular systems,
@@ -28,7 +28,7 @@ and for several relevant properties, we demonstrate its overall superior converg
 hCI inherits both the quick recovery of weak correlation and the favourable computational cost from excitation-based CI,
 as well as a better account of strong correlation from seniority-based CI.
 
-Based on the robust performance and appealing features of hCI, we expect our proposed strategy will both inspire further developments within the electronic structure community
+Based on the robust performance and appealing features of hCI, we anticipate our proposed strategy will both inspire further developments within the electronic structure community
 and foment exploratory applications to chemically relevant systems.
 Because of its novelty and impact, we expect the present study to be of interest to a wide audience within the chemistry and physics communities.
 We suggest Peter Knowles, Paul Johnson, Stijn de Baerdemacker, James Shepherd, Thomas Duguet, and Gustavo Scuseria as potential referees.

Manuscript/seniority.tex

@@ -4,12 +4,11 @@
 %\usepackage{natbib}
 %\bibliographystyle{achemso}
 
-\newcommand{\fk}[1]{\textcolor{blue}{#1}}
-
 \newcommand{\ie}{\textit{i.e.}}
 \newcommand{\eg}{\textit{e.g.}}
 \newcommand{\alert}[1]{\textcolor{black}{#1}}
 \usepackage[normalem]{ulem}
+\newcommand{\fk}[1]{\textcolor{blue}{#1}}
 \newcommand{\titou}[1]{\textcolor{red}{#1}}
 \newcommand{\trashPFL}[1]{\textcolor{red}{\sout{#1}}}
 \newcommand{\PFL}[1]{\titou{(\underline{\bf PFL}: #1)}}
@@ -25,11 +24,9 @@
 \newcommand{\tabc}[1]{\multicolumn{1}{c}{#1}}
 \newcommand{\QP}{\textsc{quantum package}}
 
-
 \newcommand{\EHF}{E_\text{HF}}
 \newcommand{\EDOCI}{E_\text{DOCI}}
 \newcommand{\EFCI}{E_\text{FCI}}
-
 \newcommand{\Ndet}{N_\text{det}}
 \newcommand{\Nbas}{N}
 
@@ -180,7 +177,7 @@ and thus one should first ensure whether including the lower-triangular blocks (
 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 $\Nbas$,
 which could make the comparison somewhat biased toward hCI.
-It is interesting to compare the lowest levels of hCI (hCI1) and excitation-based CI (CIS).
+It is also interesting to compare the lowest levels of hCI (hCI1) and excitation-based CI (CIS).
 Since single excitations do not connect with the reference (at least for HF orbitals), CIS provides the same energy as HF.
 In contrast, the paired double excitations of hCI1 do connect with the reference (and the singles contribute indirectly via the doubles).
 Therefore, while CIS based on HF orbitals does not improve with respect to the mean-field HF wave function,
@@ -203,7 +200,7 @@ being often considered when assessing novel methodologies.
 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 a 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.
+Thus, while the NPE probes the similarity regarding the shape of the PECs, the distance error measures 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).
 
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
@@ -243,7 +240,6 @@ This usually led to discontinuous PECs, meaning that distinct solutions were fou
 Then, at some geometries that seem to present the lowest lying solution,
 the optimized orbitals were employed as the guess orbitals for the neighboring geometries, and so on, until a new PEC is obtained.
 This protocol was repeated until the PEC built from the lowest lying oo-CI solution becomes continuous.
-%While we cannot guarantee that the presented solutions represent the global minima, we believe that in most cases the above protocol provides at least close enough solutions.
 We recall that saddle point solutions were purposely avoided in our orbital optimization algorithm. If that was not the case, then even more stationary solutions would have been found.
 
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
@@ -251,17 +247,11 @@ We recall that saddle point solutions were purposely avoided in our orbital opti
 %\label{sec:res}
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 
-%\subsection{Correlation energies}
-%\subsection{Potential energy curves}
-
-%\subsection{Non-parallelity errors and dissociation energies}
-%\subsection{Non-parallelity errors}
-
 While the full set of PECs and the corresponding energy differences with respect to FCI are shown in the \SupInf,
 in Fig.~\ref{fig:F2_pes} we present the PECs for \ce{F2}, which display many of the features also observed for the other systems.
 It already gives a sense of the performance of three classes of CI methods, 
 clearly showing the overall superiority of hCI over excitation-based CI.
-It further reaveals several important features which will be referenced to in the upcoming discussion.
+It further illustrates several important features which will be referenced to in the upcoming discussion.
 
