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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 get chemically accurate vertical and adiabatic excitation energies 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 monoxide, carbon dimer and ethylene).
The present study clearly evidences that special care has to be taken with 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 performance 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}.
Unless otherwise stated, atomic units are used.

%%%%%%%%%%%%%%%%%%%%%%%%
\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} \approx \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 referred 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 configuration interaction (sCI) 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.
Using near-FCI excitation energies (within a given basis set) has the indisputable advantage to remove the error inherent to the WFT method.
Indeed, in the present case, the only source of error on the excitation energies is due to basis set incompleteness.
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 up to 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.
In the following, we employ the AVXZ shorthand notations for Dunning's aug-cc-pVXZ basis sets.

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

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

Methylene is a paradigmatic system in electronic structure theory. \cite{Sch-Science-86} 
Due to its relative small size, its ground and excited states have been thoroughly studied 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}

As a first test of the present density-based basis set correction, we consider the four lowest-lying states of methylene ($1\,^{3}B_1$, $1\,^{1}A_1$, $1\,^{1}B_1$ and $2\,^{1}A_1$) at their respective equilibrium geometry and compute the corresponding adiabatic transition energies for various basis sets ranging from AVDZ to AVQZ.
We have also computed total energies at the exFCI/AV5Z level and used these alongside the quadruple-$\zeta$ ones to extrapolate the total energies to the CBS limit with the usual extrapolation formula \cite{HelJorOls-BOOK-02}
\begin{equation}
	\E{}{\text{AVXZ}} = \E{}{\CBS} + \frac{\alpha}{(\tX+1/2)^{3}}.
\end{equation}

These results are illustrated in Fig.~\ref{fig:CH2} and reported in Table \ref{tab:CH2} alongside reference values from the literature obtained with various deterministic and stochastic approaches. \cite{ChiHolAdaOttUmrShaZim-JPCA-18, SheLeiVanSch-JCP-98, JenBun-JCP-88, SheLeiVanSch-JCP-98, ZimTouZhaMusUmr-JCP-09}
Total energies for each state can be found in the {\SI}.
Figure \ref{fig:CH2} clearly shows that, for the double-$\zeta$ basis, the exFCI adiabatic energies are far from being chemically accurate with errors as high as 0.015 eV.
From the triple-$\zeta$ basis onward, the exFCI excitation energies are chemically-accurate though, and converge steadily to the CBS limit when one increases the size of the basis set.
Concerning the basis set correction, already at the double-$\zeta$ level, the PBEot correction returns chemically accurate excitation energies.
The performance of the PBE and LDA functionals (which does not require the computation of the on-top density of each state) is less impressive. 
Yet, they still yield significant reductions of the basis set incompleteness error, hence representing a good compromise between computational cost and accuracy.
Note that the results for the PBE functional are not represented in Fig.~\ref{fig:CH2} as they are very similar to the LDA ones (similar considerations apply to the other systems studied below).
It is also quite evident that, the basis set correction has the tendency of over-correcting the excitation energies via an over-stabilization of the excited states compared to the ground state.
This trend is quite systematic as we shall see below.

