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13
Manuscript/G2-srDFT.bib
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@ -12006,3 +12006,16 @@
url = {https://link.aps.org/doi/10.1103/PhysRevA.73.032506} url = {https://link.aps.org/doi/10.1103/PhysRevA.73.032506}
} }
@article{HalHelJorKloKocOls-CPL-98,
title = "Basis-set convergence in correlated calculations on Ne, N2, and H2O",
journal = "Chemical Physics Letters",
volume = "286",
number = "3",
pages = "243 - 252",
year = "1998",
issn = "0009-2614",
doi = "https://doi.org/10.1016/S0009-2614(98)00111-0",
url = "http://www.sciencedirect.com/science/article/pii/S0009261498001110",
author = "Asger Halkier and Trygve Helgaker and Poul Jørgensen and Wim Klopper and Henrik Koch and Jeppe Olsen and Angela K. Wilson",
abstract = "Valence and all-electron correlation energies of Ne, N2, and H2O at fixed experimental geometries are computed at the levels of second-order perturbation theory (MP2) and coupled cluster theory with singles and doubles excitations (CCSD), and singles and doubles excitations with a perturbative triples correction (CCSD(T)). Correlation-consistent polarized valence and core-valence basis sets up to sextuple zeta quality are employed. Guided by basis-set limits established by rij-dependent methods, a number of extrapolation schemes for use with the correlation-consistent basis sets are investigated. Among the schemes considered here, a linear least-squares procedure applied to the quintuple and sextuple zeta results yields the most accurate extrapolations."
}

71
Manuscript/G2-srDFT.tex
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@ -676,12 +676,19 @@ Therefore, we propose the following valence-only approximations for the compleme
\section{Results} \section{Results}
%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Comparison between the CIPSI and CCSD(T) models in the case of C$_2$, N$_2$, O$_2$, F$_2$} \subsection{Comparison between the CIPSI and CCSD(T) models in the case of N$_2$, O$_2$, F$_2$}
We begin the investigation of the behavior of the basis-set correction by the study of the atomization energies of the C$_2$, N$_2$, O$_2$, F$_2$ homo-nuclear diatomic molecules in the Dunning cc-pVXZ and cc-pCVXZ (X=D,T,Q,5) using both the CIPSI algorithm and the CCSD(T). All through this work, we follow the frozen core (FC) convention of Klopper \textit{et. al}\cite{HauKlo-JCP-12} which consists in all-electron calculations for Li-Be, a He core for B-Na atoms and a Ne core for the Al-Cl series. In the context of the DFT correction for the basis-set, this implies that, for a given system in a given basis set $\Bas$, the set of valence orbitals $\Basval$ involved in the definition of the valence interaction $\wbasisval$ and density $\onedmval$ refers to all MOs except the core. We begin the investigation of the behavior of the basis-set correction by the study of the atomization energies of the N$_2$, O$_2$, F$_2$ homo-nuclear diatomic molecules in the Dunning cc-pVXZ and cc-pCVXZ in the case of N$_2$ (X=D,T,Q,5) using both the CIPSI algorithm and the CCSD(T). All through this work, we follow the frozen core (FC) convention of Klopper \textit{et. al}\cite{HauKlo-JCP-12} which consists in all-electron calculations for Li-Be, a He core for B-Na atoms and a Ne core for the Al-Cl series. In the context of the DFT correction for the basis-set, this implies that, for a given system in a given basis set $\Bas$, the set of valence orbitals $\Basval$ involved in the definition of the valence interaction $\wbasisval$ and density $\onedmval$ refers to all MOs except the core.
%\subsubsection{CIPSI calculations and the basis-set correction} In order to estimate the CBS limit of each model we use the two-point extrapolation of Ref. \onlinecite{HalHelJorKloKocOls-CPL-98} for the correlation energies and report the corresponding atomization energy which are referred as $D_e^{Q5Z}$ and $D_e^{C(Q5)Z}$ for the cc-pVXZ and cc-pCVXZ basis sets, respectively. All through this work, the valence interaction and density was used when the frozen core approximation was done on the WFT model.
