\documentclass[aip,jcp,reprint,noshowkeys]{revtex4-1} \usepackage{graphicx,dcolumn,bm,xcolor,microtype,multirow,amscd,amsmath,amssymb,amsfonts,physics,mhchem,longtable} \usepackage{natbib} \bibliographystyle{achemso} \AtBeginDocument{\nocite{achemso-control}} \usepackage{mathpazo,libertine} \usepackage{hyperref} \hypersetup{ colorlinks=true, linkcolor=blue, filecolor=blue, urlcolor=blue, citecolor=blue } \newcommand{\alert}[1]{\textcolor{red}{#1}} \definecolor{darkgreen}{HTML}{009900} \usepackage[normalem]{ulem} \newcommand{\titou}[1]{\textcolor{black}{#1}} \newcommand{\jt}[1]{\textcolor{purple}{#1}} \newcommand{\manu}[1]{\textcolor{darkgreen}{#1}} \newcommand{\toto}[1]{\textcolor{brown}{#1}} \newcommand{\trashPFL}[1]{\textcolor{red}{\sout{#1}}} \newcommand{\trashJT}[1]{\textcolor{purple}{\sout{#1}}} \newcommand{\trashMG}[1]{\textcolor{darkgreen}{\sout{#1}}} \newcommand{\trashAS}[1]{\textcolor{brown}{\sout{#1}}} \newcommand{\MG}[1]{\manu{(\underline{\bf MG}: #1)}} \newcommand{\JT}[1]{\juju{(\underline{\bf JT}: #1)}} \newcommand{\PFL}[1]{\titou{(\underline{\bf PFL}: #1)}} \newcommand{\AS}[1]{\toto{(\underline{\bf TOTO}: #1)}} \usepackage{hyperref} \hypersetup{ colorlinks=true, linkcolor=blue, filecolor=blue, urlcolor=blue, citecolor=blue } \newcommand{\mc}{\multicolumn} \newcommand{\fnm}{\footnotemark} \newcommand{\fnt}{\footnotetext} \newcommand{\tabc}[1]{\multicolumn{1}{c}{#1}} \newcommand{\SI}{\textcolor{blue}{supporting information}} \newcommand{\QP}{\textsc{quantum package}} % second quantized operators \newcommand{\ai}[1]{\hat{a}_{#1}} \newcommand{\aic}[1]{\hat{a}^{\dagger}_{#1}} % units \newcommand{\IneV}[1]{#1 eV} \newcommand{\InAU}[1]{#1 a.u.} \newcommand{\InAA}[1]{#1 \AA} \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}} \newcommand{\ROHF}{\text{ROHF}} \newcommand{\LDA}{\text{LDA}} \newcommand{\PBE}{\text{PBE}} \newcommand{\FCI}{\text{FCI}} \newcommand{\CBS}{\text{CBS}} \newcommand{\exFCI}{\text{exFCI}} \newcommand{\CCSDT}{\text{CCSD(T)}} \newcommand{\lr}{\text{lr}} \newcommand{\sr}{\text{sr}} \newcommand{\Ne}{N} \newcommand{\NeUp}{\Ne^{\uparrow}} \newcommand{\NeDw}{\Ne^{\downarrow}} \newcommand{\Nb}{N_{\Bas}} \newcommand{\Ng}{N_\text{grid}} \newcommand{\nocca}{n_{\text{occ}^{\alpha}}} \newcommand{\noccb}{n_{\text{occ}^{\beta}}} \newcommand{\n}[2]{n_{#1}^{#2}} \newcommand{\Ec}{E_\text{c}} \newcommand{\E}[2]{E_{#1}^{#2}} \newcommand{\bE}[2]{\Bar{E}