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\begin{document}	

\title{Supporting Information for \\ ``Pros and Cons of the Bethe-Salpeter Formalism for Ground-State Energies''}

\author{Pierre-Fran\c{c}ois \surname{Loos}}
	\email{loos@irsamc.ups-tlse.fr}
	\affiliation{\LCPQ}
\author{Anthony \surname{Scemama}}
	\email{scemama@irsamc.ups-tlse.fr}
	\affiliation{\LCPQ}
\author{Ivan \surname{Duchemin}}
	\email{ivan.duchemin@cea.fr}
	\affiliation{\CEA}
\author{Denis \surname{Jacquemin}}
	\email{denis.jacquemin@univ-nantes.fr}
	\affiliation{\CEISAM}
\author{Xavier \surname{Blase}}
	\email{xavier.blase@neel.cnrs.fr }
	\affiliation{\NEEL}

\begin{abstract}
\end{abstract}

\maketitle

%%% TABLE I %%%
\begin{table*}
\caption{
Equilibrium bond length (in bohr) of the ground state of diatomic molecules obtained at various levels of theory and basis sets. 
The reference CC3 and corresponding BSE@{\GOWO}@HF data are highlighted in bold black and bold red for visual convenience, respectively.
When irregularities appear in the PES, the values are reported in parenthesis and they have been obtained by fitting a Morse potential to the PES.
}
\label{tab:Req}
	\begin{ruledtabular}
	\begin{tabular}{llcccccccc}
				&				&	\mc{8}{c}{Molecules}	\\
									\cline{3-10}
	Method		&	Basis		&	\ce{H2}		&	\ce{LiH}	&	\ce{LiF}	&	\ce{HCl}	&	\ce{N2}			&	\ce{CO}	&	\ce{BF}	&	\ce{F2}	\\
	\hline
	CC3				&	cc-pVDZ		&	1.438	&	3.043		&	3.012		&	2.435		&	2.114		&	2.166		&	2.444		&	2.740	\\
					&	cc-pVTZ		&	1.403	& 	3.011		&	2.961		&	2.413		&	2.079		&	2.143		&	2.392		&	2.669	\\
					&	cc-pVQZ		&\bb{1.402}	&\bb{3.019}		&\bb{2.963}		&\bb{2.403}		&\bb{2.075}		&\bb{2.136}		&\bb{2.390}		&\bb{2.663}	\\
	CCSD			&	cc-pVDZ		&	1.438	&	3.044		&	3.006		&	2.433		&	2.101		&	2.149		&	2.435		&	2.695	\\
					&	cc-pVTZ		&	1.403	&	3.012		&	2.954		&	2.409		&	2.064		&	2.126		&	2.382		&	2.629	\\
					&	cc-pVQZ		&	1.402	&	3.020		&	2.953		&	2.398		&	2.059		&	2.118		&	2.118		&	2.621	\\
	CC2				&	cc-pVDZ		&	1.426	&	3.046		&	3.026		&	2.427		&	2.146		&	2.187		&	2.444		&	2.710	\\
					&	cc-pVTZ		&	1.393	&	3.008		&	2.995		&	2.406		&	2.109		&	2.163		&	2.394		&	2.664	\\
					&	cc-pVQZ		&	1.391	&	2.989		&	2.982		&	2.396		&	2.106		&	2.156		&	2.393		&	2.665	\\
	MP2				&	cc-pVDZ		&	1.426	&	3.041		&	3.010		&	2.426		&	2.133		&	2.166		&	2.431		&	2.681	\\
					&	cc-pVTZ		&	1.393	&	3.004		&	2.968		&	2.405		&	2.095		&	2.144		&	2.383		&	2.636	\\
					&	cc-pVQZ		&	1.391	&	3.008		&	2.970		&	2.395		&	2.091		&	2.137		&	2.382		&	2.634	\\
	BSE@{\GOWO}@HF	&	cc-pVDZ		&	1.437	&	3.042		&	3.000		&	2.454		&	2.107		&	2.153		&	2.407		&	(2.698)	\\
					&	cc-pVTZ		&	1.404	&	3.023		&	(2.982)		&	2.410		&	2.068		&	2.116		&	(2.389)		&	(2.647)	\\
					&	cc-pVQZ		&\rb{1.399}	&\rb{3.017}		&\rb{(2.974)}	&\rb{2.400}		&\rb{2.065}		&\rb{2.134}		&\rb{2.383}		&\rb{(2.640)}\\
	RPA@{\GOWO}@HF	&	cc-pVDZ		&	1.426	&	3.019		&	2.994		&	2.436		&	2.083		&	2.144		&	2.403		&	(2.629)	\\
					&	cc-pVTZ		&	1.388	&	2.988		&	(2.965)		&	2.408		&	2.055		&	2.114		&	(2.370)		&	(2.584)	\\
					&	cc-pVQZ		&	1.382	&	2.997		&	(2.965)		&	2.370		&	2.043		&	2.132		&	2.367		&	(2.571)	\\
	RPAx@HF			&	cc-pVDZ		&	1.428	&	3.040		&	2.998		&	2.424		&	2.077		&	2.130		&	2.417		&	2.611	\\
					&	cc-pVTZ		&	1.395	&	3.003		&	2.943		&	2.400		&	2.046		&	2.110		&	2.368		&	2.568	\\
					&	cc-pVQZ		&	1.394	&	3.011		&	2.944		&	2.391		&	2.041		&	2.104		&	2.366		&	2.565	\\
	RPA@HF			&	cc-pVDZ		&	1.431	&	3.021		&	2.999		&	2.424		&	2.083		&	2.134		&	2.416		&	2.623	\\
					&	cc-pVTZ		&	1.388	&	2.978		&	2.939		&	2.396		&	2.045		&	2.110		&	2.362		&	2.579	\\
					&	cc-pVQZ		&	1.386	&	2.994		&	2.946		&	2.382		&	2.042		&	2.103		&	2.364		&	2.573	\\
	\end{tabular}
	\end{ruledtabular}
\end{table*}

