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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{Excitation Energies Near The Complete Basis Set Limit}

\author{Emmanuel Giner}
\affiliation{\LCT}
\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}

\begin{abstract}
By combining extrapolated selected configuration interaction (sCI) calculations performed with the CIPSI algorithm with the recently proposed short-range density-functional functional correction for basis set incompleteness [\href{https://doi.org/10.1063/1.5052714}{Giner et al., J.~Chem.~Phys.~149, 194301 (2018)}], we show that one can obtain vertical and adiabatic excitation energies with chemical accuracy with a small basis set.
\end{abstract}

\maketitle

%%%%%%%%%%%%%%%%%%%%%%%%
\section{Introduction}
\label{sec:intro}
%%%%%%%%%%%%%%%%%%%%%%%%
One of the most fundamental problem of conventional electronic structure methods is their slow energy convergence with respect to the size of the one-electron basis set.
This problem was already noticed thirty years ago by Kutzelnigg \cite{Kutzelnigg_1985} who proposed to introduce explicitly the correlation between electrons via the introduction of the interelectronic distance $r_{12} = \abs{\br_1 - \br_2}$ as a basis function. \cite{Kutzelnigg_1991, Termath_1991, Klopper_1991a, Klopper_1991b, Noga_1994}
This yields a prominent improvement of the energy convergence from $O(L^{-3})$ to $O(L^{-7})$ (where $L$ is the maximum angular momentum of the one-electron basis).
This idea was later generalised to more accurate correlation factors $f_{12} \equiv f(r_{12})$. \cite{Persson_1996, Persson_1997, May_2004, Tenno_2004b, Tew_2005, May_2005}
The resulting F12 methods achieve chemical accuracy for small organic molecules with relatively small Gaussian basis sets. \cite{Tenno_2012a, Tenno_2012b, Hattig_2012, Kong_2012}
For example, as illustrated by Tew and coworkers, one can obtain, at the CCSD(T) level, quintuple-zeta quality correlation energies with a triple-zeta basis. \cite{Tew_2007b}

In the present study, we rely on the recently proposed short-range density-functional functional correction for basis set incompleteness. \cite{Giner_2018}

%%%%%%%%%%%%%%%%%%%%%%%%
\section{Computational details}
\label{sec:compdetails}
%%%%%%%%%%%%%%%%%%%%%%%%


%%%%%%%%%%%%%%%%%%%%%%%%
\section{Results}
\label{sec:res}
%%%%%%%%%%%%%%%%%%%%%%%%
%=======================
\subsection{Water}
\label{sec:H2O}
%=======================

%=======================
\subsection{Formaldehyde}
\label{sec:CH2O}
%=======================

%=======================
\subsection{Methylene}
\label{sec:CH2}
%=======================

