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Pierre-Francois Loos 2019-04-14 17:14:43 +02:00
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Manuscript/G2-srDFT.tex
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@ -490,7 +490,7 @@ iii) vanishes in the limit of a complete basis set, hence guaranteeing an unalte
\includegraphics[width=\linewidth]{VTZ}
\includegraphics[width=\linewidth]{VQZ}
\caption{
Deviation (in \kcal) from CCSD(T)/CBS reference correlation energies obtained with various methods with the cc-pVDZ (top), cc-pVTZ (center) and cc-pVQZ (bottom) basis sets.
Deviation (in \kcal) from CCSD(T)/CBS correlation energy contribution to the atomization energy obtained with various methods with the cc-pVDZ (top), cc-pVTZ (center) and cc-pVQZ (bottom) basis sets.
The green region corresponds to chemical accuracy (i.e.~error below 1 {\kcal}).
See {\SI} for raw data.
\label{fig:G2_Ec}}
@ -498,7 +498,7 @@ iii) vanishes in the limit of a complete basis set, hence guaranteeing an unalte
%\subsection{Comparison between the CIPSI and CCSD(T) models in the case of C$_2$, N$_2$, O$_2$, F$_2$}
We begin our investigation of the performance of the basis set correction by computing the atomization energies of \ce{C2}, \ce{N2}, \ce{O2} and \ce{F2} obtained with Dunning's cc-pVXZ basis sets (X $=$ D, T, Q and 5).
In the case of \ce{C2} and \ce{N2}, we also perform calculations with the cc-pCVXZ family.
\titou{In the case of \ce{C2} and \ce{N2}, we also perform calculations with the cc-pCVXZ family.}
\ce{N2}, \ce{O2} and \ce{F2} are weakly correlated systems and belong to the G2-1 set \cite{CurRagTruPop-JCP-91} (see below), whereas \ce{C2} already contains a non-negligible amount of strong correlation. \cite{BooCleThoAla-JCP-11}
In a second time, we compute the entire correlation energies of the G2-1 set \cite{CurRagTruPop-JCP-91} composed by 55 molecules with the cc-pVXZ family of basis sets.
This molecular set has been exhausively studied in the last 20 years (see, for example, Refs.~\onlinecite{FelPetDix-JCP-08, Gro-JCP-09, FelPet-JCP-09, NemTowNee-JCP-10, FelPetHil-JCP-11, HauKlo-JCP-12, PetTouUmr-JCP-12, FelPet-JCP-13, KesSylKohTewMar-JCP-18}).
@ -509,47 +509,46 @@ We refer the interested reader to Refs.~\onlinecite{HolUmrSha-JCP-17, SceGarCafL
In the case of the CCSD(T) calculations, we have $\modZ = \HF$ as we use the restricted open-shell HF (ROHF) one-electron density to compute the complementary energy.
For exFCI, the one-electron density is computed from a very large CIPSI expansion containing several million determinants.
%For the definition of the interaction, we use a single Slater determinant built in the ROHF basis for the CCSD(T) calculation, and built with the natural orbitals of the converged variational wave function for the exFCI calculations.
The CCSD(T) calculations have been performed with Gaussian09 with standard threshold values. \cite{g09}
CCSD(T) energies are computed with Gaussian09 using standard threshold values. \cite{g09}
RS-DFT and exFCI calculations are performed with {\QP}. \cite{QP2}
For the numerical quadrature, we employ the SG-2 grid. \cite{DasHer-JCC-17}
For the numerical quadratures, we employ the SG-2 grid. \cite{DasHer-JCC-17}
Except for the carbon dimer where we have taken the experimental equilibrium bond length (\InAA{1.2425}), all geometries have been extracted from Ref.~\onlinecite{HauJanScu-JCP-09} and have been obtained at the B3LYP/6-31G(2df,p) level of theory.
Frozen core calculations are defined as such: an \ce{He} core is frozen from \ce{Li} to \ce{Ne}, while a \ce{Ne} core is frozen from \ce{Na} to \ce{Ar}.
Frozen-core calculations are defined as such: an \ce{He} core is frozen from \ce{Li} to \ce{Ne}, while a \ce{Ne} core is frozen from \ce{Na} to \ce{Ar}.
In the context of the basis set correction, the set of spinorbitals $\BasFC$ involved in the definition of the effective interaction refers to the non-frozen spinorbitals.
The FC density-based correction was used consistently when the FC approximation was applied in WFT methods.
In order to estimate the complete basis set (CBS) limit for each model, we employed the two-point extrapolation proposed in Ref.~\onlinecite{HalHelJorKloKocOlsWil-CPL-98} for the correlation energies, and we refer to these as $\CBS$.
The FC density-based correction is set consistently when the FC approximation was applied in WFT methods.
In order to estimate the complete basis set (CBS) limit for each model, we employ the two-point extrapolation proposed in Ref.~\onlinecite{HalHelJorKloKocOlsWil-CPL-98} for the correlation energies, and we refer to these as $\CBS$.
\titou{What about the HF energies?}
%\subsection{Convergence of the atomization energies with the WFT models }
As the exFCI calculations were converged with a precision of about 0.1 {\kcal}, we can consider these atomization energies as near-FCI values, and they will be our references for \ce{C2}, \ce{N2}, \ce{O2} and \ce{F2}.
As the exFCI calculations are converged with a precision of about 0.1 {\kcal}, we can consider these atomization energies as near-FCI values, and they will be our references for \ce{C2}, \ce{N2}, \ce{O2} and \ce{F2}.
