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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.}
%\bigskip
%\begin{center}
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\end{abstract}

\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 $\MO{p_\sig}$ 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.
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 $\MO{i_\sig}$ and particle orbital $\MO{a_\sig}$ have the same spin $\sig$.
In a spin-flip excitation, the hole and particle states, $\MO{i_\sig}$ and $\MO{a_\bsig}$, have opposite spins, $\sig$ and $\bsig$.

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

Within the $GW$ formalism, the dynamical screening $W(\omega)$ is computed at the RPA level using the spin-conserved neutral excitations.
In the orbital basis, the spectral representation of $W(\omega)$ read
\begin{multline}
	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 - \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{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
\end{equation}

\begin{equation}
		\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}
\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}

The spin structure of these matrices are general and reads
\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}
\\
\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-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}
\begin{align}
	\label{eq:LR_RPA-Asc}
	\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}
	\\ 
	\label{eq:LR_RPA-Bsc}
	\B{i_\sig a_\sig,j_\sigp b_\sigp}{\spc,\RPA} & = \ERI{i_\sig a_\sig}{j_\sigp b_\sigp}
\end{align}
\end{subequations}
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.

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

\begin{equation}
	\Sig{}^{\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{equation}

\begin{equation}
	\SigX{p_\sig q_\sig}(\omega) 
	= - \frac{1}{2} \sum_{i\sigp} \ERI{p_\sig i_\sigp}{i_\sigp q_\sig}	 
\end{equation}


\begin{equation}
\begin{split}
	\SigC{p_\sig q_\sig}(\omega) 
	& = \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}
	\\
	& + \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}			 
\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 equation formalism}
%================================

\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}

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
\begin{subequations}
\begin{align}
	\label{eq:LR_BSE-Asc}
	\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}
	\\ 
	\label{eq:LR_BSE-Bsc}
	\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}
\end{align}
\end{subequations}
for the spin-flip excitations.

%================================
\subsection{Dynamical correction}
%================================

\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}



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\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).}
%%%%%%%%%%%%%%%%%%%%%%%%

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\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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\bibliography{sf-BSE}
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\end{document}
