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@ -247,7 +247,7 @@ Here we apply the spin-flip technique to the BSE formalism in order to access, i
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The present BSE calculations are based on the spin-unrestricted version of both $GW$ (Sec.~\ref{sec:UGW}) and BSE (Sec.~\ref{sec:UBSE}).
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To the best of our knowledge, the present study is the first to apply the spin-flip formalism to the BSE method.
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Moreover, we also go beyond the static approximation by taking into account dynamical effects (Sec.~\ref{sec:dBSE}) via an unrestricted generalization of our recently developed (renormalized) perturbative correction which builds on the seminal work of Strinati, \cite{Strinati_1982,Strinati_1984,Strinati_1988} Romaniello and collaborators, \cite{Romaniello_2009b,Sangalli_2011} and Rohlfing and coworkers. \cite{Rohlfing_2000,Ma_2009a,Ma_2009b,Baumeier_2012b,Lettmann_2019}
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We also discuss the computation of oscillator strengths (Sec.~\ref{sec:os}) and the expectation value of the spin operator $\expval*{\hS^2}$ as a diagnostic of the spin contamination for both ground and excited states (Sec.~\ref{sec:spin}).
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We also discuss the computation of oscillator strengths (Sec.~\ref{sec:os}) and the expectation value of the spin operator $\expval{\hS^2}$ as a diagnostic of the spin contamination for both ground and excited states (Sec.~\ref{sec:spin}).
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Computational details are reported in Sec.~\ref{sec:compdet} and our results for the beryllium atom \ce{Be} (Subsec.~\ref{sec:Be}), the hydrogen molecule \ce{H2} (Subsec.~\ref{sec:H2}), and cyclobutadiene \ce{C4H4} (Subsec.~\ref{sec:CBD}) are discussed in Sec.~\ref{sec:res}.
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Finally, we draw our conclusions in Sec.~\ref{sec:ccl}.
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Unless otherwise stated, atomic units are used.
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@ -720,28 +720,28 @@ For spin-flip transitions, we have $f_{m}^{\spf} = 0$ as the transition matrix e
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\label{sec:spin}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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One of the key issues of linear response formalism based on unrestricted references is spin contamination or the artificial mixing with configurations of different spin multiplicities.
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As nicely explained in Ref.~\onlinecite{Casanova_2020}, there are two sources of spin contamination: i) spin contamination of the reference configuration for which, for example, $\expval*{\hS^2} > 2$ for high-spin triplets, and ii) spin contamination of the excited states due to spin incompleteness of the CI expansion.
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As nicely explained in Ref.~\onlinecite{Casanova_2020}, there are two sources of spin contamination: i) spin contamination of the reference configuration for which, for example, $\expval{\hS^2} > 2$ for high-spin triplets, and ii) spin contamination of the excited states due to spin incompleteness of the CI expansion.
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The latter issue is an important source of spin contamination in the present context as BSE is limited to single excitations with respect to the reference configuration.
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Specific schemes have been developed to palliate these shortcomings and we refer the interested reader to Ref.~\onlinecite{Casanova_2020} for a detailed discussion on this matter.
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In order to monitor closely how contaminated are these states, we compute
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\begin{equation}
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\expval*{\hS^2}_m = \expval*{\hS^2}_0 + \Delta \expval*{\hS^2}_m
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\expval{\hS^2}_m = \expval{\hS^2}_0 + \Delta \expval{\hS^2}_m
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\end{equation}
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where
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\begin{equation}
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\expval*{\hS^2}_{0}
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\expval{\hS^2}_{0}
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= \frac{n_{\up} - n_{\dw}}{2} \qty( \frac{n_{\up} - n_{\dw}}{2} + 1 )
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+ n_{\dw} - \sum_p (p_{\up}|p_{\dw})^2
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\end{equation}
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is the expectation value of $\hS^2$ for the reference configuration, the first term corresponding to the exact value of $\expval*{\hS^2}$, and
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is the expectation value of $\hS^2$ for the reference configuration, the first term corresponding to the exact value of $\expval{\hS^2}$, and
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\begin{equation}
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\label{eq:OV}
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(p_\sig|q_\sigp) = \int \MO{p_\sig}(\br) \MO{q_\sigp}(\br) d\br
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\end{equation}
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are overlap integrals between spin-up and spin-down orbitals.
