take 1 xavier part
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%% This BibTeX bibliography file was created using BibDesk.
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%% http://bibdesk.sourceforge.net/
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%% Created for Pierre-Francois Loos at 2020-05-21 09:37:18 +0200
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%% Created for Pierre-Francois Loos at 2020-05-26 10:43:21 +0200
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%% Saved with string encoding Unicode (UTF-8)
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@ -15,7 +15,8 @@
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School = {Universit{\'e} Pierre et Marie Curie --- Paris VI},
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Title = {Range-Separated Density-Functional Theory for Molecular Excitation Energies},
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Url = {https://tel.archives-ouvertes.fr/tel-01027522},
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Year = {2014}}
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Year = {2014},
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Bdsk-Url-1 = {https://tel.archives-ouvertes.fr/tel-01027522}}
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@article{Baumeier_2012a,
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Author = {Baumeier, Bj\"{o}rn and Andrienko, Denis and Rohlfing, Michael},
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113
BSEdyn.tex
113
BSEdyn.tex
@ -309,10 +309,10 @@ can be expressed as a function of the one- and two-body Green's functions
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\begin{subequations}
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\begin{align}
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\label{eq:G1}
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G(1,1') & = - i \mel{N}{T \hpsi(1) \hpsi^{\dagger}(1')}{N},
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G(1,2) & = - i \mel{N}{T [ \hpsi(1) \hpsi^{\dagger}(2) ] }{N},
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\\
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\label{eq:G2}
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G_2(1,2;1',2') & = - \mel{N}{T \hpsi(1) \hpsi(2) \hpsi^{\dagger}(2') \hpsi^{\dagger}(1')}{N},
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G_2(1,2;1',2') & = - \mel{N}{T [ \hpsi(1) \hpsi(2) \hpsi^{\dagger}(2') \hpsi^{\dagger}(1') ]}{N},
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\end{align}
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\end{subequations}
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and
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@ -331,17 +331,17 @@ In the optical limit of instantaneous electron-hole creation and destruction, im
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\begin{equation}
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\begin{split}
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iL(1,2; 1',2')
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& = \theta(+t_{12}) \sum_{s > 0} \chi_s(\bx_1,\bx_{1'}) \tchi_s(\bx_2,\bx_{2'}) e^{ - i \Oms t_{12} }
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& = \theta(+\tau_{12}) \sum_{s > 0} \chi_s(\bx_1,\bx_{1'}) \tchi_s(\bx_2,\bx_{2'}) e^{ - i \Oms \tau_{12} }
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\\
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& - \theta(-t_{12}) \sum_{s > 0} \chi_s(\bx_2,\bx_{2'}) \tchi_s(\bx_1,\bx_{1'}) e^{ + i \Oms t_{12} },
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& - \theta(-\tau_{12}) \sum_{s > 0} \chi_s(\bx_2,\bx_{2'}) \tchi_s(\bx_1,\bx_{1'}) e^{ + i \Oms \tau_{12} },
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\end{split}
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\end{equation}
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where $t_{12} = t_1 - t_2$, $\theta$ is the Heaviside step function, and
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where $\tau_{12} = t_1 - t_2$, $\theta$ is the Heaviside step function, and
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\begin{subequations}
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\begin{align}
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\chi_s(\bx_1,\bx_{2}) & = \mel{N}{T \hpsi(\bx_1) \hpsi^{\dagger}(\bx_{2})}{N,s},
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\chi_s(\bx_1,\bx_{2}) & = \mel{N}{T [\hpsi(\bx_1) \hpsi^{\dagger}(\bx_{2})] }{N,s},
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\\
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\tchi_s(\bx_1,\bx_{2}) & = \mel{N,s}{T \hpsi(\bx_1) \hpsi^{\dagger}(\bx_{2})}{N}.
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\tchi_s(\bx_1,\bx_{2}) & = \mel{N,s}{T [\hpsi(\bx_1) \hpsi^{\dagger}(\bx_{2})] }{N}.
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\end{align}
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\end{subequations}
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The $\Oms$'s are the neutral excitation energies of interest.
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@ -349,100 +349,101 @@ The $\Oms$'s are the neutral excitation energies of interest.
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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:
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\begin{multline} \label{eq:BSE_2}
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\mel{N}{T \hpsi(\bx_1) \hpsi^{\dagger}(\bx_{1}')}{N,s} e^{ - i \Oms t_1 }
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\theta ( t_{12} )
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\theta ( \tau_{12} )
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\\
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= \int d3456 \, L_0(1,4;1',3) \Xi(3,5;4,6)
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\\
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\times \mel{N}{T \hpsi(6) \hpsi^{\dagger}(5)}{N,s}
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\times \mel{N}{T [\hpsi(6) \hpsi^{\dagger}(5)] }{N,s}
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\theta [\min(t_5,t_6) - t_2].
