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Pierre-Francois Loos 2022-06-10 11:12:53 +02:00
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Manuscript/CBD.tex
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@ -124,11 +124,9 @@ Interestingly, the {\twoAg} and {\Aoneg} states have a strong contribution from
In order to tackle the problem of multi-configurational character and double excitations, we have explored several approaches. In order to tackle the problem of multi-configurational character and double excitations, we have explored several approaches.
The most evident way is to rely on \alert{multi-reference} methods, which are naturally designed to address such scenarios. The most evident way is to rely on \alert{multi-reference} methods, which are naturally designed to address such scenarios.
Among these methods, one can mention the complete-active-space self-consistent field (CASSCF) method, \cite{Roos_1996} its second-order perturbatively-corrected variant (CASPT2) \cite{Andersson_1990,Andersson_1992,Roos_1995a} and the second-order $n$-electron valence state perturbation theory (NEVPT2) formalism. \cite{Angeli_2001,Angeli_2001a,Angeli_2002} Among these methods, one can mention the complete-active-space self-consistent field (CASSCF) method, \cite{Roos_1996} its second-order perturbatively-corrected variant (CASPT2) \cite{Andersson_1990,Andersson_1992,Roos_1995a} and the second-order $n$-electron valence state perturbation theory (NEVPT2) formalism. \cite{Angeli_2001,Angeli_2001a,Angeli_2002}
%The exponential scaling of the computational cost (with respect to the size of the active space) associated with these methods is the principal limitation to their applicability to large molecules.
Another way to deal with double excitations and multi-reference situations is to use high level truncation of the EOM formalism \cite{Rowe_1968,Stanton_1993} of CC theory. \cite{Kucharski_1991,Kallay_2003,Kallay_2004,Hirata_2000,Hirata_2004} Another way to deal with double excitations and multi-reference situations is to use high level truncation of the EOM formalism \cite{Rowe_1968,Stanton_1993} of CC theory. \cite{Kucharski_1991,Kallay_2003,Kallay_2004,Hirata_2000,Hirata_2004}
However, to provide a correct description of these situations, one has to take into account, at the very least, contributions from the triple excitations in the CC expansion. \cite{Watson_2012,Loos_2018a,Loos_2019,Loos_2020b} However, to provide a correct description of these situations, one has to take into account, at the very least, contributions from the triple excitations in the CC expansion. \cite{Watson_2012,Loos_2018a,Loos_2019,Loos_2020b}
%Again, due to the scaling of CC methods with the number of basis functions, their applicability is limited to small molecules.
Although multi-reference CC methods have been designed, \cite{Jeziorski_1981,Mahapatra_1998,Mahapatra_1999,Lyakh_2012,Kohn_2013} they are computationally demanding and remain far from being black-box. Although multi-reference CC methods have been designed, \cite{Jeziorski_1981,Mahapatra_1998,Mahapatra_1999,Lyakh_2012,Kohn_2013} they are computationally demanding and remain far from being black-box.
In this context, an interesting alternative to \alert{multi-reference} and CC methods is provided by selected configuration interaction (SCI) methods, \cite{Bender_1969,Whitten_1969,Huron_1973,Giner_2013,Evangelista_2014,Giner_2015,Caffarel_2016b,Holmes_2016,Tubman_2016,Liu_2016,Ohtsuka_2017,Zimmerman_2017,Coe_2018,Garniron_2018} which are able to provide near full CI (FCI) ground- and excited-state energies of small molecules. \cite{Caffarel_2014,Caffarel_2016a,Scemama_2016,Holmes_2017,Li_2018,Scemama_2018,Scemama_2018b,Li_2020,Loos_2018a,Chien_2018,Loos_2019,Loos_2020b,Loos_2020c,Loos_2020e,Garniron_2019,Eriksen_2020,Yao_2020,Williams_2020,Veril_2021,Loos_2021,Damour_2021} In this context, an interesting alternative to \alert{multi-reference} and CC methods is provided by selected configuration interaction (SCI) methods, \cite{Bender_1969,Whitten_1969,Huron_1973,Giner_2013,Evangelista_2014,Giner_2015,Caffarel_2016b,Holmes_2016,Tubman_2016,Liu_2016,Ohtsuka_2017,Zimmerman_2017,Coe_2018,Garniron_2018} which are able to provide near full CI (FCI) ground- and excited-state energies of small molecules. \cite{Caffarel_2014,Caffarel_2016a,Scemama_2016,Holmes_2017,Li_2018,Scemama_2018,Scemama_2018b,Li_2020,Loos_2018a,Chien_2018,Loos_2019,Loos_2020b,Loos_2020c,Loos_2020e,Garniron_2019,Eriksen_2020,Yao_2020,Williams_2020,Veril_2021,Loos_2021,Damour_2021}
@ -203,9 +201,8 @@ For ionic excited states, like the {\sBoneg} state of CBD, it is particularly im
On top of this CASSCF treatment, CASPT2 calculations are performed within the RS2 contraction scheme, while the NEVPT2 energies are computed within both the partially contracted (PC) and strongly contracted (SC) schemes. \cite{Angeli_2001,Angeli_2001a,Angeli_2002} On top of this CASSCF treatment, CASPT2 calculations are performed within the RS2 contraction scheme, while the NEVPT2 energies are computed within both the partially contracted (PC) and strongly contracted (SC) schemes. \cite{Angeli_2001,Angeli_2001a,Angeli_2002}
Note that PC-NEVPT2 is theoretically more accurate than SC-NEVPT2 due to the larger number of external configurations and greater flexibility. Note that PC-NEVPT2 is theoretically more accurate than SC-NEVPT2 due to the larger number of external configurations and greater flexibility.
