949 lines
34 KiB
TeX
949 lines
34 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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\definecolor{fooblue}{RGB}{0,153,255}
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\definecolor{fooyellow}{RGB}{234,180,0}
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\definecolor{lavender}{rgb}{0.71, 0.49, 0.86}
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\definecolor{inchworm}{rgb}{0.7, 0.93, 0.36}
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\newcommand{\violet}[1]{\textcolor{lavender}{#1}}
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\newcommand{\orange}[1]{\textcolor{orange}{#1}}
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\newcommand{\purple}[1]{\textcolor{purple}{#1}}
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\newcommand{\blue}[1]{\textcolor{blue}{#1}}
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\newcommand{\green}[1]{\textcolor{darkgreen}{#1}}
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\newcommand{\yellow}[1]{\textcolor{fooyellow}{#1}}
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\newcommand{\red}[1]{\textcolor{red}{#1}}
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\newcommand{\highlight}[1]{\textcolor{fooblue}{#1}}
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\newcommand{\pub}[1]{\small \textcolor{purple}{#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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\newcommand{\T}[1]{#1^{\intercal}}
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% methods
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\newcommand{\evGW}{ev$GW$}
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\newcommand{\qsGW}{qs$GW$}
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\newcommand{\scGW}{sc$GW$}
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\newcommand{\GOWO}{$G_0W_0$}
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\newcommand{\GOW}{$G_0W$}
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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}{$G_0F2$}
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\newcommand{\GF}{$GF2$}
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\newcommand{\KS}{\text{KS}}
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\newcommand{\RPA}{\text{RPA}}
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\newcommand{\RPAx}{\text{RPAx}}
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\newcommand{\BSE}{\text{BSE}}
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\newcommand{\TDA}{\text{TDA}}
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\newcommand{\xc}{\text{xc}}
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\newcommand{\Ha}{\text{H}}
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\newcommand{\co}{\text{c}}
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\newcommand{\x}{\text{x}}
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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{\deKS}[1]{\Delta\epsilon^\text{KS}_{#1}}
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\newcommand{\Om}[2]{\Omega_{#1}^{#2}}
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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}[2]{A_{#1}^{#2}}
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\newcommand{\tA}[2]{\Tilde{A}_{#1}^{#2}}
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\newcommand{\B}[2]{B_{#1}^{#2}}
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\newcommand{\tB}[2]{\Tilde{B}_{#1}^{#2}}
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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{\Sig}[2]{\Sigma_{#1}^{#2}}
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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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\newcommand{\MO}[1]{\phi_{#1}}
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\newcommand{\ERI}[2]{(#1|#2)}
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\newcommand{\rbra}[1]{(#1|}
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\newcommand{\rket}[1]{|#1)}
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\newcommand{\sERI}[2]{[#1|#2]}
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\newcommand{\sig}{\sigma}
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\newcommand{\sigp}{\sigma'}
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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{\beKS}{\bm{\epsilon}^\text{KS}}
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\newcommand{\bcHF}{\bm{c}^\text{HF}}
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\newcommand{\bcKS}{\bm{c}^\text{KS}}
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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{\bO}{\bm{0}}
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\newcommand{\bI}{\bm{1}}
