diff --git a/2022/GFQC/ISTPC_Loos_QFQC.tex b/2022/GFQC/ISTPC_Loos_QFQC.tex new file mode 100644 index 0000000..8201d2c --- /dev/null +++ b/2022/GFQC/ISTPC_Loos_QFQC.tex @@ -0,0 +1,1526 @@ +\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[compat=1.1.0]{tikz-feynman} + +\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=cyan, + citecolor=purple +} +\urlstyle{same} + +\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{\GT}{$GT$} +\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}} +\renewcommand{\HF}{\text{HF}} +\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{\Eg}[1]{E_\text{g}^{#1}} +\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}[2]{\epsilon_{#1}^{#2}} +\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}[1]{\epsilon_\text{HOMO}^{#1}} +\newcommand{\eLUMO}[1]{\epsilon_\text{LUMO}^{#1}} + +\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{\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{\bC}[2]{\bm{C}_{#1}^{#2}} +\newcommand{\bc}{\bm{c}} +\newcommand{\bX}[2]{\bm{X}_{#1}^{#2}} +\newcommand{\bY}[2]{\bm{Y}_{#1}^{#2}} +\newcommand{\bZ}[2]{\bm{Z}_{#1}^{#2}} +\newcommand{\bK}[2]{\blue{\bm{K}}_{#1}^{#2}} +\newcommand{\bP}[2]{\red{\bm{P}}_{#1}^{#2}} + +\newcommand{\yo}{\yellow{\omega}} +\newcommand{\la}{\yellow{\lambda}} + +\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[Green's function-based methods in chemistry]{ + Green's function-based methods in chemistry + } + \author[PF Loos (\url{https://pfloos.github.io/WEB_LOOS})]{Pierre-Fran\c{c}ois LOOS} + \date{ISTPC 2022 --- June 24th, 2022} + \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 \textbf{Charged excitations} + \begin{itemize} + \item One-shot $GW$ (\GOWO) + \item Partially self-consistent eigenvalue $GW$ (\evGW) + \item Quasiparticle self-consistent $GW$ (\qsGW) + \item Other self-energies (GF2, SOSEX, T-matrix, etc) + \end{itemize} + \bigskip + \item \textbf{Neutral excitations} + \begin{itemize} + \item Random-phase approximation (RPA) + \item Configuration interaction with singles (CIS) + \item Time-dependent Hartree-Fock (TDHF) or RPA with exchange (RPAx) + \item Time-dependent density-functional theory (TDDFT) + \item Bethe-Salpeter equation (BSE) formalism + \end{itemize} + \bigskip + \item \textbf{Correlation energy} + \begin{itemize} + \item Plasmon (or trace) formula + \item Galitski-Migdal formulation + \item Adiabatic connection fluctuation-dissipation theorem (ACFDT) + \end{itemize} + \end{itemize} +\end{frame} +%----------------------------------------------------- + +%----------------------------------------------------- +\section{Context} +\begin{frame} +\tableofcontents[currentsection] +\end{frame} +%----------------------------------------------------- +\begin{frame}{Assumptions \& Notations} + \begin{block}{Let's talk about notations} + \begin{itemize} + \item We consider \blue{closed-shell systems} (2 opposite-spin electrons per orbital) + \item We only deal with \blue{singlet excited states} but \purple{triplets} can also be obtained + \bigskip + \item Number of \green{occupied orbitals} $O$ + \item Number of \alert{vacant orbitals} $V$ + \item \violet{Total number of orbitals} $N = O + V$ + \bigskip + \item $\MO{p}(\br)$ is a (real) \blue{spatial orbital} + \item $i,j,k,l$ are \green{occupied orbitals} + \item $a,b,c,d$ are \alert{vacant orbitals} + \item $p,q,r,s$ are \violet{arbitrary (occupied or vacant) orbitals} + \item $\mu,\nu,\lambda,\sigma$ are \purple{basis function indexes} + \bigskip + \item $m$ indexes \purple{the $OV$ single excitations} ($i \to a$) + \end{itemize} + \end{block} +\end{frame} +%----------------------------------------------------- + +%----------------------------------------------------- +\begin{frame}{Useful papers/programs} + \begin{itemize} + \item \red{mol$GW$:} 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. PRB 86 (2012) 081102 + \bigskip + \item \orange{Reviews \& Books:} + \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 + \item Martin, Reining \& Ceperley \textit{Interacting Electrons} (Cambridge University Press) + \end{itemize} + \bigskip + \item \red{$GW$100:} IPs for a set of 100 molecules. van Setten et al. JCTC 11 (2015) 5665 (\url{http://gw100.wordpress.com}) + \end{itemize} +\end{frame} +%----------------------------------------------------- + +%----------------------------------------------------- +\begin{frame}{Fundamental and optical gaps (\copyright~Bruno Senjean)} + \begin{center} + \includegraphics[width=\textwidth]{fig/gaps} + \end{center} + \begin{equation} + \underbrace{\Eg{\KS}}_{\text{KS gap}} = \eLUMO{\KS} - \eHOMO{\KS} \ll \underbrace{\green{\Eg{GW}}}_{\text{\green{{\GW} gap}}} = \eLUMO{GW} - \eHOMO{GW} + \end{equation} + \begin{equation} + \underbrace{\blue{\Eg{\text{opt}}}}_{\text{\blue{optical gap}}} = E_1^N - E_0^N = \underbrace{\red{\Eg{\text{fund}}}}_{\text{\red{fundamental gap}}} + \underbrace{\purple{E_\text{B}}}_{\text{\purple{excitonic effect}}} + \end{equation} +\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 optical 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}{The MBPT chain of actions} + \begin{center} + \includegraphics[width=0.7\textwidth]{fig/BSE-GW} + \\ + \bigskip + \pub{Blase et al. JPCL 11 (2020) 7371} + \end{center} +\end{frame} +%----------------------------------------------------- + +%----------------------------------------------------- +\begin{frame}{Photochemistry: Jablonski diagram} +% colors +\definecolor{turquoise}{rgb}{0 0.41 0.41} +\definecolor{rouge}{rgb}{0.79 0.0 0.1} +\definecolor{vert}{rgb}{0.15 0.4 0.1} +\definecolor{mauve}{rgb}{0.6 0.4 0.8} +\definecolor{violet}{rgb}{0.58 0. 