%%% TABLE 1 %%%
	\begin{squeezetable}
	\begin{table}
	\caption{
	Adiabatic transition energies $\Ead$ (in eV) of excited states of methylene for various methods and basis sets.
	The relative difference with respect to the exFCI/CBS result is reported in square brackets.
	See the {\SI} for raw data.}
	\label{tab:CH2}
	\begin{ruledtabular}
	\begin{tabular}{lllll}
				&				&	\mc{3}{c}{Transitions}	\\
	\cline{3-5}
	Method		&	Basis set	&	\tabc{$1\,^{3}B_1 \ra 1\,^{1}A_1$}		
								&	\tabc{$1\,^{3}B_1 \ra 1\,^{1}B_1$}	
								&	\tabc{$1\,^{3}B_1 \ra 2\,^{1}A_1$}		\\
	\hline
	exFCI		&	AVDZ					
								&	$0.441$ [$+0.053$]	
								&	$1.536$ [$+0.146$]
								&	$2.659$ [$+0.154$]	\\	
				&	AVTZ					
								&	$0.408$ [$+0.020$]	
								&	$1.423$ [$+0.034$]	
								&	$2.546$ [$+0.042$]	\\	
				&	AVQZ					
								&	$0.395$ [$+0.007$]	
								&	$1.399$ [$+0.010$]	
								&	$2.516$ [$+0.012$]	\\	
				&	AV5Z					
								&	$0.390$ [$+0.001$]	
								&	$1.392$ [$+0.002$]	
								&	$2.507$ [$+0.003$]	\\	
				&	CBS					
								&	$0.388$	
								&	$1.390$	
								&	$2.504$	\\	
								\\
	exFCI+PBEot	&	AVDZ					
								&	$0.347$ [$-0.042$]	
								&	$1.401$ [$+0.011$]	
								&	$2.511$ [$+0.007$]	\\	
				&	AVTZ					
								&	$0.374$ [$-0.014$]	
								&	$1.378$ [$-0.012$]	
								&	$2.491$ [$-0.013$]	\\	
				&	AVQZ					
								&	$0.379$ [$-0.009$]	
								&	$1.378$ [$-0.011$]	
								&	$2.489$ [$-0.016$]	\\	
								\\
	exFCI+PBE	&	AVDZ					
								&	$0.308$ [$-0.080$]	
								&	$1.388$ [$-0.002$]	
								&	$2.560$ [$+0.056$]	\\	
				&	AVTZ					
								&	$0.356$ [$-0.032$]	
								&	$1.371$ [$-0.019$]	
								&	$2.510$ [$+0.006$]	\\	
				&	AVQZ					
								&	$0.371$ [$-0.017$]
								&	$1.375$ [$-0.015$]	
								&	$2.498$ [$-0.006$]	\\	
								\\
	exFCI+LDA	&	AVDZ					
								&	$0.337$ [$-0.051$]	
								&	$1.420$ [$+0.030$]	
								&	$2.586$ [$+0.082$]	\\	
				&	AVTZ					
								&	$0.359$ [$-0.029$]	
								&	$1.374$ [$-0.016$]	
								&	$2.514$ [$+0.010$]	\\	
				&	AVQZ					
								&	$0.370$ [$-0.018$]	
								&	$1.375$ [$-0.015$]	
								&	$2.499$ [$-0.005$]	\\	
								\\
	SHCI\fnm[1]	&	AVQZ					
								&	$0.393$	
								&	$1.398$	
								&	$2.516$	\\
	CR-EOMCC (2,3)D\fnm[2]&	AVQZ					
								&	$0.412$	
								&	$1.460$	
								&	$2.547$	\\
	FCI\fnm[3]	&	TZ2P					
								&	$0.483$	
								&	$1.542$	
								&	$2.674$	\\
	DMC\fnm[4]	&							
								&	$0.406$	
								&	$1.416$	
								&	$2.524$	\\
	Exp.\fnm[5]	&							
								&	$0.400$	
								&	$1.411$	
	\end{tabular}
	\end{ruledtabular}
	\fnt[1]{Semistochastic heat-bath CI (SHCI) calculations from Ref.~\onlinecite{ChiHolAdaOttUmrShaZim-JPCA-18}.}
	\fnt[2]{Completely-renormalized equation-of-motion coupled cluster (CR-EOMCC) calculations from Refs.~\onlinecite{SheLeiVanSch-JCP-98, JenBun-JCP-88}.}
	\fnt[3]{Reference \onlinecite{SheLeiVanSch-JCP-98}.}
	\fnt[4]{Diffusion Monte Carlo (DMC) calculations from Ref.~\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.
	The green region corresponds to chemical accuracy (i.e., error below 1 {\kcal} or 0.043 eV).
	See Table \ref{tab:CH2} for raw data.}
	\label{fig:CH2}
\end{figure}
%%% %%% %%%

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

For the second test, we consider the water \cite{CaiTozRei-JCP-00, RubSerMer-JCP-08, LiPal-JCP-11, LooSceBloGarCafJac-JCTC-18, SceBenJacCafLoo-JCP-18, SceCafBenJacLoo-RC-19} and ammonia \cite{SchGoe-JCTC-17, BarDelPerMat-JMS-97, LooSceBloGarCafJac-JCTC-18} molecules.
They are both well-studied and possess Rydberg excited states which are highly sensitive to the radial completeness of the one-electron basis set, as evidenced in Ref.~\onlinecite{LooSceBloGarCafJac-JCTC-18}.
Table \ref{tab:Mol} reports vertical excitation energies for various singlet and triplet excited states of water and ammonia at various levels of theory (see the {\SI} for total energies).
The basis set corrected theoretical best estimates (TBEs) have been extracted from Ref.~\onlinecite{LooSceBloGarCafJac-JCTC-18} and have been obtained on the same geometries.
These results are also depicted in Figs.~\ref{fig:H2O} and \ref{fig:NH3} for \ce{H2O} and \ce{NH3}, respectively.
One would have noticed that the basis set effects are particularly strong for the third singlet excited state of water and the third and fourth singlet excited states of ammonia where this effect is even magnified.
In these cases, one really needs doubly-augmented basis sets to reach radial completeness.
The first observation worth reporting is that all three RS-DFT correlation functionals have very similar behaviors and they significantly reduce the error on the excitation energies for most of the states.
However, these results also clearly evidence 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.