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\newcommand{\bOm}[2]{\bm{\Omega}_{#1}^{#2}}
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\newcommand{\bA}[2]{\bm{A}_{#1}^{#2}}
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\newcommand{\btA}[2]{\bm{\Tilde{A}}_{#1}^{#2}}
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\newcommand{\btB}[2]{\bm{\Tilde{B}}_{#1}^{#2}}
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\newcommand{\bB}[2]{\bm{B}_{#1}^{#2}}
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\newcommand{\bX}[2]{\bm{X}_{#1}^{#2}}
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\newcommand{\bY}[2]{\bm{Y}_{#1}^{#2}}
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\newcommand{\bZ}[2]{\bm{Z}_{#1}^{#2}}
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\newcommand{\bK}[2]{\bm{K}_{#1}^{#2}}
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\newcommand{\bP}[2]{\bm{P}_{#1}^{#2}}
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\newcommand{\yo}{\yellow{\omega}}
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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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%-----------------------------------------------------
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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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%-----------------------------------------------------
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\begin{frame}{Today's program}
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\begin{itemize}
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\item Charged excitations:
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\begin{itemize}
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\item One-shot $GW$ (\GOWO)
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\item Partially self-consistent $GW$ (\evGW)
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\item Self-consistent $GW$ (\qsGW)
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\item $GW$ vs GF
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\end{itemize}
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\item Neutral excitations
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\begin{itemize}
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\item Configuration interaction with singles (CIS)
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\item Time-dependent Hartree-Fock (TDHF)
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\item Random-phase approximation (RPA)
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\item Time-dependent density-functional theory (TDDFT)
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\item Bethe-Salpeter equation (BSE) formalism
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\end{itemize}
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\item Total energies
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\begin{itemize}
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\item Plasmon formula
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\item Galitski-Migdal formulation
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\item Adiabatic connection fluctuation-dissipation theorem (ACFDT)
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\end{itemize}
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\end{itemize}
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\end{frame}
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%-----------------------------------------------------
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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 (IPs) given by occupied MO energies
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\item Electron affinities (EAs) given by virtual MO energies
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\item Fundamental (HOMO-LUMO) gap (or band gap in solids)
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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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\begin{block}{What can you calculate with BSE?}
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\begin{itemize}
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\item Singlet and triplet neutral excitations (vertical absorption energies)
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\item Oscillator strengths (absorption intensities)
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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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%-----------------------------------------------------
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\begin{frame}{Fundamental and optical gaps}
