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% units
\newcommand{\IneV}[1]{#1 eV}
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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., \textit{J.~Chem.~Phys.}~\textbf{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.

In the present study, we rely on the recently proposed short-range density-functional functional correction for basis set incompleteness. \cite{GinPraFerAssSavTou-JCP-18}

%Contemporary quantum chemistry has developed in two directions --- wave function theory (WFT) \cite{Pop-RMP-99} and density-functional theory (DFT). \cite{Koh-RMP-99}
%Although both spring from the same Schr\"odinger equation, each of these philosophies has its own \textit{pros} and \textit{cons}.
%
%WFT is attractive as it exists a well-defined path for systematic improvement as well as powerful tools, such as perturbation theory, to guide the development of new WFT \textit{ans\"atze}.
%The coupled cluster (CC) family of methods is a typical example of the WFT philosophy and is well regarded as the gold standard of quantum chemistry for weakly correlated systems. 
%By increasing the excitation degree of the CC expansion, one can systematically converge, for a given basis set, to the exact, full configuration interaction (FCI) limit, although the computational cost associated with such improvement is usually high.
%One of the most fundamental drawbacks of conventional WFT methods is the slow convergence of energies and properties with respect to the size of the one-electron basis set.
%This undesirable feature was put into light by Kutzelnigg more than thirty years ago. \cite{Kut-TCA-85}
%To palliate this, following Hylleraas' footsteps, \cite{Hyl-ZP-29} Kutzelnigg proposed to introduce explicitly the interelectronic distance $r_{12} = \abs{\br{1} - \br{2}}$ to properly describe the electronic wave function around the coalescence of two electrons. \cite{Kut-TCA-85, KutKlo-JCP-91, NogKut-JCP-94}
%The resulting F12 methods yield a prominent improvement of the energy convergence, and achieve chemical accuracy for small organic molecules with relatively small Gaussian basis sets. \cite{Ten-TCA-12, TenNog-WIREs-12, HatKloKohTew-CR-12, KonBisVal-CR-12, GruHirOhnTen-JCP-17, MaWer-WIREs-18}
%For example, at the CCSD(T) level, one can obtain quintuple-$\zeta$ quality correlation energies with a triple-$\zeta$ basis, \cite{TewKloNeiHat-PCCP-07} although computational overheads are introduced by the large auxiliary basis used to resolve three- and four-electron integrals. \cite{BarLoo-JCP-17}
%To reduce further the computational cost and/or ease the transferability of the F12 correction, approximated and/or universal schemes have recently emerged. \cite{TorVal-JCP-09, KonVal-JCP-10, KonVal-JCP-11, BooCleAlaTew-JCP-2012, IrmHumGru-arXiv-2019, IrmGru-arXiv-2019}
%
%Present-day DFT calculations are almost exclusively done within the so-called Kohn-Sham (KS) formalism, which corresponds to an exact dressed one-electron theory. \cite{KohSha-PR-65}
%The attractiveness of DFT originates from its very favorable accuracy/cost ratio as it often provides reasonably accurate energies and properties at a relatively low computational cost.
%Thanks to this, KS-DFT \cite{HohKoh-PR-64, KohSha-PR-65} has become the workhorse of electronic structure calculations for atoms, molecules and solids. \cite{ParYan-BOOK-89}
%Although there is no clear way on how to systematically improve density-functional approximations, \cite{Bec-JCP-14} climbing Perdew's ladder of DFT is potentially the most satisfactory way forward. \cite{PerSch-AIPCP-01, PerRuzTaoStaScuCso-JCP-05}
%In the context of the present work, one of the interesting feature of density-based methods is their much faster convergence with respect to the size of the basis set. \cite{FraMusLupTou-JCP-15}
%
%Progress toward unifying WFT and DFT are on-going. 
%In particular, range-separated DFT (RS-DFT) (see Ref.~\citenum{TouColSav-PRA-04} and references therein) rigorously combines these two approaches via a decomposition of the electron-electron (e-e) interaction into a non-divergent long-range part and a (complementary) short-range part treated with WFT and DFT, respectively. 
%As the WFT method is relieved from describing the short-range part of the correlation hole around the e-e coalescence points, the convergence with respect to the one-electron basis set is greatly improved. \cite{FraMusLupTou-JCP-15} 
%Therefore, a number of approximate RS-DFT schemes have been developed within single-reference \cite{AngGerSavTou-PRA-05, GolWerSto-PCCP-05, TouGerJanSavAng-PRL-09,JanHenScu-JCP-09, TouZhuSavJanAng-JCP-11, MusReiAngTou-JCP-15} or multi-reference \cite{LeiStoWerSav-CPL-97, FroTouJen-JCP-07, FroCimJen-PRA-10, HedKneKieJenRei-JCP-15, HedTouJen-JCP-18, FerGinTou-JCP-18} WFT approaches.
%Very recently, a major step forward has been taken by some of the present authors thanks to the development of a density-based basis-set correction for WFT methods. \cite{GinPraFerAssSavTou-JCP-18}
%The present work proposes an extension of this new methodological development alongside the first numerical tests on molecular systems.



