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\begin{document}
% addresses
\newcommand{\LCPQ}{Laboratoire de Chimie et Physique Quantiques (UMR 5626), Universit\'e de Toulouse, CNRS, UPS, France}
\newcommand{\CEISAM}{Universit\'e de Nantes, CNRS, CEISAM UMR 6230, F-44000 Nantes, France}
\title{Reference energies for cyclobutadiene}
\author{Enzo \surname{Monino}}
\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}
\begin{abstract}
Write an abstract
\end{abstract}
\maketitle
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Introduction}
\label{sec:intro}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
Despite the fact that excited states are involved in ubiquitious processes such as photochemistry, catalysis or in solar cell technology, none of the many methods existing is the reference in providing accurate excitation energies. Indeed, each method has its own flaws and there are so many chemical scenario that can occur, so it is still one of the biggest challenge in theoretical chemistry. Speaking of difficult task, cyclobutadiene (CBD) molecule has been a real challenge for experimental and theoretical chemists for many decades \cite{bally_1980}. Due to his antiaromaticity \cite{AromaticityAntiaromaticityElectronic,} and his large angular strain \cite{baeyer_1885} the CBD molecule presents a high reactivity which made the synthesis of this molecule a particularly difficult exercise. Hückel molecular orbital theory gives a triplet state with square ($D_{4h}$) geometry for the ground state of the CBD,with the two singly occupied frontier orbitals that are degenerated by symmetry. This degeneracy is lifted by the Jahn-Teller effect, meaning by distortion of the molecule (lowering symmetry), and gives a singlet state with rectangular ($D_{2h}$) geometry for the ground state.
Indeed, synthetic work from Pettis and co-workers \cite{reeves_1969} gives a rectangular geometry to the singlet ground state of CBD and then was confirmed by experimental works \cite{irngartinger_1983,ermer_1983,kreile_1986}.
At the ground state structrure ($D_{2h}$), the ${}^1A_g$ state has a weak multi-configurational character because of the well separated frontier orbitals and can be described by single-reference methods. But at the square ($D_{4h}$) geometry, the singlet state ${}^1B_{1g}$ has two singly occupied frontier orbitals that are degenerated so has a two-configurational character and single-reference methods are unreliable to describe it. The singlet ($D_{4h}$) is a transition state in the automerization reaction between the two rectangular structures (see Fig.\ref{fig:CBD}). The energy barrier for the automerization of the CBD was predicted, experimentally, in the range of 1.6-10 kcal.mol$^{-1}$ \cite{whitman_1982} and multi-reference calculations gave an energy barrier in the range of 6-7 kcal.mol$^{-1}$ \cite{eckert-maksic_2006}. All the specificities of the CBD molecule make it a real playground for excited-states methods.
Excited states of the CBD molecule in both geometries are represented in Fig.\ref{fig:CBD}. Are represented ${}^1A_g$ and $1{}^3B_{1g}$ states for the rectangular geometry and ${}^1B_{1g}$and $1{}^3A_{2g}$ for the square one. Due to energy scaling doubly excited states $1{}^1B_{1g}$ and $2{}^1A_{1g}$ for the $D_{2h}$ and $D_{4h}$ structures, respectively, are not drawn. Doubly excited states are known to be challenging to represent for adiabatic time-dependent density functional theory (TD-DFT) and even for state-of-the-art methods like the approximate third-order coupled-cluster (CC3) \cite{christiansen_1995,koch_1997} or equation-of-motion coupled-cluster with singles, doubles and triples (EOM-CCSDT) \cite{kucharski_1991,kallay_2004,hirata_2000,hirata_2004}.
In order to tackle the problems of multi-configurational character and double excitations several ways are explored. The most evident way that one can think about to describe multiconfigurational and double excitations are multiconfigurational methods. Among these methods, one can find complete active space self-consistent field (CASSCF) \cite{roos_1996}, the second perturbation-corrected variant (CASPT2) \cite{andersson_1990} and the second-order $n$-electron valence state perturbation theory (NEVPT2) \cite{angeli_2001b,angeli_2001a,angeli_2002}. The exponential scaling of these methods with the size of the active space is the limitation to the application of these ones to big molecules.
Another way to deal with double excitations is to use high level truncation of the equation-of-motion (EOM) formalism of coupled-cluster (CC) theory. However, to provide a correct description of doubly excited states one have to take into account contributions from the triple excitations in the CC expansion. Again, due to the scaling of CC methods with the number of basis functions the applicability of these methods is limited to small molecules.
An alternative to multiconfigurational and CC methods is the use of selected CI (SCI) methods for the computation of transition energies for singly and doubly excited states that are known to reach near full CI energies for small molecules. These methods allow to avoid an exponential increase of the size of the CI expansion by retaining the most energetically relevant determinants only, using a second-order energetic criterion to select perturbatively determinants in the FCI space.
Finally, to describe doubly excited states, one can think of spin-flip formalism established by Krylov in 2001. To briefly introduce the spin-flip idea we can present it like: instead of taking the singlet ground state as reference, the reference configuration is taken as the lowest triplet state. So one can access the singlet ground state and the singlet doubly-excited state via a spin-flip deexcitation and excitation (respectively), the difference of these two excitation energies providing an estimate of the double excitation. Obviously spin-flip methods have their own flaws, especially the spin contamination \cite{casanova_2020} (i.e., an artificial mixing of electronic states of differents spin multiplicities) due to spin incompleteness of the spin-flip expansion and by spin contamination of the reference configuration. One can adress part of this problem by expansion of the excitation order but with an increase of the computational cost or by complementing the spin-incomplete configuration set with the missing configurations.
