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Manuscript/CBD.tex
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@ -211,7 +211,7 @@
\affiliation{\LCPQ}
\begin{abstract}
Write an abstract
The cyclobutadiene (CBD) molecule represents a playground for ground state and excited states methods. Indeed due to the high symmetry of the molecule, especially at the square geometry ($D_{4h}$), the ground state and the excited states of the CBD exhibit multiconfigurational character where single reference methods such as adiabatic time-dependent density functional theory (TD-DFT) or equation-of-motion coupled cluster (EOM-CC) show difficulty to describe them. In this work we provide an extensive study of the autoisomerization barrier (AB), where the rectangular ($D_{2h}$) and the square geometry ($D_{4h}$) are needed, and of the vertical excitations energies of the CBD molecule using a large range of methods and basis set. In order to tackle the problem of multiconfigurational character presents in the AB and in the vertical excitation energies selected configuration interaction (sCI) and multi reference (CASSCF,CASPT2,NEVPT2) calculations are performed. Moreover coupled cluster calculations such as CCSD, CC3, CCSDT, CC4 and CCSDTQ are added to the set of methods. To complete the study we provide spin-flip (SF) results, which are known to give correct description of multiconfigurational character states, in the TD-DFT framework where numerous exchange-correlation functionals are considered, we also add algebraic diagrammatic construction (ADC) calculations in the SF formalism where we use the ADC(2)-s, ADC(2)-x and ADC(3) schemes. A theoretical best estimate (TBE) is given for the AB and for each vertical energies.
\end{abstract}
\maketitle
@ -240,7 +240,7 @@ In the present work we investigate ${}^1A_g$, $1{}^3B_{1g}$, $1{}^1B_{1g}$, $2{}
\begin{figure}
\includegraphics[width=0.6\linewidth]{figure2.png}
\caption{Here comes the caption of the figure.}
\caption{Pictorial representation of the excited states of the CBD molecule and its properties under investigation. Black color represent singlet ground state (S) properties and red color triplet (T) properties.}
\label{fig:CBD}
\end{figure}
@ -272,25 +272,29 @@ State-averaged complete-active-space self-consistent field (SA-CASSCF) calculati
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Spin-Flip}
\label{sec:sf}
In both structures the CBD has a singlet ground state, for the spin-flip calculations we consider the lowest triplet state as reference. Spin-flip techniques are broadly accessible and here, among them, we explore equation-of-motion coupled-cluster singles and doubles (EOM-CCSD), configuration interaction singles (CIS), algebraic-diagrammatic construction (ADC) scheme and TD-DFT. The standard and extended spin-flip ADC(2) (SF-ADC(2)-s and SF-ADC(2)-x respectively) and SF-ADC(3) are performed using Q-CHEM 5.2.1. Spin-flip TD-DFT calculations are also performed using Q-CHEM 5.2.1. The B3LYP, PBE0 and BH\&HLYP Hybrid GGA functionals are considered, they contain 20\%, 25\%, 50\% of exact exchange and are labeled, respectively, as SF-BLYP, SF-B3LYP, SF-PBE0, SF-BH\&HLYP. We also have done spin-flip TD-DFT calculations using Range-Separated Hybrid (RSH) functionals as: CAM-B3LYP, LC-$\omega$PBE08 and $\omega$B97X-V. The main difference here between these RSH 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 the Hybrid meta-GGA functional M06-2X and the RSH meta-GGA functional M11. EOM-SF-CCSD and EOM-SF-CC(2,3) are also performed with Q-CHEM 5.2.1.
