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H8_cc-pvdz/pes_ooCIo2.5_1.dat
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Manuscript/seniority.bib
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@ -245,3 +245,40 @@ volume = {141},
year = {2014}
}
@article{Allen_1962,
author = {Allen, Thomas L. and Shull, Harrison},
doi = {10.1021/j100818a001},
journal = {J. Phys. Chem.},
pages = {2281--2283},
publisher = {{Univ. of California, Davis}},
title = {Electron {{Pairs}} in the {{Beryllium Atom}}},
volume = {66},
year = {1962},
Bdsk-Url-1 = {https://doi.org/10.1021/j100818a001}}
@article{Smith_1965,
author = {Smith, Darwin W. and Fogel, Sidney J.},
doi = {10.1063/1.1701519},
journal = {J. Chem. Phys.},
pages = {S91-S96},
publisher = {{American Institute of Physics}},
title = {Natural {{Orbitals}} and {{Geminals}} of the {{Beryllium Atom}}},
volume = {43},
year = {1965},
Bdsk-Url-1 = {https://doi.org/10.1063/1.1701519}}
@article{Veillard_1967,
author = {Veillard, A. and Clementi, E.},
doi = {10.1007/BF01151915},
file = {/home/antoinem/Zotero/storage/QINAFG45/Veillard and Clementi - 1967 - Complete multi-configuration self-consistent field.pdf},
journal = {Theoret. Chim. Acta},
pages = {133--143},
title = {Complete Multi-Configuration Self-Consistent Field Theory},
volume = {7},
year = {1967},
Bdsk-Url-1 = {https://doi.org/10.1007/BF01151915}}

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Manuscript/seniority.tex
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% Abstract
\begin{abstract}
Here comes the abstract.
%aimed at recovering both static and dynamic correlation,
Here we propose a novel partitioning of the Hilbert space, hierarchy configuration interaction (hCI),
where the degree of excitation (with respect to a given reference) and the seniority number (number of unpaired electrons) are combined in a single hierarchy parameter.
The key appealing feature of hCI is that it includes all classes of determinants that share the same scaling with the number of electrons and basis functions.
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 examined how fast hCI and their excitation-based and seniority-based parents converge as
we step up towards the exact full CI limit.
We found that 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.
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 polynomical cost.
We have futher explored the role of optimizing the orbitals at several levels of CI.
For higher orders of hCI and excitation-based CI,
the additional computational burden and other known issues related to orbital optimization usually do not compensate the marginal improvements often observed,
when compared with results obtained with canonical 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,
at a very modest computational cost.
%\bigskip
%\begin{center}
% \boxed{\includegraphics[width=0.4\linewidth]{TOC}}
@ -81,8 +97,8 @@ Here comes the abstract.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Introduction}
\label{sec:intro}
%\section{Introduction}
%\label{sec:intro}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
In electronic structure theory, configuration interaction (CI) methods allow for a systematic way to obtain approximate or exact solutions of the electronic Hamiltonian,
@ -98,14 +114,16 @@ where one accounts for all determinants generated by exciting up to $e$ electron
In this way, the excitation degree $e$ parameter defines the sequence
CI with single excitations (CIS), CI with single and double excitations (CISD), CI with single, double, and triple excitations (CISDT), and so on.
Excitation-based CI manages to quickly recover weak (dynamic) correlation effects, but struggles at strong (static) correlation regimes.
Importantly, the number of determinants $N_{det}$ (which control the computational cost) scale polynomially with the number of electrons $N$ as $N^{2d}$.
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 $N^{2d}$.
%This means that the contribution of higher excitations become progressively smaller.
%In turn, seniority-based CI is specially targeted to describe static correlation.
%\fk{Still have to work in this paragraph.}
Alternatively, CI methods based on the seniority number \cite{Ring_1980} have been proposed \cite{Bytautas_2011}.
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}.
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},
which 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}.
% scaling
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.
@ -116,8 +134,8 @@ Besides CI, other methods that exploit the concpet of seniority number have been
%https://doi.org/10.1016/j.comptc.2018.08.011
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Hierarchy configuration interaction}
\label{sec:hCI}
%\section{Hierarchy configuration interaction}
%\label{sec:hCI}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
At this point, we notice the current dicothomy.