%All CIPSI calculations were performed in two steps. First, a CIPSI calculation was performed until the zeroth-order wave function reaches $10^6$ Slater determinants, from which we extracted the natural orbitals. From this set of natural orbitals, we performed CIPSI calculations until the $\EexFCIbasis$ reaches about $0.1$ mH convergence for each systems. Such convergence criterion is more than sufficient for the CIPSI densities $\dencipsi$.
%Regarding the wave function $\wf{}{\Bas}$ chosen to define the local range-separation parameter $\mur$, we take a single Slater determinant built with the natural orbitals of the first CIPSI calculation. \subsection{Convergence of the atomization energies with the WFT models }
\subsubsection{CCSD(T) calculations and the basis-set correction} As the exFCI calculations were converged with a precision of about 0.2 mH, we can consider the atomization energies computed at this level as near FCI values which we will consider as the reference for a given system in a given basis set. The results for these molecules are shown in Table \ref{tab:diatomics}.
As one can notice from the data, the convergence of the exFCI atomization energies is slow with respect to the basis set, and the chemical accuracy is barely reached for C$_2$, O$_2$ and F$_2$ even at the cc-pv5Z basis set. Also, the atomization energies are always too small, reflecting the fact that, in a given basis set, a molecule is always more poorly described than the atoms due to the larger number of interacting pairs of electrons in the molecule.
The same behaviours hold for the CCSD(T) model, 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, both for the CIPSI and CCSD(T) models, several observations can be done.
First, in a given basis set, the addition of the basis set correction, both at the LDA and PBE level, improves the result even if it can overestimates the atomization energies in some cases.
Also, the values obtained with the largest basis sets tends to converge toward a value close to the estimated CBS values. Also, one can observe that the sensitivity to the functional is quite large for the double- and triple-zeta basis sets, where clearly the PBE functional performs better. Nevertheless, from the quadruple-zeta basis set, the LDA and PBE functional agrees within a few tens of kcal/mol.
\begin{table*} \begin{table*}
\caption{ \caption{