%%%%%%%%%%%%%%%%%%%%%%%%
\section{Computational details}
\label{sec:compdetails}
%%%%%%%%%%%%%%%%%%%%%%%%
The present basis-set correction relies on the RS-DFT formalism to capture the missing part of the short-range correlation effects, a consequence of the incompleteness of the one-electron basis set. 
The present methodology is identical to the one described in Ref.~\onlinecite{LooPraSceTouGin-JPCL-19} where the main working equation are reported and discussed.
We refer the interested reader to Ref.~\onlinecite{GinPraFerAssSavTou-JCP-18} for a more formal derivation.  

exFCI stands for extrapolated FCI energies computed with the CIPSI algorithm. \cite{HurMalRan-JCP-73, GinSceCaf-CJC-13, GinSceCaf-JCP-15}
We refer the interested reader to Refs.~\citenum{HolUmrSha-JCP-17, SceGarCafLoo-JCTC-18, LooSceBloGarCafJac-JCTC-18, SceBenJacCafLoo-JCP-18, LooBogSceCafJAc-JCTC-19} for more details.
The one-electron density and on-top density is computed from a very large CIPSI expansion containing several million determinants.
All the RS-DFT and exFCI calculations have been performed with {\QP}. \cite{QP2}
For the numerical quadratures, we employ the SG-2 grid. \cite{DasHer-JCC-17}
The geometries have been extracted from Refs.~\citenum{LooSceBloGarCafJac-JCTC-18, LooBogSceCafJAc-JCTC-19} and have been obtained at the CC3/aug-cc-pVTZ level of theory.
They are also reported in the {\SI}.
Frozen-core calculations are systematically performed and defined as such: a \ce{He} core is frozen from \ce{Li} to \ce{Ne}, while a \ce{Ne} core is frozen from \ce{Na} to \ce{Ar}.
The FC density-based correction is used consistently with the FC approximation in WFT methods.


