last data benzene et other
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@ -170,6 +170,7 @@ In Sec.~\ref{sec:res}, we report our reference FCI correlation energies for the
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These reference correlation energies are then used to benchmark and study the convergence properties of various perturbative and CC methods (Sec.~\ref{sec:mpcc_res}).
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Finally, we draw our conclusions in Sec.~\ref{sec:ccl}.
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%%%%%%%%%%%%%%%%%%%%%%%%%
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\section{CIPSI with optimized orbitals}
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\label{sec:OO-CIPSI}
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@ -307,46 +308,6 @@ The addition of the level shift $\lambda \geq 0$ removes the negative eigenvalue
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By choosing the right value of $\lambda$, $\norm{\bk}$ is constrained into a hypersphere of radius $\Delta$ and is able to evolve from the Newton direction at $\lambda = 0$ to the steepest descent direction as $\lambda$ grows.
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The evolution of the trust radius during the optimization and the use of a condition to reject the step when the energy rises ensure the convergence of the algorithm.
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More details can be found in Ref.~\onlinecite{Nocedal_1999}.
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\section{Computational details}
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\label{sec:compdet}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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The geometries of the twelve systems considered in the present study were all obtained at the CC3/aug-cc-pVTZ level of theory and were extracted from a previous study. \cite{Loos_2020a}
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Note that, for the sake of consistency, the geometry of benzene considered here is different from the one of Ref.~\onlinecite{Loos_2020e} which was obtained at a lower level of theory [MP2/6-31G(d)]. \cite{Schreiber_2008}
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The MP2, MP3, MP4, CC2, CC3, CC4, CCSD, CCSDT, and CCSDTQ calculations were performed with CFOUR, \cite{Matthews_2020} the CCSD(T) and CR-CC(2,3) calculations were made with GAMESS 2014R1, \cite{gamess} and MP5 calculations were computed with GAUSSIAN 09. \cite{g09}
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The CIPSI calculations were performed with QUANTUM PACKAGE. \cite{Garniron_2019}
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In the current implementation, the selection step and the PT2 correction are computed simultaneously via a hybrid semistochastic algorithm.\cite{Garniron_2017,Garniron_2019} %(which explains the statistical error associated with the PT2 correction in the following).
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Here, we employ the renormalized version of the PT2 correction which was recently implemented and tested for a more efficient extrapolation to the FCI limit thanks to a partial resummation of the higher orders of perturbation. \cite{Garniron_2019}
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We refer the interested reader to Ref.~\onlinecite{Garniron_2019} where one can find all the details regarding the implementation of the PT2 correction and the CIPSI algorithm.
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For all these calculations, we consider Dunning's correlation-consistent double-$\zeta$ basis (cc-pVDZ).
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Although the FCI energy has the enjoyable property of being independent of the set of one-electron orbitals used to construct the many-electron Slater determinants, as a truncated CI method, the convergence properties of CIPSI strongly dependent on this orbital choice.
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In the present study, we investigate, in particular, the convergence behavior of the CIPSI energy for two sets of orbitals: natural orbitals (NOs) and optimized orbitals (OOs).
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Following our usual procedure, \cite{Scemama_2018,Scemama_2018b,Scemama_2019,Loos_2018a,Loos_2019,Loos_2020a,Loos_2020b,Loos_2020c,Loos_2020e} we perform first a preliminary SCI calculation using HF orbitals in order to generate a SCI wave function with at least $10^7$ determinants.
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Natural orbitals are computed based on this wave function and they are used to perform a new CIPSI run.
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Successive orbital optimizations are then performed, which consist in minimizing the variational CIPSI energy at each macroiteration up to approximately $2 \times 10^5$ determinants.
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When convergence is achieved in terms of orbital optimization, as our production run, we perform a new CIPSI calculation from scratch using this set of optimized orbitals.
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Using optimized orbitals has the undeniable advantage to produce, for a given variational energy, more compact CI expansions (see Sec.~\ref{sec:res}).
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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}
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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}
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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.
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Note that both localized and optimized orbitals do break the spatial symmetry.
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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.
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We have found that $\expval*{\Hat{S}^2}$ is, nonetheless, very close to zero for each system.
