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\documentclass[10pt]{book}
\usepackage{graphicx}
\usepackage{amsmath} %% pour mettre du texte en mode math
\usepackage{tcolorbox}
\newcommand\uom{\underline{\omega}}
\newcommand\Oms{{\Omega}_s}
\newcommand\hOms{\frac{{\Omega}_s}{2}}
\begin{document}
\chapter{Strinati for the big kids}
We wish to solve the self-consistent Bethe-Salpeter equation:
\begin{equation*}
L(1,2;1',2') = L_0(1,2;1',2') + \int d3456 \; L_0(1,4;1',3) \Xi[3,5;4,6] L(6,2;5,2')
\end{equation*}
where e.g. $1=(x_1,t_1)$ is a space/spin and time position and:
\begin{align*}
iL(1,2;1',2') &= -G_2(1,2 ; 1',2') + G(1,1')G(2,2') \\
iL_0(1,2;1',2') &= G(1,2')G(2,1') \\
iG(1,2') &= \langle N | T {\hat \psi}(1) {\hat \psi}^{\dagger}(2') | N \rangle \\
i^2 G_2(1,2 ; 1',2') &= \langle N | T \Big[ {\hat \psi}(1) {\hat \psi}(2) {\hat \psi}^{\dagger}(2') {\hat \psi}^{\dagger}(1') \Big] | N \rangle
\end{align*}
For optical properties, with instantaneous creation/destruction of excitations, we are interested in the collapse of the time variables starting with imposing $t_2'=t_2^{+}$ in the right-hand side $L(1,2;1',2')$ and left hand-side $L(6,2;5,2')$ response functions. Since $t_2'=t_2^{+}=(t_2 + 0^+)$ indicates that the (2,2') operators cannot be separated, we see that the ordering operator selects two possible channels depending on the times ordering, namely:
\begin{align*}
i^2 G_2^{I}(1,2 ; 1',2') &= \theta( t_{m}^{11'} -t_2 )\langle N | T [{\hat \psi}(1) {\hat \psi}^{\dagger}(1') ] T [ {\hat \psi}(2) {\hat \psi}^{\dagger}(2') ] | N \rangle \\
i^2 G_2^{II}(1,2 ; 1',2') &= \theta( t_2 - t_{M}^{11'} )\langle N | T [ {\hat \psi}(2) {\hat \psi}^{\dagger}(2') ]
T [{\hat \psi}(1) {\hat \psi}^{\dagger}(1') ] | N \rangle
\end{align*}
with $t_{M}^{11'} = \max(t_1,t_{1'}) = | \tau_{11'} |/2 + t^{11'}$ and $t_{m}^{11'} = \min(t_1,t_{1'}) = - | \tau_{11'} |/2 + t^{11'}$ where $\tau_{11'} = t_1 - t_{1'}$ and $t^{11'} = (t_1 + t_{1'})/2$.
Introducing the complete set of eigenstates of the N-electron systems $\lbrace | N,s \rangle \rbrace$, where s index the eigenstates and $| N,s=0 \rangle = | N \rangle$ the ground-state, one obtains for $G^{I}$ :
\begin{align*}
i^2 G_2^{I}(1,2 ; 1',2') &= \theta( t_{m}^{11'} -t_2 ) \sum_{s>0} \langle N | T {\hat \psi}(1) {\hat \psi}^{\dagger}(1') | N,s \rangle
\langle N,s | T {\hat \psi}(2) {\hat \psi}^{\dagger}(2') | N \rangle \\
& + \theta( t_{m}^{11'} -t_2 ) \frac{ G(1,1') }{ i } \frac{ G(2,2') }{ i } \\
\end{align*}
where in the right-hand side of the first line the sum $(s>0)$ avoids the (s=0) ground-state contribution that builds the 1-body Green's functions product in the 2nd line.
This yields a first channel for $L(1,2;1',2')$ namely:
$$
iL^{I}(1,2;1',2') = \theta( t_{m}^{11'} -t_2 ) \sum_{s>0} \langle N | T {\hat \psi}(1) {\hat \psi}^{\dagger}(1') | N,s \rangle
\langle N,s | T {\hat \psi}(2) {\hat \psi}^{\dagger}(2') | N \rangle
$$
Switching from the Heisenberg representation (time-dependent operators) to the Schr{\"o}dinger one for the field operators, with e.g.
$$
{\hat \psi}(2) = {\hat \psi}(x_2 t_2) = e^{ i H t_2} {\hat \psi}(x_2) e^{ -i H t_2}
$$
one obtains
\begin{align*}
\langle N,s | T {\hat \psi}(2) {\hat \psi}^{\dagger}(2') | N \rangle &=
- \langle N,s | e^{i H t_2^+} {\hat \psi}^{\dagger}(x_2') e^{i H (t_2 - t_2^+) } {\hat \psi}(x_2) e^{-i H t_2} | N \rangle \\
&= - \langle N,s | {\hat \psi}^{\dagger}(x_{2'}) {\hat \psi}(x_2) | N \rangle \times e^{i \Omega_s t_2 } \\
& = {\tilde \chi}_s(x_2,x_{2'}) e^{i \Omega_s t_2 }
\end{align*}
with $\;{\tilde \chi}_s(x_2,x_{2'}) = \langle N,s | {\hat \psi}(x_2) {\hat \psi}^{\dagger}(x_{2'}) | N \rangle$.
The energy $\Omega_s = E_s^N - E_0^N$ is the s-th neutral excitation energy of the system, namely the energy we are looking for.
The (-) sign in the right-hand side comes from the commutation of the creation and destruction operators with $t_2^+ > t_2$.
A similar calculation for the second channel [$t_2 > \max(t_1,t_{1'})$] leads to a frequency dependence in $e^{ -i \Omega_s t_2 }$.
