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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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\maketitle

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Introduction}
\label{sec:intro}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\alert{Here comes the introduction.}
Unless otherwise stated, atomic units are used, and we assume real quantities throughout this manuscript.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Unrestricted $GW$ formalism}
\label{sec:UGW}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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 $S^\spf = S_{\up\dw}^\spf + S_{\dw\up}^\spf = O_\up V_\dw + O_\dw V_\up$.
Let us denote as $\MO{p\si}$ the $p$th orbital of spin $\sigma$ (where $\sigma = \up$ or $\dw$).
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.
It is important to understand that, in a spin-conserved excitation the hole orbital $\MO{i\si}$ and particle orbital $\MO{a\si}$ have the same spin $\si$.
A bra and ket composed by these two orbitals will be denoted as $\rbra{ia\si}$ and $\rket{ia\si}$.
In a spin-flip excitation, the hole and the particle states have different spin.

%================================
\subsection{The dynamical screening}
%================================

Within the $GW$ formalism, the dynamical screening $W(\omega)$ is computed at the RPA level using the spin-conserved neutral excitations.
The matrix elements of $W(\omega)$ read
\begin{multline}
	W_{pq\si,rs\sip}(\omega) = \ERI{pq\si}{rs\sip} 
	+ \sum_m \sERI{pq\si}{m}\sERI{rs\sip}{m} 
	\\
	\times \qty[ \frac{1}{\omega - \Om{m}{\spc,\RPA} + i \eta} - \frac{1}{\omega + \Om{m}{\spc,\RPA} - i \eta} ]
\end{multline}
where the two-electron integrals are
\begin{equation}
		\ERI{pq\si}{rs\sip} = \iint \MO{p\si}(\br) \MO{q\si}(\br) \frac{1}{\abs{\br - \br'}}  \MO{r\sip}(\br') \MO{s\sip}(\br') d\br d\br'
\end{equation}

\begin{equation}
		\sERI{pq\si}{m} = \sum_{ia\sip} \ERI{pq\si}{rs\sip} (\bX{m}{\spc,\RPA}+\bY{m}{\spc,\RPA})_{ia\sip}
\end{equation}

\begin{equation}
\label{eq:LR-RPA}
	\begin{pmatrix}
		\bA{\spc,\RPA}	&	\bB{\spc,\RPA}	\\
		-\bB{\spc,\RPA}	&	-\bA{\spc,\RPA}	\\
	\end{pmatrix}
	\cdot
	\begin{pmatrix}
		\bX{m}{\spc,\RPA}	\\
		\bY{m}{\spc,\RPA}	\\
	\end{pmatrix}
	=
	\Om{m}{\spc,\RPA}
	\begin{pmatrix}
		\bX{m}{\spc,\RPA}	\\
		\bY{m}{\spc,\RPA}	\\
	\end{pmatrix},
\end{equation}


\begin{align}
\label{eq:LR-RPA-AB}
	\bA{\spc} & = \begin{pmatrix}
		\bA{\upup,\upup}	&	\bA{\upup,\dwdw}	\\
		\bA{\dwdw,\upup}	&	\bA{\dwdw,\dwdw}	\\
	\end{pmatrix}
	&
	\bB{\spc} & = \begin{pmatrix}
		\bB{\upup,\upup}	&	\bB{\upup,\dwdw}	\\
		\bB{\dwdw,\upup}	&	\bB{\dwdw,\dwdw}	\\
	\end{pmatrix}
\end{align}

\begin{align}
\label{eq:LR-RPA-AB}
	\bA{\spf} & = \begin{pmatrix}
		\bA{\updw,\updw}	&	\bO	\\
		\bO					&	\bA{\dwup,\dwup}	\\
	\end{pmatrix}
	&
	\bB{\spf} & = \begin{pmatrix}
		\bO			&	\bB{\updw,\dwup}	\\
		\bB{\dwup,\updw}	&	\bO	\\
	\end{pmatrix}
\end{align}
with
\begin{subequations}
\begin{align}
	\label{eq:LR_RPA-Asc}
	\A{ia\si,jb\sip}{\spc,\RPA} & = \delta_{ij} \delta_{ab} \delta_{\si\sip} (\e{a\si} - \e{i\si}) +   \ERI{ia\si}{jb\sip},
	\\ 
	\label{eq:LR_RPA-Bsc}
	\B{ia\si,jb\sip}{\spc,\RPA} & =   0,
\end{align}
\end{subequations}


%================================
\subsection{The $GW$ self-energy}
%================================

\begin{equation}
\begin{split}
	\SigC{pq\si}(\omega) 
	& = \sum_i \sum_m \frac{\sERI{pi\si}{m} \sERI{qi\si}{m}}{\omega - \e{i\si} + \Om{m}{\spc,\RPA} - i \eta}
	\\
	& + \sum_a \sum_m \frac{\sERI{pa\si}{m} \sERI{qa\si}{m}}{\omega - \e{a\si} - \Om{m}{\spc,\RPA} + i \eta}			 
\end{split}
\end{equation}

The quasiparticle energies $\eGW{p}$ are obtained by solving the frequency-dependent quasiparticle equation
\begin{equation}
	\omega = \eHF{p\sigma} + \SigC{p\sigma}(\omega)
\end{equation}

%================================
\subsection{The Bethe-Salpeter formalism}
%================================

\begin{subequations}
\begin{align}
	\label{eq:LR_BSE-Asc}
	\A{ia\si,jb\sip}{\spc,\BSE} & = \delta_{ij} \delta_{ab} \delta_{\si\sip} (\eGW{a\si} - \eGW{i\si}) +   \ERI{ia\si}{jb\sip} - W_{ij}
	\\ 
	\label{eq:LR_BSE-Bsc}
	\B{ia\si,jb\sip}{\spc,\BSE} & =   0,
\end{align}
\end{subequations}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Computational details}
\label{sec:compdet}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Conclusion}
\label{sec:ccl}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


%%%%%%%%%%%%%%%%%%%%%%%%
\acknowledgements{
We would like to thank Xavier Blase and Denis Jacquemin for insightful discussions. 
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.

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