From be611bde7eb3ca33740c81451f441fefdd8333cb Mon Sep 17 00:00:00 2001 From: Pierre-Francois Loos Date: Tue, 27 Apr 2021 21:37:10 +0200 Subject: [PATCH] final v1 and update v2 --- 2021/Lecture_1/ISTPC_Loos_1.tex | 38 ++- 2021/Lecture_1/fig/STO.pdf | Bin 0 -> 41690 bytes 2021/Lecture_2/ISTPC_Loos_2.tex | 482 ++++++++++++++++++++++++++------ 2021/Lecture_2/fig/BSE-GW.tex | 74 +++++ 2021/Lecture_2/fig/Cramer.jpg | Bin 31143 -> 0 bytes 2021/Lecture_2/fig/Helgaker.jpg | Bin 103839 -> 0 bytes 2021/Lecture_2/fig/Jensen.jpg | Bin 30315 -> 0 bytes 2021/Lecture_2/fig/Szabo.jpg | Bin 3172373 -> 0 bytes 2021/Lecture_2/fig/gaps.png | Bin 0 -> 208844 bytes 2021/Lecture_2/fig/pentagon.pdf | Bin 0 -> 47233 bytes 10 files changed, 494 insertions(+), 100 deletions(-) create mode 100644 2021/Lecture_1/fig/STO.pdf create mode 100644 2021/Lecture_2/fig/BSE-GW.tex delete mode 100644 2021/Lecture_2/fig/Cramer.jpg delete mode 100644 2021/Lecture_2/fig/Helgaker.jpg delete mode 100644 2021/Lecture_2/fig/Jensen.jpg delete mode 100644 2021/Lecture_2/fig/Szabo.jpg create mode 100644 2021/Lecture_2/fig/gaps.png create mode 100644 2021/Lecture_2/fig/pentagon.pdf diff --git a/2021/Lecture_1/ISTPC_Loos_1.tex b/2021/Lecture_1/ISTPC_Loos_1.tex index 869310b..8e7d459 100644 --- a/2021/Lecture_1/ISTPC_Loos_1.tex +++ b/2021/Lecture_1/ISTPC_Loos_1.tex @@ -191,15 +191,25 @@ decoration={snake, %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{frame}{One- and two-electron integrals} - \begin{block}{One-electron integrals: overlap \& core Hamiltonian (Appendix A)} - \begin{equation} - S_{\mu\nu} = \int \phi_\mu(\orange{\br}) \phi_\nu(\orange{\br}) d\orange{\br} - \end{equation} - \begin{equation} - H_{\mu\nu} = \int \phi_\mu(\orange{\br}) \hH^\text{c}(\orange{\br}) \phi_\nu(\orange{\br}) d\orange{\br} - \end{equation} - \end{block} - \begin{block}{Chemist/Mulliken notation for two-electron integrals (p.~68)} + \begin{columns} + \begin{column}{0.7\textwidth} + \begin{block}{One-electron integrals: overlap \& core Hamiltonian (Appendix A)} + \begin{equation} + S_{\mu\nu} + = \braket{\mu}{\nu} + = \int \phi_\mu(\orange{\br}) \phi_\nu(\orange{\br}) d\orange{\br} + \end{equation} + \begin{equation} + H_{\mu\nu} + = \mel{\mu}{\hH^\text{c}}{\nu} + = \int \phi_\mu(\orange{\br}) \hH^\text{c}(\orange{\br}) \phi_\nu(\orange{\br}) d\orange{\br} + \end{equation} + \end{block} + \end{column} + \begin{column}{0.3\textwidth} + \includegraphics[width=\textwidth]{fig/SBG} + \end{column} + \end{columns} \begin{block}{Chemist/Mulliken notation for two-electron integrals (p.~68)} \begin{equation} ( \mu \nu | \lambda \sigma ) = \iint \phi_\mu(\alert{\br_1}) \phi_\nu(\alert{\br_1}) \frac{1}{r_{12}} \phi_\lambda(\blue{\br_2}) \phi_\sigma(\blue{\br_2}) d\red{\br_1} d\blue{\br_2} @@ -234,10 +244,11 @@ decoration={snake, \phi_{\bb_1}^{\bB_1}(\br_1) \phi_{\bb_2}^{\bB_2}(\br_2) d\br_1 d\br_2 \end{split} \end{equation} + \alert{Formally, one has to compute $\order{N^4}$ ERIs!