diff --git a/2022/postHF/ISTPC_Loos_postHF.tex b/2022/postHF/ISTPC_Loos_postHF.tex index 9947454..fffbb28 100644 --- a/2022/postHF/ISTPC_Loos_postHF.tex +++ b/2022/postHF/ISTPC_Loos_postHF.tex @@ -7,6 +7,7 @@ \usetheme{Pittsburgh} \usecolortheme{seahorse} \usepackage{mathpazo,libertine} +\usepackage[normalem]{ulem} \usepackage{algorithmicx,algorithm,algpseudocode} \algnewcommand\algorithmicassert{\texttt{assert}} @@ -152,11 +153,11 @@ decoration={snake, \item Generalities on correlation methods \begin{itemize} \item Configuration Interaction (CI) \pub{[Szabo \& Ostlund, Modern Quantum Chemistry]} - \item Perturbation theory - \item Coupled-cluster (CC) theory + \item Perturbation theory \pub{[Szabo \& Ostlund, Modern Quantum Chemistry]} + \item Coupled-cluster (CC) theory \pub{[Jensen, Introduction to Computational Chemistry]} \end{itemize} \bigskip - \item Computing the 2th-order M{\o}ller-Plesset (MP2) correlation energy + \item Computing the 2nd-order M{\o}ller-Plesset (MP2) correlation energy \begin{itemize} \item Atomic orbital (AO) to molecular orbital (MO) transformation \pub{[Frisch et al. CPL 166 (1990) 281]} \item Laplace transform \pub{[Alml{\"o}f, CPL 181 (1991) 319]} @@ -200,6 +201,24 @@ decoration={snake, \end{columns} \end{frame} % +%----------------------------------------------------- +\begin{frame}{Assumptions \& Notations} + \begin{block}{Let's talk about notations} + \begin{itemize} + \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 $i,j,k,l$ 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 + \end{itemize} + \end{block} +\end{frame} +%----------------------------------------------------- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{frame}{One- and two-electron integrals} @@ -226,18 +245,18 @@ decoration={snake, ( \mu \nu | \lambda \sigma ) = \iint \phi_\mu(\alert{\br_1}) \phi_\nu(\alert{\br_1}) \frac{1}{r_{12}} \phi_\lambda(\blue{\br_2}) \phi_\sigma(\blue{\br_2}) d\red{\br_1} d\blue{\br_2} \end{equation} - \begin{equation} - ( \mu \blue{\nu} \orange{||} \lambda \red{\sigma} ) = ( \mu \blue{\nu} | \lambda \red{\sigma} ) - ( \mu \red{\sigma} | \lambda \blue{\nu} ) - \end{equation} +% \begin{equation} +% ( \mu \blue{\nu} \orange{||} \lambda \red{\sigma} ) = ( \mu \blue{\nu} | \lambda \red{\sigma} ) - ( \mu \red{\sigma} | \lambda \blue{\nu} ) +% \end{equation} \end{block} \begin{block}{Physicist/Dirac notation for two-electron integrals (p.~68)} \begin{equation} \langle \mu \nu | \lambda \sigma \rangle = \iint \phi_\mu(\alert{\br_1}) \phi_\nu(\blue{\br_2}) \frac{1}{r_{12}} \phi_\lambda(\alert{\br_1}) \phi_\sigma(\blue{\br_2}) d\red{\br_1} d\blue{\br_2} \end{equation} - \begin{equation} - \langle \mu \nu \orange{||} \blue{\lambda} \red{\sigma} \rangle = \langle \mu \nu | \blue{\lambda} \red{\sigma} \rangle - \langle \mu \nu | \red{\sigma} \blue{\lambda} \rangle - \end{equation} +% \begin{equation} +% \langle \mu \nu \orange{||} \blue{\lambda} \red{\sigma} \rangle = \langle \mu \nu | \blue{\lambda} \red{\sigma} \rangle - \langle \mu \nu | \red{\sigma} \blue{\lambda} \rangle +% \end{equation} \end{block} \end{frame} @@ -702,7 +721,7 @@ decoration={snake, \end{frame} \begin{frame}{Computation of DFT exchange} - \begin{block}{LDA exchange (in theory) = cf Julien's lectures} + \begin{block}{LDA exchange (in theory) = cf \sout{Julien's} Manu's lectures} \begin{gather} K_{\mu\nu}^\text{LDA} = \int \phi_{\mu}(\br) \violet{v_\text{x}^\text{LDA}}(\br) \phi_{\nu}(\br) d\br @@ -752,7 +771,7 @@ decoration={snake, \bigskip \item \alert{Multireference methods}: MCSCF, CASSCF, RASSCF, MRCI, MRCC, CASPT2, NEVPT2, etc. (C.~Angeli \& S. Knecht) \bigskip - \item \alert{Density-functional theory}: DFT, TDDFT, etc. (J. Toulouse, F. Sottile) + \item \alert{Density-functional theory}: DFT, TDDFT, etc. (J. Toulouse/E. Fromager, F. Sottile) \bigskip \item \alert{Quantum Monte Carlo}: VMC, DMC, FCIQMC, etc. (M.~Caffarel) \end{enumerate} @@ -1446,10 +1465,10 @@ decoration={snake, \\ & = \hI + \hT{1}{} + \qty( \underbrace{\hT{2}{}}_{\text{\green{connected}}} + \frac{1}{2} \underbrace{\hT{1}{2}}_{\text{\alert{disconnected}}} ) - + \qty( \hT{3}{} + \hT{2}{} \hT{1}{} + \frac{1}{6} \hT{3}{} ) + + \qty( \hT{3}{} + \hT{2}{} \hT{1}{} + \frac{1}{6} \hT{1}{3} ) \\ - & + \qty( \hT{4}{} + \hT{3}{} \hT{1}{} + \frac{1}{2} \underbrace{\hT{2}{2}}_{\text{\blue{two pairs of electrons}}} + \frac{1}{2} \hT{2}{} \hT{1}{2} + \frac{1}{24} \underbrace{\hT{4}{}}_{\text{ - \purple{four electrons}}} ) + & + \qty( \underbrace{\hT{4}{}}_{\text{ + \purple{four electrons}}} + \hT{3}{} \hT{1}{} + \frac{1}{2} \underbrace{\hT{2}{2}}_{\text{\blue{two pairs of electrons}}} + \frac{1}{2} \hT{2}{} \hT{1}{2} + \frac{1}{24} \hT{1}{4} ) + \ldots \end{split} \end{equation} @@ -1607,6 +1626,12 @@ decoration={snake, = \cre{p} \delta_{qa} \ani{i} - \cre{a} \delta_{ip} \ani{q} \end{equation} \end{block} + \begin{itemize} + \item At the \blue{TCC} level, the BCH expansion \blue{truncates naturally after the first five terms} + \item At the \alert{VCC} level, the BCH expansion \alert{does not truncate but terminates} + \item At the \purple{UCC} level, the BCH expansion \purple{does not terminate} + \end{itemize} + \bigskip For more details about normal-ordered operators, Wick's theorem, and diagrammatic techniques, see \pub{Crawford \& Schaefer, Reviews in Computational Chemistry, Vol.~14, Chap.~2, 2000.