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@ -224,7 +224,7 @@ Write an abstract
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. 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}. 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. 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 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. The energy of this barrier 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}. 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}.
@ -236,7 +236,7 @@ An alternative to multiconfigurational and CC methods is the use of selected CI
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. 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. 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 $D_{4h}$ geometries, respectively. 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 studied which is the autoisomerization barrier (Subsection \ref{sec:auto}) and then we study the excited states (Subsection \ref{sec:states}) of the CBD molecule for both geometries.
\begin{figure} \begin{figure}
\includegraphics[width=0.6\linewidth]{figure2.png} \includegraphics[width=0.6\linewidth]{figure2.png}
@ -283,26 +283,26 @@ In both structures the CBD has a singlet ground state, for the spin-flip calcula
\section{Results and discussion} \section{Results and discussion}
\label{sec:res} \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} \subsection{Autoisomerization barrier}
\label{sec:auto} \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. The results for the calculation of the automerization barrier energy with and without spin-flip methods are shown in Table \ref{tab:auto_standard}. Two types of methods are used with spin-flip, SF-TD-DFT with several functionals and SF-ADC with the variants SF-ADC(2)-s, SF-ADC(2)-x and SF-ADC(3). First, one can see that there is large variations of the energy between the different functionals. Indeed if we look at the hybrid functionals B3LYP and BH\&HLYP we can see that there is a difference of around 7 \kcalmol throught all the bases, the difference in energy between the B3LYP and PBE0 functionals is much smaller with around 1.5 \kcalmol throught all the bases. We find a similar behaviour regarding the RSH functionals, we find a difference of about 8.5-9 \kcalmol between the M06-2X and the CAM-B3LYP functionals for all bases. The results between the CAM-B3LYP and the $\omega$B97X-V are very close in energy with a difference of around 0.15-0.25 \kcalmol . The energy difference between the M11 and the M06-2X functionals is larger with 0.7-0.8 \kcalmol for the AVXZ bases and with 1.79 \kcalmol for the 6-31+G(d) basis. For the SF-ADC methods the energy differences are smaller with 1.76-2 \kcalmol between the ADC(2)-s and the ADC(2)-x schemes, 0.94-1.61 \kcalmol between the ADC(2)-s and the ADC(3) schemes and 0.39-0.82 \kcalmol between the ADC(2)-x and the ADC(3) schemes.
