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@ -165,9 +165,9 @@
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\affiliation{\LCPQ}
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\begin{abstract}
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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.
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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).
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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.
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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.
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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).
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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.
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\end{abstract}
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@ -196,6 +196,7 @@ This work is organized as follows.
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In Sec.~\ref{sec:theory}, the main working equations of the density-based correction are reported and discussed.
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Computational details are reported in Sec.~\ref{sec:compdetails}.
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In Sec.~\ref{sec:res}, we discuss our results for each system and draw our conclusions in Sec.~\ref{sec:ccl}.
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Unless otherwise stated, atomic units are used.
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%%%%%%%%%%%%%%%%%%%%%%%%
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\section{Theory}
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@ -330,13 +331,13 @@ This computationally-lighter functional will be referred to as PBE.
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\section{Computational details}
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\label{sec:compdetails}
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%%%%%%%%%%%%%%%%%%%%%%%%
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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}
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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}
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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}
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These energies will be labeled exFCI in the following.
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Using near-FCI excitation energies (within a given basis set) has the indisputable advantage to remove the error inherent to the WFT method.
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Indeed, in the present case, the only source of error on the excitation energies is due to basis set incompleteness.
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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.
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The one-electron and on-top densities are computed from a very large CIPSI expansion containing several million determinants.
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The one-electron and on-top densities are computed from a very large CIPSI expansion containing up to several million determinants.
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All the RS-DFT and exFCI calculations have been performed with {\QP}. \cite{QP2}
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For the numerical quadratures, we employ the SG-2 grid. \cite{DasHer-JCC-17}
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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.
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@ -369,7 +370,7 @@ We have also computed total energies at the exFCI/AV5Z level and used these alon
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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}
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Total energies for each state can be found in the {\SI}.
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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.
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From the triple-$\zeta$ basis onward, the exFCI excitation energies are chemically-accurate though, and drop steadily to the CBS limit when one increases the size of the basis set.
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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.
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Concerning the basis set correction, already at the double-$\zeta$ level, the PBEot correction returns chemically accurate excitation energies.
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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.
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Yet, they still yield significant reductions of the basis set incompleteness error, hence representing a good compromise between computational cost and accuracy.
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@ -379,14 +380,14 @@ This trend is quite systematic as we shall see below.
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%%% TABLE 1 %%%
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\begin{squeezetable}
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\begin{table*}
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\begin{table}
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\caption{
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Adiabatic transition energies $\Ead$ (in eV) of excited states of methylene for various methods and basis sets.
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The relative difference with respect to the exFCI/CBS result is reported in square brackets.
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See the {\SI} for raw data.}
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\label{tab:CH2}
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\begin{ruledtabular}
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\begin{tabular}{llddd}
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\begin{tabular}{lllll}
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& & \mc{3}{c}{Transitions} \\
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\cline{3-5}
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Method & Basis set & \tabc{$1\,^{3}B_1 \ra 1\,^{1}A_1$}
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@ -394,84 +395,84 @@ This trend is quite systematic as we shall see below.
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& \tabc{$1\,^{3}B_1 \ra 2\,^{1}A_1$} \\
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\hline
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exFCI & AVDZ
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& 0.441 [+0.053]
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& 1.536 [+0.146]
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& 2.659 [+0.154] \\
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& $0.441$ [$+0.053$]
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& $1.536$ [$+0.146$]
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& $2.659$ [$+0.154$] \\
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& AVTZ
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& 0.408 [+0.020]
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& 1.423 [+0.034]
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& 2.546 [+0.042] \\
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& $0.408$ [$+0.020$]
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& $1.423$ [$+0.034$]
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& $2.546$ [$+0.042$] \\
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& AVQZ
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& 0.395 [+0.007]
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& 1.399 [+0.010]
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& 2.516 [+0.012] \\
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& $0.395$ [$+0.007$]
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& $1.399$ [$+0.010$]
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& $2.516$ [$+0.012$] \\
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& AV5Z
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& 0.390 [+0.001]
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& 1.392 [+0.002]
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& 2.507 [+0.003] \\
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& $0.390$ [$+0.001$]
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& $1.392$ [$+0.002$]
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& $2.507$ [$+0.003$] \\
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& CBS
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& 0.388
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& 1.390
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& 2.504 \\
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& $0.388$
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& $1.390$
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& $2.504$ \\
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\\
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exFCI+PBEot & AVDZ
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& 0.347 [-0.042]
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& 1.401 [+0.011]
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& 2.511 [+0.007] \\
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& $0.347$ [$-0.042$]
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& $1.401$ [$+0.011$]
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& $2.511$ [$+0.007$] \\
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& AVTZ
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& 0.374 [-0.014]
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& 1.378 [-0.012]
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& 2.491 [-0.013] \\
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& $0.374$ [$-0.014$]
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& $1.378$ [$-0.012$]
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& $2.491$ [$-0.013$] \\
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& AVQZ
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& 0.379 [-0.009]
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& 1.378 [-0.011]
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& 2.489 [-0.016] \\
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& $0.379$ [$-0.009$]
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& $1.378$ [$-0.011$]
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& $2.489$ [$-0.016$] \\
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\\
