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\newcommand{\NEEL}{Universit\'e Grenoble Alpes, CNRS, Institut NEEL, F-38042 Grenoble, France}
\newcommand{\LCPQ}{Laboratoire de Chimie et Physique Quantiques (UMR 5626), Universit\'e de Toulouse, CNRS, UPS, France}

\begin{document}	

\title{Dynamical Correction to the Bethe-Salpeter Equation}
%\title{ \textcolor{red}{Assessing} Dynamical Correction to the Bethe-Salpeter Equation}

\author{Pierre-Fran\c{c}ois \surname{Loos}}
	\email{loos@irsamc.ups-tlse.fr}
	\affiliation{\LCPQ}
\author{Xavier \surname{Blase}}
	\email{xavier.blase@neel.cnrs.fr }
	\affiliation{\NEEL}

\begin{abstract}
Similar to the ubiquitous adiabatic approximation in time-dependent density-functional theory, the static approximation, which substitutes a dynamical (\ie, frequency-dependent) kernel by its static limit, is usually enforced in most implementations of the Bethe-Salpeter equation (BSE) formalism.
Here, going beyond the static approximation, we compute the dynamical correction in the electron-hole screening for molecular excitation energies thanks to a renormalized first-order perturbative correction to the static BSE excitation energies.
The present dynamical correction goes beyond the plasmon-pole approximation as the dynamical screening of the Coulomb interaction is computed exactly within the random-phase approximation.
Our calculations are benchmarked against high-level (coupled-cluster) calculations, allowing to assess the clear improvements brought by the dynamical correction.
%\\
%\bigskip
%\begin{center}
%	\boxed{\includegraphics[width=0.5\linewidth]{TOC}}
%\end{center}
%\bigskip
\end{abstract}

\maketitle

%%%%%%%%%%%%%%%%%%%%%%%%
\section{Introduction}
\label{sec:intro}
%%%%%%%%%%%%%%%%%%%%%%%%
The Bethe-Salpeter equation (BSE) formalism \cite{Salpeter_1951,Strinati_1988} is to the $GW$ approximation \cite{Hedin_1965,Golze_2019} of many-body perturbation theory (MBPT) \cite{Martin_2016} what time-dependent density-functional theory (TD-DFT) \cite{Runge_1984,Casida_1995} is to Kohn-Sham density-functional theory (KS-DFT), \cite{Hohenberg_1964,Kohn_1965} an affordable way of computing the neutral (or optical) excitations of a given electronic system. 
In recent years, it has been shown to be a valuable tool for computational chemists with a large number of systematic benchmark studies on large families of molecular systems appearing in the literature \cite{Boulanger_2014,Jacquemin_2015a,Bruneval_2015,Jacquemin_2015b,Hirose_2015,Jacquemin_2017a,Jacquemin_2017b,Rangel_2017,Krause_2017,Gui_2018} (see Ref.~\onlinecite{Blase_2018} for a recent review).

Taking the optical gap (\ie, the lowest optical excitation energy) as an example, BSE builds on top of a $GW$ calculation by adding up excitonic effects $\EB$ to the $GW$ HOMO-LUMO gap
\begin{equation}
	\Eg^{\GW} = \eps_{\LUMO}^{\GW} - \eps_{\HOMO}^{\GW},
\end{equation}
which is itself a corrected version of the Kohn-Sham (KS) gap
\begin{equation}
	\Eg^{\KS} = \eps_{\LUMO}^{\KS} - \eps_{\HOMO}^{\KS} \ll \Eg^{\GW} \approx \EgFun,
\end{equation}
in order to approximate the optical gap
\begin{equation}
	\EgOpt = E_1^{N} - E_0^{N} = \EgFun + \EB,
\end{equation}
where 
\begin{equation} \label{eq:Egfun}
	\EgFun = I^N - A^N
\end{equation}
is the the fundamental gap, \cite{Bredas_2014} $I^N = E_0^{N-1} - E_0^N$ and $A^N = E_0^{N} - E_0^{N+1}$ being the ionization potential and the electron affinity of the $N$-electron system, respectively.
Here, $E_S^{N}$ is the total energy of the $S$th excited state of the $N$-electron system, and $E_0^N$ corresponds to its ground-state energy.
Because the excitonic effect corresponds physically to the stabilization implied by the attraction of the excited electron and its hole left behind, we have $\EgOpt < \EgFun$.
Due to the smaller amount of screening in molecules as compared to solids, a faithful description of excitonic effects is paramount in molecular systems.

Most of BSE implementations rely on the so-called static approximation, which approximates the dynamical (\ie, frequency-dependent) BSE kernel by its static limit.
In complete analogy with the ubiquitous adiabatic approximation in TD-DFT where the exchange-correlation (xc) kernel is made static, one key consequence of the static approximation within BSE is that double (and higher) excitations are completely absent from the BSE spectrum.
Indeed, a frequency-dependent kernel has the ability to create additional poles in the response function, which describe states with a multiple-excitation character, and, in particular, double excitations.
Although these double excitations are usually experimentally dark (which means that they usually cannot be observed in photo-absorption spectroscopy), these states play, indirectly, a key role in many photochemistry mechanisms, \cite{Boggio-Pasqua_2007} as they strongly mix with the bright singly-excited states leading to the formation of satellite peaks. \cite{Helbig_2011,Elliott_2011}
They are particularly important in the faithful description of the ground state of open-shell molecules, \cite{Casida_2005,Romaniello_2009a,Huix-Rotllant_2011,Loos_2020c}
and they are, moreover, a real challenge for high-level computational methods. \cite{Loos_2018a,Loos_2019,Loos_2020b,Loos_2020c}
Double excitations play also a significant role in the correct location of the excited states of polyenes that are closely related to rhodopsin, a biological pigment found in the rods of the retina and involved in the visual transduction. \cite{Olivucci_2010,Robb_2007,Manathunga_2016}
In butadiene, for example, while the bright $1 ^1B_u$ state has a clear $\HOMO \ra \LUMO$ single-excitation character, the dark $2 ^1A_g$ state includes a substantial fraction of doubly-excited character from the $\HOMO^2 \ra \LUMO^2$ double excitation (roughly $30\%$), yet dominant contributions from the $\HOMO-1 \ra \LUMO$ and $\HOMO \ra \LUMO+1$ single excitations. \cite{Maitra_2004,Cave_2004,Saha_2006,Watson_2012,Shu_2017,Barca_2018a,Barca_2018b,Loos_2019}

Going beyond the static approximation is tricky and very few groups have dared to take the plunge. \cite{Strinati_1988,Rohlfing_2000,Sottile_2003,Myohanen_2008,Ma_2009a,Ma_2009b,Romaniello_2009b,Sangalli_2011,Huix-Rotllant_2011,Sakkinen_2012,Zhang_2013,Rebolini_2016,Olevano_2019,Lettmann_2019}
Nonetheless, it is worth mentioning the seminal work of Strinati on core excitons in semiconductors, \cite{Strinati_1982,Strinati_1984,Strinati_1988} in which the dynamical screening effects were taken into account through the dielectric matrix, and where he observed an increase of the binding energy over its value for static screening and a narrowing of the Auger width below its value for a core hole.
Following Strinati's footsteps, Rohlfing and coworkers have developed an efficient way of taking into account, thanks to first-order perturbation theory, the dynamical effects via a plasmon-pole approximation combined with the Tamm-Dancoff approximation (TDA). \cite{Rohlfing_2000,Ma_2009a,Ma_2009b,Baumeier_2012b}
With such a scheme, they have been able to compute the excited states of biological chromophores, showing that taking into account the electron-hole dynamical screening is important for an accurate description of the lowest $n \ra \pi^*$ excitations. \cite{Ma_2009a,Ma_2009b,Baumeier_2012b}
Indeed, studying PYP, retinal and GFP chromophore models, Ma \textit{et al.}~found that \textit{``the influence of dynamical screening on the excitation energies is about $0.1$ eV for the lowest $\pi \ra \pis$ transitions, but for the lowest $n \ra \pis$ transitions the influence is larger, up to $0.25$ eV.''} \cite{Ma_2009b}
A similar conclusion was reached in Ref.~\onlinecite{Ma_2009a}.
Zhang \textit{et al.}~have studied the frequency-dependent second-order Bethe-Salpeter kernel and they have observed an appreciable improvement over configuration interaction with singles (CIS), time-dependent Hartree-Fock (TDHF), and adiabatic TD-DFT results. \cite{Zhang_2013}
Rebolini and Toulouse have performed a similar investigation in a range-separated context, and they have reported a modest improvement over its static counterpart. \cite{Rebolini_2016,Rebolini_PhD} 
In these two latter studies, they also followed a (non-self-consistent) perturbative approach within the TDA with a renormalization of the first-order perturbative correction.

It is important to note that, although all the studies mentioned above are clearly going beyond the static approximation of BSE, they are not able to recover additional excitations as the perturbative treatment makes ultimately the BSE kernel static. 
However, it does permit to recover, for transitions with a dominant single-excitation character, additional relaxation effects coming from higher excitations.
These higher excitations would be explicitly present in the BSE Hamiltonian by ``unfolding'' the dynamical BSE kernel, and one would recover a linear eigenvalue problem with, nonetheless, a much larger dimension. \cite{Loos_2020f}

Based on a simple two-level model which permits to analytically solve the dynamical equations, Romaniello and coworkers \cite{Romaniello_2009b,Sangalli_2011} evidenced that one can genuinely access additional excitations by solving the non-linear, frequency-dependent eigenvalue problem. 
For this particular system, it was shown that a BSE kernel based on the random-phase approximation (RPA) produces indeed double excitations but also unphysical excitations. \cite{Romaniello_2009b} 
The appearance of these spurious excitations was attributed to the self-screening problem. \cite{Romaniello_2009a}
This was fixed in a follow-up paper by Sangalli \textit{et al.} \cite{Sangalli_2011} thanks to the design of a number-conserving approach based on the folding of the second-RPA Hamiltonian, \cite{Wambach_1988} which includes explicitly both single and double excitations.
By computing the polarizability of two unsaturated hydrocarbon chains, \ce{C8H2} and \ce{C4H6}, they showed that their approach produces the correct number of physical excitations.

Finally, let us mention efforts to borrow ingredients from BSE in order to go beyond the adiabatic approximation of TD-DFT. 
For example, Huix-Rotllant and Casida \cite{Casida_2005,Huix-Rotllant_2011} proposed a nonadiabatic correction to the xc kernel using the formalism of superoperators, which includes as a special case the dressed TD-DFT method of Maitra and coworkers, \cite{Maitra_2004,Cave_2004,Elliott_2011,Maitra_2012} where a frequency-dependent kernel is build \textit{a priori} and manually for a particular excitation.
Following a similar strategy, Romaniello \textit{et al.} \cite{Romaniello_2009b} took advantages of the dynamically-screened Coulomb potential from BSE to obtain a dynamic TD-DFT kernel.
In this regard, MBPT provides key insights about what is missing in adiabatic TD-DFT, as discussed in details by Casida and Huix-Rotllant in Ref.~\onlinecite{Casida_2016}.

