done with H2 for now
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@ -688,30 +688,24 @@ All these calculations are performed in the cc-pVQZ basis, and both the spin-con
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The top panel of Fig.~\ref{fig:H2} shows the CIS (dotted lines) and SF-CIS (dashed lines) excitation energies as a function of $R(\ce{H-H})$.
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The EOM-CCSD reference energies are represented by solid lines.
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We observe that both CIS and SF-CIS poorly describe the $\text{B}\,{}^1\Sigma_u^+$ state, especially in the dissociation limit with an error greater than $1$ eV.
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The same analysis can be done for the $\text{F}\,{}^1\Sigma_g^+$ state at dissociation.
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The EOM-CSSD curves clearly evidence the avoided crossing between the $\text{E}$ and $\text{F}$ states.
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We observe that both CIS and SF-CIS poorly describe the $\text{B}\,{}^1\Sigma_u^+$ state in the dissociation limit with an error greater than $1$ eV, while CIS, unlike SF-CIS, is much more accurate around the equilibrium geometry \titou{(spin-contamination of the SF-CIS wave function?)}.
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Similar observations can be made for the $\text{E}\,{}^1\Sigma_g^+$ state with a good description at the CIS level for all bond lengths.
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SF-CIS does not model accurately the $\text{E}\,{}^1\Sigma_g^+$ state before the avoided crossing, but the agreement between SF-CIS and EOM-CCSD is much satisfactory for bond length greater than $1.6$ \AA.
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Oppositely, SF-CIS describes better the $\text{F}\,{}^1\Sigma_g^+$ state before the avoided crossing than after.
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Nonetheless, this results in a rather good qualitative agreement with an avoided crossing placed at a slightly larger bond length than at the EOM-CCSD level.
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As mentioned earlier, CIS is unable to locate any avoided crossing as it cannot access double excitations.
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However, CIS is quite accurate for the $\text{E}\,{}^1\Sigma_g^+$.
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\titou{Spin-contamination of the E state?}
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Oppositely, SF-CIS describes better the $\text{F}\,{}^1\Sigma_g^+$ state before the avoided crossing than after \titou{(spin-contamination of the SF-CIS wave function?)}, while this state is completely absent at the CIS level.
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Indeed, as mentioned earlier, CIS is unable to locate any avoided crossing as it cannot access double excitations.
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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 center panel of Fig.~\ref{fig:H2}, we report the (SF-)TD-BH\&HLYP results.
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TD-BH\&HLYP shows bad results for all the states of interest with and without spin-flip.
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Note that \ce{H2} is a rather challenging system for (SF)-TD-DFT from a general point of view. \cite{Cohen_2008a,Cohen_2008c,Cohen_2012}
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Indeed, for the three states we have a difference in the excitation energy at the dissociation limit of several eV with and without spin-flip.
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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{Cohen_2008a,Cohen_2008c,Cohen_2012}
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In the bottom panel of Fig.~\ref{fig:H2} we have results for BSE calculation with and without spin-flip.
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SF-BSE gives a good representation of the $\text{B}\,{}^1\Sigma_u^+$ state with error of 0.05-0.3 eV.
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However SF-BSE does not describe well the $\text{E}\,{}^1\Sigma_g^+$ state with error of 0.5-1.6 eV. SF-BSE shows a good agreement with the EOM-CCSD reference for the double excitation to the $\text{F}\,{}^1\Sigma_g^+$ state, indeed we have an error of 0.008-0.6 eV. BSE results for the $\text{B}\,{}^1\Sigma_u^+$ state are close to the reference until 2.0 \AA~ and the give bad agreement for the dissociation limit.
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For the $\text{E}\,{}^1\Sigma_g^+$ state BSE gives closer results to the reference than SF-BSE.
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However we can observe that for all the methods that we compared, when the spin-flip is not used standard methods can not retrieve double excitation.
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There is no avoided crossing or perturbation in the curve for the $\text{E}\,{}^1\Sigma_g^+$ state when spin-flip is not used.
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This is because for these methods we are in the space of single excitation and de-excitation.
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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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A similar graph comparing (SF-)dBSE and EOM-CCSD excitation energies can be found in the {\SI}.
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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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However SF-BSE does not describe well the $\text{E}\,{}^1\Sigma_g^+$ state with error ranging from half an eV to $1.6$ eV \titou{(spin-contamination of the SF-BSE wave function?)}.
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Similar performances are observed at the BSE level around equilibrium with a clear improvement in the dissociation limit.
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Remarkably, SF-BSE shows a good agreement with EOM-CCSD for the $\text{F}\,{}^1\Sigma_g^+$ doubly-excited state, resulting in an avoided crossing around $R(\ce{H-H}) = 1.6$ \AA.
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%%% FIG 2 %%%
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\begin{figure}
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Notebooks/sf-BSE.nb
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Notebooks/sf-BSE.nb
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