Updates from Overleaf

BSEdyn.tex

@@ -367,7 +367,7 @@ where $\tau_{12} = t_1 - t_2$, $\theta$ is the Heaviside step function, and
 	\tchi_s(\bx_1,\bx_{2}) & = \mel{N,s}{T [\hpsi(\bx_1) \hpsi^{\dagger}(\bx_{2})] }{N}.
 \end{align}  
 \end{subequations}  
-The $\Om{s}{}$'s are the neutral excitation energies of interest. 
+The $\Om{s}{}$'s are the neutral excitation energies of interest (with $\Om{s}{} = E^N_s - E^N_0$). 
 
 Picking up the $e^{+i \Om{s}{} t_2 }$ component of both $L(1,2; 1',2')$ and $L(6,2;5,2')$, simplifying further by $\tchi_s(\bx_2,\bx_{2'})$ on both sides of the BSE [see Eq.~\eqref{eq:BSE}], we seek the $e^{-i \Om{s}{} t_1 }$ Fourier component associated with the right-hand side of a modified dynamical BSE, which reads
 \begin{multline} \label{eq:BSE_2}
@@ -435,7 +435,7 @@ leads to the following simplified BSE kernel
 where $W$ is the dynamically-screened Coulomb operator.
 The $GW$ quasiparticle energies $\eGW{p}$ are usually good approximations to the removal/addition energies $\e{p}$ introduced in Eq.~\eqref{eq:G-Lehman}. 
 
-Substituting Eqs.~\eqref{eq:iL0bis}, \eqref{eq:spectral65}, and \eqref{eq:Xi_GW} into Eq.~\eqref{eq:BSE_2},  and projecting onto $\MO{a}^*(\bx_1) \MO{i}(\bx_{1'})$, one gets after a few tedious manipulations (see {\SI}) the dynamical BSE:
+Substituting Eqs.~\eqref{eq:iL0bis}, \eqref{eq:spectral65}, and \eqref{eq:Xi_GW} into Eq.~\eqref{eq:BSE_2},  and projecting onto $\MO{a}^*(\bx_1) \MO{i}(\bx_{1'})$, one gets after a few tedious manipulations the dynamical BSE:
 \begin{equation} \label{eq:BSE-final}
 \begin{split}
 	( \eGW{a} - \eGW{i} - \Om{s}{} ) X_{ia,s}      
@@ -525,10 +525,13 @@ The analysis of the poles of the integrand in Eq.~\eqref{eq:wtilde} yields
     \times \qty[ \frac{1}{\Om{ib}{s} - \Om{m}{\RPA} + i\eta} + \frac{1}{\Om{ja}{s} - \Om{m}{\RPA} + i\eta} ].
 \end{multline} 
 One can verify that, in the static limit where $\Om{m}{\RPA} \to \infty$, the matrix elements $\widetilde{W}_{ij,ab}$ correctly reduce to the static ${W}_{ij,ab}$ ones, evidencing that the standard static BSE problem is recovered from the present dynamical formalism in this limit.
-Due to excitonic effects, the lowest BSE excitation energy, $\Om{1}{}$, stands lower than the lowest RPA excitation energy, $\Om{1}{\RPA}$, so that, $\Om{ib}{s} - \Om{m}{\RPA} < 0 $ and $\widetilde{W}_{ij,ab}(\Om{s}{})$ has no resonances. 
-Furthermore, $\Om{ib}{s}$ and $\Om{ja}{s}$ are necessarily negative quantities for in-gap low-lying BSE excitations. 
-Thus, we have $\abs*{\Omega_{ib}^{s} - \Om{m}{\RPA}}  >  \Omega_m^{\RPA}$.
+
+Due to excitonic effects, the lowest BSE excitation energy, $\Om{1}{}$, stands lower than the lowest RPA excitation energy, $\Om{1}{\RPA}$, so that, $\Om{ib}{S} - \Om{m}{\RPA} < 0 $ and $\widetilde{W}_{ij,ab}(\Om{S}{})$ has no resonances. 
+This property holds for a few low lying $\Om{s}{}$ excitations but special care must be taken for higher ones.
+Furthermore, $\Om{ib}{S}$ and $\Om{ja}{S}$ are necessarily negative quantities for in-gap low-lying BSE excitations. 
+Thus, we have $\abs*{\Om{ib}{S} - \Om{m}{\RPA}}  >  \Om{m}{\RPA}$.
 As a consequence, we observe a reduction of the electron-hole screening, \ie, an enhancement of electron-hole binding energy, as compared to the standard static BSE, and consequently smaller (red-shifted) excitation energies. 
+This will be numerically illustrated in Sec.~\ref{sec:resdis}.
 
 %================================
 
@@ -1089,7 +1092,7 @@ In this Appendix, we derive Eqs.~\eqref{eq:iL0} to \eqref{eq:iL0bis}.
 Defining the $t_1$-time Fourier transform of $L_0(1,3;4,1')$ with
 $(t_{1'} = t_1^{+})$
 \begin{align}  
-	[L_0](\bx_1,3;\bx_{1'},4 \; | \; \omega_1 ) = -i
+	[L_0](\bx_1,4;\bx_{1'},3 \; | \; \omega_1 ) = -i
 	\int dt_1 e^{i \omega_1 t_1 } G(1,3)G(4,1')
 \end{align}
  we plug-in the Fourier expansion of the Green's function, e.g.
@@ -1098,14 +1101,14 @@ $(t_{1'} = t_1^{+})$
 \end{align*}
 with $\tau_{13} = (t_1-t_3)$ to obtain:
 \begin{multline}  
-	[L_0](\bx_1,3;\bx_{1'},4 \;| \;  \omega_1 )   =
+	[L_0](\bx_1,4;\bx_{1'},3 \;| \;  \omega_1 )   =
 	\\
 	\int \frac{ d\omega }{ 2i\pi } \;  G(\bx_1,\bx_3;\omega) \; G(\bx_4,\bx_{1'};\omega-\omega_1)  
 	  e^{ i \omega t_3 }	e^{-i (\omega-\omega_1) t_4 } \nonumber
 \end{multline}
 With the change of variable $\omega \to \omega + {\omega_1}/2$ one obtains readily
 \begin{multline}  
-	[L_0](\bx_1,3;\bx_{1'},4 \; | \; \omega_1 )     = 
+	[L_0](\bx_1,4;\bx_{1'},3 \; | \; \omega_1 )     = 
 	\\
 	e^{ i  \omega_1  t^{34} }
 	\int \frac{ d\omega }{ 2i\pi } \;   G\qty(\bx_1,\bx_3;\omega+ \frac{\omega_1}{2} )  G\qty(\bx_4,\bx_{1'};\omega-\frac{\omega_1}{2} )  \;