modifications in Xav part
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BSEdyn.tex
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BSEdyn.tex
@ -180,12 +180,6 @@
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\newcommand{\pis}{\pi^*}
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\newcommand{\ra}{\rightarrow}
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\newcommand\vari{{\eps}_i}
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\newcommand\vara{{\eps}_a}
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\newcommand\varj{{\eps}_j}
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\newcommand\varb{{\eps}_b}
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\newcommand\varn{{\eps}_n}
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\newcommand\varm{{\eps}_m}
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\newcommand\Oms{{\Omega}_s}
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\newcommand\hOms{\frac{{\Omega}_s}{2}}
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\newcommand{\hpsi}{\Hat{\psi}}
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@ -285,109 +279,132 @@ More details of the derivation are provided as {\SI}.
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%=================================
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The resolution \cite{Strinati_1988} of the Bethe-Salpeter equation
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\begin{multline}
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\begin{multline} \label{eq:BSE}
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L(1,2; 1',2')
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= L_0(1,2;1',2')
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\\
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+ \int d3456 \; L_0(1,4;1',3) \Xi(3,5;4,6) L(6,2;5,2'),
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\end{multline}
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with
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\begin{align}
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iL(1,2; 1',2')
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& = - G_2(1,2;1',2') + G(1,1') G(2,2'),
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with
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\begin{gather}
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iL_0(1, 4; 1', 3) = G(1, 3)G(4, 1')
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\\
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i^2 G_2(1,2;1',2')
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& = \mel{N}{T \hpsi(1) \hpsi(2) \hpsi^{\dagger}(2') \hpsi^{\dagger}(1')}{N},
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\end{align}
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where, \eg, $1 = (\bx_1,t_1)$ 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 $\ket{N,s}$ of the $N$-electron system excited states, with $\ket{N} = \ket{N,0}$ the ground-state.
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In the optical limit of instantaneous electron-hole creation and destruction, imposing $t_{2'} = t_2^+$ and $t_{1'} = t_1^+$, with, \eg, $t_2^+ = t_2 + 0^+$ where $0^+$ is a small positive infinitesimal, one gets
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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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\\
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i^2 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{gather}
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where, \eg, $1 = (\bx_1,t_1)$ a space-spin plus time variable, starts with the expansion of the two-body Green's function $G_2$ and the response function $L$ over the complete orthonormalized set of $N$-electron excited states $\ket{N,s}$ (where $\ket{N} \equiv \ket{N,0}$ corresponds to the ground state).
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In Eq.~\eqref{eq:BSE}, the BSE kernel
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\begin{equation}
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\Xi(3,5;4,6) = i \fdv{[v_\text{H}(3) \delta(3,4) + \Sigma_\text{xc}(3,4)]}{G(6,5)},
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\end{equation}
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takes into account the self-consistent variation of the Hartree potential
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\begin{equation}
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v_\text{H}(1) = - i \int d2 v(1,2) G(2,2^+),
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\end{equation}
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[where $\delta$ is Dirac's delta function and $v$ is the bare Coulomb operator] and the exchange-correlation self-energy $ \Sigma_\text{xc}$ with respect to the variation of the one-body Green's function $G$.
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In the optical limit of instantaneous electron-hole creation and destruction, imposing $t_{2'} = t_2^+$ and $t_{1'} = t_1^+$, with, \eg, $t_2^+ = t_2 + 0^+$ where $0^+$ is a positive infinitesimal, one gets
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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(+\tau_{12}) \sum_{s > 0} \chi_s(x_1,x_{1'}) \tchi_s(x_2,x_{2'}) e^{ - i \Oms \tau_{12} }
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iL(1,2; 1',2')
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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(-\tau_{12}) \sum_{s > 0} \chi_s(x_2,x_{2'}) \tchi_s(x_1,x_{1'}) e^{ + i \Oms \tau_{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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with $\tau_{12} = t_1 - t_2$ and
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with $\tau_{12} = t_1 - t_2$, $\theta$ is the Heaviside step function, and
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\begin{subequations}
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\begin{align}
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\chi_s(x_1,x_{1'}) & = \mel{N}{T \hpsi(x_1) \hpsi^{\dagger}(x_{1'})}{N,s}
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\chi_s(\bx_1,\bx_{1'}) & = \mel{N}{T \hpsi(\bx_1) \hpsi^{\dagger}(\bx_{1'})}{N,s},
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\\
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\tchi_s(x_2,x_{2'}) & = \mel{N,s}{T \hpsi(x_2) \hpsi^{\dagger}(x_{2'})}{N}
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\tchi_s(\bx_2,\bx_{2'}) & = \mel{N,s}{T \hpsi(\bx_2) \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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Picking up the $e^{+i \Oms t_2 }$ component in $L(1,2; 1',2')$ and $L(6,2;5,2')$, simplifying further by $\tchi_s(x_2,x_{2'})$ on both side of the BSE, we are left with the search of the $e^{-i \Oms t_1 }$ Fourier component associated with the right-hand side of the BSE.
