From 7a1eeffa42e1f039f4e49fafed40921fd2e2ff19 Mon Sep 17 00:00:00 2001
From: Pierre-Francois Loos <pierrefrancois.loos@gmail.com>
Date: Mon, 22 Jun 2020 15:33:51 +0200
Subject: [PATCH] test for Antoine

---
 Notes/BSEdyn-notes.tex | 92 +++++++++++++++++++++---------------------
 1 file changed, 46 insertions(+), 46 deletions(-)

diff --git a/Notes/BSEdyn-notes.tex b/Notes/BSEdyn-notes.tex
index c8078a0..1b24ddf 100644
--- a/Notes/BSEdyn-notes.tex
+++ b/Notes/BSEdyn-notes.tex
@@ -667,53 +667,53 @@ For the double excitation, dBSE2 yields a slightly better energy, yet still in q
 %%% %%% %%% %%%
 
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-%\subsection{The forgotten kernel: Sangalli's kernel}
-%\label{sec:Sangalli}
+\subsection{The forgotten kernel: Sangalli's kernel}
+\label{sec:Sangalli}
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-%\titou{This section is experimental...}
-%In Ref.~\onlinecite{Sangalli_2011}, Sangalli proposed a dynamical kernel (based on the second RPA) without (he claims) spurious excitations thanks to the design of a number-conserving approach which correctly describes particle indistinguishability and Pauli exclusion principle.
-%We will first start by writing down explicitly this kernel as it is given in obscure physicist notations in the original article.
-%
-%The Hamiltonian with Sangalli's kernel is (I think)
-%\begin{equation}
-%	\bH_\text{S}^{\sigma}(\omega) = 
-%	\begin{pmatrix}
-%		\bR_\text{S}^{\sigma}(\omega)	&	\bC_\text{S}^{\sigma}(\omega)
-%		\\
-%		-\bC_\text{S}^{\sigma}(-\omega)	&	-\bR_\text{S}^{\sigma}(-\omega)
-%	\end{pmatrix}
-%\end{equation}
-%with
-%\begin{subequations}
-%\begin{gather}
-%	R_{ia,jb}^{\sigma}(\omega) = \delta_{ij} \delta_{ab} (\eGW{a} - \eGW{i}) + f_{ia,jb}^{\sigma} (\omega)
-%	\\
-%	C_{ia,jb}^{\sigma}(\omega) = f_{ia,bj}^{\sigma} (\omega)
-%\end{gather}
-%\end{subequations}
-%and
-%\begin{subequations}
-%\begin{gather}
-%	f_{ia,jb}^{\sigma} (\omega) = \sum_{m \neq n} \frac{ c_{ia,mn} c_{jb,mn} }{\omega - ( \omega_{m} + \omega_{n})}
-%	\\
-%	c_{ia,mn}^{\sigma} = \frac{1}{2} \sum_{jb,kc} \qty{ \qty[ \ERI{ij}{kc} \delta_{ab} + \ERI{kc}{ab} \delta_{ij} ] \qty[ R_{m,jc} R_{n,kb}
-%	+ R_{m,kb} R_{n,jc} ] }
-%\end{gather}
-%\end{subequations}
-%where $R_{m,ia}$ are the elements of the RPA eigenvectors.
-%
-%For the two-level model, Sangalli's kernel reads
-%\begin{align}
-%	R(\omega) & =  \Delta\eGW{} + f_R (\omega)
-%	\\
-%	C(\omega) & = f_C (\omega)
-%\end{align}
-%
-%\begin{gather}
-%	f_R (\omega) = 2 \frac{ [\ERI{vv}{vc} + \ERI{vc}{cc}]^2 }{\omega - 2\omega_1}
-%	\\
-%	f_C (\omega) = 0
-%\end{gather}
+\titou{This section is experimental...}
+In Ref.~\onlinecite{Sangalli_2011}, Sangalli proposed a dynamical kernel (based on the second RPA) without (he claims) spurious excitations thanks to the design of a number-conserving approach which correctly describes particle indistinguishability and Pauli exclusion principle.
+We will first start by writing down explicitly this kernel as it is given in obscure physicist notations in the original article.
+
+The Hamiltonian with Sangalli's kernel is (I think)
+\begin{equation}
+	\bH_\text{S}^{\sigma}(\omega) = 
+	\begin{pmatrix}
+		\bR_\text{S}^{\sigma}(\omega)	&	\bC_\text{S}^{\sigma}(\omega)
+		\\
+		-\bC_\text{S}^{\sigma}(-\omega)	&	-\bR_\text{S}^{\sigma}(-\omega)
+	\end{pmatrix}
+\end{equation}
+with
+\begin{subequations}
+\begin{gather}
+	R_{ia,jb}^{\sigma}(\omega) = \delta_{ij} \delta_{ab} (\eGW{a} - \eGW{i}) + f_{ia,jb}^{\sigma} (\omega)
+	\\
+	C_{ia,jb}^{\sigma}(\omega) = f_{ia,bj}^{\sigma} (\omega)
+\end{gather}
+\end{subequations}
+and
+\begin{subequations}
+\begin{gather}
+	f_{ia,jb}^{\sigma} (\omega) = \sum_{m \neq n} \frac{ c_{ia,mn} c_{jb,mn} }{\omega - ( \omega_{m} + \omega_{n})}
+	\\
+	c_{ia,mn}^{\sigma} = \frac{1}{2} \sum_{jb,kc} \qty{ \qty[ \ERI{ij}{kc} \delta_{ab} + \ERI{kc}{ab} \delta_{ij} ] \qty[ R_{m,jc} R_{n,kb}
+	+ R_{m,kb} R_{n,jc} ] }
+\end{gather}
+\end{subequations}
+where $R_{m,ia}$ are the elements of the RPA eigenvectors.
+
+For the two-level model, Sangalli's kernel reads
+\begin{align}
+	R(\omega) & =  \Delta\eGW{} + f_R (\omega)
+	\\
+	C(\omega) & = f_C (\omega)
+\end{align}
+
+\begin{gather}
+	f_R (\omega) = 2 \frac{ [\ERI{vv}{vc} + \ERI{vc}{cc}]^2 }{\omega - 2\omega_1}
+	\\
+	f_C (\omega) = 0
+\end{gather}
 
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 \section{Take-home messages}