491 lines
15 KiB
TeX
491 lines
15 KiB
TeX
\documentclass[aspectratio=169,9pt,compress]{beamer}
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% ***********
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% * PACKAGE *
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% ***********
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\usepackage{amsmath,amssymb,amsfonts,pgfpages,graphicx,subfigure,xcolor,bm,multirow,microtype,wasysym,multimedia,hyperref,tabularx,amscd,pgfgantt,mhchem}
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\usetikzlibrary{shapes.gates.logic.US,trees,positioning,arrows}
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\usetheme{Warsaw}
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%\usecolortheme{seahorse}
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\usepackage{mathpazo,libertine}
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\usepackage{algorithmicx,algorithm,algpseudocode}
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\algnewcommand\algorithmicassert{\texttt{assert}}
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\algnewcommand\Assert[1]{\State \algorithmicassert(#1)}
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%\algrenewcommand{\algorithmiccomment}[1]{$\triangleright$ #1}
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%\usepackage[version=4]{mhchem}
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\usepackage{amsmath,amsfonts,amssymb,bm,microtype,graphicx,wrapfig,geometry,physics,eurosym,multirow,pgfgantt}
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\usepackage{hyperref}
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\hypersetup{
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colorlinks=true,
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linkcolor=cyan,
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filecolor=magenta,
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urlcolor=blue,
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citecolor=purple
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}
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\definecolor{darkgreen}{RGB}{0, 180, 0}
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\newcommand{\red}[1]{\textcolor{red}{#1}}
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\newcommand{\purple}[1]{\textcolor{purple}{#1}}
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\newcommand{\orange}[1]{\textcolor{orange}{#1}}
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\newcommand{\green}[1]{\textcolor{darkgreen}{#1}}
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\newcommand{\blue}[1]{\textcolor{blue}{#1}}
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\newcommand{\pub}[1]{\textcolor{purple}{#1}}
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\newcommand{\violet}[1]{\textcolor{violet}{#1}}
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\newcommand{\cdash}{\multicolumn{1}{c}{---}}
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\newcommand{\mc}{\multicolumn}
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\newcommand{\mcc}[1]{\multicolumn{1}{c}{#1}}
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\newcommand{\mr}{\multirow}
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\newcommand{\br}{\bm{r}}
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\newcommand{\ree}{r_{12}}
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% methods
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\newcommand{\evGW}{evGW}
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\newcommand{\qsGW}{qsGW}
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\newcommand{\scGW}{scGW}
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\newcommand{\GOWO}{G$_0$W$_0$}
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\newcommand{\GOW}{G$_0$W}
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\newcommand{\GWO}{GW$_0$}
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\newcommand{\GW}{GW}
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\newcommand{\GOWOSOSEX}{{\GOWO}+SOSEX}
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\newcommand{\GWSOSEX}{{\GW}+SOSEX}
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\newcommand{\GnWn}[1]{G$_{#1}$W$_{#1}$}
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\newcommand{\GOF}{G$_0$F2}
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\newcommand{\GF}{GF2}
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% operators
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\newcommand{\hH}{\Hat{H}}
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% energies
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\newcommand{\Ec}{E_\text{c}}
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\newcommand{\EHF}{E_\text{HF}}
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\newcommand{\EcK}{E_\text{c}^\text{Klein}}
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\newcommand{\EcRPA}{E_\text{c}^\text{RPA}}
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\newcommand{\EcGM}{E_\text{c}^\text{GM}}
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\newcommand{\EcGMGW}{E_\text{c}^\text{GM@GW}}
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\newcommand{\EcGMGF}{E_\text{c}^\text{GM@GF2}}
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\newcommand{\EcGMGWSOSEX}{E_\text{c}^\text{GM@GW+SOSEX}}
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\newcommand{\EcMP}{E_c^\text{MP2}}
