saving work in S2 and f

This commit is contained in:
Pierre-Francois Loos 2020-10-24 14:19:04 +02:00
parent 413b3f37d1
commit 205603483b
2 changed files with 72 additions and 22 deletions

View File

@ -93,7 +93,6 @@
\newcommand{\Wc}[1]{W^\text{c}_{#1}} \newcommand{\Wc}[1]{W^\text{c}_{#1}}
\newcommand{\vc}[1]{v_{#1}} \newcommand{\vc}[1]{v_{#1}}
\newcommand{\Sig}[1]{\Sigma_{#1}} \newcommand{\Sig}[1]{\Sigma_{#1}}
\newcommand{\SigGW}[1]{\Sigma^{GW}_{#1}}
\newcommand{\SigC}[1]{\Sigma^\text{c}_{#1}} \newcommand{\SigC}[1]{\Sigma^\text{c}_{#1}}
\newcommand{\SigX}[1]{\Sigma^\text{x}_{#1}} \newcommand{\SigX}[1]{\Sigma^\text{x}_{#1}}
\newcommand{\SigXC}[1]{\Sigma^\text{xc}_{#1}} \newcommand{\SigXC}[1]{\Sigma^\text{xc}_{#1}}
@ -161,8 +160,8 @@
\newcommand{\bsigp}{{\Bar{\sigma}'}} \newcommand{\bsigp}{{\Bar{\sigma}'}}
\newcommand{\taup}{{\tau'}} \newcommand{\taup}{{\tau'}}
\newcommand{\up}{\downarrow} \newcommand{\up}{\uparrow}
\newcommand{\dw}{\uparrow} \newcommand{\dw}{\downarrow}
\newcommand{\upup}{\uparrow\uparrow} \newcommand{\upup}{\uparrow\uparrow}
\newcommand{\updw}{\uparrow\downarrow} \newcommand{\updw}{\uparrow\downarrow}
\newcommand{\dwup}{\downarrow\uparrow} \newcommand{\dwup}{\downarrow\uparrow}

View File

@ -116,24 +116,24 @@ The spin structure of these matrices are general and reads
\begin{align} \begin{align}
\label{eq:LR-RPA-AB} \label{eq:LR-RPA-AB}
\bA{}{\spc} & = \begin{pmatrix} \bA{}{\spc} & = \begin{pmatrix}
\bA{\upup,\upup}{} & \bA{\upup,\dwdw}{} \\ \bA{}{\upup,\upup} & \bA{}{\upup,\dwdw} \\
\bA{\dwdw,\upup}{} & \bA{\dwdw,\dwdw}{} \\ \bA{}{\dwdw,\upup} & \bA{}{\dwdw,\dwdw} \\
\end{pmatrix} \end{pmatrix}
& &
\bB{}{\spc} & = \begin{pmatrix} \bB{}{\spc} & = \begin{pmatrix}
\bB{\upup,\upup}{} & \bB{\upup,\dwdw}{} \\ \bB{}{\upup,\upup} & \bB{}{\upup,\dwdw} \\
\bB{\dwdw,\upup}{} & \bB{\dwdw,\dwdw}{} \\ \bB{}{\dwdw,\upup} & \bB{}{\dwdw,\dwdw} \\
\end{pmatrix} \end{pmatrix}
\\ \\
\label{eq:LR-RPA-AB} \label{eq:LR-RPA-AB}
\bA{}{\spf} & = \begin{pmatrix} \bA{}{\spf} & = \begin{pmatrix}
\bA{\updw,\updw}{} & \bO \\ \bA{}{\updw,\updw} & \bO \\
\bO & \bA{\dwup,\dwup}{} \\ \bO & \bA{}{\dwup,\dwup} \\
\end{pmatrix} \end{pmatrix}
& &
\bB{}{\spf} & = \begin{pmatrix} \bB{}{\spf} & = \begin{pmatrix}
\bO & \bB{\updw,\dwup}{} \\ \bO & \bB{}{\updw,\dwup} \\
\bB{\dwup,\updw}{} & \bO \\ \bB{}{\dwup,\updw} & \bO \\
\end{pmatrix} \end{pmatrix}
\end{align} \end{align}
with with
@ -171,26 +171,27 @@ for the spin-flip excitations.
%================================ %================================
\subsection{The $GW$ self-energy} \subsection{The $GW$ self-energy}
%================================ %================================
Within the acclaimed $GW$ approximation, the exchange-correlation part of the self-energy is defined as
\begin{equation} \begin{equation}
\Sig{}^{\sig}(\br_1,\br_2;\omega) \begin{split}
= \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' \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} \end{equation}
and the spectral representation of its exchange and correlation part read
\begin{equation} \begin{gather}
\SigX{p_\sig q_\sig}(\omega) \SigX{p_\sig q_\sig}
= - \frac{1}{2} \sum_{i\sigp} \ERI{p_\sig i_\sigp}{i_\sigp q_\sig} = - \frac{1}{2} \sum_{i\sigp} \ERI{p_\sig i_\sigp}{i_\sigp q_\sig}
\end{equation} \\
\begin{equation}
\begin{split} \begin{split}
\SigC{p_\sig q_\sig}(\omega) \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_{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} & + \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{split}
\end{equation} \end{gather}
The quasiparticle energies $\eGW{p}$ are obtained by solving the frequency-dependent quasiparticle equation The quasiparticle energies $\eGW{p}$ are obtained by solving the frequency-dependent quasiparticle equation
\begin{equation} \begin{equation}
@ -410,12 +411,62 @@ and
\end{equation} \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} \section{Computational details}
\label{sec:compdet} \label{sec:compdet}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Results}
\label{sec:res}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%% TABLE I %%%
%\begin{table}
%
%\end{table}
%%% %%% %%% %%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Conclusion} \section{Conclusion}
\label{sec:ccl} \label{sec:ccl}