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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 (UMR 7616), Universit\'e Pierre et Marie Curie, Sorbonne Universit\'e, CNRS, Paris, France}

\begin{document}	

\title{Chemically-Accurate Excitation Energies With Small Basis Sets}

\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) energies obtained with the CIPSI (Configuration Interaction using a Perturbative Selection made Iteratively) algorithm with the recently proposed short-range density-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, typically, augmented double-$\zeta$ basis sets.
We illustrate the present approach on various types of excited states (valence, Rydberg and double excitations) in several small organic molecules (methylene, water, ammonia, carbon dimer and ethylene).
The present study clearly evidences that special care has to be taken for very diffuse excited states where the present correction might not be enough to catch the radial incompleteness of the one-electron basis set.

\end{abstract}

\maketitle

%%%%%%%%%%%%%%%%%%%%%%%%
\section{Introduction}
\label{sec:intro}
%%%%%%%%%%%%%%%%%%%%%%%%
One of the most fundamental problems of conventional electronic structure methods is their slow energy convergence with respect to the size of the one-electron basis set.
The overall basis set incompleteness error can be, qualitatively at least, split in two contributions stemming from the radial and angular incompleteness.
Although for ground-state properties angular incompleteness is by far the main source of error, it is definitely not unusual to have a significant radial incompleteness in the case of excited states (especially for Rydberg states), which can be alleviated by using additional sets of diffuse basis functions (i.e.~augmented basis sets).

Explicitly-correlated F12 methods \cite{Kut-TCA-85, Kut-TCA-85, KutKlo-JCP-91, NogKut-JCP-94} have been specifically designed to efficiently catch angular incompleteness. \cite{Ten-TCA-12, TenNog-WIREs-12, HatKloKohTew-CR-12, KonBisVal-CR-12, GruHirOhnTen-JCP-17, MaWer-WIREs-18}
Although they have been extremely successful to speed up convergence of ground-state energies and properties, such as correlation and atomization energies, \cite{TewKloNeiHat-PCCP-07} their performances for excited states \cite{FliHatKlo-JCP-06, NeiHatKlo-JCP-06, HanKoh-JCP-09, Koh-JCP-09, ShiWer-JCP-10, ShiKniWer-JCP-11, ShiWer-JCP-11, ShiWer-MP-13} have been much more conflicting. \cite{FliHatKlo-JCP-06, NeiHatKlo-JCP-06}

Instead of F12 methods, here we propose to follow a different route and investigate the performances of the recently proposed universal density-based basis set
incompleteness correction.  \cite{GinPraFerAssSavTou-JCP-18}
Contrary to our recent study on atomization and correlation energies, \cite{LooPraSceTouGin-JPCL-19} the present contribution focuses on vertical and adiabatic excitation energies in molecular systems which is a much tougher test for the reasons mentioned above.
This density-based correction relies on short-range correlation density functionals (with multideterminant reference) from range-separated density-functional theory \cite{TouColSav-PRA-04, AngGerSavTou-PRA-05, GolWerSto-PCCP-05, TouGerJanSavAng-PRL-09,JanHenScu-JCP-09, TouZhuSavJanAng-JCP-11, MusReiAngTou-JCP-15, LeiStoWerSav-CPL-97, FroTouJen-JCP-07, FroCimJen-PRA-10, HedKneKieJenRei-JCP-15, HedTouJen-JCP-18, FerGinTou-JCP-18} (RS-DFT) to capture the missing part of the short-range correlation effects, a consequence of the incompleteness of the one-electron basis set.
Because RS-DFT combines rigorously density-functional theory (DFT) \cite{ParYan-BOOK-89} and wave function theory (WFT) \cite{SzaOst-BOOK-96} via a decomposition of the electron-electron interaction into a non-divergent long-range part and a (complementary) short-range part (treated with WFT and DFT, respectively), the WFT method is relieved from describing the short-range part of the correlation hole around the electron-electron coalescence points.
Consequently, the energy convergence with respect to the size of the basis set is significantly improved, \cite{FraMusLupTou-JCP-15} and chemical accuracy can be obtained even with small basis sets.
For example, in Ref.~\onlinecite{LooPraSceTouGin-JPCL-19}, we have shown that one can recover quintuple-$\zeta$ quality atomization and correlation energies with triple-$\zeta$ basis sets for a much lower computational cost than F12 methods.

