sfBSE/sfBSE.tex

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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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\maketitle

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

\begin{multline}
	W_{pq\si,rs\sip}(\omega) = \ERI{pq\si}{rs\sip} 
	\\
	+ \sum_m \sERI{pq\si}{m}\sERI{rs\sip}{m} \qty[ \frac{1}{\omega - \OmRPA{m} + i \eta} - \frac{1}{\omega + \OmRPA{m} - i \eta} ]
\end{multline}


\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}{\RPA}+\bY{m}{\RPA})_{ia\sip}
\end{equation}

\begin{equation}
\label{eq:LR-RPA}
	\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\si,jb\sip}{\RPA} & = \delta_{ij} \delta_{ab} \delta_{\si\sip} (\e{a} - \e{i}) +   2 \ERI{ia\si}{jb\sip},
	\\ 
	\label{eq:LR_RPA-B}
	\B{ia\si,jb\sip}{\RPA} & =   2 \ERI{ia\si}{bj\sip},
\end{align}
\end{subequations}

\subsection{The $GW$ self-energy}

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