with $\tau=\tau_{34}$ and where the $\lbrace\varepsilon_{n/m}\rbrace$ are proper addition/removal energies such that e.g.
$$
e^{ i H \tau}{\hat a}_m^{\dagger} | N \rangle = e^{ i (E_0^N + \varepsilon_m ) \tau}{\hat a}_m^{\dagger} | N \rangle
$$
Selecting (n,m)=(j,b) yields the largest components
$A_{jb}^{s}=\langle N | {\hat a}_j^{\dagger}{\hat a}_b | N,s \rangle$, while (n,m)=(b,j) yields much weaker
$B_{jb}^{s}=\langle N | {\hat a}_b^{\dagger}{\hat a}_j | N,s \rangle$ 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 $\phi_a^*(x_1)\phi_i(x_{1'})$,
one obtains after a few tedious manipulations (see Supplemental Information) the dynamical Bethe-Salpeter equation (DBSE) :
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
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
in the limit $(\omega\rightarrow0)$ of the standard adiabatic BSE . WELL, do we know the sign of
$[ij|m][ab|m]$ ?? }
\titou{This is the theory section from the previous paper.}
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}
as the linear response of the one-body Green's function $\G{}$ with respect to a general non-local external potential
\begin{equation}
\XiBSE{}(3,5,4,6) = i \fdv{[\vc{\Ha}(3) \delta(3,4) + \Sig{\xc}(3,4)]}{\G{}(6,5)},
\end{equation}
which takes into account the self-consistent variation of the Hartree potential
\begin{equation}
\vc{\Ha}(1) = - i \int d2 \vc{}(2) \G{}(2,2^+),
\end{equation}
(where $\vc{}$ is the bare Coulomb operator) and the exchange-correlation self-energy $\Sig{\xc}$.
In Eq.~\eqref{eq:BSE}, $\LBSE{0}(1,2,1',2')=- i \G{}(1,2')\G{}(2,1')$, and $(1)=(\br{1},\sigma_1,t_1)$ is a composite index gathering space, spin and time variables.
In the $GW$ approximation, \cite{Hedin_1965,Aryasetiawan_1998,Onida_2002,Martin_2016,Reining_2017} we have
\begin{equation}
\SigGW{\xc}(1,2) = i \G{}(1,2) \W{}{}(1^+,2),
\end{equation}
where $\W{}{}$ is the screened Coulomb operator, and hence the BSE reduces to
where, as commonly done, we have neglected the term $\delta\W{}{}/\delta\G{}$ in the functional derivative of the self-energy. \cite{Hanke_1980,Strinati_1984,Strinati_1982}
Finally, the static approximation is enforced, \ie, $\W{}{}(1,2)=\W{}{}(\{\br{1}, \sigma_1, t_1\},\{\br{2}, \sigma_2, t_2\})\delta(t_1-t_2)$, which corresponds to restricting $\W{}{}$ to its static limit, \ie, $\W{}{}(1,2)=\W{}{}(\{\br{1},\sigma_1\},\{\br{2},\sigma_2\}; \omega=0)$.
For a closed-shell system in a finite basis, to compute the singlet BSE excitation energies (within the static approximation) of the physical system (\ie, $\IS=1$), one must solve the following linear response problem \cite{Casida,Dreuw_2005,Martin_2016}
\begin{equation}
\label{eq:LR}
\begin{pmatrix}
\bA{\IS}&\bB{\IS}\\
-\bB{\IS}& -\bA{\IS}\\
\end{pmatrix}
\begin{pmatrix}
\bX{\IS}_m \\
\bY{\IS}_m \\
\end{pmatrix}
=
\Om{m}{\IS}
\begin{pmatrix}
\bX{\IS}_m \\
\bY{\IS}_m \\
\end{pmatrix},
\end{equation}
where $\Om{m}{\IS}$ is the $m$th excitation energy with eigenvector $\T{(\bX{\IS}_m \,\bY{\IS}_m)}$ at interaction strength $\IS$, $\T{}$ is the matrix transpose, and we assume real-valued spatial orbitals $\{\MO{p}(\br{})\}_{1\le p \le\Norb}$.
The matrices $\bA{\IS}$, $\bB{\IS}$, $\bX{\IS}$, and $\bY{\IS}$ are all of size $\Nocc\Nvir\times\Nocc\Nvir$ where $\Nocc$ and $\Nvir$ are the number of occupied and virtual orbitals (\ie, $\Norb=\Nocc+\Nvir$ is the total number of spatial orbitals), respectively.
In the following, the index $m$ labels the $\Nocc\Nvir$ single excitations, $i$ and $j$ are occupied orbitals, $a$ and $b$ are unoccupied orbitals, while $p$, $q$, $r$, and $s$ indicate arbitrary orbitals.
In the absence of instabilities (\ie, when $\bA{\IS}-\bB{\IS}$ is positive-definite), \cite{Dreuw_2005} Eq.~\eqref{eq:LR} is usually transformed into an Hermitian eigenvalue problem of smaller dimension
with $\epsilon_{\IS}$ the dielectric function at coupling constant $\IS$ and $\chi_{0}$ the non-interacting polarizability. In the occupied-to-virtual orbital product basis, the spectral representation of $\W{}{\IS}$ can be written as follows in the case of real spatial orbitals
In the case of complex orbitals, we refer the reader to Ref.~\onlinecite{Holzer_2019} for a correct use of complex conjugation in the spectral representation of $\W{}{}$.
Note that, in the case of {\GOWO}, the RPA neutral excitations in Eq.~\eqref{eq:W} are computed using the HF orbital energies.
In Eq.~\eqref{eq:W}, $\eta$ is a positive infinitesimal, and $\OmRPA{m}{\IS}$ are the direct (\ie, without exchange) RPA neutral excitation energies computed by solving the linear eigenvalue problem \eqref{eq:LR} with the following matrix elements
where $\eHF{p}$ are the Hartree-Fock (HF) orbital energies.
The relationship between the BSE formalism and the well-known RPAx (\ie, RPA with exchange) approach can be obtained by switching off the screening so that $\W{}{\IS}$ reduces to the bare Coulomb potential $\vc{}$.
In this limit, the $GW$ quasiparticle energies reduce to the HF eigenvalues, and Eqs.~\eqref{eq:LR_BSE-A} and \eqref{eq:LR_BSE-B} to the RPAx equations:
This work was performed using HPC resources from GENCI-TGCC (Grant No.~2019-A0060801738) and CALMIP (Toulouse) under allocation 2020-18005.
Funding from the \textit{``Centre National de la Recherche Scientifique''} is acknowledged.
This work has also been supported through the EUR grant NanoX ANR-17-EURE-0009 in the framework of the \textit{``Programme des Investissements d'Avenir''.}}