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\begin{center}
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\includegraphics[width=\textwidth]{fig/gaps}
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\end{center}
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\end{frame}
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%-----------------------------------------------------
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%-----------------------------------------------------
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\begin{frame}{The MBPT chain of actions}
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\begin{center}
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\includegraphics[width=0.7\textwidth]{fig/BSE-GW}
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\end{center}
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\end{frame}
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%-----------------------------------------------------
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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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\end{itemize}
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\end{block}
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\end{frame}
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%-----------------------------------------------------
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%-----------------------------------------------------
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\begin{frame}{Green's function and dynamical screening}
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\begin{block}{One-body Green's function}
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\begin{equation}
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\blue{G}(\br_1,\br_2;\yo)
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= \sum_i \frac{\MO{i}(\br_1) \MO{i}(\br_2)}{\yo - \e{i}{} - i\eta}
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+ \sum_a \frac{\MO{a}(\br_1) \MO{a}(\br_2)}{\yo - \e{a}{} + i\eta}
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\end{equation}
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\end{block}
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\begin{block}{Non-interacting polarizability}
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\begin{equation}
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P(\br_1,\br_2;\omega) = - \frac{i}{\pi} \int \blue{G}(\br_1,\br_2;\omega+\omega') \blue{G}(\br_1,\br_2;\omega') d\omega'
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\end{equation}
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\end{block}
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\begin{block}{Dielectric function}
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\begin{equation}
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\epsilon(\br_1,\br_2;\omega) = \delta(\br_1 - \br_2) - \int \frac{P(\br_1,\br_3;\omega) }{\abs{\br_2 - \br_3}} d\br_3
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\end{equation}
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\end{block}
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\begin{block}{Dynamically-screened Coulomb potential}
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\begin{equation}
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\highlight{W}(\br_1,\br_2;\omega) = \int \frac{\epsilon^{-1}(\br_1,\br_3;\omega) }{\abs{\br_2 - \br_3}} d\br_3
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\end{equation}
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\end{block}
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\end{frame}
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%-----------------------------------------------------
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%-----------------------------------------------------
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\begin{frame}{Dynamical screening in a basis}
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\begin{block}{Spectral representation of $W$}
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\begin{equation}
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\begin{split}
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\highlight{W}_{pq,rs}(\yo)
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& = \iint \MO{p}(\br_1) \MO{q}(\br_1) \highlight{W}_{pq,rs}(\br_1,\br_2;\yo) \MO{r}(\br_2) \MO{s}(\br_2) d\br_1 d\br_2
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\\
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& = \underbrace{\ERI{pq}{rs}}_{\text{(static) exchange part}}
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+ \underbrace{2 \sum_m \violet{\ERI{pq}{m}} \violet{\ERI{rs}{m}}