Putting everything together, one obtains :
\begin{align*}
iL (1,2;1',2') = & \; \theta( t_{m}^{11'} -t_2 ) \sum_{s>0} \langle N | T {\hat \psi}(1) {\hat \psi}^{\dagger}(1') | N,s \rangle
{\tilde \chi}_s(x_2,x_{2'}) \times e^{i \Omega_s t_2 } \\
- & \; \theta( t_2 - t_{M}^{11'} ) \sum_{s>0} \langle N,s | T {\hat \psi}(1) {\hat \psi}^{\dagger}(1') | N \rangle
{\chi}_s(x_2,x_{2'}) \times e^{ -i \Omega_s t_2 }
\end{align*}
with $\;{\chi}_s(x_2,x_{2'}) = \langle N | {\hat \psi}(x_2) {\hat \psi}^{\dagger}(x_{2'}) | N,s \rangle$. \\
Performing the very same analysis for $L(6,2;5,2')$ and picking up a specific $e^{ +i \Omega_s t_2 }$ contribution, dividing by
$ \langle N,s | T {\hat \psi}(2) {\hat \psi}^{\dagger}(2') | N \rangle $ both sides of the Bethe-Salpeter equation leads to: \\
\begin{tcolorbox}[ width=\textwidth, colback={white}]
\begin{align}
\langle N & | T {\hat \psi}(1) {\hat \psi}^{\dagger}(1') | N,s \rangle \theta( t_{m}^{11'} -t_2 ) = \\
&= \int d3456 \; L_0(1,4;1',3) \Xi[3,5;4,6] \langle N | T {\hat \psi}(6) {\hat \psi}^{\dagger}(5) | N,s \rangle \theta( t_{m}^{56} -t_2 ) \nonumber
\end{align}
\end{tcolorbox}
\noindent where the $L_0(1,2;1',2')$ does not contribute if \textbf{ we select some $\Omega_s$ frequency below the quasiparticle gap of the system}. This is the common situation in molecular systems where due to excitonic effects, namely electron-hole interaction, the lowest optical excitation energies are lower than the photoemission quasiparticle gap. $ L_0(1,2;1',2')$ has indeed a lowest excitation energy at the energy gap $E_g$ and cannot participate to the $e^{i \Omega_s t_2}$ response of the system if $\Omega_s < E_g$. \\
This is equation (11.3) of Strinati with e.g. $$ \langle N | T {\hat \psi}(1) {\hat \psi}^{\dagger}(1') | N,s \rangle = \chi_s(x_1,x_{1'}; \tau_1) e^{-i \Omega_s t^1}$$
(see equation G.3a of Appendix G) but keeping the Heaviside fonctions for a better definition of time integrals. \\
\noindent \textbf{Spectral representation of $G(t_1-t_3)G(t_4-t_1^+)$ } \\
We now consider the $t_1$ time. Using the same algebra as above, one obtains
$$
\langle N | T {\hat \psi}(1) {\hat \psi}^{\dagger}(1') | N,s \rangle =
\langle N | T {\hat \psi}(x_1) {\hat \psi}^{\dagger}(x_{1'}) | N,s \rangle \times e^{- i \Oms t_1 }
$$
imposing now $t_1' = t_1^+$. To extract the $e^{-i \Omega_s t_1 }$ behavior in the right hand-side of the BSE equation, one must find the Fourier components with respect to the $t_1$ variable of $L_0(1,4;1',3) = -i G(1,3) G(4,1')$ by multiplying by $ \int dt_1 e^{i \omega_1 t_1}$ both sides of the BSE equation, \textcolor{red}{ with $( \omega_1 \rightarrow \Oms + i\delta )$ where we introduce the small infinitesimal $\delta = 0^+$ for defining properly the $t_1$ time integral on the left-hand side: }
$$
\int dt_1 e^{i \omega_1 t_1} e^{ - i \Oms t_1 } \theta( t_1 -t_2 ) = e^{-\delta t_2} \int d \tau e^{- \delta \tau } \theta( \tau ) = ( \delta^{-1} )
$$
This yields an annoying but useful factor that will canceled out in the following.
Expressing again the field operators in their Schr\"{o}dinger representation, one see that the 1-body Green's functions depend on the time-differences $\tau_{13} = t_1 - t_3$ and $\tau_{41} = t_4 - t_1$, respectively, with e.g.
$$
\langle N | \psi(1) \psi^{\dagger}(3) | N \rangle = \langle N | \psi(x_1) e^{-i H \tau_{13}} \psi^{\dagger}(x_3) | N \rangle e^{+i E_N^0 \tau_{13}}
$$
Taking by convention the time difference as that of the destruction operator minus that of the creation analog and introducing our notations for the Fourier representation :
\begin{equation*}
G( \tau_{13} ) = {1 \over 2\pi} \int d \omega \; G(\omega) e^{ - i \omega \tau_{13} } \;\;\;\;\;
G(\tau_{41} ) = {1 \over 2\pi} \int d \omega' \; G(\omega') e^{ - i {\omega}' \tau_{41} }
\end{equation*}
one obtains (keeping only time/frequency variables for clarity)
$$
i L_0(1,4;1',3) = \int { d\omega \over 2\pi } { d{\omega}' \over 2\pi } \; G(\omega) G({\omega}') e^{ -i \omega \tau_{13} } e^{ - i {\omega}' \tau_{41} }
$$
The Fourier components of ($iL_0$) with respect to the ($t_1$) time are:
\begin{align*}
[iL_0]( \omega_1 ) &= \int dt_1 e^{ i \omega_1 t_1 } \int { d\omega \over 2\pi } { d{\omega}' \over 2\pi } \; G(\omega ) G( {\omega}' )
e^{- i \omega (t_1 - t_3) } e^{ -i {\omega}' (t_4 - t_1) } \\
&= {1 \over (2\pi)^2 } \int d \omega d{\omega}' \; G(\omega ) G( {\omega}' ) 2\pi \delta( -\omega + \omega_1 + {\omega}' )
e^{ i \omega t_3 } e^{ - i {\omega}' t_4 } \\
&= {1 \over 2\pi } \int d \omega \; G(\omega ) G( {\omega} - \omega_1 ) e^{ i \omega t_3 } e^{ -i ( \omega - \omega_1) t_4 } \\
&= { 1 \over 2\pi } \int d \omega \; G(\omega - {\omega_1 \over 2} ) G( {\omega} + {\omega_1 \over 2} ) e^{ i \omega \tau_{34} } e^{ i \omega_1 t^{34} }
\end{align*}
where in the last line we made the change of variable $( \omega \rightarrow \omega + {\omega_1} / 2 ) $ and introduced the average time
$t^{34} = (t_3 + t_4) /2$.
