From 205603483bb4c1c907f78260336cdecaa428d813 Mon Sep 17 00:00:00 2001 From: Pierre-Francois Loos Date: Sat, 24 Oct 2020 14:19:04 +0200 Subject: [PATCH] saving work in S2 and f --- sfBSE.rty | 5 ++-- sfBSE.tex | 89 +++++++++++++++++++++++++++++++++++++++++++------------ 2 files changed, 72 insertions(+), 22 deletions(-) diff --git a/sfBSE.rty b/sfBSE.rty index 3830d4d..a8a97d2 100644 --- a/sfBSE.rty +++ b/sfBSE.rty @@ -93,7 +93,6 @@ \newcommand{\Wc}[1]{W^\text{c}_{#1}} \newcommand{\vc}[1]{v_{#1}} \newcommand{\Sig}[1]{\Sigma_{#1}} -\newcommand{\SigGW}[1]{\Sigma^{GW}_{#1}} \newcommand{\SigC}[1]{\Sigma^\text{c}_{#1}} \newcommand{\SigX}[1]{\Sigma^\text{x}_{#1}} \newcommand{\SigXC}[1]{\Sigma^\text{xc}_{#1}} @@ -161,8 +160,8 @@ \newcommand{\bsigp}{{\Bar{\sigma}'}} \newcommand{\taup}{{\tau'}} -\newcommand{\up}{\downarrow} -\newcommand{\dw}{\uparrow} +\newcommand{\up}{\uparrow} +\newcommand{\dw}{\downarrow} \newcommand{\upup}{\uparrow\uparrow} \newcommand{\updw}{\uparrow\downarrow} \newcommand{\dwup}{\downarrow\uparrow} diff --git a/sfBSE.tex b/sfBSE.tex index e9e8f2c..8966e7d 100644 --- a/sfBSE.tex +++ b/sfBSE.tex @@ -116,24 +116,24 @@ The spin structure of these matrices are general and reads \begin{align} \label{eq:LR-RPA-AB} \bA{}{\spc} & = \begin{pmatrix} - \bA{\upup,\upup}{} & \bA{\upup,\dwdw}{} \\ - \bA{\dwdw,\upup}{} & \bA{\dwdw,\dwdw}{} \\ + \bA{}{\upup,\upup} & \bA{}{\upup,\dwdw} \\ + \bA{}{\dwdw,\upup} & \bA{}{\dwdw,\dwdw} \\ \end{pmatrix} & \bB{}{\spc} & = \begin{pmatrix} - \bB{\upup,\upup}{} & \bB{\upup,\dwdw}{} \\ - \bB{\dwdw,\upup}{} & \bB{\dwdw,\dwdw}{} \\ + \bB{}{\upup,\upup} & \bB{}{\upup,\dwdw} \\ + \bB{}{\dwdw,\upup} & \bB{}{\dwdw,\dwdw} \\ \end{pmatrix} \\ \label{eq:LR-RPA-AB} \bA{}{\spf} & = \begin{pmatrix} - \bA{\updw,\updw}{} & \bO \\ - \bO & \bA{\dwup,\dwup}{} \\ + \bA{}{\updw,\updw} & \bO \\ + \bO & \bA{}{\dwup,\dwup} \\ \end{pmatrix} & \bB{}{\spf} & = \begin{pmatrix} - \bO & \bB{\updw,\dwup}{} \\ - \bB{\dwup,\updw}{} & \bO \\ + \bO & \bB{}{\updw,\dwup} \\ + \bB{}{\dwup,\updw} & \bO \\ \end{pmatrix} \end{align} with @@ -171,26 +171,27 @@ for the spin-flip excitations. %================================ \subsection{The $GW$ self-energy} %================================ - +Within the acclaimed $GW$ approximation, the exchange-correlation part of the self-energy is defined as \begin{equation} - \Sig{}^{\sig}(\br_1,\br_2;\omega) - = \frac{i}{2\pi} \int G^{\sig}(\br_1,\br_2;\omega+\omega') W(\br_1,\br_2;\omega') e^{i \eta \omega'} d\omega' +\begin{split} + \Sig{}^{\text{xc},\sig}(\br_1,\br_2;\omega) + & = \Sig{}^{\text{x},\sig}(\br_1,\br_2) + \Sig{}^{\text{c},\sig}(\br_1,\br_2;\omega) + \\ + & = \frac{i}{2\pi} \int G^{\sig}(\br_1,\br_2;\omega+\omega') W(\br_1,\br_2;\omega') e^{i \eta \omega'} d\omega' +\end{split} \end{equation} - -\begin{equation} - \SigX{p_\sig q_\sig}(\omega) +and the spectral