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\end{multline}
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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}].
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\titou{T2: Xavier, should we mention the consequences of this more explicitly?}
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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}).
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Dropping the (space/spin) variables, the Fourier components with respect to $t_1$ of $L_0(1,4;1',3)$ reads
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\begin{align} \label{eq:iL0}
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[iL_0]( \omega_1 )
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= \int \frac{d\omega}{2\pi} \; G\qty(\omega - \frac{\omega_1}{2} ) G\qty( {\omega} + \frac{\omega_1}{2} )
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e^{ i \omega t_{34} } e^{ i \omega_1 t^{34} }
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e^{ i \omega \tau_{34} } e^{ i \omega_1 \tau^{34} },
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\end{align}
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with $t_{34} = t_3 - t_4$ and $t^{34} = (t_3 + t_4)/2$.
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with $\tau_{34} = t_3 - t_4$ and $\tau^{34} = (t_3 + t_4)/2$.
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We now adopt the Lehman representation of the one-body Green's function in the quasiparticle approximation, \ie,
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\begin{equation} \label{eq:G-Lehman}
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G(\bx_1,\bx_2 ; \omega) = \sum_p \frac{ \phi_p(\bx_1) \phi_p^*(\bx_2) } { \omega - \e{p} + i \eta \times \text{sgn} (\e{p} - \mu) }
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G(\bx_1,\bx_2 ; \omega) = \sum_p \frac{ \phi_p(\bx_1) \phi_p^*(\bx_2) } { \omega - \e{p} + i \eta \times \text{sgn} (\e{p} - \mu) },
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\end{equation}
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where $\mu$ is the chemical potential.
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The set $\e{p}$'s in Eq.~\eqref{eq:G-Lehman} are quasiparticle energies and the $\phi_p$'s are their associated one-body (spin)orbitals.
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The $\e{p}$'s in Eq.~\eqref{eq:G-Lehman} are quasiparticle energies (\ie, proper addition/removal energies) and the $\phi_p$'s are their associated one-body (spin)orbitals.
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%where the $\eps_{p}$'s are proper addition/removal energies such that
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%\begin{equation}
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% e^{i \hH \tau} \ha_p^{\dagger} \ket{N} = e^{ i (E_0^N + \e{p} ) \tau } \ha_p^{\dagger} \ket{N},
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%\end{equation}
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%$\hH$ being the exact many-body Hamiltonian.
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In the following, $i$ and $j$ are occupied orbitals, $a$ and $b$ are unoccupied orbitals, while $p$, $q$, $r$, and $s$ indicate arbitrary orbitals.
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%\titou{namely $GW$ quasiparticle energies and input Hartree-Fock molecular orbitals in the present study. (T2: shall we really mention this here?)}
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After projecting onto $\phi_a^*(\bx_1) \phi_i(\bx_{1'})$, one gets
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Projecting $L_0(1,4;1',3)$ onto $\phi_a^*(\bx_1) \phi_i(\bx_{1'})$ yields
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\begin{multline}
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\int d\bx_1 d\bx_{1'} \; \phi_a^*(\bx_1) \phi_i(\bx_{1'}) L_0(\bx_1,3;\bx_{1'},4; \Oms)
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\int d\bx_1 d\bx_{1'} \; \phi_a^*(\bx_1) \phi_i(\bx_{1'}) L_0(\bx_1,4;\bx_{1'},3; \Oms)
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\\
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=
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\frac{ \phi_a^*(\bx_3) \phi_i(\bx_4) e^{i \Oms t^{34} }} { \Oms - ( \e{a} - \e{i} ) + i \eta }
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\qty[ \theta( \tau ) e^{i ( \e{i} + \hOms) \tau } + \theta( - \tau ) e^{i (\e{a} - \hOms \tau) } ]
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\frac{ \phi_a^*(\bx_3) \phi_i(\bx_4) e^{i \Oms t^{34} }} { \Oms - ( \e{a} - \e{i} ) + i \eta }
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\qty[ \theta( \tau_{34} ) e^{i ( \e{i} + \hOms) \tau_{34} } + \theta( - \tau_{34} ) e^{i (\e{a} - \hOms \tau_{34}) } ].
