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\begin { document}
\title { Spin-Conserved and Spin-Flip Optical Excitations From the Bethe-Salpeter Equation Formalism}
\author { Enzo \surname { Monino} }
\affiliation { \LCPQ }
\author { Pierre-Fran\c { c} ois \surname { Loos} }
\email { loos@irsamc.ups-tlse.fr}
\affiliation { \LCPQ }
\begin { abstract}
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\end { abstract}
\maketitle
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\section { Introduction}
\label { sec:intro}
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\alert { Here comes the introduction.}
Unless otherwise stated, atomic units are used, and we assume real quantities throughout this manuscript.
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\section { Unrestricted $ GW $ formalism}
\label { sec:UGW}
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Let us consider an electronic system consisting of $ n = n _ \up + n _ \dw $ electrons (where $ n _ \up $ and $ n _ \dw $ are the number of spin-up and spin-down electrons respectively) and $ N $ one-electron basis functions.
The number of spin-up and spin-down occupied orbitals are $ O _ \up = n _ \up $ and $ O _ \dw = n _ \dw $ , respectively, and there is $ V _ \up = N - O _ \up $ and $ V _ \dw = N - O _ \dw $ spin-up and spin-down virtual (\ie , unoccupied) orbitals.
The number of spin-conserved single excitations is then $ S ^ \spc = S _ { \up \up } ^ \spc + S _ { \dw \dw } ^ \spc = O _ \up V _ \up + O _ \dw V _ \dw $ , while the number of spin-flip excitations is $ S ^ \spf = S _ { \up \dw } ^ \spf + S _ { \dw \up } ^ \spf = O _ \up V _ \dw + O _ \dw V _ \up $ .
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Let us denote as $ \MO { p _ \sig } $ the $ p $ th orbital of spin $ \sig $ (where $ \sig = $ $ \up $ or $ \dw $ ) and $ \e { p _ \sig } { } $ its one-electron energy.
In the present context these orbitals can originate from a HF or KS calculation.
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In the following, $ i $ and $ j $ are occupied orbitals, $ a $ and $ b $ are unoccupied orbitals, $ p $ , $ q $ , $ r $ , and $ s $ indicate arbitrary orbitals, and $ m $ labels single excitations.
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It is important to understand that, in a spin-conserved excitation the hole orbital $ \MO { i _ \sig } $ and particle orbital $ \MO { a _ \sig } $ have the same spin $ \sig $ .
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In a spin-flip excitation, the hole and particle states, $ \MO { i _ \sig } $ and $ \MO { a _ \bsig } $ , have opposite spins, $ \sig $ and $ \bsig $ .
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%================================
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\subsection { The dynamical screening}
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%================================
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The one-body Green's function is defined as
\begin { equation}
G^ { \sig } (\br _ 1,\br _ 2;\omega )
= \sum _ i \frac { \MO { i_ \sig } (\br _ 1) \MO { i_ \sig } (\br _ 2)} { \omega - \e { i_ \sig } { } - i\eta }
+ \sum _ a \frac { \MO { a_ \sig } (\br _ 1) \MO { a_ \sig } (\br _ 2)} { \omega - \e { a_ \sig } { } + i\eta }
\end { equation}
Based on this Green's function, one can easily compute the non-interacting polarizability
\begin { equation}
\chi _ 0(\br _ 1,\br _ 2;\omega ) = - \frac { i} { 2\pi } \sum _ \sig \int G^ { \sig } (\br _ 1,\br _ 2;\omega +\omega ') G^ { \sig } (\br _ 1,\br _ 2;\omega ') d\omega '
\end { equation}
and subseauently the dielectric function
\begin { equation}
\epsilon (\br _ 1,\br -2;\omega ) = \delta (\br _ 1 - \br _ 2) - \int \frac { \chi _ 0(\br _ 1,\br _ 3;\omega ) } { \abs { \br _ 2 - \br _ 3} } d\br _ 3
\end { equation}
Based on this latter ingredient, one can access the dynamically-screened Coulomb potential
\begin { equation}
W(\br _ 1,\br _ 2;\omega ) = \int \frac { \epsilon ^ { -1} (\br _ 1,\br _ 3;\omega ) } { \abs { \br _ 2 - \br _ 3} } d\br _ 3
\end { equation}
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Within the $ GW $ formalism, the dynamical screening $ W ( \omega ) $ is computed at the RPA level using the spin-conserved neutral excitations.
