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%
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\newcommand { \eGnWn } [2]{ \epsilon ^ \text { \GnWn { #2} } _ { #1} }
\newcommand { \Om } [2]{ \Omega _ { #1} ^ { #2} }
% Matrix elements
\newcommand { \A } [2]{ A_ { #1} ^ { #2} }
\newcommand { \tA } [2]{ \Tilde { A} _ { #1} ^ { #2} }
\newcommand { \B } [2]{ B_ { #1} ^ { #2} }
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\newcommand { \BRPAx } [2]{ B_ { #1} ^ { #2,\text { RPAx} } }
\newcommand { \G } [1]{ G_ { #1} }
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\newcommand { \XiBSE } [1]{ \Xi _ { #1} }
\newcommand { \Po } [1]{ P_ { #1} }
\newcommand { \W } [2]{ W_ { #1} ^ { #2} }
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% excitation energies
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\newcommand { \OmRPA } [1]{ \Omega _ { #1} ^ { \text { RPA} } }
\newcommand { \OmRPAx } [1]{ \Omega _ { #1} ^ { \text { RPAx} } }
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\newcommand \vari { { \varepsilon } _ i}
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\newcommand \varb { { \varepsilon } _ b}
\newcommand \varn { { \varepsilon } _ n}
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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}
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\title { Dynamical Correction to the Bethe-Salpeter Equation}
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\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}
%%%%%%%%%%%%%%%%%%%%%%%%
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%================================
\subsection { Theory for physics}
%=================================
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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:
\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 ] $ ?? }
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%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
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\begin { equation}
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\label { eq:LR-dyn}
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\begin { pmatrix}
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\bA { } (\omega ) & \bB { } (\omega ) \\
-\bB { } (\omega ) & -\bA { } (\omega ) \\
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\end { pmatrix}
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\cdot
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\begin { pmatrix}
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\bX { m} { } (\omega ) \\
\bY { m} { } (\omega ) \\
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\end { pmatrix}
=
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\omega
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\begin { pmatrix}
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\bX { m} { } (\omega ) \\
\bY { m} { } (\omega ) \\
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\end { pmatrix} ,
\end { equation}
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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.
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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.
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The BSE matrix elements read
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\begin { subequations}
\begin { align}
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\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 ),
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\end { align}
\end { subequations}
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where $ \eGW { ia } = \eGW { a } - \eGW { i } $ are occupied-to-virtual differences of $ GW $ quasiparticle energies,
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\begin { equation}
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\ERI { pq} { rs} = \iint \frac { \MO { p} (\br { } ) \MO { q} (\br { } ) \MO { r} (\br { } ') \MO { s} (\br { } ')} { \abs * { \br { } - \br { } '} } \dbr { } \dbr { } '
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\end { equation}
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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
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\begin { multline}
\label { eq:W}
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\W { ij,ab} { } (\omega ) = \ERI { ij} { ab} + 2 \sum _ m^ { \Nocc \Nvir } \sERI { ij} { m} \sERI { ab} { m}
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\\
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\times \qty (\frac { 1} { \omega - \OmRPA { m} { } - \eGW { ib} + i \eta } + \frac { 1} { \omega - \OmRPA { m} { } - \eGW { ja} + i \eta } ),
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\end { multline}
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where $ \eta $ is a positive infinitesimal, and
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\begin { equation}
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\label { eq:sERI}
\sERI { pq} { m} = \sum _ i^ { \Nocc } \sum _ a^ { \Nvir } \ERI { pq} { ia} (\bX { m} { \RPA } + \bY { m} { \RPA } )_ { ia}
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\end { equation}
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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
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\begin { subequations}
\begin { align}
\label { eq:LR_ RPA-A}
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\A { ia,jb} { \RPA } & = \delta _ { ij} \delta _ { ab} (\e { a} - \e { i} ) + 2 \ERI { ia} { jb} ,
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\\
\label { eq:LR_ RPA-B}
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\B { ia,jb} { \RPA } & = 2 \ERI { ia} { bj} ,
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\end { align}
\end { subequations}
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where the $ \e { p } $ 's are taken as the Hartree-Fock (HF) orbital energies in the case of $ G _ 0 W _ 0 $ or as the $ GW $ quasiparticle energies in the case of self-consistent scheme such as ev$ GW $ .
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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
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\begin { subequations}
\begin { align}
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\label { eq:BSE-1}
\A { ia,jb} { (1)} (\omega ) & = - \W { ij,ab} { } (\omega ) + \W { ij,ab} { \text { stat} } ,
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\\
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\label { eq:BSE-1}
\B { ia,jb} { (1)} (\omega ) & = - \W { ib,aj} { } (\omega ) + \W { ib,aj} { \text { stat} } ,
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\end { align}
\end { subequations}
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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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%%% FIG 1 %%%
%\begin{figure}
% \includegraphics[width=\linewidth]{}
%\caption{
%\label{fig:}
%}
%\end{figure}
%%% %%% %%%
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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}