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@ -601,76 +601,64 @@ Throughout this work, all spin-flip calculations are performed with a UHF refere
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\label{sec:Be}
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%===============================
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%%%%%%%%%%%%%%%
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%T2: I think it might be worth doing some calculations with a larger basis set (ie, aug-cc-pvqz).
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% I've done a quick check and it seems to work much better and we could get some CIPSI excitation energies as reference.
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% Also, I think we have much more spin contamination in this larger basis and it would be worth reporting it (for the reference and the excited state).
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% No need to do evGW and qsGW, G0W0 (BSE and dBSE) is enough I guess + SF-ADC and SF-TD-DFT.
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%%%%%%%%%%%%%%%
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As a first example, we consider the simple case of the beryllium atom in a small basis (6-31G) which was considered by Krylov in two of her very first papers on spin-flip methods \cite{Krylov_2001a,Krylov_2001b} and was also considered in later studies thanks to its pedagogical value. \cite{Sears_2003,Casanova_2020}
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Beryllium has a $^1S$ ground state with $1s^2 2s^2$ configuration.
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The excitation energies corresponding to the first singlet and triplet single excitations $2s \to 2p$ with $P$ spatial symmetry as well as the first singlet and triplet double excitations $2s^2 \to 2p^2$ with $D$ and $P$ spatial symmetries (respectively) are reported in Table \ref{tab:Be} and depicted in Fig.~\ref{fig:Be}.
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The left side of Fig.~\ref{fig:Be} (red lines) reports SF-TD-DFT excitation energies obtained with the BLYP, B3LYP, and BH\&HLYP functionals, which correspond to an increase of exact exchange from 0\% to 50\%.
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As mentioned in Ref.~\onlinecite{Casanova_2020}, the $^3P(1s^2 2s^1 2p^1)$ and the $^1P(1s^2 2s^1 2p^1)$ states are degenerate at the SF-TD-BLYP level and their excitation energies are given by the $2s$--$2p$ orbital energy difference due to the lack of coupling terms in the spin-flip block of the SD-TD-DFT equations (\textit{vide supra}).
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Including exact exchange, like in SF-TD-B3LYP and SF-TD-BH\&HLYP, lifts this degeneracy.
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As mentioned in Ref.~\onlinecite{Casanova_2020}, the $^3P(1s^2 2s^1 2p^1)$ and the $^1P(1s^2 2s^1 2p^1)$ states are degenerate at the SF-TD-BLYP level.
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Due to the lack of coupling terms in the spin-flip block of the SD-TD-DFT equations (see Subsec.~\ref{sec:BSE}), their excitation energies are given by the energy difference between the $2s$ and $2p$ orbitals and both states are strongly spin contaminated.
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Including exact exchange, like in SF-TD-B3LYP and SF-TD-BH\&HLYP, lifts this degeneracy and improves the description of both states.
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However, the SF-TD-BH\&HLYP excitation energy of the $^1P(1s^2 2s^1 2p^1)$ state is still off by $1.6$ eV as compared to the FCI reference.
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For the other states, the agreement between SF-TD-BH\&HLYP and FCI is significantly improved.
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The center part of Fig.~\ref{fig:Be} (blue lines) shows the SF-BSE results alongside the SF-CIS excitation energies (purple lines).
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All of these are computed with 100\% of exact exchange with the inclusion of correlation in the case of SF-BSE thanks to the introduction of the screening.
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They are close the FCI results, because of the fact that there is one hundred percent exact exchange in the BSE method, with an error of $0.02$-$0.6$ eV depending on the scheme of SF-BSE.
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For the last excited state $^1D(1s^2 2p^2)$ the largest error is $0.85$ eV with SF-dBSE@{\qsGW}, so we have a bad description of this state due to spin contamination.
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Generally we can observe that all the scheme with SF-BSE used do not increase significantly the accuracy of excitations energies.
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The center part of Fig.~\ref{fig:Be} shows the SF-(d)BSE results (blue lines) alongside the SF-CIS excitation energies (purple lines).
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All of these are computed with 100\% of exact exchange with the additional inclusion of correlation in the case of SF-BSE and SF-dBSE thanks to the introduction of the static and dynamical screening, respectively.
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Overall, the SF-CIS and SF-BSE excitation energies are closer to FCI than the SF-TD-DFT ones, except for the lowest triplet state where the SF-TD-BH\&HLYP excitation energy is more accurate probably due to error compensation.
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At the exception of the $^1D$ state, SF-BSE improves over SF-CIS with a rather small contribution from the additional dynamical effect.