In the present study, we extend the work of Rohlfing and coworkers \cite{Rohlfing_2000,Ma_2009a,Ma_2009b,Baumeier_2012b} by proposing a renormalized first-order perturbative correction to the static BSE excitation energies. 
Importantly, our correction goes beyond the plasmon-pole approximation as the dynamical screening of the Coulomb interaction is computed exactly.
%Moreover, we investigate quantitatively the effect of the TDA by computing both the resonant and anti-resonant dynamical corrections to the BSE excitation energies.
Unless otherwise stated, atomic units are used.

%%%%%%%%%%%%%%%%%%%%%%%%
\section{Theory}
\label{sec:theory}
%%%%%%%%%%%%%%%%%%%%%%%%

In this Section, following Strinati's seminal work, \cite{Strinati_1988} we first derive in some details the theoretical foundations leading to the dynamical BSE. 
%Additional details about this derivation are provided as {\SI}.
We present, in a second step, the perturbative implementation of the dynamical correction as compared to the standard static approximation.

%================================
\subsection{General dynamical BSE}
%=================================

The two-body correlation function $L(1,2; 1',2')$ --- a central quantity in the BSE formalism --- relates the variation of the one-body Green's function $G(1,1')$ with respect to an external non-local perturbation $U(2',2)$, \ie,
\begin{equation}
	iL(1,2; 1',2') = \pdv{G(1,1')}{U(2',2)},
\end{equation}
where, \eg, $1 \equiv (\bx_1 t_1)$ is a space-spin plus time composite variable. 
The relation between $G$ and the one-body charge density $\rho(1) = -i G(1,1^+)$ provides a direct connection with the density-density susceptibility $\chi(1,2) = L(1,2;1^+,2^+)$ at the core of TD-DFT.
(The notation $1^+$ means that the time $t_1$ is taken at $t_1^{+} = t_1 + 0^+$, where $0^+$ is a positive infinitesimal.)

The two-body correlation function $L$ satisfies the self-consistent BSE \cite{Strinati_1988}  
\begin{multline} \label{eq:BSE}
	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{multline}
where
\begin{subequations}
\begin{align}
	\label{eq:L0}
	iL_0(1, 4; 1', 3) & = G(1, 3)G(4, 1'),
	\\
	\label{eq:L}
	iL(1,2; 1',2') & = - G_2(1,2;1',2') + G(1,1') G(2,2'),
\end{align}
\end{subequations}
can be expressed as a function of the one- and two-body Green's functions
\begin{subequations}
\begin{align}
	\label{eq:G1}
	G(1,2) & = - i \mel{N}{T [ \hpsi(1) \hpsi^{\dagger}(2) ] }{N},
	\\
	\label{eq:G2}
	G_2(1,2;1',2') & = - \mel{N}{T [ \hpsi(1) \hpsi(2) \hpsi^{\dagger}(2') \hpsi^{\dagger}(1') ]}{N},
\end{align}
\end{subequations}
and
\begin{equation}
	\Xi(3,5;4,6) = i \fdv{[v_\text{H}(3) \delta(3,4) + \Sigma_\text{xc}(3,4)]}{G(6,5)}
\end{equation}
is the BSE kernel that takes into account the self-consistent variation of the Hartree potential 
\begin{equation}
	v_\text{H}(1) = - i \int d2 \, v(1,2) G(2,2^+),
\end{equation}
[where $\delta$ is Dirac's delta function and $v$ is the bare Coulomb operator] and the xc self-energy $ \Sigma_\text{xc}$ with respect to the variation of $G$.  
In Eqs.~\eqref{eq:G1} and \eqref{eq:G2}, the field operators $\Hat{\psi}(\bx t)$ and $\Hat{\psi}^{\dagger}(\bx't')$ remove and add (respectively) an electron to the $N$-electron ground state $\ket{N}$ in space-spin-time positions ($\bx t$) and ($\bx't'$), while $T$ is the time-ordering operator.

The resolution of the dynamical BSE starts with the expansion of $L_0$ and $L$ [see Eqs.~\eqref{eq:L0} and \eqref{eq:L}] over the complete orthonormalized set of $N$-electron excited states $\ket{N,S}$ (with $\ket{N,0} \equiv \ket{N}$). \cite{Strinati_1988} 
In the optical limit of instantaneous electron-hole creation and destruction, imposing $t_{2'} = t_2^+$ and $t_{1'} = t_1^+$, and using the relation between the field operators in their time-dependent (Heisenberg) and time-independent (Schr\"{o}dinger) representations, \eg,
\begin{equation} \label{Eisenberg}
	\hpsi(1) = e^{ i \hH t_1 } \hpsi(\bx_1) e^{-i \hH t_1 },
\end{equation}
($\hH$ being the exact many-body Hamiltonian), one gets
\begin{equation}
\begin{split}
	iL(1,2; 1',2') 
	& = \theta(+\tau_{12}) \sum_{s > 0} \chi_S(\bx_1,\bx_{1'}) \tchi_S(\bx_2,\bx_{2'}) e^{ - i \Om{S}{} \tau_{12} } 
 	\\
	& - \theta(-\tau_{12}) \sum_{s > 0} \chi_S(\bx_2,\bx_{2'}) \tchi_S(\bx_1,\bx_{1'}) e^{ + i \Om{S}{} \tau_{12} },
\end{split}
\end{equation}
where $\tau_{12} = t_1 - t_2$, $\theta$ is the Heaviside step function, and
\begin{subequations}
\begin{align}
	\chi_S(\bx_1,\bx_{1'}) & = \mel{N}{T [\hpsi(\bx_1) \hpsi^{\dagger}(\bx_{1'})] }{N,S},
	\\
	\tchi_S(\bx_1,\bx_{1'}) & = \mel{N,S}{T [\hpsi(\bx_1) \hpsi^{\dagger}(\bx_{1'})] }{N}.
\end{align}  
\end{subequations}  
The $\Om{s}{}$'s are the neutral excitation energies of interest (with $\Om{s}{} = E^N_s - E^N_0$). 

Picking up the $e^{+i \Om{S}{} t_2 }$ component of both $L(1,2; 1',2')$ and $L(6,2;5,2')$, simplifying further by $\tchi_S(\bx_2,\bx_{2'})$ on both sides of the BSE [see Eq.~\eqref{eq:BSE}], we seek the $e^{-i \Om{S}{} t_1 }$ Fourier component associated with the right-hand side of a modified dynamical BSE, which reads
\begin{multline} \label{eq:BSE_2}
    \mel{N}{T [ \hpsi(\bx_1) \hpsi^{\dagger}(\bx_{1}') ] } {N,S} e^{ - i \Om{S}{} t_1 }
	\theta ( \tau_{12} )  
    \\
	= \int d3456 \, L_0(1,4;1',3) \Xi(3,5;4,6)  
    \\
	\times \mel{N}{T [\hpsi(6) \hpsi^{\dagger}(5)] }{N,S} 
	\theta [\min(t_5,t_6) - t_2].
\end{multline}
For the neutral excitation energies falling in the fundamental gap of the system (\ie, $\Om{S}{} < \EgFun$ due to excitonic effects), $L_0(1,2;1',2')$ cannot contribute to the $e^{-i \Om{S}{} t_1 }$ response  term since its lowest excitation energy is precisely the fundamental gap [see Eq.~\eqref{eq:Egfun}].
Consequently, special care has to be taken for high-lying excited states (like core or Rydberg excitations) where additional terms have to be taken into account (see Refs.~\onlinecite{Strinati_1982,Strinati_1984}).

Dropping the space/spin variables, the Fourier components with respect to $t_1$ of $L_0(1,4;1',3)$ reads
\begin{align} \label{eq:iL0}
	[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} },
   \end{align}
with $\tau_{34} = t_3 - t_4$ and $t^{34} = (t_3 + t_4)/2$. 
We now adopt the Lehman representation of the one-body Green's function in the quasiparticle approximation, \ie,
\begin{equation} \label{eq:G-Lehman}
	G(\bx_1,\bx_2 ; \omega) = \sum_p \frac{ \MO{p}(\bx_1) \MO{p}^*(\bx_2) } { \omega - \e{p} + i \eta \times \text{sgn}  (\e{p} - \mu) },
\end{equation}
where $\eta$ is a positive infinitesimal and $\mu$ is the chemical potential. 
The $\e{p}$'s in Eq.~\eqref{eq:G-Lehman} are quasiparticle energies (\ie, proper addition/removal energies) and the $\MO{p}(\bx)$'s are their associated one-body (spin)orbitals.
In the following, $i$ and $j$ are occupied orbitals, $a$ and $b$ are unoccupied orbitals, while $p$, $q$, $r$, and $s$ indicate arbitrary orbitals.
Projecting the Fourier component $L_0(\bx_1,4;\bx_{1'},3;  \omega_1 = \Om{S}{} )$ onto $\MO{a}^*(\bx_1) \MO{i}(\bx_{1'})$ yields
\begin{multline}  \label{eq:iL0bis} 
	\iint d\bx_1 d\bx_{1'} \, \MO{a}^*(\bx_1) \MO{i}(\bx_{1'})  L_0(\bx_1,4;\bx_{1'},3; \Om{S}{}) 
	\\
	= 
	\frac{ \MO{a}^*(\bx_3) \MO{i}(\bx_4)  e^{i \Om{S}{} t^{34} }} { \Om{S}{} - ( \e{a} - \e{i} ) + i \eta }   
	\qty[ \theta( \tau_{34} ) e^{i \qty( \e{i} + \frac{\Om{S}{}}{2}) \tau_{34} } + \theta( - \tau_{34} ) e^{i \qty(\e{a} - \frac{\Om{S}{}}{2}) \tau_{34} }  ]. 
\end{multline}
%with $t^{34} = (t_3 + t_4)/2$ and $\tau_{34} = t_3 -t_4$. 
More details are provided in Appendix \ref{app:A}.
As a final step, we express the terms $\mel{N}{T [\hpsi(\bx_1) \hpsi^{\dagger}(\bx_{1}')] }{N,S}$ and $\mel{N}{T [\hpsi(6) \hpsi^{\dagger}(5)] }{N,S}$ from Eq.~\eqref{eq:BSE_2} in the standard electron-hole product (or single-excitation) space.
This is done by  expanding the field operators over a complete orbital basis of creation/destruction operators.
For example, we have (see derivation in Appendix \ref{app:B})
\begin{multline} \label{eq:spectral65}
	\mel{N}{T [\hpsi(6) \hpsi^{\dagger}(5)] }{N,S} 
	\\
	= - \qty( e^{  -i \Om{S}{} t^{65} }  ) \sum_{pq} \MO{p}(\bx_6) \MO{q}^*(\bx_5) 
	\mel{N}{\ha_q^{\dagger} \ha_p}{N,S}
	\\
	\times \qty[ \theta( \tau_{65}  ) e^{- i \qty( \e{p} - \frac{\Om{S}{}}{2} ) \tau_{65}  } + \theta( - \tau_{65}  ) e^{ - i \qty( \e{q}  + \frac{\Om{S}{}}{2}) \tau_{65}  } ], 
\end{multline}
with $t^{65} = (t_5 + t_6)/2$ and $\tau_{65} = t_6 -t_5$. The $\mel{N}{\ha_q^{\dagger} \ha_p}{N,S}$ are the unknown particle-hole amplitudes.% in the orbital product basis.