This work is organized as follows. 
In Sec.~\ref{sec:theory}, the main working equations of the density-based correction are reported and discussed. 
Computational details are reported in Sec.~\ref{sec:compdetails}.
In Sec.~\ref{sec:res}, we discuss our results for each system and draw our conclusions in Sec.~\ref{sec:ccl}.

%%%%%%%%%%%%%%%%%%%%%%%%
\section{Theory}
\label{sec:theory}
%%%%%%%%%%%%%%%%%%%%%%%%

The present basis set correction assumes that we have, in a given (finite) basis set $\Bas$, the ground-state and the $k$th excited-state energies, $\E{0}{\Bas}$ and $\E{k}{\Bas}$, their one-electron densities, $\n{k}{\Bas}$ and $\n{0}{\Bas}$, as well as their opposite-spin on-top pair densities, $\n{2,0}{\Bas}(\br{},\br{})$ and $\n{2,k}{\Bas}(\br{},\br{})$,
Therefore, the complete basis set (CBS) energy of the ground and excited states may be approximated as \cite{GinPraFerAssSavTou-JCP-18}
\begin{align}
	\label{eq:ECBS}
	\E{0}{\CBS} & \approx \E{0}{\Bas} + \bE{}{\Bas}[\n{0}{\Bas}],
	&
	\E{k}{\CBS} & \approx \E{k}{\Bas} + \bE{}{\Bas}[\n{k}{\Bas}],
\end{align}
where 
\begin{equation}
	\label{eq:E_funcbasis}
	 \bE{}{\Bas}[\n{}{}]
	= \min_{\wf{}{} \rightsquigarrow \n{}{}}  \mel*{\wf{}{}}{\hT + \hWee{}}{\wf{}{}} 
	- \min_{\wf{}{\Bas} \rightsquigarrow \n{}{}}  \mel*{\wf{}{\Bas}}{\hT + \hWee{}}{\wf{}{\Bas}}
\end{equation}
is the basis-dependent complementary density functional, 
\begin{align}
	\hT & = - \frac{1}{2} \sum_{i}^{\Ne} \nabla_i^2,
	&
	\hWee{} & = \sum_{i<j}^{\Ne} r_{ij}^{-1},
\end{align}
are the kinetic and electron-electron repulsion operators, respectively, and $\wf{}{\Bas}$ and $\wf{}{}$ are two general $\Ne$-electron normalized wave functions belonging to the Hilbert space spanned by $\Bas$ and the complete basis set, respectively.
The notation $\wf{}{} \rightsquigarrow \n{}{}$ in Eq.~\eqref{eq:E_funcbasis} states that $\wf{}{}$ yields the one-electron density $\n{}{}$.

Hence, the CBS excitation energy associated with the $k$th excited state reads
\begin{equation}
	\DE{k}{\CBS} = \E{k}{\CBS} - \E{0}{\CBS} = \DE{k}{\Bas} + \DbE{}{\Bas}[\n{0}{\Bas},\n{k}{\Bas}],
\end{equation}
where
\begin{equation}
	\label{eq:DEB}
	\DE{k}{\Bas} = \E{k}{\Bas} - \E{0}{\Bas}
\end{equation}
is the excitation energy in $\Bas$ and
\begin{equation}
	\label{eq:DbE}
	\DbE{}{\Bas}[\n{0}{\Bas},\n{k}{\Bas}] = \bE{}{\Bas}[\n{k}{\Bas}] - \bE{}{\Bas}[\n{0}{\Bas}]
\end{equation}
its basis set correction.
An important property of the present correction is
\begin{equation}
  \label{eq:limitfunc}
        \lim_{\Bas \to \CBS} \DbE{}{\Bas}[\n{0}{\Bas},\n{k}{\Bas}]  = 0.
\end{equation}
In other words, the correction vanishes in the CBS limit, hence guaranteeing an unaltered CBS limit. \cite{LooPraSceTouGin-JPCL-19}
%In the following, we will drop the state index $k$ and focus on the quantity $\bE{}{\Bas}[\n{}{}]$.

