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HF_cc-pvqz/pes_CIo3.dat
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Manuscript/seniority.tex
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@ -261,8 +261,8 @@ All CI calculations were performed for the cc-pVDZ basis set and with frozen cor
For \ce{HF} we have also 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 (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.
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.
%We have also performed orbital optimized CI (oo-CI) 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 $g_0$,
@ -406,14 +406,19 @@ The PECs and analogous results to those of Figs. \ref{fig:plot_stat}, \ref{fig:x
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.
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.
Orbital optimization does not change the overall picture.
It does, however, lead to a more monotonic convergence in the case of hCI, which is not necessarily an advantage.
In the case of \ce{N2}, hCI and excitation-based CI present similar convergence behaviours, both being superior to seniority-based CI.
Also, hCI is slightly better than excitation-based CI for HF orbitals, whereas both are equally good with orbital optimization.
%the advantages of hCI are less evident, though stil present.
Orbital optimization significantly improves the case for seniority-based CI, and leads to slightly better convergence for hCI with respect to excitation-based CI.
More often than not, the NPEs do decrease upon orbital optimization, though not always.
%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}.
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.
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...}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
@ -432,12 +437,11 @@ by comparing PECs for dissociation of 6 systems, going from single to multiple b
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),
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.
One important conclusion is that orbital optimization is not necessarily a recommended strategy, depending on the properties one is interested in.
%
One should also bear in mind that the orbital optimization is always accompanied with well-known challenges (several solutions, convergence issues)
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).
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.

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ethylene_cc-pvdz/pes_ooCIo2.dat
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