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 B3LYP, PBE0 and BH\&HLYP Hybrid GGA functionals are considered, they contain 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 Range-Separated Hybrid (RSH) functionals as: CAM-B3LYP, LC-$\omega$PBE08 and $\omega$B97X-V. The main difference here between these RSH 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 the Hybrid meta-GGA functional M06-2X and the RSH meta-GGA functional M11. EOM-SF-CCSD and EOM-SF-CC(2,3) are also performed with Q-CHEM 5.2.1.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Theoretical Best Estimate (TBE)}
For each studied quantity, i.e., the autoisomerisation barrier and the vertical excitations, we provide a theoretical best estimate (TBE). These TBEs are provided using extrapolated CCSDTQ/AVTZ values when possible and using NEVPT2(12,12) otherwise. The extrapolation of the CCSDTQ/AVTZ values is done using two schemes. The first one uses CC4 values for the extrapolation and proceed as follows
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. For each studied quantity, i.e., the autoisomerisation barrier and the vertical excitations, we provide a theoretical best estimate (TBE). These TBEs are provided using extrapolated CCSDTQ/AVTZ values when possible and using NEVPT2(12,12) otherwise. The extrapolation of the CCSDTQ/AVTZ values is done using two schemes. The first one uses CC4 values for the extrapolation and proceed as follows
\begin{equation}
\label{eq:AVTZ}
\Delta E^{\text{CCSDTQ}}_{\text{AVTZ}} = \Delta E^{\text{CCSDTQ}}_{\text{AVDZ}} + \left[ \Delta E^{\text{CC4}}_{\text{AVTZ}} - \Delta E^{\text{CC4}}_{\text{AVDZ}} \right]
\end{equation}
and we evaluate the CCSDTQ/AVTZ values as
and we evaluate the CCSDTQ/AVDZ values as
\begin{equation}
\label{eq:AVDZ}
\Delta E^{\text{CCSDTQ}}_{\text{AVDZ}} = \Delta E^{\text{CCSDTQ}}_{6-31\text{G}+\text{(d)}} + \left[ \Delta E^{\text{CC4}}_{\text{AVDZ}} - \Delta E^{\text{CC4}}_{6-31\text{G}+\text{(d)}} \right]
\end{equation}
when CC4/AVTZ values have been obtained. If it is not the case we extrapolate CC4/AVTZ values using the CCSDT ones as follows
\begin{equation}
\label{eq:CC4_AVTZ}
\Delta E^{\text{CC4}}_{\text{AVTZ}} = \Delta E^{\text{CC4}}_{\text{AVDZ}} + \left[ \Delta E^{\text{CCSDT}}_{\text{AVTZ}} - \Delta E^{\text{CCSDT}}_{\text{AVDZ}} \right]
\end{equation}
Then if the CC4 values have not been obtained then we use the second scheme which is the same as the first one but instead of the CC4 values we use CCSDT to extrapolate CCSDTQ. If none of the two schemes is possible then we use the NEVPT2(12,12) values. Note that a NEVPT2(12,12) value is used only once for one vertical excitation of the $D_{4h}$ structure.
@ -363,7 +367,7 @@ Then we compare results for multireference methods, we can see a difference of a
%SF-TD-BLYP & $23.57$ & $23.62$ & $24.23$ & $24.22$ \\
SF-TD-B3LYP & $18.59$ & $18.64$ & $19.34$ & $19.34$ \\
SF-TD-PBE0& $17.18$ & $17.19$ & $17.88$ & $17.88$ \\
SF-TD-BHHLYP & $11.90$ & $12.02$ & $12.72$ & $12.73$ \\
SF-TD-BH\&HLYP & $11.90$ & $12.02$ & $12.72$ & $12.73$ \\
SF-TD-M06-2X & $9.32$ & $9.62$ & $10.35$ & $10.37$ \\
SF-TD-CAM-B3LYP& $18.05$ & $18.10$ & $18.83$ & $18.83$ \\
SF-TD-$\omega$B97X-V & $18.26$ & $18.24$ & $18.94$ & $18.92$ \\
@ -395,13 +399,16 @@ CIPSI & $7.91\pm 0.21$ & $8.58\pm 0.14$ & & \\
\end{squeezetable}
%%% %%% %%% %%%
%%% FIGURE I %%%
%%% FIGURE II %%%
\begin{figure*}
\includegraphics[scale=0.5]{AB_AVTZ.pdf}
\caption{Autoisomerization barrier energy for the CBD molecule using the AVTZ basis. Purple histograms are for the SF-TD-DFT functionals, orange histograms are for the SF-ADC schemes, green histograms are for the multireference methods, blue histograms are for the CC methods and the black one is for the TBE.}
\caption{Autoisomerization barrier (AB) energies for the CBD molecule using the AVTZ basis. Purple histograms are for the SF-TD-DFT functionals, orange histograms are for the SF-ADC schemes, green histograms are for the multireference methods, blue histograms are for the CC methods and the black one is for the TBE.}
\label{fig:AB}
\end{figure*}
Figure \ref{fig:AB} shows the autoisomerization barrier (AB) energies for the CBD molecule for the various used methods. We see the large variations of the AB energy with the different DFT functionals with some of them giving an energy of almost 20 \kcalmol compared to the 8.93 \kcalmol of the TBE. Nevertheless, we have that some functionals, BH\&HLYP, M06-2X or M11, give comparable results to SF-ADC or to multireference methods. For SF-ADC and multireference methods we get small energy differences compared to the TBE value. Note that the SF-ADC(2)-x and SF-ADC(3) schemes do not improve the result and even increase the energy error to the TBE value. We can also notice that, as previously stated, CASSCF provide a larger energy error compared to CASPT2 and NEVPT2 due to the lack of dynamical correlation. Finally, CC methods show also good results compared to the TBE. %We see that CCSD presents a larger error and that taking into account the triple excitations improves the result.
