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\documentclass[aip,jcp,reprint,onecolumn,noshowkeys,superscriptaddress]{revtex4-1}
\usepackage{graphicx,dcolumn,bm,xcolor,microtype,multirow,amscd,amsmath,amssymb,amsfonts,physics,longtable,wrapfig,txfonts}
\usepackage[version=4]{mhchem}
\usepackage{natbib}
\bibliographystyle{achemso}
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\setcounter{figure}{0}
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\renewcommand{\theequation}{S\arabic{equation}}
\renewcommand{\thetable}{S\arabic{table}}
\renewcommand{\thesection}{S\arabic{section}}
\usepackage[utf8]{inputenc}
\usepackage[T1]{fontenc}
\usepackage{txfonts}
\usepackage{siunitx}
\usepackage{soul}
\DeclareSIUnit[number-unit-product = {\,}]
\cal{cal}
\DeclareSIUnit\kcal{\kilo\cal}
\newcommand{\kcalmol}{\si{\kcal\per\mole}}
\usepackage[
colorlinks=true,
citecolor=blue,
breaklinks=true
]{hyperref}
\urlstyle{same}
\newcommand{\ie}{\textit{i.e.}}
\newcommand{\eg}{\textit{e.g.}}
\newcommand{\alert}[1]{\textcolor{red}{#1}}
\usepackage[normalem]{ulem}
\newcommand{\titou}[1]{\textcolor{red}{#1}}
\newcommand{\trashPFL}[1]{\textcolor{red}{\sout{#1}}}
\newcommand{\trashXB}[1]{\textcolor{darkgreen}{\sout{#1}}}
\newcommand{\PFL}[1]{\titou{(\underline{\bf PFL}: #1)}}
\newcommand{\mc}{\multicolumn}
\newcommand{\fnm}{\footnotemark}
\newcommand{\fnt}{\footnotetext}
\newcommand{\tabc}[1]{\multicolumn{1}{c}{#1}}
\newcommand{\SupInf}{\textcolor{blue}{supporting information}}
\newcommand{\QP}{\textsc{quantum package}}
\newcommand{\T}[1]{#1^{\intercal}}
%% bold in Table
\newcommand{\bb}[1]{\textbf{#1}}
\newcommand{\rb}[1]{\textbf{\textcolor{red}{#1}}}
\newcommand{\gb}[1]{\textbf{\textcolor{darkgreen}{#1}}}
%geometries
\newcommand{\Dtwo}{$D_{2h}$}
\newcommand{\Dfour}{$D_{4h}$}
\sisetup{range-phrase=--}
\sisetup{range-units=single}
%states
%D2h states
\newcommand{\oneAg}{$1{}^1A_g$}
\newcommand{\tBoneg}{$1{}^3B_{1g}$}
\newcommand{\sBoneg}{$1{}^1B_{1g}$}
\newcommand{\twoAg}{$2{}^1A_g$}
%D4h states
%\newcommand{\oneBoneg}{$1{}^1B_{1g}$} same label as the D2h state
\newcommand{\Atwog}{$1{}^3A_{2g}$}
\newcommand{\Aoneg}{$1{}^1A_{1g}$}
\newcommand{\Btwog}{$1{}^1B_{2g}$}
% addresses
\newcommand{\LCPQ}{Laboratoire de Chimie et Physique Quantiques (UMR 5626), Universit\'e de Toulouse, CNRS, UPS, France}
\newcommand{\CEISAM}{Nantes Universit\'e, CNRS, CEISAM UMR-6230, Nantes F-44000, France.}
\begin{document}
\title{Supporting Information for ``Reference Energies for Cyclobutadiene: Automerization and Excited States''}
\author{Enzo \surname{Monino}*}
\email{emonino@irsamc.ups-tlse.fr}
\affiliation{\LCPQ}
\author{Martial \surname{Boggio-Pasqua}}
\affiliation{\LCPQ}
\author{Anthony \surname{Scemama}}
\affiliation{\LCPQ}
\author{Denis \surname{Jacquemin}}
\affiliation{\CEISAM}
\author{Pierre-Fran\c{c}ois \surname{Loos}*}
\email{loos@irsamc.ups-tlse.fr}
\affiliation{\LCPQ}
\maketitle
%%%%%%%%%%%%%%%%%%%%%%%%
\section{Geometries}
%%%%%%%%%%%%%%%%%%%%%%%%
Below, we provide the Cartesian coordinates (in \si{\angstrom}) of the geometries that are employed in this work.
