By combining extrapolated selected configuration interaction (sCI) energies obtained with the CIPSI (Configuration Interaction using a Perturbative Selection made Iteratively) algorithm with the recently proposed short-range density-functional correction for basis set incompleteness [\href{https://doi.org/10.1063/1.5052714}{Giner et al., \textit{J.~Chem.~Phys.}~\textbf{149}, 194301 (2018)}], we show that one can obtain vertical and adiabatic excitation energies with chemical accuracy with, typically, augmented double-$\zeta$ basis sets.
We illustrate the present approach on various types of excited states (valence, Rydberg and double excitations) in several small organic molecules (methylene, water, ammonia, carbon dimer and ethylene).
The present study clearly evidences that special care has to be taken for very diffuse excited states where the present correction might not be enough to catch the radial incompleteness of the one-electron basis set.
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).
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
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.
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 $\n{2,k}{\Bas}(\br{},\br{})$,
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 $\n{}{}$.
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
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
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),
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},
$\be{\text{c,md}}{\sr,\PBE}\qty(\n{}{},s,\zeta,\rsmu{}{})$ 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}
More recently, \cite{LooPraSceTouGin-JPCL-19} we have also proposed a simplified version of the PBEot functional where we replaced the on-top pair density by its UEG version, i.e.~$\n{2}{\Bas}(\br{},\br{})\approx\n{2}{\UEG}(\n{}{}(\br{}),\zeta(\br{}))$, where $\n{2}{\UEG}(\n{}{},\zeta)\approx\n{}{2}(1-\zeta^2) g_0(n)$ with the parametrization of the UEG on-top pair-distribution function $g_0(n)$ given in Eq.~(46) of Ref.~\citenum{GorSav-PRA-06}.
This computationally-lighter functional will be refered to as PBE.
In the present study, we compute the ground- and excited-state energies, one-electron and on-top densities with a selected CI method known as CIPSI (Configuration Interaction using a Perturbative Selection made Iteratively). \cite{HurMalRan-JCP-73, GinSceCaf-CJC-13, GinSceCaf-JCP-15}
The total energy of each state is obtained via an efficient extrapolation procedure of the sCI energies designed to reach near-FCI accuracy. \cite{QP2}
These energies will be labeled exFCI in the following.
We refer the interested reader to Refs.~\onlinecite{HolUmrSha-JCP-17, SceGarCafLoo-JCTC-18, LooSceBloGarCafJac-JCTC-18, SceBenJacCafLoo-JCP-18, LooBogSceCafJAc-JCTC-19, QP2} for more details.
The one-electron and on-top densities are computed from a very large CIPSI expansion containing several million determinants.
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.
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 $2\,^{1}A_1$) and compute the corresponding adiabatic transition energies for various basis sets ranging from AVDZ to AVQZ.
We have also computed these adiabatic energies at the exFCI/AV5Z level and used these total energies alongside the quadruple-$\zeta$ ones to extrapolate the excitation energies to the CBS limit with the usual extrapolation formula \cite{HelJorOls-BOOK-02}
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 approaches. \cite{ChiHolAdaOttUmrShaZim-JPCA-18, SheLeiVanSch-JCP-98, JenBun-JCP-88, SheLeiVanSch-JCP-98, ZimTouZhaMusUmr-JCP-09}
Water \cite{CaiTozRei-JCP-00, RubSerMer-JCP-08, LiPal-JCP-11, LooSceBloGarCafJac-JCTC-18, SceBenJacCafLoo-JCP-18} and ammonia \cite{SchGoe-JCTC-17, BarDelPerMat-JMS-97, LooSceBloGarCafJac-JCTC-18} are two interesting molecules with Rydberg excited states which are highly sensitive to the radial completeness of the one-electron basis set.
It is also interesting to study doubly-excited states. \cite{AbrShe-JCP-04, AbrShe-CPL-05, Var-JCP-08, PurZhaKra-JCP-09, AngCimPas-MP-12, BooCleThoAla-JCP-11, Sha-JCP-15, SokCha-JCP-16, VarRoc-PTRSMPES-18}
In the carbon dimer, these valence states are of $(\pi,\pi)\ra(\si,\si)$ character and they have been recently studied with state-of-the-art methods. \cite{LooBogSceCafJAc-JCTC-19}
Ethylene is an interesting molecules as it contains both valence and Rydberg excited states. \cite{SerMarNebLinRoo-JCP-93, WatGwaBar-JCP-96, WibOliTru-JPCA-02, BarPaiLis-JCP-04, Ang-JCC-08, SchSilSauThi-JCP-08, SilSchSauThi-JCP-10, SilSauSchThi-MP-10, Ang-IJQC-10, DadSmaBooAlaFil-JCTC-12, FelPetDav-JCP-14, ChiHolAdaOttUmrShaZim-JPCA-18}
We have shown that by employing the recently proposed density-based basis set correction developed by some of the authors, one can obtain chemically-accurate excitation energies with typically augmented double-$\zeta$ basis sets.
We are currently investigating the performance of the present basis set correction for strongly correlated systems and we hope to report on this in the near future.