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HF_cc-pvqz/pes_CIo2.5.dat
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Manuscript/seniority.tex
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@ -222,6 +222,28 @@ In comparison to previous approaches, our hybrid hCI scheme has two key advantag
First, it is defined by a single parameter that unifies excitation degree and seniority number (eq.\ref{eq:h}).
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.
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,
\ce{HF}, \ce{F2}, \ce{N2},
%\ce{Be2}, \ce{H2O},
ethylene, \ce{H4}, and \ce{H8},
which display a variable number of bond breaking.
For the latter two, we considered linearly arranged and equally spaced hydrogen atoms, and computed PECs along the symmetric dissociation coordinates.
%For \ce{H2O}, we considered the symmetric stretching of the O$-$H bonds,
For ethylene, we considered the C$=$C double bond stretching, while freezing the remaining internal coordinates.
%in both cases freezing the remaining internal coordinates.
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.
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).
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Computational details}
\label{sec:compdet}
@ -229,30 +251,24 @@ And second, each next level includes all classes of determinants sharing the sam
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 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.
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.
%
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).
Here we assess the performance of the hCI methods against its excitation-based and seniority-based counterparts.
To do so, we calculated the potential energy curves (PECs) for a total of 6 systems:
\ce{HF}, \ce{F2}, \ce{N2},
%\ce{Be2}, \ce{H2O},
ethylene, \ce{H4}, and \ce{H8}.
For the latter two, we considered linearly arranged and equally spaced hydrogen atoms, and computed PECs along the symmetric dissociation coordinates.
%For \ce{H2O}, we considered the symmetric stretching of the O$-$H bonds,
For ethylene, we considered the C$=$C double bond stretching, while freezing the remaining internal coordinates.
%in both cases freezing the remaining internal coordinates.
All CI calculations were performed for the cc-pVDZ basis set and with frozen core orbitals.
From the PECs, we have also extrated the equilibrium geometries and vibrational frequencies (details can be found in the SI).
It is worth mentioning that obtaining smooth PECs for the orbital optimized calculations proved to be far from trivial.
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.
@ -285,16 +301,20 @@ We recall that saddle point solutions were purposedly avoided in our orbital opt
%\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.
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 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.
\fk{in progress...}
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.
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}).
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).
While a larger number of determinants is required to achieve the same level of convergence, as expected,
the convergence profiles remain very similar for all basis sets.
We thus believe that the main findings discussed here for the other systems would be equally basis set independent.
%%% FIG 2 %%%
\begin{figure}[h!]
@ -310,60 +330,52 @@ Thus, while the NPE probes the similarity regarding the shape of the PECs, the d
%For different CI approaches, Fig.~\ref{fig:N2_pes} shows PECs and their differences with respect to FCI, as well as the NPE and distance errors.
%The associated differences with respect to the FCI result can be seen in the Supporting Information.
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.
But more importantly, the superiority of the hCI methods appear to be highlighted in the multiple bond break systems.
%Unless stated otherwise, from here on the performance of each method is probed by their NPE. Later we discuss other metrics.
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.
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.
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).
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.
%
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.
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.
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}.
For the same systems we also see the NPEs increase from hCI1.5 to hCI2, and decreasing to lower values only at the hCI3 level.
(Even than, it is important to remember that the hCI2 results remain overall superior to their excitation-based counterparts.)
Both findings are not observed for \ce{H4} and \ce{H8}.
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}).
This is because
\fk{in progress...}
Even than, it is important to remember that even the hCI2 method remains superior to its excitation-based counterpart.
%Whereas in excitation-based CI, the NPE always decrease as one moves to higher orders,
It seems that both the relative worsening of the hCI2 method and success of hCI1 and hCI1.5 methods
become less apparent as progressively more bonds are being broken (compare for instance \ce{F2}, \ce{N2}, and \ce{H8} in Fig.~\ref{fig:plot_stat}).
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.
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}
In Fig.~\ref{fig:xe} and \ref{fig:freq}, we present the convergence of the equilibrium geometries and vibrational frequencies, respectively,
with respect to the number of determinants, for the three types of CI approaches.
as functions of the number of determinants, for the three classes of CI methods.
%, vibrational frequencies, and dissociation energies,
%
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,
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 particular, oohCI1 and oohCI1.5 are less accurate than their non-optimized counterparts.
%
For \ce{HF} (results in the Supporting Information),
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.
%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.
For both observables, hCI and excitation-based CI largely outperform seniority-based CI.
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.
%
%A somewhat better convergence is also observed in the case of ethylene (see SI).
The same conclusions hold for ethylene, \ce{H4}, and \ce{H8} (see SI).
Most of the times, the convergence of hCI either exceeds or is comparable to that of excitation-based CI.
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}.
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.
%%% FIG 3 %%%
\begin{figure}[h!]
@ -386,9 +398,22 @@ Most of the times, the convergence of hCI either exceeds or is comparable to tha
\subsection{Orbital optimized configuration interaction}
Up to this point, all results and discussions have been based on CI calculations for HF orbitals.
Now we discuss the role of further optimizing the orbitals for each given CI method.
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.
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.
\fk{in progress...}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
@ -411,7 +436,7 @@ We also found surprisingly good performances for the first level of hCI (hCI1) a
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.
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.
@ -431,7 +456,11 @@ 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,
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}.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\section*{Data availability statement}

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plot_all/plot_distance.gnu
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@ -45,7 +45,7 @@ set style line 14 dt 1 lw 1.5 linecolor rgb "sea-green" pt 7 ps 1.5
set style line 18 dt 1 lw 1.5 linecolor rgb "medium-blue" pt 7 ps 1.5
set label 1 'Number of determinants' at screen 0.40,0.03 tc ls 2 #font 'Verdana,20'
set label 2 'Distance error (Hartree)' at screen 0.03,0.35 rotate by 90 tc ls 2 #font 'Verdana,20'
#set label 2 'Distance error (Hartree)' at screen 0.03,0.35 rotate by 90 tc ls 2 #font 'Verdana,20'
set label 11 'HF' at screen 0.34,0.93 tc ls 2 font 'Helvetica,26'
set label 12 'F_2' at screen 0.79,0.93 tc ls 2 font 'Helvetica,26'
set label 13 'ethylene' at screen 0.34,0.62 tc ls 2 font 'Helvetica,26'
@ -69,7 +69,7 @@ unset ylabel
unset label
set xrange[1:1e10]
set yrange[0:1.10]
set yrange[0:1.20]
set ytics 0.2
nel=14
nel=1
@ -79,13 +79,15 @@ plot '../F2_cc-pvdz/stat_CI.dat' u ($3):($5/nel) w lp ls 3 notitle, \
set xrange[1:1e11]
#set xtics 10**3
set yrange[0:1.10]
set yrange[0:1.20]
set ytics 0.2
set ylabel 'Distance error (Hartree)'
nel=12
nel=1
plot '../ethylene_cc-pvdz/stat_CI.dat' u ($3):($5/nel) w lp ls 3 notitle, \
'../ethylene_cc-pvdz/stat_CIs.dat' u ($3):($5/nel) w lp ls 8 notitle, \
'../ethylene_cc-pvdz/stat_CIo.dat' u ($3):($5/nel) w lp ls 4 notitle
unset ylabel
set xrange[1:1e9]
set yrange[0:1.40]