final v1 and update v2
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@ -191,15 +191,25 @@ decoration={snake,
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\begin{frame}{One- and two-electron integrals}
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\begin{block}{One-electron integrals: overlap \& core Hamiltonian (Appendix A)}
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\begin{equation}
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S_{\mu\nu} = \int \phi_\mu(\orange{\br}) \phi_\nu(\orange{\br}) d\orange{\br}
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\end{equation}
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\begin{equation}
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H_{\mu\nu} = \int \phi_\mu(\orange{\br}) \hH^\text{c}(\orange{\br}) \phi_\nu(\orange{\br}) d\orange{\br}
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\end{equation}
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\end{block}
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\begin{block}{Chemist/Mulliken notation for two-electron integrals (p.~68)}
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\begin{columns}
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\begin{column}{0.7\textwidth}
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\begin{block}{One-electron integrals: overlap \& core Hamiltonian (Appendix A)}
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\begin{equation}
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S_{\mu\nu}
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= \braket{\mu}{\nu}
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= \int \phi_\mu(\orange{\br}) \phi_\nu(\orange{\br}) d\orange{\br}
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\end{equation}
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\begin{equation}
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H_{\mu\nu}
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= \mel{\mu}{\hH^\text{c}}{\nu}
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= \int \phi_\mu(\orange{\br}) \hH^\text{c}(\orange{\br}) \phi_\nu(\orange{\br}) d\orange{\br}
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\end{equation}
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\end{block}
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\end{column}
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\begin{column}{0.3\textwidth}
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\includegraphics[width=\textwidth]{fig/SBG}
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\end{column}
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\end{columns} \begin{block}{Chemist/Mulliken notation for two-electron integrals (p.~68)}
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\begin{equation}
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( \mu \nu | \lambda \sigma )
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= \iint \phi_\mu(\alert{\br_1}) \phi_\nu(\alert{\br_1}) \frac{1}{r_{12}} \phi_\lambda(\blue{\br_2}) \phi_\sigma(\blue{\br_2}) d\red{\br_1} d\blue{\br_2}
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@ -234,10 +244,11 @@ decoration={snake,
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\phi_{\bb_1}^{\bB_1}(\br_1) \phi_{\bb_2}^{\bB_2}(\br_2) d\br_1 d\br_2
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\end{split}
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\end{equation}
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\alert{Formally, one has to compute $\order{N^4}$ ERIs!}
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\end{block}
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\end{column}
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\begin{column}{0.3\textwidth}
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\includegraphics[width=\textwidth]{fig/SBG}
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\includegraphics[width=\textwidth]{fig/STO}
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\end{column}
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\end{columns}
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%
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@ -315,14 +326,14 @@ decoration={snake,
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\qqtext{then}
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\end{equation}
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\begin{equation}
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\boxed{G_{\red{\alpha},\red{\bm{A}}}(\br) G_{\blue{\beta},\blue{\bm{B}}}(\br) = \green{K} \, G_{\violet{\zeta},\violet{\bm{P}}}(\br)}
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\boxed{G_{\red{\alpha},\red{\bm{A}}}(\br) G_{\blue{\beta},\blue{\bm{B}}}(\br) = \violet{K} \, G_{\violet{\zeta},\violet{\bm{P}}}(\br)}
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\qqtext{with}
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\violet{\zeta} = \red{\alpha} + \blue{\beta}
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\qqtext{and}
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\violet{\bm{P}} = \frac{\red{\alpha \bA} + \blue{\beta \bB}}{\red{\alpha} + \blue{\beta} }
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\end{equation}
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\begin{equation}
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\green{K} = \exp( -\frac{\red{\alpha} \blue{\beta}}{\red{\alpha} + \blue{\beta} } \abs{\red{\bA} - \blue{\bB}}^2)
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\violet{K} = \exp( -\frac{\red{\alpha} \blue{\beta}}{\red{\alpha} + \blue{\beta} } \abs{\red{\bA} - \blue{\bB}}^2)
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\end{equation}
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\end{block}
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\begin{block}{Gaussian product rule for ERIs}
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@ -331,9 +342,10 @@ decoration={snake,
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(\bm{\red{a}} \bm{\blue{b}}|\bm{\orange{c}} \bm{\green{d}})
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& = \iint G_{\red{\alpha},\red{\bm{A}}}(\br_1) G_{\blue{\beta},\blue{\bm{B}}}(\br_1) \frac{1}{r_{12}} G_{\orange{\gamma},\orange{\bm{C}}}(\br_2) G_{\green{\delta},\green{\bm{D}}}(\br_2) d\br_1 d\br_2
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\\
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& = \iint G_{\violet{\zeta},\violet{\bm{P}}}(\br_1) \frac{1}{r_{12}} G_{\purple{\eta},\purple{\bm{Q}}}(\br_2) d\br_1 d\br_2
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& = \violet{K} \purple{K} \iint G_{\violet{\zeta},\violet{\bm{P}}}(\br_1) \frac{1}{r_{12}} G_{\purple{\eta},\purple{\bm{Q}}}(\br_2) d\br_1 d\br_2
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\end{split}
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\end{equation}
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\alert{The number of ``significant'' ERIs in a large system is $\order{N^2}$!}
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\end{block}
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\end{frame}
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%
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BIN
2021/Lecture_1/fig/STO.pdf
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BIN
2021/Lecture_1/fig/STO.pdf
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Binary file not shown.
