srDFT_G2/Manuscript/G2-srDFT.tex

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\newcommand{\LCPQ}{Laboratoire de Chimie et Physique Quantiques (UMR 5626), Universit\'e de Toulouse, CNRS, UPS, France}
\newcommand{\LCT}{Laboratoire de Chimie Th\'eorique, Universit\'e Pierre et Marie Curie, Sorbonne Universit\'e, CNRS, Paris, France}

\begin{document}	

\title{G2 Atomization Energies With Chemical Accuracy}

\author{Bath\'elemy Pradines}
\affiliation{\LCPQ}
\author{Anthony Scemama}
\affiliation{\LCPQ}
\author{Julien Toulouse}
\affiliation{\LCT}
\author{Pierre-Fran\c{c}ois Loos}
\email[Corresponding author: ]{loos@irsamc.ups-tlse.fr}
\affiliation{\LCPQ}
\author{Emmanuel Giner}
\affiliation{\LCT}

\begin{abstract}
\end{abstract}

\maketitle

%%%%%%%%%%%%%%%%%%%%%%%%
\section{Introduction}
%%%%%%%%%%%%%%%%%%%%%%%%



%%%%%%%%%%%%%%%%%%%%%%%%
\section{Theory}
%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{The DFT basis-set correction in a nutshell}
The basis-set correction investigated here proposes to use the RSDFT formalism to capture a part of the short-range correlation effects missing in a finite one-electron basis-set.  
In a nutshell, this formalism relies on 1) the definition of a complementary density functional aiming at describing the correlation effects absent in a finite basis-set, 2) the definition of an \textit{effective non divergent interaction} as the real-space representation of the coulomb operator projected in a finite basis-set, 
3) the fit of such an effective interaction with a long-range interaction through the definition of a \textit{range-separation parameter varying in space}, 4) the use of a correlation functional from RSDFT with a \textit{multi-determinant} reference evaluated with the range-separation parameter varying in space.  
More details can be found in \cite{GinPraFerAssSavTou-JCP-18}. 

