diff --git a/dBSE.pdf b/dBSE.pdf
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Binary files a/dBSE.pdf and b/dBSE.pdf differ
diff --git a/dBSE2.pdf b/dBSE2.pdf
index d5ae500..3a9b14c 100644
Binary files a/dBSE2.pdf and b/dBSE2.pdf differ
diff --git a/dynker.bib b/dynker.bib
index 9bccf41..d09fb10 100644
--- a/dynker.bib
+++ b/dynker.bib
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 %% This BibTeX bibliography file was created using BibDesk.
 %% http://bibdesk.sourceforge.net/
 
-%% Created for Pierre-Francois Loos at 2020-08-27 21:46:59 +0200 
+%% Created for Pierre-Francois Loos at 2020-08-28 14:15:42 +0200 
 
 
 %% Saved with string encoding Unicode (UTF-8) 
@@ -42,9 +42,9 @@
 	Bdsk-Url-1 = {https://doi.org/10.1016/0022-2852(63)90151-6}}
 
 @article{Petersilka_1996,
-	Author = {M. Petersilka and U. J. Gossmann and and E. K. U. Gross},
+	Author = {M. Petersilka and U. J. Gossmann and E. K. U. Gross},
 	Date-Added = {2020-06-26 09:43:33 +0200},
-	Date-Modified = {2020-06-26 09:45:05 +0200},
+	Date-Modified = {2020-08-28 13:56:10 +0200},
 	Doi = {10.1103/PhysRevLett.76.1212},
 	Journal = {Phys. Rev. Lett.},
 	Pages = {1212},
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diff --git a/dynker.tex b/dynker.tex
index baf590c..62cdd6f 100644
--- a/dynker.tex
+++ b/dynker.tex
@@ -48,7 +48,7 @@ Linear response theory is a powerful approach that allows to directly access the
 From a practical point of view, these quantities are obtained by solving non-linear, frequency-dependent Casida-like equations in the space of single excitations and de-excitations \cite{Casida_1995}
 \begin{equation} \label{eq:LR}
 	\begin{pmatrix}
-		\bR^{\sigma}(\omega_S)		&	\bC^{\sigma}(\omega_S)
+		+\bR^{\sigma}(+\omega_S)	&	+\bC^{\sigma}(+\omega_S)
 		\\
 		-\bC^{\sigma}(-\omega_S)^*	&	-\bR^{\sigma}(-\omega_S)^*
 	\end{pmatrix}
@@ -269,7 +269,7 @@ The numerical values of the various quantities defined above are gathered in Tab
 %\end{table*}
 %%% %%% %%% %%
 
-The exact values of the singlet single and double excitations, $\omega_{1}^{\updw}$ and $\omega_{3}^{\updw}$, and the triplet single excitation, $\omega_{1}^{\upup}$,  are reported, for example, in Table \ref{tab:Maitra}.
+The exact values of the singlet single and double excitations, $\omega_{1}^{\updw}$ and $\omega_{2}^{\updw}$, and the triplet single excitation, $\omega_{1}^{\upup}$,  are reported, for example, in Table \ref{tab:Maitra}.
 We are going to use these as reference for the remaining of this study.
 
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
@@ -281,7 +281,7 @@ The kernel proposed by Maitra and coworkers \cite{Maitra_2004,Cave_2004} in the
 More specifically, D-TDDFT adds to the static kernel a frequency-dependent part by reverse-engineering the exact Hamiltonian: a single and double excitations, assumed to be strongly coupled, are isolated from among the spectrum and added manually to the static kernel.
 The very same idea was taking further by Huix-Rotllant, Casida and coworkers, \cite{Huix-Rotllant_2011} and tested on a large set of molecules.
 Here, we start instead from a HF reference.
-The static problem corresponds then to the time-dependent HF (TDHF) Hamiltonian, while in the TDA, it reduces to configuration interaction with singles (CIS).
+The static problem corresponds then to the time-dependent HF (TDHF) Hamiltonian, while in the TDA, it reduces to configuration interaction with singles (CIS). \cite{Dreuw_2005}
 
 For the two-level model, the reverse-engineering process of the exact Hamiltonian \eqref{eq:H-exact} yields
 \begin{equation} \label{eq:f-Maitra}
@@ -289,13 +289,13 @@ For the two-level model, the reverse-engineering process of the exact Hamiltonia
 \end{equation}
 while $f_\text{M}^{\co,\upup}(\omega) = 0$.
 The expression \eqref{eq:f-Maitra} can be easily obtained by folding the double excitation onto the single excitation, as explained in Sec.~\ref{sec:dyn}.
-It is clear that one must know \textit{a priori} the structure of the Hamiltonian to construct such dynamical kernel, and this obviously hampers its applicability to realistic photochemical systems where it is sometimes hard to get a clear picture of the interplay between excited states. \cite{Boggio-Pasqua_2007}
+It is clear that one must know \textit{a priori} the structure of the Hamiltonian to construct such dynamical kernel, and this obviously hampers its applicability to realistic photochemical systems where it is sometimes hard to get a clear picture of the interplay between excited states. \cite{Loos_2018a,Loos_2020b,Boggio-Pasqua_2007}
 
