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\documentclass[10pt]{letter}


\usepackage{UPS_letterhead,xcolor,mhchem,mathpazo,ragged2e}


\newcommand{\alert}[1]{\textcolor{red}{#1}}


\definecolor{darkgreen}{HTML}{009900}






\begin{document}




\begin{letter}%


{To the Editors of the Journal of Chemical Theory and Computation}




\opening{Dear Editors,}




\justifying


Please find enclosed our manuscript entitled


\begin{quote}


\textit{``Chemically Accurate Excitation Energies With Small Basis Sets''},


\end{quote}


which we would like you to consider as a Regular Article in the \textit{Journal of Chemical Theory and Computation}.


This contribution nicely fits in the \textit{``Quantum Electronic Structure''} section.




Due to their diverse nature, the faithful description of excited states within electronic structure theory methods remains one of the grand challenges of modern theoretical chemistry.


In the present article, we show that, by combining selected configuration interaction methods and the recently proposed densitybased basis set correction [see Giner et al., J. Chem. Phys. 149 (2018) 194301] one can obtain chemicallyaccurate vertical and adiabatic excitation energies with typically augmented double$\zeta$ basis sets.


This nicely complements our recent investigation on groundstate properties [see Loos et al., J. Phys. Chem. Lett. 10 (2019) 2931] which has evidenced that one recovers quintuple$\zeta$ quality atomization and correlation energies with triple$\zeta$ basis sets for a much lower computational cost than F12 methods.


The present densitybased correction relies on shortrange correlation density functionals (with multideterminant reference) from rangeseparated densityfunctional theory to capture the missing part of the shortrange correlation effects, a consequence of the incompleteness of the oneelectron basis set.




We suggest Sandeep Sharma, David Tew, Claudia Filippi, Eric Neuscamman, and Emmanuel Fromager as potential referees.


This contribution has never been submitted in total nor in parts to any other journal, and has been seen and approved by all authors.


We look forward to hearing from you.




\closing{Sincerely, the authors.}






\end{letter}


\end{document}













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% units


\newcommand{\IneV}[1]{#1 eV}


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\newcommand{\kcal}{kcal/mol}




% methods


\newcommand{\D}{\text{D}}


\newcommand{\T}{\text{T}}


\newcommand{\Q}{\text{Q}}


\newcommand{\X}{\text{X}}


\newcommand{\UEG}{\text{UEG}}


\newcommand{\HF}{\text{HF}}


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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), Sorbonne Universit\'e, CNRS, Paris, France}




\begin{document}




\title{\textcolor{blue}{Chemically accurate excitation energies with small basis sets}}




\author{Emmanuel Giner}


\email[Corresponding author: ]{emmanuel.giner@lct.jussieu.fr}


\affiliation{\LCT}


\author{Anthony Scemama}


\affiliation{\LCPQ}


\author{Julien Toulouse}


\affiliation{\LCT}


\author{PierreFran\c{c}ois Loos}


\email[Corresponding author: ]{loos@irsamc.upstlse.fr}


\affiliation{\LCPQ}




\begin{abstract}


%\begin{wrapfigure}[13]{o}[1.25cm]{0.5\linewidth}


% \centering


% \includegraphics[width=\linewidth]{TOC}


%\end{wrapfigure}


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 shortrange densityfunctional correction for basisset incompleteness [\href{https://doi.org/10.1063/1.5052714}{Giner \textit{et al.}, \textit{J.~Chem.~Phys.}~ \textbf{2018}, \textit{149}, 194301}], 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 dimer and ethylene).


The present study clearly evidences that special care has to be taken with very diffuse excited states where the present correction does not catch the radial incompleteness of the oneelectron basis set.


\end{abstract}




\maketitle




%%%%%%%%%%%%%%%%%%%%%%%%


\section{Introduction}


\label{sec:intro}


%%%%%%%%%%%%%%%%%%%%%%%%


One of the most fundamental problems of conventional wave function electronic structure methods is their slow energy convergence with respect to the size of the oneelectron basis set.


The overall basisset incompleteness error can be, qualitatively at least, split in two contributions stemming from the radial and angular incompleteness.


Although for groundstate 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).




Explicitlycorrelated F12 methods \cite{KutTCA85, KutKloJCP91, NogKutJCP94} have been specifically designed to efficiently catch angular incompleteness. \cite{TenTCA12, TenNogWIREs12, HatKloKohTewCR12, KonBisValCR12, GruHirOhnTenJCP17, MaWerWIREs18}


Although they have been extremely successful to speed up convergence of groundstate energies and properties, such as correlation and atomization energies, \cite{TewKloNeiHatPCCP07} their performance for excited states \cite{FliHatKloJCP06, NeiHatKloJCP06, HanKohJCP09, KohJCP09, ShiWerJCP10, ShiKniWerJCP11, ShiWerJCP11, ShiWerMP13} has been much more conflicting. \cite{FliHatKloJCP06, NeiHatKloJCP06}


\alert{However, very encouraging results have been reported recently using the extended explicitlycorrelated secondorder approximate coupledcluster singles and doubles ansatz suitable for response theory on systems such as methylene, formaldehyde and imidazole. \cite{HofSchKloKohJCP19}}




Instead of F12 methods, here we propose to follow a different route and investigate the performance of the recently proposed densitybased basis set


incompleteness correction. \cite{GinPraFerAssSavTouJCP18}


Contrary to our recent study on atomization and correlation energies, \cite{LooPraSceTouGinJPCL19} 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 densitybased correction relies on shortrange correlation density functionals (with multideterminant reference) from rangeseparated densityfunctional theory \cite{SavINC96, LeiStoWerSavCPL97, TouColSavPRA04, TouSavFlaIJQC04, AngGerSavTouPRA05, GolWerStoPCCP05, PazMorGorBacPRB06, FroTouJenJCP07, TouGerJanSavAngPRL09, JanHenScuJCP09, FroCimJenPRA10, TouZhuSavJanAngJCP11, MusReiAngTouJCP15, HedKneKieJenReiJCP15, HedTouJenJCP18, FerGinTouJCP19} (RSDFT) to capture the missing part of the shortrange correlation effects, a consequence of the incompleteness of the oneelectron basis set.