\begin{itemize}
\item {\Dtwo} rectangular equilibrium geometry of the {\oneAg} ground state computed at the CASPT2(12,12)/aug-cc-pVTZ level:
\begin{verbatim}
C 0.0000000000 -0.6769380253 -0.7827569236
C 0.0000000000 -0.6769380253 0.7827569236
C 0.0000000000 0.6769380253 0.7827569236
C 0.0000000000 0.6769380253 -0.7827569236
H 0.0000000000 -1.4379809006 -1.5441628360
H 0.0000000000 -1.4379809006 1.5441628360
H 0.0000000000 1.4379809006 1.5441628360
H 0.0000000000 1.4379809006 -1.5441628360
\end{verbatim}
%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%
\item {\Dfour} square planar equilibrium geometry of the {\sBoneg} ground state computed at the CASPT2(12,12)/aug-cc-pVTZ level:
\begin{verbatim}
C 1.0248323754 0.0000000000 0.0000000000
C 0.0000000000 -1.0248323754 0.0000000000
C -1.0248323754 0.0000000000 0.0000000000
C 0.0000000000 1.0248323754 0.0000000000
H 2.1005277359 0.0000000000 0.0000000000
H 0.0000000000 -2.1005277359 0.0000000000
H -2.1005277359 0.0000000000 0.0000000000
H 0.0000000000 2.1005277359 0.0000000000
\end{verbatim}
%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%
\item {\Dtwo} rectangular equilibrium geometry of the {\oneAg} ground state computed at the CC3/aug-cc-pVTZ level:
\begin{verbatim}
C -0.78248546 -0.67208001 0.00000000
C 0.78248546 -0.67208001 0.00000000
C -0.78248546 0.67208001 0.00000000
C 0.78248546 0.67208001 0.00000000
H -1.54227765 -1.43404123 -0.00000000
H 1.54227765 -1.43404123 0.00000000
H -1.54227765 1.43404123 0.00000000
H 1.54227765 1.43404123 -0.00000000
\end{verbatim}
%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%
\item {\Dfour} square planar equilibrium geometry of the {\Atwog} state computed at the (RO)-CCSD(T)/aug-cc-pVTZ level:
\begin{verbatim}
C 0.000000 1.017702 0.000000
C 1.017702 -0.000000 0.000000
C -1.017702 0.000000 0.000000
C -0.000000 -1.017702 0.000000
H 0.000000 2.092429 0.000000
H 2.092429 -0.000000 0.000000
H -0.000000 -2.092429 0.000000
H -2.092429 0.000000 0.000000
\end{verbatim}
%%%%%%%%%%%%%%%%%%%%%%%%
\end{itemize}
%%%%%%%%%%%%%%%%%%%%%%%%
\section{Comment about symmetry: standard vs non-standard orientation}
%%%%%%%%%%%%%%%%%%%%%%%%
At the $D_{4h}$ $T_1$ optimized geometry, we have used the conventional standard orientation where two $C_2$ axes run through the carbon atoms.
In this conventional orientation, the singlet ground state $1 ^1B_{1g}$ remains $1 ^1B_{1g}$ in the $D_{2h}$ point group and the singlet excited state $1 ^1A_{1g}$ becomes $1 ^1A_g$ in the $D_{2h}$ point group.
Upon rotation of the molecular framework by 45 degrees in the $xy$ plane, the two $C_2$ axes then bisect the carbon-carbon bonds.
This induces a different orbital picture.
The correlation between the orbitals and states in the new molecular framework are illustrated in Figure \ref{fig:sym} at the CASSCF(4,4) level.
In this new orientation, the two singlet states $1 ^1B_{1g}$ and $1 ^1A_{1g}$ become both $1 ^1A_{g}$ in the $D_{2h}$ point group.
Because of the different orbital picture (the frontier orbitals are localized on two carbon atoms in the standard orientation and on four carbon atoms in the other orientation), the new CI coefficients resulting from this rotation bring also a different wave function representation.
Whereas the $1 ^1B_{1g}$ ground state is described in a one-electron-excitation picture in the standard orientation (the $1 ^1A_{1g}$ excited state involves a double excitation), the corresponding $1 ^1B_{1g}$ ground state in the new orientation involves a two-electron-excitation picture (the $1 ^1A_{1g}$ excited state also involves a double excitation).
