done with H2 for now

Manuscript/sfBSE.tex

@@ -688,30 +688,24 @@ All these calculations are performed in the cc-pVQZ basis, and both the spin-con
 
 The top panel of Fig.~\ref{fig:H2} shows the CIS (dotted lines) and SF-CIS (dashed lines) excitation energies as a function of $R(\ce{H-H})$.
 The EOM-CCSD reference energies are represented by solid lines.
-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. 
-The same analysis can be done for the $\text{F}\,{}^1\Sigma_g^+$ state at dissociation. 
-The EOM-CSSD curves clearly evidence the avoided crossing between the $\text{E}$ and $\text{F}$ states. 
+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?)}.
+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.
 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. 
-Oppositely, SF-CIS describes better the $\text{F}\,{}^1\Sigma_g^+$ state before the avoided crossing than after. 
-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.
-As mentioned earlier, CIS is unable to locate any avoided crossing as it cannot access double excitations.
-However, CIS is quite accurate for the $\text{E}\,{}^1\Sigma_g^+$.
-\titou{Spin-contamination of the E state?}
+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.
+Indeed, as mentioned earlier, CIS is unable to locate any avoided crossing as it cannot access double excitations.
+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.
 
 In the center panel of Fig.~\ref{fig:H2}, we report the (SF-)TD-BH\&HLYP results.
-TD-BH\&HLYP shows bad results for all the states of interest with and without spin-flip. 
-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}
-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. 
 Similar graphs for (SF-)TD-BLYP and (SF-)TD-B3LYP are reported in the {\SI}.
+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.
+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}
 
-In the bottom panel of Fig.~\ref{fig:H2} we have results for BSE calculation with and without spin-flip. 
-SF-BSE gives a good representation of the $\text{B}\,{}^1\Sigma_u^+$ state with error of  0.05-0.3 eV. 
-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. 
-For the $\text{E}\,{}^1\Sigma_g^+$ state BSE gives closer results to the reference than SF-BSE. 
-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. 
-There is no avoided crossing or perturbation in the curve for the $\text{E}\,{}^1\Sigma_g^+$ state when spin-flip is not used. 
-This is because for these methods we are in the space of single excitation and de-excitation.
+In the bottom panel of Fig.~\ref{fig:H2}, (SF-)BSE excitation energies for the same three singlet states are represented. 
 A similar graph comparing (SF-)dBSE and EOM-CCSD excitation energies can be found in the {\SI}.
+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.
+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?)}. 
+Similar performances are observed at the BSE level around equilibrium with a clear improvement in the dissociation limit.
+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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