} \end{frame} diff --git a/2024/GFQC/ISTPC_Loos_QFQC.tex b/2024/GFQC/ISTPC_Loos_QFQC.tex new file mode 100644 index 0000000..fef792a --- /dev/null +++ b/2024/GFQC/ISTPC_Loos_QFQC.tex @@ -0,0 +1,1634 @@ +\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{\bE}{\bm{E}} +\newcommand{\bG}{\bm{G}} +\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 2024} + \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{Motivations} +\begin{frame} +\tableofcontents[currentsection] +\end{frame} +%----------------------------------------------------- +\begin{frame}{L\"owdin partitioning technique} + \begin{block}{Folding or dressing process} + \begin{equation} + \underbrace{\bH{}{} \cdot \bc = \yo \, \bc}_{\text{A large linear system with $N$ solutions\ldots}} + \qq{$\Rightarrow$} + \begin{pmatrix} + \overbrace{\bH_0}^{N_0 \times N_0} & \T{\bh} \\ + \bh & \underbrace{\bH_1}_{N_1 \times N_1} \\ + \end{pmatrix} + \cdot + \begin{pmatrix} + \bc_0 \\ + \bc_1 \\ + \end{pmatrix} + = \yo + \begin{pmatrix} + \bc_0 \\ + \bc_1 \\ + \end{pmatrix} + \qquad N = N_0 + N_1 + \end{equation} + \begin{align} + \qq*{\bf Row \#2:} + & \bh \cdot \bc_0 + \bH_1 \cdot \bc_1 = \yo \, \bc_1 + & \qq{$\Rightarrow$} + & \bc_1 = (\yo \, \bI - \bH_1)^{-1} \cdot \bh \cdot \bc_0 + \\ + \qq*{\bf Row \#1:} + & \bH_0 \cdot \bc_0 + \T{\bh} \cdot \bc_1 = \yo \, \bc_0 + & \qq{$\Rightarrow$} + & \underbrace{\Tilde{\bH}_0(\yo) \cdot \bc_0 = \yo \, \bc_0}_{\text{A smaller non-linear system with $N$ solutions\ldots}} + \end{align} + \begin{equation} + \boxed{ + \underbrace{\Tilde{\bH}_0(\yo)}_{\text{Effective Hamitonian}} + = \bH_0 + \underbrace{\T{\bh} \cdot (\yo \, \bI - \bH_1)^{-1} \cdot \bh}_{\text{Self-Energy $\bSig(\yo)$}} + } + \end{equation} + \begin{equation} + \qq*{Static approx. (e.g.~$\yo = 0$):} + \underbrace{\Tilde{\bH}_0(\yo = 0)}_{\text{A smaller linear system with $N_0$ solutions\ldots}} + = \bH_0 - \underbrace{\T{\bh} \cdot \bH_1^{-1} \cdot \bh}_{\text{approximations possible...}} + \end{equation} + \end{block} +\end{frame} + +%----------------------------------------------------- +\begin{frame}{Green's Function} + \begin{block}{Many-Body Green's Function} + \begin{equation} + \boxed{\qty( \yo \bI - \bH ) \cdot \bG = \bI} + \end{equation} + \end{block} + \begin{block}{Dyson equation} + \begin{equation} + \Tilde{\bH}_0(\yo) \cdot \bc_0 = \yo \bc_0 + \qq{$\Rightarrow$} + \qty[ \bH_0 + \bSig(\yo) ] \cdot \bc_0 = \yo \bc_0 + \qq{$\Rightarrow$} + \underbrace{\qty[ \yo \bI - \bH_0 - \bSig(\yo) ]}_{\bG^{-1}(\yo)} \cdot \bc_0 = \bO + \end{equation} + \begin{align} + \bG^{-1}(\yo) = \underbrace{\yo \bI - \bH_0}_{\bG_0^{-1}(\yo)} - \bSig(\yo) + & \qq{$\Rightarrow$} + \bG^{-1}(\yo) = \bG_0^{-1}(\yo) - \bSig(\yo) + \\ + & \qq{$\Rightarrow$} + \boxed{\bG(\yo) = \bG_0(\yo) + \bG_0(\yo) \cdot \bSig(\yo) \cdot \bG(\yo)} + \\ + & \qq{$\Rightarrow$} + \bG(\yo) = \qty[ \bI - \bG_0(\yo) \cdot \bSig(\yo) ]^{-1} \bG_0(\yo) + \end{align} + \end{block} +\end{frame} + +%----------------------------------------------------- +\begin{frame}{Non-Interacting Green's Function} + \begin{block}{Matrix representation} + \begin{equation} + \bH_0 \cdot \bc = \bc \cdot \bE + \qq{$\Rightarrow$} + \bH_0 \cdot \underbrace{\bc\cdot \bc^\dag}_{\bI} = \bc_0 \cdot \bE \cdot \bc^\dag + \qq{$\Rightarrow$} + \bH_0 = \bc \cdot \bE \cdot \bc^\dag + \end{equation} + \begin{equation} + \yo \bI - \bH_0 = \bc \cdot \qty( \yo \bI - \bE ) \cdot \bc^\dag + \qq{$\Rightarrow$} + \underbrace{\qty( \yo \bI - \bH_0 )^{-1}}_{\bG_0} = \bc \cdot \qty( \yo \bI - \bE )^{-1} \cdot \bc^\dag + \end{equation} + \begin{equation} + \bG_0 = \bc \cdot \qty( \yo \bI - \bE )^{-1} \cdot \bc^\dag + \qq{$\Rightarrow$} + (\bG_0)_{pq} = \sum_{r} \frac{c_{pr} c_{qr}^*}{\yo - E_r} + \end{equation} + \end{block} + \begin{block}{Hartree-Fock Green's function} + \begin{equation} + (\bG_\text{\HF})_{pq} + = \sum_{r} \frac{c_{pr} c_{qr}^*}{\yo - \e{r}{\HF}} + = \underbrace{\sum_{i} \frac{c_{pi} c_{qi}^*}{\yo - \e{i}{\HF}}}_{\text{removal}} + + \underbrace{\sum_{a} \frac{c_{pa} c_{qa}^*}{\yo - \e{a}{\HF}}}_{\text{addition}} + \end{equation} + \end{block} +\end{frame} + +\begin{frame}{Solving Dyson's Equation} + We're looking for the poles of $\bG(\yo)$: + \begin{equation} + \boxed{\bG^{-1}(\yo) = \bG_0^{-1}(\yo) - \bSig(\yo)} + \qq{$\Rightarrow$} + \bG_0^{-1}(\yo) - \bSig(\yo) = \bO + \qq{$\Rightarrow$} + \yo \bI - \be - \bSig(\yo) = \bO + \end{equation} + \begin{block}{Diagonal approximation} + \begin{equation} + \yo \bI - \be - \bSig(\yo) = \bO + \qq{$\Rightarrow$} + \yo - \e{p}{\HF} - \Sig{pp}{}(\yo) = 0 + \end{equation} + \end{block} + \begin{block}{Linearization} + \begin{equation} + \Sig{pp}{}(\yo) \approx \Sig{pp}{}(\yo = \e{p}{\HF}) + \qty(\yo - \e{p}{\HF}) \eval{\pdv{\Sig{pp}{}(\yo)}{\yo}}_{\yo = \e{p}{\HF}} + \qq{$\Rightarrow$} + \e{p}{} = \e{p}{\HF} + Z_p \Sig{pp}{}(\yo) + \end{equation} + \begin{equation} + \qq*{Renormalization Factor:} Z_p = \frac{1}{1 - \eval{\pdv{\Sig{pp}{}(\yo)}{\yo}}_{\yo = \e{p}{\HF}}} + \end{equation} + \end{block} +\end{frame} + +\begin{frame}{Example of Self-Energy} +\end{frame} + +\begin{frame}{Spectral Function} + \begin{equation} + \bSig(\yo) = \Re \bSig(\yo) + i \Im \bSig(\yo) + \end{equation} + \begin{equation} + \bA{}{}(\yo) + = - \frac{1}{\pi} \Im \abs{\bG(\yo)} + = - \frac{1}{\pi} \frac{\abs{\Im \bSig(\yo)}}{\qty[\yo \bI - \be - \Re \bSig(\yo)]^2 + \qty[ \Im \bSig(\yo)]^2} + \end{equation} +\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/2024/GFQC/fig/0-0.pdf b/2024/GFQC/fig/0-0.pdf new file mode 100755 index 0000000..6d83727 Binary files /dev/null and b/2024/GFQC/fig/0-0.pdf differ diff --git a/2024/GFQC/fig/BSE-GW.pdf b/2024/GFQC/fig/BSE-GW.pdf new file mode 100644 index 0000000..4124cb4 Binary files /dev/null and b/2024/GFQC/fig/BSE-GW.pdf differ diff --git a/2024/GFQC/fig/BSE-GW.tex b/2024/GFQC/fig/BSE-GW.tex new file mode 100644 index 0000000..e196337 --- /dev/null +++ b/2024/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/2024/GFQC/fig/BSE1.png b/2024/GFQC/fig/BSE1.png new file mode 100644 index 0000000..eb97b1a Binary files /dev/null and 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+\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/2024/GFQC/fig/LCPQ.pdf b/2024/GFQC/fig/LCPQ.pdf new file mode 100644 index 0000000..a21e4bd Binary files /dev/null and b/2024/GFQC/fig/LCPQ.pdf differ diff --git a/2024/GFQC/fig/QP.pdf b/2024/GFQC/fig/QP.pdf new file mode 100644 index 0000000..57c9da7 Binary files /dev/null and b/2024/GFQC/fig/QP.pdf differ diff --git a/2024/GFQC/fig/SFBSE.pdf b/2024/GFQC/fig/SFBSE.pdf new file mode 100644 index 0000000..ac075f7 Binary files /dev/null and b/2024/GFQC/fig/SFBSE.pdf differ diff --git