0.41} +\definecolor{orange}{rgb}{0.8 0.4 0.2} +\definecolor{bleu}{rgb}{0.39, 0.58, 0.93} + +\begin{center} + +\begin{tikzpicture}[scale=0.7] + + % styles + \tikzstyle{elec} = [line width=2pt,draw=black!80] + \tikzstyle{vib} = [thick,draw=black!30] + \tikzstyle{trans} = [line width=2pt,->] + \tikzstyle{transCI} = [trans,dashed,draw=vert] + \tikzstyle{transCS} = [trans,dashed,draw=violet] + \tikzstyle{relax} = [draw=orange,ultra thick,decorate,decoration=snake] + \tikzstyle{rv} = [rotate=90,text=orange,pos=0.5,yshift=3mm] + + % fondamental + \path[elec] (0,0) -- ++ (14,0) + node[below,pos=0.5,yshift=-1mm] {Ground state $S_0$}; + \path[vib] (0,0.2) -- ++ (14,0); + \path[vib] (0,0.4) -- ++ (13,0); + \foreach \i in {1,2,...,30} { + \path[vib] (0,0.4 + \i*0.2) -- ++ ({2 + 10*exp(-0.2*\i)},0); + } + + % T1 + \path[elec] (11,4) -- ++ (3,0) node[anchor=south west] {$T_1$}; + \foreach \i in {1,2,...,6} { + \path[vib] (11,4 + \i*0.2) -- ++ (3,0); + } + + % S1 + \path[elec] (4,5) node[anchor=south east] {$S_1$} -- ++ (5,0); + \foreach \i in {1,2,...,6} { + \path[vib] (4,5 + \i*0.2) -- ++ (5,0); + } + \foreach \i in {1,2,...,12} { + \path[vib] ({7.5 - 1*exp(-0.3*\i)},6.2+\i*0.2) -- (9,6.2+\i*0.2); + } + + % S2 + \path[elec] (4,8) node[anchor=south east] {$S_2$} -- ++ (2,0); + \foreach \i in {1,2,...,6} { + \path[vib] (4,8 + \i*0.2) -- ++ (2,0); + } + + % absorption + \path[trans,draw=turquoise] (4.5,0) -- ++(0,9) + node[rotate=90,pos=0.35,text=turquoise,yshift=-3mm] {\small Absorption}; + + % fluo + \path[trans,draw=rouge](7,5) -- ++(0,-4.4) + node[rotate=90,pos=0.5,text=rouge,yshift=-3mm] {\small Fluorescence}; + + % phosphorescence + \path[trans,draw=mauve] (13,4) -- ++(0,-3.4) + node[rotate=90,pos=0.5,text=mauve,yshift=-3mm] {\small Phosphorescence}; + + % Conversion interne + \path[transCI] (4,5) -- ++(-1.9,0) node[below,pos=0.5,text=vert] {\small IC}; + \path[transCI] (6,8) -- ++(1.3,0) node[above,pos=0.5,text=vert] {\small IC}; + + % Croisement intersysteme + \path[transCS] (9,5) -- ++(2,0) node[below,pos=0.5,text=violet] {\small ISC}; + \path[transCS] (11,4) -- ++(-2.5,0) node[below,pos=0.5,text=violet] {\small ISC}; + + % relaxation vib + \path[relax] (5.5,8.8) -- ++(0,-0.8) node[rv] {\small \textbf{VR}}; + \path[relax] (8,8) -- ++(0,-3) node[rv] {\small \textbf{VR}}; + \path[relax] (1,5) -- ++(0,-5) node[rv] {\small \textbf{VR}}; + \path[relax] (11.5,5) -- ++(0,-1) node[rv] {\small \textbf{VR}}; + +\end{tikzpicture} + +\end{center} + +%\tiny +%\begin{itemize} +% \item[] \tikz {\path[line width=2pt,->,dashed,draw=vert] +% (0,0) -- (1,0) node[above,pos=0.5,text=vert] {IC};} Internal Conversion, +% $S_i\,\longrightarrow\,S_j$ non radiative transition. +% +% \item[] \tikz {\path[line width=2pt,->,dashed,draw=violet] +% (0,0) -- (1,0) node[above,pos=0.5,text=violet] {ISC};} InterSystem Crossing, +% $S_i\,\longrightarrow\,T_j$ non radiative transition. +% +% \item[] \tikz {\path[line width=2pt,draw=orange,ultra thick, +% decorate,decoration=snake] (0,0) -- (1,0) node[above,pos=0.5,text=orange] {RV};} +% Vibrationnal Relaxation. +%\end{itemize} +\end{frame} +%----------------------------------------------------- + +%----------------------------------------------------- +\begin{frame}{Photochemistry: absorption, emission, and 0-0} + \begin{center} + \includegraphics[width=0.5\textwidth]{fig/0-0} + \\ + \textbf{\alert{Vertical excitation energies cannot be computed experimentally!!!}} + \end{center} +\end{frame} +%----------------------------------------------------- + +%----------------------------------------------------- +\section{Charged excitations} +\begin{frame} +\tableofcontents[currentsection] +\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) + = \underbrace{\sum_i \frac{\MO{i}(\br_1) \MO{i}(\br_2)}{\yo - \e{i}{} - i\eta}}_{\text{\green{removal part = IPs}}} + + \underbrace{\sum_a \frac{\MO{a}(\br_1) \MO{a}(\br_2)}{\yo - \e{a}{} + i\eta}}_{\text{\red{addition part = EAs}}} + \end{equation} + \end{block} + \begin{block}{Polarizability} + \begin{equation} + P(\br_1,\br_2;\yo) = - \frac{i}{\pi} \int \blue{G}(\br_1,\br_2;\yo+\omega') \blue{G}(\br_1,\br_2;\omega') d\omega' + \end{equation} + \end{block} + \begin{block}{Dielectric function and dynamically-screened Coulomb potential} + \begin{equation} + \epsilon(\br_1,\br_2;\yo) = \delta(\br_1 - \br_2) - \int \frac{P(\br_1,\br_3;\yo) }{\abs{\br_2 - \br_3}} d\br_3 + \end{equation} + \begin{equation} + \highlight{W}(\br_1,\br_2;\yo) = \int \frac{\epsilon^{-1}(\br_1,\br_3;\yo) }{\abs{\br_2 - \br_3}} d\br_3 + \end{equation} + \end{block} +\end{frame} +%----------------------------------------------------- + +%----------------------------------------------------- +\begin{frame}{Dynamical screening in the orbital 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}(\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 (ph-)RPA calculation (pseudo-hermitian linear problem)} + \begin{equation} + \begin{pmatrix} + \bA{}{\RPA} & \bB{}{\RPA} \\ + -\bB{}{\RPA} & -\bA{}{\RPA} \\ + \end{pmatrix} + \cdot + \begin{pmatrix} + \orange{\bX{m}{\RPA}} \\ + \orange{\bY{m}{\RPA}} \\ + \end{pmatrix} + = + \orange{\Om{m}{\RPA}} + \begin{pmatrix} + \orange{\bX{m}{\RPA}} \\ + \orange{\bY{m}{\RPA}} \\ + \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{m}{} + \bY{m}{}) = \Om{m}{-1/2} (\bA{}{} - \bB{}{})^{+1/2} \cdot \bZ{m}{} + \\ + (\bX{m}{} - \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 \orange{\bX{m}{}} = \orange{\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} + \red{\Sig{pp}{\xc}}(\yo) \approx \red{\Sig{pp}{\xc}}(\eKS{p}) + (\yo - \eKS{p}) \left. \pdv{\red{\Sig{pp}{\xc}}(\yo)}{\yo} \right|_{\yo = \eKS{p}} + \qq{$\Rightarrow$} + \blue{\eGW{p}} = \eKS{p} + \green{Z_{p}} [\red{\Sig{pp}{\xc}}(\eKS{p}) - V_{p}^{\xc} ] + \end{equation} + \begin{equation} + \underbrace{\green{Z_{p}}}_{\text{renormalization factor}} = \qty[ 1 - \left. \pdv{\red{\Sig{pp}{\xc}}(\yo)}{\yo} \right|_{\yo = \eKS{p}} ]^{-1} + \qq{with} 0 \le \green{Z_{p}} \le 1 + \end{equation} + \end{block} +\end{frame} +%----------------------------------------------------- + +%----------------------------------------------------- +\begin{frame}{Solutions of the non-linear QP equation: {\evGW}@HF/6-31G for \ce{H2} at $R = 1$ bohr} + \begin{columns} + \begin{column}{0.5\textwidth} + \begin{center} + \includegraphics[width=0.7\textwidth]{fig/QP} + \\ + \bigskip + \pub{V\'eril et al, JCTC 14 (2018) 5220} + \end{center} + \end{column} + \begin{column}{0.5\textwidth} + \begin{center} + \includegraphics[width=\textwidth]{fig/GWSph} + \\ + \bigskip + \pub{Loos et al, JCTC 14 (2018) 3071} + \end{center} + \end{column} + \end{columns} +\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}{Perturbative {\GW} with linearized solution} + \begin{block}{} + \begin{algorithmic} + \Procedure{{\GOWO}lin@KS}{} + \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 $\orange{\bB{}{\RPA}}$ from $\beKS$ and $\ERI{pq}{rs}$ + \State Compute RPA eigenvalues $\orange{\bOm{}{\RPA}}$ and eigenvectors $\orange{\bX{}{\RPA}+\bY{}{\RPA}}$ + \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} + \bigskip + For contour deformation technique, see, for example, \pub{Duchemin \& Blase, JCTC 16 (2020) 1742} +\end{frame} +%----------------------------------------------------- + +%----------------------------------------------------- +\begin{frame}{Example from \texttt{QuAcK} (\ce{Ne}/cc-pVDZ)} + \begin{center} + \includegraphics[width=0.55\textwidth]{fig/G0W0} + \\ + \bigskip + \pub{https://github.com/pfloos/QuAcK} + \end{center} +\end{frame} +%----------------------------------------------------- + + +%----------------------------------------------------- +\begin{frame}{Perturbative {\GW} with graphical solution} + \begin{block}{} + \begin{algorithmic} + \Procedure{{\GOWO}graph@KS}{} + \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 $\orange{\bB{}{\RPA}}$ from $\beKS$ and $\ERI{pq}{rs}$ + \State Compute RPA eigenvalues $\orange{\Om{}{\RPA}}$ and eigenvectors $\orange{\bX{}{\RPA}+\bY{}{\RPA}}$ + \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}{Newton's method} + \centering + \url{https://en.wikipedia.org/wiki/Newton\%27s_method} +\end{frame} +%----------------------------------------------------- + +%----------------------------------------------------- +\begin{frame}{Partially self-consistent eigenvalue \GW} + \begin{block}{} + \begin{algorithmic} + \Procedure{{\evGW}@KS}{} + \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 Set $\blue{\beGnWn{-1}} = \beKS$ and $n = 0$ + \While{$\max{\abs{\bDelta}} > \tau$} + \State Construct RPA matrices $\orange{\bA{}{\RPA}}$ and $\orange{\bB{}{\RPA}}$ from $\blue{\beGnWn{n-1}}$ and $\ERI{pq}{rs}$ + \State Compute RPA eigenvalues $\orange{\Om{}{\RPA}}$ and eigenvectors $\orange{\bX{}{\RPA}+\bY{}{\RPA}}$ + \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 $\blue{\eGnWn{p}{n}}$ + \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}{Example from \texttt{QuAcK} (\ce{Ne}/cc-pVDZ)} + \begin{center} + \includegraphics[width=0.5\textwidth]{fig/evGW} + \\ + \bigskip + \pub{https://github.com/pfloos/QuAcK} + \end{center} +\end{frame} +%----------------------------------------------------- + + +%----------------------------------------------------- +\begin{frame}{Quasiparticle self-consistent {\GW} (\qsGW)} + \begin{block}{} + \begin{algorithmic} + \Procedure{{\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: $\ERI{\mu\nu}{\lambda\sigma} \stackrel{\blue{\bcGnWn{n-1}}}{\rightarrow} \ERI{pq}{rs}$ + \Comment{\alert{This is a $\order*{N^5}$ step!}} + \State Construct RPA matrices $\orange{\bA{}{\RPA}}$ and $\orange{\bB{}{\RPA}}$ from $\blue{\beGnWn{n-1}}$ and $\ERI{pq}{rs}$ + \State Compute RPA eigenvalues $\orange{\Om{}{\RPA}}$ and eigenvectors $\orange{\bX{}{\RPA}+\bY{}{\RPA}}$ + \Comment{\alert{This is a $\order*{N^6}$ step!}} + \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 $\bFHF$ from $\blue{\bcGnWn{n-1}}$ and then $\purple{\Tilde{\bF}} = \bFHF + \red{\Tilde{\Sigma}^{\co}}$ + \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}{Example from \texttt{QuAcK} (\ce{Ne}/cc-pVDZ)} + \begin{center} + \includegraphics[width=0.45\textwidth]{fig/qsGW1} + \hspace{0.1\textwidth} + \includegraphics[width=0.4\textwidth]{fig/qsGW2} + \\ + \bigskip + \pub{https://github.com/pfloos/QuAcK} + \end{center} +\end{frame} +%----------------------------------------------------- + +%----------------------------------------------------- +\begin{frame}{Other self-energies} + \begin{columns} + \begin{column}{0.7\textwidth} + \begin{block}{Second-order Green's function (GF2) \pub{[Hirata et al. JCP 147 (2017) 044108]}} + \begin{equation} + \Sig{pq}{\text{GF2}}(\yo) + = \frac{1}{2} \sum_{iab} \frac{\mel{iq}{}{ab}\mel{ab}{}{ip}}{\yo + \e{i}{} - \e{a}{} - \e{b}{}} + + \frac{1}{2} \sum_{ija} \frac{\mel{aq}{}{ij}\mel{ij}{}{ap}}{\yo + \e{a}{} - \e{i}{} - \e{j}{}} + \end{equation} + \end{block} + \begin{block}{T-matrix \pub{[Romaniello et al. PRB 85 (2012) 155131; Zhang et al. JPCL 8 (2017) 3223]}} + \begin{equation} + \Sig{pq}{GT}(\omega) + = \sum_{im} \frac{\braket*{pi}{\green{\chi_m^{N+2}}} \braket*{qi}{\green{\chi_m^{N+2}}}}{\yo + \e{i}{} - \green{\Om{m}{N+2}}} + + \sum_{am} \frac{\braket*{pa}{\blue{\chi_m^{N-2}}} \braket*{qa}{\blue{\chi_m^{N-2}}}}{\yo + \e{i}{} - \blue{\Om{m}{N-2}}} + \end{equation} + \begin{gather} + \braket*{pi}{\green{\chi_m^{N+2}}} = \sum_{c=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}{Linear response} + \begin{block}{General linear response problem} + \begin{algorithmic} + \Procedure{Linear response}{} + \State Compute $\red{\bA{}{}}$ matrix at a given level of theory (RPA, RPAx, TD-DFT, BSE, etc) + \If{$\TDA$} + \State Diagonalize $\red{\bA{}{}}$ to get $\highlight{\Om{m}{\TDA}}$ and $\bX{m}{\TDA}$ + \Else + \State Compute \orange{$\bB{}{}$} matrix at a given level of theory + \State Diagonalize $\red{\bA{}{}} - \orange{\bB{}{}}$ to form $(\red{\bA{}{}} - \orange{\bB{}{}})^{1/2}$ + \State Form and diagonalize $(\red{\bA{}{}} - \orange{\bB{}{}})^{1/2} \cdot (\red{\bA{}{}} + \orange{\bB{}{}}) \cdot (\red{\bA{}{}} - \orange{\bB{}{}})^{1/2}$ + to get $\highlight{\Om{m}{2}}$ and $\bZ{m}{}$ + \State Compute $\sqrt{\highlight{\Om{m}{2}}}$ and $(\bX{m}{} + \bY{m}{}) = \highlight{\Om{m}{-1/2}} (\red{\bA{}{}} - \orange{\bB{}{}})^{1/2} \cdot \bZ{m}{}$ + \EndIf + \EndProcedure + \end{algorithmic} + \end{block} +\end{frame} +%----------------------------------------------------- + +%----------------------------------------------------- +\begin{frame}{Form linear response matrices} + \begin{block}{Linear-response matrices for BSE} + \begin{algorithmic} + \Procedure{Form $\red{\bA{}{}}$ for singlet states}{} + \State Set $\red{\bA{}{}} = \bO$ + \State $ia \gets 0$ + \For{$i=1, \ldots, O$} + \For{$a=1, \ldots, V$} + \State $ia \gets ia + 1$ + \State $jb \gets 0$ + \For{$j=1, \ldots, O$} + \For{$b=1, \ldots, V$} + \State $jb \gets jb + 1$ + \State $\red{A_{ia,jb}} = \delta_{ij} \delta_{ab} (\e{a}{\green{GW}} - \e{i}{\green{GW}}) + + 2\blue{(ia|bj)} - \yellow{(ij|ba)} + \purple{W^{\co}_{ij,ba}}(\omega = 0)$ + \EndFor + \EndFor + \EndFor + \EndFor + \EndProcedure + \end{algorithmic} + \end{block} +\end{frame} +%----------------------------------------------------- + + +%----------------------------------------------------- +\begin{frame}{Properties} + \begin{block}{Oscillator strength (length gauge)} + \begin{equation} + \boxed{\green{f_m} = \frac{2}{3} \orange{\Om{m}{}} \qty[ (\blue{\mu_m^x})^2 + (\blue{\mu_m^y})^2 + (\blue{\mu_m^z})^2 ]} + \end{equation} + \end{block} + \begin{block}{Transition dipole} + \begin{equation} + \boxed{\blue{\mu_m^x} = \sum_{ia} \red{(i|x|a)} \orange{(\bX{m}{} + \bY{m}{})_{ia}}} + \qquad + \red{(p|x|q)} = \int \MO{p}(\br) \,x\, \MO{q}(\br) d\br + \end{equation} + \end{block} + \begin{block}{Monitoring possible spin contamination \pub{[Monino \& Loos, JCTC 17 (2021) 2852]}} + \begin{equation} + \boxed{\purple{\expval{\hat{S}^2}_m} = \violet{\expval{\hat{S}^2}_0} + \underbrace{\Delta \expval{\hat{S}^2}_m}_{\text{\pub{JCP 134101 (2011) 134}}}} + \qquad + \violet{\expval{\hat{S}^2}_0} = \frac{n_\alpha - n_\beta}{2} \qty( \frac{n_\alpha - n_\beta}{2} + 1 ) + n_\beta + \sum_p (p_\alpha|p_\beta) + \end{equation} + \end{block} +\end{frame} +%----------------------------------------------------- + + +%----------------------------------------------------- +\begin{frame}{Example from \texttt{QuAcK} (\ce{H2O}/cc-pVDZ)} + \begin{center} + \includegraphics[height=0.45\textwidth]{fig/BSE1} + \hspace{0.05\textwidth} + \includegraphics[height=0.45\textwidth]{fig/BSE3} + \\ + \bigskip + \pub{https://github.com/pfloos/QuAcK} + \end{center} +\end{frame} +%----------------------------------------------------- + +%----------------------------------------------------- +\begin{frame}{Open-shell systems and double excitations} + \begin{block}{Spin-flip formalism (H2/cc-pVQZ)} + \begin{center} + \includegraphics[width=0.28\textwidth]{fig/SFBSE} + \includegraphics[width=0.4\textwidth]{fig/H2} + \includegraphics[width=0.3\textwidth]{fig/H2_QuAcK} + \\ + \bigskip + \pub{Monino \& Loos, JCTC 17 (2021) 2852} + \end{center} + \end{block} +\end{frame} +%----------------------------------------------------- + +%----------------------------------------------------- +\section{Correlation energy} +\begin{frame} +\tableofcontents[currentsection] +\end{frame} +%----------------------------------------------------- +\begin{frame}{Correlation energy at the $GW$ or BSE level} + \begin{block}{RPA@$GW$ correlation energy: plasmon (or trace) formula} + \begin{equation*} + \label{eq:Ec-RPA} + \green{\EcRPA} + = \frac{1}{2} \qty[ \sum_{p} \orange{\Om{m}{\RPA}} - \Tr(\orange{\bA{}{\RPA}}) ] + = \frac{1}{2} \sum_{m} \qty( \orange{\Om{m}{\RPA}} - \orange{\Om{m}{\TDA}} ) + \end{equation*} + \end{block} + \begin{block}{Galitskii-Migdal functional} + \begin{equation*} + \label{eq:GM} + \green{\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}{\blue{\eGW{a}} - \blue{\eGW{i}} + \orange{\Om{m}{\RPA}}} + \end{equation*} + \end{block} + \begin{block}{ACFDT@BSE@$GW$ correlation energy from the adiabatic connection} + \begin{equation} + \green{\Ec^\text{ACFDT}} = \frac{1}{2} \int_{\red{0}}^{\red{1}} \Tr( \bK{}{} \bP{}{\la}) d\la + \end{equation} + \end{block} + +\end{frame} +%----------------------------------------------------- + +%----------------------------------------------------- +\begin{frame}{Adiabatic connection fluctuation dissipation theorem (ACFDT)} + \begin{block}{Adiabatic connection} + \begin{equation} + \boxed{ + \green{\Ec^\text{ACFDT}} + = \frac{1}{2} \int_{\red{0}}^{\red{1}} \Tr( \bK{}{} \bP{}{\la}) d\la + \stackrel{\blue{\text{quad}}}{\approx} \frac{1}{2} \sum_{k=1}^{K} \purple{w_k} \Tr( \bK{}{} \bP{}{\violet{\lambda_k}}) + } + \end{equation} + $\la$ is the \textbf{strength} of the electron-electron interaction: + \begin{itemize} + \item $\la = 0$ for the \green{non-interacting system} + \item $\la = 1$ for the \alert{physical system} + \end{itemize} + \end{block} + \begin{block}{Interaction kernel} + \begin{equation} + \bK{}{} = + \begin{pmatrix} + \btA{}{} & \btB{}{} + \\ + \btB{}{} & \btA{}{} + \end{pmatrix} + \qquad + \tA{ia,jb}{} = 2\ERI{ia}{bj} + \qquad + \tB{ia,jb}{} = 2\ERI{ia}{jb} + \end{equation} + \end{block} + \begin{block}{Correlation part of the two-particle density matrix} + \begin{equation} + \bP{}{\la} = + \begin{pmatrix} + \bY{}{\la} \cdot \T{(\bY{}{\la})} & \bY{}{\la} \cdot \T{(\bX{}{\la})} + \\ + \bX{}{\la} \cdot \T{(\bY{}{\la})} & \bX{}{\la} \cdot \T{(\bX{}{\la})} + \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} + \textit{``A $K$-point \orange{Gaussian quadrature} rule is a quadrature rule constructed to yield an exact result for polynomials up to degree $2K-1$ by a suitable choice of the \violet{roots $x_k$} and \purple{weights $w_k$} for $k = 1, \ldots, K$.''} + \begin{equation} + \boxed{\int_{\red{a}}^{\red{b}} f(x) \purple{w(x)} dx \approx \sum_k^{K} \underbrace{\purple{w_k}}_{\text{\purple{weights}}} f(\underbrace{\violet{x_k}}_{\text{\violet{roots}}})} + \end{equation} + \end{block} + \begin{block}{Quadrature rules} + \begin{center} + \small + \begin{tabular}{llll} + \hline + \red{Interval $[a,b]$} & \purple{Weight function $w(x)$} & \violet{Orthogonal polynomials} & \orange{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} + \\ + \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} + \orange{\A{ia,jb}{\la,\RPA}} = \delta_{ij} \delta_{ab} (\violet{\eHF{a}} - \violet{\eHF{i}}) + 2\la\ERI{ia}{bj} + \qquad + \orange{\B{ia,jb}{\la,\RPA}} = 2\la\ERI{ia}{jb} + \end{equation} + \begin{equation} + \boxed{ + \green{\Ec^\RPA} + = \frac{1}{2} \int_{\red{0}}^{\red{1}} \Tr( \bK{}{} \bP{}{\la}) d\la + = \frac{1}{2} \qty[ \sum_{m} \orange{\Om{m}{\RPA}} - \Tr(\orange{\bA{}{\RPA}}) ] + } + \end{equation} + \end{block} + + \begin{block}{RPAx matrix elements} + \begin{equation} + \orange{\A{ia,jb}{\la,\RPAx}} = \delta_{ij} \delta_{ab} (\violet{\eHF{a}} - \violet{\eHF{i}}) + \la \qty[2 \ERI{ia}{bj} - \ERI{ij}{ab} ] + \qquad + \orange{\B{ia,jb}{\la,\RPAx}} = \la \qty[2 \ERI{ia}{jb} - \ERI{ib}{aj} ] + \end{equation} + \begin{equation} + \boxed{ + \green{\Ec^\RPAx} + = \frac{1}{2} \int_{\red{0}}^{\red{1}} \Tr( \bK{}{} \bP{}{\la}) d\la + \alert{\neq} \frac{1}{2} \qty[ \sum_{m} \orange{\Om{m}{\RPAx}} - \Tr(\orange{\bA{}{\RPAx}}) ] + } + \end{equation} + If exchange added to kernel, i.e., $\bK{}{} = \bK{}{\x}$, then \pub{[Angyan et al. JCTC 7 (2011) 3116]} + \begin{equation} + \green{\Ec^\RPAx} + = \frac{1}{4} \int_{\red{0}}^{\red{1}} \Tr( \bK{}{\x} \bP{}{\la}) d\la + \alert{=} \frac{1}{4} \qty[ \sum_{m} \orange{\Om{m}{\RPAx}} - \Tr(\orange{\bA{}{\RPAx}}) ] + \end{equation} + \end{block} +\end{frame} +%----------------------------------------------------- + + +%----------------------------------------------------- +\begin{frame}{ACFDT at the BSE level} + \begin{block}{BSE matrix elements} + \begin{equation} + \orange{\A{ia,jb}{\la,\BSE}} = \delta_{ij} \delta_{ab} (\violet{\eGW{a}} - \violet{\eGW{i}}) + \la \qty[2 \ERI{ia}{bj} - \highlight{W}_{ij,ab}^{\la}(\omega = 0) ] + \qquad + \orange{\B{ia,jb}{\la,\BSE}} = \la \qty[2 \ERI{ia}{jb} - \highlight{W}_{ib,ja}^{\la}(\omega = 0)] + \end{equation} + \begin{equation} + \boxed{ + \green{\Ec^\BSE} + = \frac{1}{2} \int_{\red{0}}^{\red{1}} \Tr( \bK{}{} \bP{}{\la}) d\la + \alert{\neq} \frac{1}{2} \qty[ \sum_{m} \orange{\Om{m}{\BSE}} - \Tr(\orange{\bA{}{\BSE}}) ] + } + \end{equation} + + \end{block} + \begin{block}{$\la$-dependent screening} + \begin{equation} + \highlight{W}_{pq,rs}^{\la}(\omega) + = \ERI{pq}{rs} + + 2 \sum_m \violet{\ERI{pq}{m}^{\la}} \violet{\ERI{rs}{m}^{\la}} + \qty[ \frac{1}{\omega - \orange{\Om{m}{\la,\RPA}} + i \eta} - \frac{1}{\omega + \orange{\Om{m}{\la,\RPA}} - i \eta} ] + \end{equation} + \begin{equation} + \violet{\ERI{pq}{m}^{\la}} = \sum_{ia} \ERI{pq}{ia} (\orange{\bX{m}{\la,\RPA}+\bY{m}{\la,\RPA}})_{ia} + \end{equation} + \end{block} +\end{frame} +%----------------------------------------------------- + +%----------------------------------------------------- +\begin{frame}{ACFDT in a computer} + \begin{block}{ACFDT correlation energy from BSE} + \begin{algorithmic} + \Procedure{ACFDT for BSE}{} + \State Compute $GW$ quasiparticle energies $\blue{\beGW}$ and interaction kernel $\bK{}{}$ + \State Get Gauss-Legendre weights and roots $\{\purple{w_k},\violet{\lambda_k}\}_{1\le k \le K}$ + \State $\green{\Ec} \gets 0$ + \For{$k=1,\ldots,K$} + \State Compute static screening elements $\highlight{W}_{pq,rs}^{\violet{\lambda_k}}(\omega = 0)$ + \State Perform BSE calculation at $\la = \violet{\lambda_k}$ to get $\bX{}{\violet{\lambda_k}}$ and $\bY{}{\violet{\lambda_k}}$ + \Comment{\alert{This is a $\order*{N^6}$ step done many times!