From fcc538b870e4dfa6d008459dd81de07b9627254e Mon Sep 17 00:00:00 2001 From: Pierre-Francois Loos Date: Sun, 25 Oct 2020 13:30:30 +0100 Subject: [PATCH] saving work --- sfBSE.bib | 13 ++++++++++++- sfBSE.tex | 30 +++++++++++++++++++++++++----- 2 files changed, 37 insertions(+), 6 deletions(-) diff --git a/sfBSE.bib b/sfBSE.bib index d3deea7..854437b 100644 --- a/sfBSE.bib +++ b/sfBSE.bib @@ -1,13 +1,24 @@ %% This BibTeX bibliography file was created using BibDesk. %% http://bibdesk.sourceforge.net/ -%% Created for Pierre-Francois Loos at 2020-10-20 14:42:08 +0200 +%% Created for Pierre-Francois Loos at 2020-10-25 13:01:52 +0100 %% Saved with string encoding Unicode (UTF-8) +@article{Casanova_2020, + Author = {D. Casanova and A. I. Krylov}, + Date-Added = {2020-10-25 13:00:27 +0100}, + Date-Modified = {2020-10-25 13:01:40 +0100}, + Doi = {10.1039/c9cp06507e}, + Journal = {Phys. Chem. Chem. Phys.}, + Pages = {4326}, + Title = {Spin-Flip Methods in Quantum Chemistry}, + Volume = {22}, + Year = {2020}} + @article{Zhang_2004, Author = {Zhang, Fan and Burke, Kieron}, Date-Added = {2020-10-20 14:41:53 +0200}, diff --git a/sfBSE.tex b/sfBSE.tex index 1e1ae9a..e000fdb 100644 --- a/sfBSE.tex +++ b/sfBSE.tex @@ -178,7 +178,7 @@ for the spin-flip excitations. %================================ \subsection{The $GW$ self-energy} %================================ -Within the acclaimed $GW$ approximation, the exchange-correlation (xc) part of the self-energy +Within the acclaimed $GW$ approximation, \cite{Hedin_1965,Golze_2019} the exchange-correlation (xc) part of the self-energy \begin{equation} \begin{split} \Sig{}^{\text{xc},\sig}(\br_1,\br_2;\omega) @@ -213,7 +213,7 @@ where $v^{\xc}(\br)$ the Kohn-Sham exchange-correlation. %================================ \subsection{The Bethe-Salpeter equation formalism} %================================ -Like its TD-DFT cousin, BSE deals with the calculation of (neutral) optical excitations as measured by absorption spectroscopy. +Like its TD-DFT cousin, BSE deals with the calculation of (neutral) optical excitations as measured by absorption spectroscopy. \cite{Salpeter_1951,Strinati_1988} Using the BSE formalism, one can access the spin-conserved and spin-flip excitations. The Dyson equation that links the generalized four-point susceptibility $L^{\sig\sigp}(\br_1,\br_2;\br_1',\br_2';\omega)$ and the BSE kernel $\Xi^{\sig\sigp}(\br_3,\br_5;\br_4,\br_6;\omega)$ is \begin{multline} @@ -232,7 +232,7 @@ where \\ = \frac{1}{2\pi} \int G^{\sig}(\br_1,\br_2';\omega+\omega') G^{\sig}(\br_1',\br_2;\omega') d\omega' \end{multline} -is the non-interacting counterpart of the two-particle correlation function $L$. +is the non-interacting analog of the two-particle correlation function $L$. Within the $GW$ approximation, the BSE kernel is \begin{multline} @@ -241,7 +241,26 @@ Within the $GW$ approximation, the BSE kernel is \\ - \delta_{\sig\sigp} W(\br_3,\br_4;\omega) \delta(\br_3 - \br_6) \delta(\br_4 - \br_6) \end{multline} - +where, as usual, we have not considered the higher-order terms in $W$ by neglecting the derivative $\partial W/\partial G$. \cite{Hanke_1980, Strinati_1982, Strinati_1984, Strinati_1988} +Within the static approximation, the spin-conserved and spin-flip optical excitation at the BSE level are obtained by solving a similar linear response problem +\begin{equation} +\label{eq:LR-RPA} + \begin{pmatrix} + \bA{}{\BSE} & \bB{}{\BSE} \\ + -\bB{}{\BSE} & -\bA{}{\BSE} \\ + \end{pmatrix} + \cdot + \begin{pmatrix} + \bX{m}{\BSE} \\ + \bY{m}{\BSE} \\ + \end{pmatrix} + = + \Om{m}{\BSE} + \begin{pmatrix} + \bX{m}{\BSE} \\ + \bY{m}{\BSE} \\ + \end{pmatrix} +\end{equation} Defining $W^{\stat}_{p_\sig q_\sig,r_\sigp s_\sigp} = W_{p_\sig q_\sig,r_\sigp s_\sigp}(\omega = 0)$, we have \begin{subequations} \begin{align} @@ -277,6 +296,7 @@ for the spin-flip excitations. %================================ \subsection{Dynamical correction} %================================ +The dynamical correction to the static BSE kernel is defined in the Tamm-Dancoff approximation as \cite{Strinati_1988,Rohlfing_2000,Ma_2009a,Ma_2009b,Romaniello_2009b,Sangalli_2011,Loos_2020e} \begin{multline} \widetilde{W}_{p_\sig q_\sig,r_\sigp s_\sigp}(\omega) = \ERI{p_\sig q_\sig}{r_\sigp s_\sigp} @@ -507,7 +527,7 @@ This project has received funding from the European Research Council (ERC) under The data that supports the findings of this study are available within the article and its supplementary material. %%%%%%%%%%%%%%%%%%%%%%%% -\bibliography{sf-BSE} +\bibliography{sfBSE} %%%%%%%%%%%%%%%%%%%%%%%% \end{document}