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Manuscript/G2-srDFT.tex
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@ -89,7 +89,7 @@
% basis sets
\newcommand{\Bas}{\mathcal{B}}
\newcommand{\BasFC}{\Bar{\mathcal{B}}}
\newcommand{\BasFC}{\mathcal{A}}
\newcommand{\FC}{\text{FC}}
\newcommand{\occ}{\text{occ}}
\newcommand{\virt}{\text{virt}}
@ -119,6 +119,9 @@
\title{A Density-Based Basis Set Correction For Wave Function Theory}
\author{Pierre-Fran\c{c}ois Loos}
\email{loos@irsamc.ups-tlse.fr}
\affiliation{\LCPQ}
\author{Bath\'elemy Pradines}
\affiliation{\LCT}
\affiliation{\ISCD}
@ -127,9 +130,6 @@
\author{Julien Toulouse}
\email{toulouse@lct.jussieu.fr}
\affiliation{\LCT}
\author{Pierre-Fran\c{c}ois Loos}
\email{loos@irsamc.ups-tlse.fr}
\affiliation{\LCPQ}
\author{Emmanuel Giner}
\email{emmanuel.giner@lct.jussieu.fr}
\affiliation{\LCT}
@ -205,7 +205,7 @@ Importantly, in the limit of a complete basis set (which we refer to as $\Bas \t
\lim_{\Bas \to \infty} \qty( \E{\modY}{\Bas} + \bE{}{\Bas}[\n{\modZ}{\Bas}] ) = \E{\modY}{} \approx E,
\end{equation}
where $\E{\modY}{}$ is the energy associated with the method $\modY$ in the complete basis set.
In the case $\modY = \FCI$ \manu{in \eqref{eq:limitfunc}}, we have a strict equality as $\E{\FCI}{} = \E{}{}$.
In the case where $\modY = \FCI$ in Eq.~\eqref{eq:limitfunc}, we have a strict equality as $\E{\FCI}{} = \E{}{}$.
Provided that the functional $\bE{}{\Bas}[\n{}{}]$ is known exactly, the only source of error at this stage lies in the potential approximate nature of the methods $\modY$ and $\modZ$.% for the FCI energy and density within $\Bas$, respectively.
The functional $\bE{}{\Bas}[\n{}{}]$ is obviously \textit{not} universal as it depends on $\Bas$.
@ -239,7 +239,7 @@ where
\n{2}{}(\br{1},\br{2})
= \sum_{pqrs \in \Bas} \SO{p}{1} \SO{q}{2} \Gam{pq}{rs} \SO{r}{1} \SO{s}{2},
\end{equation}
and $\Gam{pq}{rs} = \manu{1/2\mel*{\wf{}{\Bas}}{ \aic{r\downarrow}\aic{s\uparrow}\ai{p\uparrow}\ai{q\downarrow} + \aic{r\uparrow}\aic{s\downarrow}\ai{p\downarrow}\ai{q\uparrow} }{\wf{}{\Bas}}}$ are the opposite-spin pair density associated with $\wf{}{\Bas}$ and its corresponding tensor (respectively), $\SO{p}{}$ is a \trashMG{spinorbital}\manu{spatial orbital},
and $\Gam{pq}{rs} =\mel*{\wf{}{\Bas}}{ \aic{r_\downarrow}\aic{s_\uparrow}\ai{p_\uparrow}\ai{q_\downarrow}}{\wf{}{\Bas}}$ are the opposite-spin pair density associated with $\wf{}{\Bas}$ and its corresponding tensor (respectively), $\SO{p}{}$ is a molecular orbital (MO),
\begin{equation}
\label{eq:fbasis}
\f{\Bas}{}(\br{1},\br{2})
@ -377,8 +377,8 @@ Depending on the functional choice, the complementary functional $\bE{}{\Bas}[\n
%\subsection{Valence approximation}
%=================================================================
As most WFT calculations are performed within the frozen-core (FC) approximation, it is important to define an effective interaction within a subset of spinorbitals.