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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 ionization potential.
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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, we are left with the search of the $e^{-i \Oms t_1 }$ Fourier component associated with the right-hand side of the BSE.
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For the lowest $\Oms$ excitation energies falling in the quasiparticle gap of the system, $L_0(1,2;1',2')$ cannot contribute since its lowest excitation energy is precisely the quasiparticle gap, namely the difference between the electronic affinity and the ionization 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 ) = \int \frac{d\omega}{2\pi} \; G\qty(\omega - \frac{\omega_1}{2} ) G\qty( {\omega} + \frac{\omega_1}{2} ) e^{ i \omega \tau_{34} } e^{ i \omega_1 t^{34} }
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The Fourier components with respect to $t_1$ of $L_0$ reads, dropping the (space/spin) variables:
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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 \tau_{34} } e^{ i \omega_1 t^{34} },
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\end{align}
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with $\tau_{34} = t_3 - t_4$ and $t^{34} = (t_3 + t_4)/2$.
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Plugging now the 1-body Green's function Lehman representation, \eg,
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Plugging now the Lehman representation of the one-body Green's function
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\begin{equation}
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G(x_1,x_2 ; \omega) = \sum_n \frac{ \phi_n(x_1) \phi_n^*(x_2) } { \omega - \eps_n + i \eta \text{sgn} \times (\eps_n - \mu) }
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\end {equation}
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and projecting on $\phi_a^*(x_1) \phi_i(x_{1'})$, one obtains the $\omega_1= \Oms$ component
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G(x_1,x_2 ; \omega) = \sum_p \frac{ \phi_p(\bx_1) \phi_p^*(\bx_2) } { \omega - \e{p} + i \eta \text{sgn} \times (\e{p} - \mu) }
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\end{equation}
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\titou{T2: did you introduce a new basis here?}
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(where $\eta$ is a positive infinitesimal) into Eq.~\eqref{eq:iL0} and projecting onto \titou{$\phi_a^*(\bx_1) \phi_i(\bx_{1'})$}, one obtains the $\omega_1= \Oms$ component
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\begin{multline}
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\int dx_1 dx_{1'} \; \phi_a^*(x_1) \phi_i(x_{1'}) L_0(x_1,3;x_{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_1,3;\bx_{1'},4; \Oms)
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\\
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= e^{i \Oms t^{34} }
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\frac{ \phi_a^*(x_3) \phi_i(x_4) } { \Oms - ( \vara - \vari ) + i \eta }
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\times \qty[ \theta( \tau ) e^{i ( \vari + \hOms) \tau } + \theta( - \tau ) e^{i (\vara - \hOms \tau } ]
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\frac{ \phi_a^*(\bx_3) \phi_i(\bx_4) } { \Oms - ( \e{a} - \e{i} ) + i \eta }
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\times \qty[ \theta( \tau ) e^{i ( \e{i} + \hOms) \tau } + \theta( - \tau ) e^{i (\e{a} - \hOms \tau) } ]
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\end{multline}
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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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with $\tau = \tau_{34}$.
