\documentclass[aip,jcp,reprint,noshowkeys]{revtex4-1} \usepackage{graphicx,dcolumn,bm,xcolor,microtype,multirow,amscd,amsmath,amssymb,amsfonts,physics,mhchem,longtable} \usepackage{mathpazo,libertine} \usepackage[normalem]{ulem} \newcommand{\alert}[1]{\textcolor{red}{#1}} \definecolor{darkgreen}{RGB}{0, 180, 0} \newcommand{\beurk}[1]{\textcolor{darkgreen}{#1}} \newcommand{\trash}[1]{\textcolor{red}{\sout{#1}}} \usepackage{hyperref} \hypersetup{ colorlinks=true, linkcolor=blue, filecolor=blue, urlcolor=blue, citecolor=blue } \newcommand{\cdash}{\multicolumn{1}{c}{---}} \newcommand{\mc}{\multicolumn} \newcommand{\fnm}{\footnotemark} \newcommand{\fnt}{\footnotetext} \newcommand{\tabc}[1]{\multicolumn{1}{c}{#1}} \newcommand{\mr}{\multirow} \newcommand{\SI}{\textcolor{blue}{supporting information}} \newcommand{\br}{\mathbf{r}} % energies \newcommand{\EHF}{E_\text{HF}} \newcommand{\Ec}{E_\text{c}} \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{\ra}{\rightarrow} % units \newcommand{\IneV}[1]{#1 eV} \newcommand{\InAU}[1]{#1 a.u.} \newcommand{\InAA}[1]{#1 \AA} \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., J.~Chem.~Phys.~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. This problem was already noticed thirty years ago by Kutzelnigg \cite{Kutzelnigg_1985} who proposed to introduce explicitly the correlation between electrons via the introduction of the interelectronic distance $r_{12} = \abs{\br_1 - \br_2}$ as a basis function. \cite{Kutzelnigg_1991, Termath_1991, Klopper_1991a, Klopper_1991b, Noga_1994} This yields a prominent improvement of the energy convergence from $O(L^{-3})$ to $O(L^{-7})$ (where $L$ is the maximum angular momentum of the one-electron basis). This idea was later generalised to more accurate correlation factors $f_{12} \equiv f(r_{12})$. \cite{Persson_1996, Persson_1997, May_2004, Tenno_2004b, Tew_2005, May_2005} The resulting F12 methods achieve chemical accuracy for small organic molecules with relatively small Gaussian basis sets. \cite{Tenno_2012a, Tenno_2012b, Hattig_2012, Kong_2012} For example, as illustrated by Tew and coworkers, one can obtain, at the CCSD(T) level, quintuple-zeta quality correlation energies with a triple-zeta basis. \cite{Tew_2007b} In the present study, we rely on the recently proposed short-range density-functional functional correction for basis set incompleteness. \cite{Giner_2018} %%%%%%%%%%%%%%%%%%%%%%%% \section{Computational details} \label{sec:compdetails} %%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%% \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 and ammonia for various methods and basis sets.} \begin{ruledtabular}{} \begin{tabular}{llddddddddddddd} & & & \mc{12}{c}{Deviation with respect to TBE} \\ \cline{4-15} & & & \mc{3}{c}{exFCI} & \mc{3}{c}{exFCI+PBEot} & \mc{3}{c}{exFCI+PBE} & \mc{3}{c}{exFCI+LDA} \\ \cline{4-6} \cline{7-9} \cline{10-12} \cline{13-15} Molecule & Transition & \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 Water & $1\,^{1}A_1 \ra 1\,^{1}B_1$ & 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 \\ & $1\,^{1}A_1 \ra 1\,^{1}A_2$ & 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 \\ & $1\,^{1}A_1 \ra 2\,^{1}A_1$ & 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 \\ & $1\,^{1}A_1 \ra 1\,^{3}B_1$ & 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 \\ & $1\,^{1}A_1 \ra 1\,^{3}A_2$ & 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 \\ & $1\,^{1}A_1 \ra 1\,^{3}A_1$ & 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 \\ \\ Hydrogen sulfide & $1\,^{1}A_1 \ra 1\,^{1}A_2$ & 6.10 & 