472 lines
37 KiB
TeX
472 lines
37 KiB
TeX
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\begin{document}
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\newcommand{\LCPQ}{Laboratoire de Chimie et Physique Quantiques (UMR 5626), Universit\'e de Toulouse, CNRS, UPS, France}
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\title{Hierarchy Configuration Interaction: Combining Seniority Number and Excitation Degree}
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\author{F\'abris Kossoski}
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\email{fkossoski@irsamc.ups-tlse.fr}
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\affiliation{\LCPQ}
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\author{Yann Damour}
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\affiliation{\LCPQ}
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\author{Pierre-Fran\c{c}ois Loos}
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\email{loos@irsamc.ups-tlse.fr}
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\affiliation{\LCPQ}
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% Abstract
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\begin{abstract}
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%Aiming at recovering both static and dynamic correlation,
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We propose a novel partitioning of the Hilbert space, hierarchy configuration interaction (hCI),
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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?}
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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.}
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%number of electrons and basis functions.
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%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.
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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.
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%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.
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%The overall performance of hCI usually exceeds or at least parallels that of excitation-based CI.
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%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.
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%However, for larger systems or basis sets, and for higher accuracy, seniority-based CI becomes impractical.
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%However, some of its interesting features, particularly the small non-parallelity errors, are partially recovered with hCI, at only a polynomial cost.
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%We have further explored the role of optimizing the orbitals at several levels of CI.
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For higher orders of hCI and excitation-based CI,
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the additional computational burden related to orbital optimization usually do not compensate the marginal improvements compared with results obtained with Hartree-Fock orbitals.
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The exception is orbital-optimized CI with single excitations, a minimally correlated model displaying the qualitatively correct description of single bond breaking,
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at a very modest computational cost.
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%\bigskip
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%\begin{center}
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% \boxed{\includegraphics[width=0.4\linewidth]{TOC}}
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%\end{center}
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%\bigskip
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\end{abstract}
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% Title
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\maketitle
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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%\section{Introduction}
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%\label{sec:intro}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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In electronic structure theory, configuration interaction (CI) methods allow for a systematic way to obtain approximate and exact solutions of the electronic Hamiltonian,
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by expanding the wave function as a linear combination of Slater determinants (or configuration state functions). \cite{SzaboBook,Helgakerbook}
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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.
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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.
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The question is then how to construct an effective and computationally tractable hierarchy of truncated CI methods
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that quickly recover the correlation energy, understood as the energy difference between the FCI and the mean-field \titou{restricted?} Hartree-Fock (HF) solutions.
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Excitation-based CI is surely the most well-known and popular class of CI methods.
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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.
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In this way, the excitation degree $e$ defines the following sequence of models:
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CI with single excitations (CIS), CI with single and double excitations (CISD), CI with single, double, and triple excitations (CISDT), and so on.
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Excitation-based CI manages to quickly recover weak (dynamic) correlation effects, but struggles in strong (static) correlation regimes.
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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 $\Nbas^{2e}$.
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%This means that the contribution of higher excitations become progressively smaller.
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Alternatively, seniority-based CI methods (sCI) have been proposed in both nuclear \cite{Ring_1980} and electronic \cite{Bytautas_2011} structure calculations.
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In short, the seniority number $s$ is the number of unpaired electrons in a given determinant.
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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}
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which has been shown to be particularly effective at catching static correlation,
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while higher sectors tend to contribute progressively less. \cite{Bytautas_2011,Bytautas_2015,Alcoba_2014b,Alcoba_2014}
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However, already at the sCI0 level, $\Ndet$ scales exponentially with $\Nbas$, since excitations of all excitation degrees are included.
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Therefore, despite the encouraging successes of seniority-based CI methods, their unfavorable computational scaling restricts applications to very small systems.
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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,Chen_2015,Bytautas_2018}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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%\section{Hierarchy configuration interaction}
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%\label{sec:hCI}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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At this point, we notice the current dicothomy.
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When targeting static correlation, seniority-based CI methods tend to have a better performance than excitation-based CI, despite their higher computational cost.
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The latter class of methods, in contrast, are well-suited for recovering dynamic correlation, and only at polynomial cost with system size.
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Ideally, we aim for a method that captures most of both static and dynamic correlation, with as few determinants as possible.
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With this goal in mind, we propose a new partitioning of the Hilbert space, named \textit{hierarchy} CI (hCI).
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It combines both the excitation degree $e$ and the seniority number $s$ into one single hierarchy parameter
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\begin{equation}
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\label{eq:h}
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h = \frac{e+s/2}{2},
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\end{equation}
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which assumes half-integer values.