Because RSDFT combines rigorously densityfunctional theory (DFT) \cite{ParYanBOOK89} and wave function theory (WFT) \cite{SzaOstBOOK96} via a decomposition of the electronelectron interaction into a nondivergent longrange part and a (complementary) shortrange part (treated with WFT and DFT, respectively), the WFT method is relieved from describing the shortrange part of the correlation hole around the electronelectron coalescence points (the socalled electronelectron cusp). \cite{KatCPAM57}


Consequently, the energy convergence with respect to the size of the basis set is significantly improved, \cite{FraMusLupTouJCP15} and chemical accuracy can be obtained even with small basis sets.


For example, in Ref.~\onlinecite{LooPraSceTouGinJPCL19}, 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 densitybased correction are reported and discussed.


Computational details are given 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 basisset correction assumes that we have, in a given (finite) basis set $\Bas$, the groundstate and the $k$th excitedstate energies, $\E{0}{\Bas}$ and $\E{k}{\Bas}$, their oneelectron densities, $\n{k}{\Bas}(\br{})$ and $\n{0}{\Bas}(\br{})$, as well as their oppositespin ontop pair densities, $\n{2,0}{\Bas}(\br{})$ and $\n{2,k}{\Bas}(\br{})$,


Therefore, the completebasisset (CBS) energy of the ground and excited states may be approximated as \cite{GinPraFerAssSavTouJCP18}


\begin{subequations}


\begin{align}


\label{eq:E0CBS}


\E{0}{\CBS} & \approx \E{0}{\Bas} + \bE{}{\Bas}[\n{0}{\Bas}],


\\


\label{eq:EkCBS}


\E{k}{\CBS} & \approx \E{k}{\Bas} + \bE{}{\Bas}[\n{k}{\Bas}],


\end{align}


\end{subequations}


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 basisdependent 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 electronelectron repulsion operators, respectively, and $\wf{}{\Bas}$ and $\wf{}{}$ are two general $\Ne$electron normalized wave functions belonging to the Hilbert spaces spanned by $\Bas$ and the complete basis, respectively.


The notation $\wf{}{} \rightsquigarrow \n{}{}$ in Eq.~\eqref{eq:E_funcbasis} states that $\wf{}{}$ yields the oneelectron density $\n{}{}$.




Hence, the CBS excitation energy associated with the $k$th excited state reads


\begin{equation}


\begin{split}


\DE{k}{\CBS}


& = \E{k}{\CBS}  \E{0}{\CBS}


\\


& \approx \DE{k}{\Bas} + \DbE{}{\Bas}[\n{0}{\Bas},\n{k}{\Bas}],


\end{split}


\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 basisset 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 limit. \cite{LooPraSceTouGinJPCL19}


Note that in Eqs.~\eqref{eq:E0CBS} and \eqref{eq:EkCBS} we have assumed that the same density functional $\bE{}{\Bas}$ can be used for correcting all excitedstate energies, which seems a reasonable approximation since the electronelectron cusp effects are largely universal. \cite{KutTCA85, MorKutJPC93, KutMorZPD96, TewJCP08, LooGilMP10, LooBloGilJCP2015}




%%%%%%%%%%%%%%%%%%%%%%%%


\subsection{Rangeseparation function}


\label{sec:rs}


%%%%%%%%%%%%%%%%%%%%%%%%




As initially proposed in Ref.~\onlinecite{GinPraFerAssSavTouJCP18} and further developed in Ref.~\onlinecite{LooPraSceTouGinJPCL19}, we have shown that one can efficiently approximate $\bE{}{\Bas}[\n{}{}]$ by shortrange correlation functionals with multideterminantal (ECMD) reference borrowed from RSDFT. \cite{TouGorSavTCA05}


The ECMD functional, $\bE{\text{c,md}}{\sr}[\n{}{},\rsmu{}{}]$, is a function of the rangeseparation parameter $\mu$ and admits, for any $\n{}{}$, the following two limits


\begin{subequations}


\begin{align}


\label{eq:large_mu_ecmd}


\lim_{\mu \to \infty} \bE{\text{c,md}}{\sr}[\n{}{},\rsmu{}{}] & = 0,


\\


\label{eq:small_mu_ecmd}


\lim_{\mu \to 0} \bE{\text{c,md}}{\sr}[\n{}{},\rsmu{}{}] & = \Ec[\n{}{}],


\end{align}


\end{subequations}


which correspond to the WFT limit ($\mu \to \infty$) and the KohnSham DFT (KSDFT) limit ($\mu = 0$).


In Eq.~\eqref{eq:small_mu_ecmd}, $\Ec[\n{}{}]$ is the usual universal correlation density functional defined in KSDFT. \cite{HohKohPR64, KohShaPR65}




The key ingredient that allows us to exploit ECMD functionals for correcting the basisset incompleteness error is the rangeseparated function


\begin{equation}


\label{eq:def_mu}


\rsmu{}{\Bas}(\br{}) = \frac{\sqrt{\pi}}{2} \W{}{\Bas}(\br{},\br{}),


\end{equation}


which automatically adapts to the spatial nonhomogeneity of the basisset incompleteness error.