In the present work we investigate ${}^1A_g$, $1{}^3B_{1g}$, $1{}^1B_{1g}$, $2{}^1A_{g}$ and ${}^1B_{1g}$, $1{}^3A_{2g}$, $2{}^1A_{1g}$,$1{}^1B_{2g}$ excited states for the $D_{2h}$ and $D_{4h}$, respectively, geometries. Computational details are reported in Section \ref{sec:compmet} for SCI (Subsection \ref{sec:SCI}), EOM-CC (Subsection \ref{sec:CC}), multiconfigurational (Subsection \ref{sec:Multi}) and spin-flip (Subsection \ref{sec:sf}) methods. Section \ref{sec:res} is devoted to the discussion of our results, first we consider the ground state property which is the autoisomerization barrier (Subsection \ref{sec:auto}) and then we study the excited states (Subsection \ref{sec:states}) of the CBD molecule.
\begin{figure}
\includegraphics[width=0.6\linewidth]{figure2.png}
\caption{Here comes the caption of the figure.}
\label{fig:CBD}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Computational Details}
\label{sec:compmet}
The system under investigation in this work is the cyclobutadiene (CBD) molecule, rectangular ($D_{2h}$) and square ($D_{4h}$) geometries are considered. The ($D_{2h}$) geometry is obtained at the CC3 level without frozen core using the aug-cc-pVTZ and the ($D_{4h}$) geometry is obtained at the RO-CCSD(T) level using aug-cc-pVTZ again without frozen core. All the calculations are performed using four basis set, the 6-31+G(d) basis and the aug-cc-pVXZ with X$=$D, T, Q. In the following we will use the notation AVXZ for the aug-cc-pVXZ basis, again with X$=$D, T, Q. The $\%T_1$ metric that gives the percentage of single excitation calculated at the CC3 level in \textcolor{red}{DALTON} allows to characterize the various states.Throughout all this work, spin-flip and spin-conserved calculations are performed with a UHF reference.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Selected Configuration Interaction}
\label{sec:SCI}
States energies and excitations energies calculations in the SCI framework are performed using QUANTUM PACKAGE where the CIPSI algorithm is implemented. The CIPSI algorithm allows to avoid the exponential increase of th CI expansion. To treat electronic states in the same way we use a state-averaged formalism meaning that the ground and excited states are represented with the same number and same set of determinants but using different CI coefficients. Then the SCI energy is the sum of two terms, the variational energy obtained by diagonalization of the CI matrix in the reference space and a second-order perturbative correction which estimates the contribution of the determinants not included in the CI space (estimate error in the truncation). It is possible to estimate the FCI limit for the total energies and compute the corresponding transition energies by extrapolating this second-order correction to zero. Extrapolation brings error and we can estime this one by energy difference between excitation energies obtained with the largest SCI wave function and the FCI extrapolated value. These errors provide a rough idea of the quality of the FCI extrapolation and cannot be seen as true bar error, they are reported in the following tables.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Coupled-Cluster}
\label{sec:CC}
Different flavours of coupled-cluster (CC) calculations are performed using different codes. Indeed, CC theory provides a hierarchy of methods that provide increasingly accurate energies via the increase of the maximum excitation degree of the cluster operator. Without any truncation of the cluster operator one has the full CC (FCC) that is equivalent to the full configuration interaction (FCI) giving the exact energy and wave function of the system for a fixed atomic basis set. However, due to the computational exponential scaling with system size we have to use truncated CC methods. The CC with singles and doubles (CCSD), CC with singles, doubles and triples (CCSDT) calculations are achieved with \textcolor{red}{CFOUR}. The calculations in the context of CC response theory or ``approximate'' series (CC3,CC4) are performed with \textcolor{red}{DALTON}. The CC with singles, doubles, triples and quadruples (CCSDTQ) are done with the \textcolor{red}{CFOUR} code. The CC2, CC3 and CC4 methods can be seen as cheaper approximations of CCSD, CCSDT and CCSDTQ by skipping the most expensive terms and avoiding the storage of higher-excitations amplitudes.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Multiconfigurational methods}
\label{sec:Multi}