@ -224,7 +242,7 @@ And second, each next level includes all classes of determinants sharing the sam
Our main goal here is to assess the performance of hCI against excitation-based and seniority-based CI.
To do so, we evaluated how fast different observables converge to the FCI limit as a function of the number of determinants.
We have calculated the potential energy curves (PECs) for a total of 6 systems,
We have calculated the potential energy curves (PECs) along the dissociation of six systems,
\ce{HF}, \ce{F2}, \ce{N2},
%\ce{Be2}, \ce{H2O},
ethylene, \ce{H4}, and \ce{H8},
@ -236,17 +254,17 @@ For ethylene, we considered the C$=$C double bond stretching, while freezing the
Due to the (multiple) bond breaking, these are challenging systems for electronic structure methods,
and are often considered when assessing novel methodologies.
%
From the PECs, we evaluated the convergence of four observables: the nonparalellity error (NPE), the distance error, the equilibrium geometries, and the vibrational frequencies.
We evaluated the convergence of four observables: the nonparalellity error (NPE), the distance error, the equilibrium geometries, and the vibrational frequencies.
The NPE is defined as the maximum minus the minimum differences between the PECs obtained at given CI level and the exact FCI result.
We define the distance error as the the maximum plus and the minimum differences between a given PEC and the 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.
Thus, while the NPE probes the similarity regarding the shape of the PECs, the distance error provides a measure of how their magnitudes compare.
From the PECs, we have also extrated the equilibrium geometries and vibrational frequencies (details can be found in the SI).
From the PECs, we have also extrated the equilibrium geometries and vibrational frequencies (details can be found in the \SI).
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Computational details}
\label{sec:compdet}
%\section{Computational details}
%\label{sec:compdet}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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},
@ -258,7 +276,7 @@ Nevertheless, we decided to present the results as functions of the formal numbe
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.
For \ce{HF} we have also considered the cc-pVTZ and cc-pVQZ basis sets.
For \ce{HF} we have also tested basis set effects, by considered the cc-pVTZ and cc-pVQZ basis sets.
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 (oo-CI) method.
@ -283,31 +301,23 @@ We recall that saddle point solutions were purposedly avoided in our orbital opt
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Results and discussion}
\label{sec:res}
%\section{Results and discussion}
%\label{sec:res}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\subsection{Correlation energies}
%\ce{F2}
%\ce{N2}
%\ce{HF}
%ethylene
%Linear \ce{H4} and \ce{H8}
%\ce{Be2}
%\subsection{Potential energy curves}
%\subsection{Nonparallelity errors and dissociation energies}
\subsection{Nonparallelity errors}
%\subsection{Nonparallelity errors}
In Fig.~\ref{fig:plot_stat} we present the NPEs for the six systems studied, and for the three classes of CI methods,
as functions of the 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 corresponding PECs and the energy differences with respect to the FCI results can be found in the SI.
The corresponding PECs and the energy differences with respect to the FCI results can be found in the \SI.
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), triple (\ce{N2}), and quadruple (\ce{H4}) bond breaking.
For (\ce{H8}), hCI and excitation-based CI perform similarly.
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 the number of determinants 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}).
@ -333,7 +343,7 @@ We thus believe that the main findings discussed here for the other systems woul
%Unless stated otherwise, from here on the performance of each method is probed by their NPE. Later we discuss other metrics.
For all systems (specially 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 (except for \ce{H8}), despite its lower computational cost, whereas hCI3 is much better than CISDT, and comparable in accuracy with CISDTQ (again for all systems).
Inspection of the PECs (see SI) reveal that the lower NPE in the hCI results stem mostly from the contribution of the dissociation region.
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).
%The situation at the Franck-Condon region will be discussed later.
@ -341,7 +351,7 @@ which are accounted for in hCI but not in excitation-based CI (for a given scali
Meanwhile, the first level of seniority-based CI (sCI0, which is the same as doubly-occupied CI\cite{}) tends to offer a rather low NPE when compared 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), whereas excitation-based CI and specially hCI methods converge faster.