Then we compare results for multireference methods, we can see a difference of about 2.91-3.22 \kcalmol through all the bases between the CASSCF(12,12) and the CASPT2(12,12) methods. These differences can be explained by the well known lack of dynamical correlation for the CASSCF method that is typically important for close-shell molecules which is the case for the ground states of the rectangular and square geometries of the CBD molecule. The results between the CASPT2(12,12) and the NEVPT2(12,12) are much closer with an energy difference of around 0.12-0.23 \kcalmol for all the bases. Finally the last results shown in Table \ref{tab:auto_standard} are the CC ones, for the autoisomerization barrier energy we considered the CCSD, CCSDT, CCSDTQ methods and the approximations of CCDT and of CCSDTQ, the CC3 and the CC4 methods, respectively. We can see that when we go to a larger basis the energy barrier increase meaning that the energy differen
%%% TABLE VI %%% %%% TABLE I %%%
\begin{squeezetable} \begin{squeezetable}
\begin{table*} \begin{table*}
\caption{Autoisomerization barrier in \kcalmol.} \caption{Autoisomerization barrier energy in \kcalmol.}
\label{tab:auto_standard} \label{tab:auto_standard}
\begin{ruledtabular} \begin{ruledtabular}
\begin{tabular}{llrrrr} \begin{tabular}{llrrrr}
Method & 6-31+G(d) & AVDZ& AVTZ & AVQZ\\ Method & 6-31+G(d) & AVDZ & AVTZ & AVQZ\\
\hline \hline
SF-CIS & $2.64$ & $2.82$ & $3.43$ & $3.43$ \\ %SF-CIS & $2.64$ & $2.82$ & $3.43$ & $3.43$ \\
SF-TD-BLYP & $23.57$ & $23.62$ & $24.23$ & $24.22$ \\ %SF-TD-BLYP & $23.57$ & $23.62$ & $24.23$ & $24.22$ \\
SF-TD-B3LYP & $18.84$ & $18.93$ & $19.57$ & $19.57$ \\ SF-TD-B3LYP & $18.84$ & $18.93$ & $19.57$ & $19.57$ \\
SF-TD-PBE0 & $17.31$ & $17.36$ & $18.01$ & $18.00$ \\ 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-BH\&HLYP & $11.90$ & $12.07$ & $12.73$ & $12.73$ \\
@ -310,9 +310,9 @@ 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-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-$\omega$B97X-V & $18.46$ & $18.48$ & $19.14$ & $19.12$ \\
SF-TD-M11 & $11.13$ & $10.38$ & $11.28$ & $11.19$ \\ SF-TD-M11 & $11.13$ & $10.38$ & $11.28$ & $11.19$ \\
SF-ADC2-s & $6.69$ & $7.15$ & $8.64$ & $8.85$ \\ SF-ADC(2)-s & $6.69$ & $7.15$ & $8.64$ & $8.85$ \\
SF-ADC2-x & $8.66$ & $9.15$ & $10.40$ & \\ SF-ADC(2)-x & $8.66$ & $9.15$ & $10.40$ & \\
SF-ADC3 & $8.06$ & $8.76$ & $9.58$ & \\ SF-ADC(3) & $8.06$ & $8.76$ & $9.58$ & \\
CASSCF(12,12) & $10.19$ & $10.75$ & $11.59$ & $11.62$ \\ CASSCF(12,12) & $10.19$ & $10.75$ & $11.59$ & $11.62$ \\
CASPT2(12,12) & $7.24$ & $7.53$ & $8.51$ & $8.71$ \\ CASPT2(12,12) & $7.24$ & $7.53$ & $8.51$ & $8.71$ \\
NEVPT2(12,12) & $7.12$ & $7.33$ & $8.28$ & $8.49$ \\ NEVPT2(12,12) & $7.12$ & $7.33$ & $8.28$ & $8.49$ \\
@ -371,7 +371,9 @@ CCSDTQ & $7.51$ & $\left[ 7.89\right]$\fnm[4]& $\left[ 8.93\right]$\fnm[5]& $\le
\subsection{Excited States} \subsection{Excited States}
\label{sec:states} \label{sec:states}
%%% TABLE I %%% \subsubsection{D2h geometry}
%%% TABLE II %%%
\begin{squeezetable} \begin{squeezetable}
\begin{table} \begin{table}
\caption{ \caption{
@ -403,7 +405,7 @@ SF-TD-M06-2X & 6-31+G(d) & $1.477$ & $2.835$ & $4.378$ \\