%%% FIG 1 %%%
\begin{figure*}
	% H2
	\includegraphics[width=0.49\linewidth]{../Data/H2_GS_VDZ}
	\includegraphics[width=0.49\linewidth]{../Data/H2_GS_VTZ}	
\caption{
Ground-state potential energy surfaces of \ce{H2} around its respective equilibrium geometry obtained at various levels of theory and basis sets. 
\label{fig:PES-H2}
}
\end{figure*}
%%% %%% %%%

%%% FIG 2 %%%
\begin{figure*}
	% LiH
	\includegraphics[width=0.49\linewidth]{../Data/LiH_GS_VDZ}
	\includegraphics[width=0.49\linewidth]{../Data/LiH_GS_VDZ_FC}
	\includegraphics[width=0.49\linewidth]{../Data/LiH_GS_VTZ}
	\includegraphics[width=0.49\linewidth]{../Data/LiH_GS_VTZ_FC}
	\includegraphics[width=0.49\linewidth]{../Data/LiH_GS_VQZ_FC}
\caption{
Ground-state potential energy surfaces of \ce{LiH} around its respective equilibrium geometry obtained at various levels of theory and basis sets. 
FC stands for frozen core.
\label{fig:PES-LiH}
}
\end{figure*}
%%% %%% %%%

%%% FIG 3 %%%
\begin{figure*}
	% LiF
	\includegraphics[width=0.49\linewidth]{../Data/LiF_GS_VDZ}
	\includegraphics[width=0.49\linewidth]{../Data/LiF_GS_VDZ_FC}
	\includegraphics[width=0.49\linewidth]{../Data/LiF_GS_VTZ}
	\includegraphics[width=0.49\linewidth]{../Data/LiF_GS_VTZ_FC}
	\includegraphics[width=0.49\linewidth]{../Data/LiF_GS_VQZ_FC}
\caption{
Ground-state potential energy surfaces of \ce{LiF} around its respective equilibrium geometry obtained at various levels of theory and basis sets. 
FC stands for frozen core.
\label{fig:PES-LiF}
}
\end{figure*}
%%% %%% %%%

%%% FIG 5 %%%
\begin{figure*}
	% HCl
	\includegraphics[width=0.49\linewidth]{../Data/HCl_GS_VDZ}
	\includegraphics[width=0.49\linewidth]{../Data/HCl_GS_VDZ_FC}
	\includegraphics[width=0.49\linewidth]{../Data/HCl_GS_VTZ}
	\includegraphics[width=0.49\linewidth]{../Data/HCl_GS_VTZ_FC}
	\includegraphics[width=0.49\linewidth]{../Data/HCl_GS_VQZ_FC}
\caption{
Ground-state potential energy surfaces of \ce{HCl} around its respective equilibrium geometry obtained at various levels of theory and basis sets. 
FC stands for frozen core.
\label{fig:PES-HCl}
}
\end{figure*}
%%% %%% %%%