State-averaged complete-active-space self-consistent field (SA-CASSCF) calculations are performed with \textcolor{red}{MOLPRO}. On top of those, NEVPT2 calculations, both partially contracted (PC) and strongly contracted (SC) scheme are considered. The PC-NEVPT2 is theoretically more accurate to the strongly contracted version due to the larger number of pertubers and greater flexibility. CASPT2 is performed and extended multistate (XMS) CASPT2 for strong mixing between states with same spin and spatial symmetries is also performed.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Spin-Flip}
\label{sec:sf}
In both structures the CBD has a singlet ground state, for the spin-flip calculations we consider the lowest triplet state as reference. Spin-flip techniques are broadly accessible and here, among them, we explore equation-of-motion coupled-cluster singles and doubles (EOM-CCSD), configuration interaction singles (CIS), algebraic-diagrammatic construction (ADC) scheme and TD-DFT. The standard and extended spin-flip ADC(2) (SF-ADC(2)-s and SF-ADC(2)-x respectively) and SF-ADC(3) are performed using Q-CHEM 5.2.1. Spin-flip TD-DFT calculations are also performed using Q-CHEM 5.2.1. The BLYP, B3LYP, PBE0 and BH\&HLYP functionals are considered, they contain 0\%, 20\%, 25\%, 50\% of exact exchange and are labeled, respectively, as SF-BLYP, SF-B3LYP, SF-PBE0, SF-BH\&HLYP. We also have done spin-flip TD-DFT calculations using RSH functionals as: CAM-B3LYP, LC-$\omega$PBE08 and $\omega$B97X-V. The main difference here between these RSHs functionals is the amount of exact-exchange at long-range: 75$\%$ for CAM-B3LYP and 100$\%$ for LC-$\omega$PBE08 and $\omega$B97X-V. To complete the use of spin-flip TD-DFT we also considered GH meta-GGA functional M06-2X, RSH meta-GGA functional M11. EOM-SF-CCSD and EOM-SF-CC(2,3) are also performed with Q-CHEM 5.2.1.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Results and discussion}
\label{sec:res}
As said in \ref{sec:intro} we study both excited states and automerization barrier. The excited states of interest in this work are the $^1 Ag$, $1 ^3B_{1g}$, $1 ^1B_{1g}$ and $2 ^1A_{g}$ states for the rectangular ($D_{2h}$) structure and the $1 ^1B_{1g}$, $1 ^3A_{2g}$, $2 ^1A_{1g}$ and $1 ^1B_{2g}$ states for the square ($D_{4h}$) structure. For the excited states part we study vertical excitations, as mentioned in \ref{sec:intro} the study of the CBD molecule is a difficult task due to the multi-configurational character of some excited states and there are not reference methods for the description of those. Because of this it is important to define our reference in this work to be able to compare the results of differents methods. To do so we use the Theoretical Best Estimates (TBE)
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%================================================
\subsection{Autoisomerization barrier}
\label{sec:auto}
The autoisomerization barrier for the CBD molecule is defined as the energy difference between the singlet ground state of the square ($D_{4h}$) structure and the singlet ground state of the rectangular ($D_{2h}$) geometry. Results for the calculation of the automerization barrier are shown in Tables \ref{tab:auto_standard} and \ref{tab:auto_spin_flip}. As said in \ref{sec:intro} the range for this barrier is quite large. So again, it is important to define our reference in this work in order to be able to compare our results. Table \ref{tab:auto_standard} gives standard methods results, we can observe a large difference for the autoisomerization barrier between the multi-configurational methods. Indeed, for the CASSCF(12,12) we have a difference of the order of 3 kcal.mol$^{-1}$ with CASPT2(12,12) and NEVPT2(12,12) for all the basis. However, the difference between CASPT2(12,12) and NEVPT2(12,12) is much smaller, of the order of 0.2 kcal.mol$^{-1}$ for all the basis.
%%% TABLE VI %%%
\begin{squeezetable}
\begin{table*}
\caption{Autoisomerization barrier in \kcalmol.}
\label{tab:auto_standard}
\begin{ruledtabular}
\begin{tabular}{llrrrr}
Method & 6-31+G(d) & AVDZ& AVTZ & AVQZ\\
\hline
SF-CIS & $2.64$ & $2.82$ & $3.43$ & $3.43$ \\
SF-TD-BLYP & $23.57$ & $23.62$ & $24.23$ & $24.22$ \\
SF-TD-B3LYP & $18.84$ & $18.93$ & $19.57$ & $19.57$ \\
SF-TD-PBE0 & $17.31$ & $17.36$ & $18.01$ & $18.00$ \\
SF-TD-BH\&HLYP & $11.90$ & $12.07$ & $12.73$ & $12.73$ \\
SF-TD-M06-2X & $9.34$ & $9.68$ & $10.39$ & $10.40$ \\
SF-TD-CAM-B3LYP & $18.21$ & $18.30$ & $18.98$ & $18.97$ \\
SF-TD-$\omega$B97X-V & $18.46$ & $18.48$ & $19.14$ & $19.12$ \\
SF-TD-M11 & $11.13$ & $10.38$ & $11.28$ & $11.19$ \\
SF-ADC2-s & $6.69$ & $7.15$ & $8.64$ & $8.85$ \\
SF-ADC2-x & $8.66$ & $9.15$ & $10.40$ & \\
SF-ADC3 & $8.06$ & $8.76$ & $9.58$ & \\
CASSCF(12,12) & $10.19$ & $10.75$ & $11.59$ & $11.62$ \\
CASPT2(12,12) & $7.24$ & $7.53$ & $8.51$ & $8.71$ \\
NEVPT2(12,12) & $7.12$ & $7.33$ & $8.28$ & $8.49$ \\
CCSD & $8.31$ & $8.80$ & $9.88$ & $10.10$ \\
CC3 & $6.59$ & $6.89$ & $7.88$ & $8.06$ \\