 For the two-level model, the non-linear equations defined in Eq.~\eqref{eq:LR} provides the following effective Hamiltonian 
 \begin{equation} \label{eq:H-M}
 	\bH_\text{D-TDHF}^{\sigma}(\omega) = 
 	\begin{pmatrix}
-		R_\text{M}^{\sigma}(\omega)	&	C_\text{M}^{\sigma}(\omega)
+		+R_\text{M}^{\sigma}(+\omega)	&	+C_\text{M}^{\sigma}(+\omega)
 		\\
 		-C_\text{M}^{\sigma}(-\omega)	&	-R_\text{M}^{\sigma}(-\omega)
 	\end{pmatrix}	
@@ -312,9 +312,9 @@ with
 \end{subequations}
 yielding, for our three two-electron systems, the excitation energies reported in Table \ref{tab:Maitra} when diagonalized.
 The TDHF Hamiltonian is obtained from Eq.~\eqref{eq:H-M} by setting $f_\text{M}^{\co,\sigma}(\omega) = 0$ in Eqs.~\eqref{eq:R_M} and \eqref{eq:C_M}.
-In Fig.~\ref{fig:Maitra}, we plot $\det[\bH(\omega) - \omega \bI]$ as a function of $\omega$ for both the singlet (black and gray) and triplet (orange) manifolds in \ce{HeH+}.
+In Fig.~\ref{fig:Maitra}, we plot $\det[\bH(\omega) - \omega \bI]$ as a function of $\omega$ for both the singlet (black and gray) and triplet (orange) manifolds in \ce{HeH+}. (Very similar curves are obtained for \ce{He}.)
 The roots of $\det[\bH(\omega) - \omega \bI]$ indicate the excitation energies.
-Because, there is nothing to dress for the triplet state, only the TDHF and D-TDHF triplet excitation energies are equal.
+Because, there is nothing to dress for the triplet state, the TDHF and D-TDHF triplet excitation energies are equal.
 
 %%% TABLE II %%%
 %\begin{squeezetable}
@@ -330,15 +330,15 @@ Because, there is nothing to dress for the triplet state, only the TDHF and D-TD
 			System		&	Excitation			&	CIS		&	TDHF	&	D-CIS			&	D-TDHF			&	Exact	\\
 			\hline
 			\ce{H2}		&	$\omega_1^{\updw}$	&	$25.78$	&	$25.30$	&	$25.78[+0.00]$	&	$25.30[+0.00]$	&	$26.34$	\\
-						&	$\omega_3^{\updw}$	&			&			&					&					&	$44.04$	\\
+						&	$\omega_2^{\updw}$	&			&			&					&					&	$44.04$	\\
 						&	$\omega_1^{\upup}$	&	$15.92$	&	$15.13$	&	$15.92[+0.00]$	&	$15.13[+0.00]$	&	$16.48$	\\
-			\\
+			\hline
 			\ce{HeH+}	&	$\omega_1^{\updw}$	&	$29.68$	&	$29.42$	&	$27.75[-1.93]$	&	$27.64[-1.78]$	&	$28.05$	\\
-						&	$\omega_3^{\updw}$	&			&			&	$63.59$			&	$63.52$			&	$64.09$	\\
+						&	$\omega_2^{\updw}$	&			&			&	$63.59$			&	$63.52$			&	$64.09$	\\
 						&	$\omega_1^{\upup}$	&	$21.77$	&	$21.41$	&	$21.77[+0.00]$	&	$21.41[+0.00]$	&	$22.03$	\\
-			\\
+			\hline
 			\ce{He}		&	$\omega_1^{\updw}$	&	$52.01$	&	$51.64$	&	$51.87[-0.14]$	&	$51.52[-0.12]$	&	$52.29$	\\
-						&	$\omega_3^{\updw}$	&			&			&	$93.85$			&	$93.84$			&	$94.66$	\\
+						&	$\omega_2^{\updw}$	&			&			&	$93.85$			&	$93.84$			&	$94.66$	\\
 						&	$\omega_1^{\upup}$	&	$39.62$	&	$39.13$	&	$39.62[+0.00]$	&	$39.13[+0.00]$	&	$40.18$	\\
 		\end{tabular}
 	\end{ruledtabular}
@@ -354,7 +354,7 @@ In the case of \ce{H2} in a minimal basis, because $\mel{S}{\hH}{D} = 0$, there
 It would be, of course, a different story in a larger basis set where the coupling between singles and doubles would be non-zero.
 Table \ref{tab:Maitra} also reports the slightly improved (thanks to error compensation) CIS and D-CIS excitation energies.
 In particular, single excitations are greatly improved without altering the accuracy of the double excitation.
-Graphically, the curves obtained for CIS and D-CIS are extremely similar to the ones of TDHF and D-TDHF depicted in Fig.~\ref{fig:Maitra} for \ce{HeH+}.
+Graphically, the curves obtained for CIS and D-CIS are extremely similar to the ones of TDHF and D-TDHF depicted in Fig.~\ref{fig:Maitra}.
 
