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.
Except for very small systems, the FCI limit is unnatainable, and in practice the expansion of the CI wave function must be trunctated.
The question is then how to construct an effective and computationally tractable hierarchy of truncated CI methods
that best recover the correlation energy, understood as the energy difference between the FCI and the mean-field restricted Hartree-Fock (HF) solutions.
%that lead as fast as possible to the FCI limit.
The most well-known and popular class of CI methods is excitation-based,
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 control the computational cost) scale polynomially with the number of electrons $N$ as $N^{2d}$.
%This means that the contribution of higher excitations become progressively smaller.
%In turn, seniority-based CI is specially targeted to describe static correlation.
In short, the seniority number $s$ is the number of unpaired electrons in a given determinant.
The seniority zero ($s =0$) sector has been shown to be the most important for static correlation, while higher sectors tend to contribute progressively less ~\cite{Bytautas_2011,Bytautas_2015,Alcoba_2014b,Alcoba_2014}.
However, already at the sCI0 level the number of determinants scale exponentially with $N$, since excitations of all excitation degrees $e$ are included.
Therefore, despite the encouraging successes of seniority-based CI methods, their unfourable computational scaling restricts applications to very small systems.
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.
% tackling
Ideally, we aim for a method that captures most of both static and dynamic correlation, with as few determinants as possible.
\caption{Partionining of the full Hilbert space into blocks of specific excitation degree $e$ (with respect to a closed-shell determinant) and seniority number $s$.
Each of three classes of CI methods truncate this $e$-$s$ map differently, and each color tone represents the added determinants at a given CI level.}
We know that low degree excitations and low seniority sectors, when looked at individually, often have 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.
The second justification is computational.
%Fig.~\ref{fig:scaling} also illustrates how the number of determinants within each block scales with the number of occupied orbitals $O$ and the number of virtual orbitals $V$.
In the hCI class of methods, each next level of theory accomodates additional determinants from different excitation-seniority sectors (each block of Fig.~\ref{fig:allCI}).
The key realization 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 $h$.
%to $O$ and $V$, for all excitation-seniority sectors of a given hierarchy $h$.
Each level of excitation-based CI has a hCI counterpart with the same scaling of $N_{det}$ with respect to $N$.
%However, hCI counts with additional half-integer levels of theory, with no parallel in excitation-based CI.
For example, in both hCI2 and CISD we have $N_{det}\sim N^4$, whereas in hCI3 and CISDT, $N_{det}\sim N^6$, and so on.
%the number of determinants of hCI2 and CISD scale as $O^2V^2$, those of hCI3 and CISDT scale as $O^3V^3$, 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, the $N_{det}\sim N^6$ cost of a CISDT calculation, than one can probably afford a hCI3 calculation, which has the same computational scaling.
This gives extra flexibility in terms of choice of method.
%when evaluating the computational cost and desired accuracy of a calculation.
For a particular application with excitation-based CI, CISD might be too inaccurate, for example, while the price for the improved accuracy of CISDT might be too high.
Bytautas et al.\cite{Bytautas_2015} explored a different hybrid scheme combining determinants from a complete active space and with a maximum seniority number.
And second, 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.
The hCI method was implemented in {\QP} via a straightforward adaptation of the \textit{configuration interaction using a perturbative selection made iteratively} (CIPSI) algorithm \cite{Huron_1973,Giner_2013,Giner_2015},
by allowing only for determinants at a given hierarchy $h$.
In practice, the CI energy is converged (within a chosen threshold of) with considerably fewer determinants than the formal number of determinants at a given $h$.
The traditional excitation-based CI and the FCI calculations presented here were also performed with the CIPSI algorithm implemented in {\QP}. \cite{Huron_1973,Giner_2013,Giner_2015,Garniron_2019}
The CI calculations were performed with both canonical Hartree-Fock (HF) orbitals and optimized orbitals (oo).
In the latter case, the energy is obtained variationally in both the CI space and in the orbital parameter space, given rise to what may be called an orbital-optimized CI (ooCI) method.
%We have also performed orbital optimized CI (ooCI) calculations, where the energy is obtained variationally both in the CI space and in the orbital parameter space.
%
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 can never ensure that the obtained solutions are global minima in the orbital parameter space, we verified that in all cases surveyed here, the stationary solutions are real minima (rather than maxima or stationary points).
First, the orbital optimization started from the HF orbitals of each geometry.
This usually lead to discontinuous PECs, meaning that distinct solutions of the orbital optimization have been found with our algorithm.
