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Pierre-Francois Loos 2020-10-25 13:30:30 +01:00
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13
sfBSE.bib
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@ -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},

30
sfBSE.tex
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@ -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}