In order to avoid the intruder state problem in CASPT2, a real-valued level shift of \SI{0.3}{\hartree} is set, \cite{Roos_1995b,Roos_1996} with an additional ionization-potential-electron-affinity (IPEA) shift of \SI{0.25}{\hartree} to avoid systematic underestimation of the vertical excitation energies. \cite{Ghigo_2004,Schapiro_2013,Zobel_2017,Sarkar_2022} In order to avoid the intruder state problem in CASPT2, a real-valued level shift of \SI{0.3}{\hartree} is set, \cite{Roos_1995b,Roos_1996} with an additional ionization-potential-electron-affinity (IPEA) shift of \SI{0.25}{\hartree} to avoid systematic underestimation of the vertical excitation energies. \cite{Ghigo_2004,Schapiro_2013,Zobel_2017,Sarkar_2022}
\alert{For the sake of comparison, for the (4e,4o) active space, we have also performed multi-reference CI calculations including Davidson correction (MRCI+Q). \cite{Knowles_1988,Werner_1988}} \alert{For the sake of comparison and completeness, for the (4e,4o) active space, we also report (in the {\SupInf}) multi-reference CI calculations including Davidson correction (MRCI+Q). \cite{Knowles_1988,Werner_1988}}
All these calculations are carried out with MOLPRO. \cite{Werner_2020} All these calculations are carried out with MOLPRO. \cite{Werner_2020}
%and extended multistate (XMS) CASPT2 calculations are also performed in the cas of strong mixing between states with same spin and spatial symmetries.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
@ -341,8 +338,6 @@ One notes globally satisfying agreement between the tested methods with variatio
\cline{2-5} \cline{2-5}
Method & 6-31+G(d) & aug-cc-pVDZ & aug-cc-pVTZ & aug-cc-pVQZ\\ Method & 6-31+G(d) & aug-cc-pVDZ & aug-cc-pVTZ & aug-cc-pVQZ\\
\hline \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 & $18.59$ & $18.64$ & $19.34$ & $19.34$ \\ SF-TD-B3LYP & $18.59$ & $18.64$ & $19.34$ & $19.34$ \\
SF-TD-PBE0& $17.18$ & $17.19$ & $17.88$ & $17.88$ \\ SF-TD-PBE0& $17.18$ & $17.19$ & $17.88$ & $17.88$ \\
SF-TD-BH\&HLYP & $11.90$ & $12.02$ & $12.72$ & $12.73$ \\ SF-TD-BH\&HLYP & $11.90$ & $12.02$ & $12.72$ & $12.73$ \\
@ -369,7 +364,6 @@ CC3 & $6.59$ & $6.89$ & $7.88$ & $8.06$ \\
CCSDT & $7.26$ & $7.64$ & $8.68$ &$[ 8.86]$\fnm[1] \\ CCSDT & $7.26$ & $7.64$ & $8.68$ &$[ 8.86]$\fnm[1] \\
CC4 & $7.40$ & $7.78$ & $[ 8.82]$\fnm[2] & $[ 9.00]$\fnm[3]\\ CC4 & $7.40$ & $7.78$ & $[ 8.82]$\fnm[2] & $[ 9.00]$\fnm[3]\\
CCSDTQ & $7.51$ & $[ 7.89]$\fnm[4]& $[\bf 8.93]$\fnm[5]& $[ 9.11]$\fnm[6]\\ CCSDTQ & $7.51$ & $[ 7.89]$\fnm[4]& $[\bf 8.93]$\fnm[5]& $[ 9.11]$\fnm[6]\\
%\alert{CIPSI} & $7.91\pm 0.21$ & $8.58\pm 0.14$ & & \\
\end{tabular} \end{tabular}
\end{ruledtabular} \end{ruledtabular}
\fnt[1]{Value obtained using CCSDT/aug-cc-pVTZ corrected by the difference between CC3/aug-cc-pVQZ and CC3/aug-cc-pVTZ.} \fnt[1]{Value obtained using CCSDT/aug-cc-pVTZ corrected by the difference between CC3/aug-cc-pVQZ and CC3/aug-cc-pVTZ.}
@ -412,7 +406,6 @@ Overall, even with the best exchange-correlation functional, SF-TD-DFT is clearl