%================================
\subsection{Dynamical BSE within the $GW$ approximation}
%=================================

Adopting now the $GW$ approximation \cite{Hedin_1965} for the xc self-energy, \ie, 
\begin{equation}
	\Sigma_\text{xc}^{\GW}(1,2) = i G(1,2) W(1^+,2),
\end{equation}
leads to the following simplified BSE kernel
\begin{equation} \label{eq:Xi_GW}
	\Xi(3,5;4,6) = v(3,6) \delta(3,4) \delta(5,6) - W(3^+,4) \delta(3,6) \delta(4,5),
\end{equation}
where $W$ is the dynamically-screened Coulomb operator.
The $GW$ quasiparticle energies $\eGW{p}$ are usually good approximations to the removal/addition energies $\e{p}$ introduced in Eq.~\eqref{eq:G-Lehman}. 

Substituting Eqs.~\eqref{eq:iL0bis}, \eqref{eq:spectral65}, and \eqref{eq:Xi_GW} into Eq.~\eqref{eq:BSE_2},  and projecting onto $\MO{a}^*(\bx_1) \MO{i}(\bx_{1'})$, one gets after a few tedious manipulations the dynamical BSE:
\begin{equation} \label{eq:BSE-final}
\begin{split}
	( \eGW{a} - \eGW{i} - \Om{S}{} ) X_{ia,S}      
	& + \sum_{jb} \qty[ \kappa \ERI{ia}{jb} - \widetilde{W}_{ij,ab}(\Om{S}{}) ] X_{jb,S}  \\ 
	& + \sum_{jb} \qty[ \kappa \ERI{ia}{bj} - \widetilde{W}_{ib,aj}(\Om{S}{}) ] Y_{jb,S} 
	= 0,
\end{split}
\end{equation}
with $X_{jb,S} = \mel{N}{\ha_j^{\dagger} \ha_b}{N,S}$ and $Y_{jb,S} = \mel{N}{\ha_b^{\dagger} \ha_j}{N,S}$, and where $\kappa = 2 $ or $0$ for singlet and triplet excited states (respectively).
Neglecting the anti-resonant terms, $Y_{jb,S}$, in the dynamical BSE, which are (usually) much smaller than their resonant counterparts, $X_{jb,S}$, leads to the well-known TDA. 
In Eq.~\eqref{eq:BSE-final}, 
\begin{equation}
	\ERI{pq}{rs} = \iint d\br d\br' \, \MO{p}^*(\br) \MO{q}(\br) v(\br -\br') \MO{r}^*(\br') \MO{s}(\br'),
\end{equation}
are the bare two-electron integrals in the spatial orbital basis $\lbrace \MO{p}(\br{}) \rbrace$, and
\begin{multline} \label{eq:wtilde}
	\widetilde{W}_{pq,rs}(\Om{S}{})
	= \frac{ i }{ 2  \pi}  \int d\omega \; e^{-i \omega 0^+ } W_{pq,rs}(\omega) 
	\\
	\times \qty[ \frac{1}{ \Om{ps}{S} - \omega + i \eta } + \frac{1}{ \Om{qr}{S} + \omega + i\eta } ],
\end{multline}
is an effective dynamically-screened Coulomb potential, \cite{Romaniello_2009b} where $\Om{pq}{S} = \Om{S}{} - ( \eGW{q} - \eGW{p} )$ and
\begin{equation}
	W_{pq,rs}({\omega}) 
	 = \iint d\br d\br' \, \MO{p}(\br) \MO{q}^*(\br) W(\br ,\br'; \omega) \MO{r}^*(\br') \MO{s}(\br').
\end{equation}

%\xavier{\sout{ A second coupled equation for the $(X_{ia}^{S}, Y_{ia}^{S} )$ vector can be obtained by projecting $\mel{N}{T [ \hpsi(\bx_1) \hpsi^{\dagger}(\bx_{1}') ] } {N,S}$ and $L_0(\bx_1,4;\bx_{1'},3; \Om{S}{})$ onto $\MO{i}^*(\bx_1) \MO{a}(\bx_{1'})$ instead of $\MO{a}^*(\bx_1) \MO{i}(\bx_{1'})$. } }


%================================
\subsection{Dynamical screening}
\label{sec:dynW}
%=================================

In the present study, we consider the exact spectral representation of $W$ at the RPA level:
\begin{multline}
\label{eq:W-RPA}
	W_{ij,ab}(\omega) 
	= \ERI{ij}{ab} + 2 \sum_m \sERI{ij}{m} \sERI{ab}{m} 
	\\
	\times \qty[  \frac{1}{ \omega-\Om{m}{\RPA} + i\eta } - \frac{1}{ \omega + \Om{m}{\RPA} - i\eta } ],
\end{multline}
where $m$ labels single excitations, and
\begin{equation}
\label{eq:sERI}
	\sERI{pq}{m} =  \sum_{ia} \ERI{pq}{ia} (\bX{m}{\RPA} + \bY{m}{\RPA})_{ia}
\end{equation}
are the spectral weights.
In Eqs.~\eqref{eq:W-RPA} and \eqref{eq:sERI}, $\OmRPA{m}{}$ and $(\bX{m}{\RPA} + \bY{m}{\RPA})$ are RPA neutral excitations and their corresponding transition vectors computed by solving the (static) linear response problem 
\begin{equation}
\label{eq:LR-RPA}
	\begin{pmatrix}
		\bA{\RPA}	&	\bB{\RPA}	\\
		-\bB{\RPA}	&	-\bA{\RPA}	\\
	\end{pmatrix}
	\cdot
	\begin{pmatrix}
		\bX{m}{\RPA}	\\
		\bY{m}{\RPA}	\\
	\end{pmatrix}
	=
	\OmRPA{m}
	\begin{pmatrix}
		\bX{m}{\RPA}	\\
		\bY{m}{\RPA}	\\
	\end{pmatrix},
\end{equation}
with
\begin{subequations}
\begin{align}
	\label{eq:LR_RPA-A}
	\A{ia,jb}{\RPA} & = \delta_{ij} \delta_{ab} (\e{a} - \e{i}) +   2 \ERI{ia}{jb},
	\\ 
	\label{eq:LR_RPA-B}
	\B{ia,jb}{\RPA} & =   2 \ERI{ia}{bj},
\end{align}
\end{subequations}
where the $\e{p}$'s are taken as the HF orbital energies in the case of $G_0W_0$ \cite{Hybertsen_1985a, Hybertsen_1986} or as the $GW$ quasiparticle energies in the case of self-consistent schemes such as ev$GW$. \cite{Hybertsen_1986,Shishkin_2007,Blase_2011,Faber_2011} 
The RPA matrices $\bA{\RPA}$ and $\bB{\RPA}$ in Eq.~\eqref{eq:LR-RPA} are 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, and $\bX{m}{\RPA}$, and $\bY{m}{\RPA}$ are (eigen)vectors of length $\Nocc \Nvir$. 

The analysis of the poles of the integrand in Eq.~\eqref{eq:wtilde} yields
\begin{multline} 
	\widetilde{W}_{ij,ab}( \Om{S}{} ) 
	= \ERI{ij}{ab} + 2 \sum_m \sERI{ij}{m} \sERI{ab}{m}  
	\\ 
    \times \qty[ \frac{1}{\Om{ib}{S} - \Om{m}{\RPA} + i\eta} + \frac{1}{\Om{ja}{S} - \Om{m}{\RPA} + i\eta} ].
\end{multline} 
One can verify that, in the static limit where $\Om{m}{\RPA} \to \infty$, the matrix elements $\widetilde{W}_{ij,ab}$ correctly reduce to their static expression
\begin{equation}
\label{eq:Wstat}
	\W{ij,ab}{\text{stat}} 
	\equiv W_{ij,ab}(\omega = 0) 
	= \ERI{ij}{ab} -  4 \sum_m \frac{\sERI{ij}{m} \sERI{ab}{m}}{\OmRPA{m}{} - i \eta},
\end{equation}
evidencing that the standard static BSE problem is recovered from the present dynamical formalism in this limit.

Due to excitonic effects, the lowest BSE excitation energy, $\Om{1}{}$, stands lower than the lowest RPA excitation energy, $\Om{1}{\RPA}$, so that, $\Om{ib}{S} - \Om{m}{\RPA} < 0 $ and $\widetilde{W}_{ij,ab}(\Om{S}{})$ has no resonances. 
This property holds for low-lying excitations but special care must be taken for higher ones.
Furthermore, $\Om{ib}{S}$ and $\Om{ja}{S}$ are necessarily negative quantities for in-gap low-lying BSE excitations. 
Thus, we have $\abs*{\Om{ib}{S} - \Om{m}{\RPA}}  >  \Om{m}{\RPA}$.
As a consequence, we observe a reduction of the electron-hole screening, \ie, an enhancement of electron-hole binding energy, as compared to the standard static BSE, and consequently smaller (red-shifted) excitation energies. 
This will be numerically illustrated in Sec.~\ref{sec:resdis}.