\subsubsection{Definition of basis-set dependent complementary functional}
The 

%%%%%%%%%%%%%%%%%%%%%%%%
\section{Results}
%%%%%%%%%%%%%%%%%%%%%%%%

\subsection{The case of C$_2$ and the comparison with the F12 methods.}

\begin{table*}
\caption{Dissociation energy ($D_e$) in kcal/mol of the F$_2$ molecule computed using FCIQMC, CIPSI, FCIQMC+F$_{12}$, CIPSI+LDA$_{\rm HF}$ and CIPSI+LDA$_{\text{HF-val}}$ (valence only interaction and density) in the Dunnng cc-pVXZ (VXZ) basis sets.  $^a$ Results from Ref\cite{PetTouUmr-JCP-12} taking into account the ZPE correction. }
\begin{ruledtabular}
\begin{tabular}{lccccc}
%\hline
                 & CIPSI              & CIPSI+LDA$_{\text{HF}}$  & CIPSI+LDA${_\text{HF-val}}$ & CIPSI+PBE$_{\text{HF}}$  & CIPSI+PBE${_\text{HF-val}}$     \\
\hline                                                                                                                          
V2Z              &  27.5              &  30.8                    & 31.1                        &  32.1                    &   32.4                         \\
V3Z              &  35.4              &  37.0                    & 37.5                        &  37.5                    &   37.8                         \\
V4Z              &  37.5              &  38.7                    & 38.8                        &  38.7                    &   38.8                         \\
V5Z              &  38.0              &  38.7                    & 38.8                        &  38.7                    &   38.8                         \\
%\hline
 & \multicolumn{5}{c}{Estimated exact} \\
 & \multicolumn{5}{c}{  38.2$^a$} \\
\end{tabular}
\end{ruledtabular}
\label{conv_He_table}
\end{table*}
\begin{table*}
	\caption{
		\label{tab:diatomics}
		Dissociation energy ($\De$) in kcal/mol of the \ce{C2}, \ce{O2}, \ce{N2} and \ce{F2} molecules computed with various methods and basis sets.
		}
		\begin{ruledtabular}
			\begin{tabular}{llddddd}
							&					&	\mc{4}{c}{Dunning's basis set}													
				\\
													\cline{3-6}
				Molecule	&	Method			&	\tabc{cc-pVDZ}	&	\tabc{cc-pVTZ}	&	\tabc{cc-pVQZ}	&	\tabc{cc-pV5Z}	&	\tabc{Exp.}	
				\\
				\hline
				\ce{C2}		&	FCIQMC			&	130.0(1)	&	139.9(3)	&	143.3(2)	&				&	146.9(5)\fnm[1]		\\
							&	FCIQMC+F12		&	142.3		&	145.3		&				&				&					\\
				\hline
							&	exFCI			&	132.0		&	140.3		&	143.6		&	144.3		&					\\	
				\hline
							&	exFCI+LDA		&	141.9		&	142.8		&	145.8		&	146.2		&					\\	
							&	exFCI+LDA(FC)	&	142.9		&	145.5		&	146.2		&	146.1		&					\\	
				\hline
							&	exFCI+PBE       &	146.1		&	143.9		&	145.9		&	145.12	&					\\	
							&	exFCI+PBE       (FC)	&	147.7		&	146.3		&	146.4		&	146.0		&					\\	
				\hline
							&	exFCI+PBE-on-top&	142.7		&	142.7		&	145.3		&	144.9		&					\\	
							&	exFCI+PBE-on-top(FC)	&	143.3		&	144.7		&	145.7		&	145.6		&					\\	
				\\
				\ce{N2}		&	exFCI			&	200.9		&	217.1		&	223.5		&	225.7		&	228.5\fnm[2]			\\
				\hline
							&	exFCI+LDA		&	216.3		&	223.1		&	227.9		&	227.9		&					\\
							&	exFCI+LDA(FC)	&	218.2		&	225.8		&	228.8		&	228.4		&					\\	
				\hline
							&	exFCI+PBE       &	225.3		&	225.6		&	228.2		&	227.9		&					\\	
							&	exFCI+PBE       (FC)	&	228.6		&	228.1		&	228.9		&	228.6		&					\\	
				\hline
							&	exFCI+PBE-on-top&	222.3		&	224.6		&	227.7		&	227.7		&					\\	
							&	exFCI+PBE-on-top(FC)	&	224.8		&	226.7		&	228.3		&	228.3		&					\\	
				\\
				\ce{O2}		&	exFCI			&	105.3		&	114.6		&	118.0		&119.1				&	120.2\fnm[2]			\\
				\hline
							&	exFCI+LDA		&	111.8		&	117.2		&	120.0		&119.9				&					\\
							&	exFCI+LDA(FC)	&	112.5		&	118.5		&	120.2		&	120.2			&					\\	
				\hline
							&	exFCI+PBE       &	115.9		&	118.4		&	120.1		&119.9				&					\\	
							&	exFCI+PBE       (FC)	&	117.5		&	119.5		&	120.4		&120.3				&					\\	
				\hline
							&	exFCI+PBE-on-top&	115.0		&	118.4		&	120.2		&				&					\\	
							&	exFCI+PBE-on-top(FC)	&	116.1		&	119.4		&	120.5		&				&					\\	
				\\
				\ce{F2}		&	exFCI			&	27.5		&	35.4		&	37.5		&	38.0		&	38.2\fnm[2]			\\
				\hline
							&	exFCI+LDA		&	30.8		&	37.0		&	38.7		&	38.7		&					\\
							&	exFCI+LDA(FC)	&	31.1		&	37.5		&	38.8		&	38.8		&					\\	
				\hline
							&	exFCI+PBE       &	33.3		&	37.8		&	38.8		&       38.7				&					\\	
							&	exFCI+PBE       (FC)	&	33.9		&	38.2		&	39.0		&	38.8		&					\\	
				\hline
							&	exFCI+PBE-on-top&	32.1		&	37.5		&	38.7		&       38.7				&					\\	
							&	exFCI+PBE-on-top(FC)	&	32.4		&	37.8		&	38.8		&	38.8		&					\\	

		\end{tabular}
	\end{ruledtabular}
	\fnt[1]{Results from Ref.~\onlinecite{BytLaiRuedenJCP05}.}
	\fnt[2]{Results from Ref.~\onlinecite{PetTouUmr-JCP-12}.}
\end{table*}




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\end{document}