& AVDZ & $1.467$ & $2.785$ & $4.360$ \\ & AVDZ & $1.467$ & $2.785$ & $4.360$ \\
& AVTZ & $1.462$ & $2.771$ & $4.357$ \\ & AVTZ & $1.462$ & $2.771$ & $4.357$ \\
& AVQZ & $1.458$ & $2.771$ & $4.352$ \\[0.1cm] & AVQZ & $1.458$ & $2.771$ & $4.352$ \\[0.1cm]
SF-TD-CAM-B3LYP & 6-31+G(d) & $1.750$ & $2.337$ & $3.315$ \\ SF-TD-CAM-B3LYP & 6-31+G(d) & $1.750$ & $2.337$ & $4.140$ \\
& AVDZ & $1.745$ & $2.323$ & $4.140$ \\ & AVDZ & $1.745$ & $2.323$ & $4.140$ \\
& AVTZ & $1.742$ & $2.318$ & $4.138$ \\ & AVTZ & $1.742$ & $2.318$ & $4.138$ \\
& AVQZ & $1.743$ & $2.319$ & $4.138$ \\[0.1cm] & AVQZ & $1.743$ & $2.319$ & $4.138$ \\[0.1cm]
@ -418,30 +420,11 @@ SF-TD-LC-$\omega $PBE08 & 6-31+G(d) & $1.917$ & $2.445$ & $4.353$ \\
& AVDZ & $1.897$ & $2.415$ & $4.346$ \\ & AVDZ & $1.897$ & $2.415$ & $4.346$ \\
& AVTZ & $1.897$ & $2.415$ & $4.348$ \\ & AVTZ & $1.897$ & $2.415$ & $4.348$ \\
& AVQZ & $1.897$ & $2.415$ & $4.348$ \\[0.1cm] & AVQZ & $1.897$ & $2.415$ & $4.348$ \\[0.1cm]
%SF-CIS & 6-31+G(d) & $1.514$ & $3.854$ & $5.379$ \\
\end{tabular} %& AVDZ & $1.487$ & $3.721$ & $5.348$ \\
\end{ruledtabular} %& AVTZ & $1.472$ & $3.701$ & $5.342$ \\
\end{table} %& AVQZ & $1.471$ & $3.702$ & $5.342$ \\[0.1cm]
\end{squeezetable} SF-ADC(2)-s & 6-31+G(d) & $1.577$ & $3.303$ & $4.196$ \\
%%% %%% %%% %%%
%%% 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 & $1.513$ & $3.116$ & $4.114$ \\ & AVDZ & $1.513$ & $3.116$ & $4.114$ \\
& AVTZ & $1.531$ & $3.099$ & $4.131$ \\ & AVTZ & $1.531$ & $3.099$ & $4.131$ \\
& AVQZ & $1.544$ & $3.101$ & $4.140$ \\[0.1cm] & AVQZ & $1.544$ & $3.101$ & $4.140$ \\[0.1cm]
@ -456,6 +439,7 @@ SF-ADC(3) & 6-31+G(d) & $1.435$ & $3.352$ & $4.242$ \\
& AVTZ & $1.609$ & $3.293$ & $4.245$ \\[0.1cm] & AVTZ & $1.609$ & $3.293$ & $4.245$ \\[0.1cm]
SF-EOM-CC(2,3) & 6-31+G(d) & $1.490$ & $3.333$ & $4.061$ \\ SF-EOM-CC(2,3) & 6-31+G(d) & $1.490$ & $3.333$ & $4.061$ \\
& AVDZ & $1.464$ & $3.156$ & $4.027$ \\ & AVDZ & $1.464$ & $3.156$ & $4.027$ \\
\end{tabular} \end{tabular}
\end{ruledtabular} \end{ruledtabular}
\end{table} \end{table}
@ -463,6 +447,43 @@ SF-EOM-CC(2,3) & 6-31+G(d) & $1.490$ & $3.333$ & $4.061$ \\
%%% %%% %%% %%% %%% %%% %%% %%%
%%% TABLE III %%% %%% TABLE III %%%
%\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 IV %%%
\begin{squeezetable} \begin{squeezetable}
\begin{table*} \begin{table*}
\caption{ \caption{
@ -544,7 +565,58 @@ CIPSI &6-31+G(d)& $1.486\pm 0.005$ & $3.348\pm 0.024$ & $4.084\pm 0.012$ \\
\end{squeezetable} \end{squeezetable}
%%% %%% %%% %%% %%% %%% %%% %%%
%%% TABLE IV %%% %%% TABLE I %%%
\begin{squeezetable}
\begin{table*}
\caption{}
\label{tab:TBE_D2h}
\begin{ruledtabular}
\begin{tabular}{llrrr}
Method & $1\,{}^3B_{1g}$ & $1\,{}^1B_{1g}$ & $2\,{}^1A_{g}$ & TBE (AVTZ) \\
\hline
SF-TD-B3LYP & & & & \\
SF-TD-PBE0 & & & & \\
SF-TD-BH\&HLYP & & & & \\
SF-TD-M06-2X & & & & \\
SF-TD-CAM-B3LYP & & & & \\
SF-TD-$\omega$B97X-V & & & & \\
SF-TD-M11 & & & & \\
SF-ADC2-s & & & & \\
SF-ADC2-x & & & & \\
SF-ADC3 & & & & \\
CCSD & & & & \\
CC3 & & & & \\
CCSDT & & & & \\
CC4 & & & & \\
CCSDTQ & & & & \\