Then, at some geometry or geometries that seem to present the lowest lying solution,
the optimized orbitals were employed as the guess orbitals for the neighbouring geometries, and so on, until a new PEC is attained.
%orthonormalized
This protocol is repeated until the PEC built from the lowest lying orbital optimized solutions 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.
%Multiple solutions for the orbital optimization are usually found, meaning several local minimal in the orbital parameter landscape.
%meaning that the set of orbitals are stationary with respect to the energy.
We recall that saddle point solutions were purposedly avoided in our orbital optimization algorithm. If that was not the case, then even more stationary solutions would have been found.
%\subsection{Nonparallelity errors and dissociation energies}
\subsection{Nonparallelity errors}
In Fig.~\ref{fig:plot_stat} we present, for the six systems studied, and for the three classes of CI methods,
the nonparalellity error (NPE) as function of the formal number of determinants.
%the potential energy curves (PECs) and the corresponding differences with respect to the FCI result, as well as the nonparalellity error (NPE) and the distance error.
The NPE is defined as the maximum minus the minimum differences between the potential energy curves (PECs) obtained at given CI level and the exact FCI result.
This is an important metric because it captures the resemblance between the shape of the two PECs, which in turn determine the relevant physical observables, as equilibrium geometries, vibrational frequencies, and dissociation energies.
The corresponding PECs and the energy differences with respect to the FCI results can be found in the SI.
%
In the SI we further present the distance error, defined as the sum of the maximum and 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 magnitudes compare.
\caption{Nonparallelity 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).
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 both for single bond breaking (\ce{HF} and \ce{F2}) as well as the more challenging double (ethylene) and triple (\ce{N2}) bond breaking.
The convergence with respect to the number of determinants is slower in the latter cases, irrespective of the class of CI methods, as would be expected.
For the four systems (more so for ethylene and \ce{N2}), hCI2 is better than CISD, two methods where the number of determinants scales as $N^4$.
hCI2.5 is better than CISDT, despite its lower computational cost, whereas hCI3 is much better than CISDT, and comparable in accuracy with CISDTQ.
Inspection of the PECs (see SI) reveal that the lower NPE in the hCI results 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 with the number of determinants).
Meanwhile, the first level of seniority-based CI (sCI0, which is the same as doubly-occupied CI) tends to offer a rather low NPE when compare to the other CI methods with a similar number of determinants (hCI2.5 and CISDT).
However, convergence is clearly slower for the next levels (sCI2 and sCI4), while excitation-based CI and specially hCI methods converge faster.
For the symmetric dissociation of linear \ce{H4} and \ce{H8} the performance of hCI and excitation-based CI are similar, both being superior to seniority-based CI.
It seems that both the relative success of hCI1 and hCI1.5 methods as well as the relative worsening of the hCI2 method decrease as progressively more bonds are being broken (compare for instance \ce{F2}, \ce{N2}, and \ce{H8} in Fig.~\ref{fig:plot_stat}).
For \ce{F2}, the hCI method has an overall better convergence than the excitation-based CI counterpart, and much better than seniority-based CI.
The values oscillate around the FCI limit in hCI, whereas the convergence is monotonic in the two CI alternatives.
Interstingly, hCI1 and specially hCI1.5, two methods with a modest computational cost, provide very accurate equilibrium geometries and vibrational frequencies,
hCI and excitation-based CI are comparable to each other and superior to seniority-based CI, at least for HF orbitals.
Orbital optimization significantly improves the case for seniority-based CI, and leads to slightly better convergence for hCI with respect to excitation-based CI.
\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).
Here we have proposed a new scheme for truncating the Hilbert space in configuration interaction calculations, named hCI.
By merging the excitation degree and the seniority number into a single hierarchy parameter,
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,
by comparing PECs for dissociation of 6 systems, going from single to multiple bond breaking.
Our key finding is that the overall performance of hCI either surpasses or equals those of excitation-based CI and seniority-based CI.
The superiority of hCI methods is more noticeable for the nonparallelity errors, but and also be seen for the equilibrium geometries and vibrational frequencies.
For the challenging cases of \ce{H4} and \ce{H8}, hCI and excitation-based CI perform similarly.
We also found surprisingly good performances for the first level of hCI (hCI1) and the orbital optimized version of CIS (ooCIS),
given their very favourable computational scaling.
While optimization the orbitals will certainly improve the energy at a particular geometry, such improvement may vary largely along the PEC, which may or may not decrease the NPE.
One should also bear in mind that the orbital optimization 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, 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 cheaper alternative than optimizing the orbitals.
One interesting possibility to explore is to first employ a low order CI method to optimize the orbitals, 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}.