\alert{We observe that SF-EOM-CCSD/aug-cc-pVTZ tends to underestimate by about \SI{1.5}{\kcalmol} the energy barrier compared to the TBE, a observation in agreement with previous results by Manohar and Krylov. \cite{Manohar_2008} \alert{We observe that SF-EOM-CCSD/aug-cc-pVTZ tends to underestimate by about \SI{1.5}{\kcalmol} the energy barrier compared to the TBE, a observation in agreement with previous results by Manohar and Krylov. \cite{Manohar_2008}
This can be alleviated by including the triples correction with SF-EOM-CCSD(fT) and SF-EOM-CCSD(dT) (see {\SupInf} where we have reported the data from Ref.~\onlinecite{Manohar_2008}). This can be alleviated by including the triples correction with SF-EOM-CCSD(fT) and SF-EOM-CCSD(dT) (see {\SupInf} where we have reported the data from Ref.~\onlinecite{Manohar_2008}).
We also note that the SF-EOM-CCSD values for the energy barrier are close to the ones obtained with the more expensive (standard) CC3 method, yet less accurate than values computed with the cheaper SF-ADC(2)-s formalism. We also note that the SF-EOM-CCSD values for the energy barrier are close to the ones obtained with the more expensive (standard) CC3 method, yet less accurate than values computed with the cheaper SF-ADC(2)-s formalism.
%Our results are in agreement with previous studies \cite{Manohar_2008,Lefrancois_2015} and justify to avoid the more expensive calculations of the triples correction as we can expect a similar trend.
Note that, contrary to a previous statement, \cite{Manohar_2008} the (fT) correction performs better than the (dT) correction for the energy barrier. Note that, contrary to a previous statement, \cite{Manohar_2008} the (fT) correction performs better than the (dT) correction for the energy barrier.
However, for the excited states, the situation is reversed (see below).} However, for the excited states, the situation is reversed (see below).}
@ -449,10 +442,6 @@ Note that the introduction of the triple excitations is clearly mandatory to hav
\cline{3-5} \cline{3-5}
Method & Basis & {\tBoneg} & {\sBoneg} & {\twoAg} \\ Method & Basis & {\tBoneg} & {\sBoneg} & {\twoAg} \\
\hline \hline
% SF-TD-BLYP & 6-31+G(d) & $1.829$ & $1.926$ & $3.755$ \\
% & aug-cc-pVDZ & $1.828$ & $1.927$ & $3.586$ \\
% & aug-cc-pVTZ & $1.825$ & $1.927$ & $3.546$ \\
% & aug-cc-pVQZ & $1.825$ & $1.927$ & $3.528$ \\[0.1cm]
SF-TD-B3LYP & 6-31+G(d) & $1.706$ & $2.211$ & $3.993$ \\ SF-TD-B3LYP & 6-31+G(d) & $1.706$ & $2.211$ & $3.993$ \\
& aug-cc-pVDZ & $1.706$ & $2.204$ & $3.992$ \\ & aug-cc-pVDZ & $1.706$ & $2.204$ & $3.992$ \\
& aug-cc-pVTZ & $1.703$ & $2.199$ & $3.988$ \\ & aug-cc-pVTZ & $1.703$ & $2.199$ & $3.988$ \\
@ -505,8 +494,6 @@ SF-ADC(3) & 6-31+G(d) & $1.435$ & $3.352$ & $4.242$ \\
\alert{SF-EOM-CCSD} & \alert{6-31+G(d)} & \alert{$1.663$} & \alert{$3.515$} & \alert{$4.275$} \\ \alert{SF-EOM-CCSD} & \alert{6-31+G(d)} & \alert{$1.663$} & \alert{$3.515$} & \alert{$4.275$} \\
& \alert{aug-cc-pVDZ} & \alert{$1.611$} & \alert{$3.315$} & \alert{$4.216$} \\ & \alert{aug-cc-pVDZ} & \alert{$1.611$} & \alert{$3.315$} & \alert{$4.216$} \\
& \alert{aug-cc-pVTZ} & \alert{$1.609$} & \alert{$3.293$} & \alert{$4.245$} \\[0.1cm] & \alert{aug-cc-pVTZ} & \alert{$1.609$} & \alert{$3.293$} & \alert{$4.245$} \\[0.1cm]
%SF-EOM-CC(2,3) & 6-31+G(d) & $1.490$ & $3.333$ & $4.061$ \\
%& aug-cc-pVDZ & $1.464$ & $3.156$ & $4.027$ \\
\end{tabular} \end{tabular}
\end{ruledtabular} \end{ruledtabular}
