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@ -75,18 +75,18 @@
\title{Reference correlation energies in finite Hilbert spaces: five- and six-membered rings}
\author{Yann Damour}
\author{Yann \surname{Damour}}
\affiliation{\LCPQ}
\author{Micka\"el V\'eril}
\author{Micka\"el \surname{V\'eril}}
\affiliation{\LCPQ}
\author{Michel Caffarel}
\author{Michel \surname{Caffarel}}
\affiliation{\LCPQ}
\author{Denis Jacquemin}
\author{Denis \surname{Jacquemin}}
\affiliation{\CEISAM}
\author{Anthony Scemama}
\author{Anthony \surname{Scemama}}
\email{scemama@irsamc.ups-tlse.fr}
\affiliation{\LCPQ}
\author{Pierre-Fran\c{c}ois Loos}
\author{Pierre-Fran\c{c}ois \surname{Loos}}
\email{loos@irsamc.ups-tlse.fr}
\affiliation{\LCPQ}
@ -97,7 +97,7 @@ This corresponds to Hilbert spaces with sizes ranging from $10^{29}$ to $10^{36}
Our estimates are based on energetically optimized-orbital selected configuration interaction (SCI) calculations performed with the \textit{Configuration Interaction using a Perturbative Selection made Iteratively} (CIPSI) algorithm.
The performance and convergence properties of several series of methods are investigated.
In particular, we study the convergence properties of i) the M{\o}ller-Plesset perturbation series up to fifth-order (MP2, MP3, MP4, and MP5), ii) the iterative approximate single-reference coupled-cluster series CC2, CC3, and CC4, and iii) the single-reference coupled-cluster series CCSD, CCSDT, and CCSDTQ.
The performance of the ground-state gold standard CCSD(T) as well as the completely renormalized (CR) CC model, CR-CC(2,3), are also investigated.
The performance of the ground-state gold standard CCSD(T) as well as the completely renormalized CC model, CR-CC(2,3), are also investigated.
\end{abstract}
% Title
@ -202,8 +202,7 @@ We then linearly extrapolate, using large variational wave functions, the CIPSI
Further details concerning the extrapolation procedure are provided below (see Sec.~\ref{sec:res}).
Orbital optimization techniques at the SCI level are theoretically straightforward, but practically challenging.
\titou{
Some of the technology presented here has been borrowed from complete-active-space self-consistent-field (CASSCF) methods \cite{Werner_1980,Werner_1985,Sun_2017,Kreplin_2019,Kreplin_2020} but one of the strength of SCI methods is that one does not need to select an active space and to classify orbitals as active, inactive, and virtual orbitals.}
Some of the technology presented here has been borrowed from complete-active-space self-consistent-field (CASSCF) methods \cite{Werner_1980,Werner_1985,Sun_2017,Kreplin_2019,Kreplin_2020} but one of the strength of SCI methods is that one does not need to select an active space and to classify orbitals as active, inactive, and virtual orbitals.
Here, we detail our orbital optimization procedure within the CIPSI algorithm and we assume that the variational wave function is normalized, \ie, $\braket*{\Psivar}{\Psivar} = 1$.
As stated in Sec.~\ref{sec:intro}, $\Evar$ depends on both the CI coefficients $\{ c_I \}_{1 \le I \le \Ndet}$ [see Eq.~\eqref{eq:Psivar}] but also on the orbital rotation parameters $\{\kappa_{pq}\}_{1 \le p,q \le \Norb}$.
@ -334,7 +333,7 @@ Using optimized orbitals has the undeniable advantage to produce, for a given va
For the benzene molecule, we also explore the use of localized orbitals (LOs) which are produced with the Boys-Foster localization procedure \cite{Boys_1960} that we apply to the natural orbitals in several orbital windows in order to preserve a strict $\sigma$-$\pi$ separation in the planar systems considered here. \cite{Loos_2020e}
Because they take advantage of the local character of electron correlation, localized orbitals have been shown to provide faster convergence towards the FCI limit compared to natural orbitals. \cite{Angeli_2003,Angeli_2009,BenAmor_2011,Suaud_2017,Chien_2018,Eriksen_2020,Loos_2020e}
As we shall see below, employing optimized orbitals has the advantage to produce an even smoother and faster convergence of the SCI energy toward the FCI limit.
