saving work

sfBSE.bib

@@ -1,13 +1,24 @@
 %% This BibTeX bibliography file was created using BibDesk.
 %% http://bibdesk.sourceforge.net/
 
-%% Created for Pierre-Francois Loos at 2020-10-20 14:42:08 +0200 
+%% Created for Pierre-Francois Loos at 2020-10-25 13:01:52 +0100 
 
 
 %% Saved with string encoding Unicode (UTF-8) 
 
 
 
+@article{Casanova_2020,
+	Author = {D. Casanova and A. I. Krylov},
+	Date-Added = {2020-10-25 13:00:27 +0100},
+	Date-Modified = {2020-10-25 13:01:40 +0100},
+	Doi = {10.1039/c9cp06507e},
+	Journal = {Phys. Chem. Chem. Phys.},
+	Pages = {4326},
+	Title = {Spin-Flip Methods in Quantum Chemistry},
+	Volume = {22},
+	Year = {2020}}
+
 @article{Zhang_2004,
 	Author = {Zhang, Fan and Burke, Kieron},
 	Date-Added = {2020-10-20 14:41:53 +0200},

sfBSE.tex

@@ -178,7 +178,7 @@ for the spin-flip excitations.
 %================================
 \subsection{The $GW$ self-energy}
 %================================
-Within the acclaimed $GW$ approximation, the exchange-correlation (xc) part of the self-energy
+Within the acclaimed $GW$ approximation, \cite{Hedin_1965,Golze_2019} the exchange-correlation (xc) part of the self-energy
 \begin{equation}
 \begin{split}
 	\Sig{}^{\text{xc},\sig}(\br_1,\br_2;\omega) 
@@ -213,7 +213,7 @@ where $v^{\xc}(\br)$ the Kohn-Sham exchange-correlation.
 %================================
 \subsection{The Bethe-Salpeter equation formalism}
 %================================
-Like its TD-DFT cousin, BSE deals with the calculation of (neutral) optical excitations as measured by absorption spectroscopy.
+Like its TD-DFT cousin, BSE deals with the calculation of (neutral) optical excitations as measured by absorption spectroscopy. \cite{Salpeter_1951,Strinati_1988}
 Using the BSE formalism, one can access the spin-conserved and spin-flip excitations.
 The Dyson equation that links the generalized four-point susceptibility $L^{\sig\sigp}(\br_1,\br_2;\br_1',\br_2';\omega)$ and the BSE kernel $\Xi^{\sig\sigp}(\br_3,\br_5;\br_4,\br_6;\omega)$ is
 \begin{multline}
@@ -232,7 +232,7 @@ where
 	\\
 	= \frac{1}{2\pi} \int G^{\sig}(\br_1,\br_2';\omega+\omega') G^{\sig}(\br_1',\br_2;\omega') d\omega'
 \end{multline}
-is the non-interacting counterpart of the two-particle correlation function $L$.
+is the non-interacting analog of the two-particle correlation function $L$.
 
 Within the $GW$ approximation, the BSE kernel is
 \begin{multline}
@@ -241,7 +241,26 @@ Within the $GW$ approximation, the BSE kernel is
 	\\
 	- \delta_{\sig\sigp} W(\br_3,\br_4;\omega) \delta(\br_3 - \br_6) \delta(\br_4 - \br_6) 
 \end{multline}
-
+where, as usual, we have not considered the higher-order terms in $W$ by neglecting the derivative $\partial W/\partial G$. \cite{Hanke_1980, Strinati_1982, Strinati_1984, Strinati_1988}
+Within the static approximation, the spin-conserved and spin-flip optical excitation at the BSE level are obtained by solving a similar linear response problem
+\begin{equation}
+\label{eq:LR-RPA}
+	\begin{pmatrix}
+		\bA{}{\BSE}		&	\bB{}{\BSE}	\\
+		-\bB{}{\BSE}	&	-\bA{}{\BSE}	\\
+	\end{pmatrix}
+	\cdot
+	\begin{pmatrix}
+		\bX{m}{\BSE}	\\
+		\bY{m}{\BSE}	\\
+	\end{pmatrix}
+	=
+	\Om{m}{\BSE}
+	\begin{pmatrix}
+		\bX{m}{\BSE}	\\
+		\bY{m}{\BSE}	\\
+	\end{pmatrix}
+\end{equation}
 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}
@@ -277,6 +296,7 @@ for the spin-flip excitations.
 %================================
 \subsection{Dynamical correction}
 %================================
+The dynamical correction to the static BSE kernel is defined in the Tamm-Dancoff approximation as \cite{Strinati_1988,Rohlfing_2000,Ma_2009a,Ma_2009b,Romaniello_2009b,Sangalli_2011,Loos_2020e}
 
 \begin{multline}
 	\widetilde{W}_{p_\sig q_\sig,r_\sigp s_\sigp}(\omega) = \ERI{p_\sig q_\sig}{r_\sigp s_\sigp} 
@@ -507,7 +527,7 @@ This project has received funding from the European Research Council (ERC) under
 The data that supports the findings of this study are available within the article and its supplementary material.
 
 %%%%%%%%%%%%%%%%%%%%%%%%
-\bibliography{sf-BSE}
+\bibliography{sfBSE}
 %%%%%%%%%%%%%%%%%%%%%%%%
 
 \end{document}