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\qty[ \frac{1}{\yo - \orange{\Om{m}{\RPA}} + i \eta} - \frac{1}{\yo + \orange{\Om{m}{\RPA}} - i \eta} ]}_{\text{(dynamical) correlation part } \highlight{W}^{\co}_{pq,rs}(\yo)}
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\end{split}
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\end{equation}
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\end{block}
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\begin{block}{Electron repulsion integrals (ERIs)}
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\begin{equation}
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\ERI{pq}{rs} = \iint \frac{\MO{p}(\br_1) \MO{q}(\br_1) \MO{r}(\br_2) \MO{s}(\br_2)}{\abs{\br_1 - \br_2}} d\br_1 d\br_2
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\end{equation}
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\end{block}
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\begin{block}{Screened ERIs (or spectral weights)}
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\begin{equation}
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\violet{\ERI{pq}{m}} = \sum_{ia} \ERI{pq}{ia} (\orange{\bX{m}{\RPA}+\bY{m}{\RPA}})_{ia}
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\end{equation}
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\end{block}
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\end{frame}
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%-----------------------------------------------------
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%-----------------------------------------------------
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\begin{frame}{Computation of the dynamical screening}
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\begin{block}{Direct RPA calculation (pseudo-hermitian linear problem)}
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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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\cdot
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\begin{pmatrix}
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\orange{\bX{m}{}} \\
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\orange{\bY{m}{}} \\
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\end{pmatrix}
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=
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\orange{\Om{m}{}}
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\begin{pmatrix}
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\orange{\bX{m}{}} \\
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|
\orange{\bY{m}{}} \\
|
|
\end{pmatrix}
|
|
\end{equation}
|
|
\begin{equation}
|
|
\qq*{For singlet states:} \A{ia,jb}{\RPA} = \delta_{ij} \delta_{ab} (\e{a} - \e{i}) + 2\ERI{ia}{bj}
|
|
\qquad
|
|
\B{ia,jb}{\RPA} = 2\ERI{ia}{jb}
|
|
\end{equation}
|
|
\end{block}
|
|
\begin{block}{Non-hermitian to hermitian}
|
|
\begin{equation}
|
|
(\bA{}{} - \bB{}{})^{1/2} \cdot (\bA{}{} + \bB{}{}) \cdot (\bA{}{} - \bB{}{})^{1/2} \cdot \bZ{m}{} = \Om{m}{2} \bZ{m}{}
|
|
\end{equation}
|
|
\begin{gather}
|
|
(\bX{}{} + \bY{}{})_m = \Om{m}{-1/2} (\bA{}{} - \bB{}{})^{+1/2} \cdot \bZ{m}{}
|
|
\\
|
|
(\bX{}{} - \bY{}{})_m = \Om{m}{+1/2} (\bA{}{} - \bB{}{})^{-1/2} \cdot \bZ{m}{}
|
|
\end{gather}
|
|
\end{block}
|
|
\begin{block}{Tamm-Dancoff approximation (TDA)}
|
|
\begin{equation}
|
|
\bB{}{} = \bO \quad \Rightarrow \quad \bA{}{} \cdot \bX{m}{} = \Om{m}{\TDA} \bX{m}{}
|
|
\end{equation}
|
|
\end{block}
|
|
\end{frame}
|
|
%-----------------------------------------------------
|
|
|
|
%-----------------------------------------------------
|
|
\begin{frame}{The self-energy}
|
|
\begin{block}{$GW$ Self-energy}
|
|
\begin{equation}
|
|
\underbrace{\red{\Sig{}{\xc}}(\br_1,\br_2;\yo)}_{\text{$GW$ self-energy}}
|
|
= \underbrace{\purple{\Sig{}{\x}}(\br_1,\br_2)}_{\text{\purple{exchange}}}
|
|
+ \underbrace{\red{\Sig{}{\co}}(\br_1,\br_2;\yo)}_{\text{\red{correlation}}}
|
|
= \frac{i}{2\pi} \int \blue{G}(\br_1,\br_2;\yo+\omega') \highlight{W}(\br_1,\br_2;\omega') e^{i \eta \omega'} d\omega'
|
|
\end{equation}
|
|
\end{block}
|
|
\begin{block}{Exchange part of the (static) self-energy}
|
|
\begin{equation}
|
|
\purple{\Sig{pq}{\x}} = - \sum_{i} \ERI{pi}{iq}
|
|
\end{equation}
|
|
\end{block}
|
|
\begin{block}{Correlation part of the (dynamical) self-energy}
|
|
\begin{equation}
|
|
\red{\Sig{pq}{\co}}(\yo)
|
|
= 2 \sum_{im} \frac{\violet{\ERI{pi}{m}} \violet{\ERI{qi}{m}}}{\yo - \e{i} + \orange{\Om{m}{\RPA}} - i \eta}
|
|
+ 2 \sum_{am} \frac{\violet{\ERI{pa}{m}} \violet{\ERI{qa}{m}}}{\yo - \e{a} - \orange{\Om{m}{\RPA}} + i \eta}
|
|
\end{equation}
|
|
\end{block}
|
|
\end{frame}
|
|
%-----------------------------------------------------
|
|
|
|
%-----------------------------------------------------
|
|
\begin{frame}{Quasiparticle equation}
|
|
