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.
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.
the additional computational burden related to orbital optimization usually do not compensate the marginal improvements compared with results obtained with Hartree-Fock orbitals.
The exception is orbital-optimized CI with single excitations, a minimally correlated model displaying the qualitatively correct description of single bond breaking,
In electronic structure theory, configuration interaction (CI) methods allow for a systematic way to obtain approximate and exact solutions of the electronic Hamiltonian,
by expanding the wave function as a linear combination of Slater determinants (or configuration state functions). \cite{SzaboBook,Helgakerbook}
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.
In this context, one accounts for all determinants generated by exciting up to $e$ electrons from a given reference, which is usually the HF determinant, but does not have to.
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 $\Nbas^{2e}$.
Alternatively, seniority-based CI methods (sCI) have been proposed in both nuclear \cite{Ring_1980} and electronic \cite{Bytautas_2011} structure calculations.
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}
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}
When targeting static correlation, seniority-based CI methods tend to have a better performance than excitation-based CI, despite their higher computational cost.
\caption{Partitioning of the Hilbert space into blocks of specific excitation degree $e$ (with respect to a closed-shell determinant) and seniority number $s$.
This $e$-$s$ map is truncated differently in excitation-based CI (left), seniority-based CI (right), and hierarchy-based CI (center).
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.
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.
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),
at the same time as static correlation, by moving down (increasing the seniority number while keeping a low excitation degree).
%dynamic correlation is recovered with traditional CI.
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}).
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 $h$.
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.
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.
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.
hCI2.5 could represent an alternative, being more accurate than hCI2 and less expensive than hCI3.
For the latter two molecules, we have considered linearly arranged with equally spaced hydrogen atoms, and computed PECs along the symmetric dissociation coordinate.
For ethylene, we consider the \ce{C=C} double bond breaking, while freezing the remaining internal coordinates.
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.
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.
\textit{configuration interaction using a perturbative selection made iteratively} (CIPSI) algorithm, \cite{Huron_1973,Giner_2013,Giner_2015,Garniron_2018}
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 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.
However, the selected nature of the CIPSI algorithm means that the actual number of determinants is quite small and therefore calculations are technically feasable.}
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.
Namely, whenever the eigenvalue of the orbital rotation Hessian is negative and the corresponding gradient component $g_i$ lies below a given threshold $g_0$,
\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).
This protocol is 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.
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 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}).
\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).
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).
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.
This oscillatory behavior is particularly evident for \ce{F2}, also noticeable for \ce{HF}, becoming less apparent for ethylene, virtually absent for \ce{N2},
\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).
\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).
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).
\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.
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.
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.
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.
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.
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 on each fragment, such that multiple bond breaking become insufficiently described.
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.
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.
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.
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.
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.
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),
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.
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}
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).
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),
%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}.