%%% %%% %%% %%%
@ -410,7 +417,7 @@ CIPSI & $7.91\pm 0.21$ & $8.58\pm 0.14$ & & \\
%================================================
\subsection{Excited States}
\label{sec:states}
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.
%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.
\subsubsection{D2h geometry}
Table \ref{tab:sf_tddft_D2h} shows the results obtained for the vertical transition energies using spin-flip methods and Table \ref{tab:D2h} shows the results obtained with the standard methods. We discuss first the SF-TD-DFT values with hybrid functionals. For the B3LYP functional we can see that the energy differences for each state and throughout all bases are small with the largest one for the $1\,{}^3B_{1g}$ state with 0.012 eV. The same observation can be done for the PBE0 and BH\&HLYP functionals, we can also oberve that adding exact exchange to the functional (20\% of exact exchange for B3LYP, 25\% for PBE0 and 50\% for BH\&HLYP increase the energy difference between the $1\,{}^1B_{1g}$ and the $1\,{}^3B_{1g}$ states. Put another way, increasing the amount of exact exchange in the functional reduces the energy difference between the $1\,{}^3B_{1g}$ and the $\text{X}\,{}^1A_{g}$ states. We can also notice that the vertical energies of the different states do not vary in the same way when adding exact exchange, for instance the energy variation for the $1\,{}^3B_{1g}$ state from PBE0 to BH\&HLYP is around 0.1 eV whereas for the $1\,{}^1B_{1g}$ and the $2\,{}^1A_{1g}$ states this energy variation is about 0.4-0.5 eV and 0.34 eV respectively. For the RSH functionals we can not make the same observation, indeed we can see that for the CAM-B3LYP functional we have that the energy difference between the $1\,{}^1B_{1g}$ and the $1\,{}^3B_{1g}$ states is larger than for the $\omega$B97X-V and LC-$\omega $PBE08 functionals despite the fact that the latter ones have a bigger amount of exact exchange. However we can observe that we have a small energy difference for each state throughout the bases and for each RSH functional. The M06-2X functional is an Hybrid meta-GGA functional and contains 54\% of Hartree-Fock exchange, we can compare the energies with the BH\&HLYP functional and we can see that the energy differences are small with around 0.03-0.08 eV. We can notice that the upper bound of 0.08 eV in the energy differences is due to the $1\,{}^3B_{1g}$ state. The M11 vertical energies are close to the BH\&HLYP ones for the triplet and the first singlet excited states with 0.01-0.02 eV and 0.08-0.10 eV of energy difference, respectively. For the $2\,{}^1A_{1g}$ state the M11 energies are closer to the $\omega$B97X-V ones with 0.05-0.09 eV of energy difference.
@ -578,6 +585,17 @@ CIPSI &6-31+G(d)& $1.486\pm 0.005$ & $3.348\pm 0.024$ & $4.084\pm 0.012$ \\
\end{squeezetable}
%%% %%% %%% %%%
Figure \ref{fig:D2h} shows the vertical energies of the studied excited states described in Tables \ref{tab:sf_tddft_D2h} and \ref{tab:D2h}. We see that the SF-TD-DFT vertical energies are rather close to the TBE one for the doubly excited state $2\,{}^1A_{g}$.Indeed, as presented later in subsection \ref{sec:TBE}, SF methods are able to describe effectively states with strong multiconfigurational character. The same observation can be done for the SF-ADC values but with much better results for the two other states. Again we can notice that the SF-ADC(2)-x and SF-ADC(3) schemes do not improve the values. For the multireference methods we see that using the smallest active space do not provide a good description of the $1\,{}^1B_{1g} $ state and in the case of CASSCF(4,4), as previously said, we even have $1\,{}^1B_{1g} $ state above the $2\,{}^1A_{g}$ state. For the CC methods taking into account the quadruple excitations is necessary to have a good description of the doubly excited state $2\,{}^1A_{g}$.