%%% FIG 6 %%%
\begin{figure*}
	% N2
	\includegraphics[width=0.49\linewidth]{../Data/N2_GS_VDZ}
	\includegraphics[width=0.49\linewidth]{../Data/N2_GS_VDZ_FC}
	\includegraphics[width=0.49\linewidth]{../Data/N2_GS_VTZ}
	\includegraphics[width=0.49\linewidth]{../Data/N2_GS_VTZ_FC}
	\includegraphics[width=0.49\linewidth]{../Data/N2_GS_VQZ_FC}
\caption{
Ground-state potential energy surfaces of \ce{N2} around its respective equilibrium geometry obtained at various levels of theory and basis sets. 
FC stands for frozen core.
\label{fig:PES-N2}
}
\end{figure*}
%%% %%% %%%

%%% FIG 6 %%%
\begin{figure*}
	% CO
	\includegraphics[width=0.49\linewidth]{../Data/CO_GS_VDZ}
	\includegraphics[width=0.49\linewidth]{../Data/CO_GS_VDZ_FC}
	\includegraphics[width=0.49\linewidth]{../Data/CO_GS_VTZ}
	\includegraphics[width=0.49\linewidth]{../Data/CO_GS_VTZ_FC}
	\includegraphics[width=0.49\linewidth]{../Data/CO_GS_VQZ_FC}
\caption{
Ground-state potential energy surfaces of \ce{CO} around its respective equilibrium geometry obtained at various levels of theory and basis sets. 
FC stands for frozen core.
\label{fig:PES-CO}
}
\end{figure*}
%%% %%% %%%

%%% FIG 6 %%%
\begin{figure*}
	% N2
	\includegraphics[width=0.49\linewidth]{../Data/BF_GS_VDZ}
	\includegraphics[width=0.49\linewidth]{../Data/BF_GS_VDZ_FC}
	\includegraphics[width=0.49\linewidth]{../Data/BF_GS_VTZ}
	\includegraphics[width=0.49\linewidth]{../Data/BF_GS_VTZ_FC}
	\includegraphics[width=0.49\linewidth]{../Data/BF_GS_VQZ_FC}
\caption{
Ground-state potential energy surfaces of \ce{BF} around its respective equilibrium geometry obtained at various levels of theory and basis sets. 
FC stands for frozen core.
\label{fig:PES-BF}
}
\end{figure*}
%%% %%% %%%

%%% FIG 6 %%%
\begin{figure*}
	% N2
	\includegraphics[width=0.49\linewidth]{../Data/F2_GS_VDZ}
	\includegraphics[width=0.49\linewidth]{../Data/F2_GS_VTZ}
	\includegraphics[width=0.49\linewidth]{../Data/F2_GS_VDZ_FC}
	\includegraphics[width=0.49\linewidth]{../Data/F2_GS_VTZ_FC}
	\includegraphics[width=0.49\linewidth]{../Data/F2_GS_VQZ_FC}
\caption{
Ground-state potential energy surfaces of \ce{F2} around its respective equilibrium geometry obtained at various levels of theory and basis sets. 
FC stands for frozen core.
\label{fig:PES-F2}
}
\end{figure*}
%%% %%% %%%