0.19 & 0.08 & 0.05 & & & & & & & & & \\ & $1\,^{1}A_1 \ra 1\,^{1}B_1$ & 6.29 & -0.19 & -0.05 & 0.00 & & & & & & & & & \\ & $1\,^{1}A_1 \ra 1\,^{3}A_2$ & 5.74 & 0.16 & 0.07 & 0.05 & & & & & & & & & \\ & $1\,^{1}A_1 \ra 1\,^{3}B_1$ & 5.94 & -0.19 & -0.05 & -0.01 & & & & & & & & & \\ \\ Ammonia & $1\,^{1}A_{1} \ra 1\,^{1}A_{2}$ & 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 \\ & $1\,^{1}A_{1} \ra 2\,^{1}A_{1}$ & 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 \\ & $1\,^{1}A_{1} \ra 1\,^{3}A_{2}$ & 6.37 & -0.18 & -0.06 & -0.02 & -0.03 & 0.00 & & -0.07 & 0.02 & & -0.07 & -0.01 & \\ \\ Hydrogen chloride& ${}^1\Sigma \ra {}^1\Pi(\text{CT})$ & 7.86 & -0.04 & -0.02 & 0.02 & & & & & & & & & \end{tabular} \end{ruledtabular} \end{table*} \end{squeezetable} %%% %%% %%% %%% TABLE 3 %%% \begin{squeezetable} \begin{table*} \caption{ Vertical absorption energies $\Eabs$ (in eV) of excited states of ethylene and formaldehyde for various methods and basis sets.} \begin{ruledtabular}{} \begin{tabular}{llddddddddd} & & & \mc{8}{c}{Deviation with respect to TBE} \\ \cline{4-11} & & & \mc{2}{c}{exFCI} & \mc{2}{c}{exFCI+PBEot} & \mc{2}{c}{exFCI+PBE} & \mc{2}{c}{exFCI+LDA} \\ \cline{4-5} \cline{6-7} \cline{8-9} \cline{10-11} Molecule & Transition & \tabc{TBE} & \tabc{AVDZ} & \tabc{AVTZ} & \tabc{AVDZ} & \tabc{AVTZ} & \tabc{AVDZ} & \tabc{AVTZ} & \tabc{AVDZ} & \tabc{AVTZ} \\ \hline Acetylene & $1\,^{1}\Sigma_{g}^{+} \ra 1\,^{1}\Sigma_{u}^{-}$ & 7.10 & 0.10 & 0.00 & & & & & & \\ & $1\,^{1}\Sigma_{g}^{+} \ra 1\,^{1}\Delta_{u}$ & 7.44 & 0.07 & 0.00 & & & & & & \\ & $1\,^{1}\Sigma_{g}^{+} \ra 1\,^{3}\Sigma_{u}^{+}$ & 5.56 & -0.06 & -0.03 & & & & & & \\ & $1\,^{1}\Sigma_{g}^{+} \ra 1\,^{3}\Delta_{u}$ & 6.40 & 0.06 & 0.00 & & & & & & \\ & $1\,^{1}\Sigma_{g}^{+} \ra 1\,^{3}\Sigma_{u}^{-}$ & 7.09 & 0.05 & -0.01 & & & & & & \\ \\ Ethylene & $1\,^{1}A_{1g} \ra 1\,^{1}B_{3u}$ & 7.43 & -0.12 & -0.04 & -0.05 & -0.01 & -0.04 & -0.01 & -0.02 & 0.00 \\ & $1\,^{1}A_{1g} \ra 1\,^{1}B_{1u}$ & 7.92 & 0.01 & 0.01 & 0.00 & 0.00 & 0.06 & 0.03 & 0.06 & 0.03 \\ & $1\,^{1}A_{1g} \ra 1\,^{1}B_{1g}$ & 8.10 & -0.1 & -0.02 & -0.03 & 0.00 & -0.02 & 0.00 & 0.00 & 0.01 \\ & $1\,^{1}A_{1g} \ra 1\,^{3}B_{1u}$ & 4.54 & 0.01 & 0.00 & 0.07 & 0.03 & 0.10 & 0.04 & 0.08 & 0.04 \\ \\ Formaldehyde& $1\,^{1}A_{1} \ra 1\,^{1}A_{2}$ & 3.97 & 0.02 & 0.01 & 0.05 & 0.02 & 0.03 & 0.02 & 0.02 & 0.01 \\ & $1\,^{1}A_{1} \ra 1\,^{1}B_{2}$ & 7.30 & -0.19 & -0.07 & 0.00 & 0.00 & -0.02 & 0.00 & -0.04 & 0.00 \\ & $1\,^{1}A_{1} \ra 2\,^{1}B_{2}$ & 8.14 & -0.10 & -0.01 & 0.09 & 0.07 & 0.08 & 0.06 & 0.05 & 0.06 \\ & $1\,^{1}A_{1} \ra 2\,^{1}A_{1}$ & 8.27 & -0.15 & -0.04 & 0.03 & 0.04 & 0.02 & 0.03 & 0.00 & 0.03 \\ & $1\,^{1}A_{1} \ra 1\,^{3}A_{2}$ & 3.58 & 0.00 & 0.00 & 0.09 & 0.05 & 0.11 & 0.06 & 0.07 & 0.04 \\ & $1\,^{1}A_{1} \ra 1\,^{3}A_{1}$ & 6.07 & 0.03 & 0.01 & 0.13 & 0.04 & 0.15 & 0.05 & 0.11 & 0.04 \\ & $1\,^{1}A_{1} \ra 1\,^{3}B_{2}$ & 7.14 & -0.19 & -0.08 & 0.01 & 0.01 & 0.02 & 0.01 & -0.01 & 0.00 \\ & $1\,^{1}A_{1} \ra 2\,^{3}B_{2}$ & 7.96 & -0.09 & -0.02 & 0.13 & 0.08 & 0.14 & 0.08 & 0.10 & 0.07 \\ & $1\,^{1}A_{1} \ra 1\,^{3}A_{1}$ & 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} %%%%%%%%%%%%%%%%%%%%%%%% See {\SI} for geometries and additional information (including total energies). %%%%%%%%%%%%%%%%%%%%%%%% \begin{acknowledgements} This work was performed using HPC resources from i) GENCI-TGCC (Grant No. 2018-A0040801738), ii) CALMIP (Toulouse) under allocations 2018-0510 and 2018-12158. \end{acknowledgements} %%%%%%%%%%%%%%%%%%%%%%%% \bibliography{Ex-srDFT} \end{document}