%%% FIGURE III %%%
\begin{figure*}
%width=0.8\linewidth
\includegraphics[scale=0.5]{D2h.pdf}
\caption{Vertical energies of the $1\,{}^3B_{1g} $, $1\,{}^1B_{1g}$ and $2\,{}^1A_{g}$ states for the $D_{2h}$ geometry using the AVTZ basis. Purple lines are for the SF-TD-DFT functionals, orange lines are for the SF-ADC schemes, green lines are for the multireference methods, blue lines are for the CC methods and the black ones are for the TBE.}
\label{fig:D2h}
\end{figure*}
%%% %%% %%% %%%
\subsubsection{D4h geometry}
Table \ref{tab:sf_D4h} shows vertical energies obtained with spin-flip methods and Table \ref{tab:D4h} vertical energies obtained with standard methods. As for the previous geometry we start with the SF-TD-DFT results with hybrid functionals. We can first notice that for the B3LYP and the PBE0 functionals we have a wrong ordering of the first triplet state $1\,{}^3A_{2g}$ and the ground state $\text{X}\,{}^1B_{1g}$. We retrieve the good ordering with the BH\&HLYP functional, so adding exact exchange to the functional allows us to have the right ordering between these two states. For the B3LYP and the PBE0 functionals we have that the energy differences for each states and for all bases are small with 0.004-0.007 eV for the triplet state $1\,{}^3A_{2g}$. We have 0.015-0.021 eV of energy difference for the $2\,{}^1A_{1g}$ state through all bases, we can notice that this state is around 0.13 eV (considering all bases) higher with the PBE0 functional. We can make the same observation for the $1\,{}^1B_{2g}$ state for the B3LYP and the PBE0 functionals, indeed we have small energy differences for all bases and the state is around 0.14-0.15 eV for the PBE0 functional. For the BH\&HLYP functional the $2\,{}^1A_{1g}$ and $1\,{}^1B_{2g}$ states are higher in energy than for the two other hybrid functionals with about 0.65-0.69 eV higher for the $2\,{}^1A_{1g}$ state and 0.75-0.77 eV for the $1\,{}^1B_{2g}$ state compared to the PBE0 functional. Then, we have the RSH functionals CAM-B3LYP, $\omega$B97X-V and LC-$\omega$PBE08. For these functionals the vertical energies are similar for the $2\,{}^1A_{1g}$ and $1\,{}^1B_{2g}$ states with a maximum energy difference of 0.01-0.02 eV for the $2\,{}^1A_{1g}$ state and 0.005-0.009 eV for the $1\,{}^1B_{2g}$ state considering all bases. The maximum energy difference for the triplet state is larger with 0.047-0.057 eV for all bases. Note that the vertical energies obtained with the RSH functionals are close to the PBE0 ones except that we have the right ordering between the triplet state $1\,{}^3A_{2g}$ and the ground state $\text{X}\,{}^1B_{1g}$. The M06-2X results are closer to the BH\&HLYP ones but the M06-2X vertical energies are always higher than the BH\&HLYP ones. We can notice that the M06-2X energies for the $2\,{}^1A_{1g}$ state are close to the BH\&HLYP energies for the $1\,{}^1B_{2g}$ state. For these two states, when we compared the results obtained with the M06-2X and the BH\&HLYP functionals, we have an energy difference of 0.16-0.17 eV for the $2\,{}^1A_{1g}$ state and 0.17-0.18 eV for the $1\,{}^1B_{2g}$ state considering all bases. For the triplet state $1\,{}^3A_{2g}$ the energy differences are smaller with 0.03-0.04 eV for all bases. The M11 vertical energies are very close to the M06-2X ones for the triplet state with a maximum energy difference of 0.003 eV considering all bases, and are closer to the BH\&HLYP results for the two other states with 0.06-0.07 eV and 0.07-0.08 eV of energy difference for the $2\,{}^1A_{1g}$ and $1\,{}^1B_{2g}$ states, respectively. Then we discuss the various ADC scheme results, note that we were not able to obtain the vertical energies with the AVQZ basis due to computational resources. For the ADC(2)-s scheme we can see that the energy difference for the triplet state are smaller than for the two other states, indeed we have an energy difference of 0.09 eV for the triplet state whereas we have 0.15 eV and 0.25 eV for the $2\,{}^1A_{1g}$ and $1\,{}^1B_{2g}$ states, again when considering all bases. The energy difference for each state and through the bases are similar for the two other ADC schemes. We can notice a large variation of the vertical energies for the $2\,{}^1A_{1g}$ state between ADC(2)-s and ADC(2)-x with around 0.52-0.58 eV through all bases. The ADC(3) vertical energies are very similar to the ADC(2) ones for the $1\,{}^1B_{2g}$ state with an energy difference of 0.01-0.02 eV for all bases, whereas we have an energy difference of 0.04-0.11 eV and 0.17-0.22 eV for the $2\,{}^1A_{1g}$ and $1\,{}^1B_{2g}$ states, respectively.