_{#1}^{#2}} \newcommand{\bEc}[1]{\Bar{E}_\text{c,md}^{#1}} \newcommand{\e}[2]{\varepsilon_{#1}^{#2}} \newcommand{\be}[2]{\Bar{\varepsilon}_{#1}^{#2}} \newcommand{\bec}[1]{\Bar{e}^{#1}} \newcommand{\wf}[2]{\Psi_{#1}^{#2}} \newcommand{\W}[2]{W_{#1}^{#2}} \newcommand{\w}[2]{w_{#1}^{#2}} \newcommand{\hn}[2]{\Hat{n}_{#1}^{#2}} \newcommand{\rsmu}[2]{\mu_{#1}^{#2}} \newcommand{\V}[2]{V_{#1}^{#2}} \newcommand{\SO}[2]{\phi_{#1}(\br{#2})} \newcommand{\modY}{Y} \newcommand{\modZ}{Z} % basis sets \newcommand{\Bas}{\mathcal{B}} \newcommand{\BasFC}{\mathcal{A}} \newcommand{\FC}{\text{FC}} \newcommand{\occ}{\text{occ}} \newcommand{\virt}{\text{virt}} \newcommand{\val}{\text{val}} \newcommand{\Cor}{\mathcal{C}} % operators \newcommand{\hT}{\Hat{T}} \newcommand{\hWee}[1]{\Hat{W}_\text{ee}^{#1}} \newcommand{\updw}{\uparrow\downarrow} \newcommand{\f}[2]{f_{#1}^{#2}} \newcommand{\Gam}[2]{\Gamma_{#1}^{#2}} % coordinates \newcommand{\br}[1]{\mathbf{r}_{#1}} \newcommand{\dbr}[1]{d\br{#1}} \newcommand{\ra}{\rightarrow} % frozen core \newcommand{\WFC}[2]{\widetilde{W}_{#1}^{#2}} \newcommand{\fFC}[2]{\widetilde{f}_{#1}^{#2}} \newcommand{\rsmuFC}[2]{\widetilde{\mu}_{#1}^{#2}} \newcommand{\nFC}[2]{\widetilde{n}_{#1}^{#2}} % energies \newcommand{\EHF}{E_\text{HF}} \newcommand{\EPT}{E_\text{PT2}} \newcommand{\EFCI}{E_\text{FCI}} \newcommand{\EsCI}{E_\text{sCI}} \newcommand{\EDMC}{E_\text{DMC}} \newcommand{\EexFCI}{E_\text{exFCI}} \newcommand{\EexDMC}{E_\text{exDMC}} \newcommand{\Ead}{\Delta E_\text{ad}} \newcommand{\Eabs}{\Delta E_\text{abs}} \newcommand{\ex}[4]{$^{#1}#2_{#3}^{#4}$} \newcommand{\pis}{\pi^\star} \newcommand{\si}{\sigma} \newcommand{\sis}{\sigma^\star} \newcommand{\LCPQ}{Laboratoire de Chimie et Physique Quantiques (UMR 5626), Universit\'e de Toulouse, CNRS, UPS, France} \newcommand{\LCT}{Laboratoire de Chimie Th\'eorique, Universit\'e Pierre et Marie Curie, Sorbonne Universit\'e, CNRS, Paris, France} \begin{document} \title{Excitation Energies Near The Complete Basis Set Limit} \author{Emmanuel Giner} \affiliation{\LCT} \author{Anthony Scemama} \affiliation{\LCPQ} \author{Julien Toulouse} \affiliation{\LCT} \author{Pierre-Fran\c{c}ois Loos} \email[Corresponding author: ]{loos@irsamc.ups-tlse.fr} \affiliation{\LCPQ} \begin{abstract} 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. \end{abstract} \maketitle %%%%%%%%%%%%%%%%%%%%%%%% \section{Introduction} \label{sec:intro} %%%%%%%%%%%%%%%%%%%%%%%% 