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\begin{document}	

\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{\LCT}
\affiliation{\ISCD}
\author{Anthony Scemama}
\affiliation{\LCPQ}
\author{Julien Toulouse}
\email{toulouse@lct.jussieu.fr}
\affiliation{\LCT}
\author{Emmanuel Giner}
\email{emmanuel.giner@lct.jussieu.fr}
\affiliation{\LCT}

\begin{abstract}
\end{abstract}

\maketitle

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Local-density approximation}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
The local-density approximation (LDA) of the ECMD complementary functional is defined as 
\begin{equation}
	\label{eq:def_lda_tot}
	\bE{\LDA}{\Bas}[\n{}{},\rsmu{}{\Bas}] = \int \n{}{}(\br{}) \be{\text{c,md}}{\sr,\LDA}\qty(\n{}{}(\br{}),\zeta(\br{}),\rsmu{}{\Bas}(\br{})) \dbr{},
\end{equation}
where $\zeta = (\n{\uparrow}{} - \n{\downarrow}{})/\n{}{}$ is the spin polarization and $\be{\text{c,md}}{\sr,\LDA}(\n{}{},\zeta,\rsmu{}{})$ is the ECMD short-range correlation energy per electron of the uniform electron gas (UEG) \cite{LooGil-WIRES-16} parameterized in Ref.~\onlinecite{PazMorGorBac-PRB-06}.
The short-range LDA correlation functional relies on the transferability of the physics of the UEG which is certainly valid for large $\mu$ but is known to over correlate for small $\mu$. 

%%% TABLE I %%%
\begin{table*}
	\caption{
		\label{tab:diatomics}
		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.
		See main text for more details.
		}
		\begin{ruledtabular}
			\begin{tabular}{llddddd}
							&					&	\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		&	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]{ 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*}


