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BSEdyn.tex
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BSEdyn.tex
@ -209,7 +209,34 @@ This is the abstract
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\subsection{Theory for physics}
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%=================================
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The Fourier components with respect to time $t_1$ of $iL_0(1, 4; 1', 3) = G(1, 3)G(4, 1')$ reads, dropping the (space/spin)-variables:
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The resolution of the dynamical Bethe-Salpeter equation (dBSE) [Strinati]
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\begin{align*}
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L(1,2; & 1',2') = L_0(1,2;1',2') + \\
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&+ \int d3456 \;
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L_0(1,4;1',3) \Xi(3,5;4,6) L(6,2;5,2')
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\end{align*}
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with:
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\begin{align*}
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iL(1,2; 1',2') &= -G_2(1,2;1',2') + G(1,1')G(2,2') \\
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i^2 G_2(1,2;1',2') &= \langle N | T {\hat \psi}(1) {\hat \psi}(2) {\hat \psi}^{\dagger}(2') {\hat \psi}^{\dagger}(1') | N \rangle
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\end{align*}
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where e.g. $1 = (x_1,t_1)$ a space-spin plus time variable, starts with the expansion of the 2-body Green's function $G_2$ and response function $L$ over the complete orthonormalized set $ |N,s \rangle $ of the N-electron excited state with $| N \rangle = | N,0 \rangle$ the ground-state. In the optical limit of instantaneous electron-hole creation and destruction, imposing
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$t_{2'} = t_2^+$ and $t_{1'} = t_1^+$, one obtains:
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\begin{align*}
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iL(1,2;1',2') &= \theta(\tau_{12}) \sum_{s > 0} \chi_s(x_1,x_{1'}) {\tilde \chi}_s(x_2,x_{2'})
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e^{ +i \Oms \tau_{12} } \\
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&- \theta(-\tau_{12}) \sum_{s > 0} \chi_s(x_2,x_{2'}) {\tilde \chi}_s(x_1,x_{1'})
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e^{ - i \Oms \tau_{12} }
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\end{align*}
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with $\tau_{12} = t_1 - t_2$ and
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\begin{align*}
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\chi_s(x_1,x_{1'}) = \langle N | T {\hat \psi}(x_1) {\hat \psi}^{\dagger}(x_{1'}) | N,s \rangle \\
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{\tilde \chi}_s(x_2,x_{2'}) = \langle N,s | T {\hat \psi}(x_2) {\hat \psi}^{\dagger}(x_{2'}) | N \rangle
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\end{align*}
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The $\Oms$ are the neutral excitation energies of interest. Picking up the $e^{+i \Oms t_2 }$ component and simplifying by ${\tilde \chi}_s(x_2,x_{2'})
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e^{ i \Oms t_{2} }$ on both side of the Bethe-Salpeter equation, we are left with the search of the $e^{-i \Oms t_1 }$ Fourier component associated with the right-hand side of the BSE. For the lowest $\Oms$ excitation energies falling in the quasiparticle gap of the system, the $L_0(1,2;1',2')$ term cannot contribute since its lowest excitation energy is precisely the quasiparticle gap, namely the difference between the electronic affinity and the ionisation potential.
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The Fourier components with respect to time $t_1$ of $iL_0(1, 4; 1', 3) = G(1, 3)G(4, 1')$ reads, dropping the (space/spin)-variables:
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\begin{align*}
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[iL_0]( \omega_1 ) = \frac{ 1 }{ 2\pi } \int d \omega \; G(\omega - \frac{\omega_1}{2} ) G( {\omega} + \frac{\omega_1}{2} ) e^{ i \omega \tau_{34} } e^{ i \omega_1 t^{34} }
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\end{align*}
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@ -225,23 +252,25 @@ and projecting on $\phi_a^*(x_1) \phi_i(x_{1'})$, one obtains the $\omega_1= \Om
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\Big( \theta( \tau ) e^{i ( \vari + \hOms) \tau }
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+ \theta( - \tau ) e^{i (\vara - \hOms \tau } \Big)
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\end{align*}
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with $\tau = \tau_{34}$.