 %%% FIG 2 %%%
 \begin{figure}[h!]
@@ -295,7 +285,7 @@ But more importantly, the superiority of hCI appears to be highlighted in the on
 
 For all systems (specially ethylene and \ce{N2}), hCI2 is better than CISD, two methods where $\Ndet$ scales as $\Nbas^4$.
 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 Fig.~\ref{fig:F2_pes} for the case of \ce{F2} or the \SupInf for the other systems) reveals that the lower NPEs observed for hCI stem mostly from the contribution of the dissociation region.
+Inspection of the PECs (see Fig.~\ref{fig:F2_pes} for the case of \ce{F2} or the {\SupInf} for the other systems) 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 of $\Ndet$).
 These determinants are responsible for alleviating the size-consistency problem when going from excitation-based CI to hCI.
@@ -321,9 +311,6 @@ Most of the observations discussed for the NPE also hold for the distance error,
 The convergence is always monotonic for the latter observable (which is expected from its definition),
 and the performance of seniority-based CI is much poorer (due to the slow recovery of dynamic correlation).
 
-%\subsection{Equilibrium geometries and vibrational frequencies}
-
-%In Figs.~\ref{fig:xe} and \ref{fig:freq}, we present the convergence of the equilibrium geometries and vibrational frequencies, respectively,
 In Figs.~S6 and S7 of the \SupInf, we present the convergence of the equilibrium geometries and vibrational frequencies, respectively,
 as functions of $\Ndet$, for the three classes of CI methods.
 For the equilibrium geometries, hCI performs slightly better overall than excitation-based CI.
@@ -337,34 +324,12 @@ 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), 
 are rather accurate when evaluated at the hCI1.5 level, bearing in mind its relatively modest computational cost.
 
-%%% FIG 4 %%%
-%\begin{figure}[h!]
-%	\includegraphics[width=\linewidth]{xe}
-%	\caption{Equilibrium geometries as function of the number of determinants, for the three classes of CI methods: seniority-based CI (blue), excitation-based CI (red), and our proposed hybrid hCI (green),
-%	and according to the exact FCI result (black horizontal line).
-%        }
-%        \label{fig:xe}
-%\end{figure}
-%%% %%% %%%
-
-%%% FIG 5 %%%
-%\begin{figure}[h!]
-%\includegraphics[width=\linewidth]{freq}
-%	\caption{Vibrational frequencies (or force constants) as function of the number of determinants, for the three classes of CI methods: seniority-based CI (blue), excitation-based CI (red), and our proposed hybrid hCI (green),
-%	and according to the exact FCI result (black horizontal line).
-%	}
-%        \label{fig:freq}
-%\end{figure}
-%%% %%% %%%
-
 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 Figs.~S2 and S3 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 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}
-
 Up to this point, all results and discussions have been based on CI calculations with HF orbitals.
 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.
@@ -372,7 +337,6 @@ Similarly, hCI wave functions are not invariant under orbital rotations within e
 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.
 The PECs and convergence of properties as function of $\Ndet$ are shown in the \SupInf.
 
 Of course, at a given CI level, orbital optimization will lead to lower energies than with HF orbitals. 
@@ -380,11 +344,10 @@ However, even though the energy is lowered (thus improved) at each geometry, suc
 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},
 similar NPEs for ethylene, and smaller NPEs for \ce{N2}, \ce{H4}, and \ce{H8}.
-% oo-hCI2
 Following the same trend, oo-CISD presents smaller NPEs than HF-CISD for the multiple bond breaking systems, but very similar ones for the single bond breaking cases.
 oo-CIS has significantly smaller NPEs than HF-CIS, being comparable to oo-hCI1 for all systems except for \ce{H4} and \ce{H8}, where the latter method performs better.
 (We will come back to oo-CIS later.)
-Based on the present oo-CI results, hCI still has the upper hand when compared with excitation-based CI, though by a much smaller margin.
+Based on the present oo-CI results, hCI still has the upper hand when compared with excitation-based CI, though by a smaller margin.
 
 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{Henderson_2014}),
@@ -397,14 +360,14 @@ at least in the sense of decreasing the NPE.
 
 Optimizing the orbitals at the CI level also tends to benefit the convergence of vibrational frequencies and equilibrium geometries.
 The impact is often somewhat larger for hCI than for excitation-based CI, by a small margin.
-The large oscillations observed in the hCI convergence with HF orbitals (for \ce{HF} and \ce{F2}) are significantly suppressed upon orbital optimization.
+Also, the large oscillations observed in the hCI convergence with HF orbitals (for \ce{HF} and \ce{F2}) are significantly suppressed upon orbital optimization.
 
 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.~S12 of the \SupInf.
 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.
+Interestingly, 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 is responsible for 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}),