blush

BSEdyn.tex

@@ -345,7 +345,11 @@ 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 $\Oms$'s are the neutral excitation energies of interest. 
+The $\Oms$'s are the neutral excitation energies of interest. We have used the relation between the field operators in their time-dependent (Heisenberg)  and time-independent (Schr\"{o}dinger) representations, e.g.
+$$
+\hpsi(1) = e^{ i {\hat H} t_1 } \hpsi(\bx_1) e^{-i {\hat H} t_1 }
+$$
+with $\hat H$ the exact many-body Hamiltonian.
 
 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 a modified dynamical BSE, which reads
 \begin{multline} \label{eq:BSE_2}
@@ -381,27 +385,27 @@ The $\e{p}$'s in Eq.~\eqref{eq:G-Lehman} are quasiparticle energies (\ie, proper
 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 the Fourier component $L_0(\bx_1,4;\bx_{1'},3;  \omega_1 = \Oms )$ onto $\MO{a}^*(\bx_1) \MO{i}(\bx_{1'})$ yields
-\begin{multline}
+\begin{multline}  \label{eq:iL0bis} 
 	\iint d\bx_1 d\bx_{1'} \, \MO{a}^*(\bx_1) \MO{i}(\bx_{1'})  L_0(\bx_1,4;\bx_{1'},3; \Oms) 
 	\\
 	= 
 	\frac{ \MO{a}^*(\bx_3) \MO{i}(\bx_4)  e^{i \Oms t^{34} }} { \Oms - ( \e{a} - \e{i} ) + i \eta }   
-	\qty[ \theta( \tau_{34} ) e^{i ( \e{i} + \hOms) \tau_{34} } + \theta( - \tau_{34} ) e^{i (\e{a} - \hOms \tau_{34}) }  ]. 
+	\qty[ \theta( \tau_{34} ) e^{i ( \e{i} + \hOms) \tau_{34} } + \theta( - \tau_{34} ) e^{i (\e{a} - \hOms ) \tau_{34} }  ]. 
 \end{multline}
 % and $(i,j)$/$(a,b)$ index occupied/virtual orbitals, respectively. 
 As a final step, we express the terms $\mel{N}{T [\hpsi(\bx_1) \hpsi^{\dagger}(\bx_{1}')] }{N,s}$ and $\mel{N}{T [\hpsi(6) \hpsi^{\dagger}(5)] }{N,s}$ from Eq.~\eqref{eq:BSE_2} in the standard electron-hole product (or single-excitation) space.
 % with $(6,5) \rightarrow (5,5) \; \text{or} \; (3,4)$ when  multiplied by $\delta(5,6)$ or $\delta(3,6) \delta(4,5)$, respectively.  
 This is done by  expanding the field operators over a complete orbital basis of creation/destruction operators.
 For example, we have
-\begin{multline}
+\begin{multline} \label{eq:spectral65}
 	\mel{N}{T [\hpsi(6) \hpsi^{\dagger}(5)] }{N,s} 
 	\\
-	= - \qty( e^{  -i \Omega_s \tau^{56} }  ) \sum_{pq} \MO{p}(\bx_6) \MO{q}^*(\bx_5) 
+	= - \qty( e^{  -i \Omega_s t^{65} }  ) \sum_{pq} \MO{p}(\bx_6) \MO{q}^*(\bx_5) 
 	\mel{N}{\ha_q^{\dagger} \ha_p}{N,s}
 	\\
-	\times \qty[ \theta( \tau_{56}  ) e^{- i ( \e{p} - \hOms ) \tau_{56}  } + \theta( - \tau_{56}  ) e^{ - i ( \e{q}  + \hOms) \tau_{56}  } ] 
+	\times \qty[ \theta( \tau_{65}  ) e^{- i ( \e{p} - \hOms ) \tau_{65}  } + \theta( - \tau_{65}  ) e^{ - i ( \e{q}  + \hOms) \tau_{65}  } ] 
 \end{multline}
-with $\tau^{56} = (t_5 + t_6)/2$ and $\tau_{56} = t_5 -t_6$.
+with $t^{65} = (t_5 + t_6)/2$ and $\tau_{65} = t_6 -t_5$.
 %with a similar expression for $\mel{N}{T [\hpsi(\bx_3) \hpsi^{\dagger}(\bx_4)] }{N,s}$. 
 
 Adopting now the $GW$ approximation for the exchange-correlation self-energy, \ie, 
@@ -420,7 +424,7 @@ The $GW$ quasiparticle energies $\eGW{p}$ are good approximations to the removal
 %Neglecting the $Y_{jb}^{s}$ weights leads to the Tamm-Dancoff approximation (TDA). 
 
 %Working out similar expressions for $\mel{N}{T [\hpsi(5) \hpsi^{\dagger}(5)] }{N,s}$ and $\mel{N}{T [\hpsi(\bx_1) \hpsi^{\dagger}(\bx_{1'})] }{N,s}$, 
-Substituting Eq.~\eqref{eq:Xi_GW} into Eq.~\eqref{eq:BSE_2}, working out similar expressions for the remaining terms, and projecting onto $\MO{a}^*(\bx_1) \MO{i}(\bx_{1'})$, one gets after a few tedious manipulations (see {\SI}) the dynamical BSE (dBSE):
+Substituting Eqs.~\eqref{eq:iL0bis},\eqref{eq:spectral65},\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 (dBSE):
 \begin{equation} \label{eq:BSE-final}
 \begin{split}
 	( \eGW{a} - \eGW{i} - \Oms ) X_{ia}^{s}         
@@ -502,7 +506,7 @@ Because $\Om{m}{\RPA} > 0$, we have
     \times \qty( \frac{1}{\Om{ib}{s} - \Om{m}{\RPA} + i\eta} + \frac{1}{\Om{ja}{s} - \Om{m}{\RPA} + i\eta} )
 \end{multline} 
 \titou{Due to excitonic effects, the lowest BSE excitation energy, ${\Omega}_1$, stands lower than the lowest RPA excitation energy, $\Omega_m^{RPA}$, so that 
-e.g. $( \Omega_{ib}^{s} - \Omega_m^{RPA} )$ is strictly negative and $\widetilde{W}_{ij,ab}( \Oms )$ presents no resonances. 
+e.g. $( \Omega_{ib}^{s} - \Omega_m^{RPA} )$ is strictly negative and the $\widetilde{W}_{ij,ab}( \Oms )$ present no resonances. 
 Further, $\Omega_{ib}^{s}$ and $\Omega_{ja}^{s}$ are necessarily negative for in-gap   low lying BSE excitations, such that e.g.
 $$
 \abs{ \frac{1}{\Omega_{ib}^{s} - \Omega_m^{RPA}} }  <  \frac{1}{ \Omega_m^{RPA}}