\documentclass[aspectratio=169,9pt,compress]{beamer} % *********** % * PACKAGE * % *********** \usepackage{amsmath,amssymb,amsfonts,pgfpages,graphicx,subfigure,xcolor,bm,multirow,microtype,wasysym,multimedia,hyperref,tabularx,amscd,pgfgantt,mhchem} \usetikzlibrary{shapes.gates.logic.US,trees,positioning,arrows} \usetheme{Warsaw} %\usecolortheme{seahorse} \usepackage{mathpazo,libertine} \usepackage{algorithmicx,algorithm,algpseudocode} \algnewcommand\algorithmicassert{\texttt{assert}} \algnewcommand\Assert[1]{\State \algorithmicassert(#1)} %\algrenewcommand{\algorithmiccomment}[1]{$\triangleright$ #1} %\usepackage[version=4]{mhchem} \usepackage{amsmath,amsfonts,amssymb,bm,microtype,graphicx,wrapfig,geometry,physics,eurosym,multirow,pgfgantt} \usepackage{hyperref} \hypersetup{ colorlinks=true, linkcolor=cyan, filecolor=magenta, urlcolor=blue, citecolor=purple } \definecolor{darkgreen}{RGB}{0, 180, 0} \definecolor{fooblue}{RGB}{0,153,255} \definecolor{fooyellow}{RGB}{234,180,0} \definecolor{lavender}{rgb}{0.71, 0.49, 0.86} \definecolor{inchworm}{rgb}{0.7, 0.93, 0.36} \newcommand{\violet}[1]{\textcolor{lavender}{#1}} \newcommand{\orange}[1]{\textcolor{orange}{#1}} \newcommand{\purple}[1]{\textcolor{purple}{#1}} \newcommand{\blue}[1]{\textcolor{blue}{#1}} \newcommand{\green}[1]{\textcolor{darkgreen}{#1}} \newcommand{\yellow}[1]{\textcolor{fooyellow}{#1}} \newcommand{\red}[1]{\textcolor{red}{#1}} \newcommand{\highlight}[1]{\textcolor{fooblue}{#1}} \newcommand{\pub}[1]{\small \textcolor{purple}{#1}} \newcommand{\cdash}{\multicolumn{1}{c}{---}} \newcommand{\mc}{\multicolumn} \newcommand{\mcc}[1]{\multicolumn{1}{c}{#1}} \newcommand{\mr}{\multirow} \newcommand{\br}{\bm{r}} \newcommand{\ree}{r_{12}} \newcommand{\T}[1]{#1^{\intercal}} % methods \newcommand{\evGW}{ev$GW$} \newcommand{\qsGW}{qs$GW$} \newcommand{\scGW}{sc$GW$} \newcommand{\GOWO}{$G_0W_0$} \newcommand{\GOW}{$G_0W$} \newcommand{\GWO}{$GW_0$} \newcommand{\GW}{$GW$} \newcommand{\GOWOSOSEX}{{\GOWO}+SOSEX} \newcommand{\GWSOSEX}{{\GW}+SOSEX} \newcommand{\GnWn}[1]{$G_{#1}W_{#1}$} \newcommand{\GOF}{$G_0F2$} \newcommand{\GF}{$GF2$} \newcommand{\KS}{\text{KS}} \newcommand{\RPA}{\text{RPA}} \newcommand{\RPAx}{\text{RPAx}} \newcommand{\BSE}{\text{BSE}} \newcommand{\TDA}{\text{TDA}} \newcommand{\xc}{\text{xc}} \newcommand{\Ha}{\text{H}} \newcommand{\co}{\text{c}} \newcommand{\x}{\text{x}} % operators \newcommand{\hH}{\Hat{H}} % energies \newcommand{\Ec}{E_\text{c}} \newcommand{\EHF}{E_\text{HF}} \newcommand{\EcK}{E_\text{c}^\text{Klein}} \newcommand{\EcRPA}{E_\text{c}^\text{RPA}} \newcommand{\EcGM}{E_\text{c}^\text{GM}} \newcommand{\EcGMGW}{E_\text{c}^\text{GM@GW}} \newcommand{\EcGMGF}{E_\text{c}^\text{GM@GF2}} \newcommand{\EcGMGWSOSEX}{E_\text{c}^\text{GM@GW+SOSEX}} \newcommand{\EcMP}{E_c^\text{MP2}} \newcommand{\EcGF}{E_c^\text{\GF}} \newcommand{\EcGOF}{E_c^\text{\GOF}} \newcommand{\Egap}{E_\text{gap}} \newcommand{\IP}{\text{IP}} \newcommand{\EA}{\text{EA}} % orbital