It is defined such that the longrange interaction of RSDFT, $\w{}{\lr,\mu}(r_{12}) = \erf( \mu r_{12})/r_{12}$, coincides, at coalescence, with an effective twoelectron 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{GinPraFerAssSavTouJCP18}


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 oppositespin pair density associated with $\wf{}{\Bas}$ and its corresponding tensor, respectively, $\SO{p}{}$ is a (realvalued) 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 twoelectron Coulomb integrals.


An important feature of $\W{}{\Bas}(\br{1},\br{2})$ is that it tends to the regular Coulomb operator $r_{12}^{1}$ as $\Bas \to \CBS${, which implies that


\begin{equation}


\lim_{\Bas \rightarrow \CBS} \rsmu{}{\Bas}(\br{}) = \infty,


\end{equation}


ensuring that $\bE{}{\Bas}[\n{}{}]$ vanishes when $\Bas$ is complete.


We refer the interested readers to Refs.~\onlinecite{GinPraFerAssSavTouJCP18,LooPraSceTouGinJPCL19} for additional details.




%%%%%%%%%%%%%%%%%%%%%%%%


\subsection{Shortrange correlation functionals}


\label{sec:func}


%%%%%%%%%%%%%%%%%%%%%%%%




The localdensity 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 shortrange correlation energy per electron of the uniform electron gas (UEG) \cite{LooGilWIRES16} parameterized in Ref.~\onlinecite{PazMorGorBacPRB06}.




The functional $\be{\text{c,md}}{\sr,\LDA}$ from Eq.~\eqref{eq:def_lda_tot} presents two main defects: i) at small $\mu$, it overestimates the correlation energy, and ii) UEGbased quantities are hardly transferable when the system becomes strongly correlated.


An attempt to solve these problems was suggested by some of the authors in the context of RSDFT. \cite{FerGinTouJCP19}


They proposed to interpolate between the usual PerdewBurkeErnzerhof ($\PBE$) correlation functional \cite{PerBurErnPRL96} $\e{\text{c}}{\PBE}(\n{}{},s,\zeta)$ (where $s=\nabla n/n^{4/3}$ is the reduced density gradient) at $\mu = 0$ and the exact large$\mu$ behavior. \cite{TouColSavPRA04, GorSavPRA06, PazMorGorBacPRB06}


In the context of RSDFT, the large$\mu$ behavior corresponds to an extremely shortrange interaction in the shortrange functional.


In this regime, the ECMD energy


\begin{align}


\label{eq:exact_large_mu}


\bE{\text{c,md}}{\sr} = \frac{2\sqrt{\pi} (1  \sqrt{2})}{3\mu^3} \int \dbr{} \n{2}{}(\br{}) + \order*{\mu^{4}}


\end{align}


only depends on the \textit{exact} ontop pair density $\n{2}{}(\br{}) \equiv \n{2}{}(\br{},\br{})$ which is obtained from the \textit{exact} groundstate wave function $\Psi$ belonging to the manyelectron Hilbert space in the CBS limit.




Obviously, an exact quantity such as $\n{2}{}(\br{})$ is out of reach in practical calculations and must be approximated by a function referred here as $\tn{2}{}(\br{})$.


For a given $\tn{2}{}(\br{})$, some of the authors proposed the following functional form in order to interpolate between $\e{\text{c}}{\PBE}(\n{}{},s,\zeta)$ at $\mu = 0$ and Eq.~\eqref{eq:exact_large_mu} as $\mu \to \infty$: \cite{FerGinTouJCP19}


\begin{subequations}


\begin{gather}


\label{eq:epsilon_cmdpbe}


\be{\text{c,md}}{\sr,\PBE}(\n{}{},\tn{2}{},s,\zeta,\rsmu{}{}) = \frac{\e{\text{c}}{\PBE}(\n{}{},s,\zeta)}{1 + \beta^\PBE(\n{}{},\tn{2}{},s,\zeta) \rsmu{}{3} },


\\


\label{eq:beta_cmdpbe}


\beta^\PBE(\n{}{},\tn{2}{},s,\zeta) = \frac{3}{2\sqrt{\pi} (1  \sqrt{2})} \frac{\e{\text{c}}{\PBE}(\n{}{},s,\zeta)}{\tn{2}{}/\n{}{}}.


\end{gather}


\end{subequations}


As illustrated in the context of RSDFT, \cite{FerGinTouJCP19} such a functional form is able to treat both weakly and strongly correlated systems thanks to the explicit inclusion of $\e{\text{c}}{\PBE}$ and $\tn{2}{}$, respectively.


Therefore, in the present context, we introduce the general form of the $\PBE$based complementary functional within a given basis set $\Bas$


\begin{multline}


\bE{\PBE}{\Bas}[\n{}{},\tn{2}{},\rsmu{}{\Bas}] =


\int \n{}{}(\br{})


\\


\times \be{\text{c,md}}{\sr,\PBE}\qty(\n{}{}(\br{}),\tn{2}{}(\br{}),s(\br{}),\zeta(\br{}),\rsmu{}{\Bas}(\br{})) \dbr{},


\end{multline}


which has an explicit dependency on both the rangeseparation function $\rsmu{}{\Bas}(\br{})$ (instead of the rangeseparation parameter in RSDFT) and the approximation level of $\tn{2}{}$.




