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Pierre-Francois Loos 2019-05-08 21:30:59 +02:00
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\documentclass[aip,jcp,preprint,noshowkeys]{revtex4-1}
\usepackage{graphicx,dcolumn,bm,xcolor,microtype,multirow,amscd,amsmath,amssymb,amsfonts,physics,mhchem,longtable,xspace,wrapfig}
\usepackage{graphicx,dcolumn,bm,xcolor,microtype,multirow,amscd,amsmath,amssymb,amsfonts,physics,mhchem,longtable,xspace,float}
\usepackage{mathpazo,libertine}
\usepackage{natbib}
@ -145,14 +145,16 @@
\affiliation{\LCT}
\begin{abstract}
\begin{wrapfigure}[7]{o}[-1.2cm]{0.4\linewidth}
\centering
\includegraphics[width=\linewidth]{TOC}
\end{wrapfigure}
We report a universal density-based basis-set incompleteness correction that can be applied to any wave function method.
The present correction, which appropriately vanishes in the complete basis set (CBS) limit, relies on short-range correlation density functionals (with multi-determinant reference) from range-separated density-functional theory (RS-DFT) to estimate the basis-set incompleteness error.
Contrary to conventional RS-DFT schemes which require an \textit{ad hoc} range-separation \textit{parameter} $\mu$, the key ingredient here is a range-separation \textit{function} $\mu(\bf{r})$ that automatically adapts to the spatial non-homogeneity of the basis-set incompleteness error.
As illustrative examples, we show how this density-based correction allows us to obtain CCSD(T) atomization and correlation energies near the CBS limit for the G2 set of molecules with compact Gaussian basis sets.
\begin{figure}[H]
\centering
\includegraphics[width=0.5\linewidth]{TOC}
\\
\bf TOC Graphic
\end{figure}
\end{abstract}
\maketitle
@ -223,7 +225,7 @@ This implies that
where \titou{$\E{\CCSDT}{}$ is the $\CCSDT$ energy} in the CBS limit.
\titou{Of course, the above holds true for any method that provides a good approximation to the energy and density, not just CCSD(T) and HF.}
%In the case where $\modY = \FCI$ in Eq.~\eqref{eq:limitfunc}, we have a strict equality as $\E{\FCI}{} = \E{}{}$.
In the case where \titou{$\CCSDT$ and $\HF$ are replaced by $\FCI$} in Eq.~\eqref{eq:limitfunc}, we have a strict equality as $\E{\FCI}{} = \E{}{}$.
In the case where \titou{$\CCSDT$ is replaced by $\FCI$} 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 sources of error at this stage lie in the potential approximate nature of the methods $\modY$ and $\modZ$, and the lack of self-consistency in the present scheme.
Provided that the functional $\bE{}{\Bas}[\n{}{}]$ is known exactly, the only sources of error at this stage lie in the potential approximate nature of the \titou{$\CCSDT$ and $\HF$ methods}, and the lack of self-consistency in the present scheme.
@ -415,54 +417,12 @@ iii) vanishes in the CBS limit, hence guaranteeing an unaltered CBS limit for a
\hspace{1cm}
\includegraphics[width=0.30\linewidth]{fig1d}
\caption{
\manu{Les graphs 1a et ab sont les identiques !!}
Deviation (in \kcal) from CBS atomization energies of \ce{C2} (top left), \ce{O2} (top right), \ce{N2} (bottom left) and \ce{F2} (bottom right) obtained with various methods and basis sets.
The green region corresponds to chemical accuracy (i.e.~error below 1 {\kcal}).
See {\SI} for raw data \titou{and the corresponding LDA results}.
\label{fig:diatomics}}
\end{figure*}
%%% TABLE II %%%
\begin{table}
\caption{
Statistical analysis (in \kcal) of the G2 atomization energies depicted in Fig.~\ref{fig:G2_Ec}.
Mean absolute deviation (MAD), root-mean-square deviation (RMSD), and maximum deviation (MAX) with respect to the CCSD(T)/CBS reference atomization energies.
CA corresponds to the number of cases (out of 55) obtained with chemical accuracy.
See {\SI} for raw data \titou{and the corresponding LDA results}.
\label{tab:stats}}
\begin{ruledtabular}
\begin{tabular}{ldddd}
Method & \tabc{MAD} & \tabc{RMSD} & \tabc{MAX} & \tabc{CA} \\
\hline
CCSD(T)/cc-pVDZ & 14.29 & 16.21 & 36.95 & 2 \\
CCSD(T)/cc-pVTZ & 6.06 & 6.84 & 14.25 & 2 \\
CCSD(T)/cc-pVQZ & 2.50 & 2.86 & 6.75 & 9 \\
CCSD(T)/cc-pV5Z & 1.28 & 1.46 & 3.46 & 21 \\
% \\
% CCSD(T)+LDA/cc-pVDZ & 3.24 & 3.67 & 8.13 & 7 \\
% CCSD(T)+LDA/cc-pVTZ & 1.19 & 1.49 & 4.67 & 27 \\
% CCSD(T)+LDA/cc-pVQZ & 0.33 & 0.44 & 1.32 & 53 \\
\\
CCSD(T)+PBE/cc-pVDZ & 1.96 & 2.59 & 7.33 & 19 \\
CCSD(T)+PBE/cc-pVTZ & 0.85 & 1.11 & 2.64 & 36 \\
CCSD(T)+PBE/cc-pVQZ & 0.31 & 0.42 & 1.16 & 53 \\
\end{tabular}
\end{ruledtabular}
\end{table}
%%% FIGURE 2 %%%
\begin{figure*}
\includegraphics[width=\linewidth]{fig2a}
\includegraphics[width=\linewidth]{fig2b}
% \includegraphics[width=\linewidth]{fig2c}
\caption{
Deviation (in \kcal) from the CCSD(T)/CBS atomization energy obtained with \titou{various basis sets for CCSD(T) (top) and CCSD(T)+PBE (bottom).}
% Deviation (in \kcal) from the CCSD(T)/CBS 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}).