@ -694,29 +701,30 @@ We begin the investigation of the behavior of the basis-set correction by the st
\\ \\
\\ \\
\cline{3-6} \cline{3-6}
Molecule & Method & \tabc{cc-pVDZ} & \tabc{cc-pVTZ} & \tabc{cc-pVQZ} & \tabc{cc-pV5Z} & \tabc{$E_{QZ5Z}^{\infty}$} Molecule & Method & \tabc{cc-pVDZ} & \tabc{cc-pVTZ} & \tabc{cc-pVQZ} & \tabc{cc-pV5Z} & \tabc{$D_e^{Q5Z}$}
\\ \\
\\ \\
%\hline
%\ce{C2} & (FC)FCIQMC & 130.0(1) & 139.9(3) & 143.3(2) & & 144.9 \\
% & (FC)FCIQMC+F12 & 142.3 & 145.3 & & & \\
\hline \hline
\ce{C2} & (FC)FCIQMC & 130.0(1) & 139.9(3) & 143.3(2) & & 144.9 \\ \ce{C2} & ex (FC)FCI & 132.0 & 140.3 & 143.6 & 144.3 & 144.9 \\
& (FC)FCIQMC+F12 & 142.3 & 145.3 & & & \\
\hline
& ex (FC)FCI & 132.0 & 140.3 & 143.6 & 144.3 & \\
\hline \hline
& ex (FC)FCI+LDA-val & 143.0 & 145.4 & 146.4 & 146.0 & \\ & ex (FC)FCI+LDA-val & 143.0 & 145.4 & 146.4 & 146.0 & \\
& ex (FC)FCI+PBE-val & 147.4 & 146.1 & 146.4 & 145.9 & \\ & ex (FC)FCI+PBE-val & 147.4 & 146.1 & 146.4 & 145.9 & \\
& exFCI+PBE-on-top-val & 143.3 & 144.7 & 145.7 & 145.6 & \\ \hline
%%%%%%%% \hline \hline
%%%%%%%% & ex (FC)FCI+LDA & 141.9 & 142.8 & 145.8 & 146.2 & \\ & (FC)CCSD(T) & 129.2 & 139.1 & 143.0 & 144.2 & 145.4 \\
%%%%%%%% & ex (FC)FCI+PBE & 146.1 & 143.9 & 145.9 & 145.12 & \\ \hline
%%%%%%%% \hline & (FC)CCSD(T)+LDA-val & 139.1 & 143.7 & 145.9 & 145.9 & \\
%%%%%%%% & exFCI+PBE-on-top& 142.7 & 142.7 & 145.3 & 144.9 & \\ & (FC)CCSD(T)+PBE-val & 142.8 & 144.2 & 145.9 & 145.8 & \\
\hline
\\ \\
\cline{3-6} \cline{3-6}
& & \tabc{cc-pCVDZ} & \tabc{cc-pCVTZ} & \tabc{cc-pCVQZ} & \tabc{cc-pCV5Z} & \tabc{ $E_{CQZC5Z}^{\infty}$ } & & \tabc{cc-pCVDZ} & \tabc{cc-pCVTZ} & \tabc{cc-pCVQZ} & \tabc{cc-pCV5Z} & \tabc{ $D_e^{C(Q5)Z}$ }
\\ \\
\\ \\
\hline % \hline
%%%%%%%% & ex (FC)FC-FCI & 130.5 & 140.5 & 143.8 & 144.9 & 147.1 \\ %%%%%%%% & ex (FC)FC-FCI & 130.5 & 140.5 & 143.8 & 144.9 & 147.1 \\
%%%%%%%% \hline %%%%%%%% \hline
%%%%%%%% & ex (FC)FCI+LDA & 140.9 & 145.7 & 146.6 & 146.4 & \\ %%%%%%%% & ex (FC)FCI+LDA & 140.9 & 145.7 & 146.6 & 146.4 & \\
@ -738,22 +746,22 @@ We begin the investigation of the behavior of the basis-set correction by the st
& & \mc{4}{c}{Dunning's basis set} & & \mc{4}{c}{Dunning's basis set}
\\ \\
\cline{3-6} \cline{3-6}
Molecule & Method & \tabc{cc-pVDZ} & \tabc{cc-pVTZ} & \tabc{cc-pVQZ} & \tabc{cc-pV5Z} & \tabc{$E_{QZ5Z}^{\infty}$} Molecule & Method & \tabc{cc-pVDZ} & \tabc{cc-pVTZ} & \tabc{cc-pVQZ} & \tabc{cc-pV5Z} & \tabc{$D_e^{Q5Z}$}