We now make the identification $( \omega_1 = {\Oms} + i\delta )$ as done to properly pick-up the left-hand side $e^{-i \Oms t_1}$ contribution and obtain :
\begin{align*}
[iL_0 ]( \Omega_s)
= { 1 \over 2\pi } \int d \omega \; G(\omega - \hOms ) G( {\omega} + \hOms ) e^{ i \omega \tau_{34} } e^{ i \Oms^{+} t^{34} }
\end{align*}
with $( \Oms^{+} = \Oms + i\delta )$ keeping the infinitesimal $\delta$ where it will be needed.
Using the Lehman representation of the 1-body Green's functions, e.g.
$$
G(x_1,x_3 ; \omega) = \sum_n \frac{ \phi_n(x_1) \phi_n^*(x_3) } { \omega - \varepsilon_n + i \eta \text{sgn}(\varepsilon_n - \mu) }
$$
where we reintroduced the (space, spin) variables, one obtains by picking the residues associated with the poles of each Green's function:
\begin{align*}
\int d \omega & \; G(x_1,x_3; \omega - \hOms ) G(x_4,x_{1'}; \omega + \hOms ) e^{ i \omega \tau_{34} } = \\
& 2 {i} \pi \theta( \tau_{34} ) \sum_{nj} \frac{ \phi_n(x_1) \phi_n^*(x_3) \phi_j(x_4) \phi_j^*(x_{1'}) } { \varepsilon_j + \Omega_s - \varepsilon_n + i \eta \times \text{sgn}(\varepsilon_n - \mu) } e^{i (\varepsilon_j + {\Omega_s \over 2} ) \tau_{34} } \\
+ & 2 {i} \pi \theta( \tau_{34} ) \sum_{jn} \frac{ \phi_j(x_1) \phi_j^*(x_3) \phi_n(x_4) \phi_n^*(x_{1'}) } { \varepsilon_j - \Omega_s - \varepsilon_n + i \eta \times \text{sgn}(\varepsilon_n - \mu) } e^{i (\varepsilon_j - {\Omega_s \over 2} ) \tau_{34} } \\
- & 2 {i} \pi \theta(- \tau_{34} ) \sum_{nb} \frac{ \phi_n(x_1) \phi_n^*(x_3) \phi_b(x_4) \phi_b^*(x_{1'}) } { \varepsilon_b + \Omega_s - \varepsilon_n + i \eta \times \text{sgn}(\varepsilon_n - \mu) } e^{i (\varepsilon_b + {\Omega_s \over 2} ) \tau_{34} } \\
- & 2 {i} \pi \theta(- \tau_{34} ) \sum_{bn} \frac{ \phi_b(x_1) \phi_b^*(x_3) \phi_n(x_4) \phi_n^*(x_{1'}) } { \varepsilon_b - \Omega_s - \varepsilon_n + i \eta \times \text{sgn}(\varepsilon_n - \mu) } e^{i (\varepsilon_b - {\Omega_s \over 2} ) \tau_{34} }
\end{align*}
where for $\tau_{34} > 0$ (resp. $\tau_{34} < 0$) the contour is close in the upper (resp. lower) half complex plane.
We adopted chemist notations with (i,j) and (a,b) indexing occupied and virtual states, respectively. Projecting now on
$ \phi_a^*(x_1) \phi_i(x_{1'}) \; $ one obtains by orthonormalization :
\begin{align*}
\int dx_1 dx_{1'} \; \phi_a^*(x_1) \phi_i(x_{1'}) \int d \omega & \; G(x_1,x_3; \omega + {\Omega_s \over 2} ) G(x_4,x_{1'}; \omega - {\Omega_s \over 2} ) e^{ i \omega \tau_{34} } = \\
2 & {i} \pi \theta( \tau_{34} ) \frac{ \phi_a^*(x_3) \phi_i(x_4) } { \varepsilon_i + \Omega_s - \varepsilon_a + i \eta } e^{i (\varepsilon_i + {\Omega_s \over 2} ) \tau_{34} } \\
-2 & {i} \pi \theta( - \tau_{34} ) \frac{ \phi_a^*(x_3) \phi_i(x_4) } { \varepsilon_a - \Omega_s - \varepsilon_i - i \eta } e^{i (\varepsilon_a - {\Omega_s \over 2} ) \tau_{34} }
\end{align*}
After multiplication by $ ( \varepsilon_a - \varepsilon_i - \Omega_s -i \eta )$ (and setting $\eta \rightarrow 0$), the Bethe-Salpeter equation becomes: \\
\begin{tcolorbox}[ width=\textwidth, colback={white}]
\begin{align*}
( \varepsilon_a - \varepsilon_i & - \Omega_s ) \int dx_1 \phi_a^*(x_1) \phi_i(x_{1'}) \langle N | T {\hat \psi}(x_1) {\hat \psi}^{\dagger}(x_{1'}) | N,s \rangle ( \delta^{-1} )
= \nonumber \\
& - \int d34 \; \phi_a^*(x_3) \phi_i(x_4)
\left[ \theta( \tau_{34} ) e^{i (\varepsilon_i + {\Omega_s \over 2} ) \tau_{34} } + \theta( - \tau_{34} ) e^{i (\varepsilon_a - {\Omega_s \over 2} ) \tau_{34} } \right] \\
& \times
e^{ i \Omega_s^{+} t^{34} } \int d56 \; \Xi[3,6;4,5] \langle N | T {\hat \psi}(5) {\hat \psi}^{\dagger}(6) | N,s \rangle \times \theta( t^{56}_m - t_2 )
\end{align*}
\end{tcolorbox}
\vskip .3cm \noindent This is equation (11.5) by Strinati with the same
$$
\theta( \tau ) e^{i (\varepsilon_i + {\Omega_s \over 2} ) \tau } + \theta( - \tau ) e^{ i (\varepsilon_a - {\Omega_s \over 2} ) \tau } =
e^{i {\Omega_s \over 2} \tau } \Big( \theta( \tau ) e^{i \varepsilon_i \tau } + \theta( - \tau ) e^{i (\varepsilon_a - \Omega_s )\tau } \Big)
$$
time-structure where $\tau = \tau_{34} = t_3 - t_4 $. \\
