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?}
The key appealing feature of hCI is that each hierarchy level accounts for all classes of determinants that share the same scaling with system size.
%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.
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.
%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.
%The overall performance of hCI usually exceeds or at least parallels that of excitation-based CI.
%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.
%However, for larger systems or basis sets, and for higher accuracy, seniority-based CI becomes impractical.
%However, some of its interesting features, particularly the small non-parallelity errors, are partially recovered with hCI, at only a polynomial cost.
%We have further explored the role of optimizing the orbitals at several levels of 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 or exact solutions of the electronic Hamiltonian,
by expanding the wave function as a linear combination of Slater determinants (or configuration state functions).
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-particle basis set.
that quickly recover the correlation energy, understood as the energy difference between the FCI and the mean-field restricted Hartree-Fock (HF) solutions.
where one accounts for all determinants generated by exciting up to $e$ electrons from a given close-shell reference, which is usually the restricted HF solution, but does not have to.
In this way, the excitation degree $e$ parameter defines the sequence
Importantly, the number of determinants $N_{det}$ (which is the key parameter governing the computational cost) scales polynomially with the number of electrons $N$ as $N^{2d}$.
By truncating at the seniority zero ($s =0$) sector, one obtains the doubly-occupied CI (DOCI) method \cite{Bytautas_2011,Allen_1962,Smith_1965,Veillard_1967},
However, already at the sCI0 level, $N_{det}$ scales exponentially with $N$, since excitations of all excitation degrees $e$ are included.
Therefore, despite the encouraging successes of seniority-based CI methods, their unfavourable computational scaling restricts applications to very small systems.
Besides CI, other methods that exploit the concept of seniority number have been pursued. \cite{Henderson_2014,Chen_2015,Bytautas_2018}
When targeting static correlation, seniority-based CI methods tend to have a better performance than excitation-based CI, despite the higher computational cost.
The latter class of methods, in contrast, are well-suited for recovering dynamic correlation, and only at polynomial cost with system size.
Ideally, we aim for a method that captures most of both static and dynamic correlation, with as few determinants as possible.
\caption{Partitioning of the full 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 (top left), seniority-based CI (top right), and hierarchy-CI (bottom).
The color tones represent the determinants that are included at a given level of CI.}
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.~\ref{eq:h}, we ensure that both directions in the excitation-seniority map (see Fig.~\ref{fig:allCI}) will be contemplated.
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.
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 $N$, for all excitation-seniority sectors entering at a given hierarchy $h$.
Bytautas 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 sharing the same scaling with system size, as discussed before, thus keeping the method at a polynomial scaling.
Each level of excitation-based CI has a hCI counterpart with the same scaling of $N_{det}$ with respect to $N$.
For example, $N_{det}\sim N^4$ in both hCI2 and CISD, whereas $N_{det}\sim N^6$ in hCI3 and CISDT, and so on.
From this computational perspective, hCI can be seen as a more natural choice than the traditional excitation-based CI,
because if one can afford for, say, a CISDT calculation, than one could probably afford a hCI3 calculation, which has the same computational scaling.
Of course, in practice an integer-$h$ hCI method will have more determinants than its excitation-based counterpart (despite the same scaling),
and thus one should first ensure whether including the lower-triangular blocks (going from CISDT to hCI3 in our example)
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 $N_{det}$, rather than the formal scaling of $N_{det}$,
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).
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,
the hCI1 counterpart already represents a minimally correlated model, with the same and favourable $N_{det}\sim N^2$ scaling.
hCI also allows for half-integer values of $h$, with no parallel in excitation-based CI.
This gives extra flexibility in terms of choice of 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.
being often considered when assessing novel methodologies.
We 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.
From the PECs, we have also extracted the vibrational frequencies and equilibrium geometries (details can be found in the \SI).
The excitation-based CI, seniority-based CI, and FCI calculations presented here were also performed with the CIPSI algorithm implemented in {\QP}. \cite{Huron_1973,Giner_2013,Giner_2015,Garniron_2019}
In practice, we consider the CI energy to be converged when the second-order perturbation correction lies below $10^{-5}$ Hartree,
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).
Nevertheless, we decided to present the results as functions of the formal number of determinants,
which are not related to the particular algorithmic choices of the CIPSI calculations.
All CI calculations were performed for the cc-pVDZ basis set and with frozen core orbitals.
The CI calculations were performed with both canonical Hartree-Fock (HF) orbitals and optimized orbitals.
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.
We employed the algorithm described elsewhere \cite{Damour_2021} and also implemented in {\QP} for optimizing the orbitals within a CI wave function.
In order to avoid converging to a saddle point solution, we employed a similar strategy as recently described in Ref. \cite{Hollett_2022}.
Namely, whenever the eigenvalue of the orbital rotation Hessian is negative and the corresponding gradient component $g_i$ lies below a given threshold $g_0$,
then this gradient component is replaced by $g_0 |g_i|/g_i$.
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).
the optimized orbitals were employed as the guess orbitals for the neighbouring geometries, and so on, until a new PEC is obtained.
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 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.
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 $N_{det}$ 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}).
For \ce{HF} we also evaluated the convergence is affected by increasing the basis sets, going from cc-pVDZ to cc-pVTZ and cc-pVQZ basis sets (see Fig.Sx in the \SI).
While a larger $N_{det}$ is required to achieve the same level of convergence, as expected,
\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 oscillatory behavior is particularly evident for \ce{F2}, also noticeable for \ce{HF}, becoming less apparent for ethylene, virtually absent for \ce{N2},
Results for \ce{HF} with larger basis sets (see Fig.Sx in the \SI) show very similar convergence behaviours, though with less oscillations for the hCI methods.
\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).
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.
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.
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 \SI).
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 \SI.
At this level, the orbital rotations provide an optimized reference (different from the HF solution), from which only single excitations are performed.
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.
the hCI method ensures that all classes of determinants sharing 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,
in the sense of convergence with respect to $N_{det}$.
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 $N_{det}$, 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 favourable 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 favourable 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),
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.
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).
%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}.