@ -586,7 +604,6 @@ Now we look at Table \ref{tab:D4h} for vertical energies obtained with standard
%For the CCSD method we cannot obtain the vertical energy for the $1\,{}^1B_{2g}$ state, we can notice a small energy variation through the bases for the triplet state with around 0.06 eV. The energy variation is larger for the $2\,{}^1A_{1g}$ state with 0.20 eV.
Again, we have computed the vertical energies at the CCSDT and CCSDTQ level as well as their approximation the CC3 and CC4 methods, respectively. For the CC3 method we do not have the vertical energies for the triplet state $1\,{}^3A_{2g}$. Considering all bases for the $2\,{}^1A_{1g}$ and $1\,{}^1B_{2g}$ states we have an energy difference of about 0.15 eV and 0.12 eV, respectively. The CCSDT energies are close to the CC3 ones for the $2\,{}^1A_{1g}$ state with an energy difference of around 0.03-0.06 eV considering all bases. For the $1\,{}^1B_{2g}$ state the energy difference between the CC3 and the CCSDT values is larger with 0.18-0.27 eV. We can make a similar observation between the CC4 and the CCSDTQ values, for the $2\,{}^1A_{1g}$ state we have an energy difference of about 0.01 eV and this time we have smaller energy difference for the $1\,{}^1B_{2g}$ with 0.01 eV. Then we discuss the multireference results and this time we were able to reach the AVQZ basis. Again we consider two different active spaces, four electrons in four orbitals and twelve electrons in twelve orbitals. If we compare the CASSCF(4,4) and CASPT2(4,4) values we can see large energy differences for the $2\,{}^1A_{1g}$ and $1\,{}^1B_{2g}$ states, for the $2\,{}^1A_{1g}$ state we have an energy difference of about 0.67-0.74 eV and 1.65-1.81 eV for the $1\,{}^1B_{2g}$ state. The energy difference is smaller for the triplet state with 0.27-0.31 eV, we can notice that all the CASSCF(4,4) vertical energies are higher than the CASPT2(4,4) ones. Then we have the NEVPT2(4,4) methods (SC-NEVPT2(4,4) and PC-NEVPT2(4,4)) for which the vertical energies are quite similar, however we can notice that we have a different ordering of the $2\,{}^1A_{1g}$ and $1\,{}^1B_{2g}$ states with $1\,{}^1B_{2g}$ higher in energy than $2\,{}^1A_{1g}$ for the two NEVPT2(4,4) methods. Then we have the results for the same methods but with a larger active space. For the CASSCF(12,12) we have similar values for the triplet state with energy difference of about 0.06 eV for all bases but larger energy difference for the $2\,{}^1A_{1g}$ state with around 0.28-0.29 eV and 0.79-0.81 eV for the $1\,{}^1B_{2g}$ state. Note that from CASSCF(4,4) to CASSCF(12,12) every vertical energy is smaller. We can make the contrary observation for the CASPT2 method where from CASPT2(4,4) to CASPT2(12,12) every vertical energy is larger. For the CASPT2(12,12) method we have similar values for the triplet state and for the $2\,{}^1A_{1g}$ state, considering all bases, with an energy difference of around 0.05-0.06 eV and 0.02-0.05 eV respectively. The energy difference is larger for the $1\,{}^1B_{2g}$ state with about 0.27-0.29 eV. Next we have the NEVPT2 methods, the first observation that we can make is that by increasing the size of the active space we retrieve the right ordering of the $2\,{}^1A_{1g}$ and $1\,{}^1B_{2g}$ states. Again, every vertical energy is higher in the NEVPT2(12,12) case than for the NEVPT2(4,4) one.