\begin{block}{Dyson equation}
|
|
\begin{equation}
|
|
\qty[ \blue{G}(\br_1,\br_2;\yo) ]^{-1}
|
|
= \underbrace{\qty[ G_{\KS}(\br_1,\br_2;\yo) ]^{-1}}_{\text{KS Green's function}}
|
|
+ \red{\Sig{}{\xc}}(\br_1,\br_2;\yo) - \underbrace{\upsilon^{\xc}(\br_1)}_{\text{KS potential}} \delta(\br_1 - \br_2)
|
|
\end{equation}
|
|
\end{block}
|
|
\begin{block}{Non-linear quasiparticle (QP) equation}
|
|
\begin{equation}
|
|
\yo = \eKS{p} + \red{\Sig{pp}{\xc}}(\yo) - V_{p}^{\xc}
|
|
\qq{with}
|
|
V_{p}^{\xc} = \int \MO{p}(\br) \upsilon^{\xc}(\br) \MO{p}(\br) d\br
|
|
\end{equation}
|
|
\end{block}
|
|
\begin{block}{Linearized QP equation}
|
|
\begin{equation}
|
|
\blue{\eGW{p}} = \e{p}^{\KS} + \green{Z_{p}} [\red{\Sig{pp}{\xc}}(\e{p}^{\KS}) - V_{p}^{\xc} ]
|
|
\end{equation}
|
|
\end{block}
|
|
\begin{block}{Renormalization factor or spectral weight}
|
|
\begin{equation}
|
|
\green{Z_{p}} = \qty[ 1 - \left. \pdv{\red{\Sig{pp}{\xc}}(\yo)}{\yo} \right|_{\yo = \e{p}^{\KS}} ]^{-1}
|
|
\qq{with} 0 \le \green{Z_{p}} \le 1
|
|
\end{equation}
|
|
\end{block}
|
|
\end{frame}
|
|
%-----------------------------------------------------
|
|
|
|
%-----------------------------------------------------
|
|
\begin{frame}{Perturbative {\GW} with linearized solution}
|
|
\begin{block}{Linearized {\GOWO}~subroutine}
|
|
\begin{algorithmic}
|
|
\Procedure{{\GOWO}lin}{}
|
|
\State Perform KS calculation to get $\beKS$, $\bcKS$, and $\bm{V}^{\xc}$
|
|
\State AO to MO transformation for ERIs: $\ERI{\mu\nu}{\lambda\sigma} \stackrel{\bcKS}{\rightarrow} \ERI{pq}{rs}$
|
|
\State Construct RPA matrices $\orange{\bA{}{\RPA}}$ and $\orange{\bB{}{\RPA}}$
|
|
\State Compute RPA eigenvalues $\orange{\Om{m}{\RPA}}$ and eigenvectors $\orange{\bX{m}{\RPA}+\bY{m}{\RPA}}$
|
|
\Comment{\alert{This is a $\order*{N^6}$ step!}}
|
|
\State Form screened ERIs $\violet{\ERI{pq}{m}}$
|
|
\For{$p=1,\ldots,N$}
|
|
\State Compute diagonal of the self-energy $\red{\SigC{pp}}(\yo)$ at $\yo = \eKS{p}$
|
|
\State Compute renornalization factors \green{$\Z{p}$}
|
|
\State Evaluate $\blue{\eGOWO{p}} = \eKS{p} + \green{\Z{p}} \qty{ \Re[\red{\SigC{pp}}(\eKS{p})] - V_{p}^{\xc} }$
|
|
\EndFor
|
|
\EndProcedure
|
|
\end{algorithmic}
|
|
\end{block}
|
|
\end{frame}
|
|
%-----------------------------------------------------
|
|
|
|
%-----------------------------------------------------
|
|
\begin{frame}{Perturbative {\GW} with graphical solution}
|
|
\begin{block}{Graphical {\GOWO}~subroutine}
|
|
\begin{algorithmic}
|
|
\Procedure{{\GOWO}graph}{}
|
|
\State Perform KS calculation to get $\beKS$, $\bcKS$, and $\bm{V}^{\xc}$
|
|
\State AO to MO transformation for ERIs: $\ERI{\mu\nu}{\lambda\sigma} \stackrel{\bcKS}{\rightarrow} \ERI{pq}{rs}$
|
|
\State Construct RPA matrices $\orange{\bA{}{\RPA}}$ and $\orange{\bB{}{\RPA}}$
|
|
\State Compute RPA eigenvalues $\orange{\Om{m}{\RPA}}$ and eigenvectors $\orange{\bX{m}{\RPA}+\bY{m}{\RPA}}$
|
|
\Comment{\alert{This is a $\order*{N^6}$ step!}}
|
|
\State Form screened ERIs $\violet{\ERI{pq}{m}}$
|
|
\For{$p=1,\ldots,N$}
|
|
\State Compute diagonal of the self-energy $\red{\SigC{pp}}(\yo)$
|
|
\State Solve $\yo = \eKS{p} + \Re[\red{\SigC{pp}}(\yo)] - V_{p}^{\xc}$ to get $\blue{\eGOWO{p}}$ via Newton's method
|
|
\EndFor
|
|
\EndProcedure
|
|
\end{algorithmic}
|
|
\end{block}
|
|
\end{frame}
|
|
%-----------------------------------------------------
|
|
|
|
%-----------------------------------------------------
|
|
\begin{frame}{Partially self-consistent eigenvalue $GW$}
|
|
\begin{block}{{\evGW} subroutine}
|
|
\begin{algorithmic}
|
|
\Procedure{partially self-consistent {\evGW}}{}
|
|
\State Perform KS calculation to get $\beKS$, $\bcKS$, and $\bm{V}^{\xc}$
|
|
\State AO to MO transformation for ERIs: $\ERI{\mu\nu}{\lambda\sigma} \stackrel{\bcKS}{\rightarrow} \ERI{pq}{rs}$
|
|
\State Construct RPA matrices $\orange{\bA{}{\RPA}}$ and $\orange{\bB{}{\RPA}}$
|
|
\State Compute RPA eigenvalues $\orange{\Om{m}{\RPA}}$ and eigenvectors $\orange{\bX{m}{\RPA}+\bY{m}{\RPA}}$
|
|
\State Form screened ERIs $\violet{\ERI{pq}{m}}$
|
|
\State Set $\blue{\beGnWn{-1}} = \beKS$ and $n = 0$
|
|
\While{$\max{\abs{\bDelta}} < \tau$}
|
|
\For{$p=1,\ldots,N$}
|
|
\State Compute diagonal of the self-energy $\red{\SigC{pp}}(\yo)$
|
|
\State Solve $\yo = \eKS{p} + \Re[\red{\SigC{pp}}(\yo)] - V_{p}^{\xc}$ to get $\blue{\eGnWn{p}{n}}$
|
|
\EndFor
|
|
\State $\bDelta = \blue{\beGnWn{n}} - \blue{\beGnWn{n-1}}$
|
|
\State $n \leftarrow n + 1$
|
|
\EndWhile
|
|
\EndProcedure
|
|
\end{algorithmic}
|
|
\end{block}
|
|
\end{frame}
|
|
%-----------------------------------------------------
|
|
|
|
%-----------------------------------------------------
|
|
\begin{frame}{Quasiparticle self-consistent {\GW} (\qsGW)}
|
|
\begin{block}{{\qsGW} subroutine}
|
|
\begin{algorithmic}
|
|
\Procedure{partially self-consistent {\qsGW}}{}
