{The authors present a new formulation of Bethe-Salpeter equations (RPAx on GW reference) for spin-flip excitations. Despite the existence of many formulations of spin-flip (SF) currently, this is the first time it has been applied to GW/BSE. All the spin-flip formulations have a similar structure, changing essentially the amount of exchange and correlation effects dug in the orbital energies and the particle-hole couplings. The authors present the theory in deep detail and perform the basic applications that allow the assessment of the performance of SF-BSE.
{Figure 1/3: these show quite a relevant assessment of the performance of different SF methods. However, I think that the comparison with SF-TDDFT is unfair. None of the DFT exchange functionals is long-range corrected, whereas all other methods have the exact long-range exchange. Could the authors add the data for a long-range corrected functional?}
\alert{Following the excellent advice of Reviewer \#1, we have added data for the following range-separated hybrid functionals: CAM-B3LYP, LC-$\omega$HPBE, and $\omega$B97X-D.
{Figure 1: The similarity between SF-dBSE and SF-ADC(2)-s is more than simply the results.
I would say that the two formulations are equivalent, and should lead to the same results in Figure 1 if the authors would have used GW/SF-dBSE instead of G0W0/SF-dBSE.
Could the authors add these results based on GW and comment?}
\alert{The reviewer is right to mention similarities between the SF-dBSE and SF-ADC(2)-s schemes.
However, they are not strictly identical as ADC(2) includes second-order exchange diagrams which are not present in SF-dBSE@$GW$, even in the case of more elaborate schemes like ev$GW$ and qs$GW$.
To illustrate this and accordingly to the reviewer's suggestion, we have added the partially self-consistent SF-dBSE@ev$GW$ results as well as the fully self-consistent SF-dBSE@qs$GW$ results.
As one can see, in the case of Be, there is not much differences between these schemes and the original SF-dBSE@$G_0W_0$ which nicely illustrates that HF eigenstates are actually an excellent starting point in this particular case.
{Figure 1: The only difference between SF-ADC(2)-s and SF-ADC(2)-x is that the energy difference in the dynamic part of the BSE equation is corrected to first order.
The equivalent thing in SF-BSE would be to add in Eq. 30 the direct and exchange corrections in the orbital energy difference appearing in the denominator of the second term (i.e., similar to using the diagonal part of Eq. 29a and 29c in the orbital energy difference of Eq. 30).
We have performed SF-dBSE@$G_0W_0$ calculations replacing the $GW$ quasiparticle energies by the HF orbital energies in the denominator of Eq. (30) but it does not seem to alter much the results in the case of Be.}
{Figure 2: Could the authors discuss the kink in G0W0/SF-BSE and G0W0/SF-dBSE (in supporting) appearing at around 1.2 Angstroms between $1\Sigma_g^+$ and $1\Sigma_u^+$. It is really puzzling.
Is it due to the lack of self consistency in the G0W0 approximation?
Note that, as mentioned in our manuscript, all the calculations are performed with a UHF reference (even the ones based on a closed-shell singlet reference).
Of course, if one relies solely on the RHF solution, this kink disappears as illustrated by the new figure that we have included in the Supporting Information.
Unfortunately, it is extremely tedious to converge (partially) self-consistent $GW$ calculations with such large basis set (cc-pVQZ) for reasons discussed elsewhere [see, for example, V\'eril et al. JCTC 14, 5220 (2018)].}