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% open-shell
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Figure \ref{fig:allCI} shows how the Hilbert space is populated in excitation-based CI, seniority-based CI, and our hybrid hCI methods.
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%%% FIG 1 %%%
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\begin{figure*}%[h!]
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\includegraphics[width=0.3\linewidth]{table_exc_full}
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\hspace{0.02\linewidth}
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\includegraphics[width=0.3\linewidth]{table_hCI}
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\hspace{0.02\linewidth}
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\includegraphics[width=0.3\linewidth]{table_sen_full}
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\caption{Partitioning of the Hilbert space into blocks of specific excitation degree $e$ (with respect to a closed-shell determinant) and seniority number $s$.
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This $e$-$s$ map is truncated differently in excitation-based CI (left), seniority-based CI (right), and hierarchy-based CI (center).
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The color tones represent the determinants that are included at a given level of CI.}
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\label{fig:allCI}
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\end{figure*}
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%%% %%% %%%
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We have three key justifications for this new CI hierarchy.
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The first one is physical.
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We know that the lower degrees of excitations and lower seniority sectors, when looked at individually, often carry the most important contribution to the FCI expansion.
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By combining $e$ and $s$ as is Eq.~\eqref{eq:h}, we ensure that both directions in the excitation-seniority map (see Fig.~\ref{fig:allCI}) are contemplated.
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Rather than filling the map top-bottom (as in excitation-based CI) or left-right (as in seniority-based CI), the hCI methods fills it diagonally.
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In this sense, we hope to recover dynamic correlation by moving right in the map (increasing the excitation degree while keeping a low seniority number),
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at the same time as static correlation, by moving down (increasing the seniority number while keeping a low excitation degree).
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%dynamic correlation is recovered with traditional CI.
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The second justification is computational.
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In the hCI class of methods, each level of theory accommodates additional determinants from different excitation-seniority sectors (each block of same color tone in Fig.~\ref{fig:allCI}).
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The key insight behind hCI is that the number of additional determinants presents the same scaling with respect to $\Nbas$, for all excitation-seniority sectors entering at a given hierarchy $h$.
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This further justifies the parameter $h$ as being the simple average between $e$ and $s/2$.
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Finally, the third justification for our hCI method is empirical and closely related to the computational motivation.
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There are many possible ways to populate the Hilbert space starting from a given reference determinant,
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and one can in principle formulate any systematic recipe that includes progressively more determinants.
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Besides a physical or computational perspective, the question of what makes for a good recipe can be framed empirically.
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Does our hCI class of methods perform better than excitation-based or seniority-based CI,
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in the sense of recovering most of the correlation energy with the least computational effort?
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Hybrid approaches based on both excitation degree and seniority number have been proposed before. \cite{Alcoba_2014,Raemdonck_2015,Alcoba_2018}
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In these works, the authors established separate maximum values for the excitation and the seniority,
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and either the union or the intersection between the two sets of determinants have been considered.
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For the union case, $\Ndet$ grows exponentially with $\Nbas$,
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while in the intersection approach the Hilbert space is filled rectangle-wise in our excitation-seniority map.
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In the latter case, the scaling of $\Ndet$ would be dominated by the rightmost bottom block.
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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.
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In comparison to previous approaches, our hybrid hCI scheme has two key advantages.
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First, it is defined by a single parameter that unifies excitation degree and seniority number [see Eq.~\eqref{eq:h}].
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Second and most importantly, each next level includes all classes of determinants sharing the same scaling with system size, as discussed before, thus preserving the polynomial cost of the method.
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Each level of excitation-based CI has a hCI counterpart with the same scaling of $\Ndet$ with respect to $\Nbas$.
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For example, $\Ndet = \order*{\Nbas^4}$ in both hCI2 and CISD, whereas $\Ndet = \order*{\Nbas^6}$ in hCI3 and CISDT, and so on.
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From this computational perspective, hCI can be seen as a more natural choice than the traditional excitation-based CI,
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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}.
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Of course, in practice an integer-$h$ hCI method has more determinants than its excitation-based counterpart \titou{(despite the same scaling)},
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and thus one should first ensure whether including the lower-triangular blocks (going from CISDT to hCI3 in our example)
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is a better strategy than adding the next column (going from CISDT to CISDTQ).
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Therefore, here we decided to discuss the results in terms of $\Ndet$, rather than the formal scaling of $\Ndet$ as a function of $\Nbas$,
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which could make the comparison somewhat biased toward hCI.
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It is interesting to compare the lowest levels of hCI (hCI1) and excitation-based CI (CIS).