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In the orbital basis, the spectral representation of $ W ( \omega ) $ read
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\begin { multline}
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W_ { p_ \sig q_ \sig ,r_ \sigp s_ \sigp } (\omega ) = \ERI { p_ \sig q_ \sig } { r_ \sigp s_ \sigp }
+ \sum _ m \ERI { p_ \sig q_ \sig } { m} \ERI { r_ \sigp s_ \sigp } { m}
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\\
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\times \qty [ \frac{1}{\omega - \Om{m}{\spc,\RPA} + i \eta} - \frac{1}{\omega + \Om{m}{\spc,\RPA} - i \eta} ]
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\end { multline}
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where the two-electron integrals are
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\begin { equation}
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\ERI { p_ \sig q_ \tau } { r_ \sigp s_ \taup } = \int \frac { \MO { p_ \sig } (\br _ 1) \MO { q_ \tau } (\br _ 1) \MO { r_ \sigp } (\br _ 2) \MO { s_ \taup } (\br _ 2)} { \abs { \br _ 1 - \br _ 2} } d\br _ 1 d\br _ 2
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\end { equation}
\begin { equation}
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\ERI { p_ \sig q_ \sig } { m} = \sum _ { ia\sigp } \ERI { p_ \sig q_ \sig } { r_ \sigp s_ \sigp } (\bX { m} { \spc ,\RPA } +\bY { m} { \spc ,\RPA } )_ { i_ \sigp a_ \sigp }
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\end { equation}
\begin { equation}
\label { eq:LR-RPA}
\begin { pmatrix}
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\bA { } { \spc ,\RPA } & \bB { } { \spc ,\RPA } \\
-\bB { } { \spc ,\RPA } & -\bA { } { \spc ,\RPA } \\
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\end { pmatrix}
\cdot
\begin { pmatrix}
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\bX { m} { \spc ,\RPA } \\
\bY { m} { \spc ,\RPA } \\
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\end { pmatrix}
=
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\Om { m} { \spc ,\RPA }
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\begin { pmatrix}
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\bX { m} { \spc ,\RPA } \\
\bY { m} { \spc ,\RPA } \\
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\end { pmatrix} ,
\end { equation}
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The spin structure of these matrices are general and reads
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\begin { align}
\label { eq:LR-RPA-AB}
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\bA { } { \spc } & = \begin { pmatrix}
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\bA { } { \upup ,\upup } & \bA { } { \upup ,\dwdw } \\
\bA { } { \dwdw ,\upup } & \bA { } { \dwdw ,\dwdw } \\
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\end { pmatrix}
&
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\bB { } { \spc } & = \begin { pmatrix}
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\bB { } { \upup ,\upup } & \bB { } { \upup ,\dwdw } \\
\bB { } { \dwdw ,\upup } & \bB { } { \dwdw ,\dwdw } \\
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\end { pmatrix}
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\\
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\label { eq:LR-RPA-AB}
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\bA { } { \spf } & = \begin { pmatrix}
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\bA { } { \updw ,\updw } & \bO \\
\bO & \bA { } { \dwup ,\dwup } \\
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\end { pmatrix}
&
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\bB { } { \spf } & = \begin { pmatrix}
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\bO & \bB { } { \updw ,\dwup } \\