\\ \\
\\ \\
\ce{N2} & ex (FC)FCI & 201.1 & 217.1 & 223.5 & 225.7 & 227.8 \\ \ce{N2} & ex (FC)FCI & 201.1 & 217.1 & 223.5 & 225.7 & 227.8 \\
\hline \hline
& ex (FC)FCI+LDA-val & 217.9 & 225.9 & 228.0 & 228.6 & \\ & ex (FC)FCI+LDA-val & 217.9 & 225.9 & 228.0 & 228.6 & \\
& ex (FC)FCI+PBE-val & 227.7 & 227.8 & 228.3 & 228.5 & \\ & ex (FC)FCI+PBE-val & 227.7 & 227.8 & 228.3 & 228.5 & \\
& exFCI+PBE-on-top-val & 224.8 & 226.7 & 228.3 & 228.3 & \\ % & exFCI+PBE-on-top-val & 224.8 & 226.7 & 228.3 & 228.3 & \\
\hline \hline
\hline \hline
& (FC)CCSD(T) & 199.9 & 216.3 & 222.8 & 225.0 & 227.2 \\ & (FC)CCSD(T) & 199.9 & 216.3 & 222.8 & 225.0 & 227.2 \\
\hline \hline
%%%%%%%%& ex (FC)CCSD(T)+LDA & 214.7 & 221.9 & ----- & ----- & \\ %%%%%%%%& ex (FC)CCSD(T)+LDA & 214.7 & 221.9 & ----- & ----- & \\
%%%%%%%%& ex (FC)CCSD(T)+PBE & 223.4 & 224.3 & ----- & ----- & \\ %%%%%%%%& ex (FC)CCSD(T)+PBE & 223.4 & 224.3 & ----- & ----- & \\
& ex (FC)CCSD(T)+LDA-val & 216.3 & 224.8 & 227.2 & 227.8 & \\ & (FC)CCSD(T)+LDA-val & 216.3 & 224.8 & 227.2 & 227.8 & \\
& ex (FC)CCSD(T)+PBE-val & 225.9 & 226.7 & 227.5 & 227.8 & \\ & (FC)CCSD(T)+PBE-val & 225.9 & 226.7 & 227.5 & 227.8 & \\
\hline \hline
%%%%%%%%& ex (FC)FCI+LDA & 216.4 & 223.1 & 227.9 & 228.1 & \\ %%%%%%%%& ex (FC)FCI+LDA & 216.4 & 223.1 & 227.9 & 228.1 & \\
%%%%%%%%& ex (FC)FCI+PBE & 225.4 & 225.6 & 228.2 & 227.9 & \\ %%%%%%%%& ex (FC)FCI+PBE & 225.4 & 225.6 & 228.2 & 227.9 & \\
@ -761,7 +769,7 @@ We begin the investigation of the behavior of the basis-set correction by the st
\\ \\
\\ \\
\cline{3-6} \cline{3-6}
& & \tabc{cc-pCVDZ} & \tabc{cc-pCVTZ} & \tabc{cc-pCVQZ} & \tabc{cc-pCV5Z} & \tabc{$E_{QZ5Z}^{\infty}$} & & \tabc{cc-pCVDZ} & \tabc{cc-pCVTZ} & \tabc{cc-pCVQZ} & \tabc{cc-pCV5Z} & \tabc{$D_e^{Q5Z}$}
\\ \\
\\ \\
%%%%%%%% & ex (FC)FCI & 201.7 & 217.9 & 223.7 & 225.7 & 228.8 \\ %%%%%%%% & ex (FC)FCI & 201.7 & 217.9 & 223.7 & 225.7 & 228.8 \\
@ -796,7 +804,7 @@ We begin the investigation of the behavior of the basis-set correction by the st
\begin{tabular}{llddddd} \begin{tabular}{llddddd}
\\ \\
\cline{3-6} \cline{3-6}
Molecule & Method & \tabc{cc-pVDZ} & \tabc{cc-pVTZ} & \tabc{cc-pVQZ} & \tabc{cc-pV5Z} & \tabc{$E_{QZ5Z}^{\infty}$} Molecule & Method & \tabc{cc-pVDZ} & \tabc{cc-pVTZ} & \tabc{cc-pVQZ} & \tabc{cc-pV5Z} & \tabc{$D_e^{Q5Z}$}
\\ \\
\ce{O2} & exFCI & 105.2 & 114.5 & 118.0 &119.1 & 120.0 \\ \ce{O2} & exFCI & 105.2 & 114.5 & 118.0 &119.1 & 120.0 \\