%================================================
\subsection{Dynamical Tamm-Dancoff approximation}
%================================================
The analysis of the (off-diagonal) screened Coulomb potential matrix elements multiplying the $Y_{jb,S}$ coefficients in Eq.~\eqref{eq:BSE-final}, \ie,
\begin{multline} \label{eq:W-Y}
	\widetilde{W}_{ib,aj}(\Om{S}{}) 
	= \ERI{ib}{aj} + 2 \sum_m \sERI{ib}{m} \sERI{aj}{m}  
	\\ 
    \times \qty[ \frac{1}{\Om{ij}{S} - \Om{m}{\RPA} + i\eta} + \frac{1}{\Om{ba}{S} - \Om{m}{\RPA} + i\eta} ],
\end{multline} 
reveals strong divergences even for low-lying excitations with, for example, $\Om{ba}{S} - \Om{m}{\RPA} = \Om{S}{} - \Om{m}{\RPA} - ( \eGW{a} - \eGW{b} ) \approx 0$.
Such divergences may explain that in previous studies dynamical effects were only accounted for at the TDA level. \cite{Strinati_1988,Rohlfing_2000,Ma_2009a,Ma_2009b,Romaniello_2009b,Sangalli_2011,Zhang_2013,Rebolini_2016}
To avoid confusions here, enforcing the TDA for the dynamical correction (which corresponds to neglecting the dynamical correction originating from the anti-resonant part of the BSE Hamiltonian) will be labeled as dTDA in the following.
Going beyond the dTDA is outside the scope of the present study but shall be addressed eventually.

%=================================
\subsection{Perturbative dynamical correction}
%=================================

From a more practical point of view, Eq.~\eqref{eq:BSE-final} can be recast as an non-linear eigenvalue problem and, to compute the BSE excitation energies of a closed-shell system, one must solve the following dynamical (\ie, frequency-dependent) response problem \cite{Strinati_1988}
\begin{equation}
\label{eq:LR-dyn}
	\begin{pmatrix}
		\bA{}(\Om{S}{})		&	\bB{}(\Om{S}{})	
		\\
		-\bB{}(-\Om{S}{})	&	-\bA{}(-\Om{S}{})	
		\\
	\end{pmatrix}
	\cdot
	\begin{pmatrix}
		\bX{S}{}	\\
		\bY{S}{}	\\
	\end{pmatrix}
	=
	\Om{S}{}
	\begin{pmatrix}
		\bX{S}{}	\\
		\bY{S}{}	\\
	\end{pmatrix},
\end{equation}
where the dynamical matrices $\bA{}$ and $\bB{}$ have the same $\Nocc \Nvir \times \Nocc \Nvir$ size than their RPA counterparts, and we assume real quantities from hereon.
Same comment applies to the eigenvectors $\bX{S}{}$, and $\bY{S}{}$ of length $\Nocc \Nvir$.
Note that, due to its non-linear nature, Eq.~\eqref{eq:LR-dyn} may provide more than one solution for each value of $S$. \cite{Romaniello_2009b,Sangalli_2011,Martin_2016}

Accordingly to Eq.~\eqref{eq:BSE-final}, the BSE matrix elements in Eq.~\eqref{eq:LR-dyn} read
\begin{subequations}
\begin{align}
	\label{eq:BSE-Adyn}
	\A{ia,jb}{}(\Om{S}{}) & = \delta_{ij} \delta_{ab} (\eGW{a} - \eGW{i}) + \kappa \ERI{ia}{jb} - \tW{ij,ab}{}(\Om{S}{}),
	\\ 
	\label{eq:BSE-Bdyn}
	\B{ia,jb}{}(\Om{S}{}) & =  \kappa \ERI{ia}{bj} - \tW{ib,aj}{}(\Om{S}{}).
\end{align}
\end{subequations}

Now, let us decompose, using basic Rayleigh-Schr\"odinger perturbation theory, the non-linear eigenproblem \eqref{eq:LR-dyn} as a zeroth-order static (hence linear) reference and a first-order dynamic (hence non-linear) perturbation, such that
\begin{multline}
\label{eq:LR-PT}
	\begin{pmatrix}
		\bA{}(\Om{S}{})		&	\bB{}(\Om{S}{})	\\
		-\bB{}(-\Om{S}{})	&	-\bA{}(-\Om{S}{})	\\
	\end{pmatrix}
	\\
	=
	\begin{pmatrix}
		\bA{(0)}	&	\bB{(0)}	
		\\
		-\bB{(0)}	&	-\bA{(0)}	
		\\
	\end{pmatrix}
	+
	\begin{pmatrix}
		\bA{(1)}(\Om{S}{})		&	\bB{(1)}(\Om{S}{})	\\
		-\bB{(1)}(-\Om{S}{})	&	-\bA{(1)}(-\Om{S}{})	\\
	\end{pmatrix},
\end{multline}
with
\begin{subequations}
\begin{align}
	\label{eq:BSE-A0}
	\A{ia,jb}{(0)} & = \delta_{ij} \delta_{ab} (\eGW{a} - \eGW{i}) + \kappa \ERI{ia}{jb} - \W{ij,ab}{\text{stat}},
	\\ 
	\label{eq:BSE-B0}
	\B{ia,jb}{(0)} & =  \kappa \ERI{ia}{bj} - \W{ib,aj}{\text{stat}}.
\end{align}
\end{subequations}
and 
\begin{subequations}
\begin{align}
	\label{eq:BSE-A1}
	\A{ia,jb}{(1)}(\Om{S}{}) & = - \tW{ij,ab}{}(\Om{S}{}) + \W{ij,ab}{\text{stat}},
	\\ 
	\label{eq:BSE-B1}
	\B{ia,jb}{(1)}(\Om{S}{}) & = -  \tW{ib,aj}{}(\Om{S}{}) + \W{ib,aj}{\text{stat}}.
\end{align}
\end{subequations}

According to perturbation theory, the $S$th BSE excitation energy and its corresponding eigenvector can then expanded as
\begin{subequations}
\begin{gather}
	\Om{S}{} = \Om{S}{(0)} + \Om{S}{(1)} + \ldots,
	\\
	\begin{pmatrix}
		\bX{S}{}	\\
		\bY{S}{}	\\
	\end{pmatrix}
	= 
	\begin{pmatrix}
		\bX{S}{(0)}	\\
		\bY{S}{(0)}	\\
	\end{pmatrix}
	+
	\begin{pmatrix}
		\bX{S}{(1)}	\\
		\bY{S}{(1)}	\\
	\end{pmatrix}
	+ \ldots.
\end{gather}
\end{subequations}
Solving the zeroth-order static problem
\begin{equation}
\label{eq:LR-BSE-stat}
	\begin{pmatrix}
		\bA{(0)}	&	\bB{(0)}	\\
		-\bB{(0)}	&	-\bA{(0)}	\\
	\end{pmatrix}
	\cdot
	\begin{pmatrix}
		\bX{S}{(0)}	\\
		\bY{S}{(0)}	\\
	\end{pmatrix}
	=
	\Om{S}{(0)}
	\begin{pmatrix}
		\bX{S}{(0)}	\\
		\bY{S}{(0)}	\\
	\end{pmatrix},
\end{equation}
yields the zeroth-order (static) $\Om{S}{(0)}$ excitation energies and their corresponding eigenvectors $\bX{S}{(0)}$ and $\bY{S}{(0)}$. 
Thanks to first-order perturbation theory, the first-order correction to the $S$th excitation energy is
\begin{equation}
\label{eq:Om1}
	\Om{S}{(1)} =
	\T{\begin{pmatrix}
		\bX{S}{(0)}	\\
		\bY{S}{(0)}	\\
	\end{pmatrix}}
	\cdot
	\begin{pmatrix}
		\bA{(1)}(\Om{S}{(0)})	&	\bB{(1)}(\Om{S}{(0)})	\\
		-\bB{(1)}(-\Om{S}{(0)})	&	-\bA{(1)}(-\Om{S}{(0)})	\\
	\end{pmatrix}
	\cdot
	\begin{pmatrix}
		\bX{S}{(0)}	\\
		\bY{S}{(0)}	\\
	\end{pmatrix}.
\end{equation}
From a practical point of view, if one enforces the dTDA, we obtain the very simple expression
\begin{equation}
\label{eq:Om1-TDA}
	\Om{S}{(1)} = \T{(\bX{S}{(0)})} \cdot \bA{(1)}(\Om{S}{(0)}) \cdot \bX{S}{(0)}.
\end{equation}
This correction can be renormalized by computing, at basically no extra cost, the renormalization factor which reads, in the dTDA,
\begin{equation}
\label{eq:Z}
	Z_{S} = \qty[ 1 - \T{(\bX{S}{(0)})} \cdot \left. \pdv{\bA{(1)}(\Om{S}{})}{\Om{S}{}} \right|_{\Om{S}{} = \Om{S}{(0)}} \cdot \bX{S}{(0)} ]^{-1}.
\end{equation}
This finally yields
\begin{equation}
	\Om{S}{\text{dyn}} = \Om{S}{\text{stat}} + \Delta\Om{S}{\text{dyn}} = \Om{S}{(0)} + Z_{S} \Om{S}{(1)}.
\end{equation}
with $\Om{S}{\text{stat}} \equiv \Om{S}{(0)}$ and $\Delta\Om{S}{\text{dyn}} = Z_{S} \Om{S}{(1)}$.
This is our final expression.

%%% FIG 1 %%%
%\begin{figure}
%	\includegraphics[width=\linewidth]{}
%\caption{
%\label{fig:}
%}
%\end{figure}
%%% %%% %%%

In terms of computational cost, if one decides to compute the dynamical correction of the $M$ lowest excitation energies, one must perform, first, a conventional (static) BSE calculation and extract the $M$ lowest eigenvalues and their corresponding eigenvectors [see Eq.~\eqref{eq:LR-BSE-stat}].
These are then used to compute the first-order correction from Eq.~\eqref{eq:Om1-TDA}, which also require to construct and evaluate the dynamical part of the BSE Hamiltonian for each excitation one wants to dynamically correct. 
The static BSE Hamiltonian is computed once during the static BSE calculation and does not dependent on the targeted excitation.

Searching iteratively for the lowest eigenstates, via Davidson's algorithm for instance, can be performed in $\order*{\Norb^4}$ computational cost.
Constructing the static and dynamic BSE Hamiltonians is much more expensive as it requires the complete diagonalization of the $(\Nocc \Nvir \times \Nocc \Nvir)$ RPA linear response matrix [see Eq.~\eqref{eq:LR-RPA}], which corresponds to a $\order*{\Nocc^3 \Nvir^3} = \order*{\Norb^6}$ computational cost. 
Although it might be reduced to $\order*{\Norb^4}$ operations with standard resolution-of-the-identity techniques, \cite{Duchemin_2019,Duchemin_2020} this step is the computational bottleneck in the current implementation.