|
|
\State Perform HF calculation to get $\beHF$ and $\bcHF$ \green{(optional)}
|
|
\State Set $\blue{\beGnWn{-1}} = \beHF$, $\blue{\bcGnWn{-1}} = \bcHF$ and $n = 0$
|
|
\While{$\max{\abs{\bDelta}} < \tau$}
|
|
\State AO to MO transformation for ERIs: $\ERI{\mu\nu}{\lambda\sigma} \stackrel{\blue{\bcGnWn{n-1}}}{\rightarrow} \ERI{pq}{rs}$
|
|
\State Construct RPA matrices $\orange{\bA{}{\RPA}}$ and $\orange{\bB{}{\RPA}}$
|
|
\State Compute RPA eigenvalues $\orange{\Om{m}{\RPA}}$ and eigenvectors $\orange{\bX{m}{\RPA}+\bY{m}{\RPA}}$
|
|
\State Form screened ERIs $\violet{\ERI{pq}{m}}$
|
|
\State Evaluate $\red{\bSigC}(\blue{\beGnWn{n-1}})$ and form
|
|
$\red{\Tilde{\Sigma}^{\co}} \leftarrow \qty[ \red{\bSigC}(\blue{\beGnWn{n-1}})^\dag + \red{\bSigC}(\blue{\beGnWn{n-1}}) ]/2$
|
|
\State Form $\purple{\Tilde{\bF}} = \bFHF + \red{\Tilde{\Sigma}^{\co}}$
|
|
\State Diagonalize $\purple{\Tilde{\bF}}$ to get $\blue{\beGnWn{n}}$ and $\blue{\bcGnWn{n}}$
|
|
\State $\bDelta = \blue{\beGnWn{n}} - \blue{\beGnWn{n-1}}$
|
|
\State $n \leftarrow n + 1$
|
|
\EndWhile
|
|
\EndProcedure
|
|
\end{algorithmic}
|
|
\end{block}
|
|
\end{frame}
|
|
%-----------------------------------------------------
|
|
|
|
\begin{frame}{TD-DFT and BSE in practice: Casida-like equations}
|
|
\begin{block}{Linear response problem}
|
|
\begin{equation*}
|
|
\boxed{\begin{pmatrix}
|
|
\red{\bA{}{}} & \orange{\bB{}{}}
|
|
\\
|
|
\orange{-\bA{}{}} & \red{-\bB{}{}}
|
|
\end{pmatrix}
|
|
\cdot
|
|
\begin{pmatrix}
|
|
\bX{m}{}
|
|
\\
|
|
\bY{m}{}
|
|
\end{pmatrix}
|
|
=
|
|
\highlight{\Om{m}{}}
|
|
\begin{pmatrix}
|
|
\bX{m}{}
|
|
\\
|
|
\bY{m}{}
|
|
\end{pmatrix}}
|
|
\end{equation*}
|
|
\end{block}
|
|
%
|
|
\begin{block}{Blue pill: TD-DFT within the \alert{adiabatic} approximation}
|
|
\begin{gather}
|
|
\red{A}_{ia,jb} = \qty( \varepsilon_a^\text{\violet{KS}} - \varepsilon_i^\text{\violet{KS}} ) \delta_{ij} \delta_{ab} + 2 \blue{(ia|bj)} + \yellow{f}^{\yellow{xc}}_{ia,bj}
|
|
\qquad
|
|
\orange{B}_{ia,jb} = 2 \blue{(ia|jb)} + \yellow{f}^{\yellow{xc}}_{ia,jb}
|
|
\\
|
|
\yellow{f}^{\yellow{xc}}_{ia,bj} = \iint \phi_{i}(\br{})\phi_{a}(\br{}) \frac{\delta^2 E^{xc} }{\delta\rho(\br{}) \delta\rho(\br{}')} \phi_{b}(\br{})\phi_{j}(\br{}) d\br{} d\br{}'
|
|
\end{gather}
|
|
\end{block}
|
|
%
|
|
\begin{block}{Red pill: BSE within the \alert{static} approximation}
|
|
\begin{gather}
|
|
\red{A}_{ia,jb} = \qty( \varepsilon_a^{\green{GW}} - \varepsilon_i^{\green{GW}} ) \delta_{ij} \delta_{ab} + 2 \blue{(ia|bj)} - \purple{W}^\text{stat}_{ij,ba}
|
|
\qquad
|
|
\orange{B}_{ia,jb} = 2 \blue{(ia|jb)} - \purple{W}^\text{stat}_{ib,ja}
|
|
\\
|
|
\purple{W}^\text{stat}_{ij,ab} \equiv \purple{W}_{ij,ab} (\omega = 0) = (ij|ab) - W^{c}_{ij,ab}(\omega = 0)
|
|
\end{gather}
|
|
\end{block}
|
|
%
|
|
\end{frame}
|
|
|
|
\begin{frame}{TDHF and CIS: removing the correlation part}
|
|
\begin{block}{Linear response problem}
|
|
\begin{equation*}
|
|
\boxed{\begin{pmatrix}
|
|
\red{\bA{}{}} & \orange{\bB{}{}}
|
|
\\
|
|
\orange{-\bA{}{}} & \red{-\bB{}{}}
|
|
\end{pmatrix}
|
|
\cdot
|
|
\begin{pmatrix}
|
|
\bX{m}{}
|
|
\\
|
|
\bY{m}{}
|
|
\end{pmatrix}
|
|
=
|
|
\highlight{\Om{m}{}}
|
|
\begin{pmatrix}
|
|
\bX{m}{}
|
|
\\
|
|
\bY{m}{}
|
|
\end{pmatrix}}
|
|
\end{equation*}
|
|
\end{block}
|
|
%
|
|
\begin{block}{TDHF = RPA with exchange (RPAx)}
|
|
\begin{align}
|
|
\red{A}_{ia,jb} & = \qty( \varepsilon_a^\text{\green{HF}} - \varepsilon_i^\text{\green{HF}} ) \delta_{ij} \delta_{ab} + 2 \blue{(ia|bj)} - \yellow{(ij|ba)}
|
|
&
|
|
\orange{B}_{ia,jb} & = 2 \blue{(ia|jb)} - \yellow{(ib|ja)}
|
|
\end{align}
|
|
\end{block}
|
|
%
|
|
\begin{block}{Linear response problem within the Tamm-Dancoff approximation}
|
|
\begin{equation}
|
|
\boxed{\red{\bA{}{}} \cdot \bX{m}{} = \highlight{\Om{m}{}} \, \bX{m}{} }
|
|
\end{equation}
|
|
\end{block}
|
|
%
|
|
\begin{block}{TDHF within TDA = CIS}
|
|
\begin{equation}
|
|
\red{A}_{ia,jb}
|
|
= \qty( \varepsilon_a^\text{\green{HF}} - \varepsilon_i^\text{\green{HF}} ) \delta_{ij} \delta_{ab}
|
|
+ 2 \blue{(ia|bj)} - \yellow{(ij|ba)}
|
|
\end{equation}
|
|
\end{block}
|
|
%
|
|
\end{frame}
|
|
|
|
%-----------------------------------------------------
|
|
\begin{frame}{Linear response}
|
|
\begin{block}{General linear response problem}
|
|
\begin{algorithmic}
|
|
\Procedure{Linear response}{}
|
|
\State Compute $\bA{}{}$ matrix at a given level of theory
|
|
\If{$\TDA$}
|
|
\State Diagonalize $\bA{}{}$ to get $\Om{m}{\TDA}$ and $\bX{m}{\TDA}$
|
|
\Else
|
|
\State Compute $\bB{}{}$ matrix at a given level of theory
|
|
\State Diagonalize $\bA{}{} - \bB{}{}$ to form $(\bA{}{} - \bB{}{})^{1/2}$
|
|
\State Form and diagonalize $(\bA{}{} - \bB{}{})^{1/2} \cdot (\bA{}{} + \bB{}{}) \cdot (\bA{}{} - \bB{}{})^{1/2}$
|
|
to get $\Om{m}{2}$ and $\bZ{m}{}$
|
|
\State Compute $(\bX{}{} + \bY{}{})_m = \Om{m}{-1/2} (\bA{}{} - \bB{}{})^{+1/2} \cdot \bZ{m}{}$