%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Range-separation function}
\label{sec:rs}
%%%%%%%%%%%%%%%%%%%%%%%%

In Ref.~\onlinecite{LooPraSceTouGin-JPCL-19}, we have shown that one can efficiently approximate $\bE{}{\Bas}[\n{}{}]$ by short-range correlation functionals with multi-determinantal (ECMD) reference, \cite{TouGorSav-TCA-05} $\bE{\text{c,md}}{\sr}[\n{}{},\rsmu{}{}]$.
The ECMD functionals admit, for any $\n{}{}$, the following two limits
\begin{align}
	\label{eq:large_mu_ecmd}
	\lim_{\mu \to \infty}  \bE{\text{c,md}}{\sr}[\n{}{},\rsmu{}{}] & = 0,
	&
	\lim_{\mu \to 0}  \bE{\text{c,md}}{\sr}[\n{}{},\rsmu{}{}] & = \Ec[\n{}{}],
\end{align}
where $\Ec[\n{}{}]$ is the usual universal correlation density functional defined in Kohn-Sham DFT. \cite{HohKoh-PR-64, KohSha-PR-65}

The key ingredient --- the range-separated function $\rsmu{}{\Bas}(\br{})$ --- automatically adapts to the spatial non-homogeneity of the basis set incompleteness error.
It is defined such that the long-range interaction of RS-DFT, $\w{}{\lr,\mu}(r_{12}) = \erf( \mu r_{12})/r_{12}$, coincides, at coalescence, with an effective two-electron interaction $\W{}{\Bas}(\br{1},\br{2})$ ``mimicking'' the Coulomb operator in an incomplete basis $\Bas$, i.e.~$\w{}{\lr,\rsmu{}{\Bas}(\br{})}(0) = \W{}{\Bas}(\br{},\br{})$ at any $\br{}$. \cite{GinPraFerAssSavTou-JCP-18}
The explicit expression of $\W{}{\Bas}(\br{1},\br{2})$ is given by 
\begin{equation}
	\label{eq:def_weebasis}
	\W{}{\Bas}(\br{1},\br{2})   = 
 	\begin{cases}
       \f{}{\Bas}(\br{1},\br{2})/\n{2}{\Bas}(\br{1},\br{2}), 	& \text{if $\n{2}{\Bas}(\br{1},\br{2}) \ne 0$,}
       \\
       \infty, 												& \text{otherwise,}
    \end{cases}
\end{equation}
where
\begin{equation}
	\label{eq:n2basis}
	\n{2}{\Bas}(\br{1},\br{2})
	= \sum_{pqrs \in \Bas}  \SO{p}{1} \SO{q}{2} \Gam{pq}{rs} \SO{r}{1} \SO{s}{2},
\end{equation}
and $\Gam{pq}{rs} = 2 \mel*{\wf{}{\Bas}}{ \aic{r_\downarrow}\aic{s_\uparrow}\ai{q_\uparrow}\ai{p_\downarrow}}{\wf{}{\Bas}}$ are the opposite-spin pair density associated with $\wf{}{\Bas}$ and its corresponding tensor, respectively, $\SO{p}{}$ is a (real-valued) molecular orbital (MO),
\begin{equation}
	\label{eq:fbasis}
	\f{}{\Bas}(\br{1},\br{2}) 
	= \sum_{pqrstu \in \Bas}  \SO{p}{1} \SO{q}{2} \V{pq}{rs} \Gam{rs}{tu} \SO{t}{1} \SO{u}{2},
\end{equation}
and $\V{pq}{rs}= \braket{pq}{rs}$ are two-electron Coulomb integrals.
We refer the interested readers to Refs.~\onlinecite{GinPraFerAssSavTou-JCP-18,LooPraSceTouGin-JPCL-19} for additional details.

%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Short-range correlation functionals}
\label{sec:func}
%%%%%%%%%%%%%%%%%%%%%%%%

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.~\citenum{PazMorGorBac-PRB-06}.