%%% TABLE 2 %%%
	\begin{squeezetable}
	\begin{table*}
	\caption{
	Vertical absorption energies $\Eabs$ (in eV) of excited states of ammonia, carbon dimer, carbon monoxyde, ethylene and water for various methods and basis sets.
	The TBEs have been extracted from Refs.~\onlinecite{LooSceBloGarCafJac-JCTC-18, LooBogSceCafJac-JCTC-19} on the same geometries.
	See the {\SI} for raw data.}
	\label{tab:Mol}
	\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\fnm[1]	&	-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\fnm[1]	&	-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\fnm[1]	&	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\fnm[2]	&	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\fnm[1]	&	-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	&	$1\,^{1}\Sigma_g^+ \ra 1\,^{1}\Delta_g$		&	Val.	&	2.06\fnm[3]	&	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\fnm[3]	&	0.10	&	0.02	&	0.00
																								&	0.02	&	-0.03	&	-0.02
																								&	0.09	&	0.01	&	0.00
																								&	0.11	&	0.02	&	0.00
	\\
	\\
	Carbon monoxide	&	$1\,^{1}\Sigma^+ \ra 1\,^{1}\Pi$		&	Val.	&	8.48\fnm[1]		&	0.09	&	0.01	&	0.02
																								&	0.05	&	0.00	&	
																								&	0.07	&	0.01	&	
																								&	0.07	&	0.00	&	
	\\
	\\
	Ethylene	&	$1\,^{1}A_{1g} \ra 1\,^{1}B_{3u}$		&	Ryd.	&	7.43\fnm[3]	&	-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\fnm[3]	&	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\fnm[3]	&	-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\fnm[3]	&	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\fnm[4]	&	-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\fnm[4]	&	-0.07	&	-0.01	&	
																					&	0.01	&	0.03	&	
																					&	0.04	&	0.03	&	
																					&	0.05	&	0.04	&	
	\\
	\\
	Water		&	$1\,^{1}A_1 \ra 1\,^{1}B_1$		&	Ryd.	&	7.70\fnm[1]		&	-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\fnm[1]		&	-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\fnm[1]		&	-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\fnm[1]		&	-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\fnm[1]		&	-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\fnm[1]		&	-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]{exFCI/AVQZ data corrected with the difference between CC3/d-AV5Z and exFCI/AVQZ values. \cite{LooSceBloGarCafJac-JCTC-18}}
	\fnt[2]{exFCI/AVTZ data corrected with the difference between CC3/d-AV5Z and exFCI/AVTZ values. \cite{LooSceBloGarCafJac-JCTC-18}}
	\fnt[3]{exFCI/AVQZ data from Ref.~\onlinecite{LooBogSceCafJac-JCTC-19}.}
	\fnt[4]{exFCI/AVDZ data corrected with the difference between CC3/d-AV5Z and exFCI/AVDZ values. \cite{LooSceBloGarCafJac-JCTC-18}}
	\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.
	The green region corresponds to chemical accuracy (i.e., error below 1 {\kcal} or 0.043 eV).
	See the {\SI} for raw data.}
	\label{fig:H2O}
\end{figure}
%%% %%% %%%

%%% FIG 3 %%%
\begin{figure}
	\includegraphics[width=\linewidth]{NH3}
	\caption{Error in vertical excitation energies (in eV) of ammonia for various basis sets and methods.
	The green region corresponds to chemical accuracy (i.e., error below 1 {\kcal} or 0.043 eV).
	See the {\SI} for raw data.}
	\label{fig:NH3}
\end{figure}
%%% %%% %%%

%=======================
\subsection{Range-Separation Function in Carbon Monoxyde}
\label{sec:CO}
%=======================

\titou{It is interesting to have a look at $\rsmu{}{}(\br{})$ for the ground and excited states.
To do so, we consider the first singlet excited state of carbon monoxide (vertical excitation energies are reported in Table \ref{tab:Mol}).
Figure \ref{fig:mu} represent $\rsmu{}{}(\br{})$ for the ground and excited states for the AVDZ, AVTZ and AVQZ basis sets.}