CCSDT & $7.26$ & $7.64$ & $8.68$ &$\left[ 8.86\right]$\fnm[1] \\
CC4 & $7.40$ & $7.78$ & $\left[ 8.82\right]$\fnm[2] & $\left[ 9.00\right]$\fnm[3]\\
CCSDTQ & $7.51$ & $\left[ 7.89\right]$\fnm[4]& $\left[ 8.93\right]$\fnm[5]& $\left[ 9.11\right]$\fnm[6]\\
%CIPSI & $7.91\pm 0.21$ & $8.58\pm 0.14$ & & \\
\end{tabular}
\end{ruledtabular}
\fnt[1]{Value obtained using CCSDT/AVTZ corrected by the difference between CC3/AVQZ and CC3/AVTZ.}
\fnt[2]{Value obtained using CC4/AVDZ corrected by the difference between CCSDT/AVTZ and CCSDT/AVDZ.}
\fnt[3]{Value obtained using CC4/AVTZ corrected by the difference between CCSDT/AVQZ and CCSDT/AVTZ.}
\fnt[4]{Value obtained using CCSDTQ/6-31+G(d) corrected by the difference between CC4/AVDZ basis and CC4/6-31+G(d).}
\fnt[5]{Value obtained using CCSDTQ/AVDZ corrected by the difference between CC4/AVTZ and CC4/AVDZ.}
\fnt[6]{Value obtained using CCSDTQ/AVTZ corrected by the difference between CC4/AVQZ and CC4/AVTZ.}
\end{table*}
\end{squeezetable}
%%% %%% %%% %%%
%%% TABLE VII %%%
%\begin{squeezetable}
%\begin{table}
% \caption{Autoisomerization barrier for spin-flip methods in \kcalmol.}
%
% \label{tab:auto_spin_flip}
% \begin{ruledtabular}
% \begin{tabular}{llrrrr}
% Method & 6-31+G(d) & AVDZ& AVTZ & AVQZ\\
% \hline
%SF-CIS & $2.64$ & $2.82$ & $3.43$ & $3.43$ \\
%SF-TD-BLYP & $23.57$ & $23.62$ & $24.23$ & $24.22$ \\
%SF-TD-B3LYP & $18.84$ & $18.93$ & $19.57$ & $19.57$ \\
%SF-TD-PBE0 & $17.31$ & $17.36$ & $18.01$ & $18.00$ \\
%SF-TD-BH\&HLYP & $11.90$ & $12.07$ & $12.73$ & $12.73$ \\
%SF-TD-M06-2X & $9.34$ & $9.68$ & $10.39$ & $10.40$ \\
%SF-TD-CAM-B3LYP & $18.21$ & $18.30$ & $18.98$ & $18.97$ \\
%SF-TD-$\omega$B97X-V & $18.46$ & $18.48$ & $19.14$ & $19.12$ \\
%SF-TD-M11 & $11.13$ & $10.38$ & $11.28$ & $11.19$ \\
%SF-ADC2-s & $6.69$ & $7.15$ & $8.64$ & $8.85$ \\
%SF-ADC2-x & $8.66$ & $9.15$ & $10.40$ & \\
%SF-ADC3 & $8.06$ & $8.76$ & $9.58$ & \\
%
% \end{tabular}
% \end{ruledtabular}
%
%\end{table}
%\end{squeezetable}
%%% %%% %%% %%%
%================================================
%================================================
\subsection{Excited States}
\label{sec:states}
%%% TABLE I %%%
\begin{squeezetable}
\begin{table}
\caption{
Spin-flip TD-DFT vertical excitation energies (with respect to the singlet $\text{X}\,{}^1A_{g}$ ground state) of the $1\,{}^3B_{1g}$, $1\,{}^1B_{1g}$, and $2\,{}^1A_{g}$ states of CBD at the $D_{2h}$ rectangular equilibrium geometry of the $\text{X}\,{}^1 A_{g}$ ground state.
\label{tab:sf_tddft_D2h}}
\begin{ruledtabular}
\begin{tabular}{llrrr}
& \mc{4}{r}{Excitation energies (eV)} \hspace{0.5cm}\\
\cline{3-5}
Method & Basis & $1\,{}^3B_{1g}$ & $1\,{}^1B_{1g}$ & $2\,{}^1A_{g}$ \\
\hline
SF-TD-BLYP & 6-31+G(d) & $1.829$ & $1.926$ & $3.755$ \\
& AVDZ & $1.828$ & $1.927$ & $3.586$ \\
& AVTZ & $1.825$ & $1.927$ & $3.546$ \\
& AVQZ & $1.825$ & $1.927$ & $3.528$ \\[0.1cm]
SF-TD-B3LYP & 6-31+G(d) & $1.706$ & $2.211$ & $3.993$ \\
& AVDZ & $1.706$ & $2.204$ & $3.992$ \\
& AVTZ & $1.703$ & $2.199$ & $3.988$ \\
& AVQZ & $1.703$ & $2.199$ & $3.989$\\[0.1cm]
SF-TD-PBE0 & 6-31+G(d) & $1.687$ & $2.314$ & $4.089$ \\
& AVDZ & $1.684$ & $2.301$ & $4.085$ \\
& AVTZ & $1.682$ & $2.296$ & $4.081$ \\
& AVQZ & $1.682$ & $2.296$ & $4.079$\\[0.1cm]
SF-TD-BH\&HLYP & 6-31+G(d) & $1.552$ & $2.779$ & $4.428$ \\
& AVDZ & $1.546$ & $2.744$ & $4.422$ \\
& AVTZ & $1.540$ & $2.732$ & $4.492$ \\
& AVQZ & $1.540$ & $2.732$ & $4.415$ \\[0.1cm]
SF-TD-M06-2X & 6-31+G(d) & $1.477$ & $2.835$ & $4.378$ \\
& AVDZ & $1.467$ & $2.785$ & $4.360$ \\
& AVTZ & $1.462$ & $2.771$ & $4.357$ \\
& AVQZ & $1.458$ & $2.771$ & $4.352$ \\[0.1cm]
SF-TD-CAM-B3LYP & 6-31+G(d) & $1.750$ & $2.337$ & $3.315$ \\
& AVDZ & $1.745$ & $2.323$ & $4.140$ \\
& AVTZ & $1.742$ & $2.318$ & $4.138$ \\
& AVQZ & $1.743$ & $2.319$ & $4.138$ \\[0.1cm]
SF-TD-$\omega$B97X-V & 6-31+G(d) & $1.810$ & $2.377$ & $4.220$ \\
& AVDZ & $1.800$ & $2.356$ & $4.217$ \\
& AVTZ & $1.797$ & $2.351$ & $4.213$ \\[0.1cm]
SF-TD-M11 & 6-31+G(d) & $1.566$ & $2.687$ & $4.292$ \\
& AVDZ & $1.546$ & $2.640$ & $4.267$ \\
& AVTZ & $1.559$ & $2.651$ & $4.300$ \\
& AVQZ & $1.557$ & $2.650$ & $4.299$ \\[0.1cm]
SF-TD-LC-$\omega $PBE08 & 6-31+G(d) & $1.917$ & $2.445$ & $4.353$ \\
& AVDZ & $1.897$ & $2.415$ & $4.346$ \\
& AVTZ & $1.897$ & $2.415$ & $4.348$ \\
& AVQZ & $1.897$ & $2.415$ & $4.348$ \\[0.1cm]
\end{tabular}
\end{ruledtabular}
\end{table}
\end{squeezetable}
%%% %%% %%% %%%
%%% TABLE II %%%
\begin{squeezetable}
\begin{table}
\caption{
Spin-flip CIS, ADC and CC vertical excitation energies (with respect to the singlet $\text{X}\,{}^1A_{g}$ ground state) of the $1\,{}^3B_{1g}$, $1\,{}^1B_{1g}$, and $2\,{}^1A_{g}$ states of CBD at the $D_{2h}$ rectangular equilibrium geometry of the $\text{X}\,{}^1 A_{g}$ ground state.