\bB { } { \dwup ,\updw } & \bO \\
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\end { pmatrix}
\end { align}
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with
\begin { subequations}
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\begin { align}
\label { eq:LR_ RPA-A}
\A { i_ \sig a_ \tau ,j_ \sigp b_ \taup } { \RPA } & = \delta _ { ij} \delta _ { ab} \delta _ { \sig \sigp } \delta _ { \tau \taup } (\e { a_ \tau } - \e { i_ \sig } ) + \ERI { i_ \sig a_ \tau } { b_ \sigp j_ \taup }
\\
\label { eq:LR_ RPA-B}
\B { i_ \sig a_ \tau ,j_ \sigp b_ \taup } { \RPA } & = \ERI { i_ \sig a_ \tau } { j_ \sigp b_ \taup }
\end { align}
\end { subequations}
from which we obtain, at the RPA level, the following expressions
\begin { subequations}
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\begin { align}
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\label { eq:LR_ RPA-Asc}
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\A { i_ \sig a_ \sig ,j_ \sigp b_ \sigp } { \spc ,\RPA } & = \delta _ { ij} \delta _ { ab} \delta _ { \sig \sigp } (\e { a_ \sig } - \e { i_ \sig } ) + \ERI { i_ \sig a_ \sig } { b_ \sigp j_ \sigp }
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\\
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\label { eq:LR_ RPA-Bsc}
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\B { i_ \sig a_ \sig ,j_ \sigp b_ \sigp } { \spc ,\RPA } & = \ERI { i_ \sig a_ \sig } { j_ \sigp b_ \sigp }
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\end { align}
\end { subequations}
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for the spin-conserved excitations and
\begin { subequations}
\begin { align}
\label { eq:LR_ RPA-Asf}
\A { i_ \sig a_ \bsig ,j_ \sig b_ \bsig } { \spf ,\RPA } & = \delta _ { ij} \delta _ { ab} (\e { a_ \bsig } - \e { i_ \sig } )
\\
\label { eq:LR_ RPA-Bsf}
\B { i_ \sig a_ \bsig ,j_ \bsig b_ \sig } { \spf ,\RPA } & = 0
\end { align}
\end { subequations}
for the spin-flip excitations.
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%================================
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\subsection { The $ GW $ self-energy}
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%================================
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Within the acclaimed $ GW $ approximation, the exchange-correlation part of the self-energy is defined as
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\begin { equation}
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\begin { split}
\Sig { } ^ { \text { xc} ,\sig } (\br _ 1,\br _ 2;\omega )
& = \Sig { } ^ { \text { x} ,\sig } (\br _ 1,\br _ 2) + \Sig { } ^ { \text { c} ,\sig } (\br _ 1,\br _ 2;\omega )
\\
& = \frac { i} { 2\pi } \int G^ { \sig } (\br _ 1,\br _ 2;\omega +\omega ') W(\br _ 1,\br _ 2;\omega ') e^ { i \eta \omega '} d\omega '
\end { split}
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\end { equation}
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and the spectral representation of its exchange and correlation part read
\begin { gather}
\SigX { p_ \sig q_ \sig }
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= - \frac { 1} { 2} \sum _ { i\sigp } \ERI { p_ \sig i_ \sigp } { i_ \sigp q_ \sig }
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\\
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\begin { split}
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\SigC { p_ \sig q_ \sig } (\omega )
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& = \sum _ { im} \frac { \ERI { p_ \sig i_ \sig } { m} \ERI { q_ \sig i_ \sig } { m} } { \omega - \e { i_ \sig } + \Om { m} { \spc ,\RPA } - i \eta }
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\\