 %%% FIGURE 1 %%%
 \begin{figure}
@@ -373,16 +373,17 @@ Graphically, the curves obtained for CIS and D-CIS are extremely similar to the
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 
 As mentioned in Sec.~\ref{sec:dyn}, most of BSE calculations performed nowadays are done within the static approximation. \cite{ReiningBook,Onida_2002,Blase_2018,Blase_2020}
-However, following Strinati's footsteps, \cite{Strinati_1982,Strinati_1984,Strinati_1988} several groups have explored this formalism beyond the static approximation by retaining (or reviving) the dynamical nature of the screened Coulomb potential $W$ \cite{Sottile_2003,Romaniello_2009b,Sangalli_2011} or via a perturbative approach. \cite{Rohlfing_2000,Ma_2009a,Ma_2009b,Baumeier_2012b,Loos_2020e}
+However, following Strinati's footsteps, \cite{Strinati_1982,Strinati_1984,Strinati_1988} several groups have explored this formalism beyond the static approximation by retaining the dynamical nature of the screened Coulomb potential $W$ \cite{Sottile_2003,Romaniello_2009b,Sangalli_2011,Olevano_2019} or via a perturbative approach. \cite{Rohlfing_2000,Ma_2009a,Ma_2009b,Baumeier_2012b,Loos_2020e}
 Based on the very same two-level model that we employ here, Romaniello \textit{et al.} \cite{Romaniello_2009b} clearly evidenced that one can genuinely access additional excitations by solving the non-linear, frequency-dependent BSE eigenvalue problem. 
 For this particular system, they showed that a BSE kernel based on the random-phase approximation (RPA) produces indeed double excitations but also unphysical excitations, \cite{Romaniello_2009b} attributed to the self-screening problem. \cite{Romaniello_2009a}
 This issue was resolved in the subsequent work of Sangalli \textit{et al.} \cite{Sangalli_2011} via the design of a diagrammatic number-conserving approach based on the folding of the second-RPA Hamiltonian. \cite{Wambach_1988}
 Thanks to a careful diagrammatic analysis of the dynamical kernel, they showed that their approach produces the correct number of optically active poles, and this was further illustrated by computing the polarizability of two unsaturated hydrocarbon chains (\ce{C8H2} and \ce{C4H6}).
 Very recently, Loos and Blase have applied the dynamical correction to the BSE beyond the plasmon-pole approximation within a renormalized first-order perturbative treatment, \cite{Loos_2020e} generalizing the work of Rolhfing and coworkers on biological chromophores \cite{Ma_2009a,Ma_2009b} and dicyanovinyl-substituted oligothiophenes. \cite{Baumeier_2012b}
 They compiled a comprehensive set of vertical transitions in prototypical molecules, providing benchmark data and showing that the dynamical corrections can be sizable and improve the static BSE excitations considerably. \cite{Loos_2020e}
+Let us stress that, in all these studies, the TDA is applied to the dynamical correction (\ie, only the diagonal part of the BSE Hamiltonian is made frequency-dependent) and we shall do the same here.
 
-Within the so-called $GW$ approximation of MBPT, \cite{Golze_2019} one can easily compute the quasiparticle energies associated with the valence and conduction orbitals. 
-Assuming that $W$ has been calculated at the random-phase approximation (RPA) level and within the TDA, the expression of the $\GW$ quasiparticle energy is simply
+Within the so-called $GW$ approximation of MBPT, \cite{Aryasetiawan_1998,Onida_2002,Reining_2017,ReiningBook,Golze_2019} one can easily compute the quasiparticle energies associated with the valence and conduction orbitals. 
+Assuming that $W$ has been calculated at the random-phase approximation (RPA) level and within the TDA, the expression of the $\GW$ quasiparticle energy is simply \cite{Veril_2018}
 \begin{equation}
 	\e{p}^{\GW} = \e{p} + Z_{p}^{\GW} \SigGW{p}(\e{p})
 \end{equation}
@@ -402,9 +403,9 @@ One can now build the dynamical BSE (dBSE) Hamiltonian \cite{Strinati_1988,Roman
 \begin{equation} \label{eq:HBSE}
 	\bH_{\dBSE}^{\sigma}(\omega) = 
 	\begin{pmatrix}
-		 R_{\dBSE}^{\sigma}(\omega)		&	C_{\dBSE}^{\sigma}(\omega)
+		 +R_{\dBSE}^{\sigma}(+\omega)	&	+C_{\dBSE}^{\sigma}
 		\\
-		-C_{\dBSE}^{\sigma}(-\omega)	&	-R_{\dBSE}^{\sigma}(-\omega)
+		-C_{\dBSE}^{\sigma}	&	-R_{\dBSE}^{\sigma}(-\omega)
 	\end{pmatrix}
 \end{equation}
 with 
@@ -412,14 +413,16 @@ with
 \begin{gather}
 	R_{\dBSE}^{\sigma}(\omega) = \Delta\eGW{} + 2 \sigma \ERI{vc}{cv} - \ERI{vv}{cc} - W^{\co}_R(\omega)
 	\\
-	C_{\dBSE}^{\sigma}(\omega) = 2 \sigma \ERI{vc}{cv} -  \ERI{vc}{cv} - W^{\co}_C(\omega)
+	C_{\dBSE}^{\sigma} = 2 \sigma \ERI{vc}{cv} -  \ERI{vc}{cv} - W^{\co}_C(\omega = 0)
 \end{gather}
 \end{subequations}
 and where
 \begin{subequations}
 \begin{gather}
+	\label{eq:WR}
 	W^{\co}_R(\omega) = \frac{4 \ERI{vv}{vc} \ERI{vc}{cc}}{\omega - \Omega - \Delta\eGW{}}
 	\\
+	\label{eq:WC}
 	W^{\co}_C(\omega) = \frac{4 \ERI{vc}{cv}^2}{\omega - \Omega}
 \end{gather}
 \end{subequations}
@@ -430,7 +433,7 @@ Within the usual static approximation, the BSE Hamiltonian is simply
 \begin{equation}
 	\bH_{\BSE}^{\sigma} = 
 	\begin{pmatrix}
-		 R_{\BSE}^{\sigma} 	&	C_{\BSE}^{\sigma}
+		 +R_{\BSE}^{\sigma} &	+C_{\BSE}^{\sigma}
 		\\
 		-C_{\BSE}^{\sigma}	&	-R_{\BSE}^{\sigma}
 	\end{pmatrix}
@@ -438,9 +441,9 @@ Within the usual static approximation, the BSE Hamiltonian is simply
 with 
 \begin{subequations}
 \begin{gather}
-	R_{\BSE}^{\sigma} = \Delta\eGW{} + 2 \sigma \ERI{vc}{vc} - W_R(\omega = \Delta\eGW{})
+	R_{\BSE}^{\sigma} = \Delta\eGW{} + 2 \sigma \ERI{vc}{vc} - \ERI{vv}{cc} - W_R(\omega = \Delta\eGW{})
 	\\
-	C_{\BSE}^{\sigma} = 2 \sigma \ERI{vc}{vc} - W_C(\omega = 0)
+	C_{\BSE}^{\sigma} = C_{\dBSE}^{\sigma} = 2 \sigma \ERI{vc}{cv} -  \ERI{vc}{cv} - W^{\co}_C(\omega = 0)
 \end{gather}
 \end{subequations}
 