%%%%%%%%%%%%%%%%%%%%%%%%
\section{Computational details}
\label{sec:compdet}
%%%%%%%%%%%%%%%%%%%%%%%%
All systems under investigation have closed-shell singlet ground states.
We then adopt a restricted formalism throughout this work.
The $GW$ calculations performed to obtain the screened Coulomb operator and the quasiparticle energies are done using a (restricted) HF starting point.
Perturbative $GW$ (or {\GOWO}) \cite{Hybertsen_1985a, Hybertsen_1986} quasiparticle energies are employed as starting points to compute the BSE neutral excitations.
These quasiparticle energies are obtained by linearizing the frequency-dependent quasiparticle equation, and the entire set of orbitals is corrected.
Further details about our implementation of {\GOWO} can be found in Refs.~\onlinecite{Loos_2018b,Veril_2018}.
Note that, for the present (small) molecular systems, {\GOWO}@HF and ev$GW$@HF yield similar quasiparticle energies and fundamental gap. 
Moreover, {\GOWO} allows to avoid rather laborious iterations as well as the significant additional computational effort of ev$GW$.
As one-electron basis sets, we employ the Dunning families (cc-pVXZ and aug-cc-pVXZ) defined with cartesian Gaussian functions. 
Finally, the infinitesimal $\eta$ is set to $100$ meV for all calculations.

For comparison purposes, we employ the theoretical best estimates (TBEs) and geometries of Refs.~\onlinecite{Loos_2018a,Loos_2019,Loos_2020b} from which CIS(D), \cite{Head-Gordon_1994,Head-Gordon_1995} ADC(2), \cite{Trofimov_1997,Dreuw_2015} CC2, \cite{Christiansen_1995a} CCSD, \cite{Purvis_1982} and CC3 \cite{Christiansen_1995b} excitation energies are also extracted. 
All the static and dynamic BSE calculations have been performed with the software \texttt{QuAcK}, \cite{QuAcK} freely available on \texttt{github}, where the present perturbative correction has been implemented.

%%%%%%%%%%%%%%%%%%%%%%%%
\section{Results and Discussion}
\label{sec:resdis}
%%%%%%%%%%%%%%%%%%%%%%%%

%%% TABLE I %%%
\begin{squeezetable}
\begin{table*}
	\caption{
	Singlet and triplet excitation energies (in eV) of \ce{N2} computed at the BSE@{\GOWO}@HF level for various basis sets.
	The dynamical correction is computed in the dTDA.
	\label{tab:N2}
	}
	\begin{ruledtabular}
		\begin{tabular}{lcddddddddd}
						&					&	\mc{3}{c}{cc-pVDZ ($\Eg^{\GW} = 20.71$ eV)}	
											&	\mc{3}{c}{cc-pVTZ ($\Eg^{\GW} = 20.21$ eV)}
											&	\mc{3}{c}{cc-pVQZ ($\Eg^{\GW} = 20.05$ eV)}		\\
											\cline{3-5} \cline{6-8} \cline{9-11}
			State		&	Nature			&	\tabc{$\Om{S}{\stat}$}	&	\tabc{$\Om{S}{\dyn}$}	&	\tabc{$\Delta\Om{S}{\dyn}$} 	
											&	\tabc{$\Om{S}{\stat}$}	&	\tabc{$\Om{S}{\dyn}$}	&	\tabc{$\Delta\Om{S}{\dyn}$}	
											&	\tabc{$\Om{S}{\stat}$}	&	\tabc{$\Om{S}{\dyn}$}	&	\tabc{$\Delta\Om{S}{\dyn}$}	\\
			\hline
			$^1\Pi_g(n \ra \pis)$			&	Val.	&	9.90	&	9.58	&	-0.32	&	9.92	&	9.53	&	-0.40	&	10.01	&	9.59	&	-0.42	\\
			$^1\Sigma_u^-(\pi \ra \pis)$	&	Val.	&	9.70	&	9.37	&	-0.33	&	9.61	&	9.19	&	-0.42	&	9.69	&	9.25	&	-0.44	\\
			$^1\Delta_u(\pi \ra \pis)$		&	Val.	&	10.37	&	10.05	&	-0.31	&	10.27	&	9.88	&	-0.39	&	10.34	&	9.93	&	-0.41	\\
			$^1\Sigma_g^+$					&	Ryd.	&	15.67	&	15.50	&	-0.17	&	15.04	&	14.84	&	-0.21	&	14.72	&	14.43	&	-0.21	\\
			$^1\Pi_u$						&	Ryd.	&	15.00	&	14.79	&	-0.21	&	14.75	&	14.48	&	-0.27	&	14.80	&	14.59	&	-0.29	\\
			$^1\Sigma_u^+$					&	Ryd.	&	22.88\fnm[1]	&	22.73	&	-0.15	&	19.03	&	18.95	&	-0.08	&	16.78	&	16.71	&	-0.06	\\  
			$^1\Pi_u$						&	Ryd.	&	23.62\fnm[1]	&	23.51	&	-0.11	&	19.15	&	19.04	&	-0.11	&	16.93	&	16.85	&	-0.09	\\  
			\\
			$^3\Sigma_u^+(\pi \ra \pis)$	&	Val.	&	7.39	&	6.91	&	-0.48	&	7.46	&	6.87	&	-0.59	&	7.59	&	6.97	&	-0.62	\\  
			$^3\Pi_g(n \ra \pis)$			&	Val.	&	8.07	&	7.65	&	-0.42	&	8.14	&	7.62	&	-0.52	&	8.24	&	7.70	&	-0.54	\\  
			$^3\Delta_u(\pi \ra \pis)$		&	Val.	&	8.56	&	8.15	&	-0.41	&	8.52	&	8.00	&	-0.52	&	8.62	&	8.07	&	-0.55	\\  
			$^3\Sigma_u^-(\pi \ra \pis)$	&	Val.	&	9.70	&	9.37	&	-0.33	&	9.61	&	9.19	&	-0.42	&	9.69	&	9.25	&	-0.44	\\
			\hline
			\\
						&					&	\mc{3}{c}{aug-cc-pVDZ ($\Eg^{\GW} = 19.49$ eV)}	
											&	\mc{3}{c}{aug-cc-pVTZ ($\Eg^{\GW} = 19.20$ eV)}
											&	\mc{3}{c}{aug-cc-pVQZ ($\Eg^{\GW} = 19.00$ eV)}		\\
											\cline{3-5} \cline{6-8} \cline{9-11}
			State		&	Nature			&	\tabc{$\Om{S}{\stat}$}	&	\tabc{$\Om{S}{\dyn}$}	&	\tabc{$\Delta\Om{S}{\dyn}$} 	
											&	\tabc{$\Om{S}{\stat}$}	&	\tabc{$\Om{S}{\dyn}$}	&	\tabc{$\Delta\Om{S}{\dyn}$}	
											&	\tabc{$\Om{S}{\stat}$}	&	\tabc{$\Om{S}{\dyn}$}	&	\tabc{$\Delta\Om{S}{\dyn}$}	\\
			\hline
			$^1\Pi_g(n \ra \pis)$			&	Val.	&	10.18	&	9.77	&	-0.41	&	10.42	&	9.99	&	-0.42	&	10.52	&	10.09	&	-0.43	\\
			$^1\Sigma_u^-(\pi \ra \pis)$	&	Val.	&	9.95	&	9.51	&	-0.44	&	10.11	&	9.66	&	-0.45	&	10.20	&	9.75	&	-0.45	\\
			$^1\Delta_u(\pi \ra \pis)$		&	Val.	&	10.57	&	10.16	&	-0.41	&	10.75	&	10.33	&	-0.42	&	10.85	&	10.42	&	-0.42	\\
			$^1\Sigma_g^+$					&	Ryd.	&	13.72	&	13.68	&	-0.04	&	13.60	&	13.57	&	-0.03	&	13.54	&	13.52	&	-0.02	\\
			$^1\Pi_u$						&	Ryd.	&	14.07	&	14.02	&	-0.05	&	13.98	&	13.94	&	-0.04	&	13.96	&	13.93	&	-0.03	\\
			$^1\Sigma_u^+$					&	Ryd.	&	13.80	&	13.72	&	-0.08	&	13.98	&	13.91	&	-0.07	&	14.08	&	14.03	&	-0.06	\\  
			$^1\Pi_u$						&	Ryd.	&	14.22	&	14.19	&	-0.04	&	14.24	&	14.21	&	-0.03	&	14.26	&	14.23	&	-0.03	\\  
			\\
			$^3\Sigma_u^+(\pi \ra \pis)$	&	Val.	&	7.75	&	7.12	&	-0.63	&	8.02	&	7.38	&	-0.64	&	8.12	&	7.48	&	-0.64	\\  
			$^3\Pi_g(n \ra \pis)$			&	Val.	&	8.42	&	7.88	&	-0.54	&	8.66	&	8.10	&	-0.56	&	8.75	&	8.20	&	-0.56	\\  
			$^3\Delta_u(\pi \ra \pis)$		&	Val.	&	8.86	&	8.32	&	-0.54	&	9.04	&	8.48	&	-0.56	&	9.14	&	8.57	&	-0.56	\\  
			$^3\Sigma_u^-(\pi \ra \pis)$	&	Val.	&	9.95	&	9.51	&	-0.44	&	10.11	&	9.66	&	-0.45	&	10.20	&	9.75	&	-0.45	\\
		\end{tabular} 
	\end{ruledtabular}
		\fnt[1]{Excitation energy larger than the fundamental gap.}		
\end{table*}
\end{squeezetable}

First, we investigate the basis set dependency of the dynamical correction.
Note that, in the present calculations, the zeroth-order Hamiltonian is always the ``full'' BSE static Hamiltonian, \ie, without TDA.
The singlet and triplet excitation energies of the nitrogen molecule \ce{N2} computed at the BSE@{\GOWO}@HF level for the aug-cc-pVDZ, aug-cc-pVTZ, and aug-cc-pVQZ basis sets are reported in Table \ref{tab:N2}, where we also report the $GW$ gap, $\Eg^{\GW}$, to show that corrected transitions are usually well below this gap.
The \ce{N2} molecule is a very convenient example as it contains $n \ra \pis$ and $\pi \ra \pis$ valence excitations as well as Rydberg transitions.
As we shall further illustrate below, the magnitude of the dynamical correction is characteristic of the type of transitions.
One key result of the present investigation is that the dynamical correction is quite basis set insensitive with a maximum variation of $0.03$ eV between in smallest (aug-cc-pVDZ) and largest (aug-cc-pVQZ) basis sets.
This is quite a nice feature as one does not need to compute this more expensive correction in a very large basis.
%The second important observation extracted from the results gathered in Table \ref{tab:N2} is that the dTDA is a rather satisfactory approximation, especially for the singlet states where one observes a maximum discrepancy of $0.03$ eV between the ``full'' and dTDA excitation energies.
%The story is different for the triplet states for which deviations of the order of $0.3$ eV is observed between the two sets, the dTDA of excitation energies being, as we shall see later on, more accurate.
%Indeed, although the dynamical correction systematically red-shift the excitation energies (as anticipated in Sec.~\ref{sec:dynW}), taking into account the coupling between the resonant and anti-resonant parts of the BSE Hamiltonian [see Eq.~\eqref{eq:BSE-final}] yields a systematic blue-shift of the correction, the overall correction remaining negative but by a smaller amount.
%This outcome is similar to the conclusions of several benchmark studies \cite{Jacquemin_2017b,Rangel_2017} which clearly concluded that static BSE triplet excitations are notably too low in energy and that the use of the TDA is able to partly reduce this error. 
%In accordance with the success of the dTDA, the remaining calculations of the present study are performed within this approximation.

