added comment on SOPPA

BSEdyn.bib

@@ -1,13 +1,45 @@
 %% This BibTeX bibliography file was created using BibDesk.
 %% http://bibdesk.sourceforge.net/
 
-%% Created for Pierre-Francois Loos at 2020-06-22 10:36:09 +0200 
+%% Created for Pierre-Francois Loos at 2020-06-22 20:38:15 +0200 
 
 
 %% Saved with string encoding Unicode (UTF-8) 
 
 
 
+@article{Nielsen_1980,
+	Author = {Egon S. Nielsen and Poul Jorgensen},
+	Date-Added = {2020-06-22 20:37:32 +0200},
+	Date-Modified = {2020-06-22 20:38:12 +0200},
+	Doi = {10.1063/1.440119},
+	Journal = {J. Chem. Phys.},
+	Pages = {6238},
+	Title = {Transition moments and dynamic polarizabilities in a second order polarization propagator approach},
+	Volume = {73},
+	Year = {1980}}
+
+@article{Oddershede_1977,
+	Author = {Jens Oddershede and Poul Jorgensen},
+	Date-Added = {2020-06-22 20:36:10 +0200},
+	Date-Modified = {2020-06-22 20:36:52 +0200},
+	Doi = {10.1063/1.434118},
+	Journal = {J. Chem. Phys.},
+	Pages = {1541},
+	Title = {An order analysis of the particle--hole propagator},
+	Volume = {66},
+	Year = {1977}}
+
+@phdthesis{Huix-Rotllant_PhD,
+	Author = {M. Huix-Rotllant},
+	Date-Added = {2020-06-22 20:32:30 +0200},
+	Date-Modified = {2020-06-22 20:34:35 +0200},
+	School = {Universit{\'e} de Grenoble},
+	Title = {Improved correlation kernels for linear-response time-dependent density-functional theory},
+	Url = {https://tel.archives-ouvertes.fr/tel-00680039/},
+	Year = {2011},
+	Bdsk-Url-1 = {https://tel.archives-ouvertes.fr/tel-01027522}}
+
 @article{Loos_2020e,
 	Author = {X. Blase and Y. Duchemin and D. Jacquemin},
 	Date-Added = {2020-06-22 09:07:38 +0200},

Notes/BSEdyn-notes.tex

@@ -619,6 +619,7 @@ Note that, unlike the dBSE Hamiltonian [see Eq.~\eqref{eq:HBSE}], the BSE2 dynam
 This latter point has an important consequence as this lack of frequency dependence removes one of the spurious pole (see Fig.~\ref{fig:BSE2}).
 The singlet manifold has then the right number of excitations.
 However, one spurious triplet excitation remains.
+It is mentioned in Ref.~\onlinecite{Rebolini_2016} that the BSE2 kernel has some similarities with the second-order polarization-propagator approximation \cite{Oddershede_1977,Nielsen_1980} (SOPPA) and second RPA kernels. \cite{Huix-Rotllant_2011,Huix-Rotllant_PhD,Sangalli_2011}
 Numerical results for the two-level model are reported in Table \ref{tab:BSE2} with the usual approximations and perturbative treatments.
 In the case of BSE2, the perturbative partitioning is simply
 \begin{equation}
@@ -654,7 +655,6 @@ As compared to dBSE, dBSE2 produces much larger corrections to the static excita
 Overall, the accuracy of dBSE and dBSE2 are comparable (see Tables \ref{tab:BSE} and \ref{tab:BSE2}) for single excitations although their behavior is quite different. 
 For the double excitation, dBSE2 yields a slightly better energy, yet still in quite poor agreement with the exact value.
 
-
 %%% FIGURE 3 %%%
 \begin{figure}
 	\includegraphics[width=\linewidth]{dBSE2}
@@ -670,50 +670,50 @@ For the double excitation, dBSE2 yields a slightly better energy, yet still in q
 \subsection{The forgotten kernel: Sangalli's kernel}
 \label{sec:Sangalli}
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-%\titou{This section is experimental...}
+\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.
+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.
+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)
+The Hamiltonian with Sangalli's kernel is (I think)
-%\begin{equation}
+\begin{equation}
-%	\bH_\text{S}^{\sigma}(\omega) = 
+	\bH_\text{S}^{\sigma}(\omega) = 
-%	\begin{pmatrix}
+	\begin{pmatrix}
-%		\bR_\text{S}^{\sigma}(\omega)	&	\bC_\text{S}^{\sigma}(\omega)
+		\bR_\text{S}^{\sigma}(\omega)	&	\bC_\text{S}^{\sigma}(\omega)
-%		\\
+		\\
-%		-\bC_\text{S}^{\sigma}(-\omega)	&	-\bR_\text{S}^{\sigma}(-\omega)
+		-\bC_\text{S}^{\sigma}(-\omega)	&	-\bR_\text{S}^{\sigma}(-\omega)
-%	\end{pmatrix}
+	\end{pmatrix}
-%\end{equation}
+\end{equation}
-%with
+with
-%\begin{subequations}
+\begin{subequations}
-%\begin{gather}
+\begin{gather}
-%	R_{ia,jb}^{\sigma}(\omega) = \delta_{ij} \delta_{ab} (\eGW{a} - \eGW{i}) + f_{ia,jb}^{\sigma} (\omega)
+	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)
+	C_{ia,jb}^{\sigma}(\omega) = f_{ia,bj}^{\sigma} (\omega)
-%\end{gather}
+\end{gather}
-%\end{subequations}
+\end{subequations}
-%and
+and
-%\begin{subequations}
+\begin{subequations}
-%\begin{gather}
+\begin{gather}
-%	f_{ia,jb}^{\sigma} (\omega) = \sum_{m \neq n} \frac{ c_{ia,mn} c_{jb,mn} }{\omega - ( \omega_{m} + \omega_{n})}
+	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}
+	c_{ia,mn}^{\sigma} = \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} ] }
+	+ R_{m,kb} R_{n,jc} ] }
-%\end{gather}
+\end{gather}
-%\end{subequations}
+\end{subequations}
-%where $R_{m,ia}$ are the elements of the RPA eigenvectors.
+where $R_{m,ia}$ are the elements of the RPA eigenvectors.
-%
+
-%For the two-level model, Sangalli's kernel reads
+For the two-level model, Sangalli's kernel reads
-%\begin{align}
+\begin{align}
-%	R(\omega) & =  \Delta\eGW{} + f_R (\omega)
+	R(\omega) & =  \Delta\eGW{} + f_R (\omega)
-%	\\
+	\\
-%	C(\omega) & = f_C (\omega)
+	C(\omega) & = f_C (\omega)
-%\end{align}
+\end{align}
-%
+
-%\begin{gather}
+\begin{gather}
-%	f_R (\omega) = 2 \frac{ [\ERI{vv}{vc} + \ERI{vc}{cc}]^2 }{\omega - 2\omega_1}
+	f_R (\omega) = 2 \frac{ [\ERI{vv}{vc} + \ERI{vc}{cc}]^2 }{\omega - 2\omega_1}
-%	\\
+	\\
-%	f_C (\omega) = 0
+	f_C (\omega) = 0
-%\end{gather}
+\end{gather}
 
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 \section{Take-home messages}