\end{table} \end{table}
@ -535,10 +522,6 @@ CASPT2(4,4) &6-31+G(d)& $1.440$ & $3.162$ & $4.115$ \\
& aug-cc-pVDZ & $1.414$ & $2.971$ & $4.068$ \\ & aug-cc-pVDZ & $1.414$ & $2.971$ & $4.068$ \\
& aug-cc-pVTZ & $1.412$ & $2.923$ & $4.072$ \\ & aug-cc-pVTZ & $1.412$ & $2.923$ & $4.072$ \\
& aug-cc-pVQZ & $1.417$ & $2.911$ & $4.081$ \\[0.1cm] & aug-cc-pVQZ & $1.417$ & $2.911$ & $4.081$ \\[0.1cm]
%XMS-CASPT2(4,4) &6-31+G(d)& & & $4.151$ \\
%& aug-cc-pVDZ & & & $4.105$ \\
%& aug-cc-pVTZ & & & $4.114$ \\
%& aug-cc-pVQZ && & $4.125$ \\[0.1cm]
SC-NEVPT2(4,4) &6-31+G(d)& $1.407$ & $2.707$ & $4.145$ \\ 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-pVDZ & $1.381$ & $2.479$ & $4.109$ \\
& aug-cc-pVTZ & $1.379$ & $2.422$ & $4.108$ \\ & aug-cc-pVTZ & $1.379$ & $2.422$ & $4.108$ \\
@ -547,14 +530,6 @@ 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-pVDZ & $1.384$ & $2.424$ & $4.084$ \\
& aug-cc-pVTZ & $1.382$ & $2.368$ & $4.083$ \\ & aug-cc-pVTZ & $1.382$ & $2.368$ & $4.083$ \\
& aug-cc-pVQZ & $1.387$ & $2.353$ & $4.091$ \\[0.1cm] & 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]
\alert{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)& $1.675$ & $3.924$ & $4.220$ \\ CASSCF(12,12) &6-31+G(d)& $1.675$ & $3.924$ & $4.220$ \\
& aug-cc-pVDZ & $1.685$ & $3.856$ & $4.221$ \\ & aug-cc-pVDZ & $1.685$ & $3.856$ & $4.221$ \\
& aug-cc-pVTZ & $1.686$ & $3.844$ & $4.217$ \\ & aug-cc-pVTZ & $1.686$ & $3.844$ & $4.217$ \\
@ -563,10 +538,6 @@ CASPT2(12,12) &6-31+G(d)& $1.508$ & $3.407$ & $4.099$ \\
& aug-cc-pVDZ & $1.489$ & $3.256$ & $4.044$ \\ & aug-cc-pVDZ & $1.489$ & $3.256$ & $4.044$ \\
& aug-cc-pVTZ & $1.480$ & $3.183$ & $4.043$ \\ & aug-cc-pVTZ & $1.480$ & $3.183$ & $4.043$ \\
& aug-cc-pVQZ & $1.482$ & $3.163$ & $4.047$ \\[0.1cm] & aug-cc-pVQZ & $1.482$ & $3.163$ & $4.047$ \\[0.1cm]
%XMS-CASPT2(12,12) &6-31+G(d)& && $4.111$ \\
%& aug-cc-pVDZ & & & $4.056$ \\
%& aug-cc-pVTZ & & & $4.059$ \\
%& aug-cc-pVQZ & & & $4.065$ \\[0.1cm]
SC-NEVPT2(12,12) &6-31+G(d)& $1.522$ & $3.409$ & $4.130$ \\ 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-pVDZ & $1.511$ & $3.266$ & $4.093$ \\
& aug-cc-pVTZ & $1.501$ & $3.188$ & $4.086$ \\ & aug-cc-pVTZ & $1.501$ & $3.188$ & $4.086$ \\
@ -575,7 +546,6 @@ 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-pVDZ & $1.472$ & $3.141$ & $4.064$ \\
& aug-cc-pVTZ & $1.462$ & $3.063$ & $4.056$ \\ & aug-cc-pVTZ & $1.462$ & $3.063$ & $4.056$ \\
& aug-cc-pVQZ & $1.464$ & $3.043$ & $4.059$ \\[0.1cm] & aug-cc-pVQZ & $1.464$ & $3.043$ & $4.059$ \\[0.1cm]
%MRCI(12,12) &6-31+G(d)& & & $4.125$ \\[0.1cm]
CCSD &6-31+G(d)& $1.346$ & $3.422$ & \\ CCSD &6-31+G(d)& $1.346$ & $3.422$ & \\
& aug-cc-pVDZ & $1.319$ & $3.226$ & \\ & aug-cc-pVDZ & $1.319$ & $3.226$ & \\
& aug-cc-pVTZ & $1.317$ & $3.192$ & \\ & aug-cc-pVTZ & $1.317$ & $3.192$ & \\
@ -652,6 +622,7 @@ Regarding the \alert{multi-reference} calculations, the most striking result is
Of course, the PT2 correction is able to correct the state ordering problem but cannot provide quantitative excitation energies due to the poor zeroth-order treatment. Of course, the PT2 correction is able to correct the state ordering problem but cannot provide quantitative excitation energies due to the poor zeroth-order treatment.