a/2024/GFQC/fig/Sigma.png b/2024/GFQC/fig/Sigma.png new file mode 100644 index 0000000..2dde179 Binary files /dev/null and b/2024/GFQC/fig/Sigma.png differ diff --git a/2024/GFQC/fig/TOC_BSE.pdf b/2024/GFQC/fig/TOC_BSE.pdf new file mode 100644 index 0000000..f3e4d6a Binary files /dev/null and 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+\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/2024/GFQC/fig/qsGW1.png b/2024/GFQC/fig/qsGW1.png new file mode 100644 index 0000000..ae72804 Binary files /dev/null and b/2024/GFQC/fig/qsGW1.png differ diff --git a/2024/GFQC/fig/qsGW2.png b/2024/GFQC/fig/qsGW2.png new file mode 100644 index 0000000..359941e Binary files /dev/null and b/2024/GFQC/fig/qsGW2.png differ diff --git a/2024/postHF/ISTPC_Loos_postHF.tex b/2024/postHF/ISTPC_Loos_postHF.tex new file mode 100644 index 0000000..515cd3a --- /dev/null +++ b/2024/postHF/ISTPC_Loos_postHF.tex @@ -0,0 +1,2099 @@ +\documentclass[aspectratio=169,9pt]{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} +\usepackage{mathpazo,libertine} +\usepackage[normalem]{ulem} +\beamertemplatenavigationsymbolsempty + +\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 +} + +% operators +\newcommand{\hI}{\Hat{1}} +\newcommand{\hH}{\Hat{H}} +\newcommand{\hO}{\Hat{\mathcal{O}}} +\newcommand{\hT}[2]{\Hat{T}_{#1}^{#2}} +\newcommand{\hC}[2]{\Hat{C}_{#1}^{#2}} +\newcommand{\cre}[1]{\Hat{a}_{#1}^{\dag}} +\newcommand{\ani}[1]{\Hat{a}_{#1}} +\newcommand{\bH}{\mathbold{H}} +\newcommand{\br}{\mathbold{r}} +\newcommand{\la}{\lambda} +\newcommand{\si}{\sigma} + +\newcommand{\cJ}{\mathcal{J}} +\newcommand{\cK}{\mathcal{K}} +\newcommand{\cO}{\mathcal{O}} + +% wave functions +\newcommand{\PsiO}{\Psi_0} +\newcommand{\PsiHF}{\Psi_\text{RHF}} +\newcommand{\PsiFCI}{\Psi_\text{FCI}} +\newcommand{\PsiFCC}{\Psi_\text{FCC}} +\newcommand{\PsiCC}{\Psi_\text{CC}} +\newcommand{\PsiCCD}{\Psi_\text{CCD}} + +\newcommand{\amp}[2]{t_{#1}^{#2}} +\newcommand{\Det}[2]{\Psi_{#1}^{#2}} + +% energies +\newcommand{\EHF}{E_\text{HF}} +\newcommand{\EO}{E_\text{0}} +\newcommand{\ECC}{E_\text{CC}} +\newcommand{\ETCC}{E_\text{TCC}} +\newcommand{\EVCC}{E_\text{VCC}} +\newcommand{\EUCC}{E_\text{UCC}} +\newcommand{\ECCD}{E_\text{CCD}} + +\newcommand{\nEl}{n} +\newcommand{\nBas}{N} + +\newcommand{\ba}{\boldsymbol{a}} +\newcommand{\bb}{\boldsymbol{b}} +\newcommand{\bc}{\boldsymbol{c}} +\newcommand{\bA}{\boldsymbol{A}} +\newcommand{\bB}{\boldsymbol{B}} +\newcommand{\bo}{\boldsymbol{0}} +\newcommand{\sbra}[1]{[ #1 |} +\newcommand{\sket}[1]{| #1 ]} +\newcommand{\sexpval}[1]{[ #1 ]} +\newcommand{\sbraket}[2]{[ #1 | #2 ]} +\newcommand{\smel}[3]{[ #1 | #2 | #3 ]} + + +\definecolor{darkgreen}{RGB}{0, 180, 0} +\definecolor{fooblue}{RGB}{0,153,255} +\definecolor{fooyellow}{RGB}{234,187,0} +\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{\mc}{\multicolumn} + +\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[HF and post-HF methods]{ + Post-Hartree-Fock methods + } + + \author[PF Loos]{Pierre-Fran\c{c}ois LOOS} + \date{ISTPC 2024} + \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/peppa} + \\ + \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} + +%%% SLIDE 1 %%% +\begin{frame} + \titlepage +\end{frame} +% + +%%% SLIDE 2 %%% +\begin{frame}{Today's program} + \begin{itemize} + \item How to perform a Hartree-Fock (HF) calculation in practice? + \begin{itemize} + \item Computation of integrals \pub{[Ahlrichs, PCCP 8 (2006) 3072]} + \item Orthogonalization matrix \pub{[Szabo \& Ostlund, Modern Quantum Chemistry]} + \item Construction of the Coulomb matrix \pub{[White \& Head-Gordon, JCP 104 (1996) 2620]} + \item Resolution of the identity \pub{[Weigend et al. JCP 130 (2009) 164106]} + \item DFT exchange via quadrature \pub{[Becke, JCP 88 (1988) 2547]} + \end{itemize} + \bigskip + \item Generalities on correlation methods + \begin{itemize} + \item Configuration Interaction (CI) \pub{[Szabo \& Ostlund, Modern Quantum Chemistry]} + \item Perturbation theory \pub{[Szabo \& Ostlund, Modern Quantum Chemistry]} + \item Coupled-cluster (CC) theory \pub{[Jensen, Introduction to Computational Chemistry]} + \end{itemize} + \bigskip + \item Computing the 2nd-order M{\o}ller-Plesset (MP2) correlation energy + \begin{itemize} + \item Atomic orbital (AO) to molecular orbital (MO) transformation \pub{[Frisch et al. CPL 166 (1990) 281]} + \item Laplace transform \pub{[Alml{\"o}f, CPL 181 (1991) 319]} + \end{itemize} + \bigskip + \item Coupled cluster with doubles (CCD) + \begin{itemize} + \item Introduction to CC methods \pub{[Shavitt \& Bartlett, \textit{``Many-Body Methods in Chemistry and Physics: MBPT and Coupled-Cluster Theory''}]} + \item Algorithm to compute the CCD energy \pub{[Pople et al. IJQC 14 (1978) 545]} + \end{itemize} + \end{itemize} +\end{frame} + + + +%%% SLIDE X %%% +\begin{frame}{How to perform a HF calculation in practice?} + \begin{columns} + \begin{column}{0.7\textwidth} + \begin{block}{The SCF algorithm (p.~146)} + \begin{enumerate} + \item \orange{Specify molecule} $\{\br_A\}$ and $\{Z_A\}$ and \violet{basis set} $\{\phi_\mu\}$ + \item Calculate integrals $S_{\mu \nu}$, $H_{\mu \nu}$ and $\langle \mu \nu | \lambda \sigma \rangle$ + \item Diagonalize $\bm{S}$ and compute $\bm{X} = \bm{S}^{-1/2}$ + \item Obtain \alert{guess density matrix} for $\bm{P}$ + \begin{enumerate} + \item[1.] Calculate $\bm{J}$ and $\bm{K}$, then $\bm{F} = \bm{H} + \bm{J} + \bm{K}$ + \item[2.] Compute $\bm{F}' = \bm{X}^\dag \cdot \bm{F} \cdot \bm{X}$ + \item[3.] Diagonalize $\bm{F}'$ to obtain $\bm{C}'$ and $\bm{E}$ + \item[4.] Calculate $\bm{C}= \bm{X} \cdot \bm{C}'$ + \item[5.] Form a \blue{new density matrix} $\bm{P} = \bm{C} \cdot \bm{C}^\dag$ + \item[6.] \alert{Am I converged?