Furthermore, seniority-based CI becomes less atractive for larger basis set in view of its exponential scaling.
This can be seen in Fig.Sx of the SI, which shows that seniority-based CI has a much more pronounced increase in the number of determinants for larger basis sets.
This can be seen in Fig.Sx of the \SI, which shows that seniority-based CI has a much more pronounced increase in the number of determinants for larger basis sets.
It is worth mentioning the surprisingly good performance of the hCI1 and hCI1.5 methods.
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}.
@ -353,19 +363,17 @@ become less apparent as progressively more bonds are being broken (compare for i
This reflects the fact that higher-order excitations are needed to properly describe multiple bond breaking.
%Whereas in excitation-based CI, the NPE always decrease as one moves to higher orders.
In Fig.Sx of the SI we present the distance error, which is also found to decrease faster with the hCI methods.
In Fig.Sx of the \SI we present the distance error, which is also found to decrease faster with the hCI methods.
Most of observations discussed for the NPE also hold for the distance error, with two main differences.
The convergence is always monotonic for the latter observable (which is expected from the definition of the observable),
and the performance of seniority-based CI is much poorer (due to the slow recovery of dynamic correlation).
\subsection{Equilibrium geometries and vibrational frequencies}
%\subsection{Equilibrium geometries and vibrational frequencies}
In Fig.~\ref{fig:xe} and \ref{fig:freq}, we present the convergence of the equilibrium geometries and vibrational frequencies, respectively,
as functions of the number of determinants, for the three classes of CI methods.
%, vibrational frequencies, and dissociation energies,
%For \ce{HF}, \ce{F2}, \ce{N2}, and ethylene,
%For both observables, the overall performance of hCI either exceeds or is comparable to that of excitation-based CI,
For the equilibrium geometries, hCI performs slightly better overall than excitation-based CI.
%, being better for \ce{F2}, ethylene, and \ce{N2}, and
A more significant advantage of hCI can be seen for the vibrational frequencies.
@ -374,6 +382,7 @@ For both observables, hCI and excitation-based CI largely outperform seniority-b
Similarly to what we observed for the NPEs, the convergence of hCI was also found to be non-monotonic in some cases.
This oscillatory behavior is particularly evident for \ce{F2}, also noticeable for \ce{HF}, becoming less aparent for ethylene, virtually absent for \ce{N2},
and showing up again for \ce{H4} and \ce{H8}.
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.
Interstingly, equilibrium geometries and vibrational frequencies of \ce{HF} and \ce{F2} (single bond breaking),
are rather accurate when evaluated at the hCI1.5 level, bearing in mind its relatively modest computational cost.
@ -395,14 +404,13 @@ are rather accurate when evaluated at the hCI1.5 level, bearing in mind its rela
\end{figure}
%%% %%% %%%
\subsection{Orbital optimized configuration interaction}
%\subsection{Orbital optimized configuration interaction}
Up to this point, all results and discussions have been based on CI calculations with HF orbitals.
Now we discuss the role of further optimizing the orbitals at each given CI calculation.
Due to the significantly higher computational cost and numerical difficulties for optimizing the orbitals at higher levels of CI,
such calculations were typically limited up to oo-CISD (for excitation-based), oo-DOCI (for seniority-based), and oo-hCI2 (for hCI).
The PECs and analogous results to those of Figs. \ref{fig:plot_stat}, \ref{fig:xe}, and ~\ref{fig:freq} are shown in the SI.
The PECs and analogous results to those of Figs. \ref{fig:plot_stat}, \ref{fig:xe}, and ~\ref{fig:freq} are shown in the \SI.
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.
@ -410,46 +418,88 @@ More often than not, the NPEs do decrease upon orbital optimization, though not
%For example, oo-hCI1 presents smaller NPEs than their non-optimized counterparts for \ce{N2}, \ce{H4}, \ce{H8}, but not for \ce{HF}, \ce{F2}, and ethylene.