To go beyond the LDA and cure its over correlation at small $\mu$, some of the authors recently proposed a Perdew-Burke-Ernzerhof (PBE)-based ECMD functional \cite{FerGinTou-JCP-18}, 
\begin{equation}
	\label{eq:def_pbe_tot}
	\bE{\PBE}{\Bas}[\n{}{},\rsmu{}{\Bas}] =
	\int \n{}{}(\br{}) \be{\text{c,md}}{\sr,\PBE}\qty(\n{}{}(\br{}),s(\br{}),\zeta(\br{}),\rsmu{}{\Bas}(\br{})) \dbr{},
\end{equation}
where $s = \abs{\nabla \n{}{}}/\n{}{4/3}$ is the reduced density gradient. 
$\be{\text{c,md}}{\sr,\PBE}\qty(\n{}{},s,\zeta,\rsmu{}{})$ interpolates between the usual PBE correlation functional, \cite{PerBurErn-PRL-96} $\e{\text{c}}{\PBE}(\n{}{},s,\zeta)$, at $\rsmu{}{} = 0$ (DFT limit) and the exact large-$\rsmu{}{}$ behavior (WFT limit). \cite{TouColSav-PRA-04, GorSav-PRA-06, PazMorGorBac-PRB-06}
It reads
\begin{subequations}
\begin{gather}
	\label{eq:epsilon_cmdpbe}
	\be{\text{c,md}}{\sr,\PBE}(\n{}{},s,\zeta,\rsmu{}{}) = \frac{\e{\text{c}}{\PBE}(\n{}{},s,\zeta)}{1 + \beta(\n{}{},s,\zeta) \rsmu{}{3} },
	\\
	\label{eq:beta_cmdpbe}
	\beta(\n{}{},s,\zeta) = \frac{3}{2\sqrt{\pi} (1 - \sqrt{2} )} \frac{\e{\text{c}}{\PBE}(\n{}{},s,\zeta)}{\n{2}{}(\br{},\br{})/\n{}{}}.
\end{gather}
\end{subequations}
We will refer to this functional as the ``on top'' PBE (PBEot) ECMD functional.
More recently, \cite{LooPraSceTouGin-JPCL-19} we have also proposed a simplified version of the PBEot functional where we replaced the on-top pair density by its UEG version, i.e.~$\n{2}{\Bas}(\br{},\br{})  \approx \n{2}{\UEG}(\n{}{}(\br{}),\zeta(\br{}))$, where $\n{2}{\UEG}(\n{}{},\zeta) \approx \n{}{2} (1-\zeta^2) g_0(n)$ with the parametrization of the UEG on-top pair-distribution function $g_0(n)$ given in Eq.~(46) of Ref.~\citenum{GorSav-PRA-06}. 
This computationally-lighter functional will be refered to as PBE.


%%%%%%%%%%%%%%%%%%%%%%%%
\section{Computational details}
\label{sec:compdetails}
%%%%%%%%%%%%%%%%%%%%%%%%
In the present study, we compute the ground- and excited-state energies, one-electron and on-top densities with a selected CI method known as CIPSI (Configuration Interaction using a Perturbative Selection made Iteratively). \cite{HurMalRan-JCP-73, GinSceCaf-CJC-13, GinSceCaf-JCP-15}
The total energy of each state is obtained via an efficient extrapolation procedure of the sCI energies designed to reach near-FCI accuracy. \cite{QP2}
These energies will be labeled exFCI in the following.
We refer the interested reader to Refs.~\onlinecite{HolUmrSha-JCP-17, SceGarCafLoo-JCTC-18, LooSceBloGarCafJac-JCTC-18, SceBenJacCafLoo-JCP-18, LooBogSceCafJAc-JCTC-19, QP2} for more details.
The one-electron and on-top densities are 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}
Except for methylene for which FCI/TZVP geometries have been taken from Ref.~\onlinecite{SheLeiVanSch-JCP-98}, the other geometries have been extracted from Refs.~\onlinecite{LooSceBloGarCafJac-JCTC-18, LooBogSceCafJAc-JCTC-19} and have been obtained at the CC3/aug-cc-pVTZ level of theory.
For the sake of completeness, 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 frozen-core density-based correction is used consistently with the frozen-core approximation in WFT methods.
We refer the interested reader to Ref.~\onlinecite{LooPraSceTouGin-JPCL-19} for an explicit derivation of the equations associated with the frozen-core version of the present density-based basis set correction.
Compared to the exFCI calculations performed to compute energies and densities, the basis set correction represents, in any case, a marginal computational cost.