%%% TABLE I %%%
\begin{squeezetable}
\begin{table*}
	\caption{
	Singlet excitation energies (in eV) for various molecules obtained with the aug-cc-pVTZ basis set computed at various levels of theory.
	The dynamical correction is computed in the dTDA.
	CT stands for charge transfer.
	\label{tab:BigTabSi}
	}
	\begin{ruledtabular}
		\begin{tabular}{llcdddddddddd}
						&			&			&	\mc{5}{c}{BSE@{\GOWO}@HF}	&	\mc{5}{c}{Wave function-based methods}	\\ 
													\cline{4-8}	\cline{9-13}
			Mol.		&	State	&	Nature	&	\tabc{$\Eg^{\GW}$}	&	\tabc{$\Om{S}{\stat}$}	&	\tabc{$\Om{S}{\dyn}$}	&	\tabc{$\Delta\Om{S}{\dyn}$}	&	\tabc{$Z_{S}$}	
									&	\tabc{CIS(D)}	&	\tabc{ADC(2)}	&	\tabc{CCSD}	&	\tabc{CC2}	&	\tabc{TBE}	\\
			\hline
			\ce{HCl}	&	$^1\Pi$							&	CT		&	13.43	&	8.30	&	8.19	&	-0.11	&	1.009	
																		&	6.07	&	7.97	&	7.91	&	7.96	&	7.84	\\
			\\
			\ce{H2O}	&	$^1B_1(n \ra 3s)$				&	Ryd.	&	13.58	&	8.09	&	8.00	&	-0.09	&	1.007	
																		&	7.62	&	7.18	&	7.60	&	7.23	&	7.17	\\
						&	$^1A_2(n \ra 3p)$				&	Ryd.	&			&	9.79	&	9.72	&	-0.07	&	1.005	
																		&	9.41	&	8.84	&	9.36	&	8.89	&	8.92	\\
						&	$^1A_1(n \ra 3s)$				&	Ryd.	&			&	10.42	&	10.35	&	-0.07	&	1.006	
																		&	9.99	&	9.52	&	9.96	&	9.58	&	9.52	\\
			\\
			\ce{N2}		&	$^1\Pi_g(n \ra \pis)$			&	Val.	&	19.20	&	10.42	&	9.99	&	-0.42	&	1.031	
																		&	9.66	&	9.48	&	9.41	&	9.44	&	9.34	\\
						&	$^1\Sigma_u^-(\pi \ra \pis)$	&	Val.	&			&	10.11	&	9.66	&	-0.45	&	1.029	
																		&	10.31	&	10.26	&	10.00	&	10.32	&	9.88	\\
						&	$^1\Delta_u(\pi \ra \pis)$		&	Val.	&			&	10.75	&	10.33	&	-0.42	&	1.030	
																		&	10.85	&	10.79	&	10.44	&	10.86	&	10.29	\\
						&	$^1\Sigma_g^+$					&	Ryd.	&			&	13.60	&	13.57	&	-0.03	&	1.003	
																		&	13.67	&	12.99	&	13.15	&	12.83	&	12.98	\\
						&	$^1\Pi_u$						&	Ryd.	&			&	13.98	&	13.94	&	-0.04	&	1.004	
																		&	13.64	&	13.32	&	13.43	&	13.15	&	13.03	\\
						&	$^1\Sigma_u^+$					&	Ryd.	&			&	13.98	&	13.91	&	-0.07	&	1.008	
																		&	13.75	&	13.07	&	13.26	&	12.89	&	13.09	\\
						&	$^1\Pi_u$						&	Ryd.	&			&	14.24	&	14.21	&	-0.03	&	1.002	
																		&	14.52	&	14.00	&	13.67	&	13.96	&	13.46	\\
			\\
			\ce{CO}		&	$^1\Pi(n \ra \pis)$				&	Val.	&	16.46	&	9.54	&	9.19	&	-0.34	&	1.029	
																		&	8.78	&	8.69	&	8.59	&	8.64	&	8.49	\\
						&	$^1\Sigma^-(\pi \ra \pis)$		&	Val.	&			&	10.25	&	9.90	&	-0.35	&	1.023	
																		&	10.13	&	10.03	&	9.99	&	10.30	&	9.92	\\
						&	$^1\Delta(\pi \ra \pis)$		&	Val.	&			&	10.71	&	10.39	&	-0.32	&	1.023	
																		&	10.41	&	10.30	&	10.12	&	10.60	&	10.06	\\
						&	$^1\Sigma^+$					&	Ryd.	&			&	11.88	&	11.85	&	-0.03	&	1.005	
																		&	11.48	&	11.32	&	11.22	&	11.11	&	10.95	\\
						&	$^1\Sigma^+$					&	Ryd.	&			&	12.39	&	12.37	&	-0.02	&	1.003	
																		&	11.71	&	11.83	&	11.75	&	11.63	&	11.52	\\
						&	$^1\Pi$							&	Ryd.	&			&	12.37	&	12.32	&	-0.05	&	1.004	
																		&	12.06	&	12.03	&	11.96	&	11.83	&	11.72	\\
			\\
			\ce{HNO}	&	$^1A''(n \ra \pis)$				&	Val.	&	11.71	&	2.46	&	1.98	&	-0.48	&	1.035	
																		&	1.80	&	1.68	&	1.76	&	1.74	&	1.74	\\	
						&	$^1A'$							&	Ryd.	&			&	7.05	&	7.01	&	-0.04	&	1.003	
																		&	5.81	&	5.73	&	6.30	&	5.72	&	6.27	\\
			\\
			\ce{C2H2}	&	$^1\Sigma_{u}^-(\pi \ra \pis)$	&	Val.	&	12.28	&	7.37	&	7.05	&	-0.32	&	1.026			
																		&	7.28	&	7.24	&	7.15	&	7.26	&	7.10	\\
						&	$^1\Delta_{u}(\pi \ra \pis)$	&	Val.	&			&	7.74	&	7.46	&	-0.29	&	1.025		
																		&	7.62	&	7.56	&	7.48	&	7.59	&		7.44\\
			\\
			\ce{C2H4}	&	$^1B_{3u}(\pi \ra 3s)$			&	Ryd.	&	11.49	&	7.64	&	7.62	&	-0.03	&	1.004	
																		&	7.35	&	7.34	&	7.42	&	7.29	&	7.39	\\
						&	$^1B_{1u}(\pi \ra \pis)$		&	Val.	&			&	8.18	&	8.03	&	-0.15	&	1.022	
																		&	7.95	&	7.91	&	8.02	&	7.92	&	7.93	\\
						&	$^1B_{1g}(\pi \ra 3p)$			&	Ryd.	&			&	8.29	&	8.26	&	-0.03	&	1.003	
																		&	8.01	&	7.99	&	8.08	&	7.95	&	8.08	\\
			\\
			\ce{CH2O}	&	$^1A_2(n \ra \pis)$				&	Val.	&	12.00	&	5.03	&	4.68	&	-0.35	&	1.027	
																		&	4.04	&	3.92	&	4.01	&	4.07	&	3.98	\\
						&	$^1B_2(n \ra 3s)$				&	Ryd.	&			&	7.87	&	7.85	&	-0.02	&	1.001	
																		&	6.64	&	6.50	&	7.23	&	6.56	&	7.23	\\
						&	$^1B_2(n \ra 3p)$				&	Ryd.	&			&	8.76	&	8.72	&	-0.04	&	1.003	
																		&	7.56	&	7.53	&	8.12	&	7.57	&	8.13	\\
						&	$^1A_1(n \ra 3p)$				&	Ryd.	&			&	8.85	&	8.84	&	-0.01	&	1.000	
																		&	8.16	&	7.47	&	8.21	&	7.52	&	8.23	\\
						&	$^1A_2(n \ra 3p)$				&	Ryd.	&			&	8.87	&	8.85	&	-0.02	&	1.002	
																		&	8.04	&	7.99	&	8.65	&	8.04	&	8.67	\\
						&	$^1B_1(\si \ra \pis)$			&	Val.	&			&	10.18	&	9.77	&	-0.42	&	1.032	
																		&	9.38	&	9.17	&	9.28	&	9.32	&	9.22	\\
						&	$^1A_1(\pi \ra \pis)$			&	Val.	&			&	10.05	&	9.81	&	-0.24	&	1.026	
																		&	9.08	&	9.46	&	9.67	&	9.54	&	9.43	\\
			\hline
			MAE			&									&			&			&	0.65		&	0.50		&			&			
																		&	0.41	&	0.24		&	0.14		&	0.25	&	0.00		\\
			MSE			&									&			&			&	0.65		&	0.48		&			&			
																		&	0.12	&	0.00		&	0.13		&	0.00	&	0.00		\\
		\end{tabular} 
	\end{ruledtabular}
\end{table*}
\end{squeezetable}