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We further obtain the spectral representation of
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with $\tau = \tau_{34}$. Adopting now the $GW$ approximation for the exchange-correlation self-energy leads to a simplification of the BSE kernel:
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$$
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\Xi(3,5;4,6) = v(3,6) \delta(34) \delta(56) - W(3^+,4) \delta(36) \delta(45)
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$$
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We further obtain the needed spectral representation of
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$\langle N | T {\hat \psi}(3) {\hat \psi}^{\dagger}(4) | N,s \rangle$
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expanding the field operators over a complete orbital basis creation/destruction operators:
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\begin{align*}
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\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 \\
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\times & \Big( \theta( \tau ) e^{- i ( \varepsilon_m - \hOms ) \tau }
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+ \theta( -\tau ) e^{ - i ( \varepsilon_n + \hOms) \tau } \Big)
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\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) \times \nonumber \\
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\times & \langle N | {\hat a}_n^{\dagger} {\hat a}_m | N,s \rangle \;\Big[ \theta( \tau ) e^{- i ( \varepsilon_m - \hOms ) \tau } + \theta( -\tau ) e^{ - i ( \varepsilon_n + \hOms) \tau } \Big]
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\end{align*}
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with $\tau = \tau_{34}$ and where the $ \lbrace \varepsilon_{n/m} \rbrace$ are proper addition/removal energies such that e.g.
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$$
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e^{ i H \tau } {\hat a}_m^{\dagger} | N \rangle = e^{ i (E_0^N + \varepsilon_m ) \tau } {\hat a} _m^{\dagger} | N \rangle
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$$
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Selecting (n,m)=(j,b) yields the largest components
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The $GW$ quasiparticle energies $ \varepsilon_{n/m}^{GW}$ are good approximations to such removal/addition energies. Selecting (n,m)=(j,b) yields the largest components
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$A_{jb}^{s} = \langle N | {\hat a}_j^{\dagger} {\hat a}_b | N,s \rangle $, while (n,m)=(b,j) yields much weaker
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$B_{jb}^{s} = \langle N | {\hat a}_b^{\dagger} {\hat a}_j | N,s \rangle $ contributions. We used chemist notations with (i,j) indexing occupied orbitals and (a,b) virtual ones. Neglecting the $B_{jb}^{s}$ leads to the Tamm Dancoff approximation (TDA). Obtaining similarly the spectral representation of $ \langle N | T {\hat \psi}(1) {\hat \psi}^{\dagger}(1') | N,s \rangle$ ($t_{1'} = t_1^{+}$) projected onto $\phi_a^*(x_1) \phi_i(x_{1'})$,
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one obtains after a few tedious manipulations (see Supplemental Information) the dynamical Bethe-Salpeter equation (DBSE) :
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$B_{jb}^{s} = \langle N | {\hat a}_b^{\dagger} {\hat a}_j | N,s \rangle $ contributions. We used chemist notations with (i,j) indexing occupied orbitals and (a,b) virtual ones. Neglecting the $B_{jb}^{s}$ weights leads to the Tamm Dancoff approximation (TDA). Working out the same expansion for $ \langle N | T {\hat \psi}(5) {\hat \psi}^{\dagger}(5) | N,s \rangle$ and $ \langle N | T {\hat \psi}(x_1) {\hat \psi}^{\dagger}(x_{1'}) | N,s \rangle$, and projecting onto $\phi_a^*(x_1) \phi_i(x_{1'})$,
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one obtains after a few tedious manipulations (see Supplemental Information) the dynamical Bethe-Salpeter equation (dBSE) :
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\begin{align}
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( \varepsilon_a - \varepsilon_i - \Omega_s ) A_{ia}^{s}
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&+ \sum_{jb} \Big( v_{ai,bj} - \widetilde{W}_{ij,ab}(\Oms) \Big) A_{jb}^{s} \\
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@ -250,30 +279,29 @@ one obtains after a few tedious manipulations (see Supplemental Information) the
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\end{align}
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with an effective dynamically screened Coulomb potential (see Pina eq. 24):
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\begin{align}