}} + \State Form two-particle density matrix $\bP{}{\violet{\lambda_k}}$ + \State $\green{\Ec} \gets \green{\Ec} + \purple{w_k} \Tr( \bK{}{} \bP{}{\violet{\lambda_k}})$ + \EndFor + \EndProcedure + \end{algorithmic} + \end{block} +\end{frame} +%----------------------------------------------------- + +%----------------------------------------------------- +\begin{frame} + \begin{center} + \includegraphics[width=0.7\textwidth]{fig/TOC_BSE} + \\ + \pub{Loos et al. JPCL 11 (2020) 3536} + \end{center} +\end{frame} +%----------------------------------------------------- + +%----------------------------------------------------- +\begin{frame}{Useful papers/programs} + \begin{itemize} + \item \red{mol$GW$:} 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. PRB 86 (2012) 081102 + \bigskip + \item \orange{Reviews \& Books:} + \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 + \item Martin, Reining \& Ceperley \textit{Interacting Electrons} (Cambridge University Press) + \end{itemize} + \bigskip + \item \red{$GW$100:} IPs for a set of 100 molecules. van Setten et al. JCTC 11 (2015) 5665 (\url{http://gw100.wordpress.com}) + + \end{itemize} +\end{frame} +%----------------------------------------------------- + +\end{document} diff --git a/2022/GFQC/fig/0-0.pdf b/2022/GFQC/fig/0-0.pdf new file mode 100755 index 0000000..6d83727 Binary files /dev/null and b/2022/GFQC/fig/0-0.pdf differ diff --git a/2022/GFQC/fig/BSE-GW.pdf b/2022/GFQC/fig/BSE-GW.pdf new file mode 100644 index 0000000..4124cb4 Binary files /dev/null and b/2022/GFQC/fig/BSE-GW.pdf differ diff --git a/2022/GFQC/fig/BSE-GW.tex b/2022/GFQC/fig/BSE-GW.tex new file mode 100644 index 0000000..e196337 --- /dev/null +++ b/2022/GFQC/fig/BSE-GW.tex @@ -0,0 +1,74 @@ +\documentclass{standalone} +\usepackage{amsmath,amssymb,amsfonts,pgfpages,graphicx,subfigure,xcolor,bm,multirow,microtype,wasysym,multimedia,hyperref,tabularx,amscd,pgfgantt,mhchem,physics} +\usetikzlibrary{shapes.gates.logic.US,trees,positioning,arrows} + +\usepackage{tgchorus} +\usepackage[T1]{fontenc} + +\begin{document} + +\begin{tikzpicture} + \begin{scope}[very thick, + node distance=5cm,on grid,>=stealth', + theo1/.style={rectangle,draw,fill=red!20}, + theo2/.style={rectangle,draw,fill=orange!20}, + theo3/.style={rectangle,draw,fill=green!40}, + exp1/.style={rectangle,draw,fill=cyan!40}, + exp2/.style={rectangle,draw,fill=violet!40}] + + \node [theo1, text width=7cm, align=center] (KS) + {\textbf{\LARGE Kohn-Sham DFT} + $$ + \qty[ -\frac{\nabla^2}{2} + v_\text{ext} + V^{\text{Hxc}} ] \phi_p^{\text{KS}} = \varepsilon^{\text{KS}}_p \phi_p^{\text{KS}} + $$ + }; + + \node [theo2, text width=7cm, align=center] (GW) [below=of KS, yshift=2cm] + {\textbf{\LARGE $GW$ approximation} + $$ + \varepsilon_p^{GW} = \varepsilon_p^{\text{KS}} + + \mel{\phi_p^{\text{KS}}}{\Sigma^{GW}(\varepsilon_p^{GW}) - V^{\text{xc}}}{\phi_p^{\text{KS}}} + $$ + + }; + + \node [theo3, text width=7cm, align=center] (BSE) [below=of GW, yshift=2cm] + {\textbf{\LARGE Bethe-Salpeter equation} + $$ + \begin{pmatrix} + \bm{A} & \bm{B} \\ + -\bm{B}^* & -\bm{A}^{*} + \end{pmatrix} + \begin{pmatrix} + \bm{X}_m \\ + \bm{Y}_m + \end{pmatrix} + = + \Omega_{m} + \begin{pmatrix} + \bm{X}_m \\ + \bm{Y}_m + \end{pmatrix} + $$ + }; + + + \node [exp1, align=center] (photo) [right=of GW, xshift=3cm] + {\LARGE (Inverse) \\ \LARGE photoemission \\ \LARGE spectroscopy}; + + \node [exp2, align=center] (abs) [right=of BSE, xshift=3cm] + {\LARGE Optical \\ \LARGE spectroscopy}; + + + \path + (KS) edge [->,color=black] node [right,black] {\LARGE Fundamental gap} (GW) + (GW) edge [->,color=black] node [right,black] {\LARGE Excitonic effect} (BSE) + (photo) edge [<->,color=black] node [above,black] {Ionization potentials} node [below,black] {Electron affinities} (GW) + (abs) edge [<->,color=black] node [above,black] {Optical excitations} (BSE) + ; + + + \end{scope} +\end{tikzpicture} + +\end{document} diff --git a/2022/GFQC/fig/BSE1.png b/2022/GFQC/fig/BSE1.png new file mode 100644 index 0000000..eb97b1a Binary files /dev/null and b/2022/GFQC/fig/BSE1.png differ diff --git a/2022/GFQC/fig/BSE3.png b/2022/GFQC/fig/BSE3.png new file mode 100644 index 0000000..1e03dd0 Binary files /dev/null and b/2022/GFQC/fig/BSE3.png differ diff --git a/2022/GFQC/fig/CNRS.png b/2022/GFQC/fig/CNRS.png new file mode 100755 index 0000000..4667e1e Binary files /dev/null and b/2022/GFQC/fig/CNRS.png differ diff --git a/2022/GFQC/fig/ERC.jpg b/2022/GFQC/fig/ERC.jpg new file mode 100644 index 0000000..16c972e Binary files /dev/null and b/2022/GFQC/fig/ERC.jpg differ diff --git a/2022/GFQC/fig/G0W0.png b/2022/GFQC/fig/G0W0.png new file mode 100644 index 0000000..5833f25 Binary files /dev/null and b/2022/GFQC/fig/G0W0.png differ diff --git a/2022/GFQC/fig/GWSph.pdf b/2022/GFQC/fig/GWSph.pdf new file mode 100644 index 0000000..ae5e123 Binary files /dev/null and b/2022/GFQC/fig/GWSph.pdf differ diff --git a/2022/GFQC/fig/H2.pdf b/2022/GFQC/fig/H2.pdf new file mode 100644 index 0000000..04449c0 Binary files /dev/null and b/2022/GFQC/fig/H2.pdf differ diff --git a/2022/GFQC/fig/H2_QuAcK.png b/2022/GFQC/fig/H2_QuAcK.png new file mode 100644 index 0000000..3cfd2e0 Binary files /dev/null and b/2022/GFQC/fig/H2_QuAcK.png differ diff --git a/2022/GFQC/fig/Jablonski.pdf b/2022/GFQC/fig/Jablonski.pdf new file mode 100644 index 0000000..3c773e2 Binary files /dev/null and b/2022/GFQC/fig/Jablonski.pdf differ diff --git a/2022/GFQC/fig/Jablonski.tex b/2022/GFQC/fig/Jablonski.tex new file mode 100644 index 0000000..9b5805a --- /dev/null +++ b/2022/GFQC/fig/Jablonski.tex @@ -0,0 +1,112 @@ +\documentclass{standalone} +%\usepackage[top=3cm,left=0cm,right=0cm,bottom=3cm]{geometry} +\usepackage{mathtools,physics,bm,xcolor} + +\usepackage{tikz} +% shadows only for title +\usetikzlibrary{decorations.pathmorphing,shadows} + + +\pagestyle{empty} + +\begin{document} + +% colors +\definecolor{turquoise}{rgb}{0 0.41 0.41} +\definecolor{rouge}{rgb}{0.79 0.0 0.1} +\definecolor{vert}{rgb}{0.15 0.4 0.1} +\definecolor{mauve}{rgb}{0.6 0.4 0.8} +\definecolor{violet}{rgb}{0.58 0. 