\noindent{\textbf{The $GW$ approximation. } Adopting the $GW$ approximation, and neglecting the $(\partial W / \partial G)$ of higher order in $W$, one obtains:
\begin{equation}
\Xi[3,6;4,5] = v(3,5) \delta(3,4) \delta(5,6) - W(3^+,4) \delta(3,5) \delta(4,6)
\end{equation}
so that the Bethe-Salpeter equation becomes:
\begin{align*}
( \varepsilon_a - \varepsilon_i & - \Omega_s ) \int dx_1 \phi_a^*(x_1) \phi_i(x_{1'}) \langle N | T {\hat \psi}(x_1) {\hat \psi}^{\dagger}(x_{1'}) | N,s \rangle ( \delta^{-1} )
= \nonumber \\
& - \int d35 \; \phi_a^*(x_3) \phi_i(x_3) e^{ i \Omega_s^{+} t_3 } v(3,5) \langle N | T [{\hat \psi}(5) {\hat \psi}^{\dagger}(5) ] | N,s \rangle \times \theta( t_5 - t_2 ) \\
& + \int d34 \; \phi_a^*(x_3) \phi_i(x_4)
\left[ \theta( \tau_{34} ) e^{i (\varepsilon_i + {\Omega_s \over 2} ) \tau_{34} } + \theta( - \tau_{34} ) e^{i (\varepsilon_a - {\Omega_s \over 2} ) \tau_{34} } \right] \times \\
& \hskip 1cm \times
e^{ i \Omega_s^{+} t^{34} } W(3^+,4) \langle N | T [{\hat \psi}(3) {\hat \psi}^{\dagger}(4) ] | N,s \rangle \times \theta( t^{34}_m - t_2 )
\end{align*}
\\
\noindent\textbf{Spectral representation of $\langle N | T {\hat \psi}(3) {\hat \psi}^{\dagger}(4) | N,s \rangle$ } \\
\noindent Starting with:
\begin{align*}
\langle N | T {\hat \psi}(3) {\hat \psi}^{\dagger}(4) | N,s \rangle &= \theta( \tau_{34} ) \langle N | {\hat \psi}(3) {\hat \psi}^{\dagger}(4) | N,s \rangle \\
&- \theta( -\tau_{34} ) \langle N | {\hat \psi}^{\dagger}(4) {\hat \psi}(3) | N,s \rangle
\end{align*}
and decomposing the static (Schr\"{o}dinger) field operators over the Hartree-Fock molecular orbitals creation/destruction operators:
$$
{\hat \psi}^{\dagger}(x_4) = \sum_n \phi_n^*(x_4) {\hat a}_n^{\dagger} \;\;\; \text{ and} \;\;\; {\hat \psi}(x_3) = \sum_m \phi_m(x_3) {\hat a}_m $$
one obtains
\begin{align*}
\langle N | T {\hat \psi}(3) {\hat \psi}^{\dagger}(4) | N,s \rangle &= \sum_{mn} \phi_m(x_3) \phi_n^*(x_4) \times \\
\times & \Big( \theta( \tau_{34} ) \langle N | {\hat a}_m e^{- i H \tau_{34} } {\hat a}^{\dagger}_n | N,s \rangle e^{-i E_s^N t_4} e^{+ i E_0^N t_3} \\
&- \theta( -\tau_{34} ) \langle N | {\hat a_n}^{\dagger} e^{ i H \tau_{34} } {\hat a}_m | N,s \rangle e^{ - i E_s^N t_3} e^{ + i E_0^N t_4} \Big)
\end{align*}
Taking the conjugate of the $ {\hat a}_m e^{- i H \tau_{34} } $ and
$ {\hat a_n}^{\dagger} e^{ i H \tau_{34} } $ operators to act on the ground-state $| N \rangle$ state, one obtains:
$$
e^{ i H \tau } {\hat a}_m^{\dagger} | N \rangle = e^{ i (E_0^N + \varepsilon_m ) \tau } {\hat a} _m^{\dagger} | N \rangle
\;\;\;\;\; \text{and} \;\;\;\;
e^{ - i H \tau } {\hat a}_n | N \rangle = e^{ - i ( E_0^N - \varepsilon_n ) \tau } {\hat a}_n | N \rangle
$$
This properties hold in the case of Hartree-Fock formalism (Koopman's theorem) but holds more generally if $\lbrace \varepsilon_m , \varepsilon_n \rbrace$ are quasiparticle energies,
namely true electronic removal or addition energies such as in the $GW$ formalism.
Forming the corresponding bra, one obtains :
\begin{align*}
\langle N | T {\hat \psi}(3) & {\hat \psi}^{\dagger}(4) | N,s \rangle = - \sum_{mn} \phi_m(x_3) \phi_n^*(x_4) \langle N | {\hat a}_n^{\dagger} {\hat a}_m | N,s \rangle \times \\
\times & \Big( \theta( \tau_{34} ) e^{- i \varepsilon_m \tau_{34} } e^{ -i \Oms t_4 }
+ \theta( -\tau_{34} ) e^{ - i \varepsilon_n \tau_{34} } e^{ - i \Oms t_3 } \Big)
\end{align*}
With $t_3 = \tau_{34}/2 + t^{34}$ and $t_4 = -\tau_{34}/2 + t^{34}$, one finally obtains :
\begin{align*}
\langle N | T {\hat \psi}(3) {\hat \psi}^{\dagger}(4) & | N,s \rangle = - \Big( e^{ -i \Omega_s t^{34} } \Big) \sum_{mn} \phi_m(x_3) \phi_n^*(x_4) \langle N | {\hat a}_n^{\dagger} {\hat a}_m | N,s \rangle \times \nonumber \\
\times & \Big( \theta( \tau_{34} ) e^{- i ( \varepsilon_m - \hOms ) \tau_{34} }
+ \theta( -\tau_{34} ) e^{ - i ( \varepsilon_n + \hOms) \tau_{34} } \Big)
\end{align*}
We must now discuss the (m,n) indexes. The most natural channel for ${\hat a}_m^{\dagger} {\hat a}_n | N \rangle$ to project on an excited state occurs
for (n,m) indexing (occupied,virtual) orbitals, respectively, in order to promote an electron from the occupied to the empty orbitals.