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Since single excitations do not connect with the reference (at least for HF orbitals), CIS provides the same energy as HF.
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In contrast, the paired double excitations of hCI1 do connect with the reference (and the singles contribute indirectly via the doubles).
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Therefore, while CIS based on HF orbitals does not improve with respect to the mean-field HF wave function,
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the hCI1 counterpart already represents a minimally correlated model, with the same and favorable $\Ndet = \order*{\Nbas^2}$ scaling.
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hCI also allows for half-integer values of $h$, with no equivalent in excitation-based CI.
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This gives extra flexibility in terms of choice of method.
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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.
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hCI2.5 could represent an alternative, being more accurate than hCI2 and less expensive than hCI3.
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Our main goal here is to assess the performance of hCI against excitation-based and seniority-based CI.
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To do so, we have evaluated how fast different observables converge to the FCI limit as a function of $\Ndet$.
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We have calculated the potential energy curves (PECs) for the dissociation of six systems,
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\ce{HF}, \ce{F2}, \ce{N2}, ethylene, \ce{H4}, and \ce{H8},
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which display a variable number of bond breaking.
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For the latter two molecules, we considered linearly arranged with equally spaced hydrogen atoms, and computed PECs along the symmetric dissociation coordinate.
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For ethylene, we considered the \ce{C=C} double bond breaking, while freezing the remaining internal coordinates.
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Due to the (multiple) bond breaking, these are challenging systems for electronic structure methods,
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being often considered when assessing novel methodologies.
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We evaluated the convergence of four observables: the non-parallelity error (NPE), the distance error, the vibrational frequencies, and the equilibrium geometries.
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The NPE is defined as the maximum minus the minimum differences between the PECs obtained at given CI level and the exact FCI result.
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We define the distance error as the maximum and the minimum differences between a given PEC and the FCI result.
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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.
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From the PECs, we have also extracted the vibrational frequencies and equilibrium geometries (details can be found in the \SupInf).
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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%\section{Computational details}
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%\label{sec:compdet}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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The hCI method was implemented in {\QP} via a straightforward adaptation of the
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\textit{configuration interaction using a perturbative selection made iteratively} (CIPSI) algorithm \cite{Huron_1973,Giner_2013,Giner_2015,Garniron_2018},
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by allowing only for determinants having a given maximum hierarchy $h$.
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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}
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In practice, we consider the CI energy to be converged when the second-order perturbation correction lies below \SI{0.01}{\milli\hartree}, \cite{Garniron_2018}
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which requires 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).
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Nevertheless, we decided to present the results as functions of the formal number of determinants,
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which are not related to the particular algorithmic choices of the CIPSI calculations.
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All CI calculations were performed for the cc-pVDZ basis set and with frozen core orbitals.
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For \ce{HF} we have also tested basis set effects, by considered the cc-pVTZ and cc-pVQZ basis sets.
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\titou{Geometries? SI?}
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\titou{T2: I think it might be worth mentioning that the determinant-driven framework of {\QP} allows to include any arbitrary set of determinants.
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This would also justify why we are focusing on the number of determinants instead of the actual scaling of the method.
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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.
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We have to mention this somewhere.
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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.
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This shows that the determinant-driven algorithm is definitely not optimal.
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However, the selected nature of the CIPSI algorithm means that the actual number of determinants is quite small and therefore calculations are technically feasable.}
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The CI calculations were performed with both canonical HF orbitals and optimized orbitals.
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In the latter case, the energy is obtained variationally in the CI space and in the orbital parameter space, hence an orbital-optimized CI (oo-CI) method.
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We employed the algorithm described elsewhere \cite{Damour_2021} and also implemented in {\QP} for optimizing the orbitals within a CI wave function.
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In order to avoid converging to a saddle point solution, we employed a similar strategy as recently described in Ref.~\cite{Hollett_2022}.
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Namely, whenever the eigenvalue of the orbital rotation Hessian is negative and the corresponding gradient component $g_i$ lies below a given threshold $g_0$,
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then this gradient component is replaced by $g_0 \abs{g_i}/g_i$.
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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
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correspond to real minima (rather than maxima or saddle points).
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It is worth mentioning that obtaining smooth PECs for the orbital optimized calculations proved to be far from trivial.
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First, the orbital optimization started from the HF orbitals of each geometry.
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This usually lead to discontinuous PECs, meaning that distinct solutions of the orbital optimization have been found with our algorithm.
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Then, at some geometries that seem to present the lowest lying solution,
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the optimized orbitals were employed as the guess orbitals for the neighboring geometries, and so on, until a new PEC is obtained.