%%%%%%%%%%%%%%%%%%%%%%%%
\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 2 %%%
	\begin{squeezetable}
	\begin{table*}
	\caption{
	Vertical absorption energies $\Eabs$ (in eV) of excited states of water, carbon dimer and ammonia for various methods and basis sets.}
	\begin{ruledtabular}{}
	\begin{tabular}{lllddddddddddddd}
				&				&					&				&	\mc{12}{c}{Deviation with respect to TBE}		
	\\
	\cline{5-16}
				&				&					&				&	\mc{3}{c}{exFCI}	
																	&	\mc{3}{c}{exFCI+PBEot}	
																	&	\mc{3}{c}{exFCI+PBE}	
																	&	\mc{3}{c}{exFCI+LDA}
	\\
	\cline{5-7}	\cline{8-10}	\cline{11-13}	\cline{14-16}
	Molecule	&	Transition		&	Nature		&	\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
	Ammonia		&	$1\,^{1}A_{1} \ra 1\,^{1}A_{2}$		&	Ryd.	&	6.66	&	-0.18	&	-0.07	&	-0.02	
																				&	-0.04	&	-0.02	&	0.00	
																				&	-0.07	&	-0.03	&	0.00	
																				&	-0.07	&	-0.03	&	0.00	
	\\
				&	$1\,^{1}A_{1} \ra 1\,^{1}E$		&	Ryd.		&	8.21	&	-0.13	&	-0.05	&	0.01	
																				&	0.01	&	0.00	&	0.04	
																				&	-0.03	&	-0.01	&	0.03	
																				&	-0.03	&	0.00	&	0.03	
	\\
				&	$1\,^{1}A_{1} \ra 2\,^{1}A_{1}$		&	Ryd.	&	8.65	&	1.03	&	0.68	&	0.49	
																				&	1.17	&	0.73	&	0.75	
																				&	1.13	&	0.72	&	0.74	
																				&	1.13	&	0.71	&	0.78	
	\\
				&	$1\,^{1}A_{1} \ra 2\,^{1}A_{2}$		&	Ryd.	&	8.65	&	1.22	&	0.77	&	0.60	
																				&	1.17	&	0.73	&	0.75	
																				&	1.13	&	0.72	&	0.74	
																				&	1.13	&	0.71	&	0.78	
	\\
				&	$1\,^{1}A_{1} \ra 1\,^{3}A_{2}$		&	Ryd.	&	9.19	&	-0.18	&	-0.06	&	-0.02	
																				&	1.36	&	0.83	&	0.63	
																				&	1.33	&	0.81	&	0.62	
																				&	1.32	&	0.81	&	0.61	
	\\
	\\
	Carbon dimer\fnm[1]	&	$1\,^{1}\Sigma_g^+ \ra 1\,^{1}\Delta_g$		&	Val.	&	2.06	&	0.15	&	0.03	&	0.00
																								&	0.02	&	-0.02	&	-0.02
																								&	0.13	&	0.02	&	0.00
																								&	0.15	&	0.03	&	0.00
	\\
						&	$1\,^{1}\Sigma_g^+ \ra 2\,^{1}\Sigma_g^+$	&	Val.	&	2.40	&	0.10	&	0.02	&	0.00
																								&	0.02	&	-0.03	&	-0.02
																								&	0.09	&	0.01	&	0.00
																								&	0.11	&	0.02	&	0.00
	\\
	\\
	Hydrogen chloride&	${}^1\Sigma \ra {}^1\Pi$	&	CT\fnm[2]	&	7.86	&	-0.04	&	-0.02	&	0.02
																				&	0.13	&	0.06	&	0.06
																				&	0.11	&	0.04	&	0.05
																				&	0.10	&	0.05	&	0.06
	\\
	\\
	Hydrogen sulfide	&	$1\,^{1}A_1 \ra 1\,^{1}A_2$		&	Ryd.	&	6.10	&	0.00	&	0.08	&	0.05
																					&	0.15	&	0.12	&	0.07
																					&	0.14	&	0.11	&	0.07
																					&	0.14	&	0.11	&	0.07
	\\
						&	$1\,^{1}A_1 \ra 1\,^{1}B_1$		&	Ryd.	&	6.29	&	0.00	&	-0.05	&	0.00
																					&	-0.12	&	0.01	&	0.03
																					&	-0.14	&	0.00	&	0.03
																					&	-0.14	&	0.01	&	0.03
	\\
						&	$1\,^{1}A_1 \ra 1\,^{3}A_2$		&	Ryd.	&	5.74	&	0.01	&	0.07	&	0.05
																					&	0.18	&	0.12	&	0.08
																					&	0.20	&	0.13	&	0.08
																					&	0.19	&	0.13	&	0.08
	\\
						&	$1\,^{1}A_1 \ra 1\,^{3}B_1$		&	Ryd.	&	5.94	&	-0.04	&	-0.05	&	-0.01
																					&	0.07	&	0.02	&	0.03
																					&	0.09	&	0.03	&	0.03
																					&	0.07	&	0.04	&	0.04
	\\
	\\
	Water		&	$1\,^{1}A_1 \ra 1\,^{1}B_1$		&	Ryd.	&	7.70		&	-0.17	&	-0.07	&	-0.02	
																	&	0.01	&	0.00	&	0.02	
																	&	-0.02	&	-0.01	&	0.00	
																	&	-0.04	&	-0.01	&	0.01
	\\
				&	$1\,^{1}A_1 \ra 1\,^{1}A_2$		&	Ryd.	&	9.47		&	-0.15	&	-0.06	&	-0.01	
																	&	0.03	&	0.01	&	0.03	
																	&	0.00	&	0.00	&	0.02	
																	&	-0.03	&	0.00	&	0.00
	\\
				&	$1\,^{1}A_1 \ra 2\,^{1}A_1$		&	Ryd.	&	9.97		&	-0.03	&	0.02	&	0.06	
																	&	0.13	&	0.08	&	0.09	
																	&	0.10	&	0.07	&	0.08		
																	&	0.09	&	0.07	&	0.03
	\\
				&	$1\,^{1}A_1 \ra 1\,^{3}B_1$		&	Ryd.	&	7.33		&	-0.19	&	-0.08	&	-0.03	
																	&	0.02	&	0.00	&	0.02	
																	&	0.05	&	0.01	&	0.02	
																	&	0.00	&	0.00	&	0.04
	\\
				&	$1\,^{1}A_1 \ra 1\,^{3}A_2$		&	Ryd.	&	9.30		&	-0.16	&	-0.06	&	-0.01	
																	&	0.04	&	0.02	&	0.04	
																	&	0.07	&	0.03	&	0.04	
																	&	0.03	&	0.03	&	0.04
	\\
				&	$1\,^{1}A_1 \ra 1\,^{3}A_1$		&	Ryd.	&	9.59		&	-0.11	&	-0.05	&	-0.01	
																	&	0.07	&	0.02	&	0.03	
																	&	0.09	&	0.03	&	0.03	
																	&	0.06	&	0.03	&	0.04
	\end{tabular}
	\end{ruledtabular}
	\fnt[1]{Doubly-excited states of $(\pi,\pi) \ra (\si,\si)$ character.}
	\fnt[2]{CT stands for charge transfer.}
	\end{table*}
	\end{squeezetable}
%%% %%% %%%