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Note that the exact exchange seems to spin purified the $^3P(1s^2 2s^1 2p^1)$ state while the singlet states at the SF-BSE level are slightly more spin contaminated than their SF-CIS counterparts.
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The right side of Fig.~\ref{fig:Be} illustrates the performance of the SF-ADC methods.
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Interestingly, SF-BSE and SF-ADC(2)-s have rather similar accuracies, except again for the $^1D$ state where SF-ADC(2)-s has clearly the edge over SF-BSE.
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Finally, both SF-ADC(2)-x and SF-ADC(3) yield excitation energies very close to FCI for this simple system with significant improvements for the lowest $^3P$ state and the $^1D$ doubly-excited state.
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%%% TABLE I %%%
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\begin{squeezetable}
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%\begin{squeezetable}
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\begin{table*}
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\caption{
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Excitation energies (in eV) with respect to the $^1S(1s^2 2s^2)$ singlet ground state of \ce{Be} obtained for various methods with the 6-31G and aug-cc-pVQZ basis sets.
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Excitation energies (in eV) with respect to the $^1S(1s^2 2s^2)$ singlet ground state of \ce{Be} obtained for various methods with the 6-31G basis set.
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All the spin-flip calculations have been performed with a UHF reference.
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The $\expval{S^2}$ value associated with each state is reported in parenthesis (when available).
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The $\expval*{S^2}$ value associated with each state is reported in parenthesis (when available).
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\label{tab:Be}}
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\begin{ruledtabular}
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\begin{tabular}{lcccccccccc}
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& \mc{5}{c}{6-31G} & \mc{5}{c}{aug-cc-pVTQ} \\
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\cline{2-6} \cline{7-11}
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& \mc{5}{c}{States} \\
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% & \mc{5}{c}{6-31G} & \mc{5}{c}{aug-cc-pVQZ} \\
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\cline{2-6} %\cline{7-11}
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Method & $^1S(1s^2 2s^2)$ & $^3P(1s^2 2s^1 2p^1)$ & $^1P(1s^2 2s^1 2p^1)$
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& $^3P(1s^22 p^2)$ & $^1D(1s^22p^2)$
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& $^1S(1s^2 2s^2)$ & $^3P(1s^2 2s^1 2p^1)$ & $^1P(1s^2 2s^1 2p^1)$
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& $^3P(1s^22 p^2)$ & $^1D(1s^22p^2)$ \\
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% & $^1S(1s^2 2s^2)$ & $^3P(1s^2 2s^1 2p^1)$ & $^1P(1s^2 2s^1 2p^1)$
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% & $^3P(1s^22 p^2)$ & $^1D(1s^22p^2)$ \\
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\hline
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SF-TD-BLYP\fnm[1] & (0.002) & 3.210(1.000) & 3.210(1.000) & 6.691(1.000) & 7.598(0.013)
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& (0.012) &2.998 (1.000) & () & () & () \\
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SF-TD-B3LYP\fnm[1] & (0.001) & 3.332(1.839) & 4.275(0.164) & 6.864(1.000) & 7.762(0.006)
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& (0.009) & 3.205(1.000) & () & () & () \\
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SF-TD-BH\&HLYP\fnm[1] & (0.000) & 2.874(1.981) & 4.922(0.023) & 7.112(1.000) & 8.188(0.002)
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& (0.005) & 2.799(1.976) & () & () & () \\
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SF-CIS\fnm[2] & (0.002) & 2.111(2.000) & 6.036(0.014) & 7.480(1.000) & 8.945(0.006)
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& (0.005) & 2.059(2.000) &5.580 (0.178) & () & () \\
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SF-BSE@{\GOWO} & (0.004) & 2.399(1.999) & 6.191(0.023) & 7.792(1.000) & 9.373(0.013)