In Ref.~\onlinecite{LooPraSceTouGinJPCL19}, some of the authors introduced a version of the $\PBE$based functional, herereferred as $\PBEUEG$


\begin{equation}


\bE{\PBEUEG}{\Bas}


\equiv


\bE{\PBE}{\Bas}[\n{}{},\n{2}{\UEG},\rsmu{}{\Bas}],


\end{equation}


in which the ontop pair density was approximated by its UEG version, i.e., $\tn{2}{}(\br{}) = \n{2}{\UEG}(\br{})$, with


\begin{equation}


\label{eq:n2UEG}


\n{2}{\UEG}(\br{}) \approx n(\br{})^2 [1\zeta(\br{})^2] g_0(n(\br{})),


\end{equation}


and where $g_0(n)$ is the UEG ontop pair distribution function [see Eq.~(46) of Ref.~\onlinecite{GorSavPRA06}].


Note that in Eq.~\eqref{eq:n2UEG} the dependence on the spin polarization $\zeta$ is only approximate.


As illustrated in Ref.~\onlinecite{LooPraSceTouGinJPCL19}, the $\PBEUEG$ functional has clearly shown, for weakly correlated systems, to improve energetics over the pure UEGbased functional $\bE{\LDA}{\Bas}$ [see Eq.~\eqref{eq:def_lda_tot}] thanks to the leverage brought by the $\PBE$ functional in the small$\mu$ regime.




However, the underlying UEG ontop pair density might not be suited for the treatment of excited states and/or strongly correlated systems.


Besides, in the context of the present basisset correction, $\n{2}{\Bas}(\br{})$, the ontop pair density in $\Bas$, must be computed anyway to obtain $\rsmu{}{\Bas}(\br{})$ [see Eqs.~\eqref{eq:def_mu} and \eqref{eq:def_weebasis}].


Therefore, as in Ref.~\onlinecite{FerGinTouJCP19}, we define a better approximation of the exact ontop pair density as


\begin{equation}


\label{eq:otextrap}


\ttn{2}{\Bas}(\br{}) = \n{2}{\Bas}(\br{}) \qty( 1 + \frac{2}{\sqrt{\pi}\rsmu{}{\Bas}(\br{})})^{1}


\end{equation}


which directly follows from the large$\mu$ extrapolation of the exact ontop pair density proposed by GoriGiorgi and Savin \cite{GorSavPRA06} in the context of RSDFT.


Using this new ingredient, we propose here the ``$\PBE$ontop'' (\PBEot) functional


\begin{equation}


\bE{\PBEot}{\Bas}


\equiv


\bE{\PBE}{\Bas}[\n{}{},\ttn{2}{\Bas},\rsmu{}{\Bas}].


\end{equation}


The sole distinction between $\PBEUEG$ and $\PBEot$ is the level of approximation of the exact ontop pair density.






%%%%%%%%%%%%%%%%%%%%%%%%


\section{Computational details}


\label{sec:compdetails}


%%%%%%%%%%%%%%%%%%%%%%%%


In the present study, we compute the ground and excitedstate energies, oneelectron densities and ontop pair densities with a selected configuration interaction (sCI) method known as CIPSI (Configuration Interaction using a Perturbative Selection made Iteratively). \cite{HurMalRanJCP73, GinSceCafCJC13, GinSceCafJCP15}


Both the implementation of the CIPSI algorithm and the computational protocol for excited states is reported in Ref.~\onlinecite{SceCafBenJacLooRC19}.


The total energy of each state is obtained via an efficient extrapolation procedure of the sCI energies designed to reach nearFCI accuracy. \cite{HolUmrShaJCP17, QP2}


These energies will be labeled exFCI in the following.


Using nearFCI 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 basisset incompleteness.


We refer the interested reader to Refs.~\onlinecite{HolUmrShaJCP17, SceGarCafLooJCTC18, LooSceBloGarCafJacJCTC18, SceBenJacCafLooJCP18, LooBogSceCafJacJCTC19, QP2} for more details.


The oneelectron densities and ontop pair densities are computed from a very large CIPSI expansion containing up to several million of Slater determinants.


All the RSDFT and exFCI calculations have been performed with {\QP}. \cite{QP2}


For the numerical quadratures, we employ the SG2 grid. \cite{DasHerJCC17}


Except for methylene for which FCI/TZVP geometries have been taken from Ref.~\onlinecite{SheLeiVanSchJCP98}, the other molecular geometries have been extracted from Refs.~\onlinecite{LooSceBloGarCafJacJCTC18, LooBogSceCafJacJCTC19} and have been obtained at the CC3/augccpVTZ level of theory.


For the sake of completeness, all these geometries are reported in the {\SI}.


Frozencore 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 frozencore densitybased correction is used consistently with the frozencore approximation in WFT methods.


We refer the reader to Ref.~\onlinecite{LooPraSceTouGinJPCL19} for an explicit derivation of the equations associated with the frozencore version of the present densitybased basisset correction.


Compared to the exFCI calculations performed to compute energies and densities, the basisset correction represents, in any case, a marginal computational cost.


In the following, we employ the AVXZ shorthand notations for Dunning's augccpVXZ basis sets.