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This protocol is repeated until the PEC built from the lowest lying oo-CI solution becomes continuous.
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%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.
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%Multiple solutions for the orbital optimization are usually found, meaning several local minimal in the orbital parameter landscape.
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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.
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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%\section{Results and discussion}
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%\label{sec:res}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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%\subsection{Correlation energies}
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%\subsection{Potential energy curves}
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%\subsection{Non-parallelity errors and dissociation energies}
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%\subsection{Non-parallelity errors}
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In Fig.~\ref{fig:plot_stat} we present the NPEs for the six systems studied, and for the three classes of CI methods,
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as functions of $\Ndet$.
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The corresponding PECs and the energy differences with respect to the FCI results can be found in the \SupInf.
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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.
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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.
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For \ce{H8}, hCI and excitation-based CI perform similarly.
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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.
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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}).
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\titou{T2: Would it be a good idea to write the \ce{HF} molecule as \ce{FH}?}
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For \ce{HF} 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 in the \SupInf).
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While a larger $\Ndet$ is required to achieve the same level of convergence, as expected,
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the convergence profiles remain very similar for all basis sets.
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We thus believe that the main findings discussed here for the other systems would be equally basis set independent.
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%%% FIG 2 %%%
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\begin{figure}[h!]
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\includegraphics[width=\linewidth]{plot_stat}
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\caption{Non-parallelity errors 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).
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}
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\label{fig:plot_stat}
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\end{figure}
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%%% %%% %%%
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For all systems (specially ethylene and \ce{N2}), hCI2 is better than CISD, two methods where $\Ndet$ scales as $\Nbas^4$.
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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).
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Inspection of the PECs (see \SupInf) reveals that the lower NPEs observed for hCI stem mostly from the contribution of the dissociation region.
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This result demonstrates the importance of higher-order excitations with low seniority number in this strong correlation regime,
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which are accounted for in hCI but not in excitation-based CI (for a given scaling with $\Ndet$).
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Meanwhile, the first level of seniority-based CI (sCI0, which is the same as DOCI)
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tends to offer a rather low NPE when compared to the other CI methods with a similar $\Ndet$ (hCI2.5 and CISDT).
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However, convergence is clearly slower for the next levels (sCI2 and sCI4), whereas excitation-based CI and specially hCI methods converge faster.
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Furthermore, seniority-based CI becomes less attractive for larger basis set in view of its exponential scaling.
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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.
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It is worth mentioning the surprisingly good performance of the hCI1 and hCI1.5 methods.
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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}.
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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.
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(Even than, it is important to remember that the hCI2 results remain overall superior to their excitation-based counterparts.)
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Both findings are not observed for \ce{H4} and \ce{H8}.
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It seems that both the relative worsening of hCI2 and the success of hCI1 and hCI1.5
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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}).
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This reflects the fact that higher-order excitations are needed to properly describe multiple bond breaking,
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and also hints at some cancelation of errors in low-order hCI methods for single bond breaking.
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In Fig.~Sx of the \SupInf, we present the distance error, which is also found to decrease faster with the hCI methods.
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Most of observations discussed for the NPE also hold for the distance error, with two main differences.
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The convergence is always monotonic for the latter observable (which is expected from the definition of the observable),
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and the performance of seniority-based CI is much poorer (due to the slow recovery of dynamic correlation).
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%\subsection{Equilibrium geometries and vibrational frequencies}
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In Fig.~\ref{fig:xe} and \ref{fig:freq}, we present the convergence of the equilibrium geometries and vibrational frequencies, respectively,
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as functions of $\Ndet$, for the three classes of CI methods.
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For the equilibrium geometries, hCI performs slightly better overall than excitation-based CI.
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A more significant advantage of hCI can be seen for the vibrational frequencies.
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For both observables, hCI and excitation-based CI largely outperform seniority-based CI.
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Similarly to what we observed for the NPEs, the convergence of hCI was also found to be non-monotonic in some cases.
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This oscillatory behavior is particularly evident for \ce{F2}, also noticeable for \ce{HF}, becoming less apparent for ethylene, virtually absent for \ce{N2},
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and showing up again for \ce{H4} and \ce{H8}.
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Results for \ce{HF} with larger basis sets (see Fig.Sx in the \SupInf) show very similar convergence behaviors, though with less oscillations for the hCI methods.
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Interestingly, equilibrium geometries and vibrational frequencies of \ce{HF} and \ce{F2} (single bond breaking),
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are rather accurate when evaluated at the hCI1.5 level, bearing in mind its relatively modest computational cost.