%%% TABLE VI %%%
\begin{squeezetable}
\begin{table}
@ -652,7 +669,7 @@ SF-ADC(3) & 6-31+G(d) & $0.123$ & $1.650$ & $2.078$ \\
%%% %%% %%% %%%
%%% TABLE V %%%
%%% TABLE V %%%
\begin{squeezetable}
\begin{table*}
\caption{
@ -731,8 +748,19 @@ CIPSI & 6-31+G(d) & $0.2010\pm 0.0030$ & $1.602\pm 0.007$ & $2.13\pm 0.04$ \\
\end{squeezetable}
%%% %%% %%% %%%
\subsubsection{TBE}
Figure \ref{fig:D4h} shows the vertical energies of the studied excited states described in Tables \ref{tab:sf_D4h} and \ref{tab:D4h}. We see that all methods exhibit larger variations of the vertical energies due to the high symmetry of the $D_{4h}$ structure. For the triplet state most of the methods used are able to give a vertical energy close to the TBE one, we can although see that CASSCF, with the two different active spaces, show larger energy error than most of the DFT functionals used. Then for the, strongly multiconfigurational character, $2\,{}^1A_{1g}$ state we have a good description by the CC and multireference methods with the largest active space, except for CASSCF. The SF-ADC(2) and SF-ADC(3) schemes are also able to provide a good description of the $2\,{}^1A_{1g}$ state and even for the $1\,{}^1B_{2g} $ we see that SF-ADC(2)-x give a worst result than SF-ADC(2). Multiconfigurational methods with the smallest active space do not demonstrate a good description of the two singlet excited states especially for the CASSCF method and for the the NEVPT2 methods that give the vertical energy of the $1\,{}^1B_{2g} $ state below the $2\,{}^1A_{1g}$ one. Note that CASPT2 improve a lot the description of all the states compared to CASSCF. The SF-TD-DFT are not able to describe the two singlet excited states.
%%% FIGURE IV %%%
\begin{figure*}
%width=0.8\linewidth
\includegraphics[scale=0.5]{D4h.pdf}
\caption{Vertical energies of the $1\,{}^3A_{2g} $, $2\,{}^1A_{1g}$ and $1\,{}^1B_{2g} $ states for the $D_{4h}$ geometry using the AVTZ basis. Purple lines are for the SF-TD-DFT functionals, orange lines are for the SF-ADC schemes, green lines are for the multireference methods, blue lines are for the CC methods and the black ones are for the TBE.}
\label{fig:D4h}
\end{figure*}
%%% %%% %%% %%%
\subsubsection{TBE}
\label{sec:TBE}
Table \ref{tab:TBE} shows the energy differences, for the autoisomerization barrier (AB) and the different states considered, between the various methods treated and the Theoretical Best Estimate (TBE) at the AVTZ level for the AB and the states. First, we look at the AB energy difference. SF-TD-DFT shows large variations of the energy with errors of 1.42-10.81 \kcalmol compared to the TBE value. SF-ADC schemes provide smaller errors with 0.30-1.44 \kcalmol where we have that the SF-ADC(2)-x gives a worse error than the SF-ADC(2)-s method. CC methods also give small energy differences with 0.11-1.05 \kcalmol and where the CC4 provides an energy very close to the TBE one.
Then we look at the vertical energy errors for the $(D_{2h})$ structure. First we consider the $1\,{}^3B_{1g} $ state and we look at the SF-TD-DFT results. We see that increasing the amount of exact exchange in the functional give closer results to the TBE, indeed we have 0.24 and 0.22 eV of errors for the B3LYP and the PBE0 functionals, respectively whereas we have an error of 0.08 eV for the BH\&HLYP functional. For the other functionals we have errors of 0.10-0.43 eV, note that for this state the M06-2X functional gives the same result than the TBE. We can also notice that all the functionals considered overestimate the vertical energies. The ADC schemes give closer energies with errors of 0.04-0.08 eV, note that ADC(2)-x does not improve the result compared to ADC(2)-s and that ADC(3) understimate the vertical energy whereas ADC(2)-s and ADC(2)-x overestimate the vertical energy. The CC3 and CCSDT results provide energy errors of 0.05-0.06 eV respectively. Then we go through the multireference methods with the two different active spaces, four electrons in four orbitals and twelve electrons in twelve orbitals. For the smaller active space we have errors of 0.05-0.21 eV, the largest error comes from CASSCF(4,4) which is improved by CASPT2(4,4) that gives the smaller error. Then for the largest active space multireference methods provide energy errors of 0.02-0.22 eV with again the largest error coming from CASSCF(12,12) which is again improved by CASPT2(12,12) gives the smaller error.