We will label $A_{jb}^{s} = \langle N | {\hat a}_j^{\dagger} {\hat a}_b | N,s \rangle$ these large amplitude projections. However, in the case of a generic correlated system,
the ground-state does not project exclusively on the mono-determinant built from the occupied 1-body orbitals $\lbrace \phi_i \rbrace$, but contains also contributions from
singly, doubly, etc. excited determinants in the full $\lbrace \phi_n \rbrace$ space. As such, the $\langle N | {\hat a}_b^{\dagger} {\hat a}_j | N,s \rangle$ weights are small in general but not strictly zero.
We will label $B_{jb}^{s}$ these small amplitudes. Neglecting these $B_{jb}^{s}$ coefficients leads to the Tamm-Dancoff approximation (TDA). We start with the TDA in the following for simplicity, adding the "small B" contribution in a second step. \\
\noindent In the TDA, namely with $B_{jb}^{s} = 0$, one obtains :
\begin{align*}
\langle N | T {\hat \psi}(3) & {\hat \psi}^{\dagger}(4) | N,s \rangle \cong - \Big( e^{ -i \Omega_s t^{34} } \Big) \sum_{jb} \phi_b(x_3) \phi_j^*(x_4) A_{jb}^{s} \times \nonumber \\
\times & \Big( \theta( \tau_{34} ) e^{- i ( \varepsilon_b - \hOms ) \tau_{34} }
+ \theta( -\tau_{34} ) e^{ - i ( \varepsilon_j + \hOms) \tau_{34} } \Big)
\end{align*}
As a result, the integral associated with the screened Coulomb potential reads:
\begin{align*}
\int d34 \; \phi_a^*(x_3) & \phi_i(x_4)
\left[ \theta( \tau_{34} ) e^{i (\varepsilon_i + {\Omega_s \over 2} ) \tau_{34} } + \theta( - \tau_{34} ) e^{i (\varepsilon_a - {\Omega_s \over 2} ) \tau_{34} } \right] \times \\
& \times
e^{ i \Omega_s^{+} t^{34} } W(3^+,4) \langle N | T [{\hat \psi}(3) {\hat \psi}^{\dagger}(4) ] | N,s \rangle \times \theta( t^{34}_m - t_2 ) \\
= - \int d34 \; \phi_a^*(x_3) & \phi_i(x_4) \theta( t^{34}_m - t_2 ) W(3^+,4) \sum_{jb} \phi_b(x_3) \phi_j^*(x_4) A_{jb}^{s} e^{ - \delta t^{34} } \times \\
& \times \left[ \theta( \tau_{34} ) e^{i ( \varepsilon_i - \varepsilon_b + \Oms ) \tau_{34} } + \theta( - \tau_{34} ) e^{i (\varepsilon_a - \varepsilon_j - \Oms ) \tau_{34} } \right] \\
= - \sum_{jb} A_{jb}^{s} & \int d\tau_{34} dt^{34} e^{ - \delta t^{34} } \theta( t^{34} - | \tau_{34} |/2 - t_2 ) W_{ij,ab}( \tau_{34}^{+} ) \times \\
& \times \left[ \theta( \tau_{34} ) e^{i ( \varepsilon_i - \varepsilon_b + \Oms ) \tau_{34} } + \theta( - \tau_{34} ) e^{i (\varepsilon_a - \varepsilon_j - \Oms ) \tau_{34} } \right]
\end{align*}
where we have explicited $t^{34}_m$ and made a change of variable $(t_3,t_4) \rightarrow (\tau_{34}, t^{34})$ with unit wronskian, defining the screened Coulomb matrix elements
\begin{align*}
W_{ij,ab}( \tau_{34}^{+} )&= \int d{\bf r}_3 d{\bf r}_4 \; \phi_i(x_4) \phi_j^*(x_4) W(3^+,4) \phi_a^*(x_3) \phi_b(x_3) \;\;\;\;
\end{align*}
with $(\tau_{34}^{+} = t_3^{+} - t_4)$ and adopting Mulliken notations where we group together the index associated with orbitals taken at the same space position (and putting complexe conjugate orbitals as inner indexes).
The $t^{34}$ time integral yields again $( \delta^{-1} )$ as done previously for the $t_1$-time integral, leaving only the dependence on the time difference $\tau_{34} = t_3 - t_4$.
We can understand the origin of this problematic $( \delta^{-1} )$ factor : the screened- Coulomb potential contribution only depends on the time difference $\tau_{34}$ so that the time variable $t^{34}$ becomes problematic.
To deal now with the bare Coulomb contribution, we need
$\langle N | T {\hat \psi}(5) {\hat \psi}^{\dagger}(5) | N,s \rangle$ that in the TDA reads :
$$
\langle N | T {\hat \psi}(5) {\hat \psi}^{\dagger}(5) | N,s \rangle \cong - \Big( e^{ -i \Omega_s t_5 } \Big) \sum_{jb} \phi_b(x_5) \phi_j^*(x_5) A_{jb}^{s}
$$
leading to the following contribution :
\begin{align*}
- \int d35 \; \phi_a^*(x_3) & \phi_i(x_3) e^{ i \Omega_s^{+} t_3 } v(3,5) \langle N | T [{\hat \psi}(5) {\hat \psi}^{\dagger}(5) ] | N,s \rangle \times \theta( t_5 - t_2 ) \\
& = \sum_{jb} A_{jb}^{s} v_{ia,jb} \int dt_3 dt_5 \delta( t_3 - t_5) e^{ i \Omega_s^{+} t_3 } e^{ -i \Omega_s t_5 }\theta( t_5 - t_2 )
\end{align*}
with $v(3,5) = v( {\bf r}_3 - {\bf r}_5 ) \delta( t_3 - t_5)$ a static interaction and :
$$
v_{ia,jb} = \int d{\bf r}_3 d{\bf r}_5 \; \phi_i(x_3) \phi_a^*(x_3) v( {\bf r}_3 - {\bf r}_5 ) \phi_j^*(x_5) \phi_b(x_5)
$$
The time integral reads now $\int dt_3 e^{- \delta t_3} \theta(t_3 - t_2 )$ yielding $( \delta^{-1} )$ again.
