saving work

sfBSE.rty

@@ -127,9 +127,8 @@
 \newcommand{\bdeHF}{\mathbf{\Delta\epsilon}^\text{HF}}
 \newcommand{\bdeGW}{\mathbf{\Delta\epsilon}^{GW}}
 \newcommand{\bOm}[1]{\mathbf{\Omega}^{#1}}
-\newcommand{\bA}[1]{\mathbf{A}^{#1}}
+\newcommand{\bA}[2]{\mathbf{A}_{#1}^{#2}}
-\newcommand{\btA}[1]{\Tilde{\mathbf{A}}^{#1}}
+\newcommand{\bB}[2]{\mathbf{B}_{#1}^{#2}}
-\newcommand{\bB}[1]{\mathbf{B}^{#1}}
 \newcommand{\bX}[2]{\mathbf{X}_{#1}^{#2}}
 \newcommand{\bY}[2]{\mathbf{Y}_{#1}^{#2}}
 \newcommand{\bZ}[2]{\mathbf{Z}_{#1}^{#2}}
@@ -153,8 +152,11 @@
 \newcommand{\EgOpt}{\Eg^\text{opt}}
 \newcommand{\EB}{E_B}
 
-\newcommand{\si}{\sigma}
+\newcommand{\sig}{\sigma}
-\newcommand{\sip}{\sigma'}
+\newcommand{\bsig}{{\Bar{\sigma}}}
+\newcommand{\sigp}{{\sigma'}}
+\newcommand{\bsigp}{{\Bar{\sigma}'}}
+\newcommand{\taup}{{\tau'}}
 
 \newcommand{\up}{\downarrow}
 \newcommand{\dw}{\uparrow}
@@ -165,7 +167,6 @@
 \newcommand{\spc}{\text{sc}}
 \newcommand{\spf}{\text{sf}}
 
-
 % addresses
 \newcommand{\LCPQ}{Laboratoire de Chimie et Physique Quantiques (UMR 5626), Universit\'e de Toulouse, CNRS, UPS, France}
 

sfBSE.tex

@@ -48,11 +48,11 @@ Unless otherwise stated, atomic units are used, and we assume real quantities th
 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 $\dw$).
+Let us denote as $\MO{p_\sig}$ the $p$th orbital of spin $\sig$ (where $\sig =$ $\up$ 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 $\si$.
+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 $\rbra{ia\si}$ and $\rket{ia\si}$.
+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 and the particle states have different spin.
+In a spin-flip excitation, the hole has a spin $\sig$ and the particle has the opposite spin $\bsig$.
 
 %================================
 \subsection{The dynamical screening}
@@ -61,25 +61,25 @@ In a spin-flip excitation, the hole and the particle states have different spin.
 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} 
+	W_{p_\sig q_\sig,r_\sigp s_\sigp}(\omega) = \ERI{p_\sig q_\sig}{r_\sigp s_\sigp} 
-	+ \sum_m \sERI{pq\si}{m}\sERI{rs\sip}{m} 
+	+ \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{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'
+		\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'
 \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}
+		\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}	\\
+		\bA{}{\spc,\RPA}	&	\bB{}{\spc,\RPA}	\\
-		-\bB{\spc,\RPA}	&	-\bA{\spc,\RPA}	\\
+		-\bB{}{\spc,\RPA}	&	-\bA{}{\spc,\RPA}	\\
 	\end{pmatrix}
 	\cdot
 	\begin{pmatrix}
@@ -94,43 +94,61 @@ where the two-electron integrals are
 	\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{}{\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{}{\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}
-\end{align}
+\\
-
-\begin{align}
 \label{eq:LR-RPA-AB}
-	\bA{\spf} & = \begin{pmatrix}
+	\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}
+	\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
 \begin{subequations}
 \begin{align}
-	\label{eq:LR_RPA-Asc}
+	\label{eq:LR_RPA-A}
-	\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},
+	\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-Bsc}
+	\label{eq:LR_RPA-B}
-	\B{ia\si,jb\sip}{\spc,\RPA} & =   0,
+	\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}
@@ -138,10 +156,10 @@ with
 
 \begin{equation}
 \begin{split}
-	\SigC{pq\si}(\omega) 
+	\SigC{p_\sig q_\sig}(\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_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}
 	\\
-	& + \sum_a \sum_m \frac{\sERI{pa\si}{m} \sERI{qa\si}{m}}{\omega - \e{a\si} - \Om{m}{\spc,\RPA} + i \eta}			 
+	& + \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}			 
 \end{split}
 \end{equation}
 
@@ -151,19 +169,44 @@ The quasiparticle energies $\eGW{p}$ are obtained by solving the frequency-depen
 \end{equation}
 
 %================================
-\subsection{The Bethe-Salpeter formalism}
+\subsection{The Bethe-Salpeter equation formalism}
 %================================
-
+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{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}
+	\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{ia\si,jb\sip}{\spc,\BSE} & =   0,
+	\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{The dynamical Bethe-Salpeter equation correction}
+
+%================================
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 \section{Computational details}
 \label{sec:compdet}
@@ -178,7 +221,7 @@ The quasiparticle energies $\eGW{p}$ are obtained by solving the frequency-depen
 %%%%%%%%%%%%%%%%%%%%%%%%
 \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).
+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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