\begin{table}
\caption{
Ground-state total and correlation energies obtained at various levels of theory for diatomic molecules at their equilibrium geometry.
All the calculations have been performed without frozen-core approximation with the cc-pVQZ basis set.
\label{fig:CCSDTQ}
}
\begin{ruledtabular}
\begin{tabular}{lllccc}
System			&	Geometry	&	Method	& Energy (Ha)	&	$-\Ec$ (mHa)	&	Error wrt CC3 (\%)	\\
\hline
\ce{H2}	&	$R_{\ce{H-H}} = 1.402$		
&	HF			&	$-1.133\,458$									\\	
&	&	CC3			&	$-1.173\,840$	&	$40.4$    &					\\
&	&	CCSDT		&	$-1.173\,840$	&	$40.4$    &	$0.0\%$			\\
\\
\ce{LiH}	&	$R_{\ce{Li-H}} = 3.019$	
&	HF         &	$-7.987\,235$									\\
&	&	CC3        &	$-8.057\,209$	&	$70.0$  	&				\\
&	&	CCSDT      &	$-8.057\,233$	&	$70.0$		&	$0.0\%$		\\                
&	&	CCSDT(Q)   &	$-8.057\,233$	&	$70.0$		&	$0.0\%$		\\                 
&	&	CCSDTQ     &	$-8.057\,233$	&	$70.0$		&	$0.0\%$		\\               
\\
\ce{LiF}	&	$R_{\ce{Li-F}} = 2.963$	
&	HF			&	$-106.990\,854$									\\     
&	&	CC3			&	$-107.374\,540$	&	$383.7$						\\
&	&	CCSDT		&	$-107.373\,314$	&	$382.5$		&	$-0.3\%$ 	\\            
&	&	CCSDT(Q)	&	$-107.373\,701$	&	$382.9$		&   $-0.2\%$	\\
\\
\ce{HCl}	&	$R_{\ce{H-Cl}} = 2.403$	
&	HF         &	$-460.111\,442$									\\
&	&	CC3        &	$-460.493\,630$	&	$382.2$						\\
&	&	CCSDT      &	$-460.493\,675$	&	$382.2$		&	$+0.0\%$	\\               
&	&	CCSDT(Q)   &	$-460.494\,110$	&	$382.7$		&	$+0.1\%$	\\
\\
\ce{N2}	&	$R_{\ce{N-N}} = 2.075$	
&	HF         &	$-108.991\,326$	\\
&	&	CC3        &	$-109.485\,718$	&	$494.4$						\\          
&	&	CCSDT      &	$-109.484\,058$	&	$492.7$		&	 $-0.3\%$	\\                 
&	&	CCSDT(Q)   &	$-109.486\,040$	&	$494.7$		&	 $+0.1\%$	\\
\\
\ce{CO}	&	$R_{\ce{C-O}} = 2.136$	
&	HF         &	$-112.788\,718$	\\
&	&	CC3        &	$-113.266\,297$ &	$477.6$						\\
&	&	CCSDT      &	$-113.264\,330$	&	$475.6$		&	  $-0.4\%$	\\                
&	&	CCSDT(Q)   &	$-113.265\,613$	&	$476.9$		&	  $-0.1\%$	\\
\\
\ce{BF}	&	$R_{\ce{B-F}} = 2.390$	
&	HF         &	$-124.166\,127$	\\
&	&	CC3        &	$-124.613\,599$	&	$447.5$						\\
&	&	CCSDT      &	$-124.612\,523$	&	$446.4$		&	$-0.2\%$	\\            
&	&	CCSDT(Q)   &	$-124.613\,118$	&	$447.0$		&	$-0.1\%$	\\
\\
\ce{F2}	&	$R_{\ce{F-F}} = 2.663$
&	HF         &	$-198.769\,005$									\\
&	&	CC3        &	$-199.437\,880$	&	$668.9$						\\
&	&	CCSDT      &	$-199.437\,033$	&	$668.0$		&	$+0.1\%$	\\                 
&	&	CCSDT(Q)   &	$-199.438\,815$	&	$669.8$		&	$-0.1\%$	\\   
\end{tabular}
\end{ruledtabular}
\end{table}

\begin{table}
\caption{
Comparison between the extended BSE (XBS) and the regular BSE schemes, as defined in Ref.~\onlinecite{Holzer_2018}.
The calculations have been performed at the XBS equilibrium geometries.
Note that the methodology presented in the present paper is theoretically equivalent to the XBS scheme.
All the calculations have been performed without frozen-core approximation with the cc-pVQZ basis set.
\label{fig:XBS}
}
\begin{ruledtabular}
\begin{tabular}{lllcc}
System			&	Geometry	&	Method	&	$-\Ec$ (mHa)	&	Error wrt CC3 (\%)	\\
\hline
\ce{H2}			&	$R_{\ce{H-H}} = 1.399$		
&		BSE		&	$46.480$    &	$+15.10\%$	\\
&	&	XBS		&	$47.209$    &	$+16.91\%$	\\
\ce{LiH}		&	$R_{\ce{Li-H}} = 3.017$	
&		BSE		&	$77.973$    &	$+11.43\%$	\\
&	&	XBS		&	$78.110$    &	$+11.63\%$	\\
\ce{LiF}		&	$R_{\ce{Li-F}} = 2.974$	
&		BSE		&	$388.307$    &	$+1.20\%$	\\
&	&	XBS		&	$385.041$    &	$+0.35\%$	\\
\ce{HCl}		&	$R_{\ce{H-Cl}} = 2.400$	
&		BSE		&	$385.147$    &	$+0.77\%$	\\
&	&	XBS		&	$384.544$    &	$+0.62\%$	\\
\ce{N2}			&	$R_{\ce{N-N}} = 2.065$	
&		BSE		&	$493.742$    &	$-0.13\%$			\\
&	&	XBS		&	$497.856$    &	$+0.70\%$			\\
\ce{CO}			&	$R_{\ce{C-O}} = 2.134$	
&		BSE		&	$476.210$    &	$-0.29\%$			\\
&	&	XBS		&	$480.023$    &	$+0.51\%$			\\
\ce{BF}			&	$R_{\ce{B-F}} = 2.385$	
&		BSE		&	$x$    &	$x\%$			\\
&	&	XBS		&	$452.337$    &	$1.09\%$			\\
\ce{F2}			&	$R_{\ce{F-F}} = 2.640$
&		BSE		&	$x$    &	$x\%$			\\
&	&	XBS		&	$x$    &	$x\%$			\\
\end{tabular}
\end{ruledtabular}
\end{table}