Of course, these two representations are perfectly equivalent at the CASSCF level which describes single and double excitations on the same footing.
This is obviously not the case in linear response theory.
As mentioned in our manuscript in section II.B., for the $D_{4h}$ arrangement, we have considered the lowest closed-shell singlet state $^1A_{g}$ as reference.
Because this state has a substantial double-excitation character, we expect a significant error at the CCSD level.
The $1 ^1B_{1g}$ ground state is obtained as a singly excited state from that reference, while the $1 ^1B_{2g}$ excited state should also be a mixture involving a double excitation.
In the other (non-standard) orientation, the lowest $^1A_g$ state correlates with the $1 ^1B_{1g}$ ground state, which in this orientation has a strong double-excitation character.
Then, the $1 ^1A_{1g}$ excited state has also a strong double-excitation character, while the $1 ^1B_{2g}$ excited state is obtained by one-electron excitation.
Thus, whatever the orientation of the molecule, we will face the same problem for the reference state.
Note that in the case of the SF formalism, these three singlet states should all be described correctly if one takes the $1 ^3A_{2g}$ state as a reference high spin state, whatever the orientation.
\begin{figure}
\includegraphics[width=\textwidth]{figs1}
\caption{Standard vs non-standard orientations.}
\label{fig:sym}
\end{figure}
%\begin{squeezetable}
\begin{table*}
\caption{Energy differences between the states computed with various methods and the reference TBE values.
Note that AB stands for the automerization barrier and is reported in \si{\kcalmol}.
The numbers reported in parenthesis are the percentage of single excitations involved in the transition ($\%T_1$) calculated at the CC3/aug-cc-pVTZ level.
The values between square brackets have been obtained by extrapolation via the procedure described in the corresponding footnote.}
\label{tab:TBE}
\begin{ruledtabular}
\begin{tabular}{lrrrrrrr}
%\begin{tabular}{*{1}{*{8}{l}}}
& &\mc{3}{c}{{\Dtwo} excitation energies (eV)} & \mc{3}{c}{{\Dfour} excitation energies (eV)} \\
\cline{3-5} \cline{6-8}
Method & AB & {\tBoneg}(99\%) & {\sBoneg}(95\%)& {\twoAg}(1\%) & {\Atwog} & {\Aoneg} & {\Btwog} \\
\hline
SF-TD-B3LYP & $10.41$ & {$0.270$} & $-0.926$ & $-0.050$ & $-0.164$ & $-1.028$ & $-1.316$ \\
SF-TD-PBE0 & $8.95$ & {$0.249$} & $-0.829$ & $0.043$ & $-0.163$ & $-0.903$ &$-1.172$ \\
SF-TD-BH\&HLYP & $3.79$ & {$0.107$} & $-0.393$ & $0.454$ & $-0.099$ & $-0.251$ & $-0.418$ \\
SF-TD-M06-2X & $1.42$ & {$0.029$} & $-0.354$ & $0.319$ & $-0.066$ & $-0.097$ & $-0.247$ \\