@@ -448,14 +451,14 @@ It can be easily shown that solving the equation
 \begin{equation}
 	\det[\bH_{\dBSE}^{\sigma}(\omega) - \omega \bI] = 0
 \end{equation}
-yields 3 solutions per spin manifold (except for \ce{H2} where only 2 roots are observed), as shown in Fig.~\ref{fig:dBSE} for the case of \ce{HeH+}.
+yields 2 solutions per spin manifold (except for \ce{H2} where only one root is observed), as shown in Fig.~\ref{fig:dBSE} for the case of \ce{HeH+}.
 Their numerical values are reported in Table \ref{tab:BSE} alongside other variants discussed below.
-These numbers evidence that dBSE reproduces qualitatively well the singlet and triplet single excitations, but quite badly the double excitation which is off by several eV except in the case of \ce{H2} where the agreement is rather satisfactory ($44.30$ eV at the dBSE level compared to the exact value of $44.04$ eV).
-As mentioned in Ref.~\onlinecite{Romaniello_2009b}, spurious solutions appears due to the approximate nature of the dBSE kernel. 
+These numbers evidence that dBSE reproduces qualitatively well the singlet and triplet single excitations, but quite badly the double excitation which is off by several eV.
+As mentioned in Ref.~\onlinecite{Romaniello_2009b}, spurious solutions appear due to the approximate nature of the dBSE kernel. 
 Indeed, diagonalizing the exact Hamiltonian \eqref{eq:H-exact} produces only two singlet solutions corresponding to the singly- and doubly-excited states, and one triplet state (see Sec.~\ref{sec:exact}). 
-Therefore, there is one spurious solution for the singlet manifold ($\omega_{2}^{\dBSE,\updw}$) and two spurious solutions for the triplet manifold ($\omega_{2}^{\dBSE,\upup}$ and $\omega_{3}^{\dBSE,\upup}$).
-It is worth mentioning that, around $\omega = \omega_1^{\dBSE,\sigma}$, the slope of the curves depicted in Fig.~\ref{fig:dBSE} is small, while the two other solutions, $\omega_2^{\dBSE,\sigma}$ and $\omega_3^{\dBSE,\sigma}$, stem from poles and consequently the slope is very large around these frequency values.
-This makes these two latter solutions quite hard to locate with a method like Newton-Raphson (for example).
+Therefore, there is the right number of singlet solutions but there is one spurious solution for the triplet manifold ($\omega_{2}^{\dBSE,\upup}$).
+It is worth mentioning that, around $\omega = \omega_1^{\dBSE,\sigma}$, the slope of the curves depicted in Fig.~\ref{fig:dBSE} is small, while the other solution, $\omega_2^{\dBSE,\sigma}$, stems from a pole and consequently the slope is very large around this frequency value.
+This makes this latter solution quite hard to locate with a method like Newton-Raphson (for example).
 Let us highlight the fact that, unlike in Ref.~\onlinecite{Loos_2020e} where dynamical effects produce a systematic red-shifting of the static excitations, here we observe both blue- and red-shifted transitions.
 
