add 2 comments

dynker.bib

@@ -1,7 +1,7 @@
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-%% Created for Pierre-Francois Loos at 2020-08-28 15:45:04 +0200 
+%% Created for Pierre-Francois Loos at 2020-08-29 20:19:19 +0200 
 
 
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dynker.tex

@@ -510,6 +510,7 @@ This can be further verified by switching off gradually the electron-electron in
 
 Enforcing the TDA, which corresponds to neglecting the coupling term between the resonant and anti-resonant parts of the dBSE Hamiltonian \eqref{eq:HBSE}, does not change the situation in terms of spurious solutions: there is still one spurious excitation in the triplet manifold ($\omega_{2}^{\BSE,\upup}$), and the two solutions for the singlet manifold which corresponds to the single and double excitations.
 However, it does increase significantly the static excitations while the magnitude of the dynamical corrections is not altered by the TDA.
+The (static) BSE triplets are notably too low in energy as compared to the exact results and the TDA is able to partly reduce this error, a situation analogous in larger systems. \cite{Jacquemin_2017a,Jacquemin_2017b,Rangel_2017}
 
 Another way to access dynamical effects while staying in the static framework is to use perturbation theory, \cite{Rohlfing_2000,Ma_2009a,Ma_2009b,Baumeier_2012b,Loos_2020e} a scheme we label as perturbative BSE (pBSE).
 To do so, one must decompose the dBSE Hamiltonian into a (zeroth-order) static part and a dynamical perturbation, such that
@@ -563,7 +564,7 @@ and the renormalization factor is
 \end{equation}
 This corresponds to a dynamical perturbative correction to the static excitations. 
 
-The perturbatively-corrected values are also reported in Table \ref{tab:BSE}, which shows that this scheme is very effective at reproducing the dynamical values for the single excitations.
+The perturbatively-corrected values are also reported in Table \ref{tab:BSE}, and it shows that this scheme is very effective at reproducing the dynamical values for the single excitations.
 Because the value of $Z_{1}$ is always quite close to unity in the present systems (evidencing that the perturbative expansion behaves nicely), one could have anticipated the fact that the first-order correction is a good estimate of the non-perturbative result.
 However, because the perturbative treatment is ultimately static, one cannot access double excitations with such a scheme.
 
@@ -716,7 +717,8 @@ Moreover, because of the non-linear character of the linear response problem whe
 
 Using a simple two-model system, we have explored the physics of three dynamical kernels: i) a kernel based on the dressed TDDFT method introduced by Maitra and coworkers, \cite{Maitra_2004} ii) the dynamical kernel from the BSE formalism derived by Strinati in his hallmark 1988 paper, \cite{Strinati_1988} as well as the second-order BSE kernel derived by Zhang \textit{et al.}, \cite{Zhang_2013} and Rebolini and Toulouse. \cite{Rebolini_2016,Rebolini_PhD}
 Prototypical examples of valence, charge-transfer, and Rydberg excited states have been considered.
-From these, we have observed that, overall, the dynamical correction usually improves the static excitation energies, and that, if one has no interest in double excitations, a perturbative treatment is an excellent alternative to a non-linear resolution of the dynamical equations.
+From these, we have observed that, overall, the dynamical correction usually improves the static excitation energies, and that, although one can access double excitations, the accuracy of the BSE and BSE2 kernels for double excitations is rather average.
+If one has no interest in double excitations, a perturbative treatment is an excellent alternative to a non-linear resolution of the dynamical equations.
 
 We hope that the present contribution will foster new developments around dynamical kernels for optical excitations, in particular to access double excitations in molecular systems.