SF-TD-CAM-B3LYP & $9.90$ & {$0.309$} & $-0.807$ &$0.100$ & $-0.134$ & $-0.920$ & $-1.185$\\
SF-TD-$\omega $B97X-V & $10.01$ & {$0.364$} & $-0.774$ & $0.175$ & $-0.118$ & $-0.928$ & $-1.187$ \\
SF-TD-LC-$\omega $PBE08 & $10.81$ & {$0.464$} & $-0.710$ & $0.310$ & $-0.086$ & $-0.939$ & $-1.191$ \\
SF-TD-M11 & $2.29$ & {$0.126$} & $-0.474$ & $0.262$ & $-0.063$ & $-0.312$ & $-0.490$ \\[0.1cm]
SF-ADC(2)-s & $-0.30$ & {$0.098$} & $-0.026$ & $0.093$ & $0.112$ & $0.112$ & $-0.005$ \\
SF-ADC(2)-x & $1.44$ & {$0.106$} & $-0.094$ & $-0.335$ & $0.068$ & $-0.409$ & $-0.118$ \\
SF-ADC(2.5) & $0.18$ & {$0.042$} & $0.006$ &$0.140$ & $0.024$ & $0.094$ & $0.000$ \\
SF-ADC(3) & $0.65$ & {$-0.014$} & $0.037$ & $0.186$ & $-0.065$ & $0.075$ & $0.004$ \\
{SF-EOM-CCSD} & {$-1.53$} & {$0.176$} & {$0.168$} & {$0.207$} & {$0.210$} & {$0.268$} & {$0.211$} \\[0.1cm]
CASSCF(4,4) & $-1.55$ & {$0.237$} & $1.421$ & $0.403$& $0.290$ & $0.734$ & $1.575$ \\
CASPT2(4,4) & $-1.16$ & {$-0.021$} & $-0.202$ & $0.034$ & $-0.016$ & $0.006$ & $-0.214$ \\
%XMS-CASPT2(4,4) & & & & $-0.035$ & & & \\
SC-NEVPT2(4,4) & $0.30$ & {$-0.054$} & $-0.703$ & $0.070$ & $-0.120$ & $-0.072$ & $-0.794$ \\
PC-NEVPT2(4,4) & $0.31$ & {$-0.051$} & $-0.757$ & $0.045$ & $-0.118$ & $-0.097$ & $-0.846$ \\
{MRCI(4,4)} & & $0.135$ & $0.553$ & $0.232$ & $0.127$ & $0.324$ & $0.566$ \\
{MRCI(4,4)+Q} & & $0.086$ & $0.217$ & $0.121$ & $0.075$ & $0.167$ & $0.205$ \\
CASSCF(12,12) & $2.66$ & {$0.253$} & $0.719$ & $0.179$ & $0.226$ & $0.443$ & $0.785$ \\
CASPT2(12,12) & $-0.42$& {$0.047$} & $0.058$ & $0.005$ & $0.039$ & $0.038$ & $0.077$ \\
%XMS-CASPT2(12,12) & & & & $-0.090$ & & & \\
SC-NEVPT2(12,12) & $-0.64$ & {$0.068$} & $0.063$ & $0.048$ & $0.021$ & $0.046$ & $0.043$ \\
PC-NEVPT2(12,12) & $-0.65$ & {$0.029$} & $-0.062$ & $0.018$ & $-0.013$ & $-0.024$ & $-0.093$ \\[0.1cm]
CCSD & $0.95$ & {$-0.116$} & $0.067$ & & $-0.059$ & $0.100$ & \\
CC3 & $-1.05$ & {$-0.031$} & $-0.006$ & $0.739$ & & $0.162$ & $0.871$ \\
CCSDT & $-0.25$ & {$-0.022$} & $0.014$ & $0.391$ & $0.005$ & $0.131$ & $0.688$ \\
CC4 & $-0.11$ & {$0.000$} & $0.003$ & $0.105$ & & $0.011$ & $-0.013$ \\
CCSDTQ & $0.00$ & & $0.000$ & $0.000$ & $0.000$ & $0.000$ & $0.000$ \\[0.1cm]
%CIPSI & & $-0.001\pm 0.030$ & $0.017\pm 0.035$ & $-0.120\pm 0.090$ & $0.025\pm 0.029$ & $0.130\pm 0.050$ & \\
\bf{TBE} & $[\bf{8.93}]$\fnm[1] & {$[\bf{1.433}]$}\fnm[2] & $[\bf{3.125}]$\fnm[1] & $[\bf{4.038}]$\fnm[1] & $[\bf{0.144}]$\fnm[2] & $[\bf{1.500}]$\fnm[1] & $[\bf{1.849}]$\fnm[1] \\[0.1cm]
Literature & $8.53$\fnm[3] & $1.573$\fnm[3] & $3.208$\fnm[3] & $4.247$\fnm[3] & $0.266$\fnm[3] & $1.664$\fnm[3] & $1.910$\fnm[3] \\