%%% TABLE II %%%
\begin{squeezetable}
\begin{table*}
	\caption{
	Triplet excitation energies (in eV) for various molecules obtained with the aug-cc-pVTZ basis set computed at various levels of theory.
	The dynamical correction is computed in the dTDA.
	\label{tab:BigTabTr}
	}
	\begin{ruledtabular}
		\begin{tabular}{llcdddddddddd}
						&			&			&	\mc{5}{c}{BSE@{\GOWO}@HF}	&	\mc{5}{c}{Wave function-based methods}	\\
													\cline{4-8}	\cline{9-13}
			Mol.		&	State	&	Nature	&	\tabc{$\Eg^{\GW}$}	&	\tabc{$\Om{S}{\stat}$}	&	\tabc{$\Om{S}{\dyn}$}	&	\tabc{$\Delta\Om{S}{\dyn}$}	&	\tabc{$Z_{S}$}	
									&	\tabc{CIS(D)}	&	\tabc{ADC(2)}	&	\tabc{CCSD}	&	\tabc{CC2}	&	\tabc{TBE}	\\
			\hline
			\ce{H2O}	&	$^3B_1(n \ra 3s)$				&	Ryd.	&	13.58	&	7.62	&	7.48	&	-0.14	&	1.009	
																		&	7.25	&	6.86	&	7.20	&	6.91	&	6.92	\\
						&	$^3A_2(n \ra 3p)$				&	Ryd.	&			&	9.61	&	9.50	&	-0.11	&	1.007	
																		&	9.24	&	8.72	&	9.20	&	8.77	&	8.91	\\
						&	$^3A_1(n \ra 3s)$				&	Ryd.	&			&	9.80	&	9.66	&	-0.14	&	1.008	
																		&	9.54	&	9.15	&	9.49	&	9.20	&	9.30	\\
			\\
			\ce{N2}		&	$^3\Sigma_u^+(\pi \ra \pis)$	&	Val.	&	19.20	&	8.02	&	7.38	&	-0.64	&	1.032	
																		&	8.20	&	8.15	&	7.66	&	8.19	&	7.70	\\
						&	$^3\Pi_g(n \ra \pis)$			&	Val.	&			&	8.66	&	8.10	&	-0.56	&	1.031	
																		&	8.33	&	8.20	&	8.09	&	8.19	&	8.01	\\
						&	$^3\Delta_u(\pi \ra \pis)$		&	Val.	&			&	9.04	&	8.48	&	-0.56	&	1.031	
																		&	9.30	&	9.25	&	8.91	&	9.30	&	8.87	\\
						&	$^3\Sigma_u^-(\pi \ra \pis)$	&	Val.	&			&	10.11	&	9.66	&	-0.45	&	1.029	
																		&	10.29	&	10.23	&	9.83	&	10.29	&	9.66	\\
			\\
			\ce{CO}		&	$^3\Pi(n \ra \pis)$				&	Val.	&	16.46	&	6.80	&	6.25	&	-0.55	&	1.031	
																		&	6.51	&	6.45	&	6.36	&	6.42	&	6.28	\\
						&	$^3\Sigma^+(\pi \ra \pis)$		&	Val.	&			&	8.56	&	8.06	&	-0.50	&	1.025	
																		&	8.63	&	8.54	&	8.34	&	8.72	&	8.45	\\
						&	$^3\Delta(\pi \ra \pis)$		&	Val.	&			&	9.39	&	8.96	&	-0.43	&	1.024	
																		&	9.44	&	9.33	&	9.23	&	9.56	&	9.27	\\
						&	$^3\Sigma_u^-(\pi \ra \pis)$	&	Val.	&			&	10.25	&	9.90	&	-0.35	&	1.023	
																		&	10.10	&	10.01	&	9.81	&	10.27	&	9.80	\\
						&	$^3\Sigma_u^+$					&	Ryd.	&			&	11.17	&	11.07	&	-0.10	&	1.008	
																		&	10.98	&	10.83	&	10.71	&	10.60	&	10.47	\\
			\\
			\ce{HNO}	&	$^3A''(n \ra \pis)$				&	Val.	&	11.71	&	1.27	&	0.67	&	-0.60	&	1.036	
																		&	0.91	&	0.78	&	0.85	&	0.84	&	0.88	\\
						&	$^3A'(\pi \ra \pis)$			&	Val.	&			&	5.55	&	4.87	&	-0.69	&	1.037	
																		&	5.72	&	5.46	&	5.49	&	5.44	&	5.61	\\
			\\
			\ce{C2H2}	&	$^3\Sigma_{u}^+(\pi \ra \pis)$	&	Val.	&	12.28	&	5.83	&	5.32	&	-0.51	&	1.031	
																		&	5.79	&	5.75	&	5.45	&	5.76	&	5.53	\\
						&	$^3\Delta_{u}(\pi \ra \pis)$	&	Val.	&			&	6.64	&	6.23	&	-0.41	&	1.028	
																		&	6.62	&	6.57	&	6.41	&	6.60	&	6.40	\\
						&	$^3\Sigma_{u}^-(\pi \ra \pis)$	&	Val.	&			&	7.37	&	7.05	&	-0.32	&	1.026	
																		&	7.31	&	7.27	&	7.12	&	7.29	&	7.08	\\
			\\
			\ce{C2H4}	&	$^3B_{1u}(\pi \ra \pis)$		&	Val.	&	11.49	&	4.95	&	4.49	&	-0.46	&	1.032	
																		&	4.62	&	4.59	&	4.46	&	4.59	&	4.54	\\
						&	$^3B_{3u}(\pi \ra 3s)$			&	Ryd.	&			&	7.46	&	7.42	&	-0.04	&	1.004	
																		&	7.26	&	7.23	&	7.29	&	7.19	&	7.23	\\
						&	$^3B_{1g}(\pi \ra 3p)$			&	Ryd.	&			&	8.23	&	8.19	&	-0.04	&	1.004	
																		&	7.97	&	7.95	&	8.03	&	7.91	&	7.98	\\
			\\
			\ce{CH2O}	&	$^3A_2(n \ra \pis)$				&	Val.	&	12.00	&	4.28	&	3.87	&	-0.40	&	1.027	
																		&	3.58	&	3.46	&	3.56	&	3.59	&	3.58	\\
						&	$^3A_1(\pi \ra \pis)$			&	Val.	&			&	6.31	&	5.75	&	-0.56	&	1.033	
																		&	6.27	&	6.20	&	5.97	&	6.30	&	6.06	\\
						&	$^3B_2(n \ra 3s)$				&	Ryd.	&			&	7.60	&	7.56	&	-0.05	&	1.002	
																		&	6.66	&	6.39	&	7.08	&	6.44	&	7.06	\\
			\hline
			MAE			&									&			&			&	0.39	&	0.29	&			&			
																		&	0.25	&	0.21	&	0.09	&	0.22	&	0.00		\\
			MSE			&									&			&			&	0.39	&	0.01	&			&			
																		&	0.21	&	0.08	&	0.04	&	0.12	&	0.00		\\
		\end{tabular} 
	\end{ruledtabular}
\end{table*}
\end{squeezetable}

Tables \ref{tab:BigTabSi} and \ref{tab:BigTabTr} report, respectively, singlet and triplet excitation energies for various molecules computed at the BSE@{\GOWO}@HF level and with the aug-cc-pVTZ basis set.
For comparative purposes, excitation energies obtained with the same basis set and several second-order wave function methods [CIS(D), ADC(2), CCSD, and CC2] are also reported.
Finally, the highly-accurate TBEs of Refs.~\onlinecite{Loos_2018a,Loos_2019,Loos_2020b} will serve us as reference.
For each excitation, we report the static and dynamic excitation energies, $\Om{S}{\stat}$ and $\Om{S}{\dyn}$, as well as the value of the renormalization factor $Z_S$ defined in Eq.~\eqref{eq:Z}.
As one can see in Tables \ref{tab:BigTabSi} and \ref{tab:BigTabTr}, the value of $Z_S$ is always quite close to unity which shows that the perturbative expansion behaves nicely, and that a first-order correction is probably quite a good estimate of the non-perturbative result.
Moreover, we have observed that an iterative, self-consistent resolution [where the dynamically-corrected excitation energies are re-injected in Eq.~\eqref{eq:Om1}] yields basically the same results as its (cheaper) renormalized version.

%%% TABLE I %%%
\begin{squeezetable}
\begin{table}
	\caption{
	Singlet and triplet excitation energies (in eV) for various molecules obtained with the aug-cc-pVDZ basis set computed at various levels of theory.
	The dynamical correction is computed in the dTDA.
	\label{tab:BigMol}
	}
	\begin{ruledtabular}
		\begin{tabular}{llcdddddd}
						&		&	&	\mc{5}{c}{BSE@{\GOWO}@HF}	\\ 
													\cline{4-8}	
			Molecule			&	State		&	Nature	&	\tabc{$\Eg^{\GW}$}	&	\tabc{$\Om{S}{\stat}$}	&	\tabc{$\Om{S}{\dyn}$}	&	\tabc{$\Delta\Om{S}{\dyn}$}	&	\tabc{$Z_{S}$}	& \tabc{CC3}		\\
			\hline
			acrolein			&	$^1A''(n \ra \pis)$		&	Val.	&	11.67	&	4.62	&	4.28	&	-0.35	&	1.030	&	3.77		\\
								&	$^1A'(n \ra \pis)$		&	Val.	&			&	6.86	&	6.70	&	-0.16	&	1.023	&	6.67		\\
								&	$^1A'(n \ra 3s)$		&	Ryd.	&			&	7.57	&	7.53	&	-0.04	&	1.004	&	6.99		\\
			\\
								&	$^3A''(n \ra \pis)$		&	Val.	&			&	3.97		&	3.54	&	-0.43	&	1.031	&	3.47		\\
								&	$^3A'(\pi \ra \pis)$	&	Val.	&			&	4.03		&	3.61	&	-0.42	&	1.032	&	3.95		\\
			\\
			butadiene			&	$^1B_u(\pi \ra \pis)$	&	Val.	&	9.88	&	6.25	&	6.13	&	-0.12	&	1.019	&	6.25		\\
								&	$^1A_g(\pi \ra \pis)$	&	Val.	&			&	6.88	&	6.86	&	-0.03	&	1.003	&	6.68		\\
			\\
								&	$^3B_u(\pi \ra \pis)$	&	Val.	&			&	3.68	&	3.25	&	-0.43	&	1.032	&	3.36		\\
								&	$^3A_g(\pi \ra \pis)$	&	Val.	&			&	5.51	&	5.01	&	-0.50	&	1.040	&	5.21		\\
								&	$^3B_g(\pi \ra 3s)$		&	Ryd.	&			&	6.29	&	6.25	&	-0.04	&	1.005	&	6.20		\\
			\\
			diacetylene			&	$^1\Sigma_u^-(\pi \ra \pis)$	&	Val.	&	11.01	&	5.62	&	5.35	&	-0.28	&	1.025	&	5.44		\\
								&	$^1\Delta_u(\pi \ra \pis)$		&	Val.	&			&	5.87	&	5.63	&	-0.25	&	1.024	&	5.69		\\
			\\
								&	$^3\Sigma_u^+(\pi \ra \pis)$	&	Val.	&			&	4.30	&	3.82	&	-0.49	&	1.031	&	4.06		\\
								&	$^3\Delta_u(\pi \ra \pis)$		&	Val.	&			&	5.04	&	4.68	&	-0.36	&	1.027	&	4.86		\\
			\\
			glyoxal				&	$^1A_u(n \ra \pis)$		&	Val.		&	10.90	&	3.46	&	3.14	&	-0.33	&	1.028	&	2.90		\\
								&	$^1B_g(n \ra \pis)$		&	Val.		&			&	4.96	&	4.55	&	-0.41	&	1.034	&	4.30		\\
								&	$^1B_u(n \ra 3p)$		&	Ryd.		&			&	7.90	&	7.86	&	-0.04	&	1.004	&	7.55		\\
			\\
								&	$^3A_u(n \ra \pis)$		&	Val.		&			&	2.77	&	2.38	&	-0.39	&	1.028	&	2.49		\\
								&	$^3B_g(n \ra \pis)$		&	Val.		&			&	4.23	&	3.75	&	-0.48	&	1.034	&	3.91		\\
								&	$^3B_u(\pi \ra \pis)$	&	Val.		&			&	5.01	&	4.47	&	-0.55	&	1.034	&	5.20		\\
			\\
			streptocyanine		&	$^1B_2(\pi \ra \pis)$	&	Val.		&	13.79	&	7.66	&	7.51	&	-0.15	&	1.019	&	7.14	\\
			\hline
			MAE					&							&				&			&	0.32	&	0.30	&			&			&	0.00		\\
			MSE					&							&				&			&	0.23	&	0.00	&			&			&	0.00		\\
		\end{tabular} 
	\end{ruledtabular}
\end{table}
\end{squeezetable}


%%%%%%%%%%%%%%%%%%%%%%%%
\section{Conclusion}
\label{sec:conclusion}
%%%%%%%%%%%%%%%%%%%%%%%%
This is the conclusion

%%%%%%%%%%%%%%%%%%%%%%%%
%\section*{Supplementary material}
%%%%%%%%%%%%%%%%%%%%%%%%
%See the {\SI} for a detailed derivation of the dynamical BSE equation.