@ -1,11 +1,11 @@
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\documentclass[aspectratio=169,9pt]{beamer}
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\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{Pittsburgh}
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\usecolortheme{seahorse}
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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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@ -25,65 +25,161 @@
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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{\hI}{\Hat{1}}
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\newcommand{\hH}{\Hat{H}}
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\newcommand{\hT}[2]{\Hat{T}_{#1}^{#2}}
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\newcommand{\bH}{\mathbold{H}}
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\newcommand{\br}{\mathbold{r}}
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\newcommand{\la}{\lambda}
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\newcommand{\si}{\sigma}
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\newcommand{\cJ}{\mathcal{J}}
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\newcommand{\cK}{\mathcal{K}}
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% wave functions
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\newcommand{\PsiO}{\Psi_0}
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\newcommand{\PsiHF}{\Psi_\text{RHF}}
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\newcommand{\PsiFCI}{\Psi_\text{FCI}}
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\newcommand{\PsiCC}{\Psi_\text{CC}}
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\newcommand{\PsiCCD}{\Psi_\text{CCD}}
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\newcommand{\amp}[2]{t_{#1}^{#2}}
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\newcommand{\Det}[2]{\Psi_{#1}^{#2}}
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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{\EO}{E_\text{0}}
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\newcommand{\ECC}{E_\text{CC}}
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\newcommand{\EVCC}{E_\text{VCC}}
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\newcommand{\ECCD}{E_\text{CCD}}
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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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\newcommand{\nEl}{n}
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\newcommand{\nBas}{N}
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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{\ba}{\bm{a}}
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\newcommand{\bb}{\bm{b}}
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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{\bo}{\bm{0}}
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\newcommand{\sbra}[1]{[ #1 |}
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\newcommand{\sket}[1]{| #1 ]}
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\newcommand{\sexpval}[1]{[ #1 ]}
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\newcommand{\sbraket}[2]{[ #1 | #2 ]}
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\newcommand{\smel}[3]{[ #1 | #2 | #3 ]}
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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}{234,187,0}
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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{\mc}{\multicolumn}
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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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@ -101,8 +197,8 @@ decoration={snake,