& $10.35$\fnm[4] & $1.576$\fnm[4] & $3.141$\fnm[4] & $3.796$\fnm[4] & $0.217$\fnm[4] & $1.123$\fnm[4] & $1.799$\fnm[4]\\
& $9.58$\fnm[5]& $1.456$\fnm[5] & $3.285$\fnm[5] & $4.334$\fnm[5] & $0.083$\fnm[5] & $1.621$\fnm[5] & $1.930$\fnm[5] \\
& {$7.50$}\fnm[6] & {$1.654$}\fnm[6] & {$3.416$}\fnm[6] & {$4.360$}\fnm[6] & {$0.369$}\fnm[6] & {$1.824$}\fnm[6] & {$2.143$}\fnm[6]\\
& {$9.36$}\fnm[7] & {$1.516$}\fnm[7] & {$3.260$}\fnm[7] & {$4.205$}\fnm[7] & {$0.163$}\fnm[7] & {$1.530$}\fnm[7] & {$1.921$}\fnm[7]\\
& {$9.91$}\fnm[8] & {$1.475$}\fnm[8] & {$3.215$}\fnm[8] & {$4.176$}\fnm[8] & {$0.098$}\fnm[8] & {$1.456$}\fnm[8] & {$1.853$}\fnm[8]\\
& & {$1.403$}\fnm[9] & {$3.120$}\fnm[9] & {$4.127$}\fnm[9] & {$0.023$}\fnm[9] & {$1.406$}\fnm[9] & {$1.751$}\fnm[9] \\
& & & & & {$0.062$}\fnm[10] & &\\
& & & & & {$0.219$}\fnm[11] & & \\
\end{tabular}
\end{ruledtabular}
\fnt[1]{Value obtained using CCSDTQ/aug-cc-pVDZ corrected by the difference between CC4/aug-cc-pVTZ and CC4/aug-cc-pVDZ.}
\fnt[2]{Value obtained using CCSDTQ/aug-cc-pVDZ corrected by the difference between CCSDT/aug-cc-pVTZ and CCSDT/aug-cc-pVDZ.}
\fnt[3]{Value obtained from Ref.~\onlinecite{Lefrancois_2015} at the SF-ADC(2)-s/cc-pVTZ level with the geometry obtained at the CCSD(T)/cc-pVTZ level.}
\fnt[4]{Value obtained from Ref.~\onlinecite{Lefrancois_2015} at the SF-ADC(2)-x/cc-pVTZ level with the geometry obtained at the CCSD(T)/cc-pVTZ level.}
\fnt[5]{Value obtained from Ref.~\onlinecite{Lefrancois_2015} at the SF-ADC(3)/cc-pVTZ level with the geometry obtained at the CCSD(T)/cc-pVTZ level.}
\fnt[6]{{Value obtained from Ref.~\onlinecite{Manohar_2008} at the SF-EOM-CCSD/cc-pVTZ level with the geometry obtained at the CCSD(T)/cc-pVTZ level.}}
\fnt[7]{{Value obtained from Ref.~\onlinecite{Manohar_2008} at the SF-EOM-CCSD(fT)/cc-pVTZ level with the geometry obtained at the CCSD(T)/cc-pVTZ level.}}
\fnt[8]{{Value obtained from Ref.~\onlinecite{Manohar_2008} at the SF-EOM-CCSD(dT)/cc-pVTZ level with the geometry obtained at the CCSD(T)/cc-pVTZ level.}}
\fnt[9]{{Value obtained from Ref.~\onlinecite{Gulania_2021} at the EOM-DEA-CCSD/cc-pVTZ level with the geometry obtained at the CCSD(T)/cc-pVTZ level.}}
\fnt[10]{{Value obtained from Ref.~\onlinecite{Ajala_2017} at the DEA-EOM-CC(3p-1h)/cc-pVDZ level with the geometry obtained at the CCSD/cc-pVDZ level.}}
\fnt[11]{{Value obtained from Ref.~\onlinecite{Ajala_2017} at the DEA-EOM-CC(4p-2h)/cc-pVDZ level with the geometry obtained at the CCSD/cc-pVDZ level.}}
\end{table*}
%\end{squeezetable}
%%% %%% %%% %%%
\begin{table}
