blush

BSEdyn.tex

@@ -348,7 +348,7 @@ The $\Oms$'s are the neutral excitation energies of interest.
 
 Picking up the $e^{+i \Oms t_2 }$ component in $L(1,2; 1',2')$ and $L(6,2;5,2')$, simplifying further by $\tchi_s(\bx_2,\bx_{2'})$ on both side of the BSE [see Eq.~\eqref{eq:BSE}], we are left with the search of the $e^{-i \Oms t_1 }$ Fourier component associated with the right-hand side of the modified dynamical BSE:
 \begin{multline} \label{eq:BSE_2}
-    \mel{N}{T \hpsi(\bx_1) \hpsi^{\dagger}(\bx_{1}')}{N,s} e^{ - i \Oms t_1 }
+    \mel{N}{T [ \hpsi(\bx_1) \hpsi^{\dagger}(\bx_{1}') ] } {N,s} e^{ - i \Oms t_1 }
 	\theta ( \tau_{12} )  
     \\
 	= \int d3456 \, L_0(1,4;1',3) \Xi(3,5;4,6)  
@@ -356,7 +356,7 @@ Picking up the $e^{+i \Oms t_2 }$ component in $L(1,2; 1',2')$ and $L(6,2;5,2')$
 	\times \mel{N}{T [\hpsi(6) \hpsi^{\dagger}(5)] }{N,s} 
 	\theta [\min(t_5,t_6) - t_2].
 \end{multline}
-For the lowest neutral excitation energies falling in the fundamental gap of the system (\ie, $\Oms < \EgFun$), $L_0(1,2;1',2')$ cannot contribute to the $e^{-i \Oms t_1 }$ response due to excitonic effects since its lowest excitation energy is precisely the fundamental gap [see Eq.~\eqref{eq:Egfun}].
+For the lowest neutral excitation energies falling in the fundamental gap of the system (\ie, $\Oms < \EgFun$ due to excitonic effects), $L_0(1,2;1',2')$ cannot contribute to the $e^{-i \Oms t_1 }$ response  since its lowest excitation energy is precisely the fundamental gap [see Eq.~\eqref{eq:Egfun}].
 Consequently, special care has to be taken for high-lying excited states (like core or Rydberg excitations) where additional terms have to be taken into account (see Refs.~\onlinecite{Strinati_1982,Strinati_1984}).
 
 Dropping the (space/spin) variables, the Fourier components with respect to $t_1$ of $L_0(1,4;1',3)$ reads
@@ -379,7 +379,7 @@ The $\e{p}$'s in Eq.~\eqref{eq:G-Lehman} are quasiparticle energies (\ie, proper
 %$\hH$ being the exact many-body Hamiltonian. 
 In the following, $i$ and $j$ are occupied orbitals, $a$ and $b$ are unoccupied orbitals, while $p$, $q$, $r$, and $s$ indicate arbitrary orbitals.
 %\titou{namely $GW$ quasiparticle energies and input Hartree-Fock molecular orbitals in the present study. (T2: shall we really mention this here?)}
-Projecting $L_0(1,4;1',3)$ onto $\phi_a^*(\bx_1) \phi_i(\bx_{1'})$ yields
+Projecting the $( \omega_1 = \Oms )$ Fourier component $L_0(\bx_1,4;\bx_{1'},3; \Oms)$   onto $\phi_a^*(\bx_1) \phi_i(\bx_{1'})$ yields
 \begin{multline}
 	\int d\bx_1 d\bx_{1'} \; \phi_a^*(\bx_1) \phi_i(\bx_{1'})  L_0(\bx_1,4;\bx_{1'},3; \Oms) 
 	\\
@@ -393,15 +393,18 @@ As a final step, we express the terms $\mel{N}{T [\hpsi(\bx_1) \hpsi^{\dagger}(\
 This is done by  expanding the field operators over a complete orbital basis of creation/destruction operators.
 For example, we have
 \begin{multline}
-	\mel{N}{T [\hpsi(3) \hpsi^{\dagger}(4)] }{N,s} 
+	\mel{N}{T [\hpsi(6) \hpsi^{\dagger}(5)] }{N,s} 
 	\\
-	= - \qty( e^{  -i \Omega_s t^{34} }  ) \sum_{pq} \phi_p(\bx_3) \phi_q^*(\bx_4) 
+	= - \qty( e^{  -i \Omega_s t^{56} }  ) \sum_{pq} \phi_p(\bx_6) \phi_q^*(\bx_5) 
 	\mel{N}{\ha_q^{\dagger} \ha_p}{N,s}
 	\\
-	\times \qty[ \theta( t_{34} ) e^{- i ( \e{p} - \hOms ) t_{34} } + \theta( - t_{34} ) e^{ - i ( \e{q}  + \hOms) t_{34} } ],
+	\times \qty[ \theta( \tau_{56}  ) e^{- i ( \e{p} - \hOms ) \tau_{56}  } + \theta( - \tau_{56}  ) e^{ - i ( \e{q}  + \hOms) \tau_{56}  } ] 
 \end{multline}
-with a similar expression for $\mel{N}{T [\hpsi(\bx_3) \hpsi^{\dagger}(\bx_4)] }{N,s}$.
-
+with $t^{56} = (t_5 + t_6)/2$ and $\tau_{56} = t_5 -t_6$.
+%with a similar expression for $\mel{N}{T [\hpsi(\bx_3) \hpsi^{\dagger}(\bx_4)] }{N,s}$. 
+\xavier{ The $X_{jb}^{s} = \mel{N}{\ha_j^{\dagger} \ha_b}{N,s}$ are the largest contributions to the 
+$\mel{N}{T [\hpsi(6) \hpsi^{\dagger}(5)] }{N,s}$ weight,
+while the $Y_{jb}^{s} = \mel{N}{\ha_b^{\dagger} \ha_j}{N,s}$ are much smaller.} 
 Adopting now the $GW$ approximation for the exchange-correlation self-energy, \ie, 
 \begin{equation}
 	\Sigma_\text{xc}^{\GW}(1,2) = i G(1,2) W(1^+,2),