diff --git a/BSEdyn.bib b/BSEdyn.bib index dc6dc85..0ff0482 100644 --- a/BSEdyn.bib +++ b/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}, diff --git a/Notes/BSEdyn-notes.tex b/Notes/BSEdyn-notes.tex index 2887aa5..4cefb1a 100644 --- a/Notes/BSEdyn-notes.tex +++ b/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...} -%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} = \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}