Appendix B

BSEdyn.tex

@@ -1134,7 +1134,7 @@ PFL thanks the European Research Council (ERC) under the European Union's Horizo
 This work was performed using HPC resources from GENCI-TGCC (Grant No.~2019-A0060801738) and CALMIP (Toulouse) under allocation 2020-18005.
 Funding from the \textit{``Centre National de la Recherche Scientifique''} is acknowledged.
 This study has been (partially) supported through the EUR grant NanoX No.~ANR-17-EURE-0009 in the framework of the \textit{``Programme des Investissements d'Avenir''.}
-\titou{The authors would like to thank Elisa Rebolini for insightful discussions.}
+The authors would like to thank Elisa Rebolini, Pina Romaniello, Arjan Berger, and Julien Toulouse for insightful discussions on dynamical kernels.
 
 %%%%%%%%%%%%%%%%%%%%%%%%
 \section*{Data availability}
@@ -1194,57 +1194,57 @@ with $\Om{1}{} \to \Om{s}{}$.
 \section{ $\mel{N}{T [\hpsi(6) \hpsi^{\dagger}(5)] }{N,S}$ in the electron-hole product basis}
 \label{app:B}
 
-We now derive in some more details Eq.~\eqref{eq:spectral65}. 
+We now derive in more details Eq.~\eqref{eq:spectral65}. 
 Starting with
 \begin{equation}
 \begin{split}
 	\mel{N}{T [\hpsi(6) \hpsi^{\dagger}(5)] }{N,S}
 	& = \theta(+\tau_{65}) \mel{N}{ \hpsi(6) \hpsi^{\dagger}(5)  }{N,S} 
 	\\
-	& - \theta(-\tau_{65}) \mel{N}{  \hpsi^{\dagger}(5) \hpsi(6) }{N,S}
+	& - \theta(-\tau_{65}) \mel{N}{  \hpsi^{\dagger}(5) \hpsi(6) }{N,S},
 \end{split}
 \end{equation}
-we use the relation between operators in their Heisenberg and Schr\"{o}dinger representations [see Eq.~\eqref{Eisenberg}] to obtain
+we employ the relationship between operators in their Heisenberg and Schr\"{o}dinger representations [see Eq.~\eqref{Eisenberg}] to obtain
 \begin{equation}
 \begin{split}
 	& \mel{N}{T [\hpsi(6) \hpsi^{\dagger}(5)]}{N,S} = \\
 	& +  \theta(\tau_{65}) \mel{N}{ \hpsi(\bx_6) e^{-i\hH \tau_{65}} \hpsi^{\dagger}(\bx_5)  }{N,S} e^{ i E^N_0 t_6 } e^{ - i E^N_S t_5 }
 	\\
-	& - \theta(-\tau_{65}) \mel{N}{  \hpsi^{\dagger}(\bx_5) e^{ i\hH \tau_{65}} \hpsi(\bx_6) }{N,S} e^{ i E^N_0 t_5 } e^{ - i E^N_S t_6 }
+	& - \theta(-\tau_{65}) \mel{N}{  \hpsi^{\dagger}(\bx_5) e^{ i\hH \tau_{65}} \hpsi(\bx_6) }{N,S} e^{ i E^N_0 t_5 } e^{ - i E^N_S t_6 }.
 \end{split}
 \end{equation}
-with $E^N_0$ the $N$-electron ground-state energy and $E^N_S$ the energy of the $S$th excited state $\ket{N,S}$. 
+%with $E^N_0$ the $N$-electron ground-state energy and $E^N_S$ the energy of the $S$th excited state $\ket{N,S}$. 
-Expanding now the field operators with creation/destruction operators in the orbital basis   
+Expanding now the field operators with creation/destruction operators in the orbital basis, \ie, 
-\begin{subequations}
 \begin{align}
-    \hpsi(\bx_6)  & = \sum_p \phi_p(\bx_6) \ha_p 
+    \hpsi(\bx_6)  & = \sum_p \phi_p(\bx_6) \ha_p,
-    \\ 
+    & 
