added comment on SOPPA
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BSEdyn.bib
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BSEdyn.bib
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%% This BibTeX bibliography file was created using BibDesk.
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%% http://bibdesk.sourceforge.net/
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%% Created for Pierre-Francois Loos at 2020-06-22 10:36:09 +0200
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%% Created for Pierre-Francois Loos at 2020-06-22 20:38:15 +0200
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%% Saved with string encoding Unicode (UTF-8)
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@article{Nielsen_1980,
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Author = {Egon S. Nielsen and Poul Jorgensen},
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Date-Added = {2020-06-22 20:37:32 +0200},
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Date-Modified = {2020-06-22 20:38:12 +0200},
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Doi = {10.1063/1.440119},
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Journal = {J. Chem. Phys.},
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Pages = {6238},
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Title = {Transition moments and dynamic polarizabilities in a second order polarization propagator approach},
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Volume = {73},
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Year = {1980}}
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@article{Oddershede_1977,
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Author = {Jens Oddershede and Poul Jorgensen},
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Date-Added = {2020-06-22 20:36:10 +0200},
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Date-Modified = {2020-06-22 20:36:52 +0200},
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Doi = {10.1063/1.434118},
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Journal = {J. Chem. Phys.},
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Pages = {1541},
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Title = {An order analysis of the particle--hole propagator},
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Volume = {66},
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Year = {1977}}
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@phdthesis{Huix-Rotllant_PhD,
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Author = {M. Huix-Rotllant},
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Date-Added = {2020-06-22 20:32:30 +0200},
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Date-Modified = {2020-06-22 20:34:35 +0200},
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School = {Universit{\'e} de Grenoble},
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Title = {Improved correlation kernels for linear-response time-dependent density-functional theory},
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Url = {https://tel.archives-ouvertes.fr/tel-00680039/},
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Year = {2011},
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Bdsk-Url-1 = {https://tel.archives-ouvertes.fr/tel-01027522}}
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@article{Loos_2020e,
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Author = {X. Blase and Y. Duchemin and D. Jacquemin},
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Date-Added = {2020-06-22 09:07:38 +0200},
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@ -619,6 +619,7 @@ Note that, unlike the dBSE Hamiltonian [see Eq.~\eqref{eq:HBSE}], the BSE2 dynam
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This latter point has an important consequence as this lack of frequency dependence removes one of the spurious pole (see Fig.~\ref{fig:BSE2}).
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The singlet manifold has then the right number of excitations.
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However, one spurious triplet excitation remains.
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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}
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Numerical results for the two-level model are reported in Table \ref{tab:BSE2} with the usual approximations and perturbative treatments.
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In the case of BSE2, the perturbative partitioning is simply
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\begin{equation}
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@ -654,7 +655,6 @@ As compared to dBSE, dBSE2 produces much larger corrections to the static excita
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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.
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For the double excitation, dBSE2 yields a slightly better energy, yet still in quite poor agreement with the exact value.
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%%% FIGURE 3 %%%
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\begin{figure}
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\includegraphics[width=\linewidth]{dBSE2}
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@ -670,50 +670,50 @@ For the double excitation, dBSE2 yields a slightly better energy, yet still in q
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\subsection{The forgotten kernel: Sangalli's kernel}
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\label{sec:Sangalli}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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%\titou{This section is experimental...}
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%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.
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%We will first start by writing down explicitly this kernel as it is given in obscure physicist notations in the original article.
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%
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%The Hamiltonian with Sangalli's kernel is (I think)
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%\begin{equation}
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% \bH_\text{S}^{\sigma}(\omega) =
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% \begin{pmatrix}
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% \bR_\text{S}^{\sigma}(\omega) & \bC_\text{S}^{\sigma}(\omega)
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% \\
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% -\bC_\text{S}^{\sigma}(-\omega) & -\bR_\text{S}^{\sigma}(-\omega)
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% \end{pmatrix}
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%\end{equation}
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%with
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%\begin{subequations}
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%\begin{gather}
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% R_{ia,jb}^{\sigma}(\omega) = \delta_{ij} \delta_{ab} (\eGW{a} - \eGW{i}) + f_{ia,jb}^{\sigma} (\omega)
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% \\
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% C_{ia,jb}^{\sigma}(\omega) = f_{ia,bj}^{\sigma} (\omega)
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%\end{gather}
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%\end{subequations}
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%and
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%\begin{subequations}
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%\begin{gather}
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% f_{ia,jb}^{\sigma} (\omega) = \sum_{m \neq n} \frac{ c_{ia,mn} c_{jb,mn} }{\omega - ( \omega_{m} + \omega_{n})}
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% \\
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% 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}
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% + R_{m,kb} R_{n,jc} ] }
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%\end{gather}
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%\end{subequations}
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%where $R_{m,ia}$ are the elements of the RPA eigenvectors.
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%
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%For the two-level model, Sangalli's kernel reads
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%\begin{align}
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% R(\omega) & = \Delta\eGW{} + f_R (\omega)
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% \\
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% C(\omega) & = f_C (\omega)
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%\end{align}
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%
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%\begin{gather}
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% f_R (\omega) = 2 \frac{ [\ERI{vv}{vc} + \ERI{vc}{cc}]^2 }{\omega - 2\omega_1}
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% \\
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% f_C (\omega) = 0
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%\end{gather}
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\titou{This section is experimental...}
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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.
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We will first start by writing down explicitly this kernel as it is given in obscure physicist notations in the original article.
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The Hamiltonian with Sangalli's kernel is (I think)
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\begin{equation}
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\bH_\text{S}^{\sigma}(\omega) =
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\begin{pmatrix}
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\bR_\text{S}^{\sigma}(\omega) & \bC_\text{S}^{\sigma}(\omega)
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\\
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-\bC_\text{S}^{\sigma}(-\omega) & -\bR_\text{S}^{\sigma}(-\omega)
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\end{pmatrix}
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\end{equation}
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with
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\begin{subequations}
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\begin{gather}
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R_{ia,jb}^{\sigma}(\omega) = \delta_{ij} \delta_{ab} (\eGW{a} - \eGW{i}) + f_{ia,jb}^{\sigma} (\omega)
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\\
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C_{ia,jb}^{\sigma}(\omega) = f_{ia,bj}^{\sigma} (\omega)
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\end{gather}
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\end{subequations}
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and
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\begin{subequations}
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\begin{gather}
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f_{ia,jb}^{\sigma} (\omega) = \sum_{m \neq n} \frac{ c_{ia,mn} c_{jb,mn} }{\omega - ( \omega_{m} + \omega_{n})}
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\\
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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}
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+ R_{m,kb} R_{n,jc} ] }
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\end{gather}
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\end{subequations}
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where $R_{m,ia}$ are the elements of the RPA eigenvectors.
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For the two-level model, Sangalli's kernel reads
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\begin{align}
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R(\omega) & = \Delta\eGW{} + f_R (\omega)
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\\
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C(\omega) & = f_C (\omega)
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\end{align}
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\begin{gather}
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f_R (\omega) = 2 \frac{ [\ERI{vv}{vc} + \ERI{vc}{cc}]^2 }{\omega - 2\omega_1}
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\\
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f_C (\omega) = 0
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\end{gather}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\section{Take-home messages}
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