This factor in front of all terms can now be factorized out. \\
\noindent Finally, for the right-hand side of the Bethe-Salpeter equation, we develop:
\begin{equation*}
\langle N | {\hat \psi}^{\dagger}(x_{1'}) {\hat \psi}(x_1) | N,s \rangle = \sum_{jb} \phi_b(x_1) \phi_j^*(x_1) A_{jb}^{s}
\end{equation*}
so that by orthonormalization:
\begin{equation*}
\int dx_1 dx_{1'} \; \phi_a^*(x_1) \phi_i(x_{1'}) \langle N | {\hat \psi}^{\dagger}(x_{1'}) {\hat \psi}(x_1) | N,s \rangle =
\sum_{jb} A_{jb}^{s} \delta_{ij} \delta_{ab} = A_{ia}^{s}
\end{equation*}
Note that the projection on $\phi_a^*(x_1) \phi_i(x_{1'})$ will always cancel the small $B_{jb}^{s}$ contribution so that this last expression is always valid.
We obtain thus a much simplified reformulation of the dynamical TDA Bethe-Salpeter equation: \\
\begin{tcolorbox}[ width=\textwidth, colback={white}]
\begin{align*}
( \varepsilon_a - \varepsilon_i - \Omega_s ) & A_{ia}^{s}
+ \sum_{jb} A_{jb}^{s} \times v_{ai,bj} \stackrel{\text{TDA}}{=} \sum_{jb} A_{jb}^{s} \times \int d\tau \; W_{ij,ab}( {\tau}^{+} ) \times \\
& \times \left[ \theta( \tau ) e^{i (\varepsilon_i - \varepsilon_b+ \Oms ) \tau } + \theta( - \tau ) e^{i (\varepsilon_a - \varepsilon_j - \Oms ) \tau } \right]
\end{align*}
\end{tcolorbox}
\noindent where we dropped the index of ${\tau}_{34}$. Taking finally the Fourier representation of $W$, the time integration reads:
\begin{align*}
\widetilde{W}_{ij,ab}(\Oms) =& \int d\tau \; W_{ij,ab}( \tau^{+} )
\times \; \left[ \theta( \tau ) e^{i (\varepsilon_i - \varepsilon_b+ \Oms ) \tau } + \theta( - \tau ) e^{i (\varepsilon_a - \varepsilon_j - \Oms ) \tau } \right] \\
=& \int { d\omega \over 2\pi } W_{ij,ab}( \omega ) \int d\tau \; e^{-i \omega ( \tau + 0^+) }
\left[ \theta( \tau ) e^{i (\varepsilon_i - \varepsilon_b+ \Oms ) \tau } + \theta( - \tau ) e^{i (\varepsilon_a - \varepsilon_j - \Oms ) \tau } \right] \\
=& { i \over 2\pi} \int d\omega e^{-i \omega 0^{+}} \left[ \frac{ W_{ij,ab}( \omega ) }{ ( \Oms - \omega ) - ( \varepsilon_b - \varepsilon_i) + i\eta }
+ \frac{ W_{ij,ab}( \omega ) }{ ( \Oms + \omega) - ( \varepsilon_a - \varepsilon_j ) + i\eta } \right]
\end{align*}
where we have introduced again a small damping $( \Oms \rightarrow \Oms + i\eta)$ of the excitation for convergency. This leads to
equation (11.8) by Strinati: \\
\begin{tcolorbox}[ width=\textwidth, colback={white}]
\begin{align*}
( \varepsilon_a - \varepsilon_i - \Omega_s ) & A_{ia}^{s}
+ \sum_{jb} A_{jb}^{s} \times v_{ai,bj} \stackrel{\text{TDA}}{=} { i \over 2 \pi} \sum_{jb} A_{jb}^{s} \int d\omega e^{-i \omega 0^+ } W_{ij,ab} \times \\
\hskip 1cm &\times \left[ \frac{1}{( \Oms - \omega) - (\varepsilon_b - \varepsilon_i) +i \eta } + \frac{1}{ (\Oms + \omega) - (\varepsilon_a - \varepsilon_j) + i\eta } \right]
\end{align*}
\end{tcolorbox}
\vskip .3cm \noindent or to a form similar to the static limit : \\
\begin{tcolorbox}[ width=\textwidth, colback={white}]
\begin{align}
( \varepsilon_a - \varepsilon_i - \Omega_s ) A_{ia}^{s}
+ \sum_{jb} \Big( v_{ai,bj} - \widetilde{W}_{ij,ab}(\Oms) \Big) A_{jb}^{s} \stackrel{\text{TDA}}{=} 0
\end{align}
\end{tcolorbox}
\vskip .3cm \noindent but with an effective dynamical screened Coulomb potential (see Pina eq. 24): \\
\begin{tcolorbox}[ width=\textwidth, colback={white}]
\begin{align}
\widetilde{W}_{ij,ab}(\Oms) &= { i \over 2 \pi} \int d\omega \; e^{-i \omega 0^+ } W_{ij,ab} \times \\
\hskip 1cm &\times \left[ \frac{1}{ (\Oms - \omega) - (\varepsilon_b - \varepsilon_i) +i \eta } + \frac{1}{ (\Oms + \omega) - (\varepsilon_a - \varepsilon_j) + i\eta } \right] \nonumber
\end{align}
\end{tcolorbox}
\vskip .3cm \noindent Taking now the spectral representation for the (time-ordered) $W_{ij,ab}( \omega )$, namely:
\begin{align*}
W_{ij,ab}(\omega) &= (ij|ab) + 2 \sum_m^{OV} [ij|m] [ab|m] \times \\
& \times \Big( \frac{1}{ \omega-\Omega_m^{RPA} + i\eta } - \frac{1}{ \omega + \Omega_m^{RPA} - i\eta } \Big)
\end{align*}
the frequency integral becomes, closing in the lower half plane thanks to the $e^{-i\omega 0^+}$ factor.