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\end{multline}
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with $\tau = t_{34}$.
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\titou{T2: I think 3 and 4 have been swapped in the previous equation.}
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% and $(i,j)$/$(a,b)$ index occupied/virtual orbitals, respectively.
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Adopting now the $GW$ approximation for the exchange-correlation self-energy, \ie,
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\begin{equation}
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\Sigma_\text{xc}^{\GW}(1,2) = i G(1,2) W(1^+,2),
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\end{equation}
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leads to the following simplified BSE kernel
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\begin{equation}
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\Xi(3,5;4,6) = v(3,6) \delta(3,4) \delta(5,6) - W(3^+,4) \delta(3,6) \delta(4,5),
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\end{equation}
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where $W$ is its dynamically-screened Coulomb operator.
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\titou{T2: shall we introduce the GW approximation later on?}
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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.
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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.
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% with $(6,5) \rightarrow (5,5) \; \text{or} \; (3,4)$ when multiplied by $\delta(5,6)$ or $\delta(3,6) \delta(4,5)$, respectively.
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This is done by expanding the field operators over a complete orbital basis of creation/destruction operators.
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For example, we have
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\begin{multline}
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\mel{N}{T \hpsi(3) \hpsi^{\dagger}(4)}{N,s}
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\mel{N}{T [\hpsi(3) \hpsi^{\dagger}(4)] }{N,s}
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\\
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= - \qty( e^{ -i \Omega_s t^{34} } ) \sum_{pq} \phi_p(\bx_3) \phi_q^*(\bx_4)
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\mel{N}{\ha_q^{\dagger} \ha_p}{N,s}
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\\
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\times \qty[ \theta( t_{34} ) e^{- i ( \e{p} - \hOms ) t_{34} } + \theta( - t_{34} ) e^{ - i ( \e{q} + \hOms) t_{34} } ],
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\end{multline}
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where the $ \lbrace \eps_{p/q} \rbrace$ are proper addition/removal energies \titou{(T2: shall it be mentioned earlier around Eq. (14)?)} such that
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with a similar expression for $\mel{N}{T [\hpsi(\bx_3) \hpsi^{\dagger}(\bx_4)] }{N,s}$.
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Adopting now the $GW$ approximation for the exchange-correlation self-energy, \ie,
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\begin{equation}
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e^{i \hH \tau} \ha_p^{\dagger} \ket{N} = e^{ i (E_0^N + \e{p} ) \tau } \ha_p^{\dagger} \ket{N},
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\Sigma_\text{xc}^{\GW}(1,2) = i G(1,2) W(1^+,2),
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\end{equation}
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$\hH$ being the exact many-body Hamiltonian.
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The $GW$ quasiparticle energies $\eGW{i/a}$ are good approximations to such removal/addition energies.
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Selecting $(p,q)=(j,b)$ yields the largest components
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$X_{jb}^{s} = \mel{N}{\ha_j^{\dagger} \ha_b}{N,s}$, while $(p,q)=(b,j)$ yields much weaker
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$Y_{jb}^{s} = \mel{N}{\ha_b^{\dagger} \ha_j}{N,s}$ contributions.
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Neglecting the $Y_{jb}^{s}$ weights leads to the Tamm-Dancoff approximation (TDA).
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Working out the same expansion for $\mel{N}{T \hpsi(5) \hpsi^{\dagger}(5)}{N,s}$ and $\mel{N}{T \hpsi(\bx_1) \hpsi^{\dagger}(\bx_{1'})}{N,s}$ \titou{(where do we need these terms?)}, and projecting onto $\phi_a^*(\bx_1) \phi_i(\bx_{1'})$, one obtains after a few tedious manipulations (see {\SI}) the dynamical Bethe-Salpeter equation (dBSE):
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\begin{equation}
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leads to the following simplified BSE kernel
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\begin{equation} \label{eq:Xi_GW}
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\Xi(3,5;4,6) = v(3,6) \delta(3,4) \delta(5,6) - W(3^+,4) \delta(3,6) \delta(4,5),
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\end{equation}
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where $W$ is the dynamically-screened Coulomb operator.
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The $GW$ quasiparticle energies $\eGW{p}$ are good approximations to the removal/addition energies $\e{p}$ introduced in Eq.~\eqref{eq:G-Lehman}.
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%Selecting $(p,q)=(j,b)$ yields the largest components
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%$X_{jb}^{s} = \mel{N}{\ha_j^{\dagger} \ha_b}{N,s}$, while $(p,q)=(b,j)$ yields much weaker
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%$Y_{jb}^{s} = \mel{N}{\ha_b^{\dagger} \ha_j}{N,s}$ contributions.