} \end{block} \end{column} \begin{column}{0.3\textwidth} - \includegraphics[width=\textwidth]{fig/SBG} + \includegraphics[width=\textwidth]{fig/STO} \end{column} \end{columns} % @@ -315,14 +326,14 @@ decoration={snake, \qqtext{then} \end{equation} \begin{equation} - \boxed{G_{\red{\alpha},\red{\bm{A}}}(\br) G_{\blue{\beta},\blue{\bm{B}}}(\br) = \green{K} \, G_{\violet{\zeta},\violet{\bm{P}}}(\br)} + \boxed{G_{\red{\alpha},\red{\bm{A}}}(\br) G_{\blue{\beta},\blue{\bm{B}}}(\br) = \violet{K} \, G_{\violet{\zeta},\violet{\bm{P}}}(\br)} \qqtext{with} \violet{\zeta} = \red{\alpha} + \blue{\beta} \qqtext{and} \violet{\bm{P}} = \frac{\red{\alpha \bA} + \blue{\beta \bB}}{\red{\alpha} + \blue{\beta} } \end{equation} \begin{equation} - \green{K} = \exp( -\frac{\red{\alpha} \blue{\beta}}{\red{\alpha} + \blue{\beta} } \abs{\red{\bA} - \blue{\bB}}^2) + \violet{K} = \exp( -\frac{\red{\alpha} \blue{\beta}}{\red{\alpha} + \blue{\beta} } \abs{\red{\bA} - \blue{\bB}}^2) \end{equation} \end{block} \begin{block}{Gaussian product rule for ERIs} @@ -331,9 +342,10 @@ decoration={snake, (\bm{\red{a}} \bm{\blue{b}}|\bm{\orange{c}} \bm{\green{d}}) & = \iint G_{\red{\alpha},\red{\bm{A}}}(\br_1) G_{\blue{\beta},\blue{\bm{B}}}(\br_1) \frac{1}{r_{12}} G_{\orange{\gamma},\orange{\bm{C}}}(\br_2) G_{\green{\delta},\green{\bm{D}}}(\br_2) d\br_1 d\br_2 \\ - & = \iint G_{\violet{\zeta},\violet{\bm{P}}}(\br_1) \frac{1}{r_{12}} G_{\purple{\eta},\purple{\bm{Q}}}(\br_2) d\br_1 d\br_2 + & = \violet{K} \purple{K} \iint G_{\violet{\zeta},\violet{\bm{P}}}(\br_1) \frac{1}{r_{12}} G_{\purple{\eta},\purple{\bm{Q}}}(\br_2) d\br_1 d\br_2 \end{split} \end{equation} + \alert{The number of ``significant'' ERIs in a large system is $\order{N^2}$!} \end{block} \end{frame} % diff --git a/2021/Lecture_1/fig/STO.pdf b/2021/Lecture_1/fig/STO.pdf new file mode 100644 index 0000000000000000000000000000000000000000..33f9c8c61bc13bb14e8d2f2e156c0de372737d16 GIT binary patch literal 41690 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b/2021/Lecture_2/ISTPC_Loos_2.tex @@ -1,11 +1,11 @@ -\documentclass[aspectratio=169,9pt]{beamer} +\documentclass[aspectratio=169,9pt,compress]{beamer} % *********** % * PACKAGE * % *********** \usepackage{amsmath,amssymb,amsfonts,pgfpages,graphicx,subfigure,xcolor,bm,multirow,microtype,wasysym,multimedia,hyperref,tabularx,amscd,pgfgantt,mhchem} \usetikzlibrary{shapes.gates.logic.US,trees,positioning,arrows} -\usetheme{Pittsburgh} -\usecolortheme{seahorse} +\usetheme{Warsaw} +%\usecolortheme{seahorse} \usepackage{mathpazo,libertine} \usepackage{algorithmicx,algorithm,algpseudocode} @@ -25,65 +25,161 @@ citecolor=purple } +\definecolor{darkgreen}{RGB}{0, 180, 0} +\newcommand{\red}[1]{\textcolor{red}{#1}} +\newcommand{\purple}[1]{\textcolor{purple}{#1}} +\newcommand{\orange}[1]{\textcolor{orange}{#1}} +\newcommand{\green}[1]{\textcolor{darkgreen}{#1}} +\newcommand{\blue}[1]{\textcolor{blue}{#1}} +\newcommand{\pub}[1]{\textcolor{purple}{#1}} +\newcommand{\violet}[1]{\textcolor{violet}{#1}} + +\newcommand{\cdash}{\multicolumn{1}{c}{---}} +\newcommand{\mc}{\multicolumn} +\newcommand{\mcc}[1]{\multicolumn{1}{c}{#1}} +\newcommand{\mr}{\multirow} +\newcommand{\br}{\bm{r}} +\newcommand{\ree}{r_{12}} + +% methods +\newcommand{\evGW}{evGW} +\newcommand{\qsGW}{qsGW} +\newcommand{\scGW}{scGW} +\newcommand{\GOWO}{G$_0$W$_0$} +\newcommand{\GOW}{G$_0$W} +\newcommand{\GWO}{GW$_0$} +\newcommand{\GW}{GW} +\newcommand{\GOWOSOSEX}{{\GOWO}+SOSEX} +\newcommand{\GWSOSEX}{{\GW}+SOSEX} +\newcommand{\GnWn}[1]{G$_{#1}$W$_{#1}$} +\newcommand{\GOF}{G$_0$F2} +\newcommand{\GF}{GF2} + % operators -\newcommand{\hI}{\Hat{1}} \newcommand{\hH}{\Hat{H}} -\newcommand{\hT}[2]{\Hat{T}_{#1}^{#2}} -\newcommand{\bH}{\mathbold{H}} -\newcommand{\br}{\mathbold{r}} -\newcommand{\la}{\lambda} -\newcommand{\si}{\sigma} - -\newcommand{\cJ}{\mathcal{J}} -\newcommand{\cK}{\mathcal{K}} - -% wave functions -\newcommand{\PsiO}{\Psi_0} -\newcommand{\PsiHF}{\Psi_\text{RHF}} -\newcommand{\PsiFCI}{\Psi_\text{FCI}} -\newcommand{\PsiCC}{\Psi_\text{CC}} -\newcommand{\PsiCCD}{\Psi_\text{CCD}} - -\newcommand{\amp}[2]{t_{#1}^{#2}} -\newcommand{\Det}[2]{\Psi_{#1}^{#2}} % energies +\newcommand{\Ec}{E_\text{c}} \newcommand{\EHF}{E_\text{HF}} -\newcommand{\EO}{E_\text{0}} -\newcommand{\ECC}{E_\text{CC}} -\newcommand{\EVCC}{E_\text{VCC}} -\newcommand{\ECCD}{E_\text{CCD}} +\newcommand{\EcK}{E_\text{c}^\text{Klein}} +\newcommand{\EcRPA}{E_\text{c}^\text{RPA}} +\newcommand{\EcGM}{E_\text{c}^\text{GM}} +\newcommand{\EcGMGW}{E_\text{c}^\text{GM@GW}} +\newcommand{\EcGMGF}{E_\text{c}^\text{GM@GF2}} +\newcommand{\EcGMGWSOSEX}{E_\text{c}^\text{GM@GW+SOSEX}} +\newcommand{\EcMP}{E_c^\text{MP2}} +\newcommand{\EcGF}{E_c^\text{\GF}} +\newcommand{\EcGOF}{E_c^\text{\GOF}} +\newcommand{\Egap}{E_\text{gap}} +\newcommand{\IP}{\text{IP}} +\newcommand{\EA}{\text{EA}} -\newcommand{\nEl}{n} -\newcommand{\nBas}{N} +% orbital energies +\newcommand{\nSat}[1]{N_{#1}^\text{sat}} +\newcommand{\eSat}[2]{\epsilon_{#1,#2}} +\newcommand{\e}[1]{\epsilon_{#1}} +\newcommand{\eHF}[1]{\epsilon^\text{HF}_{#1}} +\newcommand{\eKS}[1]{\epsilon^\text{KS}_{#1}} +\newcommand{\eQP}[1]{\epsilon^\text{QP}_{#1}} +\newcommand{\eGOWO}[1]{\epsilon^\text{\GOWO}_{#1}} +\newcommand{\eGW}[1]{\epsilon^\text{\GW}_{#1}} +\newcommand{\eGnWn}[2]{\epsilon^\text{\GnWn{#2}}_{#1}} +\newcommand{\eGF}[1]{\epsilon^\text{\GF}_{#1}} +\newcommand{\eGOF}[1]{\epsilon^\text{\GOF}_{#1}} +\newcommand{\de}[1]{\Delta\epsilon_{#1}} +\newcommand{\deHF}[1]{\Delta\epsilon^\text{HF}_{#1}} +\newcommand{\Om}[1]{\Omega_{#1}} +\newcommand{\eHOMO}{\epsilon_\text{HOMO}} +\newcommand{\eLUMO}{\epsilon_\text{LUMO}} -\newcommand{\ba}{\bm{a}} -\newcommand{\bb}{\bm{b}} +\newcommand{\cHF}[1]{c^\text{HF}_{#1}} +\newcommand{\cKS}[1]{c^\text{KS}_{#1}} + + +% Matrix elements +\newcommand{\A}[1]{A_{#1}} +\newcommand{\B}[1]{B_{#1}} +\renewcommand{\S}[1]{S_{#1}} +\newcommand{\ABSE}[1]{A^\text{BSE}_{#1}} +\newcommand{\BBSE}[1]{B^\text{BSE}_{#1}} +\newcommand{\ARPA}[1]{A^\text{RPA}_{#1}} +\newcommand{\BRPA}[1]{B^\text{RPA}_{#1}} +\newcommand{\dABSE}[1]{\delta A^\text{BSE}_{#1}} +\newcommand{\dBBSE}[1]{\delta