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% * HEAD DATA *
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% *************
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\title[$GW$/BSE methods in chemistry]{
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\purple{$GW$/BSE methods in chemistry: \\
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Computational aspects}
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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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@ -111,7 +207,7 @@ decoration={snake,
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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=0.4\textwidth]{fig/jarvis}
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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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@ -139,44 +235,256 @@ decoration={snake,
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\end{frame}
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%
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%%% FINAL SLIDE %%%
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%-----------------------------------------------------
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\section{Books}
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\section{Theory}
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%-----------------------------------------------------
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\begin{frame}{Good books}
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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}{0.7\textwidth}
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\begin{itemize}
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\item Introduction to Computational Chemistry (Jensen)
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\\
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\vspace{1cm}
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\item Essentials of Computational Chemistry (Cramer)
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\\
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\vspace{1cm}
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\item Modern Quantum Chemistry (Szabo \& Ostlund)
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\\
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\vspace{1cm}
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\item Molecular Electronic Structure Theory (Helgaker, Jorgensen \& Olsen)
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\\
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\vspace{1cm}
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\end{itemize}
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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}{0.3\textwidth}
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\centering
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\includegraphics[height=0.3\textwidth]{fig/Jensen}
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\\
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\bigskip
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\includegraphics[height=0.3\textwidth]{fig/Cramer}
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\\
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\bigskip
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\includegraphics[height=0.3\textwidth]{fig/Szabo}
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\\
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\bigskip
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\includegraphics[height=0.3\textwidth]{fig/Helgaker}