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& (0.021) & 2.286(1.994) & 5.181(0.187) & 6.481(1.000) & 7.195(0.719) \\
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% SF-BSE@{\evGW} & (0.004) & 2.407(1.999) & 6.199(0.023) & 7.788(1.000) & 9.388(0.013) \\
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% \alert{SF-BSE@{\qsGW}} & (0.102) & 2.532(2.000) & 6.241(1.873) & 7.668(1.000) & 9.417(0.217) \\
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SF-dBSE@{\GOWO} & & 2.363 & 6.263 & 7.824 & 9.424
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& & 2.177 & 5.380 & 6.806 & \\
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SF-TD-BLYP\fnm[1] & (0.002) & 3.210(1.000) & 3.210(1.000) & 6.691(1.000) & 7.598(0.013) \\
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SF-TD-B3LYP\fnm[1] & (0.001) & 3.332(1.839) & 4.275(0.164) & 6.864(1.000) & 7.762(0.006) \\
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SF-TD-BH\&HLYP\fnm[1] & (0.000) & 2.874(1.981) & 4.922(0.023) & 7.112(1.000) & 8.188(0.002) \\
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SF-CIS\fnm[2] & (0.002) & 2.111(2.000) & 6.036(0.014) & 7.480(1.000) & 8.945(0.006) \\
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SF-BSE@{\GOWO} & (0.004) & 2.399(1.999) & 6.191(0.023) & 7.792(1.000) & 9.373(0.013) \\
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SF-BSE@{\evGW} & (0.004) & 2.407(1.999) & 6.199(0.023) & 7.788(1.000) & 9.388(0.013) \\
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SF-dBSE@{\GOWO} & & 2.363 & 6.263 & 7.824 & 9.424 \\
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% SF-dBSE@{\evGW} & & 2.369 & 6.273 & 7.820 & 9.441 \\
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% SF-dBSE@{\qsGW} & & 2.335 & 6.317 & 7.689 & 9.470 \\
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SF-ADC(2)-s & & 2.433 & 6.255 & 7.745 & 9.047
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& & 2.415 & 5.360 & 6.431 & 7.206 \\
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SF-ADC(2)-x & & 2.866 & 6.581 & 7.664 & 8.612
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& & 2.692 & 5.256 & 6.411 & 7.098\\
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SF-ADC(3) & & 2.863 & 6.579 & 7.658 & 8.618
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& & & 5.221 & & \\
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FCI\fnm[2] & (0.000) & 2.862(2.000) & 6.577(0.000) & 7.669(2.000) & 8.624(0.000)
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& (0.000) & 2.718(2.000) & 5.277(0.000) & 6.450(2.000) & 7.114(0.000) \\
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SF-ADC(2)-s & & 2.433 & 6.255 & 7.745 & 9.047 \\
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SF-ADC(2)-x & & 2.866 & 6.581 & 7.664 & 8.612 \\
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SF-ADC(3) & & 2.863 & 6.579 & 7.658 & 8.618 \\
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FCI\fnm[2] & (0.000) & 2.862(2.000) & 6.577(0.000) & 7.669(2.000) & 8.624(0.000) \\
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% & (0.000) & 2.718(2.000) & 5.277(0.000) & 6.450(2.000) & 7.114(0.000) \\
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\end{tabular}
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\end{ruledtabular}
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\fnt[1]{Value in the 6-31G basis taken from Ref.~\onlinecite{Casanova_2020}.}
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\fnt[2]{Value in the 6-31G basis taken from Ref.~\onlinecite{Krylov_2001a}.}
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\fnt[1]{Excitation energy taken from Ref.~\onlinecite{Casanova_2020}.}
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\fnt[2]{Excitation energy taken from Ref.~\onlinecite{Krylov_2001a}.}
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\end{table*}
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\end{squeezetable}
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%\end{squeezetable}
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%%% %%% %%% %%%
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%%% FIG 1 %%%
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@ -790,22 +778,22 @@ So here we have an example where the dynamical corrections are necessary to get
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All the spin-flip calculations have been performed with a UHF reference and the cc-pVTZ basis set.