|
|
\EndIf
|
|
\EndProcedure
|
|
\end{algorithmic}
|
|
\end{block}
|
|
\end{frame}
|
|
%-----------------------------------------------------
|
|
|
|
|
|
|
|
%-----------------------------------------------------
|
|
\begin{frame}{Correlation energy}
|
|
\begin{block}{RPA correlation energy: plasmon formula}
|
|
\begin{equation*}
|
|
\label{eq:Ec-RPA}
|
|
\EcRPA
|
|
= \frac{1}{2} \qty[ \sum_{p} \Om{m}{\RPA} - \Tr(\bA{}{\RPA}) ]
|
|
= \frac{1}{2} \sum_{m} \qty( \Om{m}{\RPA} - \Om{m}{\TDA} )
|
|
\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} \red{\SigC{pq}}(\omega) \blue{\G{pq}}(\omega) e^{i\omega\eta}
|
|
= 4 \sum_{ia} \sum_{m} \frac{\violet{\ERI{ai}{m}}^2}{\e{a} - \e{i} + \orange{\Om{m}{\RPA}}}
|
|
\end{equation*}
|
|
\end{block}
|
|
\end{frame}
|
|
%-----------------------------------------------------
|
|
|
|
%-----------------------------------------------------
|
|
\begin{frame}{Adiabatic connection fluctuation dissipation theorem (ACFDT)}
|
|
\begin{block}{Adiabatic connection}
|
|
\begin{equation}
|
|
\boxed{
|
|
\Ec^\text{ACFDT}
|
|
= \frac{1}{2} \int_0^1 \Tr( \bK{}{} \bP{}{\lambda}) d\lambda
|
|
\stackrel{\blue{\text{quad}}}{\approx} \frac{1}{2} \sum_k^{N_\text{grid}} w_k \Tr( \bK{}{} \bP{}{\lambda_k})
|
|
}
|
|
\end{equation}
|
|
\end{block}
|
|
\begin{block}{Interaction kernel}
|
|
\begin{equation}
|
|
\bK{}{} =
|
|
\begin{pmatrix}
|
|
\btA{}{} & \btB{}{}
|
|
\\
|
|
\btB{}{} & \btA{}{}
|
|
\end{pmatrix}
|
|
\qquad
|
|
\tA{ia,jb}{} = 2\lambda\ERI{ia}{bj}
|
|
\qquad
|
|
\tB{ia,jb}{} = 2\lambda\ERI{ia}{jb}
|
|
\end{equation}
|
|
\end{block}
|
|
\begin{block}{Correlation part of the two-particle density matrix}
|
|
\begin{equation}
|
|
\bP{}{\lambda} =
|
|
\begin{pmatrix}
|
|
\bY{}{\lambda} \cdot \T{(\bY{}{\lambda})} & \bY{}{\lambda} \cdot \T{(\bX{}{\lambda})}
|
|
\\
|
|
\bX{}{\lambda} \cdot \T{(\bY{}{\lambda})} & \bX{}{\lambda} \cdot \T{(\bX{}{\lambda})}
|
|
\end{pmatrix}
|
|
-
|
|
\begin{pmatrix}
|
|
\bO & \bO
|
|
\\
|
|
\bO & \bI
|
|
\end{pmatrix}
|
|
\end{equation}
|
|
\end{block}
|
|
\end{frame}
|
|
%-----------------------------------------------------
|
|
|
|
%-----------------------------------------------------
|
|
\begin{frame}{Gaussian quadrature}
|
|
\begin{block}{Numerical integration by quadrature}
|
|
\begin{equation}
|
|
\boxed{\int_a^b f(x) w(x) dx \approx \sum_k \underbrace{w_k}_{\text{weights}} f(\underbrace{x_k}_{\text{roots}})}
|
|
\end{equation}
|
|
\end{block}
|
|
\begin{block}{Quadrature rules}
|
|
\begin{center}
|
|
\begin{tabular}{llll}
|
|
\hline
|
|
Interval $[a,b]$ & Weight function $w(x)$ & Orthogonal polynomials & Name \\
|
|
\hline
|
|
$[-1,1]$ & $1$ & Legendre $P_n(x)$ & Gauss-Legendre \\
|
|
$(-1,1)$ & $(1-x)^\alpha(1+x)^\beta, \quad \alpha,\beta > -1$ & Jacobi $P_n^{\alpha,\beta}(x)$ & Gauss-Jacobi \\
|
|
$(-1,1)$ & $1/\sqrt{1-x^2}$ & Chebyshev (1st kind) $T_n(x)$ & Gauss-Chebyshev \\
|
|
$[-1,1]$ & $\sqrt{1-x^2}$ & Chebyshev (2nd kind) $U_n(x)$ & Gauss-Chebyshev \\
|
|
$[0,\infty)$ & $\exp(-x)$ & Laguerre $L_n(x)$ & Gauss-Laguerre \\
|
|
$[0,\infty)$ & $x^\alpha \exp(-x), \quad \alpha > -1$ & Generalized Laguerre $L_n^\alpha(x)$ & Gauss-Laguerre \\
|
|
$(-\infty,\infty)$ & $\exp(-x^2)$ & Hermite $H_n(x)$ & Gauss-Hermite \\
|
|
\hline
|
|
\end{tabular}
|
|
\\
|
|
\bigskip
|
|
\url{https://en.wikipedia.org/wiki/Gaussian_quadrature}
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\end{center}
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|
\end{block}
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|
\end{frame}
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|
%-----------------------------------------------------
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|
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|
%-----------------------------------------------------
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|
\begin{frame}{ACFDT at the RPA/RPAx level}
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\begin{block}{RPA matrix elements}
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\begin{equation}
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|
\A{ia,jb}{\lambda,\RPA} = \delta_{ij} \delta_{ab} (\eHF{a} - \eHF{i}) + 2\lambda\ERI{ia}{bj}
|
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\qquad
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|
\B{ia,jb}{\lambda,\RPA} = 2\lambda\ERI{ia}{jb}
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\end{equation}
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\begin{equation}
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|
\boxed{
|
|
\Ec^\RPA
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= \frac{1}{2} \int_0^1 \Tr( \bK{}{} \bP{}{\lambda}) d\lambda
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= \frac{1}{2} \qty[ \sum_{m} \Om{m}{\RPA} - \Tr(\bA{}{\RPA}) ]