We then naturally split the basis set as $\Bas = \Cor \bigcup \BasFC$, where $\Cor$ is the set of core \trashMG{spinorbitals} \manu{spatial orbitals}, and define the FC version of the effective interaction as
As most WFT calculations are performed within the frozen-core (FC) approximation, it is important to define an effective interaction within a subset of MOs.
We then naturally split the basis set as $\Bas = \Cor \bigcup \BasFC$, where $\Cor$ is the set of core MOs, and define the FC version of the effective interaction as
\begin{equation}
\W{\Bas}{\FC}(\br{1},\br{2}) =
\begin{cases}
@ -488,7 +488,6 @@ iii) vanishes in the limit of a complete basis set, hence guaranteeing an unalte
\end{figure*}
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).
\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 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 atomization energies of the entire G2 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}) \titou{and can be considered as a representative set for typical quantum chemical calculations on small organic molecules}.
@ -503,7 +502,7 @@ RS-DFT and exFCI calculations are performed with {\QP}. \cite{QP2}
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}.
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.
In the context of the basis set correction, the set of MOs $\BasFC$ involved in the definition of the effective interaction refers to the non-frozen MOs.
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, \manu{following the work of Ref.~\onlinecite{HalHelJorKloKocOlsWil-CPL-98}},
we employ the two-point extrapolation for the correlation energies \manu{for each model in quadruple- and quintuple-$\zeta$ basis sets, which is refered to as $\CBS$ correlation energies, and we add the HF energies in the largest basis sets (\textit{i.e.} quintuple-$\zeta$ quality basis sets) to the CBS correlation energies to estimate the CBS FCI energies.}

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Manuscript/SI/G2_srDFT-SI.tex
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@ -118,16 +118,18 @@
\title{Supplementary Information for ``A Density-Based Basis Set Correction For Wave Function Theory''}
\author{Pierre-Fran\c{c}ois Loos}
\email{loos@irsamc.ups-tlse.fr}
\affiliation{\LCPQ}
\author{Bath\'elemy Pradines}
\affiliation{\LCPQ}
\author{Anthony Scemama}
\affiliation{\LCPQ}
\author{Julien Toulouse}
\email{toulouse@lct.jussieu.fr}
\affiliation{\LCT}
\author{Pierre-Fran\c{c}ois Loos}
\email[Corresponding author: ]{loos@irsamc.ups-tlse.fr}
\affiliation{\LCPQ}
\author{Emmanuel Giner}
\email{emmanuel.giner@lct.jussieu.fr}
\affiliation{\LCT}
\begin{abstract}
@ -139,51 +141,51 @@
\begin{table*}
\caption{
\label{tab:diatomics}
Atomization energies (in {\kcal}) of \ce{C2}, \ce{O2}, \ce{N2} and \ce{F2} computed with various methods and basis sets.
Frozen-core atomization energies (in {\kcal}) of \ce{C2}, \ce{O2}, \ce{N2} and \ce{F2} computed with various methods and basis sets.
The deviations with respect to the corresponding CBS values are reported in parenthesis.