Another ripple effect of the unreliability of the reference wave function is the large difference between CASPT2 and NEVPT2 that differ by half an \si{\eV}. Another ripple effect of the unreliability of the reference wave function is the large difference between CASPT2 and NEVPT2 that differ by half an \si{\eV}.
This feature is characteristic of the inadequacy of the active space to model such a state. This feature is characteristic of the inadequacy of the active space to model such a state.
\alert{Additional MRCI and MRCI+Q calculations (reported in the {\SupInf}) confirm this.}
For the two other states, {\tBoneg} and {\twoAg}, the errors at the CASPT2(4,4) and NEVPT2(4,4) levels are much smaller (below \SI{0.1}{\eV}). For the two other states, {\tBoneg} and {\twoAg}, the errors at the CASPT2(4,4) and NEVPT2(4,4) levels are much smaller (below \SI{0.1}{\eV}).
Using a larger active space resolves most of these issues: CASSCF predicts the correct state ordering (though the ionic state is still badly described in term of energetics), CASPT2 and NEVPT2 excitation energies are much closer, and their accuracy is often improved (especially for the triplet and doubly-excited states) although it is difficult to reach chemical accuracy (\ie, an error below \SI{0.043}{\eV}) on a systematic basis. Using a larger active space resolves most of these issues: CASSCF predicts the correct state ordering (though the ionic state is still badly described in term of energetics), CASPT2 and NEVPT2 excitation energies are much closer, and their accuracy is often improved (especially for the triplet and doubly-excited states) although it is difficult to reach chemical accuracy (\ie, an error below \SI{0.043}{\eV}) on a systematic basis.
@ -735,7 +706,6 @@ SF-ADC(3) & 6-31+G(d) & $0.123$ & $1.650$ & $2.078$ \\
\end{squeezetable} \end{squeezetable}
%%% %%% %%% %%% %%% %%% %%% %%%
%%% TABLE V %%% %%% TABLE V %%%
\begin{squeezetable} \begin{squeezetable}
\begin{table} \begin{table}
@ -766,14 +736,6 @@ 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-pVDZ & $0.039$ & $1.440$ & $1.088$ \\
& aug-cc-pVTZ & $0.026$ & $1.403$ & $1.003$ \\ & aug-cc-pVTZ & $0.026$ & $1.403$ & $1.003$ \\
& aug-cc-pVQZ & $0.026$ & $1.395$ & $0.977$ \\[0.1cm] & 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]
\alert{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) & $0.386$ & $1.974$ & $2.736$ \\ CASSCF(12,12) & 6-31+G(d) & $0.386$ & $1.974$ & $2.736$ \\
& aug-cc-pVDZ & $0.374$ & $1.947$ & $2.649$ \\ & aug-cc-pVDZ & $0.374$ & $1.947$ & $2.649$ \\
& aug-cc-pVTZ & $0.370$ & $1.943$ & $2.634$ \\ & aug-cc-pVTZ & $0.370$ & $1.943$ & $2.634$ \\

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Manuscript/sup-CBD.tex
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@ -163,7 +163,7 @@ H -2.092429 0.000000 0.000000
\section{Comment about symmetry: standard vs non-standard orientation} \section{Comment about symmetry: standard vs non-standard orientation}
%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%
\alert{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. 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. 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. 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. This induces a different orbital picture.