%%% TABLE 1 %%%
	\begin{squeezetable}
	\begin{table*}
	\caption{
	Total energies $E$ (in hartree) and adiabatic transition energies $\Ead$ (in eV) of excited states of methylene for various methods and basis sets.}
	\begin{ruledtabular}{}
	\begin{tabular}{llddddddd}
			&					&	\mc{1}{c}{$1\,^{3}B_1$}		
								&	\mc{2}{c}{$1\,^{3}B_1 \ra 1\,^{1}A_1$}		
								&	\mc{2}{c}{$1\,^{3}B_1 \ra 1\,^{1}B_1$}	
								&	\mc{2}{c}{$1\,^{3}B_1 \ra 2\,^{1}A_1$}		\\
								\cline{3-3}					\cline{4-5}						
								\cline{6-7}					\cline{8-9}
	Method		&	Basis set	&	\tabc{$E$ (a.u.)}	
								&	\tabc{$E$ (a.u.)}	&	\tabc{$\Ead$ (eV)}	
								&	\tabc{$E$ (a.u.)}	&	\tabc{$\Ead$ (eV)}	
								&	\tabc{$E$ (a.u.)}	&	\tabc{$\Ead$ (eV)}	\\
	\hline
	exFCI		&	AVDZ		&	-39.04846(1)				
								&	-39.03225(1)	&	0.441	
								&	-38.99203(1)	&	1.536	
								&	-38.95076(1)	&	2.659	\\	
				&	AVTZ		&	-39.08064(3)				
								&	-39.06565(2)	&	0.408	
								&	-39.02833(1)	&	1.423	
								&	-38.98709(1)	&	2.546	\\	
				&	AVQZ		&	-39.08854(1)				
								&	-39.07402(2)	&	0.395	
								&	-39.03711(1)	&	1.399	
								&	-38.99607(1)	&	2.516	\\	
				&	AV5Z		&	-39.09079(1)				
								&	-39.07647(1)	&	0.390	
								&	-39.03964(3)	&	1.392	
								&	-38.99867(1)	&	2.507	\\	
				&	CBS			&	-39.09111				
								&	-39.07682		&	0.389	
								&	-39.04000		&	1.391	
								&	-38.99904		&	2.505	\\	
								\\
	exFCI+LDA	&	AVDZ		&	-39.07450(1)				
								&	-39.06213(1)	&	0.337	
								&	-39.02233(1)	&	1.420	
								&	-38.97946(1)	&	2.586	\\	
				&	AVTZ		&	-39.09099(3)				
								&	-39.07779(2)	&	0.359	
								&	-39.04051(1)	&	1.374	
								&	-38.99859(1)	&	2.514	\\	
				&	AVQZ		&	-39.09319(1)				
								&	-39.07959(2)	&	0.370	
								&	-39.04267(1)	&	1.375	
								&	-39.00135(1)	&	2.499	\\	
								\\
	exFCI+PBE	&	AVDZ		&	-39.07282(1)				
								&	-39.06150(1)	&	0.308	
								&	-39.02181(1)	&	1.388	
								&	-38.97873(1)	&	2.560	\\	
				&	AVTZ		&	-39.08948(3)				
								&	-39.07639(2)	&	0.356	
								&	-39.03911(1)	&	1.371	
								&	-38.99724(1)	&	2.510	\\	
				&	AVQZ		&	-39.09247(1)				
								&	-39.07885(2)	&	0.371
								&	-39.04193(1)	&	1.375	
								&	-39.00066(1)	&	2.498	\\	
								\\
	exFCI+PBEot	&	AVDZ		&	-39.06924(1)				
								&	-39.05651(1)	&	0.347	
								&	-39.01777(1)	&	1.401	
								&	-38.97698(1)	&	2.511	\\	
				&	AVTZ		&	-39.08805(3)				
								&	-39.07430(2)	&	0.374	
								&	-39.03742(1)	&	1.378	
								&	-38.99652(1)	&	2.491	\\	
				&	AVQZ		&	-39.09189(1)				
								&	-39.07795(2)	&	0.379	
								&	-39.04124(1)	&	1.378	
								&	-39.00044(1)	&	2.489	\\	
								\\
	SHCI		&	AVQZ		&	-39.08849(1)				
								&	-39.07404(1)	&	0.393	
								&	-39.03711(1)	&	1.398	
								&	-38.99603(1)	&	2.516	\\
	CR-EOMCC (2,3)D&	AVQZ	&	-39.08817					
								&	-39.07303		&	0.412	
								&	-39.03450		&	1.460	
								&	-38.99457		&	2.547	\\
	FCI			&	TZ2P		&	-39.066738					
								&	-39.048984		&	0.483	
								&	-39.010059		&	1.542	
								&	-38.968471		&	2.674	\\
	DMC			&				&								
								&					&	0.406	
								&					&	1.416	
								&					&	2.524	\\
	Exp.		&				&								
								&					&	0.400	
								&					&	1.411	
	\end{tabular}
	\end{ruledtabular}
	\end{table*}
	\end{squeezetable}
%%% %%% %%%