For example, compared with their non-optimized counterparts, oo-hCI1 and oo-hCI1.5 provide somewhat larger NPEs for \ce{HF} and \ce{F2},
% oo-hCI2
comparable for ethylene, and smaller for \ce{N2}, \ce{H4}, and \ce{H8}.
similar NPEs for ethylene, and smaller NPEs for \ce{N2}, \ce{H4}, and \ce{H8}.
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}. We will come back to oo-CIS latter.
oo-CIS has significantly smaller NPEs than HF-CIS, being comparable to oo-hCI1 for all systems except for \ce{H4} and \ce{H8}.
We will come back to oo-CIS latter.
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.
%Also, hCI is slightly better than excitation-based CI for HF orbitals, whereas both are equally good with orbital optimization.
Orbital optimization also reduces the NPE for seniority-based CI (in this case we only considered oo-DOCI).
The gain is specially noticeable for \ce{H4} and \ce{H8}, and much less so for \ce{HF}, ethylene, and \ce{N2}.
For \ce{F2}, we found the interesting situation where orbital optimization actually increases the NPE (though by a small amount).
\fk{in progress...}
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 the dissociating region.
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.
Since the reference is not the HF one, Briluoin's theorem no longer holds, and single excitations actually connect with the reference.
Thus, with only single excitations (and a referece 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.
% HF: 40%, F2: 30%, et: 20%, N2: 30%, H4: 30%, H8: 15%
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 dissociaion.
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, thus leading to two singly open-shell radicals.
This is enough to obtain a qualitatively correct description of single bond breaking, hence the relatively low NPEs observed for \ce{HF} and \ce{F2}.
In constrast, 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.
%For \ce{F2}, for instance,
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Conclusion and perspectives}
\label{sec:ccl}
%\section{Conclusion and perspectives}
%\label{sec:ccl}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\fk{in progress...}
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,
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 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.
by comparing PECs and derivied 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 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 (oo-CIS),
given their very favourable computational scaling.
Our key finding is that the overall performance of hCI either surpasses or equals that of excitation-based CI,
in the sense that convergence with respect to the number of determinants is usually faster.
The superiority of hCI methods is more noticeable for the nonparallelity 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 number of determinants, 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.
One important conclusion is that orbital optimization is not necessarily a recommended strategy, depending on the properties one is interested in.
One should bear in mind that the 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, Hessian, and the many iterations that are often required).
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.
%We have also performed orbital optimization at several CI levels,
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, 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 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.
One interesting possibility to explore is to first optimize the orbital at a lower level of CI, and then to employ this set of orbitals at a higher level of CI.
The hCI pathways presented here offers several interesting possibilities to explore.
One is to investigate the performance of hCI or some adaptation of it for excited states.
Another is to develop coupled cluster methods based on an analogous hybrid excitation-seniority truncation of the excitation operator.
One could also test the performance of hCI wave functions for Quantum Monte Carlo simulations.
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,
and explore hCI wave functions for Quantum Monte Carlo simulations.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{acknowledgements}
@ -462,7 +512,7 @@ This project has received funding from the European Research Council (ERC) under
\section*{Supporting information available}
\label{sec:SI}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
PECs, energy differences with respect to FCI results, NPE, closeness errors, equilibrium geometries, vibrational frequencies,
PECs, energy differences with respect to FCI results, NPE, distance errors, vibrational frequencies, and equilibrium geometries,
according to the three classes of CI methods (excitation-based CI, seniority-based CI, and hierarchy-CI),
computed for \ce{HF}, \ce{F2}, ethylene, \ce{N2}, \ce{H4}, and \ce{H8}.