@ -179,7 +179,7 @@ The $1 ^1B_{1g}$ ground state is obtained as a singly excited state from that re
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. 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. 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. 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.} 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} \begin{figure}
\includegraphics[width=\textwidth]{MOs} \includegraphics[width=\textwidth]{MOs}
@ -202,47 +202,47 @@ Note that in the case of the SF formalism, these three singlet states should all
\cline{3-5} \cline{6-8} \cline{3-5} \cline{6-8}
Method & AB & {\tBoneg}(99\%) & {\sBoneg}(95\%)& {\twoAg}(1\%) & {\Atwog} & {\Aoneg} & {\Btwog} \\ Method & AB & {\tBoneg}(99\%) & {\sBoneg}(95\%)& {\twoAg}(1\%) & {\Atwog} & {\Aoneg} & {\Btwog} \\
\hline \hline
SF-TD-B3LYP & $10.41$ & \alert{$0.270$} & $-0.926$ & $-0.050$ & $-0.164$ & $-1.028$ & $-1.316$ \\ SF-TD-B3LYP & $10.41$ & {$0.270$} & $-0.926$ & $-0.050$ & $-0.164$ & $-1.028$ & $-1.316$ \\
SF-TD-PBE0 & $8.95$ & \alert{$0.249$} & $-0.829$ & $0.043$ & $-0.163$ & $-0.903$ &$-1.172$ \\ SF-TD-PBE0 & $8.95$ & {$0.249$} & $-0.829$ & $0.043$ & $-0.163$ & $-0.903$ &$-1.172$ \\
SF-TD-BH\&HLYP & $3.79$ & \alert{$0.107$} & $-0.393$ & $0.454$ & $-0.099$ & $-0.251$ & $-0.418$ \\ SF-TD-BH\&HLYP & $3.79$ & {$0.107$} & $-0.393$ & $0.454$ & $-0.099$ & $-0.251$ & $-0.418$ \\
SF-TD-M06-2X & $1.42$ & \alert{$0.029$} & $-0.354$ & $0.319$ & $-0.066$ & $-0.097$ & $-0.247$ \\ SF-TD-M06-2X & $1.42$ & {$0.029$} & $-0.354$ & $0.319$ & $-0.066$ & $-0.097$ & $-0.247$ \\
SF-TD-CAM-B3LYP & $9.90$ & \alert{$0.309$} & $-0.807$ &$0.100$ & $-0.134$ & $-0.920$ & $-1.185$\\ SF-TD-CAM-B3LYP & $9.90$ & {$0.309$} & $-0.807$ &$0.100$ & $-0.134$ & $-0.920$ & $-1.185$\\
SF-TD-$\omega $B97X-V & $10.01$ & \alert{$0.364$} & $-0.774$ & $0.175$ & $-0.118$ & $-0.928$ & $-1.187$ \\ SF-TD-$\omega $B97X-V & $10.01$ & {$0.364$} & $-0.774$ & $0.175$ & $-0.118$ & $-0.928$ & $-1.187$ \\
SF-TD-LC-$\omega $PBE08 & $10.81$ & \alert{$0.464$} & $-0.710$ & $0.310$ & $-0.086$ & $-0.939$ & $-1.191$ \\ SF-TD-LC-$\omega $PBE08 & $10.81$ & {$0.464$} & $-0.710$ & $0.310$ & $-0.086$ & $-0.939$ & $-1.191$ \\
SF-TD-M11 & $2.29$ & \alert{$0.126$} & $-0.474$ & $0.262$ & $-0.063$ & $-0.312$ & $-0.490$ \\[0.1cm] SF-TD-M11 & $2.29$ & {$0.126$} & $-0.474$ & $0.262$ & $-0.063$ & $-0.312$ & $-0.490$ \\[0.1cm]
SF-ADC(2)-s & $-0.30$ & \alert{$0.098$} & $-0.026$ & $0.093$ & $0.112$ & $0.112$ & $-0.005$ \\ SF-ADC(2)-s & $-0.30$ & {$0.098$} & $-0.026$ & $0.093$ & $0.112$ & $0.112$ & $-0.005$ \\
SF-ADC(2)-x & $1.44$ & \alert{$0.106$} & $-0.094$ & $-0.335$ & $0.068$ & $-0.409$ & $-0.118$ \\ SF-ADC(2)-x & $1.44$ & {$0.106$} & $-0.094$ & $-0.335$ & $0.068$ & $-0.409$ & $-0.118$ \\
SF-ADC(2.5) & $0.18$ & \alert{$0.042$} & $0.006$ &$0.140$ & $0.024$ & $0.094$ & $0.000$ \\ SF-ADC(2.5) & $0.18$ & {$0.042$} & $0.006$ &$0.140$ & $0.024$ & $0.094$ & $0.000$ \\
SF-ADC(3) & $0.65$ & \alert{$-0.014$} & $0.037$ & $0.186$ & $-0.065$ & $0.075$ & $0.004$ \\ SF-ADC(3) & $0.65$ & {$-0.014$} & $0.037$ & $0.186$ & $-0.065$ & $0.075$ & $0.004$ \\
\alert{SF-EOM-CCSD} & \alert{$-1.53$} & \alert{$0.176$} & \alert{$0.168$} & \alert{$0.207$} & \alert{$0.210$} & \alert{$0.268$} & \alert{$0.211$} \\[0.1cm] {SF-EOM-CCSD} & {$-1.53$} & {$0.176$} & {$0.168$} & {$0.207$} & {$0.210$} & {$0.268$} & {$0.211$} \\[0.1cm]