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\label{tab:CBD_D2h}}
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\begin{ruledtabular}
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\begin{tabular}{lccc}
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\begin{tabular}{lrrr}
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& \mc{3}{c}{Excitation energies (eV)} \\
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\cline{2-4}
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Method & $1\,{}^3B_{1g}$ & $1\,{}^1B_{1g}$ & $2\,{}^1A_{1g}$ \\
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\hline
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SF-TD-B3LYP\fnm[3] &1.750 &2.260 &4.094 \\
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SF-TD-BH\&HLYP\fnm[3] &1.583 &2.813 & 4.528\\
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SF-CIS\fnm[1] & & & \\
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EOM-SF-CCSD\fnm[1] &1.654 & 3.416&4.360 \\
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EOM-SF-CCSD(fT)\fnm[1] & 1.516& 3.260&4.205 \\
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EOM-SF-CCSD(dT)\fnm[1] &1.475 &3.215 &4.176 \\
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SF-ADC(2)-s\fnm[2] & 1.573& 3.208& 4.247\\
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SF-ADC(2)-x\fnm[2] &1.576 &3.141 &3.796 \\
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SF-ADC(3)\fnm[2] & 1.456&3.285 &4.334 \\
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SF-BSE@{\GOWO}\fnm[3] & 1.438 & 2.704 &4.540 \\
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SF-dBSE@{\GOWO}\fnm[3] & 1.403 &2.883 &4.621 \\
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SF-TD-B3LYP\fnm[3] & $1.750$ & $2.260$ & $4.094$ \\
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SF-TD-BH\&HLYP\fnm[3] & $1.583$ & $2.813$ & $4.528$ \\
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SF-CIS\fnm[1] & $1.521$ & $3.836$ & $5.499$ \\
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EOM-SF-CCSD\fnm[1] & $1.654$ & $3.416$ & $4.360$ \\
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EOM-SF-CCSD(fT)\fnm[1] & $1.516$ & $3.260$ & $4.205$ \\
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EOM-SF-CCSD(dT)\fnm[1] & $1.475$ & $3.215$ & $4.176$ \\
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SF-ADC(2)-s\fnm[2] & $1.573$ & $3.208$ & $4.247$ \\
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SF-ADC(2)-x\fnm[2] & $1.576$ & $3.141$ & $3.796$ \\
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SF-ADC(3)\fnm[2] & $1.456$ & $3.285$ & $4.334$ \\
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SF-BSE@{\GOWO}\fnm[3] & $1.438$ & $2.704$ & $4.540$ \\
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SF-dBSE@{\GOWO}\fnm[3] & $1.403$ & $2.883$ & $4.621$ \\
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\end{tabular}
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\end{ruledtabular}
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\fnt[1]{Spin-flip EOM-CC value from Ref.~\onlinecite{Manohar_2008}.}
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@ -821,22 +809,22 @@ So here we have an example where the dynamical corrections are necessary to get
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All the spin-flip calculations have been performed with a UHF reference and the cc-pVTZ basis set.
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\label{tab:CBD_D4h}}
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\begin{ruledtabular}
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\begin{tabular}{lccc}
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\begin{tabular}{lrrr}
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& \mc{3}{c}{Excitation energies (eV)} \\
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\cline{2-4}
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Method & $1\,{}^3A_{2g}$ & $2\,{}^1A_{1g}$ & $1\,{}^1B_{2g}$ \\
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\hline
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SF-TD-B3LYP \fnm[3] &-0.020 & 0.486& 0.547\\
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SF-TD-BH\&HLYP\fnm[3] &0.048 & 1.282&1.465 \\
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SF-CIS\fnm[1] &0.317 & 3.125&2.650 \\
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EOM-SF-CCSD\fnm[1] &0.369 & 1.824& 2.143\\
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EOM-SF-CCSD(fT)\fnm[1] & 0.163&1.530 &1.921 \\
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EOM-SF-CCSD(dT)\fnm[1] &0.098 &1.456 &1.853 \\
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SF-ADC(2)-s\fnm[2] & 0.266& 1.664& 1.910 \\
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SF-ADC(2)-x\fnm[2] & 0.217&1.123 &1.799 \\
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SF-ADC(3)\fnm[2] &0.083 &1.621 &1.930 \\
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SF-BSE@{\GOWO}\fnm[3] & -0.049 & 1.189 & 1.480 \\
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SF-dBSE@{\GOWO}\fnm[3] & 0.012 & 1.507 & 1.841 \\
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SF-TD-B3LYP\fnm[3] & $-0.020$ & $0.486$ & $0.547$ \\
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SF-TD-BH\&HLYP\fnm[3] & $0.048$ & $1.282$ & $1.465$ \\
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SF-CIS\fnm[1] & $0.317$ & $3.125$ & $2.650$ \\
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EOM-SF-CCSD\fnm[1] & $0.369$ & $1.824$ & $2.143$ \\
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EOM-SF-CCSD(fT)\fnm[1] & $0.163$ & $1.530$ & $1.921$ \\
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EOM-SF-CCSD(dT)\fnm[1] & $0.098$ & $1.456$ & $1.853$ \\
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SF-ADC(2)-s\fnm[2] & $0.266$ & $1.664$ & $1.910$ \\
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SF-ADC(2)-x\fnm[2] & $0.217$ & $1.123$ & $1.799$ \\
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SF-ADC(3)\fnm[2] & $0.083$ & $1.621$ & $1.930$ \\
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SF-BSE@{\GOWO}\fnm[3] & $-0.049$ & $1.189$ & $1.480$ \\
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SF-dBSE@{\GOWO}\fnm[3] & $0.012$ & $1.507$ & $1.841$ \\
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\end{tabular}
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\end{ruledtabular}
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\fnt[1]{Spin-flip EOM-CC value from Ref.~\onlinecite{Manohar_2008}.}
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