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 \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 one-electron basis set.
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 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 performance 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} has been much more conflicting. \cite{FliHatKlo-JCP-06, NeiHatKlo-JCP-06}
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{Sav-INC-96, LeiStoWerSav-CPL-97, TouColSav-PRA-04, TouSavFla-IJQC-04, AngGerSavTou-PRA-05, GolWerSto-PCCP-05, PazMorGorBac-PRB-06, FroTouJen-JCP-07, TouGerJanSavAng-PRL-09, JanHenScu-JCP-09, FroCimJen-PRA-10, TouZhuSavJanAng-JCP-11, MusReiAngTou-JCP-15, HedKneKieJenRei-JCP-15, HedTouJen-JCP-18, FerGinTou-JCP-19} (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 (the so-called electron-electron cusp). \cite{Kat-CPAM-57}
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, $\n{k}{\Bas}(\br{})$ and $\n{0}{\Bas}(\br{})$, as well as their opposite-spin on-top pair densities, $\n{2,0}{\Bas}(\br{})$ and $\n{2,k}{\Bas}(\br{})$,
Therefore, the complete-basis-set (CBS) energy of the ground and excited states may be approximated as \cite{GinPraFerAssSavTou-JCP-18}
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 spaces spanned by $\Bas$ and the complete basis, respectively.
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 excited-state energies, which seems a reasonable approximation since the electron-electron cusp effects are largely universal. \cite{Kut-TCA-85, MoKut-JPC-93, KutMor-ZPD-96, Tew-JCP-08, LooGil-MP-10, LooGil-JCP-2015}
As initially proposed in Ref.~\onlinecite{GinPraFerAssSavTou-JCP-18} and further developed 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 borrowed from RS-DFT. \cite{TouGorSav-TCA-05}
The ECMD functional, $\bE{\text{c,md}}{\sr}[\n{}{},\rsmu{}{}]$, is a function of the range-separation parameter $\mu$ and admits, for any $\n{}{}$, the following two limits
which correspond to the WFT limit ($\mu\to\infty$) and the Kohn-Sham DFT (KS-DFT) limit ($\mu=0$).
In Eq.~\eqref{eq:small_mu_ecmd}, $\Ec[\n{}{}]$ is the usual universal correlation density functional defined in KS-DFT. \cite{HohKoh-PR-64, KohSha-PR-65}
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}
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),
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
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}.
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) UEG-based quantities are hardly transferable when the system becomes strongly correlated.
They proposed to interpolate between the usual Perdew-Burke-Ernzerhof ($\PBE$) correlation functional \cite{PerBurErn-PRL-96}$\e{\text{c}}{\PBE}(\n{}{},s,\zeta)$ (where $s=\nabla n/n^{4/3}$ is the reduced density gradient) at $\mu=0$ and the exact large-$\mu$ behavior. \cite{TouColSav-PRA-04, GorSav-PRA-06, PazMorGorBac-PRB-06}
In the context of RS-DFT, the large-$\mu$ behavior corresponds to an extremely short-range interaction in the short-range functional.
only depends on the \textit{exact} on-top pair density $\n{2}{}(\br{})\equiv\n{2}{}(\br{},\br{})$ which is obtained from the \textit{exact} ground-state wave function $\Psi$ belonging to the many-electron 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{FerGinTou-JCP-19}
As illustrated in the context of RS-DFT, \cite{FerGinTou-JCP-19} 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.
which has an explicit dependency on both the range-separation function $\rsmu{}{\Bas}(\br{})$ (instead of the range-separation parameter in RS-DFT) and the approximation level of $\tn{2}{}$.
As illustrated in Ref.~\onlinecite{LooPraSceTouGin-JPCL-19}, the $\PBEUEG$ functional has clearly shown, for weakly correlated systems, to improve energetics over the pure UEG-based 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.
Besides, in the context of the present basis-set correction, $\n{2}{\Bas}(\br{})$, the on-top 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{FerGinTou-JCP-19}, we define a better approximation of the exact on-top pair density as
which directly follows from the large-$\mu$ extrapolation of the exact on-top pair density proposed by Gori-Giorgi and Savin \cite{GorSav-PRA-06} in the context of RS-DFT.
In the present study, we compute the ground- and excited-state energies, one-electron densities and on-top pair 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}
Both the implementation of the CIPSI algorithm and the computational protocol for excited states is reported in Ref.~\onlinecite{SceCafBenJacLoo-RC-19}.
The total energy of each state is obtained via an efficient extrapolation procedure of the sCI energies designed to reach near-FCI accuracy. \cite{HolUmrSha-JCP-17, QP2}
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 densities and on-top pair densities are computed from a very large CIPSI expansion containing up to several million of Slater determinants.
Except for methylene for which FCI/TZVP geometries have been taken from Ref.~\onlinecite{SheLeiVanSch-JCP-98}, the other molecular 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.
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}.
We refer the 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.
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, GouPieWlo-MP-10, 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 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}
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}.
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{ZimTouZhaMusUmr-JCP-09} which are extremely close from the experimentally-derived adiabatic energies.
The reason of this discrepancy is probably due to the frozen-core approximation which has been applied in our case and has shown to significantly affect adiabatic energies. \cite{LooGalJac-JPCL-18, LooJac-JCTC-19}
However, the exFCI/CBS energies are in perfect agreement with the semistochastic heat-bath CI (SHCI) calculations from Ref.~\onlinecite{ChiHolAdaOttUmrShaZim-JPCA-18}, 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.
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 $\PBEUEG$ 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 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.
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.
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.
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 cannot catch the radial incompleteness of the one-electron basis set, a feature which is far from being a cusp-related effect.
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 --- $1\,^{1}\Sigma_g^+\ra1\,^{1}\Delta_g$ and $1\,^{1}\Sigma_g^+\ra2\,^{1}\Sigma_g^+$ --- 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 involve an insignificant amount of single excitations. \cite{LooBogSceCafJac-JCTC-19}
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.
In other words, the UEG on-top pair density used in the $\LDA$ and $\PBEUEG$ functionals (see Sec.~\ref{sec:func}) is a particularly bad approximation of the true on-top pair density for the present system.
It is interesting to study the behavior of the key quantities involved in the basis-set correction for different states as the basis-set incompleteness error is obviously state specific.
\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$\n{2}{}(z)$ 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 short-range correlation effects are more pronounced in the former than in the latter.
This is linked to the larger values of the excited-state on-top pair density.
\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 doubly-excited state (red curve) of \ce{C2} for various basis sets $\Bas$.
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 \textit{et al.}\cite{FelPetDav-JCP-14} for an exhaustive investigation dedicated to the excited states of ethylene using state-of-the-art CI calculations.
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.
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, using sCI methods, 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, which uses an accurate on-top pair density, is more appropriate than the $\LDA$ and $\PBEUEG$ functionals relying on the UEG on-top pair 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.
This work was performed using HPC resources from GENCI-TGCC (Grant No.~2018-A0040801738), CALMIP (Toulouse) under allocation 2019-18005 and the Jarvis-Alpha cluster from the \textit{Institut Parisien de Chimie Physique et Th\'eorique}.