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The present calculations have been performed on the AMD partition of GENCI's Irene supercomputer.
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Each Irene's AMD node is a dual-socket AMD Rome (EPYC) CPU at 2.60 GHz with 256GiB of RAM, with a total of 64 physical cores per socket.
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These nodes are connected via Infiniband HDR100.
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In total, the present calculations have required around 3~million core hours.
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All the data (geometries, energies, etc) and supplementary material associated with the present manuscript are openly available in Zenodo at \href{http://doi.org/XX.XXXX/zenodo.XXXXXXX}{http://doi.org/XX.XXXX/zenodo.XXXXXXX}.
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\section{Results and discussion}
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\label{sec:res}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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%%% FIG 2 %%%
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\begin{figure*}
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\includegraphics[width=0.24\textwidth]{Cyclopentadiene_EvsNdet}
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@ -517,11 +478,11 @@ All the data (geometries, energies, etc) and supplementary material associated w
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& 6 & $-728.242$ & $0.062$ & $-728.987$ & $0.140$ \\
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& 7 & $-728.420$ & $0.144$ & $-729.067$ & $0.117$ \\
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\hline
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Benzene & 3 & & & $-862.325$ & $0.279$ \\
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& 4 & & & $-863.024$ & $0.424$ \\
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&\bf5 & & & $\bf-862.890$ &$\bf0.266$ \\
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& 6 & & & $-862.360$ & $0.383$ \\
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& 7 & & & $-862.083$ & $0.339$ \\
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Benzene & 3 & $-860.350$ & $0.496$ & $-862.325$ & $0.279$ \\
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& 4 & $-861.949$ & $0.811$ & $-863.024$ & $0.424$ \\
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&\bf5 & $-861.807$ & $0.474$ & $\bf-862.890$ &$\bf0.266$ \\
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& 6 & $-861.110$ & $0.539$ & $-862.360$ & $0.383$ \\
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& 7 & $-861.410$ & $0.444$ & $-862.083$ & $0.339$ \\
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\hline
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Pyrazine & 3 & $-904.148$ & $0.035$ & $-904.867$ & $1.420$ \\
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& 4 & $-904.726$ & $0.377$ & $-904.588$ & $0.650$ \\
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@ -535,11 +496,11 @@ All the data (geometries, energies, etc) and supplementary material associated w
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& 6 & $-912.566$ & $1.727$ & $-908.342$ & $0.303$ \\
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& 7 & $-910.694$ & $2.210$ & $-908.368$ & $0.224$ \\
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\hline
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Pyridine & 3 & & & $-883.363$ & $0.047$ \\
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& 4 & & & $-883.413$ & $0.029$ \\
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&\bf5 & & & $\bf-882.700$ &$\bf0.405$ \\
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& 6 & & & $-882.361$ & $0.341$ \\
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& 7 & & & $-882.023$ & $0.330$ \\
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Pyridine & 3 & $-883.025$ & $3.919$ & $-883.363$ & $0.047$ \\
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& 4 & $-883.862$ & $1.869$ & $-883.413$ & $0.029$ \\
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&\bf5 & $-881.664$ & $1.760$ & $\bf-882.700$ &$\bf0.405$ \\
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& 6 & $-880.422$ & $1.456$ & $-882.361$ & $0.341$ \\
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& 7 & $-880.191$ & $1.084$ & $-882.023$ & $0.330$ \\
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\hline
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Pyrimidine & 3 & $-900.386$ & $1.884$ & $-900.817$ & $0.726$ \\
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& 4 & $-901.441$ & $0.991$ & $-900.383$ & $0.356$ \\
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@ -564,21 +525,6 @@ All the data (geometries, energies, etc) and supplementary material associated w
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\end{squeezetable}
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%%% %%% %%%
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\subsection{CIPSI estimates}
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\label{sec:cipsi_res}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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We first study the convergence of the CIPSI energy as a function of the number of determinants.
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Our motivation here is to generate FCI-quality reference correlation energies for the twelve cyclic molecules represented in Fig.~\ref{fig:mol} in order to benchmark, in a second time, the performance and convergence properties of various mainstream MP and CC methods (see Sec.~\ref{sec:mpcc_res}).
