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BSEdyn.tex
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@ -259,7 +259,7 @@ Unless otherwise stated, atomic units are used.
\subsection{Theory for physicists}
%=================================
The resolution of the dynamical Bethe-Salpeter equation (dBSE) [Strinati]
The resolution of the Bethe-Salpeter equation [Strinati]
\begin{align*}
L(1,2; & 1',2') = L_0(1,2;1',2') + \\
&+ \int d3456 \;
@ -283,8 +283,7 @@ $t_{2'} = t_2^+$ and $t_{1'} = t_1^+$, one obtains:
\chi_s(x_1,x_{1'}) = \langle N | T {\hat \psi}(x_1) {\hat \psi}^{\dagger}(x_{1'}) | N,s \rangle \\
{\tilde \chi}_s(x_2,x_{2'}) = \langle N,s | T {\hat \psi}(x_2) {\hat \psi}^{\dagger}(x_{2'}) | N \rangle
\end{align*}
The $\Oms$ are the neutral excitation energies of interest. Picking up the $e^{+i \Oms t_2 }$ component and simplifying by ${\tilde \chi}_s(x_2,x_{2'})
e^{ i \Oms t_{2} }$ 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.
The $\Oms$ 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 ${\tilde \chi}_s(x_2,x_{2'})$ 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.
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:
\begin{align*}
@ -332,7 +331,7 @@ one obtains after a few tedious manipulations (see Supplemental Information) the
\widetilde{W}_{ij,ab}(\Oms) &= \frac{ i }{ 2 \pi} \int d\omega \; e^{-i \omega 0^+ } W_{ij,ab}(\omega) \times \\
\hskip 1cm &\times \left[ \frac{1}{ \Omega_{ib}^s - \omega +i \eta } + \frac{1}{ \Omega_{ja}^{s} + \omega + i\eta } \right] \nonumber
\end{align}
with $\; \Omega_{ib}^s = \Oms - ( \varb - \vari )$ and $\; \Omega_{ja}^s = \Oms - ( \vara - \varj ).$
where $\; \Omega_{ib}^s = \Oms - ( \varb - \vari )$ and $\; \Omega_{ja}^s = \Oms - ( \vara - \varj ).$
In the present study, we use the exact spectral representation of $W(\omega)$ at the RPA level:
\begin{align*}
W_{ij,ab}(\omega) &= (ij|ab) + 2 \sum_m^{OV} [ij|m] [ab|m] \times \\
@ -345,7 +344,7 @@ W_{ij,ab}(\omega) &= (ij|ab) + 2 \sum_m^{OV} [ij|m] [ab|m] \times \\
\right] \nonumber
\end{align}
\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
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.
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.
$$
\left| \frac{ 1 }{ \Omega_{ib}^{s} - \Omega_m^{RPA} } \right|
< \frac{1}{ \Omega_m^{RPA} }