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\widetilde{W}_{ij,ab}(\Oms) &= { i \over 2 \pi} \int d\omega \; e^{-i \omega 0^+ } W_{ij,ab}(\omega) \times \\
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\hskip 1cm &\times \left[ \frac{1}{ (\Oms - \omega) - ( \varb - \vari ) +i \eta } + \frac{1}{ (\Oms + \omega) - ( \vara - \varj ) + i\eta } \right] \nonumber
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\widetilde{W}_{ij,ab}(\Oms) &= \frac{ i }{ 2 \pi} \int d\omega \; e^{-i \omega 0^+ } W_{ij,ab}(\omega) \times \\
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\hskip 1cm &\times \left[ \frac{1}{ \Omega_{ib}^s - \omega +i \eta } + \frac{1}{ \Omega_{ja}^{s} + \omega + i\eta } \right] \nonumber
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\end{align}
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with $\; \Omega_{ib}^s = \Oms - ( \varb - \vari )$ and $\; \Omega_{ja}^s = \Oms - ( \vara - \varj ).$
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In the present study, we use the exact spectral representation of $W(\omega)$ at the RPA level:
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\begin{align*}
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W_{ij,ab}(\omega) &= (ij|ab) + 2 \sum_m^{OV} [ij|m] [ab|m] \times \\
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& \times \Big( \frac{1}{ \omega-\Omega_m^{RPA} + i\eta } - \frac{1}{ \omega + \Omega_m^{RPA} - i\eta } \Big)
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\end{align*}
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so that
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($\Omega_m^{RPA} > 0 $) so that
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\begin{align}
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\widetilde{W}_{ij,ab}( \Oms ) &= (ij|ab) + 2 \sum_m^{OV} [ij|m] [ab|m] \times \\
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& \times \left[ \frac{ 1 }{ \Omega_{ib}^{s} - \Omega_m^{RPA} + i\eta } + \frac{ 1}{ \Omega_{ja}^{s} - \Omega_m^{RPA} + i\eta }
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\right] \nonumber
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\end{align}
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with e.g. $ \Omega_{ib}^{s} = \Oms - ( \varepsilon_b - \varepsilon_i) $. \textcolor{red}{Due to excitonic effects, the lowest BSE ${\Omega}_1$ excitation energy stands lower than the lowest $\Omega_m^{RPA}$ excitation energy, so that
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e.g. $( \Omega_{ib}^{s} - \Omega_m^{RPA} )$ is strictly negative and cannot diverge. Further, $\Omega_{ib}^{s}$ and $\Omega_{ja}^{s}$ are necessarily negative for in-gap low lying BSE excitations, such that
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\textcolor{red}{Due to excitonic effects, the lowest BSE ${\Omega}_1$ excitation energy stands lower than the lowest $\Omega_m^{RPA}$ excitation energy, so that
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e.g. $( \Omega_{ib}^{s} - \Omega_m^{RPA} )$ is strictly negative and cannot diverge. Further, $\Omega_{ib}^{s}$ and $\Omega_{ja}^{s}$ are necessarily negative for in-gap low lying BSE excitations, such that e.g.
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$$
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\left[ \frac{ 1 }{ \Omega_{ib}^{s} - \Omega_m^{RPA} + i\eta } + \frac{ 1}{ \Omega_{ja}^{s} - \Omega_m^{RPA} + i\eta }
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\right]
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<
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\Big( \frac{1}{ \omega-\Omega_m^{RPA} + i\eta } - \frac{1}{ \omega + \Omega_m^{RPA} - i\eta } \Big) < 0
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\left| \frac{ 1 }{ \Omega_{ib}^{s} - \Omega_m^{RPA} } \right|
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< \frac{1}{ \Omega_m^{RPA} }
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$$
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in the limit $(\omega \rightarrow 0)$ of the standard adiabatic BSE . WELL, do we know the sign of
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$[ij|m] [ab|m]$ ?? }
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This leads to reduced electron-hole screening, namely larger electron-hole stabilising binding energy, as compared to the standard adiabatic BSE, leading to smaller (blue-shifted) excitation energies. }
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%In order to compute the neutral (optical) excitations of the system and their associated oscillator strengths, the BSE expresses the two-body propagator \cite{Strinati_1988, Martin_2016}
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%\begin{multline}
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