\hline \hline
@ -805,15 +813,16 @@ We begin the investigation of the behavior of the basis-set correction by the st
%%%%%%%%\hline %%%%%%%%\hline
%%%%%%%% & exFCI+PBE & 115.9 & 118.4 & 120.1 &119.9 & \\ %%%%%%%% & exFCI+PBE & 115.9 & 118.4 & 120.1 &119.9 & \\
& exFCI+PBE-val & 117.2 & 119.4 & 120.3 &120.4 & \\ & exFCI+PBE-val & 117.2 & 119.4 & 120.3 &120.4 & \\
\hline
\hline \hline
& (FC)CCSD(T) & 103.9 & 113.6 & 117.1 & 118.6 & 120.0 \\ & (FC)CCSD(T) & 103.9 & 113.6 & 117.1 & 118.6 & 120.0 \\
& ex (FC)CCSD(T)+LDA-val & 110.6 & 117.2 & 119.2 & 119.8 & \\ & (FC)CCSD(T)+LDA-val & 110.6 & 117.2 & 119.2 & 119.8 & \\
& ex (FC)CCSD(T)+PBE-val & 115.1 & 118.0 & 119.3 & 119.8 & \\ & (FC)CCSD(T)+PBE-val & 115.1 & 118.0 & 119.3 & 119.8 & \\
\hline \hline
%%%%%%%% & exFCI+PBE-on-top & 115.0 & 118.4 & 120.2 & & \\ %%%%%%%% & exFCI+PBE-on-top & 115.0 & 118.4 & 120.2 & & \\
%%%%%%%% & exFCI+PBE-on-top-val & 116.1 & 119.4 & 120.5 & & \\ %%%%%%%% & exFCI+PBE-on-top-val & 116.1 & 119.4 & 120.5 & & \\
\\ \\
Molecule & Method & \tabc{cc-pVDZ} & \tabc{cc-pVTZ} & \tabc{cc-pVQZ} & \tabc{cc-pV5Z} & \tabc{$E_{QZ5Z}^{\infty}$} Molecule & Method & \tabc{cc-pVDZ} & \tabc{cc-pVTZ} & \tabc{cc-pVQZ} & \tabc{cc-pV5Z} & \tabc{$D_e^{Q5Z}$}
\\ \\
\ce{F2} & exFCI & 26.7 & 35.1 & 37.1 & 38.0 & 39.0 \\ \ce{F2} & exFCI & 26.7 & 35.1 & 37.1 & 38.0 & 39.0 \\
\hline \hline
@ -822,13 +831,13 @@ We begin the investigation of the behavior of the basis-set correction by the st
%%%%%%%% \hline %%%%%%%% \hline
%%%%%%%% & exFCI+PBE & 33.3 & 37.8 & 38.8 & 38.7 & \\ %%%%%%%% & exFCI+PBE & 33.3 & 37.8 & 38.8 & 38.7 & \\
& exFCI+PBE -val & 33.1 & 37.9 & 38.5 & 38.9 & \\ & exFCI+PBE -val & 33.1 & 37.9 & 38.5 & 38.9 & \\
\hline
%%%%%%%% & exFCI+PBE-on-top& 32.1 & 37.5 & 38.7 & 38.7 & \\ %%%%%%%% & exFCI+PBE-on-top& 32.1 & 37.5 & 38.7 & 38.7 & \\
%%%%%%%% & exFCI+PBE-on-top-val & 32.4 & 37.8 & 38.8 & 38.8 & \\ %%%%%%%% & exFCI+PBE-on-top-val & 32.4 & 37.8 & 38.8 & 38.8 & \\
\hline
\hline \hline
& (FC)CCSD(T) & 25.7 & 34.4 & 36.5 & 37.4 & 38.2 \\ & (FC)CCSD(T) & 25.7 & 34.4 & 36.5 & 37.4 & 38.2 \\
& ex (FC)CCSD(T)+LDA-val & 29.2 & 36.5 & 37.2 & 38.2 & \\ & (FC)CCSD(T)+LDA-val & 29.2 & 36.5 & 37.2 & 38.2 & \\
& ex (FC)CCSD(T)+PBE-val & 31.5 & 37.1 & 37.8 & 38.2 & \\ & (FC)CCSD(T)+PBE-val & 31.5 & 37.1 & 37.8 & 38.2 & \\
\end{tabular} \end{tabular}
\end{ruledtabular} \end{ruledtabular}