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& + \sum _ { am} \frac { \ERI { p_ \sig a_ \sig } { m} \ERI { q_ \sig a_ \sig } { m} } { \omega - \e { a_ \sig } - \Om { m} { \spc ,\RPA } + i \eta }
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\end { split}
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\end { gather}
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The quasiparticle energies $ \eGW { p } $ are obtained by solving the frequency-dependent quasiparticle equation
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\begin { equation}
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\omega = \eHF { p\sigma } + \SigC { p\sigma } (\omega )
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\end { equation}
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%================================
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\subsection { The Bethe-Salpeter equation formalism}
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%================================
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\begin { multline}
L^ { \sig \sigp } (\br _ 1,\br _ 2;\br _ 1',\br _ 2';\omega )
= L_ { 0} ^ { \sig \sigp } (\br _ 1,\br _ 2;\br _ 1',\br _ 2';\omega )
\\
+ \int L_ { 0} ^ { \sig \sigp } (\br _ 1,\br _ 4;\br _ 1',\br _ 3;\omega )
\Xi ^ { \sig \sigp } (\br _ 3,\br _ 5;\br _ 4,\br _ 6;\omega )
\\
\times L^ { \sig \sigp } (\br _ 6,\br _ 2;\br _ 5,\br _ 2';\omega )
d\br _ 3 d\br _ 4 d\br _ 5 d\br _ 6
\end { multline}
\begin { multline}
i \Xi ^ { \sig \sigp } (\br _ 3,\br _ 5;\br _ 4,\br _ 6;\omega )
= \frac { \delta (\br _ 3 - \br _ 4) \delta (\br _ 5 - \br _ 6) } { \abs { \br _ 3-\br _ 6} }
\\
- \delta _ { \sig \sigp } W(\br _ 3,\br _ 4;\omega ) \delta (\br _ 3 - \br _ 6) \delta (\br _ 4 - \br _ 6)
\end { multline}
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Defining $ W ^ { \stat } _ { p _ \sig q _ \sig ,r _ \sigp s _ \sigp } = W _ { p _ \sig q _ \sig ,r _ \sigp s _ \sigp } ( \omega = 0 ) $ , we have
\begin { subequations}
\begin { align}
\label { eq:LR_ BSE-A}
\A { i_ \sig a_ \tau ,j_ \sigp b_ \taup } { \BSE } & = \A { i_ \sig a_ \tau ,j_ \sigp b_ \taup } { \RPA } - \delta _ { \sig \sigp } W^ { \stat } _ { i_ \sig j_ \sigp ,b_ \taup a_ \tau }
\\
\label { eq:LR_ BSE-B}
\B { i_ \sig a_ \tau ,j_ \sigp b_ \taup } { \BSE } & = \B { i_ \sig a_ \tau ,j_ \sigp b_ \taup } { \RPA } - \delta _ { \sig \sigp } W^ { \stat } _ { i_ \sig b_ \taup ,j_ \sigp a_ \tau }
\end { align}
\end { subequations}
from which we obtain, at the BSE level, the following expressions
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\begin { subequations}
\begin { align}
\label { eq:LR_ BSE-Asc}
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\A { i_ \sig a_ \sig ,j_ \sigp b_ \sigp } { \spc ,\BSE } & = \A { i_ \sig a_ \sig ,j_ \sigp b_ \sigp } { \spc ,\RPA } - \delta _ { \sig \sigp } W^ { \stat } _ { i_ \sig j_ \sigp ,b_ \sigp a_ \sig }
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\\
\label { eq:LR_ BSE-Bsc}
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\B { i_ \sig a_ \sig ,j_ \sigp b_ \sigp } { \spc ,\BSE } & = \B { i_ \sig a_ \sig ,j_ \sigp b_ \sigp } { \spc ,\RPA } - \delta _ { \sig \sigp } W^ { \stat } _ { i_ \sig b_ \sigp ,j_ \sigp a_ \sig }
\end { align}
\end { subequations}
for the spin-conserved excitations and
\begin { subequations}
\begin { align}
\label { eq:LR_ BSE-Asf}
\A { i_ \sig a_ \bsig ,j_ \sig b_ \bsig } { \spf ,\BSE } & = \A { i_ \sig a_ \bsig ,j_ \sig b_ \bsig } { \spf ,\RPA } - W^ { \stat } _ { i_ \sig j_ \sig ,b_ \bsig a_ \bsig }
\\
\label { eq:LR_ BSE-Bsf}
\B { i_ \sig a_ \bsig ,j_ \bsig b_ \sig } { \spf ,\BSE } & = - W^ { \stat } _ { i_ \sig b_ \sig ,j_ \bsig a_ \bsig }
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\end { align}
\end { subequations}
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for the spin-flip excitations.