\titou{Note both localized and optimized orbitals do break the spatial symmetry.}
Note both localized and optimized orbitals do break the spatial symmetry.
Unlike excited-state calculations where it is important to enforce that the wave functions are eigenfunctions of the $\Hat{S}^2$ spin operator, \cite{Chilkuri_2021} the present wave functions do not fulfill this property as we aim for the lowest possible energy of a closed-shell singlet state.
We have found that $\expval*{\Hat{S}^2}$ is, nonetheless, very close to zero for each system.
@ -596,7 +595,6 @@ Figure \ref{fig:BenzenevsNdet} compares the convergence of $\Delta \Evar$ for th
As mentioned in Sec.~\ref{sec:compdet}, although both the localized and optimized orbitals break the spatial symmetry to take advantage of the local nature of electron correlation, the latter set further improve on the use of former set.
More quantitatively, optimized orbitals produce the same variational energy as localized orbitals with, roughly, a ten-fold reduction in the number of determinants.
A similar improvement is observed going from natural to localized orbitals.
\titou{Comment on PT2 for localized orbitals.}
Accordingly to these observations, all our FCI correlation energy estimates have been produced with the set of optimized orbitals.
To do so, we have then extrapolated the orbital-optimized variational CIPSI correlation energies to $\EPT = 0$ via a weighted five-point linear fit using the five largest variational wave functions (see Fig.~\ref{fig:vsEPT2}).
@ -702,19 +700,19 @@ Compared to natural orbitals, we have shown that, by using energetically optimiz
Thanks to these reference FCI energies, we have then benchmarked three families of popular electronic structure methods: i) the MP perturbation series up to fifth-order (MP2, MP3, MP4, and MP5), ii) the approximate CC series CC2, CC3, and CC4, and iii) the ``complete'' CC series CCSD, CCSDT, and CCSDTQ.
Our results have shown that, with a $\order*{N^7}$ scaling, MP4 provides an interesting accuracy/cost ratio for this particular set of weakly correlated systems, while MP5 systematically worsen the perturbative estimates of the correlation energy.
We have evidenced that CC3 (where the triples are computed iteratively) also outperforms the perturbative-triples CCSD(T) method with the same $\order*{N^7}$ scaling but also its more expensive parent, CCSDT.
We have evidenced that CC3 (where the triples are computed iteratively) also outperforms the perturbative-triples CCSD(T) method with the same $\order*{N^7}$ scaling, its completely renormalized version CR-CC(2,3), but also its more expensive parent, CCSDT.
A similar trend is observed for the methods including quadruple excitations, where the $\order*{N^9}$ CC4 model has been shown to be more accurate than CCSDTQ [which scales as $\order*{N^{10}}$].
Of course, the present trends are only valid for this particular class of (weakly-correlated) systems and it would be desirable to provide more variety in terms of systems in the future by including more challenging systems such as, for example, transition metal compounds.
Some work along this line is currently being done.
\titou{more variety in systems would be good.}
As perspectives, we are currently investigating the performance of the present approach for excited states in order to expand the QUEST database of vertical excitation energies. \cite{Veril_2021}
We hope to report on this in the near future.
The compression of the variational space brought by optimized orbitals could be also beneficial in the context of quantum Monte Carlo methods to generate compact, yet accurate multi-determinant trial wave functions. \cite{Dash_2018,Dash_2019,Scemama_2020,Dash_2021}
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\begin{acknowledgements}
This work was performed using HPC resources from GENCI-TGCC (2021-gen1738) and from CALMIP (Toulouse) under allocation 2021-18005, and was also supported by the European Centre of Excellence in Exascale Computing TREX - Targeting Real Chemical Accuracy at the Exascale. This project has received funding from the European Union's Horizon 2020 - Research and Innovation program - under grant agreement no. 95216.
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 and 952165).
This work was performed using HPC resources from GENCI-TGCC (2021-gen1738) and from CALMIP (Toulouse) under allocation 2021-18005, and was also supported by the European Centre of Excellence in Exascale Computing TREX --- Targeting Real Chemical Accuracy at the Exascale. This project has received funding from the European Union's Horizon 2020 --- Research and Innovation program --- under grant agreement no.~952165.
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).
\end{acknowledgements}
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