saving work in S2 and f
This commit is contained in:
parent
413b3f37d1
commit
205603483b
@ -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}
|
||||
|
||||
89
sfBSE.tex
89
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}
|
||||
|
||||
Loading…
Reference in New Issue
Block a user