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For a given single excitation $m$, the explicit expressions of $\Delta \expval*{\hS^2}_m^{\spc}$ and $\Delta \expval*{\hS^2}_m^{\spf}$ can be found in the Appendix of Ref.~\onlinecite{Li_2011a} for spin-conserved and spin-flip excitations, and are functions of the vectors $\bX{m}{}$ and $\bY{m}{}$ as well as the orbital overlaps defined in Eq.~\eqref{eq:OV}.
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For a given single excitation $m$, the explicit expressions of $\Delta \expval{\hS^2}_m^{\spc}$ and $\Delta \expval{\hS^2}_m^{\spf}$ can be found in the Appendix of Ref.~\onlinecite{Li_2011a} for spin-conserved and spin-flip excitations, and are functions of the vectors $\bX{m}{}$ and $\bY{m}{}$ as well as the orbital overlaps defined in Eq.~\eqref{eq:OV}.
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\section{Computational details}
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@ -804,7 +804,7 @@ Finally, both SF-ADC(2)-x and SF-ADC(3) yield excitation energies very close to
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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 basis set.
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All the spin-flip calculations have been performed with a UHF reference.
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The $\expval*{\hS^2}$ value associated with each state is reported in parenthesis (when available).
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The $\expval{\hS^2}$ 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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@ -872,9 +872,10 @@ Indeed, as mentioned earlier, CIS is unable to locate any avoided crossing as it
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At the SF-CIS level, the avoided crossing between the $\text{E}$ and $\text{F}$ states is qualitatively reproduced and placed at a slightly larger bond length than at the EOM-CCSD level.
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In the central panel of Fig.~\ref{fig:H2}, we report the (SF-)TD-BH\&HLYP results.
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Similar graphs for (SF-)TD-BLYP and (SF-)TD-B3LYP are reported in the {\SI}.
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SF-TD-BH\&HLYP shows, at best, qualitative agreement with EOM-CCSD, while the TD-BH\&HLYP excitation energies of the $\text{B}$ and $\text{E}$ states are only trustworthy around equilibrium but inaccurate at dissociation.
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Note that \ce{H2} is a rather challenging system for (SF)-TD-DFT from a general point of view. \cite{Vuckovic_2017,Cohen_2008a,Cohen_2008c,Cohen_2012}
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Similar graphs for (SF-)TD-BLYP and (SF-)TD-B3LYP are reported in the {\SI} from which one can draw similar conclusions.
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Notably, one can see that the avoided crossing is not modeled at the SF-TD-BLYP level due to the lack of Hartree-Fock exchange.
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In the bottom panel of Fig.~\ref{fig:H2}, (SF-)BSE excitation energies for the same three singlet states are represented.
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SF-BSE provides surprisingly accurate excitation energies for the $\text{B}\,{}^1\Sigma_u^+$ state with errors between $0.05$ and $0.3$ eV, outperforming in the process the standard BSE formalism.
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@ -884,27 +885,27 @@ Remarkably, SF-BSE shows a good agreement with EOM-CCSD for the $\text{F}\,{}^1\
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A similar graph comparing (SF-)dBSE and EOM-CCSD excitation energies can be found in the {\SI} where it is shown that dynamical effects do not affect the present conclusions.
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The right side of Fig.~\ref{fig:H2} shows the amount of spin contamination as a function of the bond length for SF-CIS (top), SF-TD-BH\&HLYP (center), and SF-BSE (bottom).
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Overall, one can see that $\expval*{\hS^2}$ behaves similarly for SF-CIS and SF-BSE with a small spin contamination of the $\text{B}\,{}^1\Sigma_u^+$ at short bond length. In contrast, the $\text{B}$ state is much more spin contaminated at the SF-TD-BH\&HLYP level.
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For all spin-flip methods, the $\text{E}$ is strongly spin contaminated as expected, while the $\expval*{\hS^2}$ values associated with the $\text{F}$ state
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Overall, one can see that $\expval{\hS^2}$ behaves similarly for SF-CIS and SF-BSE with a small spin contamination of the $\text{B}\,{}^1\Sigma_u^+$ at short bond length. In contrast, the $\text{B}$ state is much more spin contaminated at the SF-TD-BH\&HLYP level.
