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Pierre-Francois Loos 2020-05-26 17:13:24 +02:00
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BSEdyn.tex
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@ -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: 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} \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} ) \theta ( \tau_{12} )
\\ \\
= \int d3456 \, L_0(1,4;1',3) \Xi(3,5;4,6) = \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} \times \mel{N}{T [\hpsi(6) \hpsi^{\dagger}(5)] }{N,s}
\theta [\min(t_5,t_6) - t_2]. \theta [\min(t_5,t_6) - t_2].
\end{multline} \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}). 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 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. %$\hH$ 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. 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?)} %\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} \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) \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. This is done by expanding the field operators over a complete orbital basis of creation/destruction operators.
For example, we have For example, we have
\begin{multline} \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} \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} \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, Adopting now the $GW$ approximation for the exchange-correlation self-energy, \ie,
\begin{equation} \begin{equation}
\Sigma_\text{xc}^{\GW}(1,2) = i G(1,2) W(1^+,2), \Sigma_\text{xc}^{\GW}(1,2) = i G(1,2) W(1^+,2),