By combining extrapolated selected configuration interaction (sCI) calculations performed with the CIPSI algorithm with the recently proposed short-range density-functional functional correction for basis set incompleteness [\href{https://doi.org/10.1063/1.5052714}{Giner et al., \textit{J.~Chem.~Phys.}~\textbf{149}, 194301 (2018)}], we show that one can obtain vertical and adiabatic excitation energies with chemical accuracy with a small basis set.
One of the most fundamental problem of conventional electronic structure methods is their slow energy convergence with respect to the size of the one-electron basis set.
Explicitly-correlated F12 methods have been specifically designed to cure this problem.
Although they have extremely successful to speed up convergence of the ground state properties such as correlation and atomization energies (for example), their performances for excited states have been much more conflicting.
There are two types of basis set completeness: angular and radial completeness.
F12 is good at doing angular basis set correction.
However, radial correction are much harder to design and it is a real test for the present approach.
Instead of F12 methods, here we propose to follow a different philosophy and rely on the recently proposed short-range density-functional functional correction to reduce the basis set incompleteness error. \cite{GinPraFerAssSavTou-JCP-18}
This choice is motivated by the much faster convergence of these methods with respect to the size of the basis set. \cite{FraMusLupTou-JCP-15}
The present method is illustrated on several molecules and singly- and doubly-excited states with diffuse basis sets.
%Contemporary quantum chemistry has developed in two directions --- wave function theory (WFT) \cite{Pop-RMP-99} and density-functional theory (DFT). \cite{Koh-RMP-99}
%Although both spring from the same Schr\"odinger equation, each of these philosophies has its own \textit{pros} and \textit{cons}.
%
%WFT is attractive as it exists a well-defined path for systematic improvement as well as powerful tools, such as perturbation theory, to guide the development of new WFT \textit{ans\"atze}.
%The coupled cluster (CC) family of methods is a typical example of the WFT philosophy and is well regarded as the gold standard of quantum chemistry for weakly correlated systems.
%By increasing the excitation degree of the CC expansion, one can systematically converge, for a given basis set, to the exact, full configuration interaction (FCI) limit, although the computational cost associated with such improvement is usually high.
%One of the most fundamental drawbacks of conventional WFT methods is the slow convergence of energies and properties with respect to the size of the one-electron basis set.
%This undesirable feature was put into light by Kutzelnigg more than thirty years ago. \cite{Kut-TCA-85}
%To palliate this, following Hylleraas' footsteps, \cite{Hyl-ZP-29} Kutzelnigg proposed to introduce explicitly the interelectronic distance $r_{12} = \abs{\br{1} - \br{2}}$ to properly describe the electronic wave function around the coalescence of two electrons. \cite{Kut-TCA-85, KutKlo-JCP-91, NogKut-JCP-94}
%The resulting F12 methods yield a prominent improvement of the energy convergence, and achieve chemical accuracy for small organic molecules with relatively small Gaussian basis sets. \cite{Ten-TCA-12, TenNog-WIREs-12, HatKloKohTew-CR-12, KonBisVal-CR-12, GruHirOhnTen-JCP-17, MaWer-WIREs-18}
%For example, at the CCSD(T) level, one can obtain quintuple-$\zeta$ quality correlation energies with a triple-$\zeta$ basis, \cite{TewKloNeiHat-PCCP-07} although computational overheads are introduced by the large auxiliary basis used to resolve three- and four-electron integrals. \cite{BarLoo-JCP-17}
%To reduce further the computational cost and/or ease the transferability of the F12 correction, approximated and/or universal schemes have recently emerged. \cite{TorVal-JCP-09, KonVal-JCP-10, KonVal-JCP-11, BooCleAlaTew-JCP-2012, IrmHumGru-arXiv-2019, IrmGru-arXiv-2019}
%
%Present-day DFT calculations are almost exclusively done within the so-called Kohn-Sham (KS) formalism, which corresponds to an exact dressed one-electron theory. \cite{KohSha-PR-65}
%The attractiveness of DFT originates from its very favorable accuracy/cost ratio as it often provides reasonably accurate energies and properties at a relatively low computational cost.