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For all spin-flip methods, the $\text{E}$ is strongly spin contaminated as expected, while the $\expval{\hS^2}$ values associated with the $\text{F}$ state
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only deviate significantly from zero for short bond length and around the avoided crossing where it strongly couples with the spin contaminated $\text{E}$ state.
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%%% FIG 2 %%%
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\begin{figure*}
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\includegraphics[width=0.4\linewidth]{H2_CIS}
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\includegraphics[width=0.45\linewidth]{H2_CIS}
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\hspace{0.05\linewidth}
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\includegraphics[width=0.4\linewidth]{H2_CIS_S2}
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\includegraphics[width=0.45\linewidth]{H2_CIS_S2}
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\vspace{0.025\linewidth}
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\\
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\includegraphics[width=0.4\linewidth]{H2_BHHLYP}
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\includegraphics[width=0.45\linewidth]{H2_BHHLYP}
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\hspace{0.05\linewidth}
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\includegraphics[width=0.4\linewidth]{H2_BHHLYP_S2}
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\includegraphics[width=0.45\linewidth]{H2_BHHLYP_S2}
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\vspace{0.025\linewidth}
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\\
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\includegraphics[width=0.4\linewidth]{H2_BSE}
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\includegraphics[width=0.45\linewidth]{H2_BSE}
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\hspace{0.05\linewidth}
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\includegraphics[width=0.4\linewidth]{H2_BSE_S2}
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\includegraphics[width=0.45\linewidth]{H2_BSE_S2}
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\caption{
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Excitation energies with respect to the $\text{X}\,{}^1 \Sigma_g^+$ ground state (left) and expectation value of the spin operator $\expval*{\hS^2}$ (right) of the $\text{B}\,{}^1\Sigma_u^+$ (red), $\text{E}\,{}^1\Sigma_g^+$ (black), and $\text{F}\,{}^1\Sigma_g^+$ (blue) states of \ce{H2} obtained with the cc-pVQZ basis at the (SF-)CIS (top), (SF-)TD-BH\&HLYP (middle), and (SF-)BSE (bottom) levels of theory.
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Excitation energies with respect to the $\text{X}\,{}^1 \Sigma_g^+$ ground state (left) and expectation value of the spin operator $\expval{\hS^2}$ (right) of the $\text{B}\,{}^1\Sigma_u^+$ (red), $\text{E}\,{}^1\Sigma_g^+$ (black), and $\text{F}\,{}^1\Sigma_g^+$ (blue) states of \ce{H2} obtained with the cc-pVQZ basis at the (SF-)CIS (top), (SF-)TD-BH\&HLYP (middle), and (SF-)BSE (bottom) levels of theory.
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The reference EOM-CCSD excitation energies are represented as solid lines, while the results obtained with and without spin-flip are represented as dashed and dotted lines, respectively.
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All the spin-conserved and spin-flip calculations have been performed with an unrestricted reference.
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The raw data are reported in the {\SI}.
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@ -949,9 +950,9 @@ This issue does not appear at the SF-BSE, SF-ADC, and SF-EOM-SF-CCSD levels.
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%%% FIG 3 %%%
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\begin{figure*}
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\includegraphics[width=0.4\linewidth]{CBD_D2h}
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\includegraphics[width=0.45\linewidth]{CBD_D2h}
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\hspace{0.05\linewidth}
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\includegraphics[width=0.4\linewidth]{CBD_D4h}
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\includegraphics[width=0.45\linewidth]{CBD_D4h}
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\caption{
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Vertical excitation energies of CBD.
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Left: $1\,{}^3B_{1g}$, $1\,{}^1B_{1g}$, and $2\,{}^1A_{1g}$ states at the $D_{2h}$ rectangular equilibrium geometry of the $\text{X}\,{}^1 A_{g}$ ground state (see Table \ref{tab:CBD_D2h} for the raw data).
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@ -973,22 +974,23 @@ This issue does not appear at the SF-BSE, SF-ADC, and SF-EOM-SF-CCSD levels.