}
\begin{ruledtabular}
\begin{tabular}{llddddd}
& & \mc{4}{c}{Dunning's basis set}
& & \mc{4}{c}{Dunning's basis set cc-pVXZ}
\\
\cline{3-6}
Molecule & Method & \tabc{$\X = \D$} & \tabc{$\X = \T$} & \tabc{$\X = \Q$} & \tabc{$\X = 5$} & \tabc{CBS}
\\
\hline
\ce{C2} & exFCI\fnm[1] & 132.0 (-13.7 ) & 140.3 (-5.4 ) & 143.6 (-2.1 ) & 144.7 (-1.0 ) & 145.7 \\
(cc-pVXZ) & exFCI+LDA\fnm[1] & 141.3 (-4.4 ) & 145.1 (-0.6 ) & 146.4 (+0.7 ) & 146.3 (+0.6 ) & \\
& exFCI+PBE\fnm[1] & 145.7 (+0.0 ) & 145.7 (+0.0 ) & 146.3 (+0.6 ) & 146.2 (+0.5 ) & \\
& CCSD(T)\fnm[1] & 129.2 (-16.2 ) & 139.1 (-6.3 ) & 143.0 (-2.4 ) & 144.2 (-1.2 ) & 145.4 \\
& CCSD(T)+LDA\fnm[1] & 139.1 (-6.3 ) & 143.7 (-1.7 ) & 145.9 (+0.5 ) & 145.9 (+0.5 ) & \\
& CCSD(T)+PBE\fnm[1] & 142.8 (-2.6 ) & 144.2 (-1.2 ) & 145.9 (+0.5 ) & 145.8 (+0.4 ) & \\ \\
\ce{C2} & exFCI\fnm[2] & 131.0 (-16.1 ) & 141.5 (-5.6 ) & 145.1 (-2.0 ) & 146.1 (-1.0 ) & 147.1 \\
(cc-pCVXZ) & exFCI+LDA\fnm[2] & 141.4 (-5.7 ) & 146.7 (-0.4 ) & 147.8 (+0.7 ) & 147.6 (+0.5 ) & \\
& exFCI+PBE\fnm[2] & 145.1 (-2.0 ) & 147.0 (-0.1 ) & 147.7 (+0.6 ) & 147.5 (+0.4 ) & \\ \\
\ce{N2} & exFCI\fnm[1] & 201.1 (-26.7 ) & 217.1 (-10.7 ) & 223.5 (-4.3 ) & 225.7 (-2.1 ) & 227.8 \\
(cc-pVXZ) & exFCI+LDA\fnm[1] & 217.9 (-9.9 ) & 225.9 (-1.9 ) & 228.0 (+0.2 ) & 228.6 (+0.8 ) & \\
& exFCI+PBE\fnm[1] & 227.7 (-0.1 ) & 227.8 (+0.0 ) & 228.3 (+0.5 ) & 228.5 (+0.7 ) & \\
& CCSD(T)\fnm[1] & 199.9 (-27.3 ) & 216.3 (-10.9 ) & 222.8 (-4.4 ) & 225.0 (-2.2 ) & 227.2 \\
& CCSD(T)+LDA\fnm[1] & 216.3 (-10.9 ) & 224.8 (-2.4 ) & 227.2 (-0.0 ) & 227.8 (+0.6 ) & \\
& CCSD(T)+PBE\fnm[1] & 225.9 (-1.3 ) & 226.7 (-0.5 ) & 227.5 (+0.3 ) & 227.8 (+0.6 ) & \\ \\
\ce{N2} & exFCI\fnm[2] & 202.2 (-26.6 ) & 218.5 (-10.3 ) & 224.4 (-4.4 ) & 226.6 (-2.2 ) & 228.8 \\
(cc-pCVXZ) & exFCI+LDA\fnm[2] & 218.0 (-10.8 ) & 226.8 (-2.0 ) & 229.1 (+0.3 ) & 229.4 (+0.6 ) & \\
& exFCI+PBE\fnm[2] & 226.4 (-2.4 ) & 228.2 (-0.6 ) & 229.1 (+0.3 ) & 229.2 (+0.4 ) & \\ \\
\ce{O2} & exFCI\fnm[1] & 105.2 (-14.8 ) & 114.5 (-5.5 ) & 118.0 (-2.0 ) & 119.1 (-0.9 ) & 120.0 \\
(cc-pVXZ) & exFCI+LDA\fnm[1] & 112.4 (-7.6 ) & 118.4 (-1.6 ) & 120.2 (+0.2 ) & 120.4 (+0.4 ) & \\
& exFCI+PBE\fnm[1] & 117.2 (-2.8 ) & 119.4 (-0.6 ) & 120.3 (+0.3 ) & 120.4 (+0.4 ) & \\