representation of its exchange and correlation part read +\begin{gather} + \SigX{p_\sig q_\sig} = - \frac{1}{2} \sum_{i\sigp} \ERI{p_\sig i_\sigp}{i_\sigp q_\sig} -\end{equation} - - -\begin{equation} + \\ \begin{split} \SigC{p_\sig q_\sig}(\omega) & = \sum_{im} \frac{\ERI{p_\sig i_\sig}{m} \ERI{q_\sig i_\sig}{m}}{\omega - \e{i_\sig} + \Om{m}{\spc,\RPA} - i \eta} \\ & + \sum_{am} \frac{\ERI{p_\sig a_\sig}{m} \ERI{q_\sig a_\sig}{m}}{\omega - \e{a_\sig} - \Om{m}{\spc,\RPA} + i \eta} \end{split} -\end{equation} +\end{gather} The quasiparticle energies $\eGW{p}$ are obtained by solving the frequency-dependent quasiparticle equation \begin{equation} @@ -410,12 +411,62 @@ and \end{equation} +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +\subsection{Oscillator strengths} +\label{sec:os} +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +For the spin-conserved transition, the transition dipole moment in the $x$ direction is +\begin{equation} + \mu_{x,m}^{\spc} = \sum_{ia\sig} (i_\sig|x|a_\sig)(\bX{m}{\spc}+\bY{m}{\spc})_{i_\sig a_\sig} +\end{equation} +with +\begin{equation} + (p_\sig|x|q_\sigp) = \int \MO{p_\sig}(\br) \, x \, \MO{q_\sigp}(\br) d\br +\end{equation} +and the total oscillator strength is given by +\begin{equation} + f_{m}^{\spc} = \frac{2}{3} \Om{m}{\spc} \qty[ \qty(\mu_{x,m}^{\spc})^2 + \qty(\mu_{x,m}^{\spc})^2 + \qty(\mu_{x,m}^{\spc})^2 ] +\end{equation} +For spin-flip transitions, we have $f_{m}^{\spf} = 0$ as the transition matrix elements $(i_\sig|x|a_\bsig)$ vanish. + +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +\subsection{Spin contamination} +\label{sec:spin} +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + +\begin{equation} + \expval{S^2}_m = \expval{S^2}_0 + \Delta \expval{S^2}_m +\end{equation} + +\begin{equation} + \expval{S^2}_{0} + = \frac{n_{\up} - n_{\dw}}{2} \qty( \frac{n_{\up} - n_{\dw}}{2} + 1 ) + + n_{\dw} - \sum_p (p_{\up}|p_{\dw})^2 +\end{equation} +where +\begin{equation} + (p_\sig|q_\sigp) = \int \MO{p_\sig}(\br) \MO{q_\sigp}(\br) d\br +\end{equation} +is the overlap between spin-up and spin-down orbitals. + +The explicit expressions of $\Delta \expval{S^2}_m^{\spc}$ and $\Delta \expval{S^2}_m^{\spf}$ are given in Ref.~\onlinecite{Li_2010}. +As explained in Ref.~\onlinecite{Casanova_2020}, there are two sources of spin contamination: i) spin contamination of the reference, and ii) spin-contamination of the excited states due to the spin incompleteness. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \section{Computational details} \label{sec:compdet} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +\section{Results} +\label{sec:res} +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + +%%% TABLE I %%% +%\begin{table} +% +%\end{table} +%%% %%% %%% %%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \section{Conclusion} \label{sec:ccl}