\textcolor{red}{(This is the only place where the (+) in $3^+$ makes a difference as for the $GW$ operator ... check carefully origin ...)}:
\begin{align*}
\int d\omega e^{-i \omega 0^{+}} & \Big( \frac{ W_{ij,ab}( \omega ) }{ ( \Oms - \omega ) - ( \varepsilon_b - \varepsilon_i) + i\eta }
+ \frac{ W_{ij,ab}( \omega ) }{ ( \Oms + \omega) - ( \varepsilon_a - \varepsilon_j ) + i\eta } \Big)= \\
= & (-2i \pi) W_{ij,ab}( - \Oms + (\varepsilon_a - \varepsilon_j) ) + (-2i \pi) \times 2 \sum_m^{OV} [ij|m] [ab|m] \times \\
& \times \Big( \frac{ 1 }{ ( \Oms - \Omega_m^{RPA} ) - ( \varepsilon_b - \varepsilon_i) + i\eta }
+ \frac{ 1 }{ ( \Oms + \Omega_m^{RPA} ) - ( \varepsilon_a - \varepsilon_j ) + i\eta } \Big)
\end{align*}
that is :
\begin{align*}
\widetilde{W}_{ij,ab}( \Oms ) = & (ij|ab) + 2 \sum_m^{OV} [ij|m] [ab|m] \times \\
\times & \Big( \frac{ 1}{ -\Oms + (\varepsilon_a - \varepsilon_j) -\Omega_m^{RPA} + i\eta } - \frac{ 1}{ -\Oms + (\varepsilon_a - \varepsilon_j) + \Omega_m^{RPA} - i\eta } \\
+ & \frac{ 1 }{ ( \Oms - \Omega_m^{RPA} ) - ( \varepsilon_b - \varepsilon_i) + i\eta }
+ \frac{ 1 }{ ( \Oms + \Omega_m^{RPA} ) - ( \varepsilon_a - \varepsilon_j ) + i\eta } \Big)
\end{align*} \\
or
\begin{align*}
\widetilde{W}_{ij,ab}( \Oms ) = & (ij|ab) + 2 \sum_m^{OV} [ij|m] [ab|m] \times \\
\times & \Big( \frac{ 1 }{ ( \Oms - \Omega_m^{RPA} ) - ( \varepsilon_b - \varepsilon_i) + i\eta } + \frac{ 1}{ ( \Oms - \Omega_m^{RPA} ) - (\varepsilon_a - \varepsilon_j) + i\eta } \\
+ & \frac{ 1 }{ ( \Oms + \Omega_m^{RPA} ) - ( \varepsilon_a - \varepsilon_j ) + i\eta } - \frac{ 1}{ ( \Oms +\Omega_m^{RPA} ) - (\varepsilon_a - \varepsilon_j) - i\eta } \Big)
\end{align*} \\
Keeping only \textbf{ the two resonant contributions : } (ACTUALLY the other ones in the last line cancel when $\eta \rightarrow 0$ !! check ...) one obtains Pina's equation (27) \\
\begin{tcolorbox}[ width=\textwidth, colback={white}]
\begin{align}
\widetilde{W}_{ij,ab}&( \Oms ) = (ij|ab) + 2 \sum_m^{OV} [ij|m] [ab|m] \times \\
\times & \left[ \frac{ 1 }{ ( \Oms - \Omega_m^{RPA} ) - ( \varepsilon_b - \varepsilon_i) + i\eta } + \frac{ 1}{ ( \Oms - \Omega_m^{RPA} ) - (\varepsilon_a - \varepsilon_j) + i\eta }
\right] \nonumber
\end{align}
\end{tcolorbox}
\noindent This dynamical $\widetilde{W}( \Oms )$ kernel can be rewritten:
\begin{align}
\widetilde{W}_{ij,ab}( \Oms ) &= (ij|ab) + 2 \sum_m^{OV} [ij|m] [ab|m] \times \\
& \times \left[ \frac{ 1 }{ \Omega_{ib}^{s} - \Omega_m^{RPA} + i\eta } + \frac{ 1}{ \Omega_{ja}^{s} - \Omega_m^{RPA} + i\eta }
\right] \nonumber
\end{align}
with e.g. $ \Omega_{ib}^{s} = \Oms - ( \varepsilon_b - \varepsilon_i) $. This formulation is similar to the spectral representation of $W_{ij,ab}(\omega)$ but with
$( \omega \rightarrow \Omega_{ib}^{s} )$ and $( \Omega \rightarrow \Omega_{ja}^{s} )$.