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For the natural and optimized orbital sets, we report, in Fig.~\ref{fig:vsNdet}, the evolution of the variational correlation energy $\Delta \Evar = \Evar - \EHF$ (where $\EHF$ is the HF energy) and its perturbatively corrected value $\Delta \Evar + \EPT$ with respect to the number of determinants $\Ndet$ for each cyclic molecule.
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As compared to natural orbitals (solid red lines), one can see that, for a given number of determinants, the use of optimized orbitals greatly lowers $\Delta \Evar$ (solid blue lines).
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Adding the perturbative correction $\EPT$ yields similar curves for both sets of orbitals (dashed lines).
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This indicates that, for a given number of determinants, $\EPT$ (which, we recall, provides a qualitative idea to the distance to the FCI limit) is much smaller for optimized orbitals than for natural orbitals.
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This is further evidenced in Fig.~\ref{fig:vsEPT2} where we show the behavior of $\Delta \Evar$ as a function of $\EPT$ for both sets of orbitals.
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From Fig.~\ref{fig:vsEPT2}, it is clear that, using optimized orbitals, the behavior of $\Delta \Evar$ is much more linear and produces smaller $\EPT$ values, hence facilitating the extrapolation procedure to the FCI limit (see below).
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The five-point weighted linear fit using the five largest variational wave functions are also represented (dashed black lines), while the FCI estimate of the correlation energy (solid black line) is reported for reference in Figs.~\ref{fig:vsNdet} and \ref{fig:vsEPT2}.
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%%% FIG 4 %%%
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\begin{figure}
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\includegraphics[width=\linewidth]{Benzene_EvsNdetLO}
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@ -589,31 +535,6 @@ The five-point weighted linear fit using the five largest variational wave funct
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\end{figure}
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%%% %%% %%%
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Figure \ref{fig:BenzenevsNdet} compares the convergence of $\Delta \Evar$ for the natural, localized, and optimized sets of orbitals in the particular case of benzene.
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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.
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More quantitatively, optimized orbitals produce the same variational energy as localized orbitals with, roughly, a ten-fold reduction in the number of determinants.
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A similar improvement is observed going from natural to localized orbitals.
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According to these observations, all our FCI correlation energy estimates have been produced with the set of optimized orbitals.
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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}).
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The fitting weights have been taken as the inverse square of the perturbative corrections.
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Our final FCI correlation energy estimates are reported in Tables \ref{tab:Tab5-VDZ} and \ref{tab:Tab6-VDZ} for the five- and six-membered rings, respectively, alongside their corresponding fitting error.
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The stability of these estimates are illustrated by the results gathered in Table \ref{tab:fit} where we report, for each system, the extrapolated correlation energies $\Delta \Eextrap$ and their associated fitting errors obtained via weighted linear fits varying the number of fitting points from $3$ to $7$.
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Although we cannot provide a mathematically rigorous error bar, the data provided by Table \ref{tab:fit} show that the extrapolation procedure is robust and that our FCI estimates are very likely accurate to a few tenths of a millihartree.
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Logically, the FCI estimates for the five-membered rings seem slightly more accurate than for the (larger) six-membered rings.
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Note that it is pleasing to see that, although different geometries are considered, our present estimate of the frozen-core correlation energy of the benzene molecule in the cc-pVDZ basis is very close to the one reported in Refs.~\onlinecite{Eriksen_2020,Loos_2020e}.
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Table \ref{tab:fit} does report extrapolated correlation energies and fitting errors for both natural and optimized orbitals.
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Again, the superiority of the latter set is clear as the variation in extrapolated values and fitting error is much larger with the natural set.
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Taking cyclopentadiene as an example, the extrapolated values vary by almost \SI{1}{\milli\hartree} with natural orbitals and less than \SI{0.1}{\milli\hartree} with the optimized set.
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The fitting errors follow the same trend.
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\subsection{Benchmark of CC and MP methods}
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\label{sec:mpcc_res}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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%%% FIG 5 %%%
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\begin{figure*}
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\includegraphics[width=0.32\textwidth]{Cyclopentadiene_MPCC}
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@ -670,6 +591,85 @@ The fitting errors follow the same trend.