B^\text{BSE}_{#1}} +\newcommand{\G}[1]{G_{#1}} +\newcommand{\Po}[1]{P_{#1}} +\newcommand{\W}[1]{W_{#1}} +\newcommand{\Wc}[1]{W^\text{c}_{#1}} +\newcommand{\vc}[1]{v_{#1}} +\newcommand{\SigX}[1]{\Sigma^\text{x}_{#1}} +\newcommand{\SigC}[1]{\Sigma^\text{c}_{#1}} +\newcommand{\SigGW}[1]{\Sigma^\text{\GW}_{#1}} +\newcommand{\SigGWSOSEX}[1]{\Sigma^\text{\GWSOSEX}_{#1}} +\newcommand{\SigGF}[1]{\Sigma^\text{\GF}_{#1}} +\newcommand{\Z}[1]{Z_{#1}} + +% excitation energies +\newcommand{\OmRPA}[1]{\Omega^\text{RPA}_{#1}} +\newcommand{\OmCIS}[1]{\Omega^\text{CIS}_{#1}} +\newcommand{\OmTDHF}[1]{\Omega^\text{TDHF}_{#1}} +\newcommand{\OmBSE}[1]{\Omega^\text{BSE}_{#1}} + +\newcommand{\spinup}{\downarrow} +\newcommand{\spindw}{\uparrow} +\newcommand{\singlet}{\uparrow\downarrow} +\newcommand{\triplet}{\uparrow\uparrow} + +\newcommand{\Oms}[1]{{}^{1}\Omega_{#1}} +\newcommand{\OmsRPA}[1]{{}^{1}\Omega^\text{RPA}_{#1}} +\newcommand{\OmsCIS}[1]{{}^{1}\Omega^\text{CIS}_{#1}} +\newcommand{\OmsTDHF}[1]{{}^{1}\Omega^\text{TDHF}_{#1}} +\newcommand{\OmsBSE}[1]{{}^{1}\Omega^\text{BSE}_{#1}} + +\newcommand{\Omt}[1]{{}^{3}\Omega_{#1}} +\newcommand{\OmtRPA}[1]{{}^{3}\Omega^\text{RPA}_{#1}} +\newcommand{\OmtCIS}[1]{{}^{3}\Omega^\text{CIS}_{#1}} +\newcommand{\OmtTDHF}[1]{{}^{3}\Omega^\text{TDHF}_{#1}} +\newcommand{\OmtBSE}[1]{{}^{3}\Omega^\text{BSE}_{#1}} + +% Wigner symbols +\newcommand{\WJ}[3]{ +\begin{pmatrix} +#1 & #2 & #3 \\ +0 & 0 & 0 \\ +\end{pmatrix} +} +\newcommand{\ERI}[3]{\qty(#1 #2 #3)} +\newcommand{\sERI}[3]{\qty[#1 #2 #3]} + +% Matrices +\newcommand{\bF}{\bm{F}} +\newcommand{\bFHF}{\bm{F}^\text{HF}} +\newcommand{\bH}{\bm{H}} +\newcommand{\bvc}{\bm{v}} +\newcommand{\bSig}{\bm{\Sigma}} +\newcommand{\bSigX}{\bm{\Sigma}^\text{x}} +\newcommand{\bSigC}{\bm{\Sigma}^\text{c}} +\newcommand{\bSigGW}{\bm{\Sigma}^\text{\GW}} +\newcommand{\bSigGWSOSEX}{\bm{\Sigma}^\text{\GWSOSEX}} +\newcommand{\bSigGF}{\bm{\Sigma}^\text{\GF}} +\newcommand{\be}{\bm{\epsilon}} +\newcommand{\bDelta}{\bm{\Delta}} +\newcommand{\beHF}{\bm{\epsilon}^\text{HF}} +\newcommand{\bcHF}{\bm{c}^\text{HF}} +\newcommand{\beGW}{\bm{\epsilon}^\text{\GW}} +\newcommand{\beGnWn}[1]{\bm{\epsilon}^\text{\GnWn{#1}}} +\newcommand{\bcGnWn}[1]{\bm{c}^\text{\GnWn{#1}}} +\newcommand{\beGF}{\bm{\epsilon}^\text{\GF}} +\newcommand{\bde}{\bm{\Delta\epsilon}} +\newcommand{\bdeHF}{\bm{\Delta\epsilon}^\text{HF}} +\newcommand{\bdeGW}{\bm{\Delta\epsilon}^\text{GW}} +\newcommand{\bdeGF}{\bm{\Delta\epsilon}^\text{GF2}} +\newcommand{\bOm}{\bm{\Omega}} \newcommand{\bA}{\bm{A}} \newcommand{\bB}{\bm{B}} -\newcommand{\bo}{\bm{0}} -\newcommand{\sbra}[1]{[ #1 |} -\newcommand{\sket}[1]{| #1 ]} -\newcommand{\sexpval}[1]{[ #1 ]} -\newcommand{\sbraket}[2]{[ #1 | #2 ]} -\newcommand{\smel}[3]{[ #1 | #2 | #3 ]} - - -\definecolor{darkgreen}{RGB}{0, 180, 0} -\definecolor{fooblue}{RGB}{0,153,255} -\definecolor{fooyellow}{RGB}{234,187,0} -\definecolor{lavender}{rgb}{0.71, 0.49, 0.86} -\definecolor{inchworm}{rgb}{0.7, 0.93, 