%%%%%%%%%%%%%%%%%%%%%%%%


\section{Results and Discussion}


\label{sec:res}


%%%%%%%%%%%%%%%%%%%%%%%%




%=======================


\subsection{Methylene}


\label{sec:CH2}


%=======================




Methylene is a paradigmatic system in electronic structure theory. \cite{SchScience86}


Due to its relative small size, its ground and excited states have been thoroughly studied with highlevel ab initio methods. \cite{SchScience86, BauTayJCP86, JenBunJCP88, SheVanYamSchJMS97, SheLeiVanSchJCP98, AbrSheJCP04, AbrSheCPL05, ZimTouZhaMusUmrJCP09, GouPieWloMP10, ChiHolAdaOttUmrShaZimJPCA18}




As a first test of the present densitybased basisset correction, we consider the four lowestlying 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 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{HelJorOlsBOOK02}


\begin{equation}


\E{}{\text{AVXZ}}(\tX) = \E{}{\CBS} + \alpha \, \tX^{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{ChiHolAdaOttUmrShaZimJPCA18, SheLeiVanSchJCP98, JenBunJCP88, SheLeiVanSchJCP98, ZimTouZhaMusUmrJCP09}


Total energies for each state can be found in the {\SI}.


The exFCI/CBS values are still off by a few tenths of a {\kcal} compared to the DMC results of Zimmerman \textit{et al.} \cite{ZimTouZhaMusUmrJCP09} which are extremely close from the experimentallyderived adiabatic energies.


The reason of this discrepancy is probably due to the frozencore approximation which has been applied in our case and has shown to significantly affect adiabatic energies. \cite{LooGalJacJPCL18, LooJacJCTC19}


However, the exFCI/CBS energies are in perfect agreement with the semistochastic heatbath CI (SHCI) calculations from Ref.~\onlinecite{ChiHolAdaOttUmrShaZimJPCA18}, as expected.




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 0.15 eV.


From the triple$\zeta$ basis onward, the exFCI excitation energies are chemically accurate though (i.e. error below 1 {\kcal} or 0.043 eV), and converge steadily to the CBS limit when one increases the size of the basis set.


Concerning the basisset correction, already at the double$\zeta$ level, the $\PBEot$ correction returns chemically accurate excitation energies.


The performance of the $\PBEUEG$ and $\LDA$ functionals is less impressive.


Yet, they still yield significant reductions of the basisset incompleteness error, hence representing a good compromise between computational cost and accuracy.


Note that the results for the $\PBEUEG$ 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 basisset correction has the tendency of overcorrecting the excitation energies via an overstabilization 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 (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 {\SI} for total energies.}


\label{tab:CH2}


\begin{ruledtabular}


\begin{tabular}{lllll}


& & \mc{3}{c}{Transitions} \\


\cline{35}


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


& $0.441$ [$+0.057$]


& $1.536$ [$+0.152$]


& $2.659$ [$+0.162$] \\


& AVTZ


& $0.408$ [$+0.024$]


& $1.423$ [$+0.040$]


& $2.546$ [$+0.049$] \\


& AVQZ


& $0.395$ [$+0.011$]


& $1.399$ [$+0.016$]


& $2.516$ [$+0.020$] \\


& AV5Z


& $0.390$ [$+0.006$]


& $1.392$ [$+0.008$]


& $2.507$ [$+0.010$] \\


& CBS


& $0.384$


& $1.384$


& $2.497$ \\


\\


exFCI+$\PBEot$ & AVDZ


& $0.347$ [$0.037$]


& $1.401$ [$+0.017$]


& $2.511$ [$+0.014$] \\


& AVTZ


& $0.374$ [$0.010$]


& $1.378$ [$0.006$]


& $2.491$ [$0.006$] \\


& AVQZ


& $0.379$ [$0.005$]


& $1.378$ [$0.006$]


& $2.489$ [$0.008$] \\


\\


exFCI+$\PBEUEG$ & AVDZ


& $0.308$ [$0.076$]


& $1.388$ [$+0.004$]


& $2.560$ [$+0.064$] \\


& AVTZ


& $0.356$ [$0.028$]


& $1.371$ [$0.013$]


& $2.510$ [$+0.013$] \\


& AVQZ


& $0.371$ [$0.013$]


& $1.375$ [$0.009$]


& $2.498$ [$+0.002$] \\


\\


exFCI+$\LDA$ & AVDZ


& $0.337$ [$0.047$]


& $1.420$ [$+0.036$]


& $2.586$ [$+0.089$] \\


& AVTZ


& $0.359$ [$0.025$]


& $1.374$ [$0.010$]


& $2.514$ [$+0.017$] \\


& AVQZ


& $0.370$ [$0.014$]


& $1.375$ [$0.009$]


& $2.499$ [$0.002$] \\


\\


SHCI\fnm[1] & AVQZ


& $0.393$


& $1.398$


& $2.516$ \\


CREOMCC (2,3)D\fnm[2]& AV5Z


& $0.430$


& $1.464$


& $2.633$ \\


FCI\fnm[3] & TZ2P


& $0.483$


& $1.542$


& $2.674$ \\


DMC\fnm[4] &


& $0.406$


& $1.416$


& $2.524$ \\


Exp.\fnm[5] &


& $0.406$


& $1.415$


\end{tabular}


\end{ruledtabular}


\fnt[1]{Semistochastic heatbath CI (SHCI) calculations from Ref.~\onlinecite{ChiHolAdaOttUmrShaZimJPCA18}.}


\fnt[2]{Completelyrenormalized equationofmotion coupled cluster (CREOMCC) calculations from Refs.~\onlinecite{GouPieWloMP10}.}


\fnt[3]{Reference \onlinecite{SheLeiVanSchJCP98}.}


\fnt[4]{Diffusion Monte Carlo (DMC) calculations from Ref.~\onlinecite{ZimTouZhaMusUmrJCP09} obtained with a CAS(6,6) trial wave function.}


\fnt[5]{Experimentallyderived values. See footnotes of Table II from Ref.~\onlinecite{GouPieWloMP10} for additional details.}


\end{table}


\end{squeezetable}


%%% %%% %%%




%%% FIG 1 %%%


\begin{figure}


\includegraphics[width=\linewidth]{fig1}


\caption{Error in adiabatic excitation energies (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:H2ONH3}


%=======================




For the second test, we consider the water \cite{CaiTozReiJCP00, RubSerMerJCP08, LiPalJCP11, LooSceBloGarCafJacJCTC18, SceBenJacCafLooJCP18, SceCafBenJacLooRC19} and ammonia \cite{SchGoeJCTC17, BarDelPerMatJMS97, LooSceBloGarCafJacJCTC18} molecules.