%Thanks to this, KS-DFT \cite{HohKoh-PR-64, KohSha-PR-65} has become the workhorse of electronic structure calculations for atoms, molecules and solids. \cite{ParYan-BOOK-89}
%Although there is no clear way on how to systematically improve density-functional approximations, \cite{Bec-JCP-14} climbing Perdew's ladder of DFT is potentially the most satisfactory way forward. \cite{PerSch-AIPCP-01, PerRuzTaoStaScuCso-JCP-05}
%In the context of the present work, one of the interesting feature of density-based methods is their much faster convergence with respect to the size of the basis set. \cite{FraMusLupTou-JCP-15}
%
%Progress toward unifying WFT and DFT are on-going.
%In particular, range-separated DFT (RS-DFT) (see Ref.~\citenum{TouColSav-PRA-04} and references therein) rigorously combines these two approaches via a decomposition of the electron-electron (e-e) interaction into a non-divergent long-range part and a (complementary) short-range part treated with WFT and DFT, respectively.
%As the WFT method is relieved from describing the short-range part of the correlation hole around the e-e coalescence points, the convergence with respect to the one-electron basis set is greatly improved. \cite{FraMusLupTou-JCP-15}
%Therefore, a number of approximate RS-DFT schemes have been developed within single-reference \cite{AngGerSavTou-PRA-05, GolWerSto-PCCP-05, TouGerJanSavAng-PRL-09,JanHenScu-JCP-09, TouZhuSavJanAng-JCP-11, MusReiAngTou-JCP-15} or multi-reference \cite{LeiStoWerSav-CPL-97, FroTouJen-JCP-07, FroCimJen-PRA-10, HedKneKieJenRei-JCP-15, HedTouJen-JCP-18, FerGinTou-JCP-18} WFT approaches.
%Very recently, a major step forward has been taken by some of the present authors thanks to the development of a density-based basis-set correction for WFT methods. \cite{GinPraFerAssSavTou-JCP-18}
%The present work proposes an extension of this new methodological development alongside the first numerical tests on molecular systems.
The present basis-set correction relies on the RS-DFT formalism to capture the missing part of the short-range correlation effects, a consequence of the incompleteness of the one-electron basis set.
The present methodology is identical to the one described in Ref.~\onlinecite{LooPraSceTouGin-JPCL-19} where the main working equation are reported and discussed.
We refer the interested reader to Ref.~\onlinecite{GinPraFerAssSavTou-JCP-18} for a more formal derivation.
exFCI stands for extrapolated FCI energies computed with the CIPSI algorithm. \cite{HurMalRan-JCP-73, GinSceCaf-CJC-13, GinSceCaf-JCP-15}
We refer the interested reader to Refs.~\citenum{HolUmrSha-JCP-17, SceGarCafLoo-JCTC-18, LooSceBloGarCafJac-JCTC-18, SceBenJacCafLoo-JCP-18, LooBogSceCafJAc-JCTC-19} for more details.
The one-electron density and on-top density is computed from a very large CIPSI expansion containing several million determinants.
All the RS-DFT and exFCI calculations have been performed with {\QP}. \cite{QP2}
For the numerical quadratures, we employ the SG-2 grid. \cite{DasHer-JCC-17}
The geometries have been extracted from Refs.~\citenum{LooSceBloGarCafJac-JCTC-18, LooBogSceCafJAc-JCTC-19} and have been obtained at the CC3/aug-cc-pVTZ level of theory.
They are also reported in the {\SI}.
Frozen-core calculations are systematically performed and defined as such: a \ce{He} core is frozen from \ce{Li} to \ce{Ne}, while a \ce{Ne} core is frozen from \ce{Na} to \ce{Ar}.
The FC density-based correction is used consistently with the FC approximation in WFT methods.