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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}

\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}
This is the abstract
%\\
%\bigskip
%\begin{center}
%	\boxed{\includegraphics[width=0.5\linewidth]{TOC}}
%\end{center}
%\bigskip
\end{abstract}

\maketitle

%%%%%%%%%%%%%%%%%%%%%%%%
\section{Introduction}
\label{sec:intro}
%%%%%%%%%%%%%%%%%%%%%%%%

%%%%%%%%%%%%%%%%%%%%%%%%
\section{Theory}
\label{sec:theory}
%%%%%%%%%%%%%%%%%%%%%%%%
%================================
\subsection{Theory for physics}
%=================================

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*}
[iL_0]( \omega_1 )  = \frac{ 1 }{  2\pi } \int d \omega   \; G(\omega - \frac{\omega_1}{2} ) G( {\omega} + \frac{\omega_1}{2}  )   e^{  i  \omega  \tau_{34}  } e^{  i  \omega_1   t^{34}   }   
   \end{align*}
with $\tau_{34} = t_3 - t_4$  and 
$t^{34} = (t_3 + t_4)/2$. Plugging now the 1-body Green's function Lehman representation, e.g.
$$
G(x_1,x_3 ; \omega) = \sum_n \frac{ \phi_n(x_1) \phi_n^*(x_3) } { \omega - \varepsilon_n + i \eta \text{sgn}(\varepsilon_n - \mu) }
$$
and projecting on $\phi_a^*(x_1) \phi_i(x_{1'})$, one obtains the $\omega_1= \Oms$ component  
\begin{align*}
\int dx_1 dx_{1'} \; & \phi_a^*(x_1) \phi_i(x_{1'})  L_0(x_1,3;x_{1'},4; \Oms) = e^{i \Oms t^{34} } \times    \\
     &       \frac{ \phi_a^*(x_3) \phi_i(x_4)  } {   \Oms - (  \vara - \vari )  + i \eta }   
       \Big( \theta( \tau ) e^{i ( \vari + \hOms) \tau }   
     +   \theta( - \tau  )   e^{i (\vara -  \hOms \tau }  \Big)  
\end{align*}
with $\tau = \tau_{34}$.
We further obtain the spectral representation of
$\langle N  | T {\hat \psi}(3) {\hat \psi}^{\dagger}(4) | N,s \rangle$
expanding the field operators  over a complete orbital basis creation/destruction operators:
\begin{align*}
\langle N  | T   {\hat \psi}(3)     {\hat \psi}^{\dagger}(4) & | N,s \rangle = - \Big( e^{  -i \Omega_s t^{34} }  \Big)    \sum_{mn} \phi_m(x_3) \phi_n^*(x_4)  \langle N | {\hat a}_n^{\dagger} {\hat a}_m  | N,s \rangle    \times \nonumber \\
\times & \Big( \theta( \tau )   e^{- i   ( \varepsilon_m  -  \hOms ) \tau  }  
 + \theta( -\tau )   e^{ - i   ( \varepsilon_n  + \hOms)  \tau  }      \Big)
\end{align*}
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) :
 \begin{align}
      ( \varepsilon_a - \varepsilon_i    - \Omega_s    )       A_{ia}^{s}         
       &+ \sum_{jb}  \Big(  v_{ai,bj} - \widetilde{W}_{ij,ab}(\Oms)    \Big)  A_{jb}^{s}  \\ 
       &+  \sum_{bj}  \Big(  v_{ai,jb} - \widetilde{W}_{ib,aj}(\Oms)    \Big)  B_{jb}^{s} 
       = 0
\end{align}
 with an effective dynamically  screened Coulomb potential (see Pina  eq. 24): 
 \begin{align}
 \widetilde{W}_{ij,ab}(\Oms)  &= { i \over 2  \pi}  \int d\omega \; e^{-i \omega 0^+ } W_{ij,ab}(\omega) \times \\
    \hskip 1cm &\times \left[ \frac{1}{ (\Oms - \omega) - ( \varb - \vari ) +i \eta } + \frac{1}{ (\Oms + \omega) - ( \vara - \varj ) + i\eta } \right] \nonumber
\end{align}
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 \\
   & \times \Big(  \frac{1}{ \omega-\Omega_m^{RPA} + i\eta }  -   \frac{1}{ \omega + \Omega_m^{RPA} - i\eta } \Big) 
\end{align*}
so that
\begin{align}
 \widetilde{W}_{ij,ab}( \Oms ) &=   (ij|ab) + 2 \sum_m^{OV}    [ij|m] [ab|m] \times \\ 
    & \times     \left[ \frac{  1 }{    \Omega_{ib}^{s} - \Omega_m^{RPA}    + i\eta   }   +   \frac{ 1}{   \Omega_{ja}^{s}   - \Omega_m^{RPA}   + i\eta }
     \right]   \nonumber
\end{align} 
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
$$
\left[ \frac{  1 }{    \Omega_{ib}^{s} - \Omega_m^{RPA}    + i\eta   }   +   \frac{ 1}{   \Omega_{ja}^{s}   - \Omega_m^{RPA}   + i\eta }
     \right] 
     < 
  \Big(  \frac{1}{ \omega-\Omega_m^{RPA} + i\eta }  -   \frac{1}{ \omega + \Omega_m^{RPA} - i\eta } \Big) < 0   
$$
in the limit $(\omega \rightarrow 0)$ of the standard adiabatic BSE . WELL, do we know the sign of 
$[ij|m] [ab|m]$ ?? }