\label{tab:sf_adc_D2h}}
\begin{ruledtabular}
\begin{tabular}{llrrr}
& \mc{4}{r}{Excitation energies (eV)} \hspace{0.5cm}\\
\cline{3-5}
Method & Basis & $1\,{}^3B_{1g}$ & $1\,{}^1B_{1g}$ & $2\,{}^1A_{g}$ \\
\hline
SF-CIS & 6-31+G(d) & $1.514$ & $3.854$ & $5.379$ \\
& AVDZ & $1.487$ & $3.721$ & $5.348$ \\
& AVTZ & $1.472$ & $3.701$ & $5.342$ \\
& AVQZ & $1.471$ & $3.702$ & $5.342$ \\[0.1cm]
SF-ADC(2)-s & 6-31+G(d) & $1.577$ & $3.303$ & $4.196$ \\
& AVDZ & $1.513$ & $3.116$ & $4.114$ \\
& AVTZ & $1.531$ & $3.099$ & $4.131$ \\
& AVQZ & $1.544$ & $3.101$ & $4.140$ \\[0.1cm]
SF-ADC(2)-x & 6-31+G(d) & $1.557$ & $3.232$ & $3.728$ \\
& AVDZ & $1.524$ & $3.039$ & $3.681$ \\
& AVTZ & $1.539$ & $3.031$ & $3.703$ \\[0.1cm]
SF-ADC(3) & 6-31+G(d) & $1.435$ & $3.352$ & $4.242$ \\
& AVDZ & $1.422$ & $3.180$ & $4.208$ \\
& AVTZ & $1.419$ & $3.162$ & $4.224$ \\[0.1cm]
SF-EOM-CCSD & 6-31+G(d) & $1.663$ & $3.515$ & $4.275$ \\
& AVDZ & $1.611$ & $3.315$ & $3.856$ \\
& AVTZ & $1.609$ & $3.293$ & $4.245$ \\[0.1cm]
SF-EOM-CC(2,3) & 6-31+G(d) & $1.490$ & $3.333$ & $4.061$ \\
& AVDZ & $1.464$ & $3.156$ & $4.027$ \\
\end{tabular}
\end{ruledtabular}
\end{table}
\end{squeezetable}
%%% %%% %%% %%%
%%% TABLE III %%%
\begin{squeezetable}
\begin{table*}
\caption{
Standard vertical excitation energies (with respect to the singlet $\text{X}\,{}^1A_{g}$ ground state) of the $1\,{}^3B_{1g}$, $1\,{}^1B_{1g}$, and $2\,{}^1A_{g}$ states of CBD at the $D_{2h}$ rectangular equilibrium geometry of the $\text{X}\,{}^1 A_{g}$ ground state.
