done for now with Be results

Manuscript/sfBSE.tex

@@ -613,16 +613,23 @@ Beryllium has a $^1S$ ground state with $1s^2 2s^2$ configuration.
 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}.
 
 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\%. 
-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 energy is given by the energy difference between the $2s$ and $2p$ orbitals due to the lack of coupling terms in the spin-flip block of the SD-TD-DFT equations (see Subsec.~\ref{sec:BSE}). 
-Including exact exchange, like in SF-TD-B3LYP and SF-TD-BH\&HLYP, lifts this degeneracy. 
+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.
+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.
+Including exact exchange, like in SF-TD-B3LYP and SF-TD-BH\&HLYP, lifts this degeneracy and improves the description of both states.
 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. 
 For the other states, the agreement between SF-TD-BH\&HLYP and FCI is significantly improved. 
 
-The center part of Fig.~\ref{fig:Be} (blue lines) shows the SF-BSE results alongside the SF-CIS excitation energies (purple lines). 
-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.
-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. 
-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. 
-Generally we can observe that all the scheme with SF-BSE used do not increase significantly the accuracy of excitations energies.
+The center part of Fig.~\ref{fig:Be} shows the SF-(d)BSE results (blue lines) alongside the SF-CIS excitation energies (purple lines). 
+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.
+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.
+At the exception of the $^1D$ state, SF-BSE improves over SF-CIS with a rather small contribution from the additional dynamical effect.
+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.
+
+The right side of Fig.~\ref{fig:Be} illustrates the performance of the SF-ADC methods.
+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.
+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.
+
+\titou{Here comes the discussion for the larger basis.}
 
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