\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{\DE}[2]{\Delta E_{#1}^{#2}} \newcommand{\bE}[2]{\Bar{E}_{#1}^{#2}} \newcommand{\DbE}[2]{\Delta \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 (UMR 7616), Universit\'e Pierre et Marie Curie, Sorbonne Universit\'e, CNRS, Paris, France} \begin{document} \title{Chemically-Accurate Excitation Energies With Small Basis Sets} \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) 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. \end{abstract} \maketitle %%%%%%%%%%%%%%%%%%%%%%%% \section{Introduction} \label{sec:intro} %%%%%%%%%%%%%%%%%%%%%%%% 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). Explicitly-correlated F12 methods \cite{Kut-TCA-85, Kut-TCA-85, KutKlo-JCP-91, NogKut-JCP-94} have been specifically designed to efficiently catch angular incompleteness. \cite{Ten-TCA-12, TenNog-WIREs-12, HatKloKohTew-CR-12, KonBisVal-CR-12, GruHirOhnTen-JCP-17, MaWer-WIREs-18} 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 performances of the recently proposed universal density-based basis set incompleteness correction. \cite{GinPraFerAssSavTou-JCP-18} 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. This work is organized as follows. In Sec.~\ref{sec:theory}, the main working equations of the density-based correction are reported and discussed. Computational details are reported in Sec.~\ref{sec:compdetails}. In Sec.~\ref{sec:res}, we discuss our results for each system and draw our conclusions in Sec.~\ref{sec:ccl}. %%%%%%%%%%%%%%%%%%%%%%%% \section{Theory} \label{sec:theory} %%%%%%%%%%%%%%%%%%%%%%%% 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 $\n{2,k}{\Bas}(\br{},\br{})$, Therefore, the complete basis set (CBS) energy of the ground and excited states may be approximated as \cite{GinPraFerAssSavTou-JCP-18} \begin{align} \label{eq:ECBS} \E{0}{\CBS} & \approx \E{0}{\Bas} + \bE{}{\Bas}[\n{0}{\Bas}], & \E{k}{\CBS} & \approx \E{k}{\Bas} + \bE{}{\Bas}[\n{k}{\Bas}], \end{align} where \begin{equation} \label{eq:E_funcbasis} \bE{}{\Bas}[\n{}{}] = \min_{\wf{}{} \rightsquigarrow \n{}{}} \mel*{\wf{}{}}{\hT + \hWee{}}{\wf{}{}} - \min_{\wf{}{\Bas} \rightsquigarrow \n{}{}} \mel*{\wf{}{\Bas}}{\hT + \hWee{}}{\wf{}{\Bas}} \end{equation} is the basis-dependent complementary density functional, \begin{align} \hT & = - \frac{1}{2} \sum_{i}^{\Ne} \nabla_i^2, & \hWee{} & = \sum_{i