saving work in S2 and f

sfBSE.rty

@@ -93,7 +93,6 @@
 \newcommand{\Wc}[1]{W^\text{c}_{#1}}
 \newcommand{\vc}[1]{v_{#1}}
 \newcommand{\Sig}[1]{\Sigma_{#1}}
-\newcommand{\SigGW}[1]{\Sigma^{GW}_{#1}}
 \newcommand{\SigC}[1]{\Sigma^\text{c}_{#1}}
 \newcommand{\SigX}[1]{\Sigma^\text{x}_{#1}}
 \newcommand{\SigXC}[1]{\Sigma^\text{xc}_{#1}}
@@ -161,8 +160,8 @@
 \newcommand{\bsigp}{{\Bar{\sigma}'}}
 \newcommand{\taup}{{\tau'}}
 
-\newcommand{\up}{\downarrow}
+\newcommand{\up}{\uparrow}
-\newcommand{\dw}{\uparrow}
+\newcommand{\dw}{\downarrow}
 \newcommand{\upup}{\uparrow\uparrow}
 \newcommand{\updw}{\uparrow\downarrow}
 \newcommand{\dwup}{\downarrow\uparrow}

sfBSE.tex

@@ -116,24 +116,24 @@ 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{}{\upup,\upup}	&	\bA{}{\upup,\dwdw}	\\
-		\bA{\dwdw,\upup}{}	&	\bA{\dwdw,\dwdw}{}	\\
+		\bA{}{\dwdw,\upup}	&	\bA{}{\dwdw,\dwdw}	\\
 	\end{pmatrix}
 	&
 	\bB{}{\spc} & = \begin{pmatrix}
-		\bB{\upup,\upup}{}	&	\bB{\upup,\dwdw}{}	\\
+		\bB{}{\upup,\upup}	&	\bB{}{\upup,\dwdw}	\\
-		\bB{\dwdw,\upup}{}	&	\bB{\dwdw,\dwdw}{}	\\
+		\bB{}{\dwdw,\upup}	&	\bB{}{\dwdw,\dwdw}	\\
 	\end{pmatrix}
 \\
 \label{eq:LR-RPA-AB}
 	\bA{}{\spf} & = \begin{pmatrix}
-		\bA{\updw,\updw}{}	&	\bO	\\
+		\bA{}{\updw,\updw}	&	\bO	\\
-		\bO					&	\bA{\dwup,\dwup}{}	\\
+		\bO					&	\bA{}{\dwup,\dwup}	\\
 	\end{pmatrix}
 	&
 	\bB{}{\spf} & = \begin{pmatrix}
-		\bO			&	\bB{\updw,\dwup}{}	\\
+		\bO					&	\bB{}{\updw,\dwup}	\\
-		\bB{\dwup,\updw}{}	&	\bO	\\
+		\bB{}{\dwup,\updw}	&	\bO	\\
 	\end{pmatrix}
 \end{align}
 with
@@ -171,26 +171,27 @@ for the spin-flip excitations.
 %================================
 \subsection{The $GW$ self-energy}
 %================================
-
+Within the acclaimed $GW$ approximation, the exchange-correlation part of the self-energy is defined as
 \begin{equation}
-	\Sig{}^{\sig}(\br_1,\br_2;\omega) 
+\begin{split}
-	= \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'
+	\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}
 \end{equation}
-
+and the spectral representation of its exchange and correlation part read
-\begin{equation}
+\begin{gather}
-	\SigX{p_\sig q_\sig}(\omega) 
+	\SigX{p_\sig q_\sig}
 	= - \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}
+\end{gather}
 
 The quasiparticle energies $\eGW{p}$ are obtained by solving the frequency-dependent quasiparticle equation
 \begin{equation}
@@ -410,12 +411,62 @@ and
 \end{equation}
 
 
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\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.
 
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 \section{Computational details}
 \label{sec:compdet}
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\section{Results}
+\label{sec:res}
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+%%% TABLE I %%%
+%\begin{table}
+%
+%\end{table}
+%%% %%% %%% %%%
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 \section{Conclusion}
 \label{sec:ccl}