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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
|
||||
\item Electron affinities (EA) given by virtual MO energies
|
||||
\bigskip
|
||||
\item HOMO-LUMO gap (or band gap in solids)
|
||||
\bigskip
|
||||
\item Singlet and triplet neutral excitations (vertical absorption energies)
|
||||
\bigskip
|
||||
\item Correlation and total energies
|
||||
\end{itemize}
|
||||
\end{block}
|
||||
\end{column}
|
||||
\end{columns}
|
||||
\end{frame}
|
||||
|
||||
%-----------------------------------------------------
|
||||
\subsection{GW flavours}
|
||||
%-----------------------------------------------------
|
||||
\begin{frame}{GW flavours}
|
||||
\begin{block}{Acronyms}
|
||||
\begin{itemize}
|
||||
\bigskip
|
||||
\item perturbative GW one-shot GW, or \green{\GOWO}
|
||||
\bigskip
|
||||
\item \orange{\evGW} or eigenvalue-only (partially) self-consistent GW
|
||||
\bigskip
|
||||
\item \red{\qsGW} or quasiparticle (partially) self-consistent GW
|
||||
\bigskip
|
||||
\item \violet{\scGW} or (fully) self-consistent GW
|
||||
\bigskip
|
||||
\item \purple{BSE} or Bethe-Salpeter equation for neutral excitations
|
||||
\bigskip
|
||||
\end{itemize}
|
||||
\end{block}
|
||||
\end{frame}
|
||||
%-----------------------------------------------------
|
||||
\subsection{Literature}
|
||||
%-----------------------------------------------------
|
||||
\begin{frame}{useful papers for chemists}
|
||||
\begin{itemize}
|
||||
\item \red{molGW:} Bruneval et al. Comp. Phys. Comm. 208 (2016) 149
|
||||
\bigskip
|
||||
\item \green{Turbomole:} van Setten et al. JCTC 9 (2013) 232; Kaplan et al. JCTC 12 (2016) 2528
|
||||
\bigskip
|
||||
\item \violet{Fiesta:} Blase et al. Chem. Soc. Rev. 47 (2018) 1022
|
||||
\bigskip
|
||||
\item \purple{FHI-AIMS:} Caruso et al. 86 (2012) 081102
|
||||
\bigskip
|
||||
\item \orange{Review:} Reining, WIREs Comput Mol Sci 2017, e1344. doi: 10.1002/wcms.1344; Onida et al. Rev. Mod. Phys. 74 (2002) 601.
|
||||
\bigskip
|
||||
\item \red{GW100:} Data set of 100 molecules. van Setten et al. JCTC 11 (2015) 5665
|
||||
\end{itemize}
|
||||
\end{frame}
|
||||
|
||||
%-----------------------------------------------------
|
||||
\section{Implementation}
|
||||
%-----------------------------------------------------
|
||||
\subsection{\GOWO}
|
||||
%-----------------------------------------------------
|
||||
\begin{frame}{\GOWO}
|
||||
\begin{block}{{\GOWO}~subroutine}
|
||||
\begin{algorithmic}
|
||||
\Procedure{Perturbative {\GW}}{}
|
||||
\State Perform HF calculation to get $\beHF$ and $\bcHF$
|
||||
\For{$p=1,\ldots,N$}
|
||||
\State Compute \red{$\SigC{pp}(\omega)$} and \green{$\Z{p}(\omega)$}
|
||||
\State $\eGOWO{p} = \eHF{p} + \green{\Z{p}(\eHF{p})} \Re[\red{\SigC{pp}(\eHF{p})}]$
|
||||
\State \Comment{This is the linearized version of the}
|
||||
\State \Comment{quasiparticle (QP) equation
|
||||
$\omega = \eHF{p} + \Re[\red{\SigC{pp}(\omega)}]$}
|
||||
\EndFor
|
||||
\If{BSE}
|
||||
\State Compute BSE excitations energies
|
||||
\EndIf
|
||||
\EndProcedure
|
||||
\end{algorithmic}
|
||||
\end{block}
|
||||
\end{frame}
|
||||
|
||||
\begin{frame}{\GOWO}
|
||||
\begin{block}{\red{Correlation part of the self-energy:}}
|
||||
\begin{equation*}
|
||||
\red{\SigC{pq}(\omega)}
|
||||
= 2 \sum_{ix}\frac{\violet{[pi|x] [qi|x]}}{\omega - \eHF{i} + \orange{\Om{x}} - i \eta}
|
||||
+ 2 \sum_{ax}\frac{\violet{[pa|x] [qa|x]}}{\omega - \eHF{a} - \orange{\Om{x}} + i \eta}
|
||||
\end{equation*}
|
||||
\end{block}
|
||||
\begin{block}{\green{Renormalization factor}}
|
||||
\begin{equation*}
|
||||
\green{\Z{p}(\omega)} = \qty[ 1 - \pdv{\Re[\red{\SigGW{pp}(\omega)}]}{\omega} ]^{-1}
|
||||
\end{equation*}
|
||||
\end{block}
|
||||