\label{tab:D2h}}
\begin{ruledtabular}
\begin{tabular}{llrrr}
& \mc{4}{r}{Excitation energies (eV)} \hspace{0.5cm}\\
\cline{3-5}
Method & Basis & $1\,{}^3B_{1g}$ & $1\,{}^1B_{1g}$ & $2\,{}^1A_{g}$ \\
\hline
CC3 &6-31+G(d)& $1.420$ & $3.341$ & $4.658$ \\
& AVDZ & $1.396$ & $3.158$ & $4.711$ \\
& AVTZ & $1.402$ & $3.119$ & $4.777$ \\
& AVQZ & $1.409$ & $3.113$ & $4.774$ \\[0.1cm]
CCSDT &6-31+G(d)& $1.442$ & $3.357$ & $4.311$ \\
& AVDZ & $1.411$ & $3.175$ & $4.327$ \\
& AVTZ & $1.411$ & $3.139$ & $4.429$ \\[0.1cm]
CC4 &6-31+G(d)& & $3.343$ & $4.067$ \\
& AVDZ & & $3.164$ & $4.041$ \\
& AVTZ & & $\left[3.128\right]$\fnm[1] & $\left[4.143\right]$\fnm[1]\\[0.1cm]
CCSDTQ &6-31+G(d)& & $3.340$ & $4.073$ \\
& AVDZ & & $\left[3.161\right]$\fnm[2]& $\left[4.047\right]$\fnm[2] \\
& AVTZ & & $\left[3.125\right]$\fnm[3]& $\left[4.149\right]$\fnm[3]\\[0.1cm]
SA2-CASSCF(4,4) &6-31+G(d)& $1.662$ & $4.657$ & $4.439$ \\
& AVDZ & $1.672$ & $4.563$ & $4.448$ \\
& AVTZ & $1.670$ & $4.546$ & $4.441$ \\
& AVQZ & $1.671$ & $4.549$ & $4.440$ \\[0.1cm]
CASPT2(4,4) &6-31+G(d)& $1.440$ & $3.162$ & $4.115$ \\
& AVDZ & $1.414$ & $2.971$ & $4.068$ \\
& AVTZ & $1.412$ & $2.923$ & $4.072$ \\
& AVQZ & $1.417$ & $2.911$ & $4.081$ \\[0.1cm]
XMS-CASPT2(4,4) &6-31+G(d)& & & $4.151$ \\
& AVDZ & & & $4.105$ \\
& AVTZ & & & $4.114$ \\
& AVQZ && & $4.125$ \\[0.1cm]
SC-NEVPT2(4,4) &6-31+G(d)& $1.407$ & $2.707$ & $4.145$ \\
& AVDZ & $1.381$ & $2.479$ & $4.109$ \\
& AVTZ & $1.379$ & $2.422$ & $4.108$ \\
& AVQZ & $1.384$ & $2.408$ & $4.125$ \\[0.1cm]
PC-NEVPT2(4,4) &6-31+G(d)& $1.409$ & $2.652$ & $4.120$ \\
& AVDZ & $1.384$ & $2.424$ & $4.084$ \\
& AVTZ & $1.382$ & $2.368$ & $4.083$ \\
& AVQZ & $1.387$ & $2.353$ & $4.091$ \\[0.1cm]
MRCI(4,4) &6-31+G(d)& $1.564$ & $3.802$ & $4.265$ \\
& AVDZ & $1.558$ & $3.670$ & $4.254$ \\
& AVTZ & $1.568$ & $3.678$ & $4.270$ \\
& AVQZ & $1.574$ & $3.681$ & $4.280$ \\[0.1cm]
SA2-CASSCF(12,12) &6-31+G(d)& $1.675$ & $3.924$ & $4.220$ \\
& AVDZ & $1.685$ & $3.856$ & $4.221$ \\
& AVTZ & $1.686$ & $3.844$ & $4.217$ \\
& AVQZ & $1.687$ & $3.846$ & $4.216$ \\[0.1cm]
CASPT2(12,12) &6-31+G(d)& $1.508$ & $3.407$ & $4.099$ \\
& AVDZ & $1.489$ & $3.256$ & $4.044$ \\
& AVTZ & $1.480$ & $3.183$ & $4.043$ \\
& AVQZ & $1.482$ & $3.163$ & $4.047$ \\[0.1cm]
XMS-CASPT2(12,12) &6-31+G(d)& && $4.111$ \\
& AVDZ & & & $4.056$ \\
& AVTZ & & & $4.059$ \\
& AVQZ & & & $4.065$ \\[0.1cm]
SC-NEVPT2(12,12) &6-31+G(d)& $1.522$ & $3.409$ & $4.130$ \\
& AVDZ & $1.511$ & $3.266$ & $4.093$ \\
& AVTZ & $1.501$ & $3.188$ & $4.086$ \\
& AVQZ & $1.503$ & $3.167$ & $4.088$ \\[0.1cm]
PC-NEVPT2(12,12) &6-31+G(d)& $1.487$ & $3.296$ & $4.103$ \\
& AVDZ & $1.472$ & $3.141$ & $4.064$ \\
& AVTZ & $1.462$ & $3.063$ & $4.056$ \\
& AVQZ & $1.464$ & $3.043$ & $4.059$ \\[0.1cm]
MRCI(12,12) &6-31+G(d)& & & $4.125$ \\[0.1cm]
CIPSI &6-31+G(d)& $1.486\pm 0.005$ & $3.348\pm 0.024$ & $4.084\pm 0.012$ \\
& AVDZ & $1.458\pm 0.009$ & $3.187\pm 0.035$ & $4.04\pm 0.04$ \\
& AVTZ & $1.461\pm 0.030$ & $3.142\pm 0.035$ & $4.03\pm 0.09$ \\
\end{tabular}
\end{ruledtabular}
\fnt[1]{Value obtained using CC4/AVDZ corrected by the difference between CCSDT/AVTZ and CCSDT/AVDZ.}
\fnt[2]{Value obtained using CCSDTQ/6-31+G(d) corrected by the difference between CC4/AVDZ and CC4/6-31+G(d).}
\fnt[3]{Value obtained using CCSDTQ/AVDZ corrected by the difference between CC4/AVTZ and CC4/AVDZ.}
\end{table*}
\end{squeezetable}
%%% %%% %%% %%%
%%% TABLE IV %%%
\begin{squeezetable}
\begin{table*}
\caption{
Standard vertical excitation energies (with respect to the singlet $\text{X}\,{}^1B_{1g}$ ground state) of the $1\,{}^3A_{2g}$, $2\,{}^1A_{1g}$, and $1\,{}^1B_{2g}$ states of CBD at the $D_{4h}$ square-planar equilibrium geometry of the $1\,{}^3A_{2g}$ state.