\caption{Automerization energy (in \si{\kcalmol}) of CBD computed at various levels of theory.}
\begin{ruledtabular}
\begin{tabular}{lcr}
Level of theory & Automerization barrier & Reference \\
& (\kcalmol) & \\
\hline
CCSDTQ/aug-cc-pVTZ & $8.93$ & This work \\
ic-MRCISD+Q/cc-pVTZ & $8.93$ & Ref.~\onlinecite{Zhang_2019}\\
Mk-MRCCSD/cc-pVTZ & $10.09$ & Ref.~\onlinecite{Zhang_2019}\\
Mk-MRCCSD(T)/cc-pVTZ & $8.56$ & Ref.~\onlinecite{Zhang_2019}\\
SUCCSD/cc-pVTZ & $8.7$ & Ref.~\onlinecite{Li_2009}\\
MkCCSD/cc-pVTZ & $9.6$ & Ref.~\onlinecite{Li_2009}\\
RMRCCSD(T)/cc-pVTZ & $9.5$ & Ref.~\onlinecite{Li_2009}\\
MRCISD/cc-pVTZ & $8.4$ & Ref.~\onlinecite{Eckert-Maksic_2006}\\
MRCISD + Q/cc-pVTZ & $8.8$ & Ref.~\onlinecite{Eckert-Maksic_2006}\\
MRAQCC/cc-pVTZ & $8.9$ & Ref.~\onlinecite{Eckert-Maksic_2006}\\
CCSDt/cc-pVTZ & $9.5$ & Ref.~\onlinecite{Shen_2012}\\
CCSD(T)-h/cc-pVTZ & $6.8$ & Ref.~\onlinecite{Shen_2012}\\
CC(t;3)/cc-pVTZ & $10.0$ & Ref.~\onlinecite{Shen_2012}\\
CC(P;Q)/cc-pVDZ &$8.65$ & Ref.~\onlinecite{Gururangan_2021}\\
\end{tabular}
\end{ruledtabular}
\end{table}
%%% %%% %%% %%%
%%%%%%%%%%%%%%%%%%%%%%%%
\begin{table}
\caption{$\expval*{S^2}$ values for the different excited states computed at the SF-TD-DFT/aug-cc-pVTZ level for the {\Dtwo} and {\Dfour} structures.}
% \label{tab:Ssquare}
\begin{ruledtabular}
\begin{tabular}{lrrrrrr}
&\mc{3}{c}{$\expval*{S^2}$ for {\Dtwo} geometry} & \mc{3}{c}{{$\expval*{S^2}$ for {\Dfour} geometry}} \\
\cline{2-4} \cline{5-7}
Method & {\tBoneg} & {\sBoneg} & {\twoAg} & {\Atwog} & {\Aoneg} & {\Btwog} \\
\hline
SF-TD-B3LYP
& $1.989$ & $0.030$ & $0.017$ & $2.007$ & $0.014$ & $0.012$\\[0.1cm]
SF-TD-PBE0
& $2.001$ & $0.021$ & $0.019$ & $2.009$ & $0.018$ & $0.012$ \\[0.1cm]
SF-TD-BH\&HLYP
& $2.017$ & $0.026$ & $0.041$ & $2.020$ & $0.021$ & $0.018$\\[0.1cm]
SF-TD-M06-2X
& $2.014$ & $0.017$ & $0.040$ & $2.014$ & $0.015$ & $0.012$\\[0.1cm]
SF-TD-CAM-B3LYP
& $1.990$ & $0.033$ & $0.024$ & $2.008$ & $0.013$ & $0.012$\\[0.1cm]
SF-TD-$\omega$B97X-V
& $1.986$ & $0.035$ & $0.024$ & $2.008$ & $0.012$ & $0.010$\\[0.1cm]
SF-TD-LC-$\omega $PBE08
& $1.984$ & $0.044$ & $0.031$ & $2.012$ & $0.015$ & $0.012$\\[0.1cm]
SF-TD-M11
& $2.011$ & $0.023$ & $0.045$ & $2.012$ & $0.016$ & $0.014$\\
\end{tabular}
\end{ruledtabular}
\end{table}
%%% %%% %%% %%%
%%% TABLE IV %%%
\begin{table}
\caption{
Vertical excitation energies (with respect to the {\oneAg} ground state) obtained with multireference methods for the {\tBoneg}, {\sBoneg}, and {\twoAg} states of CBD at the {\Dtwo} rectangular equilibrium geometry of the {\oneAg} ground state.}
\label{tab:D2h}