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. In the present study, we rely on the recently proposed short-range density-functional functional correction for basis set incompleteness. \cite{GinPraFerAssSavTou-JCP-18} %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. %%%%%%%%%%%%%%%%%%%%%%%% \section{Computational details} \label{sec:compdetails} %%%%%%%%%%%%%%%%%%%%%%%% 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. %%%%%%%%%%%%%%%%%%%%%%%% \section{Results} \label{sec:res} %%%%%%%%%%%%%%%%%%%%%%%% %======================= \subsection{Water} \label{sec:H2O} %======================= %======================= \subsection{Formaldehyde} \label{sec:CH2O} %======================= %======================= \subsection{Methylene} \label{sec:CH2} %======================= %%% TABLE 1 %%% \begin{squeezetable} \begin{table*} \caption{ Total energies $E$ (in hartree) and adiabatic transition energies $\Ead$ (in eV) of excited states of methylene for various methods and basis sets.} \begin{ruledtabular}{} \begin{tabular}{llddddddd} & & \mc{1}{c}{$1\,^{3}B_1$} & \mc{2}{c}{$1\,^{3}B_1 \ra 1\,^{1}A_1$} & \mc{2}{c}{$1\,^{3}B_1 \ra 1\,^{1}B_1$} & \mc{2}{c}{$1\,^{3}B_1 \ra 2\,^{1}A_1$} \\ \cline{3-3} \cline{4-5} \cline{6-7} \cline{8-9} Method & Basis set & \tabc{$E$ (a.u.)} & \tabc{$E$ (a.u.)} & \tabc{$\Ead$ (eV)} & \tabc{$E$ (a.u.)} & \tabc{$\Ead$ (eV)} & \tabc{$E$ (a.u.)} & \tabc{$\Ead$ (eV)} \\ \hline exFCI & AVDZ & -39.04846(1) & -39.03225(1) & 0.441 & -38.99203(1) & 1.536 & -38.95076(1) & 2.659 \\ & AVTZ & -39.08064(3) & -39.06565(2) & 0.408 & -39.02833(1) & 1.423 & -38.98709(1) & 2.546 \\ & AVQZ & -39.08854(1) & -39.07402(2) & 0.395 & -39.03711(1) & 1.399 & -38.99607(1) & 2.516 \\ & AV5Z & -39.09079(1) & -39.07647(1) & 0.390 & -39.03964(3) & 1.392 & -38.99867(1) & 2.507 \\ & CBS & -39.09111 & -39.07682 & 0.389 & -39.04000 & 1.391 & -38.99904 & 2.505 \\ \\ exFCI+LDA & AVDZ & -39.07450(1) & -39.06213(1) & 0.337 & -39.02233(1) & 1.420 & -38.97946(1) & 2.586 \\ & AVTZ & -39.09099(3) & -39.07779(2) & 0.359 & -39.04051(1) & 1.374 & -38.99859(1) & 2.514 \\ & AVQZ & -39.09319(1) & -39.07959(2) & 0.370 & -39.04267(1) & 1.375 & -39.00135(1) & 2.499 \\ \\ exFCI+PBE & AVDZ & -39.07282(1) & -39.06150(1) & 0.308 & -39.02181(1) & 1.388 & -38.97873(1) & 2.560 \\ & AVTZ & -39.08948(3) & -39.07639(2) & 0.356 & -39.03911(1) & 1.371 & -38.99724(1) & 