%%% %%% %%%
%%% TABLE I %%%
%\begin{table*}
%\caption{
%Equilibrium distances (in bohr) of the ground state of diatomic molecules obtained at various levels of theory and basis sets. 
%All these values have been obtained within the frozen-core approximation.
%The reference CC3 and corresponding BSE@{\GOWO}@HF data are highlighted in bold black and bold red for visual convenience, respectively.
%The values in parenthesis have been obtained by fitting a Morse potential to the PES.
%}
%\label{tab:Req-FC}
%	
%	\begin{ruledtabular}
%	\begin{tabular}{llcccccccc}
%				&				&	\mc{8}{c}{Molecules}	\\
%									\cline{3-10}
%	Method		&	Basis		&	\ce{H2}		&	\ce{LiH}	&	\ce{LiF}	&	\ce{HCl}	&	\ce{N2}			&	\ce{CO}	&	\ce{BF}	&	\ce{F2}	\\
%	\hline
%	CC3				&	cc-pVDZ		&	1.438	&	3.052	&	3.014	&	2.115	&	2.167	&	2.447	&	2.741	&	2.438	\\
%					&	cc-pVTZ		&	1.403	&	3.036	&	2.985	&	2.087	&	2.150	&	2.405	&	2.672	&	2.414	\\
%					&	cc-pVQZ		&	1.402	&	3.037	&	2.985	&	2.080	&	2.142	&	2.398	&	2.667	&	2.413	\\
%	CCSD			&	cc-pVDZ		&	1.438	&	3.044	&	3.006	&	2.101	&	2.149	&	2.435	&	2.695	&	2.433	\\
%					&	cc-pVTZ		&	1.403	&	3.012	&	2.954	&	2.064	&	2.126	&	2.382	&	2.629	&	2.409	\\
%					&	cc-pVQZ		&	1.402	&	3.020	&	2.953	&	2.059	&	2.118	&	2.380	&	2.621	&	2.398	\\
%	CC2				&	cc-pVDZ		&	1.426	&			&		&		&		&		&		&			\\
%					&	cc-pVTZ		&	1.393	&			&		&		&		&		&		&			\\
%					&	cc-pVQZ		&	1.391	&			&		&		&		&		&		&			\\
%	MP2				&	cc-pVDZ		&	1.426	&	3.049	&	3.012	&	2.134	&	2.167	&	2.433	&	2.681	&	2.429		\\
%					&	cc-pVTZ		&	1.393	&	3.026	&	2.990	&	2.104	&	2.151	&	2.395	&	2.640	&	2.407		\\
%					&	cc-pVQZ		&	1.391	&	3.026	&	2.990	&	2.098	&	2.144	&	2.389	&	2.638	&	2.405		\\
%	BSE@{\GOWO}@HF	&	cc-pVDZ		&	1.437	&			&		&		&		&		&		&			\\
%					&	cc-pVTZ		&	1.404	&			&		&		&		&		&		&			\\
%					&	cc-pVQZ		&	1.399	&			&		&		&		&		&		&			\\
%	RPA@{\GOWO}@HF	&	cc-pVDZ		&	1.426	&			&		&		&		&		&		&			\\
%					&	cc-pVTZ		&	1.388	&			&		&		&		&		&		&			\\
%					&	cc-pVQZ		&	1.382	&			&		&		&		&		&		&			\\
%	RPAx@HF			&	cc-pVDZ		&	1.428	&			&		&		&		&		&		&			\\
%					&	cc-pVTZ		&	1.395	&			&		&		&		&		&		&			\\
%					&	cc-pVQZ		&	1.394	&			&		&		&		&		&		&			\\
%	RPA@HF			&	cc-pVDZ		&	1.431	&			&		&		&		&		&		&			\\
%					&	cc-pVTZ		&	1.388	&			&		&		&		&		&		&			\\
%					&	cc-pVQZ		&	1.386	&			&		&		&		&		&		&			\\
%	\end{tabular}
%	\end{ruledtabular}
%\end{table*}


\bibliography{BSE-PES,BSE-PES-control}

\end{document}