They are both well studied and possess Rydberg excited states which are highly sensitive to the radial completeness of the oneelectron basis set, as evidenced in Ref.~\onlinecite{LooSceBloGarCafJacJCTC18}.


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 basisset corrected theoretical best estimates (TBEs) have been extracted from Ref.~\onlinecite{LooSceBloGarCafJacJCTC18} 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 basisset 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 other words, substantial error remains in these cases even with the largest AVQZ basis set.


In these cases, one really needs doubly augmented basis sets to reach radial completeness.


The first observation worth reporting is that all three RSDFT 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 cannot catch the radial incompleteness of the oneelectron basis set, a feature which is far from being a cusprelated effect.


\alert{In other words, the DFTbased correction recovers dynamic correlation effects only and one must ensure that the basis set includes enough diffuse functions in order to describe Rydberg states.}




%%% TABLE 2 %%%


\begin{squeezetable}


\begin{table*}


\caption{


Vertical excitation energies (in eV) of excited states of water, ammonia, carbon dimer and ethylene for various methods and basis sets.


The TBEs have been extracted from Refs.~\onlinecite{LooSceBloGarCafJacJCTC18, LooBogSceCafJacJCTC19} on the same geometries.


See the {\SI} for total energies.}


\label{tab:Mol}


\begin{ruledtabular}


\begin{tabular}{lllddddddddddddd}


& & & & \mc{12}{c}{Deviation with respect to TBE}


\\


\cline{516}


& & & & \mc{3}{c}{exFCI}


& \mc{3}{c}{exFCI+$\PBEot$}


& \mc{3}{c}{exFCI+$\PBEUEG$}


& \mc{3}{c}{exFCI+$\LDA$}


\\


\cline{57} \cline{810} \cline{1113} \cline{1416}


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


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


\\


\\


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. & 2.04\fnm[3] & 0.17 & 0.05 & 0.02


& 0.04 & 0.00 & 0.00


& 0.15 & 0.04 & 0.02


& 0.17 & 0.05 & 0.02


\\


& $1\,^{1}\Sigma_g^+ \ra 2\,^{1}\Sigma_g^+$ & Val. & 2.38\fnm[3] & 0.12 & 0.04 & 0.02


& 0.00 & 0.00 & 0.00


& 0.11 & 0.03 & 0.02


& 0.13 & 0.04 & 0.02


\\


\\


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 &


& 0.05 & 0.03 &


& 0.08 & 0.04 &


& 0.07 & 0.04 &


\\


& $1\,^{1}A_{1g} \ra 1\,^{3}B_{3u}$ & Val. & 7.28\fnm[4] & 0.12 & 0.04 &


& 0.04 & 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 &


& 0.00 & 0.03 &


& 0.04 & 0.03 &


& 0.05 & 0.04 &


\end{tabular}


\end{ruledtabular}


\fnt[1]{exFCI/AVQZ data corrected with the difference between CC3/dAV5Z and exFCI/AVQZ values. \cite{LooSceBloGarCafJacJCTC18}


dAV5Z is the doubly augmented V5Z basis set.}


\fnt[2]{exFCI/AVTZ data corrected with the difference between CC3/dAV5Z and exFCI/AVTZ values. \cite{LooSceBloGarCafJacJCTC18}}


\fnt[3]{exFCI/CBS obtained from the exFCI/AVTZ and exFCI/AVQZ data of Ref.~\onlinecite{LooBogSceCafJacJCTC19}.}


\fnt[4]{exFCI/AVDZ data corrected with the difference between CC3/dAV5Z and exFCI/AVDZ values. \cite{LooSceBloGarCafJacJCTC18}}


\end{table*}


\end{squeezetable}


%%% %%% %%%




%%% FIG 2 %%%


\begin{figure}


\includegraphics[width=\linewidth]{fig2}


\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 Table \ref{tab:Mol} for raw data.}


\label{fig:H2O}


\end{figure}


%%% %%% %%%




%%% FIG 3 %%%


\begin{figure}


\includegraphics[width=\linewidth]{fig3}


\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 Table \ref{tab:Mol} for raw data.}


\label{fig:NH3}


\end{figure}


%%% %%% %%%




%=======================


\subsection{DoublyExcited 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 doublyexcited states of the carbon dimer \ce{C2}, a prototype system for strongly correlated and multireference systems. \cite{AbrSheJCP04, AbrSheCPL05, VarJCP08, PurZhaKraJCP09, AngCimPasMP12, BooCleThoAlaJCP11, ShaJCP15, SokChaJCP16, HolUmrShaJCP17, VarRocPTRSMPES18}


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 stateoftheart methods, and have been shown to be ``pure'' doublyexcited states as they involve an insignificant amount of single excitations. \cite{LooBogSceCafJacJCTC19}


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 to consider the $\PBEot$ functional to get chemically accurate excitation 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 ontop pair density used in the $\LDA$ and $\PBEUEG$ functionals (see Sec.~\ref{sec:func}) is a particularly bad approximation of the true ontop pair density for the present system.