%%%%%%%%%%%%%%%%%%%%%%%%
\acknowledgements{
%%%%%%%%%%%%%%%%%%%%%%%%
PFL thanks the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme (Grant agreement No.~863481) for financial support.
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 study has been (partially) supported through the EUR grant NanoX No.~ANR-17-EURE-0009 in the framework of the \textit{``Programme des Investissements d'Avenir''.}
\titou{The authors would like to thank Elisa Rebolini for insightful discussions.}

%%%%%%%%%%%%%%%%%%%%%%%%
\section*{Data availability}
%%%%%%%%%%%%%%%%%%%%%%%%
The data that support the findings of this study are available within the article and its {\SI}.

\appendix

\section{$L_0(1,4; 1',3)$ $(t_1)$-time Fourier transform}
\label{app:A}

In this Appendix, we derive Eqs.~\eqref{eq:iL0} to \eqref{eq:iL0bis}.
Defining the $t_1$-time Fourier transform of $L_0(1,3;4,1')$ with
$(t_{1'} = t_1^{+})$
\begin{align}  
	[L_0](\bx_1,4;\bx_{1'},3 \; | \; \omega_1 ) = -i
	\int dt_1 e^{i \omega_1 t_1 } G(1,3)G(4,1')
\end{align}
 we plug-in the Fourier expansion of the Green's function, e.g.
\begin{align*}
     G(1,3) =  \int \frac{ d\omega }{ 2\pi } G(\bx_1,\bx_3;\omega) e^{-i \omega \tau_{13} }
\end{align*}
with $\tau_{13} = (t_1-t_3)$ to obtain:
\begin{multline}  
	[L_0](\bx_1,4;\bx_{1'},3 \;| \;  \omega_1 )   =
	\\
	\int \frac{ d\omega }{ 2i\pi } \;  G(\bx_1,\bx_3;\omega) \; G(\bx_4,\bx_{1'};\omega-\omega_1)  
	  e^{ i \omega t_3 }	e^{-i (\omega-\omega_1) t_4 } \nonumber
\end{multline}
With the change of variable $\omega \to \omega + {\omega_1}/2$ one obtains readily
\begin{multline}  
	[L_0](\bx_1,4;\bx_{1'},3 \; | \; \omega_1 )     = 
	\\
	e^{ i  \omega_1  t^{34} }
	\int \frac{ d\omega }{ 2i\pi } \;   G\qty(\bx_1,\bx_3;\omega+ \frac{\omega_1}{2} )  G\qty(\bx_4,\bx_{1'};\omega-\frac{\omega_1}{2} )  \;
  e^{ i \omega \tau_{34} }	
\end{multline}
   with $\tau_{34} = t_3 - t_4$ and $t^{34}= (t_3+t_4)/2$.
  Using now the Lehman representation of the Green's functions (Eq.~\eqref{eq:G-Lehman}), and picking up the poles associated with the occupied (virtual) states in the upper (lower)  half-plane for $\tau_{34} > 0$ ($\tau_{34} < 0$), one obtains using the residue theorem (with $\tau = \tau_{34})$
\begin{equation}
\begin{split}
    & \int \frac{ d \omega }{2i\pi}    \; G\qty(\bx_1,\bx_3; \omega +  \homu  )    G\qty(\bx_4,\bx_{1'}; \omega - \homu  )  e^{  i    \omega  \tau    } 
      \\
         & = \sum_{bj}   
     \frac{ \phi_b(\bx_1) \phi_b^*(\bx_3) \phi_j(\bx_4) \phi_j^*(\bx_{1'})} { \omega_1   - (\e{b} - \e{j}) + i\eta    }  
   \qty[  \theta(\tau) e^{i ( \e{j} + \homu ) \tau }  + \theta(-\tau) e^{i ( \e{b} - \homu ) \tau } ] 
  \\
  & -  \sum_{bj} \frac{ \phi_j(\bx_1) \phi_j^*(\bx_3) \phi_b(\bx_4) \phi_b^*(\bx_{1'})} {  \omega_1   + (\e{b} - \e{j} ) -i\eta } 
     \qty[  \theta(\tau)  e^{i ( \e{j} - \homu ) \tau } + \theta(-\tau)  e^{i ( \e{b} + \homu ) \tau }    ] 
     \\
 & +  \sum_{ab} \text{ pp terms }  + \sum_{ij} \text{ hh terms }
\end{split}
\end{equation}
where (pp) and (hh) labels particle-particle and hole-hole channels neglected here. 
Projecting onto $\phi_a^*(\bx_1) \phi_i(\bx_{1'})$   selects the first line  of the RHS, leading to Eq.~\eqref{eq:iL0bis}
with $ (\Om{1}{} \to \Om{s}{} )$.

\section{ $\mel{N}{T [\hpsi(6) \hpsi^{\dagger}(5)] }{N,S}$ in the electron/hole product basis  }
\label{app:B}

We now derive in some more details Eq.~\eqref{eq:spectral65}. 
Starting with
\begin{equation}
\begin{split}
	\mel{N}{T [\hpsi(6) \hpsi^{\dagger}(5)] }{N,S}
	& = \theta(+\tau_{65}) \mel{N}{ \hpsi(6) \hpsi^{\dagger}(5)  }{N,S} 
	\\
	& - \theta(-\tau_{65}) \mel{N}{  \hpsi^{\dagger}(5) \hpsi(6) }{N,S}
\end{split}
\end{equation}
we use the relation between operators in their Heisenberg and Schr\"{o}dinger representations [see Eq.~\eqref{Eisenberg}] to obtain
\begin{equation}
\begin{split}
	& \mel{N}{T [\hpsi(6) \hpsi^{\dagger}(5)]}{N,S} = \\
	& +  \theta(\tau_{65}) \mel{N}{ \hpsi(\bx_6) e^{-i\hH \tau_{65}} \hpsi^{\dagger}(\bx_5)  }{N,S} e^{ i E^N_0 t_6 } e^{ - i E^N_S t_5 }
	\\
	& - \theta(-\tau_{65}) \mel{N}{  \hpsi^{\dagger}(\bx_5) e^{ i\hH \tau_{65}} \hpsi(\bx_6) }{N,S} e^{ i E^N_0 t_5 } e^{ - i E^N_S t_6 }
\end{split}
\end{equation}
with $E^N_0$ the $N$-electron ground-state energy and $E^N_S$ the energy of the $S$th excited state $\ket{N,S}$. 
Expanding now the field operators with creation/destruction operators in the orbital basis   
\begin{subequations}
\begin{align}
    \hpsi(\bx_6)  & = \sum_p \phi_p(\bx_6) \ha_p 
    \\ 
    \hpsi^{\dagger}(\bx_5)  & = \sum_q \phi_q^{*}(\bx_5) \ha^{\dagger}_q
\end{align}
\end{subequations}
one gets
\begin{equation}
\begin{split}
     \mel{N}{T [\hpsi(6) \hpsi^{\dagger}(5)]}{N,S}
     \\
     = \sum_{pq} \phi_p(\bx_6)  \phi_q^{*}(\bx_5) 
     [ & \theta(+\tau_{65}) \mel{N}{ \ha_p e^{-i \hH \tau_{65}} \ha^{\dagger}_q  }{N,S} e^{ i E^N_0 t_6 } e^{ - i E^N_S t_5 }  
     \\
      - & \theta(-\tau_{65}) \mel{N}{  \ha^{\dagger}_q e^{ i \hH \tau_{65}} \ha_p }{N,S} e^{ i E^N_0 t_5 } e^{ - i E^N_S t_6 } ]
\end{split}
\end{equation}
We now act on the $N$-electron ground-state with
\begin{subequations}
\begin{align}
  e^{+i\hH \tau_{65} } \ha^{\dagger}_p \ket{N} &=
    e^{+i \qty( E^N_0 + \e{p} ) \tau_{65} } \ket{N} 
    \\
      e^{ -i\hH \tau_{65} } \ha_q \ket{N} &=
    e^{-i \qty( E^N_0 - \e{q} ) \tau_{65} } \ket{N}
\end{align}
\end{subequations}
where $\lbrace \e{p/q} \rbrace$ are quasiparticle energies, such as the $GW$ ones, namely proper addition/removal energies. 
Taking the associated  bras that we plug into the orbital product basis expansion of $\mel{N}{T [\hpsi(6) \hpsi^{\dagger}(5)]}{N,S}$  one obtains:
\begin{equation}
\begin{split}
	\mel{N}{T [\hpsi(6) \hpsi^{\dagger}(5)]}{N,S} 
	\\
	= \sum_{pq} \phi_p(\bx_6)  \phi_q^{*}(\bx_5) 
    [  & \theta(+ \tau_{65}) \mel{N}{ \ha_p   \ha^{\dagger}_q  }{N,S} e^{ -i \e{p} \tau_{65} } e^{ - i \Om{S}{} t_5 } 
     \\
     - & \theta(-\tau_{65}) \mel{N}{  \ha^{\dagger}_q \ha_p }{N,S} e^{ -i \e{q} \tau_{65}  } e^{ - i \Om{S}{} t_6 } ]
\end{split}
\end{equation}
leading to Eq.~\eqref{eq:spectral65} with $\Om{S}{} = (E^N_S - E^N_0)$, $t_6 = \tau_{65}/2 + t^{65}$ and $t_5 = - \tau_{65}/2 + t^{65}$.
  

\bibliography{BSEdyn}

\end{document}