\titou{Note the difference in scaling of the vertical axes.}
See {\SI} for raw data \titou{and the corresponding LDA results}.
\label{fig:G2_Ec}}
\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 (X $=$ D, T, Q and 5).
\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}
@ -499,6 +459,59 @@ In most cases, the basis-set corrected triple-$\zeta$ atomization energies are o
%However, from the quadruple-$\zeta$ basis, the LDA and PBE functionals agree within a few tenths of a {\kcal}.
%Such weak sensitivity to the density-functional approximation when reaching large basis sets shows the robustness of the approach.
%%% FIGURE 2 %%%
\begin{figure}
\includegraphics[width=0.5\linewidth]{fig2}
\caption{
$\rsmu{}{}(z)$ along the molecular axis ($z$) for \ce{N2} in various basis sets.
The two nitrogen atoms are located at $z=0$ and $z=2.076$ bohr.
\label{fig:N2}}
\end{figure}
Figure \ref{fig:N2} shows $\rsmu{}{}(z)$ along the molecular axis ($z$) for \ce{N2} in various basis sets.
%%% TABLE II %%%
\begin{table}
\caption{
Statistical analysis (in \kcal) of the G2 atomization energies depicted in Fig.~\ref{fig:G2_Ec}.
Mean absolute deviation (MAD), root-mean-square deviation (RMSD), and maximum deviation (MAX) with respect to the CCSD(T)/CBS reference atomization energies.
CA corresponds to the number of cases (out of 55) obtained with chemical accuracy.
See {\SI} for raw data \titou{and the corresponding LDA results}.
\label{tab:stats}}
\begin{ruledtabular}
\begin{tabular}{ldddd}
Method & \tabc{MAD} & \tabc{RMSD} & \tabc{MAX} & \tabc{CA} \\
\hline
CCSD(T)/cc-pVDZ & 14.29 & 16.21 & 36.95 & 2 \\
CCSD(T)/cc-pVTZ & 6.06 & 6.84 & 14.25 & 2 \\
CCSD(T)/cc-pVQZ & 2.50 & 2.86 & 6.75 & 9 \\
CCSD(T)/cc-pV5Z & 1.28 & 1.46 & 3.46 & 21 \\
% \\
% CCSD(T)+LDA/cc-pVDZ & 3.24 & 3.67 & 8.13 & 7 \\
% CCSD(T)+LDA/cc-pVTZ & 1.19 & 1.49 & 4.67 & 27 \\
% CCSD(T)+LDA/cc-pVQZ & 0.33 & 0.44 & 1.32 & 53 \\
\\
CCSD(T)+PBE/cc-pVDZ & 1.96 & 2.59 & 7.33 & 19 \\
CCSD(T)+PBE/cc-pVTZ & 0.85 & 1.11 & 2.64 & 36 \\
CCSD(T)+PBE/cc-pVQZ & 0.31 & 0.42 & 1.16 & 53 \\
\end{tabular}
\end{ruledtabular}
\end{table}
%%% FIGURE 2 %%%
\begin{figure*}
\includegraphics[width=\linewidth]{fig3a}
\includegraphics[width=\linewidth]{fig3b}
% \includegraphics[width=\linewidth]{fig2c}
\caption{
Deviation (in \kcal) from the CCSD(T)/CBS atomization energy obtained with \titou{various basis sets for CCSD(T) (top) and CCSD(T)+PBE (bottom).}
% Deviation (in \kcal) from the CCSD(T)/CBS 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}).
\titou{Note the difference in scaling of the vertical axes.}
See {\SI} for raw data \titou{and the corresponding LDA results}.
\label{fig:G2_Ec}}
\end{figure*}
%As a second set of numerical examples, we compute the error (with respect to the CBS values) of the atomization energies from the G2 test set with $\modY=\CCSDT$, $\modZ=\ROHF$ and the cc-pVXZ basis sets.
As a second set of numerical examples, we compute the error (with respect to the CBS values) of the atomization energies from the G2 test set \titou{with $\CCSDT$} and the cc-pVXZ basis sets.
Here, all atomization energies have been computed with the same near-CBS HF/cc-pV5Z energies; only the correlation energy contribution varies from one method to the other.

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