%%% TABLE 3 %%%
	\begin{squeezetable}
	\begin{table*}
	\caption{
	Vertical absorption energies $\Eabs$ (in eV) of excited states of acetylene, ethylene and formaldehyde for various methods and basis sets.}
	\begin{ruledtabular}{}
	\begin{tabular}{lllddddddddd}
				&			&							&				&	\mc{8}{c}{Deviation with respect to TBE}		
	\\
	\cline{5-12}
				&			&							&				&	\mc{2}{c}{exFCI}	
																		&	\mc{2}{c}{exFCI+PBEot}	
																		&	\mc{2}{c}{exFCI+PBE}	
																		&	\mc{2}{c}{exFCI+LDA}
	\\
	\cline{5-6}	\cline{7-8}	\cline{9-10}	\cline{11-12}
	Molecule	&	Transition	&	Nature				&	\tabc{TBE}	&	\tabc{AVDZ}	&	\tabc{AVTZ}		
																		&	\tabc{AVDZ}	&	\tabc{AVTZ}		
																		&	\tabc{AVDZ}	&	\tabc{AVTZ}		
																		&	\tabc{AVDZ}	&	\tabc{AVTZ}		
	\\
	\hline
	Acetylene	&	$1\,^{1}\Sigma_{g}^{+} \ra 1\,^{1}\Sigma_{u}^{-}$	&	Val.	&	7.10	&	0.10	&	0.00
																								&	0.07	&	0.00	
																								&	0.11	&	0.00	
																								&	0.11	&	0.00	
	\\
				&	$1\,^{1}\Sigma_{g}^{+} \ra 1\,^{1}\Delta_{u}$		&	Val.	&	7.44	&	0.07	&	0.00	
																								&	0.04	&	-0.01	
																								&	0.12	&	0.02	
																								&	0.11	&	0.02	
	\\
				&	$1\,^{1}\Sigma_{g}^{+} \ra 1\,^{3}\Sigma_{u}^{+}$	&	Val.	&	5.56	&	-0.06	&	-0.03		
																								&	0.07	&	0.02	
																								&	0.04	&	0.00	
																								&	0.02	&	0.00	
	\\
				&	$1\,^{1}\Sigma_{g}^{+} \ra 1\,^{3}\Delta_{u}$		&	Val.	&	6.40	&	0.06	&	0.00	
																								&	0.10	&	0.02	
																								&	0.14	&	0.03	
																								&	0.12	&	0.03	
	\\
				&	$1\,^{1}\Sigma_{g}^{+} \ra 1\,^{3}\Sigma_{u}^{-}$	&	Val.	&	7.09	&	0.05	&	-0.01
																								&	0.08	&	0.00	
																								&	0.16	&	0.04	
																								&	0.14	&	0.04	
	\\
	\\
	Ethylene	&	$1\,^{1}A_{1g} \ra 1\,^{1}B_{3u}$		&	Ryd.	&	7.43	&	-0.12	&	-0.04	
																					&	-0.05	&	-0.01	
																					&	-0.04	&	-0.01	
																					&	-0.02	&	0.00	
	\\
				&	$1\,^{1}A_{1g} \ra 1\,^{1}B_{1u}$		&	Val.	&	7.92	&	0.01	&	0.01	
																					&	0.00	&	0.00	
																					&	0.06	&	0.03	
																					&	0.06	&	0.03	
	\\
				&	$1\,^{1}A_{1g} \ra 1\,^{1}B_{1g}$		&	Ryd.	&	8.10	&	-0.1	&	-0.02	
																					&	-0.03	&	0.00	
																					&	-0.02	&	0.00		
																					&	0.00	&	0.01	
	\\
				&	$1\,^{1}A_{1g} \ra 1\,^{3}B_{1u}$		&	Val.	&	4.54	&	0.01	&	0.00
																					&	0.07	&	0.03	
																					&	0.10	&	0.04	
																					&	0.08	&	0.04	