\begin{ruledtabular}
\begin{tabular}{llrrr}
& & \mc{3}{c}{Excitation energies (eV)} \\
\cline{3-5}
Method & Basis & {\tBoneg} & {\sBoneg} & {\twoAg} \\
\hline
CASSCF(4,4) &6-31+G(d)& $1.662$ & $4.657$ & $4.439$ \\
& aug-cc-pVDZ & $1.672$ & $4.563$ & $4.448$ \\
& aug-cc-pVTZ & $1.670$ & $4.546$ & $4.441$ \\
& aug-cc-pVQZ & $1.671$ & $4.549$ & $4.440$ \\[0.1cm]
CASPT2(4,4) &6-31+G(d)& $1.440$ & $3.162$ & $4.115$ \\
& aug-cc-pVDZ & $1.414$ & $2.971$ & $4.068$ \\
& aug-cc-pVTZ & $1.412$ & $2.923$ & $4.072$ \\
& aug-cc-pVQZ & $1.417$ & $2.911$ & $4.081$ \\[0.1cm]
SC-NEVPT2(4,4) &6-31+G(d)& $1.407$ & $2.707$ & $4.145$ \\
& aug-cc-pVDZ & $1.381$ & $2.479$ & $4.109$ \\
& aug-cc-pVTZ & $1.379$ & $2.422$ & $4.108$ \\
& aug-cc-pVQZ & $1.384$ & $2.408$ & $4.116$ \\[0.1cm]
PC-NEVPT2(4,4) &6-31+G(d)& $1.409$ & $2.652$ & $4.120$ \\
& aug-cc-pVDZ & $1.384$ & $2.424$ & $4.084$ \\
& aug-cc-pVTZ & $1.382$ & $2.368$ & $4.083$ \\
& aug-cc-pVQZ & $1.387$ & $2.353$ & $4.091$ \\[0.1cm]
MRCI(4,4) &6-31+G(d)& $1.564$ & $3.802$ & $4.265$ \\
& aug-cc-pVDZ & $1.558$ & $3.670$ & $4.254$ \\
& aug-cc-pVTZ & $1.568$ & $3.678$ & $4.270$ \\
& aug-cc-pVQZ & $1.574$ & $3.681$ & $4.280$ \\[0.1cm]
MRCI(4,4)+Q & 6-31+G(d) & $1.525$ & $3.515$ & $4.165$ \\
& aug-cc-pVDZ & $1.510$ & $3.347$ & $4.142$ \\
& aug-cc-pVTZ & $1.519$ & $3.342$ & $4.159$ \\
& aug-cc-pVQZ & $1.525$ & $3.342$ & $4.169$ \\[0.1cm]
CASSCF(12,12) &6-31+G(d)& $1.675$ & $3.924$ & $4.220$ \\
& aug-cc-pVDZ & $1.685$ & $3.856$ & $4.221$ \\
& aug-cc-pVTZ & $1.686$ & $3.844$ & $4.217$ \\
& aug-cc-pVQZ & $1.687$ & $3.846$ & $4.216$ \\[0.1cm]
CASPT2(12,12) &6-31+G(d)& $1.508$ & $3.407$ & $4.099$ \\
& aug-cc-pVDZ & $1.489$ & $3.256$ & $4.044$ \\
& aug-cc-pVTZ & $1.480$ & $3.183$ & $4.043$ \\
& aug-cc-pVQZ & $1.482$ & $3.163$ & $4.047$ \\[0.1cm]
SC-NEVPT2(12,12) &6-31+G(d)& $1.522$ & $3.409$ & $4.130$ \\
& aug-cc-pVDZ & $1.511$ & $3.266$ & $4.093$ \\
& aug-cc-pVTZ & $1.501$ & $3.188$ & $4.086$ \\
& aug-cc-pVQZ & $1.503$ & $3.167$ & $4.088$ \\[0.1cm]
PC-NEVPT2(12,12) &6-31+G(d)& $1.487$ & $3.296$ & $4.103$ \\
& aug-cc-pVDZ & $1.472$ & $3.141$ & $4.064$ \\
& aug-cc-pVTZ & $1.462$ & $3.063$ & $4.056$ \\
\end{tabular}
\end{ruledtabular}
\end{table}
%%% %%% %%% %%%
%%% TABLE V %%%
\begin{table}
\caption{
Vertical excitation energies (with respect to the {\sBoneg} ground state) obtained with multireference methods for the {\Atwog}, {\Aoneg}, and {\Btwog} states of CBD at the {\Dfour} square-planar equilibrium geometry of the {\Atwog} state.
The values in square brackets have been obtained by extrapolation via the procedure described in the corresponding footnote.