\vskip 1cm
\noindent {\textbf {Non-resonant contributions.}} We now add the $B_{jb}^{s}$ small contributions to the $\langle N | T {\hat \psi}(3) {\hat \psi}^{\dagger}(4) | N,s \rangle $ weight:
\begin{align*}
\langle N | T {\hat \psi}(3) {\hat \psi}^{\dagger}(4) | N,s \rangle^{\text{corr}} &= - \Big( e^{ -i \Omega_s t^{34} } \Big) \sum_{bj} \phi_j(x_3) \phi_b^*(x_4) B_{jb}^{s} \times \nonumber \\
\times & \Big( \theta( \tau_{34} ) e^{- i ( \varepsilon_j - \hOms ) \tau_{34} }
+ \theta( -\tau_{34} ) e^{ - i ( \varepsilon_b + \hOms) \tau_{34} } \Big)
\end{align*}
where ``corr" stands for correction. As a result, the integral associated with the screened Coulomb potential must be corrected by:
\begin{align*}
\int d34 \; \phi_a^*(x_3) & \phi_i(x_4)
\left[ \theta( \tau_{34} ) e^{i (\varepsilon_i + {\Omega_s \over 2} ) \tau_{34} } + \theta( - \tau_{34} ) e^{i (\varepsilon_a - {\Omega_s \over 2} ) \tau_{34} } \right] \times \\
& \times
e^{ i \Omega_s^{+} t^{34} } W(3^+,4) \langle N | T [{\hat \psi}(3) {\hat \psi}^{\dagger}(4) ] | N,s \rangle \times \theta( t^{34}_m - t_2 ) \\
= - \int d34 \; \phi_a^*(x_3) & \phi_i(x_4) \theta( t^{34}_m - t_2 ) W(3^+,4) \sum_{bj} \phi_j(x_3) \phi_b^*(x_4) B_{jb}^{s} e^{ - \delta t^{34} } \times \\
& \times \left[ \theta( \tau_{34} ) e^{i ( \varepsilon_i - \varepsilon_j + \Oms ) \tau_{34} } + \theta( - \tau_{34} ) e^{i (\varepsilon_a - \varepsilon_b - \Oms ) \tau_{34} } \right] \\
= - ({ \delta}^{-1} ) \sum_{jb} B_{jb}^{s} & \int d\tau \; W_{ib,aj}( {\tau}^{+} ) \times \\
& \times \left[ \theta( \tau ) e^{i ( \varepsilon_i - \varepsilon_j + \Oms ) \tau } + \theta( - \tau ) e^{i (\varepsilon_a - \varepsilon_b - \Oms ) \tau } \right]
\end{align*}
The time integration yields now:
\begin{align*}
\widetilde{W}_{ib,aj}(\Oms) =& \int d\tau \; W_{ib,aj}( \tau^{+} )
\times \; \left[ \theta( \tau ) e^{i (\varepsilon_i - \varepsilon_j+ \Oms ) \tau } + \theta( - \tau ) e^{i (\varepsilon_a - \varepsilon_b - \Oms ) \tau } \right] \\
=& \int { d\omega \over 2\pi } W_{ib,aj}( \omega ) \int d\tau \; e^{-i \omega ( \tau + 0^+) }
\left[ \theta( \tau ) e^{i (\varepsilon_i - \varepsilon_j+ \Oms ) \tau } + \theta( - \tau ) e^{i (\varepsilon_a - \varepsilon_b - \Oms ) \tau } \right] \\
=& { i \over 2\pi} \int d\omega e^{-i \omega 0^{+}} \left[ \frac{ W_{ib,aj}( \omega ) }{ ( \Oms - \omega ) - ( \varepsilon_j - \varepsilon_i) + i\eta }
+ \frac{ W_{ib,aj}( \omega ) }{ ( \Oms + \omega) - ( \varepsilon_a - \varepsilon_b ) + i\eta } \right]
\end{align*} \\
Plugging now the spectral representation of $W_{ib,aj}( \omega )$:
\begin{align*}
\widetilde{W}_{ib,aj}( \Oms ) = & (ib|aj) + 2 \sum_m^{OV} [ib|m] [aj|m] \times \\
\times & \Big( \frac{ 1 }{ ( \Oms - \Omega_m^{RPA} ) - ( \varepsilon_j - \varepsilon_i) + i\eta } + \frac{ 1}{ ( \Oms - \Omega_m^{RPA} ) - (\varepsilon_a - \varepsilon_b) + i\eta } \\
+ & \frac{ 1 }{ ( \Oms + \Omega_m^{RPA} ) - ( \varepsilon_a - \varepsilon_b ) + i\eta } - \frac{ 1}{ ( \Oms +\Omega_m^{RPA} ) - (\varepsilon_a - \varepsilon_b) - i\eta } \Big)
\end{align*}
where again the last line contribution cancel. \\
\noindent The correction to the bare Coulomb contribution stems from the correction to
$\langle N | T {\hat \psi}(5) {\hat \psi}^{\dagger}(5) | N,s \rangle$, namely :
$$
\langle N | T {\hat \psi}(5) {\hat \psi}^{\dagger}(5) | N,s \rangle^{\text{corr}} = - \Big( e^{ -i \Omega_s t_5 } \Big) \sum_{bj} \phi_j(x_5) \phi_b^*(x_5) B_{jb}^{s}
$$
leading to the following correction :
\begin{align*}
- \int d35 \; \phi_a^*(x_3) & \phi_i(x_3) e^{ i \Omega_s^{+} t_3 } v(3,5) \langle N | T [{\hat \psi}(5) {\hat \psi}^{\dagger}(5) ] | N,s \rangle \times \theta( t_5 - t_2 ) \\
& = \sum_{bj} B_{bj}^{s} v_{ia,bj} ( \delta^{-1} )
\end{align*}
As discussed above, there is no correction to the diagonal part of the BSE Hamiltonian so that one obtains beyond TDA so that: \\
\begin{tcolorbox}[ width=\textwidth, colback={white}]
\begin{align}
( \varepsilon_a - \varepsilon_i - \Omega_s ) A_{ia}^{s}
+& \sum_{jb} \Big( v_{ai,bj} - \widetilde{W}_{ij,ab}(\Oms) \Big) A_{jb}^{s} \nonumber \\
+ & \sum_{bj}\Big( v_{ai,jb} - \widetilde{W}_{ib,aj}(\Oms) \Big) B_{jb}^{s} = 0
\end{align}
\end{tcolorbox}
\noindent with:
\begin{align*}
\widetilde{W}_{ib,aj}( \Oms ) = & (ib|aj) + 2 \sum_m^{OV} [ib|m] [aj|m] \times \\
\times & \left[ \frac{ 1 }{ ( \Oms - \Omega_m^{RPA} ) - ( \varepsilon_j - \varepsilon_i) + i\eta } + \frac{ 1}{ ( \Oms - \Omega_m^{RPA} ) - (\varepsilon_a - \varepsilon_b) + i\eta } \right]
\end{align*}
%%%%%%%%%%
\end{document}
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