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\newcommand{\EcGF}{E_c^\text{\GF}}
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\newcommand{\EcGOF}{E_c^\text{\GOF}}
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\newcommand{\Egap}{E_\text{gap}}
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\newcommand{\IP}{\text{IP}}
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\newcommand{\EA}{\text{EA}}
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% orbital energies
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\newcommand{\nSat}[1]{N_{#1}^\text{sat}}
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\newcommand{\eSat}[2]{\epsilon_{#1,#2}}
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\newcommand{\e}[1]{\epsilon_{#1}}
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\newcommand{\eHF}[1]{\epsilon^\text{HF}_{#1}}
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\newcommand{\eKS}[1]{\epsilon^\text{KS}_{#1}}
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\newcommand{\eQP}[1]{\epsilon^\text{QP}_{#1}}
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\newcommand{\eGOWO}[1]{\epsilon^\text{\GOWO}_{#1}}
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\newcommand{\eGW}[1]{\epsilon^\text{\GW}_{#1}}
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\newcommand{\eGnWn}[2]{\epsilon^\text{\GnWn{#2}}_{#1}}
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\newcommand{\eGF}[1]{\epsilon^\text{\GF}_{#1}}
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\newcommand{\eGOF}[1]{\epsilon^\text{\GOF}_{#1}}
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\newcommand{\de}[1]{\Delta\epsilon_{#1}}
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\newcommand{\deHF}[1]{\Delta\epsilon^\text{HF}_{#1}}
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\newcommand{\Om}[1]{\Omega_{#1}}
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\newcommand{\eHOMO}{\epsilon_\text{HOMO}}
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\newcommand{\eLUMO}{\epsilon_\text{LUMO}}
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\newcommand{\cHF}[1]{c^\text{HF}_{#1}}
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\newcommand{\cKS}[1]{c^\text{KS}_{#1}}
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% Matrix elements
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\newcommand{\A}[1]{A_{#1}}
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\newcommand{\B}[1]{B_{#1}}
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\renewcommand{\S}[1]{S_{#1}}
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\newcommand{\ABSE}[1]{A^\text{BSE}_{#1}}
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\newcommand{\BBSE}[1]{B^\text{BSE}_{#1}}
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\newcommand{\ARPA}[1]{A^\text{RPA}_{#1}}
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\newcommand{\BRPA}[1]{B^\text{RPA}_{#1}}
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\newcommand{\dABSE}[1]{\delta A^\text{BSE}_{#1}}
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\newcommand{\dBBSE}[1]{\delta B^\text{BSE}_{#1}}
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\newcommand{\G}[1]{G_{#1}}
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\newcommand{\Po}[1]{P_{#1}}
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\newcommand{\W}[1]{W_{#1}}
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\newcommand{\Wc}[1]{W^\text{c}_{#1}}
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\newcommand{\vc}[1]{v_{#1}}
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\newcommand{\SigX}[1]{\Sigma^\text{x}_{#1}}
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\newcommand{\SigC}[1]{\Sigma^\text{c}_{#1}}
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\newcommand{\SigGW}[1]{\Sigma^\text{\GW}_{#1}}
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\newcommand{\SigGWSOSEX}[1]{\Sigma^\text{\GWSOSEX}_{#1}}
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\newcommand{\SigGF}[1]{\Sigma^\text{\GF}_{#1}}
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\newcommand{\Z}[1]{Z_{#1}}
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% excitation energies
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\newcommand{\OmRPA}[1]{\Omega^\text{RPA}_{#1}}
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\newcommand{\OmCIS}[1]{\Omega^\text{CIS}_{#1}}
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\newcommand{\OmTDHF}[1]{\Omega^\text{TDHF}_{#1}}
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\newcommand{\OmBSE}[1]{\Omega^\text{BSE}_{#1}}
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\newcommand{\spinup}{\downarrow}
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\newcommand{\spindw}{\uparrow}
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\newcommand{\singlet}{\uparrow\downarrow}
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\newcommand{\triplet}{\uparrow\uparrow}
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\newcommand{\Oms}[1]{{}^{1}\Omega_{#1}}
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\newcommand{\OmsRPA}[1]{{}^{1}\Omega^\text{RPA}_{#1}}
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\newcommand{\OmsCIS}[1]{{}^{1}\Omega^\text{CIS}_{#1}}
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\newcommand{\OmsTDHF}[1]{{}^{1}\Omega^\text{TDHF}_{#1}}
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\newcommand{\OmsBSE}[1]{{}^{1}\Omega^\text{BSE}_{#1}}