The TBE/aug-cc-pVTZ values are highlighted in bold.}
\label{tab:D4h}
\begin{ruledtabular}
\begin{tabular}{llrrr}
& \mc{3}{r}{Excitation energies (eV)} \hspace{0.1cm}\\
\cline{3-5}
Method & Basis & {\Atwog} & {\Aoneg} & {\Btwog} \\
\hline
CASSCF(4,4) & 6-31+G(d) & $0.447$ & $2.257$ & $3.549$ \\
& aug-cc-pVDZ & $0.438$ & $2.240$ & $3.443$ \\
& aug-cc-pVTZ & $0.434$ & $2.234$ & $3.424$ \\
& aug-cc-pVQZ & $0.435$ & $2.235$ & $3.427$ \\[0.1cm]
CASPT2(4,4) & 6-31+G(d) & $0.176$ & $1.588$ & $1.899$ \\
& aug-cc-pVDZ & $0.137$ & $1.540$ & $1.708$ \\
& aug-cc-pVTZ & $0.128$ & $1.506$ & $1.635$ \\
& aug-cc-pVQZ & $0.128$ & $1.498$ & $1.612$ \\[0.1cm]
SC-NEVPT2(4,4) & 6-31+G(d) & $0.083$ & $1.520$ & $1.380$ \\
& aug-cc-pVDZ & $0.037$ & $1.465$ & $1.140$ \\
& aug-cc-pVTZ & $0.024$ & $1.428$ & $1.055$ \\
& aug-cc-pVQZ & $0.024$ & $1.420$ & $1.030$ \\[0.1cm]
PC-NEVPT2(4,4) & 6-31+G(d) & $0.085$ & $1.496$ & $1.329$ \\
& aug-cc-pVDZ & $0.039$ & $1.440$ & $1.088$ \\
& aug-cc-pVTZ & $0.026$ & $1.403$ & $1.003$ \\
& aug-cc-pVQZ & $0.026$ & $1.395$ & $0.977$ \\[0.1cm]
MRCI(4,4) & 6-31+G(d) & $0.297$ & $1.861$ & $2.571$ \\
& aug-cc-pVDZ & $0.273$ & $1.823$ & $2.419$ \\
& aug-cc-pVTZ & $0.271$ & $1.824$ & $2.415$ \\
& aug-cc-pVQZ & $0.273$ & $1.825$ & $2.413$ \\[0.1cm]
MRCI(4,4)+Q & 6-31+G(d) & $0.260$ & $1.728$ & $2.272$ \\
& aug-cc-pVDZ & $0.225$ & $1.669$ & $2.073$ \\
& aug-cc-pVTZ & $0.219$ & $1.667$ & $2.054$ \\
& aug-cc-pVQZ & $0.220$ & $1.667$ & $2.048$ \\[0.1cm]
CASSCF(12,12) & 6-31+G(d) & $0.386$ & $1.974$ & $2.736$ \\
& aug-cc-pVDZ & $0.374$ & $1.947$ & $2.649$ \\
& aug-cc-pVTZ & $0.370$ & $1.943$ & $2.634$ \\
& aug-cc-pVQZ & $0.371$ & $1.945$ & $2.637$ \\[0.1cm]
CASPT2(12,12) & 6-31+G(d) & $0.235$ & $1.635$ & $2.170$ \\
& aug-cc-pVDZ & $0.203$ & $1.588$ & $2.015$ \\
& aug-cc-pVTZ & $0.183$ & $1.538$ & $1.926$ \\
& aug-cc-pVQZ & $0.179$ & $1.522$ & $1.898$ \\[0.1cm]
SC-NEVPT2(12,12) & 6-31+G(d) & $0.218$ & $1.644$ & $2.143$ \\
& aug-cc-pVDZ & $0.189$ & $1.600$ & $1.991$ \\
& aug-cc-pVTZ & $0.165$ & $1.546$ & $1.892$ \\
& aug-cc-pVQZ & $0.160$ & $1.529$ & $1.862$ \\[0.1cm]
PC-NEVPT2(12,12) & 6-31+G(d) & $0.189$ & $1.579$ & $2.020$ \\
& aug-cc-pVDZ & $0.156$ & $1.530$ & $1.854$ \\
& aug-cc-pVTZ & $0.131$ & $1.476$ & $1.756$ \\
& aug-cc-pVQZ & $0.126$ & $1.460$ & $1.727$ \\[0.1cm]
\end{tabular}
\end{ruledtabular}
\end{table}
%%% %%% %%% %%%
\clearpage
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\bibliography{CBD}
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\end{document}
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