} If not go back to 1. + \end{enumerate} + \item Calculate stuff that you want, like $E_\text{HF}$ for example + \end{enumerate} + \end{block} + \end{column} + \begin{column}{0.3\textwidth} + \includegraphics[width=\textwidth]{fig/Szabo} + \end{column} + \end{columns} +\end{frame} +% +%----------------------------------------------------- +\begin{frame}{Assumptions \& Notations} + \begin{block}{Let's talk about notations} + \begin{itemize} + \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 $i,j,k,l$ 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 + \end{itemize} + \end{block} +\end{frame} +%----------------------------------------------------- + +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +\begin{frame}{One- and two-electron integrals} + \begin{columns} + \begin{column}{0.7\textwidth} + \begin{block}{One-electron integrals: overlap \& core Hamiltonian (Appendix A)} + \begin{equation} + S_{\mu\nu} + = \braket{\mu}{\nu} + = \int \phi_\mu(\orange{\br}) \phi_\nu(\orange{\br}) d\orange{\br} + \end{equation} + \begin{equation} + H_{\mu\nu} + = \mel{\mu}{\hH^\text{c}}{\nu} + = \int \phi_\mu(\orange{\br}) \hH^\text{c}(\orange{\br}) \phi_\nu(\orange{\br}) d\orange{\br} + \end{equation} + \end{block} + \end{column} + \begin{column}{0.3\textwidth} + \includegraphics[width=\textwidth]{fig/SBG} + \end{column} + \end{columns} \begin{block}{Chemist/Mulliken notation for two-electron integrals (p.~68)} + \begin{equation} + ( \mu \nu | \lambda \sigma ) + = \iint \phi_\mu(\alert{\br_1}) \phi_\nu(\alert{\br_1}) \frac{1}{r_{12}} \phi_\lambda(\blue{\br_2}) \phi_\sigma(\blue{\br_2}) d\red{\br_1} d\blue{\br_2} + \end{equation} +% \begin{equation} +% ( \mu \blue{\nu} \orange{||} \lambda \red{\sigma} ) = ( \mu \blue{\nu} | \lambda \red{\sigma} ) - ( \mu \red{\sigma} | \lambda \blue{\nu} ) +% \end{equation} + \end{block} + \begin{block}{Physicist/Dirac notation for two-electron integrals (p.~68)} + \begin{equation} + \langle \mu \nu | \lambda \sigma \rangle + = \iint \phi_\mu(\alert{\br_1}) \phi_\nu(\blue{\br_2}) \frac{1}{r_{12}} \phi_\lambda(\alert{\br_1}) \phi_\sigma(\blue{\br_2}) d\red{\br_1} d\blue{\br_2} + \end{equation} +% \begin{equation} +% \langle \mu \nu \orange{||} \blue{\lambda} \red{\sigma} \rangle = \langle \mu \nu | \blue{\lambda} \red{\sigma} \rangle - \langle \mu \nu | \red{\sigma} \blue{\lambda} \rangle +% \end{equation} + \end{block} +\end{frame} + + +\begin{frame}{Computing the electron repulsion integrals (ERIs)} + \begin{columns} + \begin{column}{0.7\textwidth} + \begin{block}{Four-center two-electron integrals} + \small + \begin{equation} + \begin{split} + \braket{\ba_1\ba_2}{\bb_1\bb_2} + & \equiv \mel{\ba_1\ba_2}{\alert{r_{12}^{-1}}}{\bb_1\bb_2} + \\ + & = \iint \phi_{\ba_1}^{\bA_1}(\br_1) \phi_{\ba_2}^{\bA_2}(\br_2) \,\alert{\frac{1}{r_{12}}} \, + \phi_{\bb_1}^{\bB_1}(\br_1) \phi_{\bb_2}^{\bB_2}(\br_2) d\br_1 d\br_2 + \end{split} + \end{equation} + \alert{Formally, one has to compute $\order{N^4}$ ERIs!} + \end{block} + \end{column} + \begin{column}{0.3\textwidth} + \includegraphics[width=\textwidth]{fig/STO} + \end{column} + \end{columns} + % + \begin{block}{Gaussian-type orbital (GTO)} + \small + \begin{align*} + \text{\violet{Contracted} GTO} & = \ket{\ba} + \equiv \phi_{\ba}^{\bA}(\br) + = \sum_k^K D_k \sket{\ba}_k + \\ + \text{\blue{Primitive} GTO} & = \sket{\ba} + = (x-A_x)^{a_x} (y-A_y)^{a_y} (z-A_z)^{a_z} e^{-\alpha \abs{ \br -\bA }^2} + \end{align*} + \end{block} + \begin{itemize} + \item \textbf{\purple{Exponent:}} $\alpha$ + \item \textbf{\purple{Center:}} $\bA = (A_x, A_y, A_z)$ + \item \textbf{\purple{Angular momentum:}} $\ba = (a_x, a_y, a_z)$ and total angular momentum $a=a_x + a_y + a_z$ + \end{itemize} + % +\end{frame} + +\begin{frame}{The contraction problem} + \begin{columns} + \begin{column}{0.7\textwidth} + \begin{block}{Primitive vs Contracted} + \begin{itemize} + \item Same center $\bA$ + \item Same angular momentum $\ba$ + \item Different exponent $\violet{\alpha_k}$ + \item Contraction coefficient $\blue{D_k}$ and degree $K$ + \end{itemize} + \begin{equation} + \underbrace{\braket{\ba_1\ba_2}{\bb_1\bb_2}}_{\text{\green{contracted ERI}}} + = \sum_{k_1}^{K_1} \sum_{k_2}^{K_2} \sum_{k_3}^{K_3} \sum_{k_4}^{K_4} + \blue{D_{k_1} D_{k_2} D_{k_3} D_{k_4}} + \underbrace{\sbraket{\ba_{1,k_1}\ba_{2,k_2}}{\bb_{1,k_3}\bb_{2,k_4}}}_{\text{\red{primitive ERI}}} + \end{equation} + \centering + \green{One} contracted ERI required \red{$K_1 \times K_2 \times K_3 \times K_4$} primitive ERIs! + \end{block} + \begin{block}{Dunning's cc-pVTZ basis for the carbon atom} + \begin{equation} + \green{\braket{1s1s}{1s1s}} + = \sum_{k_1}^{10} \sum_{k_2}^{10} \sum_{k_3}^{10} \sum_{k_4}^{10} + \blue{D_{k_1} D_{k_2} D_{k_3} D_{k_4}} + \red{\sbraket{s_{k_1}^{\violet{\alpha_{k_1}}} s_{k_2}^{\violet{\alpha_{k_2}}}} {s_{k_3}^{\violet{\alpha_{k_3}}} s_{k_4}^{\violet{\alpha_{k_4}}} }} + \end{equation} + \centering + The $\green{\braket{1s1s}{1s1s}}$ integral requires $10^4$ \red{$s$-type integrals}! + \end{block} + \end{column} + \begin{column}{0.3\textwidth} + \begin{equation} + \boxed{\green{\ket{\ba}} = \sum_k^K \blue{D_k} \red{\sket{\ba_k}}} + \end{equation} + \\ + \bigskip + \begin{block}{https://www.basissetexchange.org} + \bigskip + \centering + \includegraphics[width=\textwidth]{fig/C} + \end{block} + \end{column} + \end{columns} +\end{frame} + +%%% SLIDE X %%% +\begin{frame}{Properties of Gaussian functions} + \begin{block}{Gaussian product rule: \textit{``The product of two gaussians is a gaussian''}} + \begin{equation} + G_{\red{\alpha},\red{\bm{A}}}(\br) = \exp(-\red{\alpha} \abs{\br - \red{\bA}}^2) + \qqtext{and} + G_{\blue{\beta},\blue{\bm{B}}}(\br) = \exp(-\blue{\beta} \abs{\br - \blue{\bB}}^2) + \qqtext{then} + \end{equation} + \begin{equation} + \boxed{G_{\red{\alpha},\red{\bm{A}}}(\br) G_{\blue{\beta},\blue{\bm{B}}}(\br) = \violet{K} \, G_{\violet{\zeta},\violet{\bm{P}}}(\br)} + \qqtext{with} + \violet{\zeta} = \red{\alpha} + \blue{\beta} + \qqtext{and} + \violet{\bm{P}} = \frac{\red{\alpha \bA} + \blue{\beta \bB}}{\red{\alpha} + \blue{\beta} } + \end{equation} + \begin{equation} + \violet{K} = \exp( -\frac{\red{\alpha} \blue{\beta}}{\red{\alpha} + \blue{\beta} } \abs{\red{\bA} - \blue{\bB}}^2) + \end{equation} + \end{block} + \begin{block}{Gaussian product rule for ERIs} + \begin{equation} + \begin{split} + (\bm{\red{a}} \bm{\blue{b}}|\bm{\orange{c}} \bm{\green{d}}) + & = \iint G_{\red{\alpha},\red{\bm{A}}}(\br_1) G_{\blue{\beta},\blue{\bm{B}}}(\br_1) \frac{1}{r_{12}} G_{\orange{\gamma},\orange{\bm{C}}}(\br_2) G_{\green{\delta},\green{\bm{D}}}(\br_2) d\br_1 d\br_2 + \\ + & = \violet{K} \purple{K} \iint G_{\violet{\zeta},\violet{\bm{P}}}(\br_1) \frac{1}{r_{12}} G_{\purple{\eta},\purple{\bm{Q}}}(\br_2) d\br_1 d\br_2 + \end{split} + \end{equation} + \alert{The number of ``significant'' ERIs in a large system is $\order{N^2}$!