 %%% TABLE III %%%
@@ -465,29 +468,23 @@ Let us highlight the fact that, unlike in Ref.~\onlinecite{Loos_2020e} where dyn
 	\label{tab:BSE}
 	}
 	\begin{ruledtabular}
-		\begin{tabular}{lcllllllll}
-						&						&	\mc{8}{c}{Method}	\\
-													\cline{3-10}	
-			System		&	Excitation			&	BSE		&	pBSE		&	pBSE(dTDA)	&	dBSE		&	BSE(TDA)	&	pBSE(TDA)	&	dBSE(TDA)	&	Exact	\\
+		\begin{tabular}{lclllllll}
+						&						&	\mc{7}{c}{Method}	\\
+													\cline{3-9}	
+			System		&	Excitation			&	BSE		&	pBSE			&	dBSE			&	BSE(TDA)	&	pBSE(TDA)		&	dBSE(TDA)	&	Exact	\\
 			\hline
-			\ce{H2}		&	$\omega_1^{\updw}$	&	$26.06$	&	$25.52[-0.54]$	&	$26.06[+0.00]$	&	$25.78[-0.28]$	&	$27.02$		&	$27.02[+0.00]$	&	$27.02[+0.00]$	&	$26.34$	\\
-						&	$\omega_3^{\updw}$	&			&					&					&	$44.30$			&				&					&				&	$44.04$	\\
-						&	$\omega_1^{\upup}$	&	$16.94$	&	$17.10[+0.16]$	&	$16.94[+0.00]$	&	$17.03[+0.09]$	&	$17.16$		&	$17.16[+0.00]$	&	$17.16[+0.00]$	&	$16.48$	\\
-						&	$\omega_3^{\upup}$	&			&					&					&	$43.61$			&				&					&				&			\\
-			\\
-			\ce{HeH+}	&	$\omega_1^{\updw}$	&	$28.56$	&	$28.41[-0.15]$	&	$28.63[+0.07]$	&	$28.52[-0.04]$	&	$29.04$		&	$29.11[+0.07]$	&	$29.11[+0.07]$	&	$28.05$	\\
-						&	$\omega_2^{\updw}$	&			&					&					&	$47.85$			&				&					&				&			\\
-						&	$\omega_3^{\updw}$	&			&					&					&	$87.47$			&				&					&	$87.47$		&	$64.09$	\\
-						&	$\omega_1^{\upup}$	&	$20.96$	&	$21.16[+0.20]$	&	$21.07[+0.11]$	&	$21.12[+0.16]$	&	$21.13$		&	$21.24[+0.11]$	&	$21.24[+0.11]$	&	$22.03$	\\
-						&	$\omega_2^{\upup}$	&			&					&					&	$47.54$			&				&					&				&			\\
-						&	$\omega_3^{\upup}$	&			&					&					&	$87.43$			&				&					&	$87.43$		&			\\
-			\\
-			\ce{He}		&	$\omega_1^{\updw}$	&	$52.46$	&	$51.71[-0.75]$	&	$52.12[-0.33]$	&	$51.85[-0.61]$	&	$53.10$		&	$52.79[-0.31]$	&	$52.79[-0.31]$	&	$52.29$	\\
-						&	$\omega_2^{\updw}$	&			&					&					&	$75.75$			&				&					&				&			\\
-						&	$\omega_3^{\updw}$	&			&					&					&	$133.37$		&				&					&	$133.37$		&	$94.66$	\\
-						&	$\omega_1^{\upup}$	&	$40.50$	&	$39.96[-0.53]$	&	$39.80[-0.70]$	&	$39.90[-0.60]$	&	$40.71$		&	$40.02[-0.69]$	&	$40.02[-0.69]$	&	$40.18$	\\
-						&	$\omega_2^{\upup}$	&			&					&					&	$75.15$			&				&					&				&			\\
-						&	$\omega_3^{\upup}$	&			&					&					&	$133.76$		&				&					&	$133.75$		&			\\
+			\ce{H2}		&	$\omega_1^{\updw}$	&	$26.06$	&	$26.06[+0.00]$	&	$26.06[+0.00]$	&	$27.02$		&	$27.02[+0.00]$	&	$27.02[+0.00]$	&	$26.34$	\\
+						&	$\omega_1^{\upup}$	&	$16.94$	&	$16.94[+0.00]$	&	$16.94[+0.00]$	&	$17.16$		&	$17.16[+0.00]$	&	$17.16[+0.00]$	&	$16.48$	\\
+			\hline
+			\ce{HeH+}	&	$\omega_1^{\updw}$	&	$28.56$	&	$28.63[+0.07]$	&	$28.63[+0.07]$	&	$29.04$		&	$29.11[+0.07]$	&	$29.11[+0.07]$	&	$28.05$	\\
+						&	$\omega_2^{\updw}$	&			&					&	$87.47$			&				&					&	$87.47$			&	$64.09$	\\
+						&	$\omega_1^{\upup}$	&	$20.96$	&	$21.07[+0.11]$	&	$21.07[+0.11]$	&	$21.13$		&	$21.24[+0.11]$	&	$21.24[+0.11]$	&	$22.03$	\\
+						&	$\omega_2^{\upup}$	&			&					&	$87.43$			&				&					&	$87.43$			&			\\
+			\hline
+			\ce{He}		&	$\omega_1^{\updw}$	&	$52.46$	&	$52.12[-0.34]$	&	$52.11[-0.35]$	&	$53.10$		&	$52.79[-0.31]$	&	$52.79[-0.31]$	&	$52.29$	\\
+						&	$\omega_2^{\updw}$	&			&					&	$133.38$		&				&					&	$133.37$		&	$94.66$	\\
+						&	$\omega_1^{\upup}$	&	$40.50$	&	$39.80[-0.70]$	&	$39.79[-0.71]$	&	$40.71$		&	$40.02[-0.69]$	&	$40.02[-0.69]$	&	$40.18$	\\
+						&	$\omega_2^{\upup}$	&			&					&	$133.75$		&				&					&	$133.75$		&			\\
 		\end{tabular}
 	\end{ruledtabular}
 \end{table*}
@@ -504,25 +501,14 @@ Let us highlight the fact that, unlike in Ref.~\onlinecite{Loos_2020e} where dyn
 \end{figure}
 %%% %%% %%% %%%
 