%%% FIG 5 %%%


\begin{figure}


\includegraphics[width=\linewidth]{fig4}


\caption{Error in vertical excitation energies (in eV) for two doublyexcited 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 Table \ref{tab:Mol} for raw data.}


\label{fig:C2}


\end{figure}


%%% %%% %%%




It is interesting to study the behavior of the key quantities involved in the basisset correction for different states as the basisset incompleteness error is obviously state specific.


In Fig.~\ref{fig:C2_mu}, we report $\rsmu{}{\Bas}(z)$, $\n{}{\Bas}(z) \be{\text{c,md}}{\sr,\PBEot}(z)$, and $\n{2}{\Bas}(z)$


along the nuclear axis ($z$) for the two $^1 \Sigma_g^+$ electronic states of \ce{C2} computed with the AVDZ, AVTZ, and AVQZ basis sets.


The graphs gathered in Fig.~\ref{fig:C2_mu} illustrate several general features regarding the present basisset correction:


\begin{itemize}


\item the maximal values of $\rsmu{}{\Bas}(z)$ are systematically close to the nuclei, a signature of the atomcentered basis set;


\item the overall magnitude of $\rsmu{}{\Bas}(z)$ increases with the basis set, which reflects the improvement of the description of the correlation effects when enlarging the basis set;


\item the absolute value of the energetic correction decreases when the size of the basis set increases;


\item there is a clear correspondence between the values of the energetic correction and the ontop pair density.


\end{itemize}


Regarding now the differential effect of the basisset correction in the special case of the two $^1 \Sigma_g^+$ states studied here, we observe that:


\begin{itemize}


\item $\rsmu{}{\Bas}(z)$ has the same overall behavior for the two states, with slightly more fine structure in the case of the ground state.


Such feature is consistent with the fact that the two states considered are both of $\Sigma_g^+$ symmetry and of valence character.


\item $\n{2}{}(z)$ is overall larger in the excited state, specially in the bonding and outer regions.


This is can be explained by the nature of the electronic transition which qualitatively corresponds to a double excitation from $\pi$ to $\sigma$ orbitals, therefore increasing the overall electronic population on the bond axis.


\item The energetic correction clearly stabilizes preferentially the excited state rather than the ground state, illustrating that shortrange correlation effects are more pronounced in the former than in the latter.


This is linked to the larger values of the excitedstate ontop pair density.


\end{itemize}




%%% FIG 4 %%%


\begin{figure*}


\includegraphics[height=0.35\linewidth]{fig5a}


\includegraphics[height=0.35\linewidth]{fig5b}


\includegraphics[height=0.35\linewidth]{fig5c}


\caption{$\rsmu{}{\Bas}$ (left), $\n{}{\Bas} \be{\text{c,md}}{\sr,\PBEot}$ (center) and $\n{2}{\Bas}$ (right) along the molecular axis ($z$) for the ground state (black curve) and second doublyexcited state (red curve) of \ce{C2} for various basis sets $\Bas$.


The two electronic states are both of $\Sigma_g^+$ symmetry.


The carbon nuclei are located at $z= \pm 1.180$ bohr and are represented by the thin black lines.}


\label{fig:C2_mu}


\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 highlevel ab initio methods. \cite{SerMarNebLinRooJCP93, WatGwaBarJCP96, WibOliTruJPCA02, BarPaiLisJCP04, AngJCC08, SchSilSauThiJCP08, SilSchSauThiJCP10, SilSauSchThiMP10, AngIJQC10, DadSmaBooAlaFilJCTC12, FelPetDavJCP14, ChiHolAdaOttUmrShaZimJPCA18}


We refer the interested reader to the work of Feller \textit{et al.}\cite{FelPetDavJCP14} for an exhaustive investigation dedicated to the excited states of ethylene using stateoftheart CI calculations.


In the present context, ethylene is a particularly interesting system as it contains a mixture of valence and Rydberg excited states.


Our basisset corrected vertical excitation energies are gathered in Table \ref{tab:Mol} and depicted in Fig.~\ref{fig:C2H4}.


Note that exFCI/AVQZ calculations are inaccessible for ethylene.


The exFCI+$\PBEot$/AVDZ excitation energies are at near chemical accuracy and the errors drop further when one goes to the triple$\zeta$ basis.


Consistently with the previous examples, the $\LDA$ and $\PBEUEG$ functionals are slightly less accurate, although they still correct the excitation energies in the right direction.




%%% FIG 6 %%%


\begin{figure}


\includegraphics[width=\linewidth]{fig6}


\caption{Error in vertical excitation energies (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 Table \ref{tab:Mol} for raw data.}


\label{fig:C2H4}


\end{figure}


%%% %%% %%%




%%%%%%%%%%%%%%%%%%%%%%%%


\section{Conclusion}


\label{sec:ccl}


%%%%%%%%%%%%%%%%%%%%%%%%


We have shown that, by employing the recently proposed densitybased basisset correction developed by some of the authors, \cite{GinPraFerAssSavTouJCP18} one can obtain, using sCI methods, chemically accurate excitation energies with typically augmented double$\zeta$ basis sets.4


This nicely complements our recent investigation on groundstate properties, \cite{LooPraSceTouGinJPCL19} 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 shortrange correlation functionals from RSDFT might not be enough to catch the radial incompleteness of the oneelectron basis set.