2.510 \\ & AVQZ & -39.09247(1) & -39.07885(2) & 0.371 & -39.04193(1) & 1.375 & -39.00066(1) & 2.498 \\ \\ exFCI+PBEot & AVDZ & -39.06924(1) & -39.05651(1) & 0.347 & -39.01777(1) & 1.401 & -38.97698(1) & 2.511 \\ & AVTZ & -39.08805(3) & -39.07430(2) & 0.374 & -39.03742(1) & 1.378 & -38.99652(1) & 2.491 \\ & AVQZ & -39.09189(1) & -39.07795(2) & 0.379 & -39.04124(1) & 1.378 & -39.00044(1) & 2.489 \\ \\ SHCI & AVQZ & -39.08849(1) & -39.07404(1) & 0.393 & -39.03711(1) & 1.398 & -38.99603(1) & 2.516 \\ CR-EOMCC (2,3)D& AVQZ & -39.08817 & -39.07303 & 0.412 & -39.03450 & 1.460 & -38.99457 & 2.547 \\ FCI & TZ2P & -39.066738 & -39.048984 & 0.483 & -39.010059 & 1.542 & -38.968471 & 2.674 \\ DMC & & & & 0.406 & & 1.416 & & 2.524 \\ Exp. & & & & 0.400 & & 1.411 \end{tabular} \end{ruledtabular} \end{table*} \end{squeezetable} %%% %%% %%% %%% TABLE 2 %%% \begin{squeezetable} \begin{table*} \caption{ Vertical absorption energies $\Eabs$ (in eV) of excited states of water, carbon dimer and ammonia for various methods and basis sets.} \begin{ruledtabular}{} \begin{tabular}{lllddddddddddddd} & & & & \mc{12}{c}{Deviation with respect to TBE} \\ \cline{5-16} & & & & \mc{3}{c}{exFCI} & \mc{3}{c}{exFCI+PBEot} & \mc{3}{c}{exFCI+PBE} & \mc{3}{c}{exFCI+LDA} \\ \cline{5-7} \cline{8-10} \cline{11-13} \cline{14-16} 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 Ammonia & $1\,^{1}A_{1} \ra 1\,^{1}A_{2}$ & Ryd. & 6.66 & -0.18 & -0.07 & -0.02 & -0.04 & -0.02 & 0.00 & -0.07 & -0.03 & 0.00 & -0.07 & -0.03 & 0.00 \\ & $1\,^{1}A_{1} \ra 1\,^{1}E$ & Ryd. & 8.21 & -0.13 & -0.05 & 0.01 & 0.01 & 0.00 & 0.04 & -0.03 & -0.01 & 0.03 & -0.03 & 0.00 & 0.03 \\ & $1\,^{1}A_{1} \ra 2\,^{1}A_{1}$ & Ryd. & 8.65 & 1.03 & 0.68 & 0.49 & 1.17 & 0.73 & 0.75 & 1.13 & 0.72 & 0.74 & 1.13 & 0.71 & 0.78 \\ & $1\,^{1}A_{1} \ra 2\,^{1}A_{2}$ & Ryd. & 8.65 & 1.22 & 0.77 & 0.60 & 1.17 & 0.73 & 0.75 & 1.13 & 0.72 & 0.74 & 1.13 & 0.71 & 0.78 \\ & $1\,^{1}A_{1} \ra 1\,^{3}A_{2}$ & Ryd. & 9.19 & -0.18 & -0.06 & -0.02 & 1.36 & 0.83 & 0.63 & 1.33 & 0.81 & 0.62 & 1.32 & 0.81 & 0.61 \\ \\ Carbon dimer\fnm[1] & $1\,^{1}\Sigma_g^+ \ra 1\,^{1}\Delta_g$ & Val. & 2.06 & 0.15 & 0.03 & 0.00 & 0.02 & -0.02 & -0.02 & 0.13 & 0.02 & 0.00 & 0.15 & 0.03 & 0.00 \\ & $1\,^{1}\Sigma_g^+ \ra 2\,^{1}\Sigma_g^+$ & Val. & 2.40 & 0.10 & 0.02 & 0.00 & 0.02 & -0.03 & -0.02 & 0.09 & 0.01 & 0.00 & 0.11 & 0.02 & 0.00 \\ \\ Hydrogen chloride& ${}^1\Sigma \ra {}^1\Pi$ & CT\fnm[2] & 7.86 & -0.04 & -0.02 & 0.02 & 0.13 & 0.06 & 0.06 & 0.11 & 0.04 & 0.05 & 0.10 & 0.05 & 0.06 \\ \\ Hydrogen sulfide & $1\,^{1}A_1 \ra 1\,^{1}A_2$ & Ryd. & 6.10 & 0.00 & 0.08 & 0.05 & 0.15 & 0.12 & 0.07 & 0.14 & 0.11 & 0.07 & 0.14 & 0.11 & 0.07 \\ & $1\,^{1}A_1 \ra 1\,^{1}B_1$ & Ryd. & 6.29 & 0.00 & -0.05 & 0.00 & -0.12 & 0.01 & 0.03 & -0.14 & 0.00 & 0.03 & -0.14 & 0.01 & 0.03 \\ & $1\,^{1}A_1 \ra 1\,^{3}A_2$ & Ryd. & 5.74 & 0.01 & 0.07 & 0.05 & 0.18 & 0.12 & 0.08 & 0.20 & 0.13 & 0.08 & 0.19 & 0.13 & 0.08 \\ & $1\,^{1}A_1 \ra 1\,^{3}B_1$ & Ryd. & 5.94 & -0.04 & -0.05 & -0.01 & 0.07 & 0.02 & 0.03 & 0.09 & 0.03 & 0.03 & 0.07 & 0.04 & 0.04 \\ \\ Water & $1\,^{1}A_1 \ra 1\,^{1}B_1$ & Ryd. & 7.70 & -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 & -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 & -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 & -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 & -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 & -0.11 & -0.05 & -0.01 & 0.07 & 0.02 & 0.03 & 0.09 & 0.03 & 0.03 & 0.06 & 0.03 & 0.04 \end{tabular} \end{ruledtabular} \fnt[1]{Doubly-excited states of $(\pi,\pi) \ra (\si,\si)$ character.} \fnt[2]{CT stands for charge transfer.} \end{table*} \end{squeezetable} %%% %%% %%% %%% TABLE 3 %%% \begin{squeezetable} \begin{table*} \caption{ Vertical absorption energies $\Eabs$ (in eV) of excited states of acetylene, ethylene and formaldehyde for various methods and basis sets.} \begin{ruledtabular}{} \begin{tabular}{lllddddddddd} & & & & \mc{8}{c}{Deviation with respect to TBE} \\ \cline{5-12} & & & & \mc{2}{c}{exFCI} & \mc{2}{c}{exFCI+PBEot} & \mc{2}{c}{exFCI+PBE} & \mc{2}{c}{exFCI+LDA} \\ \cline{5-6} \cline{7-8} \cline{9-10} \cline{11-12} Molecule & Transition & Nature & \tabc{TBE} & \tabc{AVDZ} & \tabc{AVTZ} & \tabc{AVDZ} & \tabc{AVTZ} & \tabc{AVDZ} & \tabc{AVTZ} & \tabc{AVDZ} & \tabc{AVTZ} \\ \hline Acetylene & $1\,^{1}\Sigma_{g}^{+} \ra 1\,^{1}\Sigma_{u}^{-}$ & Val. & 7.10 & 0.10 & 0.00 & 0.07 & 0.00 & 0.11 & 0.00 & 0.11 & 0.00 \\ & $1\,^{1}\Sigma_{g}^{+} \ra 1\,^{1}\Delta_{u}$ & Val. & 7.44 & 0.07 & 0.00 & 0.04 & -0.01 & 0.12 & 0.02 & 0.11 & 0.02 \\ & $1\,^{1}\Sigma_{g}^{+} \ra 1\,^{3}\Sigma_{u}^{+}$ & Val. & 5.56 & -0.06 & -0.03 & 0.07 & 0.02 & 0.04 & 0.00 & 0.02 & 0.00 \\ & $1\,^{1}\Sigma_{g}^{+} \ra 1\,^{3}\Delta_{u}$ & Val. & 6.40 & 0.06 & 0.00 & 0.10 & 0.02 & 0.14 & 