& CCSD(T)\fnm[1] & 103.9 (-16.1 ) & 113.6 (-6.0 ) & 117.1 (-2.5 ) & 118.6 (-1.0 ) & 119.6 \\
& CCSD(T)+LDA\fnm[1] & 110.6 (-9.0 ) & 117.2 (-2.4 ) & 119.2 (-0.4 ) & 119.8 (+0.2 ) & \\
& CCSD(T)+PBE\fnm[1] & 115.1 (-4.5 ) & 118.0 (-1.6 ) & 119.3 (-0.3 ) & 119.8 (+0.2 ) & \\ \\
\ce{F2} & exFCI\fnm[1] & 26.7 (-12.3 ) & 35.1 (-3.9 ) & 37.1 (-1.9 ) & 38.0 (-1.0 ) & 39.0 \\
(cc-pVXZ) & exFCI+LDA\fnm[1] & 30.4 (-8.6 ) & 37.2 (-1.8 ) & 38.4 (-0.6 ) & 38.9 (-0.1 ) & \\
& exFCI+PBE\fnm[1] & 33.1 (-5.9 ) & 37.9 (-1.1 ) & 38.5 (-0.5 ) & 38.9 (-0.1 ) & \\
& CCSD(T)\fnm[1] & 25.7 (-12.5 ) & 34.4 (-3.8 ) & 36.5 (-1.7 ) & 37.4 (-0.8 ) & 38.2 \\
& CCSD(T)+LDA\fnm[1] & 29.2 (-9.0 ) & 36.5 (-1.7 ) & 37.2 (-1.0 ) & 38.2 (+0.0 ) & \\
& CCSD(T)+PBE\fnm[1] & 31.5 (-6.7 ) & 37.1 (-1.1 ) & 37.8 (-0.4 ) & 38.2 (+0.0 ) & \\
\ce{C2} & exFCI & 132.0 (-13.7 ) & 140.3 (-5.4 ) & 143.6 (-2.1 ) & 144.7 (-1.0 ) & 145.7 \\
& exFCI+LDA & 141.3 (-4.4 ) & 145.1 (-0.6 ) & 146.4 (+0.7 ) & 146.3 (+0.6 ) & \\
& exFCI+PBE & 145.7 (+0.0 ) & 145.7 (+0.0 ) & 146.3 (+0.6 ) & 146.2 (+0.5 ) & \\
& CCSD(T) & 129.2 (-16.2 ) & 139.1 (-6.3 ) & 143.0 (-2.4 ) & 144.2 (-1.2 ) & 145.4 \\
& CCSD(T)+LDA & 139.1 (-6.3 ) & 143.7 (-1.7 ) & 145.9 (+0.5 ) & 145.9 (+0.5 ) & \\
& CCSD(T)+PBE & 142.8 (-2.6 ) & 144.2 (-1.2 ) & 145.9 (+0.5 ) & 145.8 (+0.4 ) & \\ \\
% \ce{C2} & exFCI\fnm[2] & 131.0 (-16.1 ) & 141.5 (-5.6 ) & 145.1 (-2.0 ) & 146.1 (-1.0 ) & 147.1 \\
% (cc-pCVXZ) & exFCI+LDA\fnm[2] & 141.4 (-5.7 ) & 146.7 (-0.4 ) & 147.8 (+0.7 ) & 147.6 (+0.5 ) & \\
% & exFCI+PBE\fnm[2] & 145.1 (-2.0 ) & 147.0 (-0.1 ) & 147.7 (+0.6 ) & 147.5 (+0.4 ) & \\ \\
\ce{N2} & exFCI & 201.1 (-26.7 ) & 217.1 (-10.7 ) & 223.5 (-4.3 ) & 225.7 (-2.1 ) & 227.8 \\
& exFCI+LDA & 217.9 (-9.9 ) & 225.9 (-1.9 ) & 228.0 (+0.2 ) & 228.6 (+0.8 ) & \\
& exFCI+PBE & 227.7 (-0.1 ) & 227.8 (+0.0 ) & 228.3 (+0.5 ) & 228.5 (+0.7 ) & \\
& CCSD(T) & 199.9 (-27.3 ) & 216.3 (-10.9 ) & 222.8 (-4.4 ) & 225.0 (-2.2 ) & 227.2 \\
& CCSD(T)+LDA & 216.3 (-10.9 ) & 224.8 (-2.4 ) & 227.2 (-0.0 ) & 227.8 (+0.6 ) & \\
& CCSD(T)+PBE & 225.9 (-1.3 ) & 226.7 (-0.5 ) & 227.5 (+0.3 ) & 227.8 (+0.6 ) & \\ \\
% \ce{N2} & exFCI\fnm[2] & 202.2 (-26.6 ) & 218.5 (-10.3 ) & 224.4 (-4.4 ) & 226.6 (-2.2 ) & 228.8 \\
