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saving work 1st part of Xav part

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Pierre-Francois Loos 2020-05-26 10:01:32 +02:00
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BSEdyn.tex
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@ -346,7 +346,7 @@ where $t_{12} = t_1 - t_2$, $\theta$ is the Heaviside step function, and
\end{subequations}
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 Bethe-Salpeter equation:
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 }
\theta ( t_{12} )
@ -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 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, $\Oms < \EgFun$), $L_0(1,2;1',2')$ 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}].
\titou{T2: Xavier, should we mention the consequences of this more explicitly?}
Dropping the (space/spin) variables, the Fourier components with respect to $t_1$ of $L_0(1,4;1',3)$ reads
@ -371,7 +371,7 @@ We now adopt the Lehman representation of the one-body Green's function in the q
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) }
\end{equation}
where $\mu$ is the chemical potential.
The set $\lbrace \e{p} \rbrace$ in Eq.~\eqref{eq:G-Lehman} are quasiparticle energies and $\lbrace \phi_p \rbrace$ is their associated one-body (spin)orbitals.
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.
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?)}
After projecting onto $\phi_a^*(\bx_1) \phi_i(\bx_{1'})$, one gets
@ -382,7 +382,9 @@ After projecting onto $\phi_a^*(\bx_1) \phi_i(\bx_{1'})$, one gets
\frac{ \phi_a^*(\bx_3) \phi_i(\bx_4) e^{i \Oms t^{34} }} { \Oms - ( \e{a} - \e{i} ) + i \eta }
\qty[ \theta( \tau ) e^{i ( \e{i} + \hOms) \tau } + \theta( - \tau ) e^{i (\e{a} - \hOms \tau) } ]
\end{multline}
with $\tau = t_{34}$. % and $(i,j)$/$(a,b)$ index occupied/virtual orbitals, respectively.
with $\tau = t_{34}$.
\titou{T2: I think 3 and 4 have been swapped in the previous equation.}
% and $(i,j)$/$(a,b)$ index occupied/virtual orbitals, respectively.
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),
@ -392,22 +394,24 @@ leads to the following simplified BSE kernel
\Xi(3,5;4,6) = v(3,6) \delta(3,4) \delta(5,6) - W(3^+,4) \delta(3,6) \delta(4,5),
\end{equation}
where $W$ is its dynamically-screened Coulomb operator.
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 creation/destruction operators, with \eg,
\titou{T2: shall we introduce the GW approximation later on?}
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}
\mel{N}{T \hpsi(3) \hpsi^{\dagger}(4)}{N,s}
\\
= - \qty( e^{ -i \Omega_s t^{34} } ) \sum_{pq} \phi_p(\bx_3) \phi_q^*(\bx_4)
\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( t_{34} ) e^{- i ( \e{p} - \hOms ) t_{34} } + \theta( - t_{34} ) e^{ - i ( \e{q} + \hOms) t_{34} } ],
\end{multline}
where the $ \lbrace \eps_{p/q} \rbrace$ are proper addition/removal energies \titou{(T2: shall it be mentioned earlier around Eq. (14)?)} such that
\begin{equation}
e^{i \hH \tau} \ha_p^{\dagger} \ket{N} = e^{ i (E_0^N + \e{p} ) \tau } \ha_p^{\dagger} \ket{N}
e^{i \hH \tau} \ha_p^{\dagger} \ket{N} = e^{ i (E_0^N + \e{p} ) \tau } \ha_p^{\dagger} \ket{N},
\end{equation}
with $\hH$ the exact many-body Hamiltonian.
$\hH$ being the exact many-body Hamiltonian.
The $GW$ quasiparticle energies $\eGW{i/a}$ are good approximations to such removal/addition energies.
Selecting $(p,q)=(j,b)$ yields the largest components
$X_{jb}^{s} = \mel{N}{\ha_j^{\dagger} \ha_b}{N,s}$, while $(p,q)=(b,j)$ yields much weaker
@ -422,7 +426,7 @@ Working out the same expansion for $\mel{N}{T \hpsi(5) \hpsi^{\dagger}(5)}{N,s}$
= 0
\end{split}
\end{equation}
with an effective dynamically screened Coulomb potential (see Pina eq. 24):
with an effective dynamically-screened Coulomb potential \cite{Romaniello_2009b}
\begin{multline}
\widetilde{W}_{ij,ab}(\Oms)
= \frac{ i }{ 2 \pi} \int d\omega \; e^{-i \omega 0^+ } W_{ij,ab}(\omega)