-Enforcing the TDA, which corresponds to neglecting the coupling term between the resonant and anti-resonant part of the dBSE Hamiltonian \eqref{eq:HBSE}, allows to remove some of these spurious excitations.
-There is thus only one spurious excitation in the triplet manifold ($\omega_{3}^{\BSE,\upup}$), the two solutions of the singlet manifold corresponding now to the single and double excitations.
-The spin blindness of the dBSE kernel is probably to blame for the existence of this spurious triplet excitation.
-Figure \ref{fig:dBSE-TDA} shows the same curves as Fig.~\ref{fig:dBSE} but in the TDA.
-
-%%% FIGURE 3 %%%
-\begin{figure}
-	\includegraphics[width=\linewidth]{dBSE-TDA}
-	\caption{
-	$\det[\bH(\omega) - \omega \bI]$ as a function of $\omega$ (in hartree) for both the singlet (gray and black) and triplet (orange and red) manifolds of \ce{HeH+} within the TDA.
-	The static BSE Hamiltonian (dashed) and dynamic dBSE Hamiltonian (solid) are considered.
-	\label{fig:dBSE-TDA}
-	}
-\end{figure}
-%%% %%% %%% %%%
-
 In the static approximation, only one solution per spin manifold is obtained by diagonalizing $\bH_{\BSE}^{\sigma}$ (see Fig.~\ref{fig:dBSE} and Table \ref{tab:BSE}).
 Therefore, the static BSE Hamiltonian misses the (singlet) double excitation (as it should), and it shows that the physical single excitation stemming from the dBSE Hamiltonian is indeed the lowest in energy for each spin manifold, \ie, $\omega_1^{\dBSE,\updw}$ and $\omega_1^{\dBSE,\upup}$.
 
+Enforcing the TDA, which corresponds to neglecting the coupling term between the resonant and anti-resonant part 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 dynamical correction are not altered by the TDA.
+%Figure \ref{fig:dBSE-TDA} shows the same curves as Fig.~\ref{fig:dBSE} but in the TDA.
+%The spin blindness of the dBSE kernel is probably to blame for the existence of this spurious triplet excitation.
+
 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
 \begin{equation}
@@ -574,18 +560,19 @@ and the renormalization factor is
 	 }^{-1}
 \end{equation}
 This corresponds to a dynamical perturbative correction to the static excitations. 
-Obviously, the TDA can be applied to the dynamical correction as well, a scheme we label as dTDA in the following.
 
-The perturbatively-corrected values are also reported in Table \ref{tab:BSE}, which shows that this scheme is very effective at reproducing the dynamical value for the single excitations, especially in the TDA where pBSE(TDA) is an excellent approximation of dBSE(TDA).
+The perturbatively-corrected values are also reported in Table \ref{tab:BSE}, which shows that this scheme is very effective at reproducing the dynamical value for the single excitations.
 However, because the perturbative treatment is ultimately static, one cannot access double excitations with such a scheme.
 
+For \ce{H2}, there is no dynamical corrections at the BSE, pBSE or dBSE levels.
+Indeed, as $\ERI{vv}{vc} = \ERI{vc}{cc} = 0$ (see Table \ref{tab:params}), we have $W^{\co}_R(\omega) = 0$ [see Eq.~\eqref{eq:WR}].
 
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 \subsection{Second-order BSE kernel}
 \label{sec:BSE2}
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 
-The third and final dynamical kernel that we consider here is the second-order BSE (BSE2) kernel derived by Yang and collaborators in the TDA, \cite{Zhang_2013} and by Rebolini and Toulouse in a range-separated context \cite{Rebolini_2016,Rebolini_PhD} (see also Refs.~\onlinecite{Myohanen_2008,Sakkinen_2012}).
+The third and final dynamical kernel that we consider here is the second-order BSE (BSE2) kernel derived by Yang and collaborators in the TDA, \cite{Zhang_2013} and by Rebolini and Toulouse in a range-separated context \cite{Rebolini_2016,Rebolini_PhD} (see also Refs.~\onlinecite{Myohanen_2008,Sakkinen_2012,Olevano_2019}).
 Note that a beyond-TDA BSE2 kernel was also derived in Ref.~\onlinecite{Rebolini_2016}, but was not tested.
 In a nutshell, the BSE2 scheme applies second-order perturbation theory to optical excitations within the Green's function framework by taking the functional derivative of the second-order self-energy $\SigGF{}$ with respect to the one-body Green's function. 
 Because $\SigGF{}$ is a proper functional derivative, it was claimed in Ref.~\onlinecite{Zhang_2013} that BSE2 does not produce spurious excitations.
@@ -598,19 +585,19 @@ Like BSE requires $GW$ quasiparticle energies, BSE2 requires the second-order Gr
 where the second-order self-energy is
 \begin{equation} 
 	\label{eq:SigGF}
-	\SigGF{p}(\omega) = \frac{2 \ERI{pv}{vc}^2}{\omega - \e{v} + \e{c} - \e{v}} + \frac{2 \ERI{pc}{cv}^2}{\omega - \e{c} - (\e{c} - \e{v})}
+	\SigGF{p}(\omega) = \frac{\ERI{pv}{vc}^2}{\omega - \e{v} + \e{c} - \e{v}} + \frac{\ERI{pc}{cv}^2}{\omega - \e{c} - (\e{c} - \e{v})}
 \end{equation}
 and
 \begin{equation}
 	Z_{p}^{\GF} = \qty( 1 - \left. \pdv{\SigGF{p}(\omega)}{\omega} \right|_{\omega = \e{p}} )^{-1}
 \end{equation}
-The expression of the GF2 self-energy \eqref{eq:SigGF} can be easily obtained from its $GW$ counterpart \eqref{eq:SigGW} via the substitution $\Omega \to \e{c} - \e{v}$, which shows that there is no screening within GF2 and BSE2.
+The expression of the GF2 self-energy \eqref{eq:SigGF} can be easily obtained from its $GW$ counterpart \eqref{eq:SigGW} via the substitution $\Omega \to \e{c} - \e{v}$ and by dividing the numerator by a factor two. This shows that there is no screening within GF2 and BSE2, but that second-order exchange is properly taken into account. \cite{Zhang_2013,Loos_2018b}
 