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\newcommand{\Omt}[1]{{}^{3}\Omega_{#1}}
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\newcommand{\OmtRPA}[1]{{}^{3}\Omega^\text{RPA}_{#1}}
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\newcommand{\OmtCIS}[1]{{}^{3}\Omega^\text{CIS}_{#1}}
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\newcommand{\OmtTDHF}[1]{{}^{3}\Omega^\text{TDHF}_{#1}}
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\newcommand{\OmtBSE}[1]{{}^{3}\Omega^\text{BSE}_{#1}}
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% Wigner symbols
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\newcommand{\WJ}[3]{
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\begin{pmatrix}
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#1 & #2 & #3 \\
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0 & 0 & 0 \\
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\end{pmatrix}
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}
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\newcommand{\ERI}[3]{\qty(#1 #2 #3)}
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\newcommand{\sERI}[3]{\qty[#1 #2 #3]}
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% Matrices
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\newcommand{\bF}{\bm{F}}
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\newcommand{\bFHF}{\bm{F}^\text{HF}}
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\newcommand{\bH}{\bm{H}}
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\newcommand{\bvc}{\bm{v}}
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\newcommand{\bSig}{\bm{\Sigma}}
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\newcommand{\bSigX}{\bm{\Sigma}^\text{x}}
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\newcommand{\bSigC}{\bm{\Sigma}^\text{c}}
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\newcommand{\bSigGW}{\bm{\Sigma}^\text{\GW}}
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\newcommand{\bSigGWSOSEX}{\bm{\Sigma}^\text{\GWSOSEX}}
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\newcommand{\bSigGF}{\bm{\Sigma}^\text{\GF}}
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\newcommand{\be}{\bm{\epsilon}}
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\newcommand{\bDelta}{\bm{\Delta}}
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\newcommand{\beHF}{\bm{\epsilon}^\text{HF}}
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\newcommand{\bcHF}{\bm{c}^\text{HF}}
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\newcommand{\beGW}{\bm{\epsilon}^\text{\GW}}
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\newcommand{\beGnWn}[1]{\bm{\epsilon}^\text{\GnWn{#1}}}
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\newcommand{\bcGnWn}[1]{\bm{c}^\text{\GnWn{#1}}}
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\newcommand{\beGF}{\bm{\epsilon}^\text{\GF}}
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\newcommand{\bde}{\bm{\Delta\epsilon}}
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\newcommand{\bdeHF}{\bm{\Delta\epsilon}^\text{HF}}
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\newcommand{\bdeGW}{\bm{\Delta\epsilon}^\text{GW}}
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\newcommand{\bdeGF}{\bm{\Delta\epsilon}^\text{GF2}}
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\newcommand{\bOm}{\bm{\Omega}}
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\newcommand{\bA}{\bm{A}}
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\newcommand{\bB}{\bm{B}}
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\newcommand{\bX}{\bm{X}}
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\newcommand{\bY}{\bm{Y}}
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\newcommand{\bZ}{\bm{Z}}
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\newcommand{\mycirc}[1][black]{\Large\textcolor{#1}{\ensuremath\bullet}}
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\usepackage{tikz}
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\usetikzlibrary{arrows,positioning,shapes.geometric}
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\usetikzlibrary{decorations.pathmorphing}
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\tikzset{snake it/.style={
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decoration={snake,
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amplitude = .4mm,
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segment length = 2mm},decorate}}
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% *************
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% * HEAD DATA *
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% *************
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\title[$GW$/BSE methods in chemistry]{
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$GW$/BSE methods in chemistry:
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Computational aspects
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}
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\author[PF Loos]{Pierre-Fran\c{c}ois LOOS}
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\date{Online ISTPC 2021 school --- April 27th, 2021}
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\institute[CNRS@LCPQ]{
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Laboratoire de Chimie et Physique Quantiques (UMR 5626),\\
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Universit\'e de Toulouse, CNRS, UPS, Toulouse, France.