\begin{turnpage}
\begin{squeezetable}
\begin{table}
	\caption{
	\label{tab:AE}
	Deviation from the reference CCSD(T)/CBS atomization energies (in {\kcal}) for various methods and basis sets.
	See main text for more details.}
		\begin{ruledtabular}
			\begin{tabular}{lddddddddddd}
							&	
							&	\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}
				Molecule	&	\tabc{CCSD(T)/CBS}
							&	\tabc{cc-pVDZ} 	&	\tabc{cc-pVTZ} 	&	\tabc{cc-pVQZ}	&	\tabc{cc-pV5Z}	
							&	\tabc{cc-pVDZ} 	&	\tabc{cc-pVTZ} 	&	\tabc{cc-pVQZ}	
							&	\tabc{cc-pVDZ} 	&	\tabc{cc-pVTZ} 	&	\tabc{cc-pVQZ}	
				\\
				\hline	
\ce{BeH}  & 50.12 & -3.38 & -1.01 & -0.36 & -0.19 & -0.75 & -0.03 & 0.04 & -0.75 & -0.03 & 0.04 \\
\ce{C2H2}  & 403.00 & -22.99 & -8.69 & -3.43 & -1.76 & -5.12 & -1.08 & 0.37 & -5.12 & -1.08 & 0.37 \\
\ce{C2H4}  & 561.69 & -27.16 & -9.44 & -3.59 & -1.84 & -4.88 & -0.56 & 0.58 & -4.88 & -0.56 & 0.58 \\
\ce{C2H6}  & 710.81 & -32.62 & -10.65 & -3.97 & -2.03 & -5.28 & -0.20 & 0.76 & -5.28 & -0.20 & 0.76 \\
\ce{CH}  & 83.89 & -5.96 & -1.97 & -0.75 & -0.39 & -0.81 & -0.03 & 0.15 & -0.81 & -0.03 & 0.15 \\
\ce{CH2 ^1A_1}  & 180.61 & -10.99 & -3.57 & -1.34 & -0.68 & -1.55 & 0.00 & 0.29 & -1.55 & 0.00 & 0.29 \\
\ce{CH2 ^3B_1}  & 189.94 & -8.82 & -2.78 & -1.0 & -0.51 & -1.87 & -0.38 & 0.00 & -1.87 & -0.38 & -0.00 \\
\ce{CH3}  & 306.81 & -14.33 & -4.47 & -1.62 & -0.83 & -2.33 & -0.08 & 0.27 & -2.33 & -0.08 & 0.27 \\
\ce{CH3Cl}  & 395.02 & -20.77 & -8.34 & -3.35 & -1.72 & -4.38 & -1.60 & -0.16 & -4.38 & -1.60 & -0.16 \\
\ce{CH4}  & 419.19 & -18.35 & -5.64 & -2.01 & -1.03 & -2.55 & 0.18 & 0.51 & -2.55 & 0.18 & 0.51 \\
\ce{CN}  & 179.32 & -14.98 & -7.26 & -3.18 & -1.63 & -2.16 & -0.82 & 0.20 & -2.16 & -0.82 & 0.20 \\
\ce{CO}  & 258.64 & -11.01 & -5.94 & -2.60 & -1.33 & -0.93 & -0.61 & 0.44 & -0.93 & -0.61 & 0.44 \\
\ce{CO2}  & 388.29 & -20.79 & -10.95 & -4.67 & -2.39 & -3.19 & -1.69 & 0.60 & -3.19 & -1.69 & 0.60 \\
\ce{CS}  & 170.82 & -10.72 & -6.25 & -2.88 & -1.47 & -1.94 & -1.58 & -0.16 & -1.94 & -1.58 & -0.16 \\
\ce{Cl2}  & 59.33 & -8.26 & -5.23 & -2.49 & -1.28 & -3.63 & -2.46 & -0.90 & -3.63 & -2.46 & -0.90 \\
\ce{ClF}  & 62.43 & -8.51 & -4.83 & -2.33 & -1.19 & -4.02 & -2.23 & -0.81 & -4.02 & -2.23 & -0.81 \\
\ce{ClO}  & 64.35 & -11.86 & -6.35 & -3.16 & -1.62 & -5.99 & -3.08 & -1.25 & -5.99 & -3.08 & -1.25 \\
\ce{F2}  & 38.24 & -6.62 & -3.52 & -1.72 & -0.88 & -3.17 & -1.54 & -0.48 & -3.17 & -1.54 & -0.48 \\
\ce{H2CO}  & 373.18 & -19.31 & -8.07 & -3.31 & -1.70 & -2.84 & -0.70 & 0.50 & -2.84 & -0.70 & 0.50 \\
\ce{H2O}  & 232.78 & -15.21 & -5.83 & -2.37 & -1.21 & -4.03 & -1.07 & -0.14 & -4.03 & -1.07 & -0.14 \\
\ce{H2O2}  & 268.77 & -22.78 & -9.44 & -4.08 & -2.09 & -6.64 & -2.06 & -0.30 & -6.64 & -2.06 & -0.30 \\
\ce{H2S}  & 183.36 & -10.66 & -4.33 & -1.55 & -0.79 & -2.50 & -1.16 & -0.12 & -2.50 & -1.16 & -0.12 \\
\ce{H3COH}  & 512.25 & -28.32 & -10.40 & -4.10 & -2.10 & -5.95 & -1.17 & 0.31 & -5.95 & -1.17 & 0.31 \\
\ce{H3CSH}  & 473.92 & -25.84 & -10.02 & -3.88 & -1.98 & -5.49 & -1.79 & -0.01 & -5.49 & -1.79 & -0.01 \\
\ce{HCN}  & 311.54 & -20.86 & -9.06 & -3.80 & -1.95 & -3.59 & -0.85 & 0.37 & -3.59 & -0.85 & 0.37 \\