%%%%%%%%%%%%%%%%%%%%%%%%
\section{Results}
\label{sec:res}
%%%%%%%%%%%%%%%%%%%%%%%%

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

As a first test of the present basis set correction, we consider the adiabatic transitions of methylene which have been thourhoughly studied in the literature with high-level ab initio methods. \cite{Sch-Science-86, BauTay-JCP-86, JenBun-JCP-88, SheVanYamSch-JMS-97, SheLeiVanSch-JCP-98, AbrShe-JCP-04, AbrShe-CPL-05, ZimTouZhaMusUmr-JCP-09, ChiHolAdaOttUmrShaZim-JPCA-18}

%%% 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\fnm[1]	&	AVQZ		&	-39.08849(1)				
								&	-39.07404(1)	&	0.393	
								&	-39.03711(1)	&	1.398	
								&	-38.99603(1)	&	2.516	\\
	CR-EOMCC (2,3)D\fnm[2]&	AVQZ	&	-39.08817					
								&	-39.07303		&	0.412	
								&	-39.03450		&	1.460	
								&	-38.99457		&	2.547	\\
	FCI\fnm[3]			&	TZ2P		&	-39.066738					
								&	-39.048984		&	0.483	
								&	-39.010059		&	1.542	
								&	-38.968471		&	2.674	\\
	DMC\fnm[4]			&				&								
								&					&	0.406	
								&					&	1.416	
								&					&	2.524	\\
	Exp.\fnm[5]		&				&								
								&					&	0.400	
								&					&	1.411	
	\end{tabular}
	\end{ruledtabular}
	\fnt[1]{Reference \onlinecite{ChiHolAdaOttUmrShaZim-JPCA-18}.}
	\fnt[2]{References \onlinecite{SheLeiVanSch-JCP-98, JenBun-JCP-88}.}
	\fnt[3]{Reference \onlinecite{SheLeiVanSch-JCP-98}.}
	\fnt[4]{Reference \onlinecite{ZimTouZhaMusUmr-JCP-09}.}
	\fnt[5]{References \onlinecite{SheLeiVanSch-JCP-98, JenBun-JCP-88}.}
	\end{table*}
	\end{squeezetable}
%%% %%% %%%

%%% FIG 1 %%%
\begin{figure}
	\includegraphics[width=\linewidth]{CH2}
	\caption{Error in adiabatic excitation energies $\Ead$ (in eV) of methylene for various basis sets and methods.}
	\label{fig:CH2}
\end{figure}
%%% %%% %%%

%=======================
\subsection{Rydberg States of Water and Ammonia}
\label{sec:H2O-NH3}
%=======================

Water \cite{CaiTozRei-JCP-00, RubSerMer-JCP-08, LiPal-JCP-11, LooSceBloGarCafJac-JCTC-18, SceBenJacCafLoo-JCP-18} and ammonia \cite{SchGoe-JCTC-17, BarDelPerMat-JMS-97, LooSceBloGarCafJac-JCTC-18} are two interesting molecules with Rydberg excited states which are highly sensitive to the radial completeness of the one-electron basis set.

%%% TABLE 2 %%%
	\begin{squeezetable}
	\begin{table*}
	\caption{
	Vertical absorption energies $\Eabs$ (in eV) of excited states of ammonia, carbon dimer, water and ethylene for various methods and basis sets.
	The TBEs have been extracted from Refs.~\onlinecite{LooSceBloGarCafJac-JCTC-18, LooBogSceCafJAc-JCTC-19}.}
	\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	&	-0.04	
																				&	-0.04	&	-0.02	&	-0.01	
																				&	-0.07	&	-0.03	&	-0.02	
																				&	-0.07	&	-0.03	&	-0.02	
	\\
				&	$1\,^{1}A_{1} \ra 1\,^{1}E$		&	Ryd.		&	8.21	&	-0.13	&	-0.05	&	-0.02	
																				&	0.01	&	0.00	&	0.01	
																				&	-0.03	&	-0.01	&	0.00	
																				&	-0.03	&	0.00	&	0.00	
	\\
				&	$1\,^{1}A_{1} \ra 2\,^{1}A_{1}$		&	Ryd.	&	8.65	&	1.03	&	0.68	&	0.47	
																				&	1.17	&	0.73	&	0.50	
																				&	1.12	&	0.72	&	0.49	
																				&	1.11	&	0.71	&	0.49	
	\\
				&	$1\,^{1}A_{1} \ra 2\,^{1}A_{2}$		&	Ryd.	&	8.65	&	1.22	&	0.77	&	0.59	
																				&	1.36	&	0.83	&	0.62	
																				&	1.33	&	0.81	&	0.61	
																				&	1.32	&	0.81	&	0.61	
	\\
				&	$1\,^{1}A_{1} \ra 1\,^{3}A_{2}$		&	Ryd.	&	9.19	&	-0.18	&	-0.06	&	-0.03	
																				&	-0.03	&	0.00	&	-0.02	
																				&	-0.07	&	-0.02	&	-0.03	
																				&	-0.07	&	-0.01	&	-0.03	
	\\
	\\
	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
	\\
	\\
%	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}
	\fnt[1]{Doubly-excited states of $(\pi,\pi) \ra (\si,\si)$ character.}
%	\fnt[2]{CT stands for charge transfer.}
	\end{table*}
	\end{squeezetable}
%%% %%% %%%