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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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\Xi(3,5;4,6) = v(3,6) \delta(34) \delta(56) - W(3^+,4) \delta(36) \delta(45)
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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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We further obtain the needed spectral representation of $\mel{N}{T \hpsi(3) \hpsi^{\dagger}(4)}{N,s}$ expanding the field operators over a complete orbital basis creation/destruction operators:
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\begin{multline}
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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_{mn} \phi_m(x_3) \phi_n^*(x_4)
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\mel{N}{\ha_n^{\dagger} \ha_m}{N,s}
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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( \tau ) e^{- i ( \eps_m - \hOms ) \tau } + \theta( -\tau ) e^{ - i ( \eps_n + \hOms) \tau } ]
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\times \qty[ \theta( \tau ) e^{- i ( \e{p} - \hOms ) \tau } + \theta( -\tau ) e^{ - i ( \e{q} + \hOms) \tau } ]
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\end{multline}
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with $\tau = \tau_{34}$ and where the $ \lbrace \eps_{n/m} \rbrace$ are proper addition/removal energies such that
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\begin{equation}
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e^{i H \tau} \ha_m^{\dagger} \ket{N} = e^{ i (E_0^N + \eps_m ) \tau } \ha _m^{\dagger} \ket{N}
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e^{i H \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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The $GW$ quasiparticle energies $\eps{n/m}^{GW}$ are good approximations to such removal/addition energies.
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Selecting $(n,m)=(j,b)$ yields the largest components
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$A_{jb}^{s} = \mel{N}{\ha_j^{\dagger} \ha_b}{N,s}$, while $(n,m)=(b,j)$ yields much weaker
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\titou{T2: what is $H$?}
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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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$A_{jb}^{s} = \mel{N}{\ha_j^{\dagger} \ha_b}{N,s}$, while $(p,q)=(b,j)$ yields much weaker
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$B_{jb}^{s} = \mel{N}{\ha_b^{\dagger} \ha_j}{N,s}$ contributions. We used chemist notations with $(i,j)$ indexing occupied orbitals and $(a,b)$ virtual ones.
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Neglecting the $B_{jb}^{s}$ 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 $\mel{N}{T \hpsi(x_1) \hpsi^{\dagger}(x_{1'})}{N,s}$, and projecting onto $\phi_a^*(x_1) \phi_i(x_{1'})$, one obtains after a few tedious manipulations (see {\SI}) the dynamical Bethe-Salpeter equation (dBSE) :
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Working out the same expansion for $\mel{N}{T \hpsi(5) \hpsi^{\dagger}(5)}{N,s}$ and $\mel{N}{T \hpsi(\bx_1) \hpsi^{\dagger}(\bx_{1'})}{N,s}$, 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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\begin{split}
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( \eps_a - \eps_i - \Omega_s ) A_{ia}^{s}
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( \e{a} - \e{i} - \Oms ) A_{ia}^{s}
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& + \sum_{jb} \qty[ v_{ai,bj} - \widetilde{W}_{ij,ab}(\Oms) ] A_{jb}^{s} \\
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& + \sum_{jb} \qty[ v_{ai,jb} - \widetilde{W}_{ib,aj}(\Oms) ] B_{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 (see Pina eq. 24):
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with an effective dynamically screened Coulomb potential (see Pina eq. 24):
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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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\end{multline}
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where $\Omega_{ib}^s = \Oms - ( \varb - \vari )$ and $\Omega_{ja}^s = \Oms - ( \vara - \varj )$.
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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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In the present study, we use the exact spectral representation of $W(\omega)$ at the RPA level:
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\begin{multline}
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W_{ij,ab}(\omega)
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= (ij|ab) + 2 \sum_m^{OV} [ij|m] [ab|m]
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\\
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\times \qty( \frac{1}{ \omega-\Omega_m^{RPA} + i\eta } - \frac{1}{ \omega + \Omega_m^{RPA} - i\eta } )
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\times \qty( \frac{1}{ \omega-\Om{m}{\RPA} + i\eta } - \frac{1}{ \omega + \Om{m}{\RPA} - i\eta } )
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\end{multline}
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($\Omega_m^{RPA} > 0 $) so that
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($\Om{m}{\RPA} > 0 $) so that
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\begin{multline}
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\widetilde{W}_{ij,ab}( \Oms )
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= (ij|ab) + 2 \sum_m^{OV} [ij|m] [ab|m]
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\\
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\times \qty( \frac{1}{\Omega_{ib}^{s} - \Omega_m^{RPA} + i\eta} + \frac{1}{\Omega_{ja}^{s} - \Omega_m^{RPA} + i\eta} )
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\times \qty( \frac{1}{\Om{ib}{s} - \Om{m}{\RPA} + i\eta} + \frac{1}{\Om{ja}{s} - \Om{m}{\RPA} + i\eta} )
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\end{multline}
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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 $\widetilde{W}_{ij,ab}( \Oms )$ presents no resonances. 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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