CASSCF(4,4) & $-1.55$ & \alert{$0.237$} & $1.421$ & $0.403$& $0.290$ & $0.734$ & $1.575$ \\ CASSCF(4,4) & $-1.55$ & {$0.237$} & $1.421$ & $0.403$& $0.290$ & $0.734$ & $1.575$ \\
CASPT2(4,4) & $-1.16$ & \alert{$-0.021$} & $-0.202$ & $0.034$ & $-0.016$ & $0.006$ & $-0.214$ \\ CASPT2(4,4) & $-1.16$ & {$-0.021$} & $-0.202$ & $0.034$ & $-0.016$ & $0.006$ & $-0.214$ \\
%XMS-CASPT2(4,4) & & & & $-0.035$ & & & \\ %XMS-CASPT2(4,4) & & & & $-0.035$ & & & \\
SC-NEVPT2(4,4) & $0.30$ & \alert{$-0.054$} & $-0.703$ & $0.070$ & $-0.120$ & $-0.072$ & $-0.794$ \\ SC-NEVPT2(4,4) & $0.30$ & {$-0.054$} & $-0.703$ & $0.070$ & $-0.120$ & $-0.072$ & $-0.794$ \\
PC-NEVPT2(4,4) & $0.31$ & \alert{$-0.051$} & $-0.757$ & $0.045$ & $-0.118$ & $-0.097$ & $-0.846$ \\ PC-NEVPT2(4,4) & $0.31$ & {$-0.051$} & $-0.757$ & $0.045$ & $-0.118$ & $-0.097$ & $-0.846$ \\
\alert{MRCI(4,4)} & & $0.135$ & $0.553$ & $0.232$ & $0.127$ & $0.324$ & $0.566$ \\ {MRCI(4,4)} & & $0.135$ & $0.553$ & $0.232$ & $0.127$ & $0.324$ & $0.566$ \\
\alert{MRCI(4,4)+Q} & & $0.086$ & $0.217$ & $0.121$ & $0.075$ & $0.167$ & $0.205$ \\ {MRCI(4,4)+Q} & & $0.086$ & $0.217$ & $0.121$ & $0.075$ & $0.167$ & $0.205$ \\
CASSCF(12,12) & $2.66$ & \alert{$0.253$} & $0.719$ & $0.179$ & $0.226$ & $0.443$ & $0.785$ \\ CASSCF(12,12) & $2.66$ & {$0.253$} & $0.719$ & $0.179$ & $0.226$ & $0.443$ & $0.785$ \\
CASPT2(12,12) & $-0.42$& \alert{$0.047$} & $0.058$ & $0.005$ & $0.039$ & $0.038$ & $0.077$ \\ CASPT2(12,12) & $-0.42$& {$0.047$} & $0.058$ & $0.005$ & $0.039$ & $0.038$ & $0.077$ \\
%XMS-CASPT2(12,12) & & & & $-0.090$ & & & \\ %XMS-CASPT2(12,12) & & & & $-0.090$ & & & \\
SC-NEVPT2(12,12) & $-0.64$ & \alert{$0.068$} & $0.063$ & $0.048$ & $0.021$ & $0.046$ & $0.043$ \\ SC-NEVPT2(12,12) & $-0.64$ & {$0.068$} & $0.063$ & $0.048$ & $0.021$ & $0.046$ & $0.043$ \\
PC-NEVPT2(12,12) & $-0.65$ & \alert{$0.029$} & $-0.062$ & $0.018$ & $-0.013$ & $-0.024$ & $-0.093$ \\[0.1cm] PC-NEVPT2(12,12) & $-0.65$ & {$0.029$} & $-0.062$ & $0.018$ & $-0.013$ & $-0.024$ & $-0.093$ \\[0.1cm]
CCSD & $0.95$ & \alert{$-0.116$} & $0.067$ & & $-0.059$ & $0.100$ & \\ CCSD & $0.95$ & {$-0.116$} & $0.067$ & & $-0.059$ & $0.100$ & \\
CC3 & $-1.05$ & \alert{$-0.031$} & $-0.006$ & $0.739$ & & $0.162$ & $0.871$ \\ CC3 & $-1.05$ & {$-0.031$} & $-0.006$ & $0.739$ & & $0.162$ & $0.871$ \\
CCSDT & $-0.25$ & \alert{$-0.022$} & $0.014$ & $0.391$ & $0.005$ & $0.131$ & $0.688$ \\ CCSDT & $-0.25$ & {$-0.022$} & $0.014$ & $0.391$ & $0.005$ & $0.131$ & $0.688$ \\
CC4 & $-0.11$ & \alert{$0.000$} & $0.003$ & $0.105$ & & $0.011$ & $-0.013$ \\ 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] CCSDTQ & $0.00$ & & $0.000$ & $0.000$ & $0.000$ & $0.000$ & $0.000$ \\[0.1cm]
%CIPSI & & $-0.001\pm 0.030$ & $0.017\pm 0.035$ & $-0.120\pm 0.090$ & $0.025\pm 0.029$ & $0.130\pm 0.050$ & \\ %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] & \alert{$[\bf{1.433}]$}\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] \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] \\ 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]\\ & $10.35$\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] \\ & $9.58$\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] \\
& \alert{$7.50$}\fnm[6] & \alert{$1.654$}\fnm[6] & \alert{$3.416$}\fnm[6] & \alert{$4.360$}\fnm[6] & \alert{$0.369$}\fnm[6] & \alert{$1.824$}\fnm[6] & \alert{$2.143$}\fnm[6]\\ & {$7.50$}\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]\\