\label{tab:D4h}}
\begin{ruledtabular}
\begin{tabular}{llrrr}
& \mc{3}{r}{Excitation energies (eV)} \hspace{0.1cm}\\
\cline{3-5}
Method & Basis & $1\,{}^3A_{2g}$ & $2\,{}^1A_{1g}$ & $1\,{}^1B_{2g}$ \\
\hline
CCSD & 6-31+G(d) & $0.148$ & $1.788$ & \\
& AVDZ & $0.100$ & $1.650$ & \\
& AVTZ & $0.085$ & $1.600$ & \\
& AVQZ & $0.084$ & $1.588$ & \\[0.1cm]
CC3 & 6-31+G(d) & & $1.809$ & $2.836$ \\
& AVDZ & & $1.695$ & $2.646$ \\
& AVTZ & & $1.662$ & $2.720$ \\[0.1cm]
CCSDT & 6-31+G(d) & $0.210$ & $1.751$ & $2.565$ \\
& AVDZ & $0.165$ & $1.659$ & $2.450$ \\
& AVTZ & $0.149$ & $1.631$ & $2.537$ \\[0.1cm]
CC4 & 6-31+G(d) & & $1.604$ & $2.121$ \\
& AVDZ & & $1.539$ & $1.934$ \\
& AVTZ & & $\left[1.511 \right]$\fnm[1] &$\left[2.021 \right]$\fnm[1] \\[0.1cm]
CCSDTQ & 6-31+G(d) & $0.205$ & $1.593$ & $2.134$ \\
& AVDZ & & $\left[1.528 \right]$\fnm[2]&$\left[1.947\right]$\fnm[2] \\
& AVTZ & & $\left[1.500 \right]$\fnm[3]&$\left[2.034\right]$\fnm[3] \\ [0.1cm]
SA2-CASSCF(4,4) & 6-31+G(d) & $0.447$ & $2.257$ & $3.549$ \\
& AVDZ & $0.438$ & $2.240$ & $3.443$ \\
& AVTZ & $0.434$ & $2.234$ & $3.424$ \\
& AVQZ & $0.435$ & $2.235$ & $3.427$ \\[0.1cm]
CASPT2(4,4) & 6-31+G(d) & $0.176$ & $1.588$ & $1.899$ \\
& AVDZ & $0.137$ & $1.540$ & $1.708$ \\
& AVTZ & $0.128$ & $1.506$ & $1.635$ \\
& AVQZ & $0.128$ & $1.498$ & $1.612$ \\[0.1cm]
SC-NEVPT2(4,4) & 6-31+G(d) & $0.083$ & $1.520$ & $1.380$ \\
& AVDZ & $0.037$ & $1.465$ & $1.140$ \\
& AVTZ & $0.024$ & $1.428$ & $1.055$ \\
& AVQZ & $0.024$ & $1.420$ & $1.030$ \\[0.1cm]
PC-NEVPT2(4,4) & 6-31+G(d) & $0.085$ & $1.496$ & $1.329$ \\
& AVDZ & $0.039$ & $1.440$ & $1.088$ \\
& AVTZ & $0.026$ & $1.403$ & $1.003$ \\
& AVQZ & $0.026$ & $1.395$ & $0.977$ \\[0.1cm]
MRCI(4,4) & 6-31+G(d) & $0.297$ & $1.861$ & $2.571$ \\
& AVDZ & $0.273$ & $1.823$ & $2.419$ \\
& AVTZ & $0.271$ & $1.824$ & $2.415$ \\
& AVQZ & $0.273$ & $1.825$ & $2.413$ \\[0.1cm]
SA2-CASSCF(12,12) & 6-31+G(d) & $0.386$ & $1.974$ & $2.736$ \\
& AVDZ & $0.374$ & $1.947$ & $2.649$ \\
& AVTZ & $0.370$ & $1.943$ & $2.634$ \\
& AVQZ & $0.371$ & $1.945$ & $2.637$ \\[0.1cm]
CASPT2(12,12) & 6-31+G(d) & $0.235$ & $1.635$ & $2.170$ \\
& AVDZ & $0.203$ & $1.588$ & $2.015$ \\
& AVTZ & $0.183$ & $1.538$ & $1.926$ \\
& AVQZ & $0.179$ & $1.522$ & $1.898$ \\[0.1cm]
SC-NEVPT2(12,12) & 6-31+G(d) & $0.218$ & $1.644$ & $2.143$ \\
& AVDZ & $0.189$ & $1.600$ & $1.991$ \\
& AVTZ & $0.165$ & $1.546$ & $1.892$ \\
& AVQZ & $0.160$ & $1.529$ & $1.862$ \\[0.1cm]
PC-NEVPT2(12,12) & 6-31+G(d) & $0.189$ & $1.579$ & $2.020$ \\
& AVDZ & $0.156$ & $1.530$ & $1.854$ \\
& AVTZ & $0.131$ & $1.476$ & $1.756$ \\
& AVQZ & $0.126$ & $1.460$ & $1.727$ \\[0.1cm]
CIPSI & 6-31+G(d) & $0.2010\pm 0.0030$ & $1.602\pm 0.007$ & $2.13\pm 0.04$ \\
& AVDZ & $0.1570\pm 0.0030$ & $1.587\pm 0.005$ & $2.102\pm 0.027$ \\
& AVTZ & $0.169\pm 0.029$ & $1.63\pm 0.05$ & \\
\end{tabular}
\end{ruledtabular}
\fnt[1]{Value obtained using CC4/AVDZ corrected by the difference between CCSDT/AVTZ and CCSDT/AVDZ.}
\fnt[2]{Value obtained using CCSDTQ/6-31+G(d) corrected by the difference between CCSDT/AVDZ and CCSDT/6-31+G(d).}
\fnt[3]{Value obtained using CCSDTQ/AVDZ corrected by the difference between CCSDT/AVTZ and CCSDT/AVDZ.}
\end{table*}
\end{squeezetable}
%%% %%% %%% %%%
%%% TABLE V %%%
\begin{squeezetable}
\begin{table}
\caption{
Standard vertical excitation energies (with respect to the singlet $\text{X}\,{}^1B_{1g}$ ground state) of the $1\,{}^3A_{2g}$, $2\,{}^1A_{1g}$, and $1\,{}^1B_{2g}$ states of CBD at the $D_{4h}$ square-planar equilibrium geometry of the $1\,{}^3A_{2g}$ state.