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Manuscript/table_exc_full.pdf
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Manuscript/table_hCI.pdf
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Manuscript/table_sen_full.pdf
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2
ethylene_cc-pvdz/pes_ooCIo2.dat
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@ -52,4 +52,4 @@
13.0 -77.90419730
14.0 -77.90387539
15.0 -77.90359805
16.0 -77.90336366
16.0 -78.00927763

18
plot_all/plot_pes.gnu
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@ -50,9 +50,10 @@ set ytics 0.2
ymin=-100.25
ymax=-99.85
plot '../HF_cc-pvdz/pes_rhf.dat' w l ls 1 notitle, \
'../HF_cc-pvdz/pes_CIo1.dat' w l ls 4 notitle, \
'../HF_cc-pvdz/pes_ooCIS.dat' w l ls 3 notitle, \
'../HF_cc-pvdz/pes_fci.dat' w l ls 2 notitle
# '../HF_cc-pvdz/pes_CISD.dat' w l ls 3 notitle, \
# '../HF_cc-pvdz/pes_CIo1.dat' w l ls 4 notitle, \
unset ylabel
unset label
@ -68,9 +69,10 @@ set ytics 0.2
ymin=-199.11
ymax=-198.6
plot '../F2_cc-pvdz/pes_rhf.dat' w l ls 1 notitle, \
'../F2_cc-pvdz/pes_CIo1.dat' w l ls 4 notitle, \
'../F2_cc-pvdz/pes_ooCIS.dat' w l ls 3 notitle, \
'../F2_cc-pvdz/pes_fci.dat' w l ls 2 notitle
# '../F2_cc-pvdz/pes_CISD.dat' w l ls 3 notitle, \
# '../F2_cc-pvdz/pes_CIo1.dat' w l ls 4 notitle, \
set xrange[1.5:16.0]
xmin=1.5
@ -83,9 +85,10 @@ ymin=-78.40
ymax=-77.7
set ylabel 'Energy (Hartree)'
plot '../ethylene_cc-pvdz/pes_rhf.dat' w l ls 1 notitle, \
'../ethylene_cc-pvdz/pes_CIo1.dat' w l ls 4 notitle, \
'../ethylene_cc-pvdz/pes_ooCIS.dat' w l ls 3 notitle, \
'../ethylene_cc-pvdz/pes_fci.dat' w l ls 2 notitle
# '../ethylene_cc-pvdz/pes_CISD.dat' w l ls 3 notitle, \
# '../ethylene_cc-pvdz/pes_CIo1.dat' w l ls 4 notitle, \
unset ylabel
set xrange[0.7:4.0]
@ -100,9 +103,10 @@ set ytics 0.4
ymin=-109.30
ymax=-108.30
plot '../N2_cc-pvdz/pes_rhf.dat' w l ls 1 notitle, \
'../N2_cc-pvdz/pes_CIo1.dat' w l ls 4 notitle, \
'../N2_cc-pvdz/pes_ooCIS.dat' w l ls 3 notitle, \
'../N2_cc-pvdz/pes_fci.dat' w l ls 2 notitle
# '../N2_cc-pvdz/pes_CISD.dat' w l ls 3 notitle, \
# '../N2_cc-pvdz/pes_CIo1.dat' w l ls 4 notitle, \
set xrange[1.0:10.0]
xmin=1.0
@ -114,9 +118,10 @@ set ytics 0.2
ymin=-2.3
ymax=-1.7
plot '../H4_cc-pvdz/pes_rhf.dat' w l ls 1 notitle, \
'../H4_cc-pvdz/pes_CIo1.dat' w l ls 4 notitle, \
'../H4_cc-pvdz/pes_ooCIS.dat' w l ls 3 notitle, \
'../H4_cc-pvdz/pes_fci.dat' w l ls 2 notitle
# '../H4_cc-pvdz/pes_CISD.dat' w l ls 3 notitle, \
# '../H4_cc-pvdz/pes_CIo1.dat' w l ls 4 notitle, \
set xrange[1.0:10.0]
xmin=1.0
@ -127,6 +132,7 @@ set ytics 0.4
ymin=-4.6
ymax=-3.0
plot '../H8_cc-pvdz/pes_rhf.dat' w l ls 1 notitle, \
'../H8_cc-pvdz/pes_CIo1.dat' w l ls 4 notitle, \
'../H8_cc-pvdz/pes_ooCIS.dat' w l ls 3 notitle, \
'../H8_cc-pvdz/pes_fci.dat' w l ls 2 notitle
# '../H8_cc-pvdz/pes_CISD.dat' w l ls 3 notitle, \
# '../H8_cc-pvdz/pes_CIo1.dat' w l ls 4 notitle, \