% (cc-pCVXZ) & exFCI+LDA\fnm[2] & 218.0 (-10.8 ) & 226.8 (-2.0 ) & 229.1 (+0.3 ) & 229.4 (+0.6 ) & \\
% & exFCI+PBE\fnm[2] & 226.4 (-2.4 ) & 228.2 (-0.6 ) & 229.1 (+0.3 ) & 229.2 (+0.4 ) & \\ \\
\ce{O2} & exFCI & 105.2 (-14.8 ) & 114.5 (-5.5 ) & 118.0 (-2.0 ) & 119.1 (-0.9 ) & 120.0 \\
& exFCI+LDA & 112.4 (-7.6 ) & 118.4 (-1.6 ) & 120.2 (+0.2 ) & 120.4 (+0.4 ) & \\
& exFCI+PBE & 117.2 (-2.8 ) & 119.4 (-0.6 ) & 120.3 (+0.3 ) & 120.4 (+0.4 ) & \\
& CCSD(T) & 103.9 (-16.1 ) & 113.6 (-6.0 ) & 117.1 (-2.5 ) & 118.6 (-1.0 ) & 119.6 \\
& CCSD(T)+LDA & 110.6 (-9.0 ) & 117.2 (-2.4 ) & 119.2 (-0.4 ) & 119.8 (+0.2 ) & \\
& CCSD(T)+PBE & 115.1 (-4.5 ) & 118.0 (-1.6 ) & 119.3 (-0.3 ) & 119.8 (+0.2 ) & \\ \\
\ce{F2} & exFCI & 26.7 (-12.3 ) & 35.1 (-3.9 ) & 37.1 (-1.9 ) & 38.0 (-1.0 ) & 39.0 \\
& exFCI+LDA & 30.4 (-8.6 ) & 37.2 (-1.8 ) & 38.4 (-0.6 ) & 38.9 (-0.1 ) & \\
& exFCI+PBE & 33.1 (-5.9 ) & 37.9 (-1.1 ) & 38.5 (-0.5 ) & 38.9 (-0.1 ) & \\
& CCSD(T) & 25.7 (-12.5 ) & 34.4 (-3.8 ) & 36.5 (-1.7 ) & 37.4 (-0.8 ) & 38.2 \\
& CCSD(T)+LDA & 29.2 (-9.0 ) & 36.5 (-1.7 ) & 37.2 (-1.0 ) & 38.2 (+0.0 ) & \\
& CCSD(T)+PBE & 31.5 (-6.7 ) & 37.1 (-1.1 ) & 37.8 (-0.4 ) & 38.2 (+0.0 ) & \\
\end{tabular}
\end{ruledtabular}
\fnt[1]{Frozen core calculations. Only valence spinorbitals are taken into account in the basis set correction.}
\fnt[2]{``Full'' calculation, i.e., all electrons are correlated. All spinorbitals are taken into account in the basis set correction.}
% \fnt[1]{ calculations. Only valence orbitals are taken into account in the basis set correction.}
% \fnt[2]{``Full'' calculation, i.e., all electrons are correlated. All spinorbitals are taken into account in the basis set correction.}
\end{table*}
@ -192,11 +194,11 @@
\begin{table}
\caption{
\label{tab:AE}
Deviation from the reference CBS correlation energies (in {\kcal}) for various methods and basis sets.}
Deviation from the reference CBS atomization energies (in {\kcal}) for various methods and basis sets.}
\begin{ruledtabular}
\begin{tabular}{lddddddddddd}
&
& \mc{10}{c}{Deviation from CBS correlation energies} \\
& \mc{10}{c}{Deviation from CBS atomization energies} \\
\cline{3-12}
& & \mc{4}{c}{CCSD(T)} & \mc{3}{c}{CCSD(T)+LDA} & \mc{3}{c}{CCSD(T)+PBE} \\
\cline{3-6} \cline{7-9} \cline{10-12}