Also, in the case of multireference systems, we have evidenced that the $\PBEot$ functional, which uses an accurate ontop pair density, is more appropriate than the $\LDA$ and $\PBEUEG$ functionals relying on the UEG ontop pair density.


We are currently investigating the performance of the present basisset correction for strongly correlated systems and we hope to report on this in the near future.




%%%%%%%%%%%%%%%%%%%%%%%%


\section*{Supplementary material}


%%%%%%%%%%%%%%%%%%%%%%%%


See {\SI} for geometries and additional information (including total energies and energetic correction of the various functionals).




%%%%%%%%%%%%%%%%%%%%%%%%


\begin{acknowledgements}


PFL would like to thank Denis Jacquemin for numerous discussions on excited states.


This work was performed using HPC resources from GENCITGCC (Grant No.~2018A0040801738), CALMIP (Toulouse) under allocation 201918005 and the JarvisAlpha cluster from the \textit{Institut Parisien de Chimie Physique et Th\'eorique}.


\end{acknowledgements}


%%%%%%%%%%%%%%%%%%%%%%%%




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\affiliation{\LCPQ}




\begin{abstract}


\begin{wrapfigure}[13]{o}[1.25cm]{0.5\linewidth}


\centering


\includegraphics[width=\linewidth]{TOC}


\end{wrapfigure}


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 shortrange densityfunctional correction for basisset incompleteness [\href{https://doi.org/10.1063/1.5052714}{Giner \textit{et al.}, \textit{J.~Chem.~Phys.}~ \textbf{2018}, \textit{149}, 194301}], 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 dimer and ethylene).


The present study clearly evidences that special care has to be taken with very diffuse excited states where the present correction does not catch the radial incompleteness of the oneelectron basis set.



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\begin{document}




\title{Supporting Information for ``Chemically Accurate Excitation Energies With Small Basis Sets''}


\title{Supplementary material for``Chemically accurate excitation energies with small basis sets''}




\author{Emmanuel Giner}


\email[Corresponding author: ]{emmanuel.giner@lct.jussieu.fr}



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\begin{document}




\begin{letter}%


{To the Editors of the Journal of Chemical Physics}




\opening{Dear Editors,}




\justifying


Please find attached a revised version of the manuscript entitled


\begin{quote}


\textit{``Chemically accurate excitation energies with small basis sets''},


\end{quote}


We would to thank the reviewer for his/her constructive comments.


Our detailed responses to their comments can be found below.


For convenience, changes are highlighted in red in the revised version of the manuscript.


We hope that you will agree that our manuscript is now suitable for publication in JCP.




We look forward to hearing from you.




\closing{Sincerely, the authors.}




%%% REVIEWER 1 %%%


\noindent \textbf{\large Authors' answer to Reviewer \#1}




\begin{itemize}


\item


{The present paper deals with a recently proposed aposteriori densitybased correction for basisset incompleteness.


This correction is derived by projecting the 1/r12 operator onto a finite basis set and on mimicking it pointwise by damped erf(mu*r12)/r12 potentials.


The effects of the missing shortrange potentials are recovered by rangeseparated correlationenergy density functionals of the LDA or PBE type, modified by asymptotic expressions for large mu.


In the present paper, the asymptotic expressions are slightly modified, with respect to previous work, and the method is applied to excitation energies of small molecules (CH2 etc.).


The study is done in a systematic and accurate way, coupling the basisset corrections to near fullCI calculations for basis sets of increasing quality.


The results are interesting, and the discussion is clear and physically well motivated.


On the whole, I feel that the paper is well suited for readers of JCP. }


\\


\alert{We thank the reviewer for his/her support.}




\item


{The method may be considered as a simple substitute for F12 methods. The latter are


briefly mentioned, in the Introduction, hinting at difficulties of F12 for excited states.


It is somewhat unfortunate that no direct comparison to these methods has been attempted


in the present paper. To my knowledge, there is a recent paper (JCP 150, 184110 (2019)),


not cited by the present authors, which shows that excitation energies for CH2 (and other


molecules) can be reliably determined in an F12 context. }


\\


\alert{We have cited the recent paper mentioned by the reviewer in the Introduction of our manuscript and added the following sentence:


\textit{``However, very encouraging results have been reported recently using the extended explicitlycorrelated secondorder approximate coupledcluster singles and doubles ansatz suitable for response theory on systems such as methylene, formaldehyde and imidazole.''}


Unfortunately, the excitation energies provided in the paper mentioned above (obtained at the CC2 level) cannot be directly compared to our results which are obtained at the FCI level.


We agree with the reviewer that it is unfortunate that no direct comparison to these methods has been yet attempted.


However, we would like to mention that this is part of our future plan.}




\item


{The correction seems to be inappropriate for certain Rydberg excitations (see Fig. 3).


Is there a possibility to predict such cases? Is the correction meant to recover genuine


(dynamic) correlation effects only? If yes, then the basis should include enough diffuse


functions to at least describe the excited Rydberg states at the HF or MCSCF levels. }


\\


\alert{Yes, we also agree with the referee that the present correction is inappropriate for certain Rydberg excitations.


Note that we have explicitly mentioned this several times in our manuscript where we mention that the present correction cannot catch the radial incompleteness of the oneelectron basis set, a feature which is far from being a cusprelated effect.


We have added a sentence to clarify this point: \textit{``In other words, the DFTbased correction recovers dynamic correlation effects only and one must ensure that the basis set includes enough diffuse functions in order to describe Rydberg states.''}}


\end{itemize}




\end{letter}


\end{document}













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