The results for these diatomics are reported in Fig.~\ref{fig:diatomics}.
The corresponding numerical data can be found as {\SI}.
As one can see, the convergence of the exFCI atomization energies is, as expected, slow with respect to the basis set: chemical accuracy (error below 1 {\kcal}) is barely reached for \ce{C2}, \ce{O2} and \ce{F2} even with a cc-pV5Z basis set.
Also, the atomization energies are consistently underestimated, reflecting that, in a given basis, the atom is always better described than the molecule due to the larger number of interacting electron pairs in the molecular system.
The corresponding numerical data can be found in the {\SI}.
As one can see, the convergence of the exFCI atomization energies is, as expected, slow with respect to the basis set: chemical accuracy (error below 1 {\kcal}) is barely reached for \ce{C2}, \ce{O2} and \ce{F2} even with a cc-pV5Z basis set, and the atomization energies are consistently underestimated.
%Also, the atomization energies are consistently underestimated, reflecting that, in a given basis, the atom is always better described than the molecule due to the larger number of interacting electron pairs in the molecular system.
A similar trend holds for CCSD(T).
%, and one can notice that the atomization energies of the CCSD(T) are always slightly underestimated with respect to the CIPSI ones, showing that the CCSD(T) ansatz is better suited for the atoms than for the molecule.
%\subsection{The effect of the basis set correction within the LDA and PBE approximation}
Regarding the effect of the basis set correction, several general observations can be made for both exFCI and CCSD(T).
First, in a given basis set, the basis set correction systematically improves the result (both at the LDA and PBE level).
First, in a given basis set, the basis set correction systematically improves the atomization energies (both at the LDA and PBE level).
A small overestimation can occur compared to the CBS values by a few tenths of a {\kcal} (the largest deviation being 0.6 {\kcal} for \ce{N2} at the CCSD(T)+PBE/cc-pV5Z level).
Nevertheless, the deviation observed for the largest basis set is typically within the extrapolation error of the CBS atomization energies, which is highly satisfactory knowing the marginal computation cost of the present correction.
In most cases, the basis set corrected triple-$\zeta$ results are on par with the uncorrected quintuple-$\zeta$ ones.
%Also, the values obtained with the largest basis sets tends to converge toward a value close to the estimated CBS values.
Importantly, the sensitivity with respect to the SR-DFT functional is quite large for the double- and triple-$\zeta$ basis sets, where clearly the PBE functional performs better.
However, from the quadruple-$\zeta$ basis, the LDA and PBE functionals agree within a few tenths of a {\kcal}.
Such weak sensitivity to the approximated functionals in the DFT part when reaching large basis sets shows the robustness of the approach.
Such weak sensitivity when reaching large basis sets shows the robustness of the approach.
We have computed the correlation energies of the G2-1 test sets with $\modY=\CCSDT$, $\modZ=\HF$ and the cc-pVXZ basis sets.
As a second set of numerical examples, we compute the correlation energy contribution to the atomization energies of the G2-1 test sets with $\modY=\CCSDT$, $\modZ=\HF$ and the cc-pVXZ basis sets.
The ``plain'' CCSD(T) correlation energies as well as the corrected CCSD(T)+LDA and CCSD(T)+PBE values are depicted in Fig.~\ref{fig:G2_Ec}.
The raw data can be found in the {\SI}.
A statistical analysis of these data is also provided in Table \ref{tab:stats}, where we have reported the mean absolute deviation (MAD), root-mean-square deviation (RMSD), and maximum deviation (MAX) with respect to the CCSD(T)/CBS correlation energies.
From double-$\zeta$ to quintuple-$\zeta$ basis, the MAD associated with the CCSD(T) correlation energies goes down slowly from 14.29 to 1.28 {\kcal}.
A statistical analysis of these data is also provided in Table \ref{tab:stats}, where we report the mean absolute deviation (MAD), root-mean-square deviation (RMSD), and maximum deviation (MAX) with respect to the CCSD(T)/CBS correlation energies.
From double-$\zeta$ to quintuple-$\zeta$ basis, the MAD associated with the CCSD(T) correlation energy contribution to the atomization energies goes down slowly from 14.29 to 1.28 {\kcal}.
For a commonly-used basis like cc-pVTZ, the MAD of CCSD(T) is still 6.06 {\kcal}.
Applying the basis set correction drastically reduces the basis set incompleteness error.
Already at the CCSD(T)+LDA/cc-pVDZ and CCSD(T)+PBE/cc-pVDZ levels, the MAD is reduced to 3.24 and 1.96 {\kcal}.
With the triple-$\zeta$ basis, the MAD of CCSD(T)+PBE/cc-pVTZ is already below 1 {\kcal} with 36 cases (out of 55) where we achieve chemical accuracy.
CCSD(T)+LDA/cc-pVQZ and CCSD(T)+PBE/cc-pVQZ return MAD of 0.33 and 0.31 kcal/mol (respectively) while CCSD(T)/cc-pVQZ still yields a fairly large MAD of 2.50 {\kcal}.
Therefore, similar to F12 methods, \cite{TewKloNeiHat-PCCP-07} we can safely claim that the present basis set correction recovers quintuple-$\zeta$ quality correlation energies with triple-$\zeta$ basis sets for a much cheaper computational cost.
\titou{Therefore, similar to F12 methods, \cite{TewKloNeiHat-PCCP-07} we can safely claim that the present basis set correction recovers quintuple-$\zeta$ quality correlation energies with triple-$\zeta$ basis sets for a much cheaper computational cost.}
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\section*{Supporting information}