%%% FIG 3 %%%
\begin{figure}
	\includegraphics[width=\linewidth]{CO}
	\caption{$\rsmu{}{\Bas}(z)$ along the molecular axis ($z$) for the ground state and first singlet excited state of \ce{CO} for various basis sets $\Bas$.
	The carbon and oxygen nuclei are located at $z=-1.249$ and $z=0.893$ bohr, respectively.}
	\label{fig:CO}
\end{figure}
%%% %%% %%%

%=======================
\subsection{Doubly-Excited States of the Carbon Dimer}
\label{sec:C2}
%=======================
In order to have a miscellaneous test set of excitations, in a third time, we propose to study some doubly-excited states of the carbon dimer \ce{C2}, a prototype system for strongly correlated and multireference systems. \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, HolUmrSha-JCP-17, VarRoc-PTRSMPES-18}
These two valence excitations --- $1\,^{1}\Sigma_g^+ \ra 1\,^{1}\Delta_g$ and $1\,^{1}\Sigma_g^+ \ra 2\,^{1}\Sigma_g^+$ --- are both of $(\pi,\pi) \ra (\si,\si)$ character.
They have been recently studied with state-of-the-art methods, and have been shown to be ``pure'' doubly-excited states as they do not involve single excitations. \cite{LooBogSceCafJac-JCTC-19} 
The vertical excitation energies associated with these transitions are reported in Table \ref{tab:Mol} and represented in Fig.~\ref{fig:C2}.
An interesting point here is that one really needs the PBEot to get chemically-accurate absorption energies with the AVDZ atomic basis set.
We believe that the present result is a direct consequence of the multireference character of the \ce{C2} molecule.
In other words, the UEG on-top density used in the LDA and PBE functionals (see Sec.~\ref{sec:func}) is a particularly bad approximation of the true on-top density.

%%% 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.
	The green region corresponds to chemical accuracy (i.e., error below 1 {\kcal} or 0.043 eV).
	See the {\SI} for raw data.}
	\label{fig:C2}
\end{figure}
%%% %%% %%%

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

As a final example, we consider the ethylene molecule, yet another system which has been particularly scrutinized theoretically using high-level ab initio methods. \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}
We refer the interested reader to the work of Feller et al.\cite{FelPetDav-JCP-14} for an exhaustive investigation dedicated to the excited states of ethylene using state-of-the-art CI calculations.
In the present context, ethylene is a particularly interesting system as it contains a mixture of valence and Rydberg excited states. 
Our basis set corrected vertical excitation energies are gathered in Table \ref{tab:Mol} and depicted in Fig.~\ref{fig:C2H4}.
Except for one particular excitation (the lowest singlet-triplet excitation $1\,^{1}A_{1g} \ra 1\,^{3}B_{1u}$), the exFCI+PBEot/AVDZ excitation energies are chemically accurate and the errors drop further when one goes to the triple-$\zeta$ basis.
Consistently with the previous examples, the LDA and PBE functionals are slightly less accurate, although they still correct the excitation energies in the right direction.

%%% FIG 5 %%%
\begin{figure}
	\includegraphics[width=\linewidth]{C2H4}
	\caption{Error in vertical excitation energies $\Eabs$ (in eV) of ethylene for various basis sets and methods.
	The green region corresponds to chemical accuracy (i.e., error below 1 {\kcal} or 0.043 eV).
	See the {\SI} for raw data.}
	\label{fig:C2H4}
\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, \cite{GinPraFerAssSavTou-JCP-18} one can obtain chemically-accurate excitation energies with typically augmented double-$\zeta$ basis sets.
This nicely complements our recent investigation on ground-state properties, \cite{LooPraSceTouGin-JPCL-19} which has evidenced that one recovers quintuple-$\zeta$ quality atomization and correlation energies with triple-$\zeta$ basis sets.
The present study clearly shows that, for very diffuse excited states, the present correction relying on short-range correlation functionals from RS-DFT might not be enough to catch the radial incompleteness of the one-electron basis set.
Also, in the case of multireference systems, we have evidenced that the PBEot functional is more appropriate than the LDA and PBE functionals relying on the UEG on-top density.
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 and energetic correction of the various functionals).

%%%%%%%%%%%%%%%%%%%%%%%%
\begin{acknowledgements}
The authors would like to thank the \textit{Centre National de la Recherche Scientifique} (CNRS) for funding. 
PFL would like to thank Denis Jacquemin for numerous discussions on excited states.
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}