0.36} -\newcommand{\violet}[1]{\textcolor{lavender}{#1}} -\newcommand{\orange}[1]{\textcolor{orange}{#1}} -\newcommand{\purple}[1]{\textcolor{purple}{#1}} -\newcommand{\blue}[1]{\textcolor{blue}{#1}} -\newcommand{\green}[1]{\textcolor{darkgreen}{#1}} -\newcommand{\yellow}[1]{\textcolor{fooyellow}{#1}} -\newcommand{\red}[1]{\textcolor{red}{#1}} -\newcommand{\highlight}[1]{\textcolor{fooblue}{#1}} -\newcommand{\pub}[1]{\small \textcolor{purple}{#1}} -\newcommand{\mc}{\multicolumn} +\newcommand{\bX}{\bm{X}} +\newcommand{\bY}{\bm{Y}} +\newcommand{\bZ}{\bm{Z}} \newcommand{\mycirc}[1][black]{\Large\textcolor{#1}{\ensuremath\bullet}} @@ -101,8 +197,8 @@ decoration={snake, % * HEAD DATA * % ************* \title[$GW$/BSE methods in chemistry]{ - \purple{$GW$/BSE methods in chemistry: \\ - Computational aspects} + $GW$/BSE methods in chemistry: + Computational aspects } \author[PF Loos]{Pierre-Fran\c{c}ois LOOS} \date{Online ISTPC 2021 school --- April 27th, 2021} @@ -111,7 +207,7 @@ decoration={snake, Universit\'e de Toulouse, CNRS, UPS, Toulouse, France. } \titlegraphic{ - \includegraphics[width=0.4\textwidth]{fig/jarvis} + \includegraphics[width=0.3\textwidth]{fig/jarvis} \\ \vspace{0.05\textheight} \includegraphics[height=0.05\textwidth]{fig/UPS} @@ -139,44 +235,256 @@ decoration={snake, \end{frame} % - -%%% FINAL SLIDE %%% %----------------------------------------------------- -\section{Books} +\section{Theory} %----------------------------------------------------- -\begin{frame}{Good books} +\subsection{Hedin's pentagon} +%----------------------------------------------------- +\begin{frame}{Hedin's pentagon} \begin{columns} - \begin{column}{0.7\textwidth} - \begin{itemize} - \item Introduction to Computational Chemistry (Jensen) - \\ - \vspace{1cm} - \item Essentials of Computational Chemistry (Cramer) - \\ - \vspace{1cm} - \item Modern Quantum Chemistry (Szabo \& Ostlund) - \\ - \vspace{1cm} - \item Molecular Electronic Structure Theory (Helgaker, Jorgensen \& Olsen) - \\ - \vspace{1cm} - \end{itemize} + \begin{column}{0.4\textwidth} + \centering + \includegraphics[width=0.8\linewidth]{fig/pentagon} + \\ + \pub{Hedin, Phys Rev 139 (1965) A796} \end{column} - \begin{column}{0.3\textwidth} - \centering - \includegraphics[height=0.3\textwidth]{fig/Jensen} - \\ - \bigskip - \includegraphics[height=0.3\textwidth]{fig/Cramer} - \\ - \bigskip - \includegraphics[height=0.3\textwidth]{fig/Szabo} - \\ - \bigskip - \includegraphics[height=0.3\textwidth]{fig/Helgaker} + \begin{column}{0.6\textwidth} + \begin{block}{What can you calculate with GW?} + \begin{itemize} + \item Ionization potentials (IP) given by occupied MO energies + \bigskip + \item Electron affinities (EA) given by virtual MO energies + \bigskip + \item HOMO-LUMO gap (or band gap in solids) + \bigskip + \item Singlet and triplet neutral excitations (vertical absorption energies) + \bigskip + \item Correlation and total energies + \end{itemize} + \end{block} \end{column} \end{columns} \end{frame} +%----------------------------------------------------- +\subsection{GW flavours} +%----------------------------------------------------- +\begin{frame}{GW flavours} + \begin{block}{Acronyms} + \begin{itemize} + \bigskip + \item perturbative GW one-shot GW, or \green{\GOWO} + \bigskip + \item \orange{\evGW} or eigenvalue-only (partially) self-consistent GW + \bigskip + \item \red{\qsGW} or quasiparticle (partially) self-consistent GW + \bigskip + \item \violet{\scGW} or (fully) self-consistent GW + \bigskip + \item \purple{BSE} or Bethe-Salpeter equation for neutral excitations + \bigskip + \end{itemize} + \end{block} +\end{frame} +%----------------------------------------------------- +\subsection{Literature} +%----------------------------------------------------- +\begin{frame}{useful papers for chemists} + \begin{itemize} + \item \red{molGW:} Bruneval et al. Comp. Phys. Comm. 208 (2016) 149 + \bigskip + \item \green{Turbomole:} van Setten et al. JCTC 9 (2013) 232; Kaplan et al. JCTC 12 (2016) 2528 + \bigskip + \item \violet{Fiesta:} Blase et al. Chem. Soc. Rev. 47 (2018) 1022 + \bigskip + \item \purple{FHI-AIMS:} Caruso et al. 86 (2012) 081102 + \bigskip + \item \orange{Review:} Reining, WIREs Comput Mol Sci 2017, e1344. doi: 10.1002/wcms.1344; Onida et al. Rev. Mod. Phys. 74 (2002) 601. + \bigskip + \item \red{GW100:} Data set of 100 molecules. van Setten et al. JCTC 11 (2015) 5665 + \end{itemize} +\end{frame} + +%----------------------------------------------------- +\section{Implementation} +%----------------------------------------------------- +\subsection{\GOWO} +%----------------------------------------------------- +\begin{frame}{\GOWO} + \begin{block}{{\GOWO}~subroutine} + \begin{algorithmic} + \Procedure{Perturbative {\GW}}{} + \State Perform HF calculation to get $\beHF$ and $\bcHF$ + \For{$p=1,\ldots,N$} + \State Compute \red{$\SigC{pp}(\omega)$} and \green{$\Z{p}(\omega)$} + \State $\eGOWO{p} = \eHF{p} + \green{\Z{p}(\eHF{p})} \Re[\red{\SigC{pp}(\eHF{p})}]$ + \State \Comment{This is the linearized version of the} + \State \Comment{quasiparticle (QP) equation + $\omega = \eHF{p} + \Re[\red{\SigC{pp}(\omega)}]$} + \EndFor + \If{BSE} + \State Compute BSE excitations energies + \EndIf + \EndProcedure + \end{algorithmic} + \end{block} +\end{frame} + +\begin{frame}{\GOWO} + \begin{block}{\red{Correlation part of the self-energy:}} + \begin{equation*} + \red{\SigC{pq}(\omega)} + = 2 \sum_{ix}\frac{\violet{[pi|x] [qi|x]}}{\omega - \eHF{i} + \orange{\Om{x}} - i \eta} + + 2 \sum_{ax}\frac{\violet{[pa|x] [qa|x]}}{\omega - \eHF{a} - \orange{\Om{x}} + i \eta} + \end{equation*} + \end{block} + \begin{block}{\green{Renormalization factor}} + \begin{equation*} + \green{\Z{p}(\omega)} = \qty[ 1 - \pdv{\Re[\red{\SigGW{pp}(\omega)}]}{\omega} ]^{-1} + \end{equation*} + \end{block} + \begin{block}{\violet{Screened two-electron MO integrals}} + \begin{equation*} + \violet{[pq|x]} = \sum_{ia} (pq|ia) \orange{(\bX+\bY)_{ia}^{x}} + \end{equation*} + \end{block} + \begin{block}{\orange{RPA excitation energies}} + \small + \begin{equation*} + \begin{pmatrix} + \bA & \bB \\ + \bB & \bA \\ + \end{pmatrix} + \orange{\begin{pmatrix} + \bX \\ + \bY \\ + \end{pmatrix}} + = + \orange{\bOm} + \begin{pmatrix} + \bm{1} & 