@ -801,20 +829,6 @@ PC-NEVPT2(12,12) & & $0.000$ & $-0.062$ & $-0.093$ & $-0.013$ & $-0.024$ & $-0.
%%% %%% %%% %%%
\begin{figure*}
%width=0.8\linewidth
\includegraphics[scale=0.5]{D2h.pdf}
\caption{Vertical energies of the $1\,{}^3B_{1g} $, $1\,{}^1B_{1g} $ and $2\,{}^1A_{g} $ states for the $D_{2h}$ geometry using the AVTZ basis. Purple lines are for the SF-TD-DFT functionals, orange lines are for the SF-ADC schemes, green lines are for the multireference methods, blue lines are for the CC methods and the black ones are for the TBE.}
\label{fig:D2h}
\end{figure*}
\begin{figure*}
%width=0.8\linewidth
\includegraphics[scale=0.5]{D4h.pdf}
\caption{Vertical energies of the $1\,{}^3A_{2g} $, $2\,{}^1A_{1g} $ and $1\,{}^1B_{2g} $ states for the $D_{2h}$ geometry using the AVTZ basis. Purple lines are for the SF-TD-DFT functionals, orange lines are for the SF-ADC schemes, green lines are for the multireference methods, blue lines are for the CC methods and the black ones are for the TBE.}
\label{fig:D4h}
\end{figure*}
%================================================
@ -823,7 +837,11 @@ PC-NEVPT2(12,12) & & $0.000$ & $-0.062$ & $-0.093$ & $-0.013$ & $-0.024$ & $-0.
\label{sec:conclusion}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
We have considered the automerization barrier (AB) energy and the vertical energies of the cyclobutadiene (CBD) molecule in the square ($D_{4h}$) and rectangular ($D_{2h}$) geometries. For the AB and vertical energies we have defined theoretical best estimates (TBEs) by using the CCSDTQ/AVTZ values when we were able to obtain them. Otherwise we got the CCSDTQ/AVTZ values by correcting the CCSDTQ/AVDZ values by the difference between CC4/AVTZ and CC4/AVDZ (Eq.~\eqref{eq:AVTZ}) and we obtain the CCSDTQ/AVDZ values by correcting the CCSDTQ/6-31G+(d) values by the difference between CC4/AVDZ and CC4/6-31G+(d) (Eq.~\eqref{eq:AVDZ}). When the CC4/AVTZ values were not obtained we corrected the CC4/AVDZ values by the difference between CCSDT/AVTZ and CCSDT/AVDZ to obtain them (Eq.~\eqref{eq:CC4_AVTZ}). If the CC4 values have not been obtained then we used the same scheme that we just described but by using the CCSDT values. If neither the CC4 and CCSDTQ values were not available then we used the NEVPT2(12,12)/AVTZ values.
In order to provide a benchmark of the AB and vertical energies we used coupled-cluster (CC) methods with doubles (CCSD), with triples (CCSDT and CC3) and with quadruples (CCSTQ and CC4). Due to the presence of multiconfigurational states we used multireference methods (CASSCF, CASPT2 and NEVPT2) with two active spaces ((4,4) and (12,12)). We also used spin-flip (SF-) within two frameworks, in TD-DFT with various global and range-separated hybrids functionals, and in ADC with the ADC(2)-s, ADC(2)-x and ADC(3) schemes. The CC methods provide good results for the AB and vertical energies, however in the case of multiconfigurational states CC with only triples is not sufficient and we have to include the quadruples to correctly describe these states. Multiconfigurational methods also provide very solid results for the largest active space with second order correction (CASPT2 and NEVPT2).
With SF-TD-DFT the quality of the results are, of course, dependent on the functional but for the doubly excited states we have solid results. In SF-ADC we have very good results compared to the TBEs even for the doubly excited states, nevertheless the ADC(2)-x scheme give almost systematically worse results than the ADC(2)-s ones and using the ADC(3) scheme does not always provide better values.