0.03 & 0.12 & 0.03 \\ & $1\,^{1}\Sigma_{g}^{+} \ra 1\,^{3}\Sigma_{u}^{-}$ & Val. & 7.09 & 0.05 & -0.01 & 0.08 & 0.00 & 0.16 & 0.04 & 0.14 & 0.04 \\ \\ Ethylene & $1\,^{1}A_{1g} \ra 1\,^{1}B_{3u}$ & Ryd. & 7.43 & -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 & 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 & -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 & 0.01 & 0.00 & 0.07 & 0.03 & 0.10 & 0.04 & 0.08 & 0.04 \\ & $1\,^{1}A_{1g} \ra 1\,^{3}B_{3u}$ & Val. & 7.28 & -0.12 & -0.04 & -0.03 & 0.00 & 0.00 & 0.00 & 0.00 & 0.02 \\ & $1\,^{1}A_{1g} \ra 1\,^{3}B_{1g}$ & Val. & 8.00 & -0.07 & -0.01 & 0.01 & 0.03 & 0.04 & 0.03 & 0.05 & 0.04 \\ \\ Formaldehyde& $1\,^{1}A_{1} \ra 1\,^{1}A_{2}$ & Val. & 3.97 & 0.02 & 0.01 & 0.05 & 0.02 & 0.03 & 0.02 & 0.02 & 0.01 \\ & $1\,^{1}A_{1} \ra 1\,^{1}B_{2}$ & Ryd. & 7.30 & -0.19 & -0.07 & 0.00 & 0.00 & -0.02 & 0.00 & -0.04 & 0.00 \\ & $1\,^{1}A_{1} \ra 2\,^{1}B_{2}$ & Ryd. & 8.14 & -0.10 & -0.01 & 0.09 & 0.07 & 0.08 & 0.06 & 0.05 & 0.06 \\ & $1\,^{1}A_{1} \ra 2\,^{1}A_{1}$ & Ryd. & 8.27 & -0.15 & -0.04 & 0.03 & 0.04 & 0.02 & 0.03 & 0.00 & 0.03 \\ & $1\,^{1}A_{1} \ra 1\,^{3}A_{2}$ & Val. & 3.58 & 0.00 & 0.00 & 0.09 & 0.05 & 0.11 & 0.06 & 0.07 & 0.04 \\ & $1\,^{1}A_{1} \ra 1\,^{3}A_{1}$ & Val. & 6.07 & 0.03 & 0.01 & 0.13 & 0.04 & 0.15 & 0.05 & 0.11 & 0.04 \\ & $1\,^{1}A_{1} \ra 1\,^{3}B_{2}$ & Ryd. & 7.14 & -0.19 & -0.08 & 0.01 & 0.01 & 0.02 & 0.01 & -0.01 & 0.00 \\ & $1\,^{1}A_{1} \ra 2\,^{3}B_{2}$ & Ryd. & 7.96 & -0.09 & -0.02 & 0.13 & 0.08 & 0.14 & 0.08 & 0.10 & 0.07 \\ & $1\,^{1}A_{1} \ra 1\,^{3}A_{1}$ & Ryd. & 8.15 & -0.14 & -0.05 & 0.07 & 0.05 & 0.07 & 0.04 & 0.04 & 0.04 \\ \end{tabular} \end{ruledtabular} \end{table*} \end{squeezetable} %%% %%% %%% %%%%%%%%%%%%%%%%%%%%%%%% \section{Conclusion} \label{sec:ccl} %%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%% \section*{Supporting Information Available} %%%%%%%%%%%%%%%%%%%%%%%% See {\SI} for geometries and additional information (including total energies). %%%%%%%%%%%%%%%%%%%%%%%% \begin{acknowledgements} The authors would like to thank the \textit{Centre National de la Recherche Scientifique} (CNRS) for funding. This work was performed using HPC resources from GENCI-TGCC (Grant No.~2018-A0040801738) and CALMIP (Toulouse) under allocation 2019-18005. \end{acknowledgements} %%%%%%%%%%%%%%%%%%%%%%%% \bibliography{Ex-srDFT,Ex-srDFT-control} \end{document}