%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} 
%\begin{multline}
%\label{eq:BSE}
%	\LBSE{}(1,2,1',2') = \LBSE{0}(1,2,1',2') 
%	\\
%	+ \int d3 d4 d5  d6  \LBSE{0}(1,4,1',3) \XiBSE{}(3,5,4,6) \LBSE{}(6,2,5,2')
%\end{multline}
%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 
%\begin{equation}
%	\XiBSE{}(3,5,4,6) = \delta(3,4) \delta(5,6) \vc{}(3,6) - \delta(3,6) \delta(4,5) \W{}{}(3,4),
%\end{equation}
%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, $\W{}{}(1,2) = \W{}{}(\{\br{1},\sigma_1\},\{\br{2},\sigma_2\}; \omega=0)$.

%================================
\subsection{Theory for chemists}
%=================================

For a closed-shell system in a finite basis, to compute the BSE excitation energies, one must solve the following (non-linear) dynamical (\ie, frequency-dependent) response problem 
\begin{equation}
\label{eq:LR-dyn}
	\begin{pmatrix}
		\bA{}(\omega)	&	\bB{}(\omega)	\\
		-\bB{}(\omega)	&	-\bA{}(\omega)	\\
	\end{pmatrix}
	\cdot
	\begin{pmatrix}
		\bX{m}{}(\omega)	\\
		\bY{m}{}(\omega)	\\
	\end{pmatrix}
	=
	\omega
	\begin{pmatrix}
		\bX{m}{}(\omega)	\\
		\bY{m}{}(\omega)	\\
	\end{pmatrix},
\end{equation}
where the dynamical matrices $\bA{}(\omega)$, $\bB{}(\omega)$, $\bX{}{}(\omega)$, and $\bY{}{}(\omega)$ 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.

The BSE matrix elements read
\begin{subequations}
\begin{align}
	\label{eq:BSE-Adyn}
	\A{ia,jb}{}(\omega) & = \delta_{ij} \delta_{ab} \eGW{ia} + 2 \ERI{ia}{jb} - \W{ij,ab}{}(\omega),
	\\ 
	\label{eq:BSE-Bdyn}
	\B{ia,jb}{}(\omega) & =  2 \ERI{ia}{bj} -   \W{ib,aj}{}(\omega),
\end{align}
\end{subequations}
where $\eGW{ia} = \eGW{a} - \eGW{i}$ are occupied-to-virtual differences of $GW$ quasiparticle energies, 
\begin{equation}
	\ERI{pq}{rs} = \iint \frac{\MO{p}(\br{}) \MO{q}(\br{}) \MO{r}(\br{}') \MO{s}(\br{}')}{\abs*{\br{} - \br{}'}} \dbr{} \dbr{}'
\end{equation}
are the bare two-electron integrals in the molecular orbital basis $\lbrace \MO{p}(\br{}) \rbrace_{1 \le p \le \Norb}$, and the dynamically-screened Coulomb potential reads
\begin{multline}
\label{eq:W}
	\W{ij,ab}{}(\omega) = \ERI{ij}{ab} +  2 \sum_m^{\Nocc \Nvir} \sERI{ij}{m} \sERI{ab}{m}
	\\
	\times  \qty(\frac{1}{\omega - \OmRPA{m}{} - \eGW{ib} + i \eta} + \frac{1}{\omega - \OmRPA{m}{} - \eGW{ja} + i \eta}),
\end{multline}
where $\eta$ is a positive infinitesimal, and
\begin{equation}
\label{eq:sERI}
	\sERI{pq}{m} =  \sum_i^{\Nocc} \sum_a^{\Nvir} \ERI{pq}{ia} (\bX{m}{\RPA} + \bY{m}{\RPA})_{ia}
\end{equation}
are the spectral weights.
In Eqs.~\eqref{eq:W} and \eqref{eq:sERI}, $\OmRPA{m}{}$ and $(\bX{m}{\RPA} + \bY{m}{\RPA})$ are direct (\ie, without exchange) RPA neutral excitations and their corresponding transition vectors computed by solving the (linear) static response problem 
\begin{equation}
\label{eq:LR-stat}
	\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 Hartree-Fock (HF) orbital energies in the case of $G_0W_0$ or as the $GW$ quasiparticle energies in the case of self-consistent scheme such as ev$GW$.