	\\
				&	$1\,^{1}A_{1g} \ra 1\,^{3}B_{3u}$		&	Val.	&	7.28	&	-0.12	&	-0.04
																					&	-0.03	&	0.00	
																					&	0.00	&	0.00	
																					&	0.00	&	0.02	
	\\
				&	$1\,^{1}A_{1g} \ra 1\,^{3}B_{1g}$		&	Val.	&	8.00	&	-0.07	&	-0.01
																					&	0.01	&	0.03	
																					&	0.04	&	0.03	
																					&	0.05	&	0.04	
	\\
	\\
	Formaldehyde&	$1\,^{1}A_{1} \ra 1\,^{1}A_{2}$			&	Val.	&	3.97	&	0.02	&	0.01	
																			&	0.05	&	0.02	
																			&	0.03	&	0.02	
																			&	0.02	&	0.01	
	\\
				&	$1\,^{1}A_{1} \ra 1\,^{1}B_{2}$			&	Ryd.	&	7.30	&	-0.19	&	-0.07	
																			&	0.00	&	0.00	
																			&	-0.02	&	0.00	
																			&	-0.04	&	0.00	
	\\
				&	$1\,^{1}A_{1} \ra 2\,^{1}B_{2}$			&	Ryd.	&	8.14	&	-0.10	&	-0.01	
																			&	0.09	&	0.07	
																			&	0.08	&	0.06	
																			&	0.05	&	0.06	
	\\
				&	$1\,^{1}A_{1} \ra 2\,^{1}A_{1}$			&	Ryd.	&	8.27	&	-0.15	&	-0.04	
																			&	0.03	&	0.04	
																			&	0.02	&	0.03	
																			&	0.00	&	0.03	
	\\
				&	$1\,^{1}A_{1} \ra 1\,^{3}A_{2}$			&	Val.	&	3.58	&	0.00	&	0.00	
																			&	0.09	&	0.05	
																			&	0.11	&	0.06	
																			&	0.07	&	0.04	
	\\
				&	$1\,^{1}A_{1} \ra 1\,^{3}A_{1}$			&	Val.	&	6.07	&	0.03	&	0.01	
																			&	0.13	&	0.04	
																			&	0.15	&	0.05	
																			&	0.11	&	0.04	
	\\
				&	$1\,^{1}A_{1} \ra 1\,^{3}B_{2}$			&	Ryd.	&	7.14	&	-0.19	&	-0.08	
																			&	0.01	&	0.01	
																			&	0.02	&	0.01	
																			&	-0.01	&	0.00	
	\\
				&	$1\,^{1}A_{1} \ra 2\,^{3}B_{2}$			&	Ryd.	&	7.96	&	-0.09	&	-0.02	
																			&	0.13	&	0.08	
																			&	0.14	&	0.08	
																			&	0.10	&	0.07	
	\\
				&	$1\,^{1}A_{1} \ra 1\,^{3}A_{1}$			&	Ryd.	&	8.15	&	-0.14	&	-0.05	
																			&	0.07	&	0.05	
																			&	0.07	&	0.04	
																			&	0.04	&	0.04	
	\\
	\end{tabular}
	\end{ruledtabular}
	\end{table*}
	\end{squeezetable}
%%% %%% %%%


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

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

%%%%%%%%%%%%%%%%%%%%%%%%
\begin{acknowledgements}
The authors would like to thank the \textit{Centre National de la Recherche Scientifique} (CNRS) for funding. 
This work was performed using HPC resources from GENCI-TGCC (Grant No.~2018-A0040801738) and CALMIP (Toulouse) under allocation 2019-18005.
\end{acknowledgements}
%%%%%%%%%%%%%%%%%%%%%%%%



\bibliography{Ex-srDFT,Ex-srDFT-control}

\end{document}
