From 893ce4905d9b6503d41af1387c9e385b65a12041 Mon Sep 17 00:00:00 2001 From: Pierre-Francois Loos Date: Tue, 20 Oct 2020 23:14:57 +0200 Subject: [PATCH] saving work --- sfBSE.rty | 94 +++++++++++++++++++++++++++++++++++++------------------ sfBSE.tex | 50 +++++++++++++++++++++++++++-- 2 files changed, 111 insertions(+), 33 deletions(-) diff --git a/sfBSE.rty b/sfBSE.rty index 635a22f..a4624de 100644 --- a/sfBSE.rty +++ b/sfBSE.rty @@ -4,6 +4,7 @@ \usepackage[normalem]{ulem} \newcommand{\titou}[1]{\textcolor{red}{#1}} \newcommand{\trashPFL}[1]{\textcolor{red}{\sout{#1}}} +\newcommand{\trashXB}[1]{\textcolor{darkgreen}{\sout{#1}}} \newcommand{\PFL}[1]{\titou{(\underline{\bf PFL}: #1)}} \newcommand{\mc}{\multicolumn} @@ -23,11 +24,9 @@ \newcommand{\qsGW}{qs$GW$} \newcommand{\GOWO}{$G_0W_0$} \newcommand{\Hxc}{\text{Hxc}} -\newcommand{\Hx}{\text{Hx}} \newcommand{\xc}{\text{xc}} \newcommand{\Ha}{\text{H}} -\newcommand{\co}{\text{c}} -\newcommand{\ex}{\text{x}} +\newcommand{\co}{\text{x}} % \newcommand{\Norb}{N_\text{orb}} @@ -37,21 +36,14 @@ % operators \newcommand{\hH}{\Hat{H}} +\newcommand{\ha}{\Hat{a}} % methods \newcommand{\KS}{\text{KS}} \newcommand{\HF}{\text{HF}} \newcommand{\RPA}{\text{RPA}} -\newcommand{\RPAx}{\text{RPAx}} -\newcommand{\TDHF}{\text{TDHF}} -\newcommand{\dRPAx}{\text{dRPAx}} \newcommand{\BSE}{\text{BSE}} -\newcommand{\TDABSE}{\text{BSE(TDA)}} -\newcommand{\dBSE}{\text{dBSE}} -\newcommand{\pBSE}{\text{pBSE}} -\newcommand{\TDAdBSE}{\text{dBSE(TDA)}} \newcommand{\GW}{GW} -\newcommand{\GF}{\text{GF2}} \newcommand{\stat}{\text{stat}} \newcommand{\dyn}{\text{dyn}} \newcommand{\TDA}{\text{TDA}} @@ -65,23 +57,52 @@ \newcommand{\EcBSE}{E_\text{c}^\text{BSE}} % orbital energies -\newcommand{\e}[2]{\eps_{#1}^{#2}} +\newcommand{\e}[1]{\eps_{#1}} \newcommand{\eHF}[1]{\eps^\text{HF}_{#1}} \newcommand{\eKS}[1]{\eps^\text{KS}_{#1}} \newcommand{\eQP}[1]{\eps^\text{QP}_{#1}} +\newcommand{\eGOWO}[1]{\eps^\text{\GOWO}_{#1}} \newcommand{\eGW}[1]{\eps^{GW}_{#1}} -\newcommand{\eGF}[1]{\eps^{\text{GF2}}_{#1}} +\newcommand{\eevGW}[1]{\eps^\text{\evGW}_{#1}} +\newcommand{\eGnWn}[2]{\eps^\text{\GnWn{#2}}_{#1}} \newcommand{\Om}[2]{\Omega_{#1}^{#2}} +\newcommand{\tOm}[2]{\Tilde{\Omega}_{#1}^{#2}} + +\newcommand{\homu}{\frac{{\omega}_1}{2}} + + % Matrix elements -\newcommand{\Z}[1]{Z_{#1}} +\newcommand{\A}[2]{A_{#1}^{#2}} +\newcommand{\tA}[2]{\Tilde{A}_{#1}^{#2}} +\newcommand{\B}[2]{B_{#1}^{#2}} +\renewcommand{\S}[1]{S_{#1}} +\newcommand{\ABSE}[2]{A_{#1}^{#2,\text{BSE}}} +\newcommand{\BBSE}[2]{B_{#1}^{#2,\text{BSE}}} +\newcommand{\ARPA}[2]{A_{#1}^{#2,\text{RPA}}} +\newcommand{\BRPA}[2]{B_{#1}^{#2,\text{RPA}}} +\newcommand{\ARPAx}[2]{A_{#1}^{#2,\text{RPAx}}} +\newcommand{\BRPAx}[2]{B_{#1}^{#2,\text{RPAx}}} +\newcommand{\G}[1]{G_{#1}} +\newcommand{\LBSE}[1]{L_{#1}} +\newcommand{\XiBSE}[1]{\Xi_{#1}} +\newcommand{\Po}[1]{P_{#1}} +\newcommand{\W}[2]{W_{#1}^{#2}} +\newcommand{\tW}[2]{\widetilde{W}_{#1}^{#2}} +\newcommand{\Wc}[1]{W^\text{c}_{#1}} +\newcommand{\vc}[1]{v_{#1}} \newcommand{\Sig}[1]{\Sigma_{#1}} -\newcommand{\SigGW}[1]{\Sigma^{\GW}_{#1}} -\newcommand{\SigGF}[1]{\Sigma^{\GF}_{#1}} +\newcommand{\SigGW}[1]{\Sigma^{GW}_{#1}} +\newcommand{\Z}[1]{Z_{#1}} \newcommand{\MO}[1]{\phi_{#1}} \newcommand{\ERI}[2]{(#1|#2)} \newcommand{\sERI}[2]{[#1|#2]} +%% bold in Table +\newcommand{\bb}[1]{\textbf{#1}} +\newcommand{\rb}[1]{\textbf{\textcolor{red}{#1}}} +\newcommand{\gb}[1]{\textbf{\textcolor{darkgreen}{#1}}} + % excitation energies \newcommand{\OmRPA}[1]{\Omega_{#1}^{\text{RPA}}} \newcommand{\OmRPAx}[1]{\Omega_{#1}^{\text{RPAx}}} @@ -94,19 +115,27 @@ % Matrices \newcommand{\bO}{\mathbf{0}} -\newcommand{\bH}{\mathbf{H}} -\newcommand{\bR}{\mathbf{R}} -\newcommand{\bS}{\mathbf{S}} -\newcommand{\bX}{\mathbf{X}} -\newcommand{\bY}{\mathbf{Y}} -\newcommand{\bV}{\mathbf{V}} \newcommand{\bI}{\mathbf{1}} -\newcommand{\bb}{\mathbf{b}} -\newcommand{\bA}{\mathbf{A}} -\newcommand{\bB}{\mathbf{B}} -\newcommand{\bC}{\mathbf{C}} -\newcommand{\bc}{\mathbf{c}} -\newcommand{\bx}{\mathbf{x}} +\newcommand{\bvc}{\mathbf{v}} +\newcommand{\bSig}{\mathbf{\Sigma}} +\newcommand{\bSigX}{\mathbf{\Sigma}^\text{x}} +\newcommand{\bSigC}{\mathbf{\Sigma}^\text{c}} +\newcommand{\bSigGW}{\mathbf{\Sigma}^{GW}} +\newcommand{\be}{\mathbf{\epsilon}} +\newcommand{\beGW}{\mathbf{\epsilon}^{GW}} +\newcommand{\beGnWn}[1]{\mathbf{\epsilon}^\text{\GnWn{#1}}} +\newcommand{\bde}{\mathbf{\Delta\epsilon}} +\newcommand{\bdeHF}{\mathbf{\Delta\epsilon}^\text{HF}} +\newcommand{\bdeGW}{\mathbf{\Delta\epsilon}^{GW}} +\newcommand{\bOm}[1]{\mathbf{\Omega}^{#1}} +\newcommand{\bA}[1]{\mathbf{A}^{#1}} +\newcommand{\btA}[1]{\Tilde{\mathbf{A}}^{#1}} +\newcommand{\bB}[1]{\mathbf{B}^{#1}} +\newcommand{\bX}[2]{\mathbf{X}_{#1}^{#2}} +\newcommand{\bY}[2]{\mathbf{Y}_{#1}^{#2}} +\newcommand{\bZ}[2]{\mathbf{Z}_{#1}^{#2}} +\newcommand{\bK}{\mathbf{K}} +\newcommand{\bP}[1]{\mathbf{P}^{#1}} % units \newcommand{\IneV}[1]{#1 eV} @@ -115,9 +144,7 @@ \newcommand{\kcal}{kcal/mol} % orbitals, gaps, etc -\newcommand{\updw}{\uparrow\downarrow} -\newcommand{\upup}{\uparrow\uparrow} -\newcommand{\eps}{\epsilon} +\newcommand{\eps}{\varepsilon} \newcommand{\IP}{I} \newcommand{\EA}{A} \newcommand{\HOMO}{\text{HOMO}} @@ -127,6 +154,11 @@ \newcommand{\EgOpt}{\Eg^\text{opt}} \newcommand{\EB}{E_B} +\newcommand{\si}{\sigma} +\newcommand{\sip}{\sigma'} + + +% addresses \newcommand{\LCPQ}{Laboratoire de Chimie et Physique Quantiques (UMR 5626), Universit\'e de Toulouse, CNRS, UPS, France} diff --git a/sfBSE.tex b/sfBSE.tex index c231098..cf05480 100644 --- a/sfBSE.tex +++ b/sfBSE.tex @@ -48,11 +48,57 @@ Unless otherwise stated, atomic units are used, and we assume real quantities th \subsection{The dynamical screening} +Within the $GW$ formalism, the dynamical screening $W(\omega)$ is computed at the RPA level using the spin-conserved neutral excitations. + +\begin{multline} + W_{pq\si,rs\sip}(\omega) = \ERI{pq\si}{rs\sip} + \\ + + \sum_m \sERI{pq\si}{m}\sERI{rs\sip}{m} \qty[ \frac{1}{\omega - \OmRPA{m} + i \eta} - \frac{1}{\omega + \OmRPA{m} - i \eta} ] +\end{multline} + + +\begin{equation} + \ERI{pq\si}{rs\sip} = \iint \MO{p\si}(\br) \MO{q\si}(\br) \frac{1}{\abs{\br - \br'}} \MO{r\sip}(\br') \MO{s\sip}(\br') d\br d\br' +\end{equation} + +\begin{equation} + \sERI{pq\si}{m} = \sum_{ia\sip} \ERI{pq\si}{rs\sip} (\bX{m}{\RPA}+\bY{m}{\RPA})_{ia\sip} +\end{equation} + +\begin{equation} +\label{eq:LR-RPA} + \begin{pmatrix} + \bA{\RPA} & \bB{\RPA} \\ + -\bB{\RPA} & -\bA{\RPA} \\ + \end{pmatrix} + \cdot + \begin{pmatrix} + \bX{m}{\RPA} \\ + \bY{m}{\RPA} \\ + \end{pmatrix} + = + \OmRPA{m} + \begin{pmatrix} + \bX{m}{\RPA} \\ + \bY{m}{\RPA} \\ + \end{pmatrix}, +\end{equation} +with +\begin{subequations} +\begin{align} + \label{eq:LR_RPA-A} + \A{ia\si,jb\sip}{\RPA} & = \delta_{ij} \delta_{ab} \delta_{\si\sip} (\e{a} - \e{i}) + 2 \ERI{ia\si}{jb\sip}, + \\ + \label{eq:LR_RPA-B} + \B{ia\si,jb\sip}{\RPA} & = 2 \ERI{ia\si}{bj\sip}, +\end{align} +\end{subequations} + \subsection{The $GW$ self-energy} -The quasiparticle energies $\eGW{p}$ are obtained by solving the +The quasiparticle energies $\eGW{p}$ are obtained by solving the frequency-dependent quasiparticle equation \begin{equation} - \omega = \eHF{p} + \Z{p} \SigGW{p}(\omega) + \omega = \eHF{p\sigma} + \SigGW{p\sigma}(\omega) \end{equation} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \subsection{Computational details}