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|
}
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|
\end{equation}
|
|
\end{block}
|
|
|
|
\begin{block}{RPAx matrix elements}
|
|
\begin{equation}
|
|
\A{ia,jb}{\lambda,\RPAx} = \delta_{ij} \delta_{ab} (\eHF{a} - \eHF{i}) + \underbrace{\lambda \qty[2 \ERI{ia}{bj} - \ERI{ij}{ab} ]}_{\tA{ia,jb}{\lambda,\RPAx}}
|
|
\qquad
|
|
\B{ia,jb}{\lambda,\RPAx} = \lambda \qty[2 \ERI{ia}{jb} - \ERI{ib}{aj} ]
|
|
\end{equation}
|
|
\end{block}
|
|
|
|
\end{frame}
|
|
%-----------------------------------------------------
|
|
|
|
|
|
%-----------------------------------------------------
|
|
\begin{frame}{ACFDT at the BSE level}
|
|
\begin{block}{BSE matrix elements}
|
|
\begin{equation}
|
|
\A{ia,jb}{\lambda,\BSE} = \delta_{ij} \delta_{ab} (\eGW{a} - \eGW{i}) + \underbrace{\lambda \qty[2 \ERI{ia}{bj} - W_{ij,ab}^{\lambda}(\omega = 0) ]}_{\tA{ia,jb}{\lambda,\BSE}}
|
|
\qquad
|
|
\B{ia,jb}{\lambda,\BSE} = \lambda \qty[2 \ERI{ia}{jb} - W_{ib,ja}^{\lambda}(\omega = 0)]
|
|
\end{equation}
|
|
\end{block}
|
|
\begin{block}{$\lambda$-dependent screening}
|
|
\begin{equation}
|
|
\highlight{W}_{pq,rs}^{\lambda}(\yo)
|
|
= \ERI{pq}{rs}
|
|
+ 2 \sum_m \violet{\ERI{pq}{m}^{\lambda}} \violet{\ERI{rs}{m}^{\lambda}}
|
|
\qty[ \frac{1}{\yo - \orange{\Om{m}{\lambda,\RPA}} + i \eta} - \frac{1}{\yo + \orange{\Om{m}{\lambda,\RPA}} - i \eta} ]
|
|
\end{equation}
|
|
\begin{equation}
|
|
\violet{\ERI{pq}{m}^{\lambda}} = \sum_{ia} \ERI{pq}{ia} (\orange{\bX{m}{\lambda,\RPA}+\bY{m}{\lambda,\RPA}})_{ia}
|
|
\end{equation}
|
|
\end{block}
|
|
\end{frame}
|
|
%-----------------------------------------------------
|
|
|
|
|
|
|
|
%-----------------------------------------------------
|
|
\begin{frame}{The bridge between TD-DFT and BSE}
|
|
\begin{center}
|
|
\begin{tabular}{lcr}
|
|
\hline
|
|
\bf \red{TD-DFT} & \bf \purple{Connection} & \bf \violet{BSE}
|
|
\\
|
|
\hline
|
|
\\
|
|
\red{One-point density} & & \violet{Two-point Green's function}
|
|
\\
|
|
$\rho(1)$ & $\rho(1) = -iG(11^{+})$ & $G(12)$
|
|
\\
|
|
\\
|
|
\red{Two-point susceptibility} & & \violet{Four-point susceptibility}
|
|
\\
|
|
$\chi(12) = \pdv{\rho(1)}{U(2)}$ & $\chi(12) = -i L(12;1^+2^+)$ & $L(12;34) = \pdv{G(13)}{U(42)}$
|
|
\\
|
|
\\
|
|
\red{Two-point kernel} & & \violet{Four-point kernel}
|
|
\\
|
|
$K(12) = v(12) + \pdv{V^{xc}(1)}{\rho(2)}$ & & $i \Xi(1234) = v(13) \delta(12) \delta(34) - \pdv{\Sigma^{xc}(12)}{G(34)}$ \\
|
|
\hline
|
|
\end{tabular}
|
|
\end{center}
|
|
\end{frame}
|
|
%-----------------------------------------------------
|
|
|
|
%-----------------------------------------------------
|
|
|
|
%-----------------------------------------------------
|
|
|
|
%-----------------------------------------------------
|
|
\begin{frame}{Relationship between CIS, TDHF, DFT and TDDFT}
|
|
\center
|
|
\begin{tikzpicture}
|
|
\usetikzlibrary{shapes.misc}
|
|
\begin{scope}[very thick,
|
|
node distance=3cm,on grid,>=stealth',
|
|
box/.style={rectangle,draw,fill=green!40}],
|
|
\node [box, align=center] (CIS) {\textbf{CIS}};
|
|
\node [box, align=center] (HF) [left=of CIS, yshift=1cm] {\textbf{HF}};
|
|
\node [box, align=center] (TDHF) [right=of CIS, yshift=1cm] {\textbf{TDHF}};
|
|
\node [box, align=center] (DFT) [below=of HF] {\textbf{DFT}};
|
|
\node [box, align=center] (TDDFT) [below=of TDHF] {\textbf{TDDFT}};
|
|
\node [box, align=center] (TDA) [below=of CIS] {\textbf{TDA}};
|
|
\path
|
|
(CIS) edge [<-] node[below,sloped]{CI} (HF)
|
|
(CIS) edge [<-] node[below,sloped]{$\bB{}{}=\bO$} (TDHF)
|
|
(HF) edge [->] node[above]{linear response} (TDHF)
|
|
(HF) edge [<->] node[left]{$\upsilon_\text{x}^\text{HF}$ vs $\upsilon_\text{xc}$} (DFT)
|
|
(TDHF) edge [<->] node[right]{$\upsilon_\text{x}^\text{HF}$ vs $\upsilon_\text{xc}$} (TDDFT)
|
|
(DFT) edge [->] node[above]{linear response} (TDDFT)
|
|
(DFT) edge [->] node[below,sloped]{CI} node[strike out,sloped]{\alert{$\cross$}} (TDA)
|
|
(TDDFT) edge [->] node[below,sloped]{$\bB{}{}=\bO{}{}$} (TDA)
|
|
;
|
|
\end{scope}
|
|
\end{tikzpicture}
|
|
\end{frame}
|
|
%-----------------------------------------------------
|
|
|
|
%-----------------------------------------------------
|
|
\begin{frame}{Useful papers}
|
|
\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:}
|
|
\begin{itemize}
|
|
\item Reining, WIREs Comput Mol Sci 2017, e1344. doi: 10.1002/wcms.1344
|
|
\item Onida et al. Rev. Mod. Phys. 74 (2002) 601
|
|
\item Blase et al. Chem. Soc. Rev. , 47 (2018) 1022
|
|
\item Golze et al. Front. Chem. 7 (2019) 377
|
|
\item Blase et al. JPCL 11 (2020) 7371
|
|
\end{itemize}
|
|
\bigskip
|
|
\item \red{GW100:} Data set of 100 molecules. van Setten et al. JCTC 11 (2015) 5665
|
|
\end{itemize}
|
|
\end{frame}
|
|
%-----------------------------------------------------
|
|
|
|
\end{document}
|