 The static Hamiltonian of BSE2 is just the usual TDHF Hamiltonian where one substitutes the HF orbital energies by the GF2 quasiparticle energies, \ie, 
 \begin{equation}
 	\bH_{\BSE2}^{\sigma} = 
 	\begin{pmatrix}
-		 R_{\BSE2}^{\sigma}	&	C_{\BSE2}^{\sigma}
+		+R_{\BSE2}^{\sigma}	&	+C_{\BSE2}^{\sigma}
 		\\
 		-C_{\BSE2}^{\sigma}	&	-R_{\BSE2}^{\sigma}
 	\end{pmatrix}
@@ -629,7 +616,7 @@ The correlation part of the dynamical kernel for BSE2 is a bit cumbersome \cite{
 \begin{equation}
 	\bH_{\dBSE2}^{\sigma} = \bH_{\BSE2}^{\sigma} +
 	\begin{pmatrix}
-		 f_{\dBSE2}^{\co,\sigma}(\omega)	&	f_{\dBSE2}^{\co,\sigma}
+		 +f_{\dBSE2}^{\co,\sigma}(+\omega)	&	+f_{\dBSE2}^{\co,\sigma}
 		\\
 		-f_{\dBSE2}^{\co,\sigma}	&	-f_{\BSE2}^{\co,\sigma}(-\omega)
 	\end{pmatrix}
@@ -650,9 +637,8 @@ and
 	f_{\dBSE2}^{\co,\upup} = \frac{\ERI{cc}{cc} \ERI{vc}{cv} + \ERI{vv}{vv} \ERI{vc}{cv} }{2 \Delta\eGF{}}
 \end{gather}
 \end{subequations}
-Note that, unlike the dBSE Hamiltonian [see Eq.~\eqref{eq:HBSE}], the BSE2 dynamical kernel is spin aware with distinct expressions for singlets and triplets, \cite{Rebolini_PhD} and the coupling block $C_{\dBSE2}^{\sigma}$ is frequency independent.
-This latter point has an important consequence as this lack of frequency dependence removes one of the spurious pole (see Fig.~\ref{fig:BSE2}).
-The singlet manifold has then the right number of excitations.
+Note that, unlike the dBSE Hamiltonian [see Eq.~\eqref{eq:HBSE}], the BSE2 dynamical kernel is spin aware with distinct expressions for singlets and triplets. \cite{Rebolini_PhD} 
+Like in dBSE, dBSE2 generates the right number of excitations for the singlet manifold (see Fig.~\ref{fig:BSE2}).
 However, one spurious triplet excitation remains.
 It is mentioned in Ref.~\onlinecite{Rebolini_2016} that the BSE2 kernel has some similarities with the second-order polarization-propagator approximation \cite{Oddershede_1977,Nielsen_1980} (SOPPA) and second RPA kernels. \cite{Huix-Rotllant_2011,Huix-Rotllant_PhD,Sangalli_2011}
 