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%%% FIG 3 %%%
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\begin{figure}[h!]
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\includegraphics[width=\linewidth]{xe}
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\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).
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}
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\label{fig:xe}
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\end{figure}
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%%% %%% %%%
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%%% FIG 4 %%%
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\begin{figure}[h!]
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\includegraphics[width=\linewidth]{freq}
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\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).
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}
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\label{fig:freq}
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\end{figure}
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%%% %%% %%%
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%\subsection{Orbital optimized configuration interaction}
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\titou{T2: Would it be a good idea to have mentioned that seniority-based schemes are not invariant with respect to orbital rotations?}
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Up to this point, all results and discussions have been based on CI calculations with HF orbitals.
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Now we discuss the role of further optimizing the orbitals at each given CI calculation.
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Due to the significantly higher computational cost and numerical difficulties for optimizing the orbitals at higher levels of CI,
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such calculations were typically limited up to oo-CISD (for excitation-based), oo-DOCI (for seniority-based), and oo-hCI2 (for hCI).
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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.
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At a given CI level, orbital optimization will lead to lower energies than with HF orbitals.
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|
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.
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|
More often than not, the NPEs do decrease upon orbital optimization, though not always.
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|
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}.
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|
% oo-hCI2
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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 latter.
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|
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.
|
|
%It does, however, lead to a more monotonic convergence in the case of hCI, which is not necessarily an advantage.
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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).
|
|
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.
|
|
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.
|
|
|
|
Optimizing the orbitals at the CI level also tends to benefit the convergence of vibrational frequencies and equilibrium geometries (shown in Fig.~Sx of the \SupInf).
|
|
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.
|
|
|
|
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.
|
|
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.
|
|
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}.
|
|
|
|
The above findings can be understood by looking at the character of the oo-CIS orbitals.
|
|
At dissociation, the closed-shell reference is actually ionic, with orbitals assuming localized atomic-like characters.
|
|
The reference has a decreasing weight in the CI expansion as the bond is stretched, becoming virtually zero at dissociation.
|
|
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.
|
|
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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|
%\section{Conclusion and perspectives}
|
|
%\label{sec:ccl}
|
|
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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|
|
|
In summary, here 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.
|
|
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.
|
|
|
|
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.
|
|
Finally, the exponential scaling of seniority-based CI in practice precludes this approach for larger systems and larger basis sets,
|
|
while the favorable polynomial scaling and encouraging performance of hCI as 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,
|
|
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)
|
|
and may imply in a significant computational burden (associated with the calculations of the orbital gradient and Hessian, and the many iterations that are often required),
|
|
specially for larger CI spaces.
|
|
In this sense, stepping up in the CI hierarchy might be a more straightforward and possibly a cheaper alternative than optimizing the orbitals.
|
|
One interesting possibility to explore is to first optimize the orbitals at a lower level of CI, and then to employ this set of orbitals at a higher level of CI.
|
|
|
|
The hCI pathway presented here offers several interesting possibilities to pursue.
|
|
One could generalize and adapt hCI for excited states and open-shell systems,
|
|
develop coupled-cluster methods based on an analogous excitation-seniority truncation of the excitation operator, \cite{Aroeira_2021,Magoulas_2021,Lee_2021}
|
|
and explore the accuracy of hCI trial wave functions for quantum Monte Carlo simulations. \cite{Dash_2019,Dash_2021,Cuzzocrea_2022}
|
|
|
|
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
\begin{acknowledgements}
|
|
This work was performed using HPC resources from CALMIP (Toulouse) under allocation 2021-18005.
|
|
This project has received funding from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme (Grant agreement No.~863481).
|
|
\end{acknowledgements}
|
|
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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|
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|
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
\section*{Supporting information available}
|
|
\label{sec:SI}
|
|
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
Potential energy curves, energy differences with respect to FCI results, non-parallelity errors, distance errors, vibrational frequencies, and equilibrium geometries,
|
|
according to the three classes of CI methods (excitation-based CI, seniority-based CI, and hierarchy-based CI),
|
|
computed for \ce{HF}, \ce{F2}, ethylene, \ce{N2}, \ce{H4}, and \ce{H8}.
|
|
|
|
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
%\section*{Data availability statement}
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|
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
%The data that support the findings of this study are openly available in Zenodo at \href{http://doi.org/XX.XXXX/zenodo.XXXXXXX}{http://doi.org/XX.XXXX/zenodo.XXXXXXX}.
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|
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|
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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|
\bibliography{seniority}
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|
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\end{document}
|