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Pierre-Francois Loos 2020-06-22 15:43:10 +02:00
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Notes/BSEdyn-notes.tex
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@ -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...}
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}