} + \end{block} +\end{frame} +% + +\begin{frame}{Upper bounds for ERIs} + \begin{columns} + \begin{column}{0.35\textwidth} + \begin{block}{A ``good'' upper bound must be} + \begin{itemize} + \item tight (i.e., a good estimate) + \item simple (i.e, cheap to compute) + \end{itemize} + \end{block} + \end{column} + \begin{column}{0.65\textwidth} + \begin{equation} + \boxed{\abs{(\bm{\red{a}} \bm{\blue{b}}|\bm{\orange{c}} \bm{\green{d}})} \le B} + \end{equation} + \end{column} + \end{columns} + \bigskip + \begin{block}{Cauchy-Schwartz bound} + \begin{equation} + \abs{(\bm{\red{a}} \bm{\blue{b}}|\bm{\orange{c}} \bm{\green{d}})} + \le + \sqrt{(\bm{\red{a}} \bm{\blue{b}}|\bm{\red{a}} \bm{\blue{b}})} + \sqrt{(\bm{\orange{c}} \bm{\green{d}}|\bm{\orange{c}} \bm{\green{d}})} + \qqtext{or} + \abs{(\bm{\violet{P}}|\bm{\purple{Q}})} + \le + \sqrt{(\bm{\violet{P}}|\bm{\violet{P}})} + \sqrt{(\bm{\purple{Q}}|\bm{\purple{Q}})} + \end{equation} + \end{block} + \begin{block}{The family of generalized H\"older bounds} + \begin{equation} + \abs{(\bm{\red{a}} \bm{\blue{b}}|\bm{\orange{c}} \bm{\green{d}})} + \le + \qty[ (\bm{\red{a}} \bm{\blue{b}}|\bm{\red{a}} \bm{\blue{b}}) ]^{1/\purple{m}} + \qty[ (\bm{\orange{c}} \bm{\green{d}}|\bm{\orange{c}} \bm{\green{d}}) ]^{1/\violet{n}} + \qqtext{with} + \frac{1}{\purple{m}} + \frac{1}{\violet{n}} = 1 + \qqtext{and} + \purple{m},\violet{n} > 1 + \end{equation} + \end{block} +\end{frame} + + +\begin{frame}{Asymptotic scaling of two-electron integrals} + \begin{block}{Number of significant two-electron integrals} + \begin{equation} + (\bm{\red{a}} \bm{\blue{b}}|\bm{\orange{c}} \bm{\green{d}}) \equiv (\bm{\red{a}} \bm{\blue{b}}| \mathcal{O}_2 | \bm{\orange{c}} \bm{\green{d}}) + \end{equation} + \end{block} + \bigskip + \begin{block}{Long-range vs short-range operators} + \begin{equation} + N_\text{sig} = c\,N^{\alpha} + \end{equation} + \center + \begin{tabular}{lcrccrc} + \hline + \hline + Molecule & $N$ & \mc{2}{c}{\red{$\hO = r_{12}^{-1}$}} && \mc{2}{c}{\orange{$\hO = e^{-r_{12}^2}$}} \\ + \cline{3-4} \cline{6-7} + & & \mc{1}{c}{$N_\text{sig}$} & $\alpha$ && \mc{1}{c}{$N_\text{sig}$} & $\alpha$ \\ + \hline + propene & 12 & 1\,625 & --- && 1\,650 & --- \\ + butadiene & 16 & 5\,020 & 3.9 && 5\,020 & 3.9 \\ + hexatriene & 24 & 24\,034 & 3.9 && 23\,670 & 3.8 \\ + octatetraene & 32 & 63\,818 & 3.4 && 52\,808 & 2.8 \\ + decapentaene & 40 & 119\,948 & 2.8 && 81\,404 & 1.9 \\ + dodecaexaene & 48 & 192\,059 & 2.6 && 109\,965 & 1.6 \\ + \hline + \hline + \end{tabular} + \bigskip + \end{block} +\end{frame} + + +\begin{frame}{Recipe for computing two-electron integrals} +\center +\begin{tikzpicture} + \begin{scope}[very thick, + node distance=4cm,on grid,>=stealth', + boxRR/.style={rectangle,draw,fill=green!40}, + boxUB/.style={rectangle,draw,fill=orange!40}, + boxFI/.style={rectangle,draw,fill=red!40}, + integral/.style={rectangle,draw,fill=violet!40}], + \node [integral, align=center] (1) {\textbf{The cake:} \\ Two-electron integrals \\ $\braket{\ba_1 \ba_2}{ \bb_1 \bb_2}$}; + \node [boxUB, align=center] (2A) [below=of 1] {\textbf{Ingredient number 2:} \\ Recurrence relations \\ $\expval*{\ba_1^+} = \expval*{\ba_1} + \expval*{\ba_1^-}$}; + \node [boxRR, align=center] (2B) [right=of 2A] {\textbf{Ingredient number 3:} \\ Upper bounds \\ $\abs{\braket{\ba_1 \ba_2}{ \bb_1 \bb_2}} \le B$}; + \node [boxFI, align=center] (2C) [left=of 2A] {\textbf{Ingredient number 1:} \\ Fundamental integrals \\ $\braket{\bo\bo}{\bo\bo}^{\bm{m}}$}; + \path + (1) edge [<-] (2A) + (1) edge [<-,bend right] (2B) + (1) edge [<-,bend left] (2C) + ; + \end{scope} +\end{tikzpicture} +\end{frame} + +\begin{frame}{Late-contraction path algorithm (Head-Gordon-Pople \& PRISM inspired)} + \begin{tikzpicture} + \begin{scope}[ + very thick, + node distance=1.5cm,on grid,>=stealth', + boxSP/.style={rectangle,draw,fill=purple!40}, + box0m/.style={rectangle,draw,fill=red!40}, + boxCm/.style={rectangle,draw,fill=gray!40}, + boxA/.style={rectangle,draw,fill=red!40}, + boxAA/.style={rectangle,draw,fill=red!40}, + boxAAA/.style={rectangle,draw,fill=red!40}, + boxC/.style={rectangle,draw,fill=gray!40}, + boxCC/.style={rectangle,draw,fill=gray!40}, + boxCCC/.style={rectangle,draw,fill=orange!40}, + boxCCCCCC/.style={rectangle,draw,fill=green!40}, + ], + \node [boxSP, align=center] (SP) {Shell-pair \\ data}; + \node [box0m, align=center] (0m) [right=of 1,xshift=1.25cm] {$\sbraket{00}{00}^{\bm{m}}$}; + \node [boxCm, align=center] (Cm) [right=of 0m,xshift=1.75cm] {$\braket{00}{00}^{\bm{m}}$}; + \node [boxA, align=center] (A) [below=of 0m] {$\sbraket{0 a_2}{00}^{\bm{m}}$}; + \node [boxC, align=center] (C) [right=of A,xshift=1.75cm] {$\braket{0 a_2}{00}^{\bm{m}}$}; + \node [boxAA, align=center] (AA) [below=of A] {$\sbraket{a_1 a_2}{00}$}; + \node [boxCC, align=center] (CC) [right=of AA,xshift=1.75cm] {$\braket{a_1 a_2}{00}$}; + \node [boxCCCCCC, align=center] (CCCC) [right=of CC,xshift=2cm] {$\braket{a_1 a_2}{b_1 b_2}$}; + \path + (SP) edge[->] node[below,blue]{T$_0$} (0m) + (0m) edge[->] node[left,orange]{T$_1$} node [right,red]{VRR$_1$} (A) + (0m) edge[->,gray!70] (Cm) + (A) edge[->] node[left,orange]{T$_2$} node [right,red]{VRR$_2$} (AA) + (A) edge[->,gray!70] (C) + (AA) edge[->] node [below,blue]{CC} (CC) + (Cm) edge[->,gray!70] (C) + (C) edge[->,gray!70] (CC) + (CC) edge[->] node [above,orange]{T$_3$} node [below,red]{HRR} (CCCC) + ; + \end{scope} + \end{tikzpicture} + \bigskip + \begin{itemize} + \item \red{HRR} = horizontal recurrence relation [Obara-Saika] + \item \red{VRR} = vertical recurrence relation + \item \blue{CC} = bra contraction + \end{itemize} +\end{frame} + +%\begin{frame}{Screening algorithm for two-electron integrals} +% +%\resizebox{\textwidth}{!