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%================================
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\subsection { Dynamical correction}
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%================================
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\begin { multline}
\widetilde { W} _ { p_ \sig q_ \sig ,r_ \sigp s_ \sigp } (\omega ) = \ERI { p_ \sig q_ \sig } { r_ \sigp s_ \sigp }
+ \sum _ m \ERI { p_ \sig q_ \sig } { m} \ERI { r_ \sigp s_ \sigp } { m}
\\
\times \qty [ \frac{1}{\omega - (\e{s_\sigp}{} - \e{q_\sig}{}) - \Om{m}{\spc,\RPA} + i \eta} + \frac{1}{\omega - (\e{r_\sigp}{} - \e{p_\sig}{}) - \Om{m}{\spc,\RPA} + i \eta} ]
\end { multline}
\begin { equation}
\label { eq:LR-dyn}
\begin { pmatrix}
\bA { } { \dBSE } (\omega ) & \bB { } { \dBSE } (\omega )
\\
-\bB { } { \dBSE } (-\omega ) & -\bA { } { \dBSE } (-\omega )
\\
\end { pmatrix}
\cdot
\begin { pmatrix}
\bX { m} { \dBSE } \\
\bY { m} { \dBSE } \\
\end { pmatrix}
=
\Om { m} { \dBSE }
\begin { pmatrix}
\bX { m} { \dBSE } \\
\bY { m} { \dBSE } \\
\end { pmatrix}
\end { equation}
\begin { subequations}
\begin { align}
\label { eq:LR_ dBSE-A}
\A { i_ \sig a_ \tau ,j_ \sigp b_ \taup } { \dBSE } (\omega ) & = \A { i_ \sig a_ \tau ,j_ \sigp b_ \taup } { \RPA } - \delta _ { \sig \sigp } \widetilde { W} _ { i_ \sig j_ \sigp ,b_ \taup a_ \tau } (\omega )
\\
\label { eq:LR_ dBSE-B}
\B { i_ \sig a_ \tau ,j_ \sigp b_ \taup } { \dBSE } (\omega ) & = \B { i_ \sig a_ \tau ,j_ \sigp b_ \taup } { \RPA } - \delta _ { \sig \sigp } \widetilde { W} _ { i_ \sig b_ \taup ,j_ \sigp a_ \tau } (\omega )
\end { align}
\end { subequations}
\begin { multline}
\label { eq:LR-PT}
\begin { pmatrix}
\bA { } { \dBSE } (\omega ) & \bB { } { \dBSE } (\omega ) \\
-\bB { } { \dBSE } (-\omega ) & -\bA { } { \dBSE } (-\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 { multline}
with
\begin { subequations}
\begin { align}
\label { eq:BSE-A0}
\A { i_ \sig a_ \tau ,j_ \sigp b_ \taup } { (0)} & = \A { i_ \sig a_ \tau ,j_ \sigp b_ \taup } { \BSE }
\\
\label { eq:BSE-B0}
\B { i_ \sig a_ \tau ,j_ \sigp b_ \taup } { (0)} & = \B { i_ \sig a_ \tau ,j_ \sigp b_ \taup } { \BSE }
\end { align}
\end { subequations}
and
\begin { subequations}
\begin { align}
\label { eq:BSE-A1}
\A { i_ \sig a_ \tau ,j_ \sigp b_ \taup } { (1)} (\omega ) & = - \delta _ { \sig \sigp } \widetilde { W} _ { i_ \sig j_ \sigp ,b_ \taup a_ \tau } (\omega ) + \delta _ { \sig \sigp } W^ { \stat } _ { i_ \sig j_ \sigp ,b_ \taup a_ \tau }
\\
\label { eq:BSE-B1}
\B { i_ \sig a_ \tau ,j_ \sigp b_ \taup } { (1)} (\omega ) & = - \delta _ { \sig \sigp } \widetilde { W} _ { i_ \sig b_ \taup ,j_ \sigp a_ \tau } (\omega ) + \delta _ { \sig \sigp } W^ { \stat } _ { i_ \sig b_ \taup ,j_ \sigp a_ \tau }
\end { align}
\end { subequations}
\begin { subequations}
\begin { gather}
\Om { m} { \dBSE } = \Om { m} { (0)} + \Om { m} { (1)} + \ldots
\\
\begin { pmatrix}
\bX { m} { \dBSE } \\
\bY { m} { \dBSE } \\
\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}
\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 { m} { (0)}
\begin { pmatrix}
\bX { m} { (0)} \\
\bY { m} { (0)} \\
\end { pmatrix}
\end { equation}
\begin { equation}
\label { eq:Om1}
\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}
\begin { equation}
\label { eq:Om1-TDA}
\Om { S} { (1)} = \T { (\bX { m} { (0)} )} \cdot \bA { } { (1)} (\Om { m} { (0)} ) \cdot \bX { m} { (0)}