%%% TABLE 1 %%%
	\begin{squeezetable}
	\begin{table*}
	\caption{
	Vertical absorption energies $\Ead$ (in eV) of excited states of water for various methods and basis sets.}
	\begin{ruledtabular}{}
	\begin{tabular}{llddddddddddddd}
				&										&				&	\mc{12}{c}{Deviation with respect to TBE}		
	\\
	\cline{4-15}
				&										&				&	\mc{3}{c}{exFCI}	
																		&	\mc{3}{c}{exFCI+PBEot}	
																		&	\mc{3}{c}{exFCI+PBE}	
																		&	\mc{3}{c}{exFCI+LDA}
	\\
	\cline{4-6}	\cline{7-9}	\cline{10-12}	\cline{13-15}
	Molecule	&	Transition							&	\tabc{TBE}	&	\tabc{AVDZ}	&	\tabc{AVTZ}	&	\tabc{AVQZ}	
																		&	\tabc{AVDZ}	&	\tabc{AVTZ}	&	\tabc{AVQZ}	
																		&	\tabc{AVDZ}	&	\tabc{AVTZ}	&	\tabc{AVQZ}	
																		&	\tabc{AVDZ}	&	\tabc{AVTZ}	&	\tabc{AVQZ}	
	\\
	\hline
	Water		&	\tabc{$1\,^{1}A_1 \ra 1\,^{1}B_1$}	&	7.70		&	-0.17	&	-0.07	&		
																		&	-0.19	&	+0.00	&		
																		&	-0.02	&	-0.01	&		
																		&	-0.04	&	-0.01	&
	\\
				&	\tabc{$1\,^{1}A_1 \ra 1\,^{1}A_2$}	&	9.47		&	-0.15	&	-0.06	&		
																		&	+0.03	&	+0.01	&		
																		&	+0.00	&	+0.00	&		
																		&	-0.03	&	+0.00	&
	\\
				&	\tabc{$1\,^{1}A_1 \ra 1\,^{1}A_1$}	&	9.97		&	-0.03	&	+0.02	&		
																		&	+0.13	&	+0.08	&		
																		&	+0.10	&	+0.07	&			
																		&	+0.09	&	0.07	&
	\\
				&	\tabc{$1\,^{1}A_1 \ra 3\,^{1}B_1$}	&	7.33		&	-0.19	&	-0.08	&		
																		&	+0.02	&	+0.00	&		
																		&	+0.05	&	+0.01	&		
																		&	0.00	&	+0.00	&
	\\
				&	\tabc{$1\,^{1}A_1 \ra 3\,^{1}A_2$}	&	9.30		&	-0.16	&	-0.06	&		
																		&	+0.04	&	+0.02	&		
																		&	+0.07	&	+0.03	&		
																		&	+0.03	&	0.03	&
	\\
				&	\tabc{$1\,^{1}A_1 \ra 3\,^{1}A_1$}	&	9.59		&	-0.11	&	-0.05	&		
																		&	+0.07	&	+0.02	&		
																		&	+0.09	&	+0.03	&		
																		&	+0.06	&	0.03	&
	\\
	\end{tabular}
	\end{ruledtabular}
	\end{table*}
	\end{squeezetable}
%%% %%% %%%


%%%%%%%%%%%%%%%%%%%%%%%%
\section{Conclusion}
\label{sec:ccl}
%%%%%%%%%%%%%%%%%%%%%%%%

%%%%%%%%%%%%%%%%%%%%%%%%
\section*{Supporting Information}
%%%%%%%%%%%%%%%%%%%%%%%%
See {\SI} for geometries and additional information (including total energies).

%%%%%%%%%%%%%%%%%%%%%%%%
\begin{acknowledgements}
This work was performed using HPC resources from 
i) GENCI-TGCC (Grant No. 2018-A0040801738),
ii) CALMIP (Toulouse) under allocations 2018-0510 and 2018-12158. 
\end{acknowledgements}
%%%%%%%%%%%%%%%%%%%%%%%%



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