0 \\ + 0 & \bm{-1} \\ + \end{pmatrix} + \orange{\begin{pmatrix} + \bX \\ + \bY \\ + \end{pmatrix}} + \end{equation*} + \begin{align*} + A^\text{RPA}_{ia,jb} & = \delta_{ij} \delta_{ab} (\epsilon_a - \epsilon_i) + 2 (ia|jb) + & + B^\text{RPA}_{ia,jb} & = 2 (ia|bj) + \end{align*} + \end{block} +\end{frame} + +%----------------------------------------------------- +\subsection{\evGW} +%----------------------------------------------------- +\begin{frame}{\evGW} + \begin{block}{{\evGW} subroutine} + \begin{algorithmic} + \Procedure{Partially self-consistent {\evGW}}{} + \State Perform HF calculation to get $\beHF$ and $\bcHF$ + \State Set $\beGnWn{-1} = \beHF$ and $n = 0$ + \While{$\max{\abs{\bDelta}} < \tau$} + \For{$p=1,\ldots,N$} + \State Compute \red{$\SigC{pp}(\omega)$} + \State Solve $\omega = \eHF{p} + \Re[\red{\SigC{pp}(\omega)}]$ to obtain $\eGnWn{p}{n}$ + \EndFor + \State $\bDelta = \beGnWn{n} - \beGnWn{n-1}$ + \State $n \leftarrow n + 1$ + \EndWhile + \If{BSE} + \State Compute BSE excitations energies + \EndIf + \EndProcedure + \end{algorithmic} + \end{block} +\end{frame} + +%----------------------------------------------------- +\subsection{\qsGW} +%----------------------------------------------------- +\begin{frame}{\qsGW} + \begin{block}{{\qsGW} subroutine} + \begin{algorithmic} + \Procedure{Partially self-consistent {\qsGW}}{} + \State Perform HF calculation to get $\beHF$ and $\bcHF$ + \State Set $\beGnWn{-1} = \beHF$, $\bcGnWn{-1} = \bcHF$ and $n = 0$ + \While{$\max{\abs{\bDelta}} < \tau$} + \State Form \red{$\bSigC(\omega)$} and symmetrize it: $\red{\bSigC(\omega)} \leftarrow (\red{\bSigC(\omega)}^\dag + \red{\bSigC(\omega)})/2$ + \State Form $\green{\bF(\omega)} = \bFHF + \red{\bSigC(\omega)}$ + \State Diagonalize $\green{\bF(\beGnWn{n-1})}$ to get $\beGnWn{n}$ and $\bcGnWn{n}$ + \State $\bDelta = \beGnWn{n} - \beGnWn{n-1}$ + \State $n \leftarrow n + 1$ + \EndWhile + \If{BSE} + \State Compute BSE excitations energies + \EndIf + \EndProcedure + \end{algorithmic} + \end{block} +\end{frame} + +%----------------------------------------------------- +\subsection{BSE} +%----------------------------------------------------- +\begin{frame}{BSE} + \begin{block}{Bethe-Salpeter equation} + \begin{equation*} + \begin{pmatrix} + \bA & \bB \\ + \bB & \bA \\ + \end{pmatrix} + \purple{\begin{pmatrix} + \bX \\ + \bY \\ + \end{pmatrix}} + = + \purple{\bOm} + \begin{pmatrix} + \bm{1} & 0 \\ + 0 & \bm{-1} \\ + \end{pmatrix} + \purple{\begin{pmatrix} + \bX \\ + \bY \\ + \end{pmatrix}} + \end{equation*} + \begin{equation*} + (\bA - \bB)^{1/2} (\bA + \bB) (\bA - \bB)^{1/2} \bZ = \bOm^2 \bZ, + \end{equation*} + \begin{equation*} + \bX + \bY = \bOm^{-1/2} (\bA - \bB)^{1/2} \bZ. + \end{equation*} + \begin{align*} + A^\text{BSE}_{ia,jb} & = A^\text{RPA}_{ia,jb} - (ij|ab) + 4 \sum_{x} \frac{[ij|x][ab|x]}{\Om{x}} + \\ + B^\text{BSE}_{ia,jb} & = B^\text{RPA}_{ia,jb} - (ib|aj) + 4 \sum_{x} \frac{[ib|x][aj|x]}{\Om{x}} + \end{align*} + \end{block} +\end{frame} + +%----------------------------------------------------- +\subsection{$\Ec$} +%----------------------------------------------------- +\begin{frame}{Correlation