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exFCI+PBE & AVDZ
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& 0.308 [-0.080]
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& 1.388 [-0.002]
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& 2.560 [+0.056] \\
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& $0.308$ [$-0.080$]
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& $1.388$ [$-0.002$]
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& $2.560$ [$+0.056$] \\
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& AVTZ
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& 0.356 [-0.032]
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& 1.371 [-0.019]
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& 2.510 [+0.006] \\
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& $0.356$ [$-0.032$]
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& $1.371$ [$-0.019$]
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& $2.510$ [$+0.006$] \\
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& AVQZ
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& 0.371 [-0.017]
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& 1.375 [-0.015]
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& 2.498 [-0.006] \\
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& $0.371$ [$-0.017$]
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& $1.375$ [$-0.015$]
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& $2.498$ [$-0.006$] \\
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\\
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exFCI+LDA & AVDZ
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& 0.337 [-0.051]
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& 1.420 [+0.030]
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& 2.586 [+0.082] \\
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& $0.337$ [$-0.051$]
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& $1.420$ [$+0.030$]
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& $2.586$ [$+0.082$] \\
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& AVTZ
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& 0.359 [-0.029]
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& 1.374 [-0.016]
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& 2.514 [+0.010] \\
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& $0.359$ [$-0.029$]
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& $1.374$ [$-0.016$]
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& $2.514$ [$+0.010$] \\
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& AVQZ
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& 0.370 [-0.018]
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& 1.375 [-0.015]
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& 2.499 [-0.005] \\
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& $0.370$ [$-0.018$]
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& $1.375$ [$-0.015$]
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& $2.499$ [$-0.005$] \\
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\\
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SHCI\fnm[1] & AVQZ
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& 0.393
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& 1.398
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& 2.516 \\
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& $0.393$
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& $1.398$
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& $2.516$ \\
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CR-EOMCC (2,3)D\fnm[2]& AVQZ
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& 0.412
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& 1.460
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& 2.547 \\
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& $0.412$
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& $1.460$
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& $2.547$ \\
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FCI\fnm[3] & TZ2P
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& 0.483
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& 1.542
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& 2.674 \\
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& $0.483$
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& $1.542$
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& $2.674$ \\
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DMC\fnm[4] &
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& 0.406
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& 1.416
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& 2.524 \\
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& $0.406$
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& $1.416$
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& $2.524$ \\
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Exp.\fnm[5] &
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& 0.400
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& 1.411
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& $0.400$
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& $1.411$
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\end{tabular}
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\end{ruledtabular}
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\fnt[1]{Semistochastic heat-bath CI (SHCI) calculations from Ref.~\onlinecite{ChiHolAdaOttUmrShaZim-JPCA-18}.}
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@ -479,7 +480,7 @@ This trend is quite systematic as we shall see below.
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\fnt[3]{Reference \onlinecite{SheLeiVanSch-JCP-98}.}
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\fnt[4]{Diffusion Monte Carlo (DMC) calculations from Ref.~\onlinecite{ZimTouZhaMusUmr-JCP-09}.}
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\fnt[5]{References \onlinecite{SheLeiVanSch-JCP-98, JenBun-JCP-88}.}
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\end{table*}
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\end{table}
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\end{squeezetable}
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%%% %%% %%%
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@ -503,7 +504,7 @@ They are both well-studied and possess Rydberg excited states which are highly s
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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).
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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.
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These results are also depicted in Figs.~\ref{fig:H2O} and \ref{fig:NH3} for \ce{H2O} and \ce{NH3}, respectively.
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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 is effect is even magnified.
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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.
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In these cases, one really needs doubly-augmented basis sets to reach radial completeness.
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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.
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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.
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@ -669,6 +670,15 @@ However, these results also clearly evidence that special care has to be taken f
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\end{figure}
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%%% %%% %%%
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%=======================
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\subsection{Range-Separation Function in Carbon Monoxyde}
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\label{sec:CO}
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%=======================
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\titou{It is interesting to have a look at $\rsmu{}{}(\br{})$ for the ground and excited states.
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To do so, we consider the first singlet excited state of carbon monoxide (vertical excitation energies are reported in Table \ref{tab:Mol}).
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Figure \ref{fig:mu} represent $\rsmu{}{}(\br{})$ for the ground and excited states for the AVDZ, AVTZ and AVQZ basis sets.}
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%=======================
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\subsection{Doubly-Excited States of the Carbon Dimer}
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\label{sec:C2}
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@ -678,7 +688,8 @@ These two valence excitations --- $1\,^{1}\Sigma_g^+ \ra 1\,^{1}\Delta_g$ and $1
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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}
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The vertical excitation energies associated with these transitions are reported in Table \ref{tab:Mol} and represented in Fig.~\ref{fig:C2}.
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An interesting point here is that one really needs the PBEot to get chemically-accurate absorption energies with the AVDZ atomic basis set.
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\titou{New figure represented $\rsmu{}{}(\br{})$ to understand what's going on here. Maybe it is because strongly correlated?}
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We believe that the present result is a direct consequence of the multireference character of the \ce{C2} molecule.
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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.
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%%% FIG 4 %%%
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\begin{figure}
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@ -719,6 +730,7 @@ Consistently with the previous examples, the LDA and PBE functionals are slightl
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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.
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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.
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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.
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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.
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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.
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%%%%%%%%%%%%%%%%%%%%%%%%
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