energies \newcommand{\nSat}[1]{N_{#1}^\text{sat}} \newcommand{\eSat}[2]{\epsilon_{#1,#2}} \newcommand{\e}[1]{\epsilon_{#1}} \newcommand{\eHF}[1]{\epsilon^\text{HF}_{#1}} \newcommand{\eKS}[1]{\epsilon^\text{KS}_{#1}} \newcommand{\eQP}[1]{\epsilon^\text{QP}_{#1}} \newcommand{\eGOWO}[1]{\epsilon^\text{\GOWO}_{#1}} \newcommand{\eGW}[1]{\epsilon^\text{\GW}_{#1}} \newcommand{\eGnWn}[2]{\epsilon^\text{\GnWn{#2}}_{#1}} \newcommand{\eGF}[1]{\epsilon^\text{\GF}_{#1}} \newcommand{\eGOF}[1]{\epsilon^\text{\GOF}_{#1}} \newcommand{\de}[1]{\Delta\epsilon_{#1}} \newcommand{\deHF}[1]{\Delta\epsilon^\text{HF}_{#1}} \newcommand{\deKS}[1]{\Delta\epsilon^\text{KS}_{#1}} \newcommand{\Om}[2]{\Omega_{#1}^{#2}} \newcommand{\eHOMO}{\epsilon_\text{HOMO}} \newcommand{\eLUMO}{\epsilon_\text{LUMO}} \newcommand{\cHF}[1]{c^\text{HF}_{#1}} \newcommand{\cKS}[1]{c^\text{KS}_{#1}} % Matrix elements \newcommand{\A}[2]{A_{#1}^{#2}} \newcommand{\tA}[2]{\Tilde{A}_{#1}^{#2}} \newcommand{\B}[2]{B_{#1}^{#2}} \newcommand{\tB}[2]{\Tilde{B}_{#1}^{#2}} \renewcommand{\S}[1]{S_{#1}} \newcommand{\ABSE}[1]{A^\text{BSE}_{#1}} \newcommand{\BBSE}[1]{B^\text{BSE}_{#1}} \newcommand{\ARPA}[1]{A^\text{RPA}_{#1}} \newcommand{\BRPA}[1]{B^\text{RPA}_{#1}} \newcommand{\dABSE}[1]{\delta A^\text{BSE}_{#1}} \newcommand{\dBBSE}[1]{\delta B^\text{BSE}_{#1}} \newcommand{\G}[1]{G_{#1}} \newcommand{\Po}[1]{P_{#1}} \newcommand{\W}[1]{W_{#1}} \newcommand{\Wc}[1]{W^\text{c}_{#1}} \newcommand{\vc}[1]{v_{#1}} \newcommand{\SigX}[1]{\Sigma^\text{x}_{#1}} \newcommand{\SigC}[1]{\Sigma^\text{c}_{#1}} \newcommand{\Sig}[2]{\Sigma_{#1}^{#2}} \newcommand{\SigGW}[1]{\Sigma^\text{\GW}_{#1}} \newcommand{\SigGWSOSEX}[1]{\Sigma^\text{\GWSOSEX}_{#1}} \newcommand{\SigGF}[1]{\Sigma^\text{\GF}_{#1}} \newcommand{\Z}[1]{Z_{#1}} % excitation energies \newcommand{\OmRPA}[1]{\Omega^\text{RPA}_{#1}} \newcommand{\OmCIS}[1]{\Omega^\text{CIS}_{#1}} \newcommand{\OmTDHF}[1]{\Omega^\text{TDHF}_{#1}} \newcommand{\OmBSE}[1]{\Omega^\text{BSE}_{#1}} \newcommand{\spinup}{\downarrow} \newcommand{\spindw}{\uparrow} \newcommand{\singlet}{\uparrow\downarrow} \newcommand{\triplet}{\uparrow\uparrow} \newcommand{\Oms}[1]{{}^{1}\Omega_{#1}} \newcommand{\OmsRPA}[1]{{}^{1}\Omega^\text{RPA}_{#1}} \newcommand{\OmsCIS}[1]{{}^{1}\Omega^\text{CIS}_{#1}} \newcommand{\OmsTDHF}[1]{{}^{1}\Omega^\text{TDHF}_{#1}} \newcommand{\OmsBSE}[1]{{}^{1}\Omega^\text{BSE}_{#1}} \newcommand{\Omt}[1]{{}^{3}\Omega_{#1}} \newcommand{\OmtRPA}[1]{{}^{3}\Omega^\text{RPA}_{#1}} \newcommand{\OmtCIS}[1]{{}^{3}\Omega^\text{CIS}_{#1}} \newcommand{\OmtTDHF}[1]{{}^{3}\Omega^\text{TDHF}_{#1}} \newcommand{\OmtBSE}[1]{{}^{3}\Omega^\text{BSE}_{#1}} \newcommand{\MO}[1]{\phi_{#1}} \newcommand{\ERI}[2]{(#1|#2)} \newcommand{\rbra}[1]{(#1|} \newcommand{\rket}[1]{|#1)} \newcommand{\sERI}[2]{[#1|#2]} \newcommand{\sig}{\sigma} \newcommand{\sigp}{\sigma'} % Matrices \newcommand{\bF}{\bm{F}} \newcommand{\bFHF}{\bm{F}^\text{HF}} \newcommand{\bH}{\bm{H}} \newcommand{\bvc}{\bm{v}} \newcommand{\bSig}{\bm{\Sigma}} \newcommand{\bSigX}{\bm{\Sigma}^\text{x}} \newcommand{\bSigC}{\bm{\Sigma}^\text{c}} \newcommand{\bSigGW}{\bm{\Sigma}^\text{\GW}} \newcommand{\bSigGWSOSEX}{\bm{\Sigma}^\text{\GWSOSEX}} \newcommand{\bSigGF}{\bm{\Sigma}^\text{\GF}} \newcommand{\be}{\bm{\epsilon}} \newcommand{\bDelta}{\bm{\Delta}} \newcommand{\beHF}{\bm{\epsilon}^\text{HF}} \newcommand{\beKS}{\bm{\epsilon}^\text{KS}} \newcommand{\bcHF}{\bm{c}^\text{HF}} \newcommand{\bcKS}{\bm{c}^\text{KS}} \newcommand{\beGW}{\bm{\epsilon}^\text{\GW}} \newcommand{\beGnWn}[1]{\bm{\epsilon}^\text{\GnWn{#1}}} \newcommand{\bcGnWn}[1]{\bm{c}^\text{\GnWn{#1}}} \newcommand{\beGF}{\bm{\epsilon}^\text{\GF}} \newcommand{\bde}{\bm{\Delta\epsilon}} \newcommand{\bdeHF}{\bm{\Delta\epsilon}^\text{HF}} \newcommand{\bdeGW}{\bm{\Delta\epsilon}^\text{GW}} \newcommand{\bdeGF}{\bm{\Delta\epsilon}^\text{GF2}} \newcommand{\bO}{\bm{0}} \newcommand{\bI}{\bm{1}} \newcommand{\bOm}[2]{\bm{\Omega}_{#1}^{#2}} \newcommand{\bA}[2]{\bm{A}_{#1}^{#2}} \newcommand{\btA}[2]{\bm{\Tilde{A}}_{#1}^{#2}} \newcommand{\btB}[2]{\bm{\Tilde{B}}_{#1}^{#2}} \newcommand{\bB}[2]{\bm{B}_{#1}^{#2}} \newcommand{\bX}[2]{\bm{X}_{#1}^{#2}} \newcommand{\bY}[2]{\bm{Y}_{#1}^{#2}} \newcommand{\bZ}[2]{\bm{Z}_{#1}^{#2}} \newcommand{\bK}[2]{\bm{K}_{#1}^{#2}} \newcommand{\bP}[2]{\bm{P}_{#1}^{#2}} \newcommand{\yo}{\yellow{\omega}} \newcommand{\mycirc}[1][black]{\Large\textcolor{#1}{\ensuremath\bullet}} \usepackage{tikz} \usetikzlibrary{arrows,positioning,shapes.geometric} \usetikzlibrary{decorations.pathmorphing} \tikzset{snake it/.style={ decoration={snake, amplitude = .4mm, segment length = 2mm},decorate}} % ************* % * HEAD DATA * % ************* \title[$GW$/BSE methods in chemistry]{ $GW$/BSE methods in chemistry: Computational aspects } \author[PF Loos]{Pierre-Fran\c{c}ois LOOS} \date{Online ISTPC 2021 school --- April 27th, 2021} \institute[CNRS@LCPQ]{ Laboratoire de Chimie et Physique Quantiques (UMR 5626),\\ Universit\'e de Toulouse, CNRS, UPS, Toulouse, France. } \titlegraphic{ \includegraphics[width=0.3\textwidth]{fig/jarvis} \\ \vspace{0.05\textheight} \includegraphics[height=0.05\textwidth]{fig/UPS} \hspace{0.2\textwidth} \includegraphics[height=0.05\textwidth]{fig/ERC} \hspace{0.2\textwidth} \includegraphics[height=0.05\textwidth]{fig/LCPQ} \hspace{0.2\textwidth} \includegraphics[height=0.05\textwidth]{fig/CNRS} } \begin{document} %----------------------------------------------------- \begin{frame} \titlepage \end{frame} %----------------------------------------------------- %----------------------------------------------------- \begin{frame}{Today's program} \begin{itemize} \item Charged excitations: \begin{itemize} \item One-shot $GW$ (\GOWO) \item Partially self-consistent $GW$ (\evGW) \item Self-consistent $GW$ (\qsGW) \item $GW$ vs GF \end{itemize} \item Neutral excitations \begin{itemize} \item Configuration interaction with singles (CIS) \item Time-dependent Hartree-Fock (TDHF) \item Random-phase approximation (RPA) \item Time-dependent density-functional theory (TDDFT) \item Bethe-Salpeter equation (BSE) formalism \end{itemize} \item Total energies \begin{itemize} \item Plasmon formula \item Galitski-Migdal formulation \item Adiabatic connection fluctuation-dissipation theorem (ACFDT) \end{itemize} \end{itemize} \end{frame} %----------------------------------------------------- %----------------------------------------------------- \begin{frame}{Hedin's pentagon} \begin{columns} \begin{column}{0.4\textwidth} \centering \includegraphics[width=0.8\linewidth]{fig/pentagon} \\ \pub{Hedin, Phys Rev 139 (1965) A796} \end{column} \begin{column}{0.6\textwidth} \begin{block}{What can you calculate with $GW$?} \begin{itemize} \item Ionization potentials (IPs) given by occupied MO energies \item Electron affinities (EAs) given by virtual MO energies \item Fundamental (HOMO-LUMO) gap (or band gap in solids) \item Correlation and total energies \end{itemize} \end{block} \begin{block}{What can you calculate with BSE?} \begin{itemize} \item Singlet and triplet neutral excitations (vertical absorption energies) \item Oscillator strengths (absorption intensities) \item Correlation and total energies \end{itemize} \end{block} \end{column} \end{columns} \end{frame} %----------------------------------------------------- %----------------------------------------------------- \begin{frame}{Fundamental and optical gaps} \begin{center} \includegraphics[width=\textwidth]{fig/gaps} \end{center} \end{frame} %----------------------------------------------------- %----------------------------------------------------- \begin{frame}{The MBPT chain of actions} \begin{center} \includegraphics[width=0.7\textwidth]{fig/BSE-GW} \end{center} \end{frame} %----------------------------------------------------- %----------------------------------------------------- \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 \end{itemize} \end{block} \end{frame} %----------------------------------------------------- %----------------------------------------------------- \begin{frame}{Green's function and dynamical screening} \begin{block}{One-body