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}
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\titlegraphic{
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\includegraphics[width=0.3\textwidth]{fig/jarvis}
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\\
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\vspace{0.05\textheight}
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\includegraphics[height=0.05\textwidth]{fig/UPS}
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\hspace{0.2\textwidth}
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\includegraphics[height=0.05\textwidth]{fig/ERC}
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\hspace{0.2\textwidth}
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\includegraphics[height=0.05\textwidth]{fig/LCPQ}
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\hspace{0.2\textwidth}
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\includegraphics[height=0.05\textwidth]{fig/CNRS}
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}
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\begin{document}
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%%% SLIDE 1 %%%
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\begin{frame}
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\titlepage
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\end{frame}
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%
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%%% SLIDE 2 %%%
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\begin{frame}{Today's program}
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\begin{itemize}
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\item
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\end{itemize}
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\end{frame}
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%
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%-----------------------------------------------------
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\section{Theory}
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%-----------------------------------------------------
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\subsection{Hedin's pentagon}
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%-----------------------------------------------------
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\begin{frame}{Hedin's pentagon}
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\begin{columns}
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\begin{column}{0.4\textwidth}
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\centering
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\includegraphics[width=0.8\linewidth]{fig/pentagon}
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\\
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\pub{Hedin, Phys Rev 139 (1965) A796}
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\end{column}
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\begin{column}{0.6\textwidth}
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\begin{block}{What can you calculate with GW?}
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\begin{itemize}
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\item Ionization potentials (IP) given by occupied MO energies
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\bigskip
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\item Electron affinities (EA) given by virtual MO energies
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\bigskip
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\item HOMO-LUMO gap (or band gap in solids)
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\bigskip
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\item Singlet and triplet neutral excitations (vertical absorption energies)
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\bigskip
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\item Correlation and total energies
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\end{itemize}
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\end{block}
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\end{column}
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\end{columns}
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\end{frame}
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%-----------------------------------------------------
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\subsection{GW flavours}
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%-----------------------------------------------------
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\begin{frame}{GW flavours}
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\begin{block}{Acronyms}
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\begin{itemize}
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\bigskip
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\item perturbative GW one-shot GW, or \green{\GOWO}
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\bigskip
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\item \orange{\evGW} or eigenvalue-only (partially) self-consistent GW
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\bigskip
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\item \red{\qsGW} or quasiparticle (partially) self-consistent GW
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\bigskip
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\item \violet{\scGW} or (fully) self-consistent GW
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\bigskip
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\item \purple{BSE} or Bethe-Salpeter equation for neutral excitations
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\bigskip
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\end{itemize}
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\end{block}
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\end{frame}
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%-----------------------------------------------------
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\subsection{Literature}
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%-----------------------------------------------------
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\begin{frame}{useful papers for chemists}
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\begin{itemize}
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\item \red{molGW:} Bruneval et al. Comp. Phys. Comm. 208 (2016) 149
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\bigskip
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\item \green{Turbomole:} van Setten et al. JCTC 9 (2013) 232; Kaplan et al. JCTC 12 (2016) 2528
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\bigskip
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\item \violet{Fiesta:} Blase et al. Chem. Soc. Rev. 47 (2018) 1022
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\bigskip
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\item \purple{FHI-AIMS:} Caruso et al. 86 (2012) 081102
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\bigskip
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\item \orange{Review:} Reining, WIREs Comput Mol Sci 2017, e1344. doi: 10.1002/wcms.1344; Onida et al. Rev. Mod. Phys. 74 (2002) 601.
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\bigskip
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\item \red{GW100:} Data set of 100 molecules. van Setten et al. JCTC 11 (2015) 5665
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\end{itemize}
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\end{frame}
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%-----------------------------------------------------
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\section{Implementation}
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%-----------------------------------------------------
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\subsection{\GOWO}
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%-----------------------------------------------------
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\begin{frame}{\GOWO}
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\begin{block}{{\GOWO}~subroutine}
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\begin{algorithmic}
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\Procedure{Perturbative {\GW}}{}
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\State Perform HF calculation to get $\beHF$ and $\bcHF$
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\For{$p=1,\ldots,N$}
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\State Compute \red{$\SigC{pp}(\omega)$} and \green{$\Z{p}(\omega)$}
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\State $\eGOWO{p} = \eHF{p} + \green{\Z{p}(\eHF{p})} \Re[\red{\SigC{pp}(\eHF{p})}]$
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\State \Comment{This is the linearized version of the}
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\State \Comment{quasiparticle (QP) equation
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$\omega = \eHF{p} + \Re[\red{\SigC{pp}(\omega)}]$}