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%Neglecting the $Y_{jb}^{s}$ weights leads to the Tamm-Dancoff approximation (TDA).
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%Working out similar expressions for $\mel{N}{T [\hpsi(5) \hpsi^{\dagger}(5)] }{N,s}$ and $\mel{N}{T [\hpsi(\bx_1) \hpsi^{\dagger}(\bx_{1'})] }{N,s}$,
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Substituting Eq.~\eqref{eq:Xi_GW} into Eq.~\eqref{eq:BSE_2}, working out similar expressions for the remaining terms, and projecting onto $\phi_a^*(\bx_1) \phi_i(\bx_{1'})$, one gets after a few tedious manipulations (see {\SI}) the dynamical Bethe-Salpeter equation (dBSE):
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\begin{equation} \label{eq:BSE-final}
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\begin{split}
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( \e{a} - \e{i} - \Oms ) X_{ia}^{s}
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& + \sum_{jb} \qty[ v_{ai,bj} - \widetilde{W}_{ij,ab}(\Oms) ] X_{jb}^{s} \\
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& + \sum_{jb} \qty[ v_{ai,jb} - \widetilde{W}_{ib,aj}(\Oms) ] Y_{jb}^{s}
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( \eGW{a} - \eGW{i} - \Oms ) X_{ia}^{s}
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& + \sum_{jb} \qty[ (ia|jb) - \widetilde{W}_{ij,ab}(\Oms) ] X_{jb}^{s} \\
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& + \sum_{jb} \qty[ (ia|bj) - \widetilde{W}_{ib,aj}(\Oms) ] Y_{jb}^{s}
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= 0
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\end{split}
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\end{equation}
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with an effective dynamically-screened Coulomb potential \cite{Romaniello_2009b}
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with
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\begin{equation}
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(pq|rs) = \int d\br d\br' \, \phi_p^*(\br) \phi_q(\br) v(\br -\br') \phi_r^*(\br') \phi_s(\br'),
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\end{equation}
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and an effective dynamically-screened Coulomb potential \cite{Romaniello_2009b}
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\begin{multline}
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\widetilde{W}_{ij,ab}(\Oms)
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= \frac{ i }{ 2 \pi} \int d\omega \; e^{-i \omega 0^+ } W_{ij,ab}(\omega)
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\\
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\times \qty[ \frac{1}{ \Omega_{ib}^s - \omega + i \eta } + \frac{1}{ \Omega_{ja}^{s} + \omega + i\eta } ]
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\times \qty[ \frac{1}{ \Omega_{ib}^s - \omega + i \eta } + \frac{1}{ \Omega_{ja}^{s} + \omega + i\eta } ],
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\end{multline}
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where $\Om{ib}{s} = \Oms - ( \e{b} - \e{i} )$ and $\Om{ja}{s} = \Oms - ( \e{a} - \e{j} )$.
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Following Mulliken's notations, the Coulomb matrix elements are defined as
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\begin{align}
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v_{ai,bj}
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& = \int d\br d\br' \; \phi_a(\br) \phi_i^*(\br) v(\br -\br') \phi_b^*(\br') \phi_j(\br'),
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\\
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where $\Om{ib}{s} = \Oms - ( \eGW{b} - \eGW{i} )$, $\Om{ja}{s} = \Oms - ( \eGW{a} - \eGW{j} )$, and
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\begin{equation}
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W_{ij,ab}({\omega})
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& = \int d\br d\br' \; \phi_i(\br) \phi_j^*(\br) W(\br ,\br'; \omega) \phi_a^*(\br') \phi_b(\br'),
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\end{align}
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where we group together the indices of orbitals taken at the same space position, taking further as inner indices those associated with orbitals with complex conjugation.
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= \int d\br d\br' \, \phi_i(\br) \phi_j^*(\br) W(\br ,\br'; \omega) \phi_a^*(\br') \phi_b(\br'),
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\end{equation}
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Neglecting the terms $Y_{jb}^{s}$ in Eq.~\eqref{eq:BSE-final} leads to the well-known Tamm-Dancoff approximation (TDA).
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\xavier{A second coupled equation for the $(X_{ia}^{s}, Y_{ia}^{s} )$ vector can be obtained by projecting now onto the $\mel{N}{T \hpsi(\bx_1) \hpsi^{\dagger}(\bx_{1'})}{N,s}$ left-hand side and right-hand-side of the BSE, leading to : }
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