0.41} +\definecolor{orange}{rgb}{0.8 0.4 0.2} +\definecolor{bleu}{rgb}{0.39, 0.58, 0.93} + + +\begin{center} + +\begin{tikzpicture} + + % styles + \tikzstyle{elec} = [line width=2pt,draw=black!80] + \tikzstyle{vib} = [thick,draw=black!30] + \tikzstyle{trans} = [line width=2pt,->] + \tikzstyle{transCI} = [trans,dashed,draw=vert] + \tikzstyle{transCS} = [trans,dashed,draw=violet] + \tikzstyle{relax} = [draw=orange,ultra thick,decorate,decoration=snake] + \tikzstyle{rv} = [rotate=90,text=orange,pos=0.5,yshift=3mm] + + % fondamental + \path[elec] (0,0) -- ++ (14,0) + node[below,pos=0.5,yshift=-1mm] {\large Ground state $S_0$}; + \path[vib] (0,0.2) -- ++ (14,0); + \path[vib] (0,0.4) -- ++ (13,0); + \foreach \i in {1,2,...,30} { + \path[vib] (0,0.4 + \i*0.2) -- ++ ({2 + 10*exp(-0.2*\i)},0); + } + + % T1 + \path[elec] (11,4) -- ++ (3,0) node[anchor=south west] {\large $T_1$}; + \foreach \i in {1,2,...,6} { + \path[vib] (11,4 + \i*0.2) -- ++ (3,0); + } + + % S1 + \path[elec] (4,5) node[anchor=south east] {\large $S_1$} -- ++ (5,0); + \foreach \i in {1,2,...,6} { + \path[vib] (4,5 + \i*0.2) -- ++ (5,0); + } + \foreach \i in {1,2,...,12} { + \path[vib] ({7.5 - 1*exp(-0.3*\i)},6.2+\i*0.2) -- (9,6.2+\i*0.2); + } + + % S2 + \path[elec] (4,8) node[anchor=south east] {\large $S_2$} -- ++ (2,0); + \foreach \i in {1,2,...,6} { + \path[vib] (4,8 + \i*0.2) -- ++ (2,0); + } + + % absorption + \path[trans,draw=turquoise] (4.5,0) -- ++(0,9) + node[rotate=90,pos=0.35,text=turquoise,yshift=-3mm] {\large Absorption}; + + % fluo + \path[trans,draw=rouge](7,5) -- ++(0,-4.4) + node[rotate=90,pos=0.5,text=rouge,yshift=-3mm] {\large Fluorescence}; + + % phosphorescence + \path[trans,draw=mauve] (13,4) -- ++(0,-3.4) + node[rotate=90,pos=0.5,text=mauve,yshift=-3mm] {\large Phosphorescence}; + + % Conversion interne + \path[transCI] (4,5) -- ++(-1.9,0) node[below,pos=0.5,text=vert] {\large IC}; + \path[transCI] (6,8) -- ++(1.3,0) node[above,pos=0.5,text=vert] {\large IC}; + + % Croisement intersysteme + \path[transCS] (9,5) -- ++(2,0) node[below,pos=0.5,text=violet] {\large ISC}; + \path[transCS] (11,4) -- ++(-2.5,0) node[below,pos=0.5,text=violet] {\large ISC}; + + % relaxation vib + \path[relax] (5.5,8.8) -- ++(0,-0.8) node[rv] {\textbf{VR}}; + \path[relax] (8,8) -- ++(0,-3) node[rv] {\textbf{VR}}; + \path[relax] (1,5) -- ++(0,-5) node[rv] {\textbf{VR}}; + \path[relax] (11.5,5) -- ++(0,-1) node[rv] {\textbf{VR}}; + +\end{tikzpicture} + +\end{center} + + +\begin{itemize} + \item[] \tikz {\path[line width=2pt,->,dashed,draw=vert] + (0,0) -- (1,0) node[above,pos=0.5,text=vert] {IC};} Internal Conversion, + $S_i\,\longrightarrow\,S_j$ non radiative transition. + + \item[] \tikz {\path[line width=2pt,->,dashed,draw=violet] + (0,0) -- (1,0) node[above,pos=0.5,text=violet] {ISC};} InterSystem Crossing, + $S_i\,\longrightarrow\,T_j$ non radiative transition. + + \item[] \tikz {\path[line width=2pt,draw=orange,ultra thick, + decorate,decoration=snake] (0,0) -- (1,0) node[above,pos=0.5,text=orange] {RV};} + Vibrationnal Relaxation. +\end{itemize} + +\end{document} \ No newline at end of file diff --git a/2022/GFQC/fig/LCPQ.pdf b/2022/GFQC/fig/LCPQ.pdf new file mode 100644 index 0000000..a21e4bd Binary files /dev/null and b/2022/GFQC/fig/LCPQ.pdf differ diff --git a/2022/GFQC/fig/QP.pdf b/2022/GFQC/fig/QP.pdf new file mode 100644 index 0000000..57c9da7 Binary files /dev/null and b/2022/GFQC/fig/QP.pdf differ diff --git a/2022/GFQC/fig/SFBSE.pdf b/2022/GFQC/fig/SFBSE.pdf new file mode 100644 index 0000000..ac075f7 Binary files /dev/null and b/2022/GFQC/fig/SFBSE.pdf differ diff --git a/2022/GFQC/fig/Sigma.png b/2022/GFQC/fig/Sigma.png new file mode 100644 index 0000000..2dde179 Binary files /dev/null and b/2022/GFQC/fig/Sigma.png differ diff --git a/2022/GFQC/fig/TOC_BSE.pdf b/2022/GFQC/fig/TOC_BSE.pdf new file mode 100644 index 0000000..f3e4d6a Binary files /dev/null and b/2022/GFQC/fig/TOC_BSE.pdf differ diff --git a/2022/GFQC/fig/Tmatrix.png b/2022/GFQC/fig/Tmatrix.png new file mode 100644 index 0000000..cda328c Binary files /dev/null and b/2022/GFQC/fig/Tmatrix.png differ diff --git a/2022/GFQC/fig/UPS.pdf b/2022/GFQC/fig/UPS.pdf new file mode 100644 index 0000000..ca9ecee Binary files /dev/null and b/2022/GFQC/fig/UPS.pdf differ diff --git a/2022/GFQC/fig/dyn.pdf b/2022/GFQC/fig/dyn.pdf new file mode 100644 index 0000000..027e20b Binary files /dev/null and b/2022/GFQC/fig/dyn.pdf differ diff --git a/2022/GFQC/fig/evGW.png b/2022/GFQC/fig/evGW.png new file mode 100644 index 0000000..11b5a96 Binary files /dev/null and b/2022/GFQC/fig/evGW.png differ diff --git a/2022/GFQC/fig/gaps.png b/2022/GFQC/fig/gaps.png new file mode 100644 index 0000000..c220d3f Binary files /dev/null and b/2022/GFQC/fig/gaps.png differ diff --git a/2022/GFQC/fig/jarvis.jpg b/2022/GFQC/fig/jarvis.jpg new file mode 100644 index 0000000..aaadfa3 Binary files /dev/null and b/2022/GFQC/fig/jarvis.jpg differ diff --git a/2022/GFQC/fig/pentagon.pdf