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\EndFor
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\If{BSE}
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\State Compute BSE excitations energies
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\EndIf
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\EndProcedure
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\end{algorithmic}
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\end{block}
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\end{frame}
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\begin{frame}{\GOWO}
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\begin{block}{\red{Correlation part of the self-energy:}}
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\begin{equation*}
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\red{\SigC{pq}(\omega)}
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= 2 \sum_{ix}\frac{\violet{[pi|x] [qi|x]}}{\omega - \eHF{i} + \orange{\Om{x}} - i \eta}
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+ 2 \sum_{ax}\frac{\violet{[pa|x] [qa|x]}}{\omega - \eHF{a} - \orange{\Om{x}} + i \eta}
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\end{equation*}
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\end{block}
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\begin{block}{\green{Renormalization factor}}
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\begin{equation*}
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\green{\Z{p}(\omega)} = \qty[ 1 - \pdv{\Re[\red{\SigGW{pp}(\omega)}]}{\omega} ]^{-1}
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\end{equation*}
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\end{block}
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\begin{block}{\violet{Screened two-electron MO integrals}}
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\begin{equation*}
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\violet{[pq|x]} = \sum_{ia} (pq|ia) \orange{(\bX+\bY)_{ia}^{x}}
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\end{equation*}
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\end{block}
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\begin{block}{\orange{RPA excitation energies}}
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\small
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\begin{equation*}
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\begin{pmatrix}
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\bA & \bB \\
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\bB & \bA \\
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\end{pmatrix}
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\orange{\begin{pmatrix}
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\bX \\
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\bY \\
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\end{pmatrix}}
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=
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\orange{\bOm}
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\begin{pmatrix}
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\bm{1} & 0 \\
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0 & \bm{-1} \\
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\end{pmatrix}
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\orange{\begin{pmatrix}
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\bX \\
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\bY \\
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\end{pmatrix}}
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\end{equation*}
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\begin{align*}
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A^\text{RPA}_{ia,jb} & = \delta_{ij} \delta_{ab} (\epsilon_a - \epsilon_i) + 2 (ia|jb)
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&
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B^\text{RPA}_{ia,jb} & = 2 (ia|bj)
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\end{align*}
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\end{block}
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\end{frame}
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%-----------------------------------------------------
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\subsection{\evGW}
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%-----------------------------------------------------
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\begin{frame}{\evGW}
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\begin{block}{{\evGW} subroutine}
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\begin{algorithmic}
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\Procedure{Partially self-consistent {\evGW}}{}
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\State Perform HF calculation to get $\beHF$ and $\bcHF$
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\State Set $\beGnWn{-1} = \beHF$ and $n = 0$
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\While{$\max{\abs{\bDelta}} < \tau$}
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\For{$p=1,\ldots,N$}
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\State Compute \red{$\SigC{pp}(\omega)$}
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\State Solve $\omega = \eHF{p} + \Re[\red{\SigC{pp}(\omega)}]$ to obtain $\eGnWn{p}{n}$
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\EndFor
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\State $\bDelta = \beGnWn{n} - \beGnWn{n-1}$
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\State $n \leftarrow n + 1$
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\EndWhile
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\If{BSE}
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\State Compute BSE excitations energies
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\EndIf
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\EndProcedure
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\end{algorithmic}
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\end{block}
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\end{frame}
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%-----------------------------------------------------
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\subsection{\qsGW}
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%-----------------------------------------------------
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\begin{frame}{\qsGW}
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\begin{block}{{\qsGW} subroutine}
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\begin{algorithmic}
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\Procedure{Partially self-consistent {\qsGW}}{}
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\State Perform HF calculation to get $\beHF$ and $\bcHF$
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\State Set $\beGnWn{-1} = \beHF$, $\bcGnWn{-1} = \bcHF$ and $n = 0$
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\While{$\max{\abs{\bDelta}} < \tau$}
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\State Form \red{$\bSigC(\omega)$} and symmetrize it: $\red{\bSigC(\omega)} \leftarrow (\red{\bSigC(\omega)}^\dag + \red{\bSigC(\omega)})/2$
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\State Form $\green{\bF(\omega)} = \bFHF + \red{\bSigC(\omega)}$
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\State Diagonalize $\green{\bF(\beGnWn{n-1})}$ to get $\beGnWn{n}$ and $\bcGnWn{n}$
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\State $\bDelta = \beGnWn{n} - \beGnWn{n-1}$
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\State $n \leftarrow n + 1$
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\EndWhile
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\If{BSE}
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\State Compute BSE excitations energies
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\EndIf
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\EndProcedure
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\end{algorithmic}
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\end{block}
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\end{frame}
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%-----------------------------------------------------
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\subsection{BSE}
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%-----------------------------------------------------
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\begin{frame}{BSE}
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\begin{block}{Bethe-Salpeter equation}
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\begin{equation*}
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\begin{pmatrix}
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\bA & \bB \\
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\bB & \bA \\
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\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}
|
|
|
|
%-----------------------------------------------------
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|
\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}
|