%%% FIG 2 %%%
\begin{figure}
	\includegraphics[width=\linewidth]{H2O}
	\caption{Error in vertical excitation energies (in eV) of water for various basis sets and methods.}
	\label{fig:H2O}
\end{figure}
%%% %%% %%%

%%% FIG 4 %%%
\begin{figure}
	\includegraphics[width=\linewidth]{NH3}
	\caption{Error in vertical excitation energies (in eV) of ammonia for various basis sets and methods.}
	\label{fig:NH3}
\end{figure}
%%% %%% %%%

%=======================
\subsection{Doubly-Excited States of the Carbon Dimer}
\label{sec:C2}
%=======================
It is also interesting to study doubly-excited states. \cite{AbrShe-JCP-04, AbrShe-CPL-05, Var-JCP-08, PurZhaKra-JCP-09, AngCimPas-MP-12, BooCleThoAla-JCP-11, Sha-JCP-15, SokCha-JCP-16, VarRoc-PTRSMPES-18}
In the carbon dimer, these valence states are of $(\pi,\pi) \ra (\si,\si)$ character and they have been recently studied with state-of-the-art methods. \cite{LooBogSceCafJAc-JCTC-19}

%%% FIG 4 %%%
\begin{figure}
	\includegraphics[width=\linewidth]{C2}
	\caption{Error in vertical excitation energies $\Eabs$ (in eV) for two doubly-excited states of the carbon dimer for various basis sets and methods.}
	\label{fig:C2}
\end{figure}
%%% %%% %%%


%=======================
\subsection{Ethylene}
\label{sec:C2H4}
%=======================

Ethylene is an interesting molecules as it contains both valence and Rydberg excited states. \cite{SerMarNebLinRoo-JCP-93, WatGwaBar-JCP-96, WibOliTru-JPCA-02, BarPaiLis-JCP-04, Ang-JCC-08, SchSilSauThi-JCP-08, SilSchSauThi-JCP-10, SilSauSchThi-MP-10, Ang-IJQC-10, DadSmaBooAlaFil-JCTC-12, FelPetDav-JCP-14, ChiHolAdaOttUmrShaZim-JPCA-18}

%\begin{figure}
%	\includegraphics[width=\linewidth]{C2H2}
%	\caption{Error in vertical excitation energies (in eV) of acetylene for various basis sets and methods.}
%	\label{fig:C2H2}
%\end{figure}

\begin{figure}
	\includegraphics[width=\linewidth]{C2H4}
	\caption{Error in vertical excitation energies $\Eabs$ (in eV) of ethylene for various basis sets and methods.}
	\label{fig:C2H4}
\end{figure}

%\begin{figure}
%	\includegraphics[width=\linewidth]{CH2O}
%	\caption{Error in vertical excitation energies $\Eabs$ (in eV) of formaldehyde for various basis sets and methods.}
%	\label{fig:CH2O}
%\end{figure}

%%%%%%%%%%%%%%%%%%%%%%%%
\section{Conclusion}
\label{sec:ccl}
%%%%%%%%%%%%%%%%%%%%%%%%
We have shown that by employing the recently proposed density-based basis set correction developed by some of the authors, one can obtain chemically-accurate excitation energies with typically augmented double-$\zeta$ basis sets.
We are currently investigating the performance of the present basis set correction for strongly correlated systems and we hope to report on this in the near future.

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

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\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}
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