diff --git a/JPCL_revision/G2-srDFT.tex b/JPCL_revision/G2-srDFT.tex index 321f8ed..80cbdeb 100644 --- a/JPCL_revision/G2-srDFT.tex +++ b/JPCL_revision/G2-srDFT.tex @@ -1,5 +1,5 @@ \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. 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