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\cline{2-4}
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Method & $1\,{}^3B_{1g}$ & $1\,{}^1B_{1g}$ & $2\,{}^1A_{g}$ \\
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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] & $1.521$ & $3.836$ & $5.499$ \\
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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-TD-B3LYP\fnm[1] & $1.750$ & $2.260$ & $4.094$ \\
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SF-TD-BH\&HLYP\fnm[1] & $1.583$ & $2.813$ & $4.528$ \\
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SF-CIS\fnm[2] & $1.521$ & $3.836$ & $5.499$ \\
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EOM-SF-CCSD\fnm[3] & $1.654$ & $3.416$ & $4.360$ \\
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EOM-SF-CCSD(fT)\fnm[3] & $1.516$ & $3.260$ & $4.205$ \\
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EOM-SF-CCSD(dT)\fnm[3] & $1.475$ & $3.215$ & $4.176$ \\
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SF-ADC(2)-s\fnm[4] & $1.573$ & $3.208$ & $4.247$ \\
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SF-ADC(2)-x\fnm[4] & $1.576$ & $3.141$ & $3.796$ \\
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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-BSE@{\GOWO}\fnm[1] & $1.438$ & $2.704$ & $4.540$ \\
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SF-dBSE@{\GOWO}\fnm[1] & $1.403$ & $2.883$ & $4.621$ \\
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\end{tabular}
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\end{ruledtabular}
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\fnt[1]{Spin-flip EOM-CC values from Ref.~\onlinecite{Manohar_2008}.}
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\fnt[2]{Values from Ref.~\onlinecite{Lefrancois_2015}.}
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\fnt[3]{This work.}
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\fnt[1]{This work.}
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\fnt[2]{Values from Ref.~\onlinecite{Casanova_2020}.}
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\fnt[3]{Values from Ref.~\onlinecite{Manohar_2008}.}
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\fnt[4]{Values from Ref.~\onlinecite{Lefrancois_2015}.}
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\end{table}
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%%% %%% %%% %%%
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@ -1004,22 +1006,23 @@ This issue does not appear at the SF-BSE, SF-ADC, and SF-EOM-SF-CCSD levels.
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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] & $-0.020$ & $0.547$ & $0.486$ \\
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SF-TD-BH\&HLYP\fnm[3] & $0.048$ & $1.465$ & $1.282$ \\
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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[1] & $-0.020$ & $0.547$ & $0.486$ \\
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SF-TD-BH\&HLYP\fnm[1] & $0.048$ & $1.465$ & $1.282$ \\
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SF-CIS\fnm[2] & $0.317$ & $3.125$ & $2.650$ \\
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EOM-SF-CCSD\fnm[3] & $0.369$ & $1.824$ & $2.143$ \\
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EOM-SF-CCSD(fT)\fnm[3] & $0.163$ & $1.530$ & $1.921$ \\
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EOM-SF-CCSD(dT)\fnm[3] & $0.098$ & $1.456$ & $1.853$ \\
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SF-ADC(2)-s\fnm[4] & $0.266$ & $1.664$ & $1.910$ \\
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SF-ADC(2)-x\fnm[4] & $0.217$ & $1.123$ & $1.799$ \\
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SF-ADC(3)\fnm[4] & $0.083$ & $1.621$ & $1.930$ \\
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SF-BSE@{\GOWO}\fnm[1] & $-0.049$ & $1.189$ & $1.480$ \\
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SF-dBSE@{\GOWO}\fnm[1] & $0.012$ & $1.507$ & $1.841$ \\
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\end{tabular}
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\end{ruledtabular}
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\fnt[1]{Spin-flip EOM-CC values from Ref.~\onlinecite{Manohar_2008}.}
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\fnt[2]{Values from Ref.~\onlinecite{Lefrancois_2015}.}
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\fnt[3]{This work.}
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\fnt[1]{This work.}
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\fnt[2]{Values from Ref.~\onlinecite{Casanova_2020}.}
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\fnt[3]{Values from Ref.~\onlinecite{Manohar_2008}.}
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\fnt[4]{Values from Ref.~\onlinecite{Lefrancois_2015}.}
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\end{table}
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%%% %%% %%% %%%
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2325
Notebooks/sf-BSE.nb
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