Now, let us decompose, using basis perturbation theory, the eigenproblem \eqref{eq:LR-dyn} as a zeroth-order static part and a first-order dynamic part, such that
\begin{equation}
\label{eq:LR-dyn}
	\begin{pmatrix}
		\bA{}(\omega)	&	\bB{}(\omega)	\\
		-\bB{}(\omega)	&	-\bA{}(\omega)	\\
	\end{pmatrix}
	=
	\begin{pmatrix}
		\bA{(0)}	&	\bB{(0)}	\\
		-\bB{(0)}	&	-\bA{(0)}	\\
	\end{pmatrix}
	+
	\begin{pmatrix}
		\bA{(1)}(\omega)	&	\bB{(1)}(\omega)	\\
		-\bB{(1)}(\omega)	&	-\bA{(1)}(\omega)	\\
	\end{pmatrix}
\end{equation}
where
\begin{subequations}
\begin{align}
	\label{eq:BSE-0}
	\A{ia,jb}{(0)} & = \delta_{ij} \delta_{ab} \eGW{ia} + 2 \ERI{ia}{jb} - \W{ij,ab}{\text{stat}},
	\\ 
	\label{eq:BSE-0}
	\B{ia,jb}{(0)} & =  2 \ERI{ia}{bj} -   \W{ib,aj}{\text{stat}},
\end{align}
\end{subequations}
and 
\begin{subequations}
\begin{align}
	\label{eq:BSE-1}
	\A{ia,jb}{(1)}(\omega) & = - \W{ij,ab}{}(\omega) + \W{ij,ab}{\text{stat}},
	\\ 
	\label{eq:BSE-1}
	\B{ia,jb}{(1)}(\omega) & = -  \W{ib,aj}{}(\omega) + \W{ib,aj}{\text{stat}},
\end{align}
\end{subequations}
The static version of the screened Coulomb potential reads 
\begin{equation}
\label{eq:Wstat}
	\W{ij,ab}{\text{stat}} = \ERI{ij}{ab} -  4 \sum_m^{\Nocc \Nvir} \frac{\sERI{ij}{m} \sERI{ab}{m}}{\OmRPA{m}{} - i \eta}.
\end{equation}
The $m$th BSE excitation energy and its corresponding eigenvector can then decomposed as
\begin{subequations}
\begin{gather}
	\Om{m}{} = \Om{m}{(0)} + \Om{m}{(1)} + \ldots
	\\
	\begin{pmatrix}
		\bX{m}{}	\\
		\bY{m}{}	\\
	\end{pmatrix}
	= 
	\begin{pmatrix}
		\bX{m}{(0)}	\\
		\bY{m}{(0)}	\\
	\end{pmatrix}
	+
	\begin{pmatrix}
		\bX{m}{(1)}	\\
		\bY{m}{(1)}	\\
	\end{pmatrix}
	+ \ldots		
\end{gather}
\end{subequations}
Solving the zeroth-order static problem yields
\begin{equation}
	\begin{pmatrix}
		\bA{(0)}	&	\bB{(0)}	\\
		-\bB{(0)}	&	-\bA{(0)}	\\
	\end{pmatrix}
	\cdot
	\begin{pmatrix}
		\bX{m}{(0)}	\\
		\bY{m}{(0)}	\\
	\end{pmatrix}
	=
	\Om{m}{(0)}
	\begin{pmatrix}
		\bX{m}{(0)}	\\
		\bY{m}{(0)}	\\
	\end{pmatrix},
\end{equation}
Thanks to first-order perturbation theory, the first-order correction to the $m$th excitation energy is
\begin{equation}
	\Om{m}{(1)} =
	\T{\begin{pmatrix}
		\bX{m}{(0)}	\\
		\bY{m}{(0)}	\\
	\end{pmatrix}}
	\cdot
	\begin{pmatrix}
		\bA{(1)}(\Om{m}{(0)})	&	\bB{(1)}(\Om{m}{(0)})	\\
		-\bB{(1)}(\Om{m}{(0)})	&	-\bA{(1)}(\Om{m}{(0)})	\\
	\end{pmatrix}
	\cdot
	\begin{pmatrix}
		\bX{m}{(0)}	\\
		\bY{m}{(0)}	\\
	\end{pmatrix}.
\end{equation}
From a practical point of view, if one enforces the Tamm-Dancoff approximation (TDA), we obtain the very simple expression
\begin{equation}
	\Om{m}{(1)} = \T{(\bX{m}{(0)})} \cdot \bA{(1)}(\Om{m}{(0)}) \cdot \bX{m}{(0)}.
\end{equation}

This correction can be renormalized by computing, at basically no extra cost, the renormalization factor
\begin{equation}
	Z_{m} = \qty[ \T{(\bX{m}{(0)})} \cdot \left. \pdv{\bA{(1)}(\omega)}{\omega} \right|_{\omega = \Om{m}{(0)}} \cdot \bX{m}{(0)} ]^{-1}.
\end{equation}
which finally yields
\begin{equation}
	\Om{m}{} \approx \Om{m}{(0)} + Z_{m} \Om{m}{(1)}.
\end{equation}
This is our final expression.

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

%%%%%%%%%%%%%%%%%%%%%%%%
\acknowledgements{
%%%%%%%%%%%%%%%%%%%%%%%%
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''.}}

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\section*{Supporting Information}
%%%%%%%%%%%%%%%%%%%%%%%%
See {\SI} for plenty of stuff

\bibliography{BSEdyn}

\end{document}