@@ -671,32 +657,28 @@ In the case of BSE2, the perturbative partitioning is simply
 	\label{tab:BSE2}
 	}
 	\begin{ruledtabular}
-		\begin{tabular}{lcllllllll}
-			System		&	Excitation			&	BSE2	&	pBSE2		&	pBSE2(dTDA)	&	dBSE2		&	BSE2(TDA)	&	pBSE2(TDA)	&	dBSE2(TDA)	&	Exact	\\
+		\begin{tabular}{lclllllll}
+						&						&	\mc{7}{c}{Method}	\\
+													\cline{3-9}
+			System		&	Excitation			&	BSE2	&	pBSE2			&	dBSE2		&	BSE2(TDA)	&	pBSE2(TDA)	&	dBSE2(TDA)	&	Exact	\\
 			\hline
-			\ce{H2}		&	$\omega_1^{\updw}$	&	$26.03$	&	$26.03[+0.00$]&	$26.03[+0.00]$&	$26.24[+0.21]$&	$26.49$		&	$26.49[+0.00]$&	$26.49[+0.00]$&	$26.34$	\\
-						&	$\omega_1^{\upup}$	&	$15.88$	&	$15.88[+0.00]$&	$15.88[+0.00]$&	$16.47[+0.59]$&	$16.63$		&	$16.63[+0.00]$&	$16.63[+0.00]$&	$16.48$	\\
-			\\
-			\ce{HeH+}	&	$\omega_1^{\updw}$	&	$29.23$	&	$28.40[-0.83]$&	$28.40[-0.83]$&	$28.56[-0.67]$&	$29.50$		&	$28.66[-0.84]$&	$28.66[-0.84]$&	$28.05$	\\
-						&	$\omega_3^{\updw}$	&			&				&				&	$79.94$		&				&				&	$79.94$		&	$64.09$	\\
-						&	$\omega_1^{\upup}$	&	$21.22$	&	$21.63[+0.41]$&	$21.63[+0.41]$&	$21.93[+0.71]$&	$21.59$		&	$21.99[+0.40]$&	$21.99[+0.40]$&	$22.03$	\\
-						&	$\omega_3^{\upup}$	&			&				&				&	$78.70$		&				&				&	$78.70$		&			\\
-			\\
-			\ce{He}		&	$\omega_1^{\updw}$	&	$50.31$	&	$51.96[+1.64]$&	$51.96[+1.65]$&	$52.10[+1.79]$&	$50.69$		&	$52.34[+1.65]$&	$52.34[+1.65]$&	$52.29$	\\
-						&	$\omega_3^{\updw}$	&			&				&				&	$121.67$	&				&				&	$121.66$	&	$94.66$	\\
-						&	$\omega_1^{\upup}$	&	$37.80$	&	$39.26[+1.46]$&	$39.27[+1.47]$&	$39.59[+1.79]$&	$38.30$		&	$39.77[+1.47]$&	$39.77[+1.47]$&	$40.18$	\\
-						&	$\omega_3^{\upup}$	&			&				&				&	$121.85$		&				&				&	$121.84$		&	\\
+			\ce{H2}		&	$\omega_1^{\updw}$	&	$26.03$	&	$26.03[+0.00$]	&	$26.24[+0.21]$	&	$26.49$		&	$26.49[+0.00]$	&	$26.49[+0.00]$	&	$26.34$	\\
+						&	$\omega_1^{\upup}$	&	$15.88$	&	$15.88[+0.00]$	&	$16.47[+0.59]$	&	$16.63$		&	$16.63[+0.00]$	&	$16.63[+0.00]$	&	$16.48$	\\
+			\hline
+			\ce{HeH+}	&	$\omega_1^{\updw}$	&	$29.23$	&	$28.40[-0.83]$	&	$28.56[-0.67]$	&	$29.50$		&	$28.66[-0.84]$	&	$28.66[-0.84]$	&	$28.05$	\\
+						&	$\omega_2^{\updw}$	&			&					&	$79.94$			&				&					&	$79.94$			&	$64.09$	\\
+						&	$\omega_1^{\upup}$	&	$21.22$	&	$21.63[+0.41]$	&	$21.93[+0.71]$	&	$21.59$		&	$21.99[+0.40]$	&	$21.99[+0.40]$	&	$22.03$	\\
+						&	$\omega_2^{\upup}$	&			&					&	$78.70$			&				&					&	$78.70$			&			\\
+			\hline
+			\ce{He}		&	$\omega_1^{\updw}$	&	$50.31$	&	$51.96[+1.64]$	&	$52.10[+1.79]$	&	$50.69$		&	$52.34[+1.65]$	&	$52.34[+1.65]$	&	$52.29$	\\
+						&	$\omega_2^{\updw}$	&			&					&	$121.67$		&				&					&	$121.66$		&	$94.66$	\\
+						&	$\omega_1^{\upup}$	&	$37.80$	&	$39.26[+1.46]$	&	$39.59[+1.79]$	&	$38.30$		&	$39.77[+1.47]$	&	$39.77[+1.47]$	&	$40.18$	\\
+						&	$\omega_2^{\upup}$	&			&					&	$121.85$		&				&					&	$121.84$		&	\\
 		\end{tabular}
 	\end{ruledtabular}
 \end{table*}
 %%% %%% %%% %%%
 
-As compared to dBSE, dBSE2 produces much larger dynamical corrections to the static excitation energies (see values in square brackets in Table \ref{tab:BSE2}) probably due to the poorer quality of its static reference (TDHF or CIS).
-Similarly to what has been observed in Sec.~\ref{sec:Maitra}, the TDA vertical excitations are slightly more accurate due to error compensations.
-Note also that the perturbative treatment is a remarkably good approximation to the dynamical scheme for single excitations, except for \ce{H2} (see below).
-Overall, the accuracy of dBSE and dBSE2 are comparable for single excitations although their behavior is quite different (see Tables \ref{tab:BSE} and \ref{tab:BSE2}). 
-For the double excitation, dBSE2 yields a slightly better energy, yet still in quite poor agreement with the exact value.
-
 %%% FIGURE 4 %%%
 \begin{figure}
 	\includegraphics[width=\linewidth]{dBSE2}
@@ -708,7 +690,14 @@ For the double excitation, dBSE2 yields a slightly better energy, yet still in q
 \end{figure}
 %%% %%% %%% %%%
 
-Again, the case of \ce{H2} is a bit peculiar as the perturbative treatment (pBSE2) does not provide dynamical corrections, while its dynamical version (dBSE2) does yields sizable corrections and makes the singlet and triplet excitation energies very accurate.
+As compared to dBSE, dBSE2 produces much larger dynamical corrections to the static excitation energies (see values in square brackets in Table \ref{tab:BSE2}) probably due to the poorer quality of its static reference (TDHF or CIS).
+Similarly to what has been observed in Sec.~\ref{sec:Maitra}, the TDA vertical excitations are slightly more accurate due to error compensations.
+Note also that the perturbative treatment is a remarkably good approximation to the dynamical scheme for single excitations (except for \ce{H2}, see below), especially in the TDA which justifies the use of the perturbative treatment in Ref.~\onlinecite{Zhang_2013,Rebolini_2016}.
+Overall, the accuracy of dBSE and dBSE2 are comparable for single excitations although their behavior is quite different (see Tables \ref{tab:BSE} and \ref{tab:BSE2}). 
+For the double excitation, dBSE2 yields a slightly better energy, yet still in quite poor agreement with the exact value.
+
+Again, the case of \ce{H2} is a bit peculiar as the perturbative treatment (pBSE2) does not provide dynamical corrections, while its dynamical version (dBSE2) does yield sizable corrections originating from the coupling term $f_{\dBSE2}^{\co,\sigma}$ which is non-zero in the case of dBSE2.
+Although frequency-independent, this additional term makes the singlet and triplet excitation energies very accurate.
 However, one cannot access the double excitation.
 
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%