\end { equation}
\begin { equation}
\label { eq:Z}
Z_ { m} = \qty [ 1 - \T{(\bX{m}{(0)})} \cdot \left. \pdv{\bA{}{(1)}(\Om{m}{})}{\Om{S}{}} \right|_{\Om{m}{} = \Om{m}{(0)}} \cdot \bX{m}{(0)} ] ^ { -1}
\end { equation}
\begin { equation}
\Om { m} { \dBSE } = \Om { m} { (0)} + Z_ { m} \Om { m} { (1)}
\end { equation}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection { Oscillator strengths}
\label { sec:os}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
For the spin-conserved transition, the transition dipole moment in the $ x $ direction is
\begin { equation}
\mu _ { x,m} ^ { \spc } = \sum _ { ia\sig } (i_ \sig |x|a_ \sig )(\bX { m} { \spc } +\bY { m} { \spc } )_ { i_ \sig a_ \sig }
\end { equation}
with
\begin { equation}
(p_ \sig |x|q_ \sigp ) = \int \MO { p_ \sig } (\br ) \, x \, \MO { q_ \sigp } (\br ) d\br
\end { equation}
and the total oscillator strength is given by
\begin { equation}
f_ { m} ^ { \spc } = \frac { 2} { 3} \Om { m} { \spc } \qty [ \qty(\mu_{x,m}^{\spc})^2 + \qty(\mu_{x,m}^{\spc})^2 + \qty(\mu_{x,m}^{\spc})^2 ]
\end { equation}
For spin-flip transitions, we have $ f _ { m } ^ { \spf } = 0 $ as the transition matrix elements $ ( i _ \sig |x|a _ \bsig ) $ vanish.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection { Spin contamination}
\label { sec:spin}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin { equation}
\expval { S^ 2} _ m = \expval { S^ 2} _ 0 + \Delta \expval { S^ 2} _ m
\end { equation}
\begin { equation}
\expval { S^ 2} _ { 0}
= \frac { n_ { \up } - n_ { \dw } } { 2} \qty ( \frac { n_ { \up } - n_ { \dw } } { 2} + 1 )
+ n_ { \dw } - \sum _ p (p_ { \up } |p_ { \dw } )^ 2
\end { equation}
where
\begin { equation}
(p_ \sig |q_ \sigp ) = \int \MO { p_ \sig } (\br ) \MO { q_ \sigp } (\br ) d\br
\end { equation}
is the overlap between spin-up and spin-down orbitals.
The explicit expressions of $ \Delta \expval { S ^ 2 } _ m ^ { \spc } $ and $ \Delta \expval { S ^ 2 } _ m ^ { \spf } $ are given in Ref.~\onlinecite { Li_ 2010} .
As explained in Ref.~\onlinecite { Casanova_ 2020} , there are two sources of spin contamination: i) spin contamination of the reference, and ii) spin-contamination of the excited states due to the spin incompleteness.
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\section { Computational details}
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\label { sec:compdet}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section { Results}
\label { sec:res}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%% TABLE I %%%
%\begin{table}
%
%\end{table}
%%% %%% %%% %%%
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section { Conclusion}
\label { sec:ccl}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%
\acknowledgements {
We would like to thank Xavier Blase and Denis Jacquemin for insightful discussions.
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This project has received funding from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme (Grant agreement No.~863481).}
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%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section * { Data availability statement}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
The data that supports the findings of this study are available within the article and its supplementary material.
%%%%%%%%%%%%%%%%%%%%%%%%
\bibliography { sf-BSE}
%%%%%%%%%%%%%%%%%%%%%%%%
\end { document}