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\end{squeezetable}
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%%% %%% %%%
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\section{Computational details}
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\label{sec:compdet}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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The geometries of the twelve systems considered in the present study were all obtained at the CC3/aug-cc-pVTZ level of theory and were extracted from a previous study. \cite{Loos_2020a}
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Note that, for the sake of consistency, the geometry of benzene considered here is different from the one of Ref.~\onlinecite{Loos_2020e} which was obtained at a lower level of theory [MP2/6-31G(d)]. \cite{Schreiber_2008}
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The MP2, MP3, MP4, CC2, CC3, CC4, CCSD, CCSDT, and CCSDTQ calculations were performed with CFOUR, \cite{Matthews_2020} the CCSD(T) and CR-CC(2,3) calculations were made with GAMESS 2014R1, \cite{gamess} and MP5 calculations were computed with GAUSSIAN 09. \cite{g09}
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The CIPSI calculations were performed with QUANTUM PACKAGE. \cite{Garniron_2019}
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In the current implementation, the selection step and the PT2 correction are computed simultaneously via a hybrid semistochastic algorithm.\cite{Garniron_2017,Garniron_2019} %(which explains the statistical error associated with the PT2 correction in the following).
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Here, we employ the renormalized version of the PT2 correction which was recently implemented and tested for a more efficient extrapolation to the FCI limit thanks to a partial resummation of the higher orders of perturbation. \cite{Garniron_2019}
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We refer the interested reader to Ref.~\onlinecite{Garniron_2019} where one can find all the details regarding the implementation of the PT2 correction and the CIPSI algorithm.
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For all these calculations, we consider Dunning's correlation-consistent double-$\zeta$ basis (cc-pVDZ).
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Although the FCI energy has the enjoyable property of being independent of the set of one-electron orbitals used to construct the many-electron Slater determinants, as a truncated CI method, the convergence properties of CIPSI strongly dependent on this orbital choice.
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In the present study, we investigate, in particular, the convergence behavior of the CIPSI energy for two sets of orbitals: natural orbitals (NOs) and optimized orbitals (OOs).
|
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Following our usual procedure, \cite{Scemama_2018,Scemama_2018b,Scemama_2019,Loos_2018a,Loos_2019,Loos_2020a,Loos_2020b,Loos_2020c,Loos_2020e} we perform first a preliminary SCI calculation using HF orbitals in order to generate a SCI wave function with at least $10^7$ determinants.
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Natural orbitals are computed based on this wave function and they are used to perform a new CIPSI run up to $8 \times 10^7$ determinants.
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Successive orbital optimizations are then performed, which consist in minimizing the variational CIPSI energy at each macroiteration up to approximately $2 \times 10^5$ determinants.
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When convergence is achieved in terms of orbital optimization, as our production run, we perform a new CIPSI calculation from scratch using this set of optimized orbitals to $8 \times 10^7$ determinants.
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Using optimized orbitals has the undeniable advantage to produce, for a given variational energy, more compact CI expansions (see Sec.~\ref{sec:res}).
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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}
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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}
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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.
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Note that both localized and optimized orbitals do break the spatial symmetry.
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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.
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We have found that $\expval*{\Hat{S}^2}$ is, nonetheless, very close to zero for each system.
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The present calculations have been performed on the AMD partition of GENCI's Irene supercomputer.
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Each Irene's AMD node is a dual-socket AMD Rome (EPYC) CPU at 2.60 GHz with 256GiB of RAM, with a total of 64 physical cores per socket.
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These nodes are connected via Infiniband HDR100.
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In total, the present calculations have required around 3~million core hours.
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All the data (geometries, energies, etc) and supplementary material associated with the present manuscript are openly available in Zenodo at \href{http://doi.org/XX.XXXX/zenodo.XXXXXXX}{http://doi.org/XX.XXXX/zenodo.XXXXXXX}.
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\section{Results and discussion}
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\label{sec:res}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\subsection{CIPSI estimates}
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\label{sec:cipsi_res}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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We first study the convergence of the CIPSI energy as a function of the number of determinants.
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Our motivation here is to generate FCI-quality reference correlation energies for the twelve cyclic molecules represented in Fig.~\ref{fig:mol} in order to benchmark, in a second time, the performance and convergence properties of various mainstream MP and CC methods (see Sec.~\ref{sec:mpcc_res}).