\label{tab:sf_D4h}}
\begin{ruledtabular}
\begin{tabular}{llrrr}
& \mc{4}{r}{Excitation energies (eV)} \hspace{0.5cm}\\
\cline{3-5}
Method & Basis & $1\,{}^3A_{2g}$ & $2\,{}^1A_{1g}$ & $1\,{}^1B_{2g}$ \\
\hline
SF-CIS & 6-31+G(d) & $0.355$ & $2.742$ & $3.101$ \\
& AVDZ & $0.318$ & $2.593$ & $3.052$ \\
& AVTZ & $0.305$ & $2.576$ & $3.053$ \\
& AVQZ & $0.306$ & $2.577$ & $3.056$ \\[0.1cm]
SF-TD-B3LYP & 6-31+G(d) & $-0.016$ & $0.487$ & $0.542$ \\
& AVDZ & $-0.019$ & $0.477$ & $0.536$ \\
& AVTZ & $-0.020$ & $0.472$ & $0.533$ \\
& AVQZ & $-0.020$ & $0.473$ & $0.533$ \\[0.1cm]
SF-TD-PBE0 & 6-31+G(d) & $-0.012$ & $0.618$ & $0.689$ \\
& AVDZ & $-0.016$ & $0.602$ & $0.680$ \\
& AVTZ & $-0.019$ & $0.597$ & $0.677$ \\
& AVQZ & $-0.018$ & $0.597$ & $0.677$ \\[0.1cm]
SF-TD-BH\&HLYP& 6-31+G(d) & $0.064$ & $1.305$ & $1.458$ \\
& AVDZ & $0.051$ & $1.260$ & $1.437$ \\
& AVTZ & $0.045$ & $1.249$ & $1.431$ \\
& AVQZ & $0.046$ & $1.250$ & $1.432$ \\[0.1cm]
SF-TD-M06-2X & 6-31+G(d) & $0.102$ & $1.476$ & $1.640$ \\
& AVDZ & $0.086$ & $1.419$ & $1.611$ \\
& AVTZ & $0.078$ & $1.403$ & $1.602$ \\
& AVQZ & $0.079$ & $1.408$ & $1.607$ \\[0.1cm]
SF-TD-CAM-B3LYP & 6-31+G(d) & $0.021$ & $0.603$ & $0.672$ \\
& AVDZ & $0.012$ & $0.585$ & $0.666$ \\
& AVTZ & $0.010$ & $0.580$ & $0.664$ \\
& AVQZ & $0.010$ & $0.580$ & $0.664$ \\[0.1cm]
SF-TD-$\omega $B97X-V & 6-31+G(d) & $0.040$ & $0.600$ & $0.670$ \\
& AVDZ & $0.029$ & $0.576$ & $0.664$ \\
& AVTZ & $0.026$ & $0.572$ & $0.662$ \\
& AVQZ & $0.026$ & $0.572$ & $0.662$ \\[0.1cm]
SF-TD-M11 & 6-31+G(d) & $0.102$ & $1.236$ & $1.374$ \\
& AVDZ & $0.087$ & $1.196$ & $1.362$ \\
& AVTZ & $0.081$ & $1.188$ & $1.359$ \\
& AVQZ & $0.080$ & $1.185$ & $1.357$ \\[0.1cm]
SF-TD-LC-$\omega $PBE08 & 6-31+G(d) & $0.078$ & $0.593$ & $0.663$ \\
& AVDZ & $0.060$ & $0.563$ & $0.659$ \\
& AVTZ & $0.058$ & $0.561$ & $0.658$ \\
& AVQZ & $0.058$ & $0.561$ & $0.659$ \\[0.1cm]
SF-ADC(2)-s & 6-31+G(d) & $0.345$ & $1.760$ & $2.096$ \\
& AVDZ & $0.269$ & $1.656$ & $1.894$ \\
& AVTZ & $0.256$ & $1.612$ & $1.844$ \\[0.1cm]
SF-ADC(2)-x & 6-31+G(d) & $0.264$ & $1.181$ & $1.972$ \\
& AVDZ & $0.216$ & $1.107$ & $1.760$ \\
& AVTZ & $0.212$ & $1.091$ & $1.731$ \\[0.1cm]
SF-ADC(3) & 6-31+G(d) & $0.123$ & $1.650$ & $2.078$ \\
& AVDZ & $0.088$ & $1.571$ & $1.878$ \\
& AVTZ & $0.079$ & $1.575$ & $1.853$ \\
\end{tabular}
\end{ruledtabular}
\end{table}
\end{squeezetable}
%%% %%% %%% %%%
%================================================
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Conclusion}
\label{sec:conclusion}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%
\acknowledgements{
EM, AS, and PFL acknowledge funding from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme (Grant agreement No.~863481).}
%%%%%%%%%%%%%%%%%%%%%%%%
\bibliography{CBD}
\end{document}
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