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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.
 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.