\begin{block}{\violet{Screened two-electron MO integrals}}
|
||||
\begin{equation*}
|
||||
\violet{[pq|x]} = \sum_{ia} (pq|ia) \orange{(\bX+\bY)_{ia}^{x}}
|
||||
\end{equation*}
|
||||
\end{block}
|
||||
\begin{block}{\orange{RPA excitation energies}}
|
||||
\small
|
||||
\begin{equation*}
|
||||
\begin{pmatrix}
|
||||
\bA & \bB \\
|
||||
\bB & \bA \\
|
||||
\end{pmatrix}
|
||||
\orange{\begin{pmatrix}
|
||||
\bX \\
|
||||
\bY \\
|
||||
\end{pmatrix}}
|
||||
=
|
||||
\orange{\bOm}
|
||||
\begin{pmatrix}
|
||||
\bm{1} & 0 \\
|
||||
0 & \bm{-1} \\
|
||||
\end{pmatrix}
|
||||
\orange{\begin{pmatrix}
|
||||
\bX \\
|
||||
\bY \\
|
||||
\end{pmatrix}}
|
||||
\end{equation*}
|
||||
\begin{align*}
|
||||
A^\text{RPA}_{ia,jb} & = \delta_{ij} \delta_{ab} (\epsilon_a - \epsilon_i) + 2 (ia|jb)
|
||||
&
|
||||
B^\text{RPA}_{ia,jb} & = 2 (ia|bj)
|
||||
\end{align*}
|
||||
\end{block}
|
||||
\end{frame}
|
||||
|
||||
%-----------------------------------------------------
|
||||
\subsection{\evGW}
|
||||
%-----------------------------------------------------
|
||||
\begin{frame}{\evGW}
|
||||
\begin{block}{{\evGW} subroutine}
|
||||
\begin{algorithmic}
|
||||
\Procedure{Partially self-consistent {\evGW}}{}
|
||||
\State Perform HF calculation to get $\beHF$ and $\bcHF$
|
||||
\State Set $\beGnWn{-1} = \beHF$ and $n = 0$
|
||||
\While{$\max{\abs{\bDelta}} < \tau$}
|
||||
\For{$p=1,\ldots,N$}
|
||||
\State Compute \red{$\SigC{pp}(\omega)$}
|
||||
\State Solve $\omega = \eHF{p} + \Re[\red{\SigC{pp}(\omega)}]$ to obtain $\eGnWn{p}{n}$
|
||||
\EndFor
|
||||
\State $\bDelta = \beGnWn{n} - \beGnWn{n-1}$
|
||||
\State $n \leftarrow n + 1$
|
||||
\EndWhile
|
||||
\If{BSE}
|
||||
\State Compute BSE excitations energies
|
||||
\EndIf
|
||||
\EndProcedure
|
||||
\end{algorithmic}
|
||||
\end{block}
|
||||
\end{frame}
|
||||
|
||||
%-----------------------------------------------------
|
||||
\subsection{\qsGW}
|
||||
%-----------------------------------------------------
|
||||
\begin{frame}{\qsGW}
|
||||
\begin{block}{{\qsGW} subroutine}
|
||||
\begin{algorithmic}
|
||||
\Procedure{Partially self-consistent {\qsGW}}{}
|
||||
\State Perform HF calculation to get $\beHF$ and $\bcHF$
|
||||
\State Set $\beGnWn{-1} = \beHF$, $\bcGnWn{-1} = \bcHF$ and $n = 0$
|
||||
\While{$\max{\abs{\bDelta}} < \tau$}
|
||||
\State Form \red{$\bSigC(\omega)$} and symmetrize it: $\red{\bSigC(\omega)} \leftarrow (\red{\bSigC(\omega)}^\dag + \red{\bSigC(\omega)})/2$
|
||||
\State Form $\green{\bF(\omega)} = \bFHF + \red{\bSigC(\omega)}$
|
||||
\State Diagonalize $\green{\bF(\beGnWn{n-1})}$ to get $\beGnWn{n}$ and $\bcGnWn{n}$
|
||||
\State $\bDelta = \beGnWn{n} - \beGnWn{n-1}$
|
||||
\State $n \leftarrow n + 1$
|
||||
\EndWhile
|
||||
\If{BSE}
|
||||
\State Compute BSE excitations energies
|
||||
\EndIf
|
||||
\EndProcedure
|
||||
\end{algorithmic}
|
||||
\end{block}
|
||||
\end{frame}
|
||||
|
||||
%-----------------------------------------------------
|
||||
\subsection{BSE}
|
||||
%-----------------------------------------------------
|
||||
\begin{frame}{BSE}
|
||||
\begin{block}{Bethe-Salpeter equation}
|
||||
\begin{equation*}
|
||||
\begin{pmatrix}
|
||||
\bA & \bB \\
|
||||
\bB & \bA \\
|
||||
\end{pmatrix}
|
||||
\purple{\begin{pmatrix}
|
||||
\bX \\
|
||||
\bY \\
|
||||
\end{pmatrix}}
|
||||
=
|
||||
\purple{\bOm}
|
||||
\begin{pmatrix}
|
||||
\bm{1} & 0 \\
|
||||
0 & \bm{-1} \\
|
||||
\end{pmatrix}
|
||||
\purple{\begin{pmatrix}
|
||||
\bX \\
|
||||
\bY \\
|
||||
\end{pmatrix}}
|
||||
\end{equation*}
|
||||
\begin{equation*}
|
||||
(\bA - \bB)^{1/2} (\bA + \bB) (\bA - \bB)^{1/2} \bZ = \bOm^2 \bZ,
|
||||
\end{equation*}
|
||||
\begin{equation*}
|
||||
\bX + \bY = \bOm^{-1/2} (\bA - \bB)^{1/2} \bZ.
|
||||
\end{equation*}
|
||||
\begin{align*}
|