energy} + + \begin{block}{RPA correlation energy or Klein functional} + \begin{equation*} + \label{eq:Ec-RPA} + \EcRPA = -\sum_{p} \qty(\ARPA{pp} - \Om{p}) + \end{equation*} + \end{block} + + \begin{block}{Galitskii-Migdal functional} + \begin{equation*} + \label{eq:GM} + \EcGM = \frac{-i}{2}\sum_{pq}^{\infty}\int \frac{d\omega}{2\pi} \SigC{pq}(\omega) \G{pq}(\omega) e^{i\omega\eta} + \end{equation*} + \end{block} + +\end{frame} \end{document} diff --git a/2021/Lecture_2/fig/BSE-GW.tex b/2021/Lecture_2/fig/BSE-GW.tex new file mode 100644 index 0000000..f5cda29 --- /dev/null +++ b/2021/Lecture_2/fig/BSE-GW.tex @@ -0,0 +1,74 @@ +\documentclass{standalone} +\usepackage{amsmath,amssymb,amsfonts,pgfpages,graphicx,subfigure,xcolor,bm,multirow,microtype,wasysym,multimedia,hyperref,tabularx,amscd,pgfgantt,mhchem,physics} +\usetikzlibrary{shapes.gates.logic.US,trees,positioning,arrows} + +\usepackage{tgchorus} +\usepackage[T1]{fontenc} + +\begin{document} + +\begin{tikzpicture} + \begin{scope}[very thick, + node distance=5cm,on grid,>=stealth', + theo1/.style={rectangle,draw,fill=red!20}, + theo2/.style={rectangle,draw,fill=orange!20}, + theo3/.style={rectangle,draw,fill=green!40}, + exp1/.style={rectangle,draw,fill=cyan!40}, + exp2/.style={rectangle,draw,fill=violet!40}] + + \node [theo1, text width=7cm, align=center] (KS) + {\textbf{\LARGE Kohn-Sham DFT} + $$ + \qty[ -\frac{\nabla^2}{2} + v_\text{ext} + V^{\text{Hxc}} ] \phi_p^{\text{KS}} = \varepsilon^{\text{KS}}_p \phi_p^{\text{KS}} + $$ + }; + + \node [theo2, text width=7cm, align=center] (GW) [below=of KS, yshift=2cm] + {\textbf{\LARGE $GW$ approximation} + $$ + \varepsilon_p^{GW} = \varepsilon_p^{\text{KS}} + + \mel{\phi_p^{\text{KS}}}{\Sigma^{GW}(\varepsilon_p^{GW}) - V^{\text{xc}}}{\phi_p^{\text{KS}}} + $$ + + }; + + \node [theo3, text width=7cm, align=center] (BSE) [below=of GW, yshift=2cm] + {\textbf{\LARGE Bethe-Salpeter equation} + $$ + \begin{pmatrix} + R & C \\ + -C^* & -R^{*} + \end{pmatrix} + \begin{pmatrix} + X_m \\ + Y_m + \end{pmatrix} + = + \Omega_{m} + \begin{pmatrix} + X_m \\ + Y_m + \end{pmatrix} + $$ + }; + + + \node [exp1, align=center] (photo) [right=of GW, xshift=3cm] + {\LARGE (Inverse) \\ \LARGE photoemission \\ \LARGE spectroscopy}; + + \node [exp2, align=center] (abs) [right=of BSE, xshift=3cm] + {\LARGE Optical \\ \LARGE spectroscopy}; + + + \path + (KS) edge [->,color=black] node [right,black] {\LARGE Fundamental gap} (GW) + (GW) edge [->,color=black] node [right,black] {\LARGE Excitonic effect} (BSE) + (photo) edge [<->,color=black] node [above,black] {Ionization potentials} node [below,black] {Electron affinities} (GW) + (abs) edge [<->,color=black] node [above,black] {Optical excitations} (BSE) + ; + + + \end{scope} +\end{tikzpicture} + +\end{document} diff --git a/2021/Lecture_2/fig/Cramer.jpg b/2021/Lecture_2/fig/Cramer.jpg deleted file mode 100644 index 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