& \alert{$9.36$}\fnm[7] & \alert{$1.516$}\fnm[7] & \alert{$3.260$}\fnm[7] & \alert{$4.205$}\fnm[7] & \alert{$0.163$}\fnm[7] & \alert{$1.530$}\fnm[7] & \alert{$1.921$}\fnm[7]\\ & {$9.36$}\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]\\
& \alert{$9.91$}\fnm[8] & \alert{$1.475$}\fnm[8] & \alert{$3.215$}\fnm[8] & \alert{$4.176$}\fnm[8] & \alert{$0.098$}\fnm[8] & \alert{$1.456$}\fnm[8] & \alert{$1.853$}\fnm[8]\\ & {$9.91$}\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]\\
& & \alert{$1.403$}\fnm[9] & \alert{$3.120$}\fnm[9] & \alert{$4.127$}\fnm[9] & \alert{$0.023$}\fnm[9] & \alert{$1.406$}\fnm[9] & \alert{$1.751$}\fnm[9] \\ & & {$1.403$}\fnm[9] & {$3.120$}\fnm[9] & {$4.127$}\fnm[9] & {$0.023$}\fnm[9] & {$1.406$}\fnm[9] & {$1.751$}\fnm[9] \\
& & & & & \alert{$0.062$}\fnm[10] & &\\ & & & & & {$0.062$}\fnm[10] & &\\
& & & & & \alert{$0.219$}\fnm[11] & & \\ & & & & & {$0.219$}\fnm[11] & & \\
\end{tabular} \end{tabular}
\end{ruledtabular} \end{ruledtabular}
@ -253,12 +253,12 @@ Literature & $8.53$\fnm[3] & $1.573$\fnm[3] & $3.208$\fnm[3] & $4.247$\fnm[3] &
\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[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[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[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]{\alert{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[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]{\alert{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[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]{\alert{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[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]{\alert{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[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]{\alert{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[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]{\alert{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.}} \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{table*}
%\end{squeezetable} %\end{squeezetable}
@ -287,15 +287,12 @@ Literature & $8.53$\fnm[3] & $1.573$\fnm[3] & $3.208$\fnm[3] & $4.247$\fnm[3] &
CCSDt/cc-pVTZ & $9.5$ & Ref.~\onlinecite{Shen_2012}\\ CCSDt/cc-pVTZ & $9.5$ & Ref.~\onlinecite{Shen_2012}\\
CCSD(T)-h/cc-pVTZ & $6.8$ & 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(t;3)/cc-pVTZ & $10.0$ & Ref.~\onlinecite{Shen_2012}\\
\alert{ CC(P;Q)/cc-pVDZ} &\alert{$8.65$} & Ref.~\onlinecite{Gururangan_2021}\\ CC(P;Q)/cc-pVDZ &$8.65$ & Ref.~\onlinecite{Gururangan_2021}\\
\end{tabular} \end{tabular}
\end{ruledtabular} \end{ruledtabular}
\end{table} \end{table}
%%% %%% %%% %%% %%% %%% %%% %%%
%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%
\begin{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.} \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.}
@ -336,6 +333,116 @@ SF-TD-M11
\end{ruledtabular} \end{ruledtabular}
\end{table} \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]
\end{tabular}
\end{ruledtabular}
\end{table}
%%% %%% %%% %%%
%%% TABLE V %%%
\begin{squeezetable}
\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}
\end{squeezetable}
%%% %%% %%% %%%
\clearpage \clearpage
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