b/2022/GFQC/fig/pentagon.pdf new file mode 100644 index 0000000..3da5b82 Binary files /dev/null and b/2022/GFQC/fig/pentagon.pdf differ diff --git a/2022/GFQC/fig/photochemistry.pdf b/2022/GFQC/fig/photochemistry.pdf new file mode 100644 index 0000000..ba4e0e1 Binary files /dev/null and b/2022/GFQC/fig/photochemistry.pdf differ diff --git a/2022/GFQC/fig/photochemistry.tex b/2022/GFQC/fig/photochemistry.tex new file mode 100644 index 0000000..bd40ba2 --- /dev/null +++ b/2022/GFQC/fig/photochemistry.tex @@ -0,0 +1,120 @@ +\documentclass{standalone} +\usepackage{tikz} +\usepackage{siunitx} +\usepackage{mathtools,physics,bm,xcolor} +%\usetikzlibrary{arrows.meta} +%\tikzset{myarr/.style={ +% {Triangle[width=4pt, length=4pt]}-{Triangle[width=4pt, length=4pt]}, +%}} +\usepackage{tikz} + \usetikzlibrary{intersections} +\usepackage{pgfplots} + \usepgfplotslibrary{fillbetween} + \definecolor{darkgreen}{RGB}{0, 180, 0} + +\begin{document} + +\begin{tikzpicture}[scale=2] + + % x axis + \draw [->] (0,0) -- (0,4); + \node [left] at (0,4) {Energy}; + % x axis + \draw [->] (0,0) -- (4,0); + \node [below] at (4,0) {Nuclear coordinates}; + % absorption + \draw [thick, blue, <->] (1,1) -- (1,3); + % emission + \draw [thick, red, <->] (2,1.5) node[along]{$E^{fluo}$} -- (2,2.5); + % adiabatic + \draw [thick, darkgreen, <->] (2.5,1) node[right]{$E^\text{adia}$} -- (2.5,2.5); + % 0-0 + \draw [thick, blue, <->] (3,1.1) node[along]{$E^{0-0}$} -- (3,2.7); + +% \node [right] at (2.65,-2) {$\theta$}; +% \draw [thick] (-2,-2.05) node[below]{\SI{70}{\degree}} -- (-2,-1.95); +% \draw [thick] (0,-2.05) node[below]{\SI{90}{\degree}} -- (0,-1.95); +% \draw [thick] (2,-2.05) node[below]{\SI{110}{\degree}} -- (2,-1.95); + + % Theta = 75 rectangular +% \node[draw,circle,minimum size=85pt,opacity=0.3,thick] (a) at (-2,-1) {}; +% \node [above] at (-2,-0.2) {$D_{2h}$}; +% \draw [thick] (a.35) -- (a.145) -- (a.-145) -- (a.-35) -- (a.35); +% \node[fill,draw,circle, label=above right:H, minimum size=3pt,inner sep=0pt] at (a.35) {}; +% \node[fill,draw,circle, label=below right:H, minimum size=3pt,inner sep=0pt] at (a.-35) {}; +% \node[fill,draw,circle, label=below left:H, minimum size=3pt,inner sep=0pt] at (a.-145) {}; +% \node[fill,draw,circle, label=above left:H, minimum size=3pt,inner sep=0pt] at (a.145) {}; +% \draw [dashed,thick] (-2,-1) -- (a.35); +% \draw [dashed,thick] (-2,-1) -- (a.-35); +% \draw [thick] (-1.8,-1) arc [start angle=0,end angle=35,radius=0.2]; +% \draw [thick] (-1.8,-1) arc [start angle=0,end angle=-35,radius=0.2]; +% \node [right] at (-1.8,-1) {$\theta$}; + + % Theta = 90 square +% \node[draw,circle,minimum size=85pt,opacity=0.3,thick] (b) at (0,-1) {}; +% \node [above] at (0,-0.2) {$D_{4h}$}; +% \draw [thick] (b.45) -- (b.135) -- (b.-135) -- (b.-45) -- (b.45); +% \node[fill,draw,circle, label=above right:H, minimum size=3pt,inner sep=0pt] at (b.45) {}; +% \node[fill,draw,circle, label=below right:H, minimum size=3pt,inner sep=0pt] at (b.-45) {}; +% \node[fill,draw,circle, label=below left:H, minimum size=3pt,inner sep=0pt] at (b.-135) {}; +% \node[fill,draw,circle, label=above left:H, minimum size=3pt,inner sep=0pt] at (b.135) {}; +% \draw [thick,myarr,dashed] (b.-135) -- (b.45); +% \node [above left] at (0,-1) {$d$}; + + % Theta = 105 rectangular +% \node[draw,circle,minimum size=85pt,opacity=0.3,thick] (c) at (2,-1) {}; +% \node [above] at (2,-0.2) {$D_{2h}$}; +% \draw [thick] (c.55) -- (c.125) -- (c.-125) -- (c.-55) -- (c.55); +% \node[fill,draw,circle, label=above right:H, minimum size=3pt,inner sep=0pt] at (c.55) {}; +% \node[fill,draw,circle, label=below right:H, minimum size=3pt,inner sep=0pt] at (c.-55) {}; +% \node[fill,draw,circle, label=below left:H, minimum size=3pt,inner sep=0pt] at (c.-125) {}; +% \node[fill,draw,circle, label=above left:H, minimum size=3pt,inner sep=0pt] at (c.125) {}; +% \draw [dashed,thick] (2,-1) -- (c.55); +% \draw [dashed,thick] (2,-1) -- (c.-55); +% \draw [thick] (2.2,-1) arc [start angle=0,end angle=55,radius=0.2]; +% \draw [thick] (2.2,-1) arc [start angle=0,end angle=-55,radius=0.2]; +% \node [right] at (2.2,-1) {$\theta$}; + %%%%%%% Define Potential Function %%%%%%% +% \pgfmathsetmacro{\DeGS}{1} +% \pgfmathsetmacro{\RoGS}{1} +% \pgfmathsetmacro{\alphaGS}{1} +% \pgfmathsetmacro{\DeES}{1.2} +% \pgfmathsetmacro{\RoES}{1.2} +% \pgfmathsetmacro{\alphaES}{1.2} +% \pgfmathdeclarefunction{GS}{1}{% +% \pgfmathparse{% +% \DeGS*((1-exp(-\alphaGS*(#1-\RoGS)))^2-1)% +% }% +% }% +% \pgfmathdeclarefunction{ES}{1}{% +% \pgfmathparse{% +% \DeES*((1-exp(-\alphaES*(#1-\RoES)))^2-1)% +% }% +% }% +%%%%%%%% Energy Levels %%%%%%% +% \pgfmathdeclarefunction{energyGS}{1}{% +% \pgfmathparse{% +% -\DeGS+(#1+.5) - (#1+.5)^2/(1*\DeGS) +% }% +% }% +% \pgfmathdeclarefunction{energyES}{1}{% +% \pgfmathparse{% +% -\DeES+(#1+.5) - (#1+.5)^2/(1*\DeES) +% }% +% }% +% +% \begin{axis}[ +% axis lines=none, +% smooth, +% no markers, +% domain=0:4, +% xmax=10, +% ymax=10, +% scale=1 +% ] +% \addplot [black, samples=50, name path global=GSCurve] {GS(x)}; +% \addplot [black, samples=50, name path global=ESCurve] {ES(x)}; +% \end{axis} +\end{tikzpicture} + +\end{document} diff --git a/2022/GFQC/fig/qsGW1.png b/2022/GFQC/fig/qsGW1.png new file mode 100644 index 0000000..ae72804 Binary files /dev/null and b/2022/GFQC/fig/qsGW1.png differ diff --git a/2022/GFQC/fig/qsGW2.png b/2022/GFQC/fig/qsGW2.png new file mode 100644 index 0000000..359941e Binary files /dev/null and b/2022/GFQC/fig/qsGW2.png differ