SA2-CASSCF(4,4) & & & & \\
CASPT2(4,4) & & & & \\
XMS-CASPT2(4,4) & & & & \\
SC-NEVPT2(4,4) & & & & \\
PC-NEVPT2(4,4) & & & & \\
MRCI(4,4) & & & & \\
SA2-CASSCF(12,12) & & & & \\
CASPT2(12,12) & & & & \\
XMS-CASPT2(12,12) & & & & \\
SC-NEVPT2(12,12) & & & & \\
PC-NEVPT2(12,12) & & & & \\
MRCI(12,12) & & & & \\
CIPSI & & & & \\
\end{tabular}
\end{ruledtabular}
\end{table*}
\end{squeezetable}
%%% %%% %%% %%%
\subsubsection{D4h geometry}
%%% TABLE V %%%
\begin{squeezetable} \begin{squeezetable}
\begin{table*} \begin{table*}
\caption{ \caption{
@ -620,7 +692,7 @@ CIPSI & 6-31+G(d) & $0.2010\pm 0.0030$ & $1.602\pm 0.007$ & $2.13\pm 0.04$ \\
\end{squeezetable} \end{squeezetable}
%%% %%% %%% %%% %%% %%% %%% %%%
%%% TABLE V %%% %%% TABLE VI %%%
\begin{squeezetable} \begin{squeezetable}
\begin{table} \begin{table}
\caption{ \caption{
@ -632,10 +704,10 @@ CIPSI & 6-31+G(d) & $0.2010\pm 0.0030$ & $1.602\pm 0.007$ & $2.13\pm 0.04$ \\
\cline{3-5} \cline{3-5}
Method & Basis & $1\,{}^3A_{2g}$ & $2\,{}^1A_{1g}$ & $1\,{}^1B_{2g}$ \\ Method & Basis & $1\,{}^3A_{2g}$ & $2\,{}^1A_{1g}$ & $1\,{}^1B_{2g}$ \\
\hline \hline
SF-CIS & 6-31+G(d) & $0.355$ & $2.742$ & $3.101$ \\ %SF-CIS & 6-31+G(d) & $0.355$ & $2.742$ & $3.101$ \\
& AVDZ & $0.318$ & $2.593$ & $3.052$ \\ %& AVDZ & $0.318$ & $2.593$ & $3.052$ \\
& AVTZ & $0.305$ & $2.576$ & $3.053$ \\ %& AVTZ & $0.305$ & $2.576$ & $3.053$ \\
& AVQZ & $0.306$ & $2.577$ & $3.056$ \\[0.1cm] %& AVQZ & $0.306$ & $2.577$ & $3.056$ \\[0.1cm]
SF-TD-B3LYP & 6-31+G(d) & $-0.016$ & $0.487$ & $0.542$ \\ SF-TD-B3LYP & 6-31+G(d) & $-0.016$ & $0.487$ & $0.542$ \\
& AVDZ & $-0.019$ & $0.477$ & $0.536$ \\ & AVDZ & $-0.019$ & $0.477$ & $0.536$ \\
& AVTZ & $-0.020$ & $0.472$ & $0.533$ \\ & AVTZ & $-0.020$ & $0.472$ & $0.533$ \\
@ -684,13 +756,57 @@ SF-ADC(3) & 6-31+G(d) & $0.123$ & $1.650$ & $2.078$ \\
\end{squeezetable} \end{squeezetable}
%%% %%% %%% %%% %%% %%% %%% %%%
%%% TABLE I %%%
\begin{squeezetable}
\begin{table*}
\caption{}
\label{tab:TBE_D4h}
\begin{ruledtabular}
\begin{tabular}{llrrr}
Method & $1\,{}^3A_{2g}$ & $2\,{}^1A_{1g}$ & $1\,{}^1B_{2g}$ & TBE (AVTZ) \\
\hline
SF-TD-B3LYP & & & & \\
SF-TD-PBE0 & & & & \\
SF-TD-BH\&HLYP & & & & \\
SF-TD-M06-2X & & & & \\
SF-TD-CAM-B3LYP & & & & \\
SF-TD-$\omega$B97X-V & & & & \\
SF-TD-M11 & & & & \\
SF-ADC2-s & & & & \\
SF-ADC2-x & & & & \\
SF-ADC3 & & & & \\
CCSD & & & & \\
CC3 & & & & \\
CCSDT & & & & \\
CC4 & & & & \\
CCSDTQ & & & & \\
SA2-CASSCF(4,4) & & & & \\
CASPT2(4,4) & & & & \\
XMS-CASPT2(4,4) & & & & \\
SC-NEVPT2(4,4) & & & & \\
PC-NEVPT2(4,4) & & & & \\
MRCI(4,4) & & & & \\
SA2-CASSCF(12,12) & & & & \\
CASPT2(12,12) & & & & \\
XMS-CASPT2(12,12) & & & & \\
SC-NEVPT2(12,12) & & & & \\
PC-NEVPT2(12,12) & & & & \\
MRCI(12,12) & & & & \\
CIPSI & & & & \\
\end{tabular}
\end{ruledtabular}
\end{table*}
\end{squeezetable}
%%% %%% %%% %%%
%================================================ %================================================
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Conclusion} \section{Conclusion}
\label{sec:conclusion} \label{sec:conclusion}