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For the natural and optimized orbital sets, we report, in Fig.~\ref{fig:vsNdet}, the evolution of the variational correlation energy $\Delta \Evar = \Evar - \EHF$ (where $\EHF$ is the HF energy) and its perturbatively corrected value $\Delta \Evar + \EPT$ with respect to the number of determinants $\Ndet$ for each cyclic molecule.
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As compared to natural orbitals (solid red lines), one can see that, for a given number of determinants, the use of optimized orbitals greatly lowers $\Delta \Evar$ (solid blue lines).
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Adding the perturbative correction $\EPT$ yields similar curves for both sets of orbitals (dashed lines).
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This indicates that, for a given number of determinants, $\EPT$ (which, we recall, provides a qualitative idea to the distance to the FCI limit) is much smaller for optimized orbitals than for natural orbitals.
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This is further evidenced in Fig.~\ref{fig:vsEPT2} where we show the behavior of $\Delta \Evar$ as a function of $\EPT$ for both sets of orbitals.
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From Fig.~\ref{fig:vsEPT2}, it is clear that, using optimized orbitals, the behavior of $\Delta \Evar$ is much more linear and produces smaller $\EPT$ values, hence facilitating the extrapolation procedure to the FCI limit (see below).
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The five-point weighted linear fit using the five largest variational wave functions are also represented (dashed black lines), while the FCI estimate of the correlation energy (solid black line) is reported for reference in Figs.~\ref{fig:vsNdet} and \ref{fig:vsEPT2}.
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Figure \ref{fig:BenzenevsNdet} compares the convergence of $\Delta \Evar$ for the natural, localized, and optimized sets of orbitals in the particular case of benzene.
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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.
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More quantitatively, optimized orbitals produce the same variational energy as localized orbitals with, roughly, a ten-fold reduction in the number of determinants.
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A similar improvement is observed going from natural to localized orbitals.
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According to these observations, all our FCI correlation energy estimates have been produced with the set of optimized orbitals.
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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}).
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The fitting weights have been taken as the inverse square of the perturbative corrections.
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Our final FCI correlation energy estimates are reported in Tables \ref{tab:Tab5-VDZ} and \ref{tab:Tab6-VDZ} for the five- and six-membered rings, respectively, alongside their corresponding fitting error.
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The stability of these estimates are illustrated by the results gathered in Table \ref{tab:fit} where we report, for each system, the extrapolated correlation energies $\Delta \Eextrap$ and their associated fitting errors obtained via weighted linear fits varying the number of fitting points from $3$ to $7$.
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Although we cannot provide a mathematically rigorous error bar, the data provided by Table \ref{tab:fit} show that the extrapolation procedure is robust and that our FCI estimates are very likely accurate to a few tenths of a millihartree.
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Logically, the FCI estimates for the five-membered rings seem slightly more accurate than for the (larger) six-membered rings.
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Note that it is pleasing to see that, although different geometries are considered, our present estimate of the frozen-core correlation energy of the benzene molecule in the cc-pVDZ basis is very close to the one reported in Refs.~\onlinecite{Eriksen_2020,Loos_2020e}.
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Table \ref{tab:fit} does report extrapolated correlation energies and fitting errors for both natural and optimized orbitals.
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Again, the superiority of the latter set is clear as the variation in extrapolated values and fitting error is much larger with the natural set.
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Taking cyclopentadiene as an example, the extrapolated values vary by almost \SI{1}{\milli\hartree} with natural orbitals and less than \SI{0.1}{\milli\hartree} with the optimized set.
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The fitting errors follow the same trend.
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\subsection{Benchmark of CC and MP methods}
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\label{sec:mpcc_res}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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Using the CIPSI estimates of the FCI correlation energy produced in Sec.~\ref{sec:cipsi_res}, we now study the performance and convergence properties of three series of methods: i) MP2, MP3, MP4, and MP5, ii) CC2, CC3, and CC4, and iii) CCSD, CCSDT, and CCSDTQ.
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Additionally, we also report CCSD(T) and CR-CC(2,3) correlation energies.
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All these data are reported in Tables \ref{tab:Tab5-VDZ} and \ref{tab:Tab6-VDZ} for the five- and six-membered rings, respectively.
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