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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}
\alert { Here comes the 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 $ ).
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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 $ .
A bra and ket composed by these two orbitals will be denoted as $ \rbra { ia \sig } $ and $ \rket { ia \sig } $ .
In a spin-flip excitation, the hole has a spin $ \sig $ and the particle has the opposite spin $ \bsig $ .
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%================================
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\subsection { The dynamical screening}
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%================================
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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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The matrix elements 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 } = \iint \MO { p_ \sig } (\br ) \MO { q_ \tau } (\br ) \frac { 1} { \abs { \br - \br '} } \MO { r_ \sigp } (\br ') \MO { s_ \taup } (\br ') d\br d\br '
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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}
\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}
\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}
\bA { \updw ,\updw } { } & \bO \\
\bO & \bA { \dwup ,\dwup } { } \\
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\end { pmatrix}
&
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\bB { } { \spf } & = \begin { pmatrix}
\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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\begin { equation}
\begin { split}
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\SigC { p_ \sig q_ \sig } (\omega )
& = \sum _ i \sum _ m \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 _ a \sum _ m \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}
\end { equation}
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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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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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%================================
\subsection { The dynamical Bethe-Salpeter equation correction}
%================================
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\section { Computational details}
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\label { sec:compdet}
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\section { Conclusion}
\label { sec:ccl}
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\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}
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The data that supports the findings of this study are available within the article and its supplementary material.
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\bibliography { sf-BSE}
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\end { document}