||||
A^\text{BSE}_{ia,jb} & = A^\text{RPA}_{ia,jb} - (ij|ab) + 4 \sum_{x} \frac{[ij|x][ab|x]}{\Om{x}}
|
||||
\\
|
||||
B^\text{BSE}_{ia,jb} & = B^\text{RPA}_{ia,jb} - (ib|aj) + 4 \sum_{x} \frac{[ib|x][aj|x]}{\Om{x}}
|
||||
\end{align*}
|
||||
\end{block}
|
||||
\end{frame}
|
||||
|
||||
%-----------------------------------------------------
|
||||
\subsection{$\Ec$}
|
||||
%-----------------------------------------------------
|
||||
\begin{frame}{Correlation energy}
|
||||
|
||||
\begin{block}{RPA correlation energy or Klein functional}
|
||||
\begin{equation*}
|
||||
\label{eq:Ec-RPA}
|
||||
\EcRPA = -\sum_{p} \qty(\ARPA{pp} - \Om{p})
|
||||
\end{equation*}
|
||||
\end{block}
|
||||
|
||||
\begin{block}{Galitskii-Migdal functional}
|
||||
\begin{equation*}
|
||||
\label{eq:GM}
|
||||
\EcGM = \frac{-i}{2}\sum_{pq}^{\infty}\int \frac{d\omega}{2\pi} \SigC{pq}(\omega) \G{pq}(\omega) e^{i\omega\eta}
|
||||
\end{equation*}
|
||||
\end{block}
|
||||
|
||||
\end{frame}
|
||||
|
||||
\end{document}
|
||||
|
||||
74
2021/Lecture_2/fig/BSE-GW.tex
Normal file
74
2021/Lecture_2/fig/BSE-GW.tex
Normal file
@ -0,0 +1,74 @@
|
||||
\documentclass{standalone}
|
||||
\usepackage{amsmath,amssymb,amsfonts,pgfpages,graphicx,subfigure,xcolor,bm,multirow,microtype,wasysym,multimedia,hyperref,tabularx,amscd,pgfgantt,mhchem,physics}
|
||||
\usetikzlibrary{shapes.gates.logic.US,trees,positioning,arrows}
|
||||
|
||||
\usepackage{tgchorus}
|
||||
\usepackage[T1]{fontenc}
|
||||
|
||||
\begin{document}
|
||||
|
||||
\begin{tikzpicture}
|
||||
\begin{scope}[very thick,
|
||||
node distance=5cm,on grid,>=stealth',
|
||||
theo1/.style={rectangle,draw,fill=red!20},
|
||||
theo2/.style={rectangle,draw,fill=orange!20},
|
||||
theo3/.style={rectangle,draw,fill=green!40},
|
||||
exp1/.style={rectangle,draw,fill=cyan!40},
|
||||
exp2/.style={rectangle,draw,fill=violet!40}]
|
||||
|
||||
\node [theo1, text width=7cm, align=center] (KS)
|
||||
{\textbf{\LARGE Kohn-Sham DFT}
|
||||
$$
|
||||
\qty[ -\frac{\nabla^2}{2} + v_\text{ext} + V^{\text{Hxc}} ] \phi_p^{\text{KS}} = \varepsilon^{\text{KS}}_p \phi_p^{\text{KS}}
|
||||
$$
|
||||
};
|
||||
|
||||
\node [theo2, text width=7cm, align=center] (GW) [below=of KS, yshift=2cm]
|
||||
{\textbf{\LARGE $GW$ approximation}
|
||||
$$
|
||||
\varepsilon_p^{GW} = \varepsilon_p^{\text{KS}} +
|
||||
\mel{\phi_p^{\text{KS}}}{\Sigma^{GW}(\varepsilon_p^{GW}) - V^{\text{xc}}}{\phi_p^{\text{KS}}}
|
||||
$$
|
||||
|
||||
};
|
||||
|
||||
\node [theo3, text width=7cm, align=center] (BSE) [below=of GW, yshift=2cm]
|
||||
{\textbf{\LARGE Bethe-Salpeter equation}
|
||||
$$
|
||||
\begin{pmatrix}
|
||||
R & C \\
|
||||
-C^* & -R^{*}
|
||||
\end{pmatrix}
|
||||
\begin{pmatrix}
|
||||
X_m \\
|
||||
Y_m
|
||||
\end{pmatrix}
|
||||
=
|
||||
\Omega_{m}
|
||||
\begin{pmatrix}
|
||||
X_m \\
|
||||
Y_m
|
||||
\end{pmatrix}
|
||||
$$
|
||||
};
|
||||
|
||||
|
||||
\node [exp1, align=center] (photo) [right=of GW, xshift=3cm]
|
||||
{\LARGE (Inverse) \\ \LARGE photoemission \\ \LARGE spectroscopy};
|
||||
|
||||
\node [exp2, align=center] (abs) [right=of BSE, xshift=3cm]
|
||||
{\LARGE Optical \\ \LARGE spectroscopy};
|
||||
|
||||
|
||||
\path
|
||||
(KS) edge [->,color=black] node [right,black] {\LARGE Fundamental gap} (GW)
|
||||
(GW) edge [->,color=black] node [right,black] {\LARGE Excitonic effect} (BSE)
|
||||
(photo) edge [<->,color=black] node [above,black] {Ionization potentials} node [below,black] {Electron affinities} (GW)
|
||||
(abs) edge [<->,color=black] node [above,black] {Optical excitations} (BSE)
|
||||
;
|
||||
|
||||
|
||||
\end{scope}
|
||||
\end{tikzpicture}
|
||||
|
||||
\end{document}
|
||||
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Loading…
Reference in New Issue
Block a user