\ce{HCO}  & 277.92 & -15.04 & -6.87 & -2.84 & -1.46 & -2.38 & -0.89 & 0.37 & -2.38 & -0.89 & 0.37 \\
\ce{HCl}  & 107.31 & -5.64 & -2.46 & -0.93 & -0.48 & -1.37 & -0.81 & -0.22 & -1.37 & -0.81 & -0.22 \\
\ce{HF}  & 141.67 & -8.45 & -3.48 & -1.42 & -0.73 & -2.70 & -0.96 & -0.23 & -2.70 & -0.96 & -0.23 \\
\ce{HOCl}  & 165.85 & -15.82 & -7.34 & -3.28 & -1.68 & -5.39 & -2.29 & -0.62 & -5.39 & -2.29 & -0.62 \\
\ce{Li2}  & 24.10 & -0.78 & -0.30 & -0.07 & -0.04 & 0.39 & 0.13 & 0.1 & 0.39 & 0.13 & 0.1 \\
\ce{LiF}  & 138.33 & -10.78 & -4.56 & -1.95 & -1.0 & -3.30 & -0.91 & 0.02 & -3.30 & -0.91 & 0.02 \\
\ce{LiH}  & 57.70 & -4.15 & -1.05 & -0.38 & -0.20 & -0.71 & 0.12 & 0.11 & -0.71 & 0.12 & 0.11 \\
\ce{N2}  & 227.18 & -18.66 & -9.41 & -4.20 & -2.15 & -2.26 & -0.90 & 0.20 & -2.26 & -0.90 & 0.20 \\
\ce{N2H4}  & 437.39 & -36.95 & -14.25 & -5.83 & -2.99 & -8.13 & -1.60 & -0.01 & -8.13 & -1.60 & -0.01 \\
\ce{NH}  & 82.86 & -8.10 & -3.02 & -1.24 & -0.63 & -1.51 & -0.22 & -0.06 & -1.51 & -0.22 & -0.06 \\
\ce{NH2}  & 182.14 & -15.31 & -5.61 & -2.26 & -1.16 & -3.16 & -0.50 & -0.03 & -3.16 & -0.50 & -0.03 \\
\ce{NH3}  & 297.46 & -21.21 & -7.62 & -3.01 & -1.54 & -4.42 & -0.62 & 0.07 & -4.42 & -0.62 & 0.07 \\
\ce{NO}  & 151.73 & -13.87 & -7.26 & -3.25 & -1.66 & -2.20 & -1.10 & 0.08 & -2.20 & -1.10 & 0.08 \\
\ce{Na2}  & 16.44 & -0.71 & -0.14 & -0.06 & -0.03 & 0.20 & 0.15 & 0.08 & 0.20 & 0.15 & 0.08 \\
\ce{NaCl}  & 99.61 & -11.48 & -5.41 & -2.26 & -1.16 & -4.34 & -2.13 & -0.52 & -4.34 & -2.13 & -0.52 \\
\ce{O2}  & 119.64 & -9.86 & -5.34 & -2.40 & -1.23 & -3.08 & -1.72 & -0.30 & -3.08 & -1.72 & -0.30 \\
\ce{OH}  & 107.08 & -8.37 & -3.27 & -1.34 & -0.69 & -2.23 & -0.63 & -0.08 & -2.23 & -0.63 & -0.08 \\
\ce{P2}  & 115.68 & -18.05 & -9.34 & -4.00 & -2.05 & -3.96 & -1.89 & 0.26 & -3.96 & -1.89 & 0.26 \\
\ce{PH2}  & 153.91 & -11.50 & -4.32 & -1.51 & -0.77 & -2.56 & -0.82 & 0.16 & -2.56 & -0.82 & 0.16 \\
\ce{PH3}  & 241.49 & -15.43 & -5.80 & -2.03 & -1.04 & -3.12 & -1.03 & 0.20 & -3.12 & -1.03 & 0.20 \\
\ce{S2}  & 103.07 & -12.76 & -7.34 & -3.28 & -1.68 & -5.69 & -3.18 & -0.95 & -5.69 & -3.18 & -0.95 \\
\ce{SO}  & 125.12 & -10.77 & -6.14 & -2.94 & -1.51 & -3.33 & -2.10 & -0.59 & -3.33 & -2.10 & -0.59 \\
\ce{SO2}  & 258.09 & -22.35 & -13.90 & -6.75 & -3.46 & -5.16 & -4.67 & -1.32 & -5.16 & -4.67 & -1.32 \\
\ce{Si2}  & 75.74 & -10.01 & -4.47 & -1.82 & -0.93 & -4.19 & -1.49 & -0.17 & -4.19 & -1.49 & -0.17 \\
\ce{Si2H6}  & 535.58 & -27.32 & -10.13 & -3.46 & -1.77 & -5.59 & -1.86 & 0.30 & -5.59 & -1.86 & 0.30 \\
\ce{SiH2 ^1A_1}  & 153.66 & -8.72 & -2.98 & -0.98 & -0.50 & -3.51 & -1.46 & -0.34 & -3.51 & -1.46 & -0.34 \\
\ce{SiH2 ^3B_1}  & 133.17 & -6.68 & -2.40 & -0.73 & -0.37 & -3.02 & -1.57 & -0.49 & -3.02 & -1.57 & -0.49 \\
\ce{SiH3}  & 227.99 & -10.53 & -3.72 & -1.15 & -0.59 & -1.82 & -0.74 & 0.12 & -1.82 & -0.74 & 0.12 \\
\ce{SiH4}  & 324.59 & -14.10 & -4.88 & -1.52 & -0.78 & -1.91 & -0.58 & 0.33 & -1.91 & -0.58 & 0.33 \\
\ce{SiO}  & 191.48 & -11.34 & -6.72 & -3.15 & -1.61 & -0.76 & -1.18 & 0.12 & -0.76 & -1.18 & 0.12 \\
  \end{tabular}
  \end{ruledtabular}
\end{table}
\end{squeezetable}
\end{turnpage}

\bibliography{../G2_srDFT,../G2_srDFT-control}




\end{document}