Green's function} \begin{equation} \blue{G}(\br_1,\br_2;\yo) = \sum_i \frac{\MO{i}(\br_1) \MO{i}(\br_2)}{\yo - \e{i}{} - i\eta} + \sum_a \frac{\MO{a}(\br_1) \MO{a}(\br_2)}{\yo - \e{a}{} + i\eta} \end{equation} \end{block} \begin{block}{Non-interacting polarizability} \begin{equation} 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' \end{equation} \end{block} \begin{block}{Dielectric function} \begin{equation} \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 \end{equation} \end{block} \begin{block}{Dynamically-screened Coulomb potential} \begin{equation} \highlight{W}(\br_1,\br_2;\omega) = \int \frac{\epsilon^{-1}(\br_1,\br_3;\omega) }{\abs{\br_2 - \br_3}} d\br_3 \end{equation} \end{block} \end{frame} %----------------------------------------------------- %----------------------------------------------------- \begin{frame}{Dynamical screening in a basis} \begin{block}{Spectral representation of $W$} \begin{equation} \begin{split} \highlight{W}_{pq,rs}(\yo) & = \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 \\ & = \underbrace{\ERI{pq}{rs}}_{\text{(static) exchange part}} + \underbrace{2 \sum_m \violet{\ERI{pq}{m}} \violet{\ERI{rs}{m}} \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)} \end{split} \end{equation} \end{block} \begin{block}{Electron repulsion integrals (ERIs)} \begin{equation} \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 \end{equation} \end{block} \begin{block}{Screened ERIs (or spectral weights)} \begin{equation} \violet{\ERI{pq}{m}} = \sum_{ia} \ERI{pq}{ia} (\orange{\bX{m}{\RPA}+\bY{m}{\RPA}})_{ia} \end{equation} \end{block} \end{frame} %----------------------------------------------------- %----------------------------------------------------- \begin{frame}{Computation of the dynamical screening} \begin{block}{Direct RPA calculation (pseudo-hermitian linear problem)} \begin{equation} \begin{pmatrix} \bA{}{} & \bB{}{} \\ -\bB{}{} & -\bA{}{} \\ \end{pmatrix} \cdot \begin{pmatrix} \orange{\bX{m}{}} \\ \orange{\bY{m}{}} \\ \end{pmatrix} = \orange{\Om{m}{}} \begin{pmatrix} \orange{\bX{m}{}} \\ \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} \end{center} \end{block} \end{frame} %----------------------------------------------------- %----------------------------------------------------- \begin{frame}{ACFDT at the RPA/RPAx level} \begin{block}{RPA matrix elements} \begin{equation} \A{ia,jb}{\lambda,\RPA} = \delta_{ij} \delta_{ab} (\eHF{a} - \eHF{i}) + 2\lambda\ERI{ia}{bj} \qquad \B{ia,jb}{\lambda,\RPA} = 2\lambda\ERI{ia}{jb} \end{equation} \begin{equation} \boxed{ \Ec^\RPA = \frac{1}{2} \int_0^1 \Tr( \bK{}{} \bP{}{\lambda}) d\lambda = \frac{1}{2} \qty[ \sum_{m} \Om{m}{\RPA} - \Tr(\bA{}{\RPA}) ] } \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}