-    \hpsi^{\dagger}(\bx_5)  & = \sum_q \phi_q^{*}(\bx_5) \ha^{\dagger}_q
+    \hpsi^{\dagger}(\bx_5)  & = \sum_q \phi_q^{*}(\bx_5) \ha^{\dagger}_q,
 \end{align}
-\end{subequations}
 one gets
-\begin{equation}
+\begin{equation} \label{eq:N65NS}
 \begin{split}
      \mel{N}{T [\hpsi(6) \hpsi^{\dagger}(5)]}{N,S}
      \\
      = \sum_{pq} \phi_p(\bx_6)  \phi_q^{*}(\bx_5) 
      [ & \theta(+\tau_{65}) \mel{N}{ \ha_p e^{-i \hH \tau_{65}} \ha^{\dagger}_q  }{N,S} e^{ i E^N_0 t_6 } e^{ - i E^N_S t_5 }  
      \\
-      - & \theta(-\tau_{65}) \mel{N}{  \ha^{\dagger}_q e^{ i \hH \tau_{65}} \ha_p }{N,S} e^{ i E^N_0 t_5 } e^{ - i E^N_S t_6 } ]
+      - & \theta(-\tau_{65}) \mel{N}{  \ha^{\dagger}_q e^{ i \hH \tau_{65}} \ha_p }{N,S} e^{ i E^N_0 t_5 } e^{ - i E^N_S t_6 } ].
 \end{split}
 \end{equation}
-We now act on the $N$-electron ground-state with
+%We now act on the $N$-electron ground-state wave function with
+Substituting the following identities 
 \begin{subequations}
 \begin{align}
   e^{+i\hH \tau_{65} } \ha^{\dagger}_p \ket{N} &=
-    e^{+i \qty( E^N_0 + \e{p} ) \tau_{65} } \ket{N} 
+    e^{+i \qty( E^N_0 + \e{p} ) \tau_{65} } \ket{N},
     \\
       e^{ -i\hH \tau_{65} } \ha_q \ket{N} &=
-    e^{-i \qty( E^N_0 - \e{q} ) \tau_{65} } \ket{N}
+    e^{-i \qty( E^N_0 - \e{q} ) \tau_{65} } \ket{N},
 \end{align}
 \end{subequations}
-where $\lbrace \e{p/q} \rbrace$ are quasiparticle energies, such as the $GW$ ones, namely proper addition/removal energies. 
+into Eq.~\eqref{eq:N65NS} yields
-Taking the associated  bras that we plug into the orbital product basis expansion of $\mel{N}{T [\hpsi(6) \hpsi^{\dagger}(5)]}{N,S}$  one obtains:
+%where $\lbrace \e{p/q} \rbrace$ are quasiparticle energies, such as the $GW$ ones, namely proper addition/removal energies. 
+%Taking the associated  bras that we plug into the orbital product basis expansion of $\mel{N}{T [\hpsi(6) \hpsi^{\dagger}(5)]}{N,S}$  one obtains:
 \begin{equation}
 \begin{split}
 	\mel{N}{T [\hpsi(6) \hpsi^{\dagger}(5)]}{N,S} 
@@ -1252,11 +1252,10 @@ Taking the associated  bras that we plug into the orbital product basis expansio
 	= \sum_{pq} \phi_p(\bx_6)  \phi_q^{*}(\bx_5) 
     [  & \theta(+ \tau_{65}) \mel{N}{ \ha_p   \ha^{\dagger}_q  }{N,S} e^{ -i \e{p} \tau_{65} } e^{ - i \Om{S}{} t_5 } 
      \\
-     - & \theta(-\tau_{65}) \mel{N}{  \ha^{\dagger}_q \ha_p }{N,S} e^{ -i \e{q} \tau_{65}  } e^{ - i \Om{S}{} t_6 } ]
+     - & \theta(-\tau_{65}) \mel{N}{  \ha^{\dagger}_q \ha_p }{N,S} e^{ -i \e{q} \tau_{65}  } e^{ - i \Om{S}{} t_6 } ],
 \end{split}
 \end{equation}
-leading to Eq.~\eqref{eq:spectral65} with $\Om{S}{} = (E^N_S - E^N_0)$, $t_6 = \tau_{65}/2 + t^{65}$ and $t_5 = - \tau_{65}/2 + t^{65}$.
+leading to Eq.~\eqref{eq:spectral65} with $\Om{S}{} = E^N_S - E^N_0$, $t_6 = \tau_{65}/2 + t^{65}$, and $t_5 = - \tau_{65}/2 + t^{65}$.
-  
 
 \bibliography{BSEdyn}