}{ +%\begin{tikzpicture} +% \begin{scope}[very thick, +% node distance=2.5cm,on grid,>=stealth', +% bound2/.style={diamond,draw,fill=blue!40}, +% bound4/.style={diamond,draw,fill=blue!40}, +% bound6/.style={diamond,draw,fill=blue!40}, +% shell/.style={circle,draw,fill=green!40}, +% shellpair/.style={circle,draw,fill=green!40}, +% shellquartet/.style={circle,draw,fill=green!40}, +% shell1/.style={rectangle,draw,fill=yellow!40}, +% shell2/.style={rectangle,draw,fill=orange!40}, +% shell3/.style={rectangle,draw,fill=red!40}, +% integral/.style={rectangle,draw,fill=violet!40}], +% \node [shell1, align=center] (1) {Primitive\\shells\\$\sket{a}$}; +% \node [bound2, align=center] (B2) [right=of 1] {$\sexpval{B_2}$}; +% \node [shell, align=center] (S1T) [above=of B2, yshift=-0.5cm] {$\sket{a}$}; +% \node [shell, align=center] (S1B) [below=of B2, yshift=0.5cm] {$\sket{b}$}; +% \node [shell2, align=center] (2) [right=of B2,xshift=0.75cm] {Contracted\\shell-pairs\\$\ket{ab}$}; +% \node [bound4, align=center] (B4) [right=of 2] {$\expval{B_4}$} ; +% \node [shellpair, align=center] (S2T) [above=of B4, yshift=-0.5cm] {$\ket{a_1b_1}$}; +% \node [shellpair, align=center] (S2B) [below=of B4, yshift=0.5cm] {$\ket{a_2b_2}$}; +% \node [shell3, align=center] (3) [right=of B4] {Two-Electron\\integrals\\$\braket{a_1b_1}{a_2b_2}$}; +% \path +% (1) edge [->,bend left] (S1T) +% (1) edge [->,bend right] (S1B) +% (S1T) edge [snake it] (B2) +% (S1B) edge [snake it] (B2) +% (B2) edge [->,color=red] node [below] {\small Contraction} (2) +% (2) edge [->,bend left] (S2T) +% (2) edge [->,bend right] (S2B) +% (S2T) edge [snake it] (B4) +% (S2B) edge [snake it] (B4) +% (B4) edge [->] (3) +% ; +% \end{scope} +%\end{tikzpicture} +%} +%\end{frame} + +\begin{frame}{Orthogonalization matrix} + \red{\bf We are looking for a matrix in order to orthogonalize the AO basis, i.e.~$\bm{X}^\dag \cdot \bm{S} \cdot \bm{X} = \bm{1}$} + \\ + \bigskip + \begin{block}{Symmetric (or L\"owdin) orthogonalization} + \begin{equation} + \text{$\bm{X} =\bm{S}^{-1/2} = \bm{U} \cdot \bm{s}^{-1/2} \cdot \bm{U}^\dag$ is one solution...} + \end{equation} + \purple{\bf Is it working?} + \begin{equation} + \bm{X}^\dag \cdot \bm{S} \cdot \bm{X} + = \bm{S}^{-1/2} \cdot \bm{S} \cdot \bm{S}^{-1/2} + = \bm{S}^{-1/2} \cdot \bm{S} \cdot \bm{S}^{-1/2} + = \bm{I} \quad \green{\checkmark} + \end{equation} + \end{block} + \begin{block}{Canonical orthogonalization} + \begin{equation} + \text{$\bm{X} =\bm{U} \cdot \bm{s}^{-1/2}$ is another solution (when you have linear dependencies)...} + \end{equation} + \purple{\bf Is it working?} + \begin{equation} + \bm{X}^\dag \cdot \bm{S} \cdot \bm{X} + = \bm{s}^{-1/2} \cdot \underbrace{\bm{U}^{\dag} \cdot \bm{S} \cdot \bm{U}}_{\bm{s}} \cdot \bm{s}^{-1/2} + = \bm{I} \quad \green{\checkmark} + \end{equation} + \end{block} +\end{frame} + +\begin{frame}{Computation of the Fock matrix and energy} + \begin{block}{Density matrix (closed-shell system)} + \begin{equation} + P_{\red{\mu \nu}} = 2 \sum_{i}^\text{occ} C_{\red{\mu} i} C_{\red{\nu} i} + \qqtext{or} + \boxed{\bm{P} = \bm{C} \cdot \bm{C}^{\dag}} + \end{equation} + \end{block} + \begin{block}{Fock matrix in the AO basis (closed-shell system)} + \begin{equation} + F_{\red{\mu\nu}} + = H_{\red{\mu\nu}} + + \underbrace{\sum_{\blue{\la \si}} P_{\blue{\la\si}} (\red{\mu\nu}|\blue{\la\si})}_{J_{\red{\mu \nu}} = \text{ Coulomb}} + \underbrace{ - \frac{1}{2} \sum_{\blue{\la \si}} P_{\blue{\la\si}} (\red{\mu}\blue{\si}|\blue{\la}\red{\nu})}_{K_{\red{\mu \nu}} = \text{ exchange}} + \end{equation} + \end{block} + \begin{block}{HF energy in the AO basis (closed-shell system)} + \begin{equation} + E_\text{HF} = \sum_{\red{\mu \nu}} P_{\red{\mu \nu}} H_{\red{\mu \nu}} + + \frac{1}{2} \sum_{\red{\mu \nu} \blue{\la\si}} P_{\red{\mu \nu}} \qty[ (\red{\mu \nu} | \blue{\lambda \sigma}) - \frac{1}{2} (\red{\mu} \blue{\sigma} | \red{\lambda} \blue{\nu}) ] P_{\blue{\lambda\sigma}} + \qqtext{or} + \boxed{E_\text{HF} = \frac{1}{2} \text{Tr}{\qty[\bm{P} \cdot (\bm{H} + \bm{F})]}} + \end{equation} + \end{block} +\end{frame} + +\begin{frame}{Computation of the Fock matrix and energy} + \begin{algorithmic} + \Procedure{Computing the Coulomb matrix}{} + \For{$\red{\mu}=1,N$} + \For{$\blue{\nu}=1,N$} + \State $J_{\red{\mu}\blue{\nu}} = 0$ \Comment{Initialization of the array} + \For{$\orange{\la}=1,N$} + \For{$\violet{\si}=1,N$} + \State $J_{\red{\mu}\blue{\nu}} + = J_{\red{\mu}\blue{\nu}} + + P_{\orange{\la}\violet{\si}} (\red{\mu}\blue{\nu}|\orange{\la}\violet{\si})$ + \Comment{Accumulation step} + \EndFor + \EndFor + \EndFor + \EndFor + \EndProcedure + \Comment{\bf \red{This is a $\order{N^4}$ algorithm as it involves four loops}} + \end{algorithmic} +\end{frame} + +%%% SLIDE X %%% +\begin{frame}{Resolution of the identity} + \begin{block}{Resolution of the identity (RI)} + \begin{equation} + \sum_{\green{A}=1}^{\red{\infty}} \dyad{\green{A}} = \hI + \qq{with} + \braket{\green{A}}{\green{B}} = \delta_{AB} + \qq{$\Leftrightarrow$} + \sum_{\green{A}=1}^{\red{\infty}} \green{A}(\br_1) \green{A}(\br_2) + = \delta(\br_1 - \br_2) + \end{equation} + \end{block} + \begin{block}{Generalization to a two-body operator $\hO$} + \begin{equation} + \sum_{\green{\Tilde{A}}=1}^{\red{\infty}} \dyad{\green{\Tilde{A}}} = \hO + \qq{with} + \mel{\green{A}}{\hO}{\green{B}} = \delta_{AB} + \qq{and} + \hO \ket{\green{A}} = \ket{\green{\Tilde{A}}} + \qq{$\Leftrightarrow$} + \sum_{\green{\Tilde{A}}=1}^{\red{\infty}} \green{\Tilde{A}}(\br_1) \green{\Tilde{A}}(\br_2) + = \hO(\br_1,\br_2) + \end{equation} + \end{block} +\end{frame} +% + +%%% SLIDE X %%% +\begin{frame}{Resolution of the Coulomb operator} + \begin{block}{RI in practice = RI \alert{approximation}} + \begin{equation} + \boxed{\sum_{\green{A}=1}^{\red{\infty}} \dyad{\green{A}} = \hI + \qqtext{and, in practice, } + \sum_{\green{A}=1}^{\red{K}} \dyad{\green{A}} \approx \hI} + \end{equation} + \end{block} + \begin{block}{Computing the Coulomb matrix within the RI approximation} + \begin{equation} + \begin{split} + J_{\red{\mu\nu}} + & = \sum_{\blue{\la \si}} P_{\blue{\la\si}} (\red{\mu\nu}|\blue{\la\si}) + \\ + & \stackrel{\text{\green{RI}}}{=} \sum_{\blue{\la \si}} P_{\blue{\la\si}} \sum_{\green{A}} (\red{\mu\nu}|\green{A}) (\green{A}|\blue{\la\si}) + \\ + & = \sum_{\green{A}} (\red{\mu\nu}|\green{A}) + \underbrace{\sum_{\blue{\la \si}} P_{\blue{\la\si}} (\green{A}|\blue{\la\si})}_{\order{KN^2} \text{ and $K$ storage}} + = \underbrace{\sum_{\green{A}} (\red{\mu\nu}|\green{A}) \rho_{\green{A}}}_{\order{KN^2}} + \end{split} + \end{equation} + \\ + Similar (more effective) approaches are named Cholesky decomposition, low-rank approximation, etc. + \end{block} +\end{frame} +% + +\begin{frame}{Computation of exact exchange} + \begin{algorithmic} + \Procedure{Computing the exchange matrix}{} + \For{$\red{\mu}=1,N$} + \For{$\blue{\nu}=1,N$} + \State $K_{\red{\mu}\blue{\nu}} = 0$ \Comment{Initialization of the array} + \For{$\orange{\la}=1,N$} + \For{$\violet{\si}=1,N$} + \State $K_{\red{\mu}\blue{\nu}} + = K_{\red{\mu}\blue{\nu}} + + P_{\orange{\la}\violet{\si}} (\red{\mu}\violet{\si}|\orange{\la}\blue{\nu})$ + \Comment{Accumulation step} + \EndFor + \EndFor + \EndFor + \EndFor + \EndProcedure + \Comment{\bf \red{This is a $\order{N^4}$ algorithm and it's hard to play games...}} + \end{algorithmic} +\end{frame} + +\begin{frame}{Computation of DFT exchange} + \begin{block}{LDA exchange (in theory) = cf \sout{Julien's} Manu's lectures} + \begin{gather} + K_{\mu\nu}^\text{LDA} + = \int \phi_{\mu}(\br) \violet{v_\text{x}^\text{LDA}}(\br) \phi_{\nu}(\br) d\br + = \frac{4}{3} C_\text{x} \overbrace{\int \phi_{\mu}(\br) \blue{\rho^{1/3}}(\br) \phi_{\nu}(\br) d\br}^{\text{\alert{no closed-form expression in general}}} + \\ + \blue{\rho}(\br) = \sum_{\mu \nu} \phi_{\mu}(\br) \blue{P_{\mu \nu}} \phi_{\nu}(\br) + \end{gather} + \end{block} + \begin{block}{LDA exchange (in practice) = \alert{numerical integration via quadrature} = $\int f(x) dx \approx \sum_k w_k f(x_k)$} + \begin{gather} + \underbrace{K_{\mu\nu}^\text{LDA}}_{\green{\order{N_\text{grid} N^2}}} + \approx \sum_{k=1}^{\purple{N_\text{grid}}} + \underbrace{\orange{w_k}}_{\orange{\text{weights}}} \phi_{\mu}(\red{\br_k}) \violet{v_\text{x}^\text{LDA}}(\underbrace{\red{\br_k}}_{\text{\red{roots}}}) \phi_{\nu}(\red{\br_k}) + = \frac{4}{3} C_\text{x} \sum_{k=1}^{\purple{N_\text{grid}}} \orange{w_k} \phi_{\mu}(\red{\br_k}) \blue{\rho^{1/3}}(\red{\br_k}) \phi_{\nu}(\red{\br_k}) + \\ + \underbrace{\blue{\rho}(\red{\br_k})}_{\green{\order{N_\text{grid} N^2}}} = \sum_{\mu \nu} \phi_{\mu}(\red{\br_k}) \blue{P_{\mu \nu}} \phi_{\nu}(\red{\br_k}) + \end{gather} + \end{block} +\end{frame} + + +\begin{frame}{The correlation energy} + \begin{itemize} + \item HF replaces the e-e interaction by an \green{averaged interaction} + \bigskip + \item The error in the HF method is called the \purple{correlation energy} + $$\boxed{E_c = E - E_\text{HF}} $$ + \item The correlation energy is small \orange{but cannot but neglected!} + \bigskip + \item HF energy \blue{roughly 99\%} of total but \blue{chemistry very sensitive to remaining 1\%} + \bigskip + \item The correlation energy is \alert{always negative} + \bigskip + \item Computing $E_c$ is one of the \violet{central problem of quantum chemistry} + \bigskip + \item In quantum chemistry, we usually \alert{``freeze'' the core electrons} for correlated calculations + \end{itemize} +\end{frame} + +\begin{frame}{Most common correlation methods in quantum chemistry} + \begin{enumerate} + \item \alert{Configuration Interaction}: CID, CIS, CISD, CISDTQ, etc. + \bigskip + \item \alert{Coupled Cluster}: CCD, CCSD, CCSD(T), CCSDT, CCSDTQ, etc. + \bigskip + \item \alert{M{\o}ller-Plesset perturbation theory}: MP2, MP3, MP4, MP5, etc. + \bigskip + \item \alert{Multireference methods}: MCSCF, CASSCF, RASSCF, MRCI, MRCC, CASPT2, NEVPT2, etc. (C.~Angeli \& S. Knecht) + \bigskip + \item \alert{Density-functional theory}: DFT, TDDFT, etc. (J. Toulouse/E. Fromager, F. Sottile) + \bigskip + \item \alert{Quantum Monte Carlo}: VMC, DMC, FCIQMC, etc. (M.~Caffarel) + \end{enumerate} +\end{frame} + +\begin{frame}{Configuration Interaction (CI)} + \begin{itemize} + \item This is the \blue{oldest} and perhaps the \blue{easiest method to understand} + \bigskip + \item CI is based on the \orange{variational principle} (like HF) + \bigskip + \item The CI wave function is a \blue{linear combination of determinants} + \bigskip + \item CI methods use \violet{excited determinants} to ``improve'' the reference (usually HF) wave function + \begin{equation} + \ket{\Phi_0} + = \underbrace{c_0 \ket*{\Psi_0}}_{\text{reference}} + + \underbrace{\violet{\sum_{\substack{i \\ a}} c_i^a \ket*{\Psi_i^a}}}_{\text{singles}} + + \underbrace{\purple{\sum_{\substack{i < j \\ a < b}} c_{ij}^{ab} \ket*{\Psi_{ij}^{ab}}}}_{\text{doubles}} + + \underbrace{\orange{\sum_{\substack{i < j < k \\ a < b < c}} c_{ijk}^{abc} \ket*{\Psi_{ijk}^{abc}}}}_{\text{triples}} + + \underbrace{\blue{\sum_{\substack{i < j < k < l \\ a < b < c < d}} c_{ijkl}^{abcd} \ket*{\Psi_{ijkl}^{abcd}}}}_{\text{quadruples}} + + \ldots + \end{equation} + \end{itemize} +\end{frame} + +\begin{frame}{CI method and Excited determinants} + \begin{block}{Excited determinants} + \center + \includegraphics[width=0.7\textwidth]{fig/det} + \end{block} + \begin{block}{CI wave function} + \begin{equation} + \boxed{ + \ket{\Phi_0} + = c_0 \ket{\text{0}} + + \violet{c_\text{S} \ket{\text{S}}} + + \purple{c_\text{D} \ket{\text{D}}} + + \orange{c_\text{T} \ket{\text{T}}} + + \blue{c_\text{Q} \ket{\text{Q}}} + + \ldots + } + \end{equation} + \end{block} +\end{frame} + + +\begin{frame}{Excited determinants} + \begin{block}{Reference determinant} + \begin{equation} + \qq*{\green{The electrons are in the $N$ lowest spinorbitals (Aufbau principle):}} \ket{\Psi_0} \equiv \ket{0} = \ket{\chi_1 \ldots \chi_{\green{i}} \chi_{\green{j}} \ldots \chi_N} + \end{equation} + \end{block} + \begin{block}{Singly-excited determinants} + \begin{equation} + \qq*{Electron in $\green{i}$ promoted in $\red{a}$:} \ket{\Psi_{\green{i}}^{\red{a}}} = \ket{\chi_1 \ldots \chi_{\red{a}} \chi_{\green{j}} \ldots \chi_N} + \end{equation} + \end{block} + \begin{block}{Doubly-excited determinants} + \begin{equation} + \qq*{Electrons in $\green{i}$ and $\green{j}$ promoted in $\red{a}$ and $\red{b}$:} \ket{\Psi_{\green{ij}}^{\red{ab}}} = \ket{\chi_1 \ldots \chi_{\red{a}} \chi_{\red{b}} \ldots \chi_N} + \end{equation} + \end{block} +% \begin{center} +% \includegraphics[width=0.15\textwidth]{fig/GS} +% \hspace{0.2\textwidth} +% \includegraphics[width=0.15\textwidth]{fig/single} +% \hspace{0.2\textwidth} +% \includegraphics[width=0.15\textwidth]{fig/double} +% \end{center} +\end{frame} + + +\begin{frame}{Truncated CI} + \begin{itemize} + \item When $\ket{\text{S}}$ (\violet{singles}) are taken into account: \textbf{CIS} + \begin{equation} + \violet{\ket{\Phi_\text{CIS}} = c_0 \ket{\text{0}} + c_\text{S} \ket{\text{S}}} + \end{equation} + \textbf{NB:} CIS is an \violet{excited state method} + \item When $\ket{\text{D}}$ (\alert{doubles}) are taken into account: \textbf{CID} + \begin{equation} + \alert{\ket{\Phi_\text{CID}} = c_0 \ket{\text{0}} + c_\text{D} \ket{\text{D}}} + \end{equation} + \textbf{NB:} CID is the \alert{cheapest CI method} + \item When $\ket{\text{S}}$ and $\ket{\text{D}}$ are taken into account: \textbf{CISD} + \begin{equation} + \purple{\ket{\Phi_\text{CISD}} = c_0 \ket{\text{0}} + c_\text{S} \ket{\text{S}} + c_\text{D} \ket{\text{D}} } + \end{equation} + \textbf{NB:} CISD is the \purple{most commonly-used} CI method + \item When $\ket{\text{S}}$, $\ket{\text{D}}$ and $\ket{\text{T}}$ (\orange{triples}) are taken into account: \textbf{CISDT} + \begin{equation} + \orange{\ket{\Phi_\text{CISDT}} = c_0 \ket{\text{0}} + c_\text{S} \ket{\text{S}} + c_\text{D} \ket{\text{D}} + c_\text{T} \ket{\text{T}}} + \end{equation} + \item \textbf{CISDTQ}, etc. + \end{itemize} +\end{frame} + +\begin{frame}{Full CI} + \begin{itemize} + \item When all possible excitations are taken into account, + \alert{this is called a Full CI calculation} (\textbf{FCI}) + \begin{equation} + \alert{\ket{\Phi_\text{FCI}} + = c_0 \ket{\text{0}} + + c_\text{S} \ket{\text{S}} + + c_\text{D} \ket{\text{D}} + + c_\text{T} \ket{\text{T}} + + c_\text{Q} \ket{\text{Q}} + + \ldots} + \end{equation} + \item FCI gives the \violet{exact solution of the Schr\"odinger equation within a given basis} + \bigskip + \item FCI is becoming more and more fashionable these days (e.g. \orange{FCIQMC and SCI methods}) + \bigskip + \item \blue{So, why do we care about other methods?} + \bigskip + \item \alert{Because FCI is super computationally expensive!} + \bigskip + \end{itemize} +\end{frame} + + +\begin{frame}{Size of CI Matrix} + \violet{\textit{``Assume we have 10 electrons in 38 spin MOs: 10 are occupied and 28 are empty''}} + \bigskip + \begin{columns} + \begin{column}{0.65\textwidth} + \begin{itemize} + \item There is $C_{10}^k$ possible ways of selecting $k$ electrons out of the 10 occupied orbitals + $$ C_{n}^k = \frac{n!}{k!(n-k)!} $$ + \item There is $C_{28}^k$ ways of distributing them out in the 28 virtual orbitals + \item For a given excitation level $k$, \alert{there is $C_{10}^k C_{28}^k$ excited determinants} + \item \violet{The total number of possible excited determinant} is + $$ \sum_{k=0}^{10}C_{10}^k C_{28}^k = C_{38}^{10} = 472,733,756$$ + \item \alert{This is a lot...} + \end{itemize} + \end{column} + \begin{column}{0.35\textwidth} + \small + \orange{For $n = 10$ and $N = 38$:} + \\ + \bigskip + \begin{tabular}{cr} + \hline \hline + $k$ & Num. of excitations \\ + \hline + 0 & 1 \\ + 1 & 280 \\ + 2 & 17,010 \\ + 3 & 393,120 \\ + 4 & 4,299,750 \\ + 5 & 24,766,560 \\ + 6 & 79,115,400 \\ + 7 & 142,084,800 \\ + 8 & 139,864,725 \\ + 9 & 69,069,000 \\ + 10 & 13,123,110 \\ + \hline + Tot. & 472,733,756 \\ + \hline \hline + \end{tabular} + \end{column} + \end{columns} +\end{frame} +\begin{frame}{Pople diagram} + \centering + \includegraphics[width=0.7\textwidth]{fig/pople_CI} +\end{frame} + +\begin{frame}{CI Lagrangian} + The CI Lagrangian is + \begin{equation} + L = \mel{\Phi_\text{CI}}{\hH}{\Phi_\text{CI}} - \la \qty( \braket{\Phi_\text{CI}}{\Phi_\text{CI}} - 1) + \qq{with} + \ket{\Phi_\text{CI}} = \sum_I c_I \ket{I} + \end{equation} + with + \begin{gather} + \mel{\Phi_\text{CI}}{\hH}{\Phi_\text{CI}} + = \sum_{IJ} c_I c_J \mel{I}{\hH}{J} + = \sum_I c_I^2 \underbrace{\mel{I}{\hH}{I}}_{H_{II}} + \sum_{I \neq J} \underbrace{\mel{I}{\hH}{J}}_{H_{IJ}} + \\ + \braket{\Phi_\text{CI}}{\Phi_\text{CI}} + = \sum_{IJ} c_I c_J \braket{I}{J} + = \sum_{I} c_I^2 + \end{gather} + Following the variational procedure, we get + \begin{equation} + \pdv{L}{c_I} = 2 \sum_J c_J H_{IJ} - 2 \la c_I = 0 + \qq{or} + \boxed{\qty( H_{II} - \la) c_I + \sum_{J \neq I} H_{IJ} c_J = 0} + \end{equation} +\end{frame} + +\begin{frame}{CI secular equations} + \begin{equation} + \begin{pmatrix} + H_{00} - E & H_{01} & \hdots & H_{0J} & \hdots \\ + H_{10} & H_{11} - E & \hdots & H_{1J} & \hdots \\ + \vdots & \vdots & \ddots & \vdots & \hdots \\ + H_{J0} & \vdots & \hdots & H_{JJ} - E & \hdots \\ + \vdots & \vdots & \hdots & \vdots & \ddots \\ + \end{pmatrix} + \begin{pmatrix} + c_{0} \\ + c_{1} \\ + \vdots \\ + c_{J} \\ + \vdots \\ + \end{pmatrix} + = + \begin{pmatrix} + 0 \\ + 0 \\ + \vdots \\ + 0 \\ + \vdots \\ + \end{pmatrix} + \qq{or} + \boxed{\bH \cdot \bc = E \bc} + \end{equation} +\end{frame} + +\begin{frame}{The FCI matrix: \alert{before pruning}} + \begin{equation} + \boxed{ + \ket{\Phi_0} + = c_0 \ket{\text{HF}} + + c_\text{S} \ket{\text{S}} + + c_\text{D} \ket{\text{D}} + + c_\text{T} \ket{\text{T}} + + c_\text{Q} \ket{\text{Q}} + + \ldots + } + \end{equation} + \bigskip + \begin{equation} + \bH = + \begin{array}{ccccccc} + & | \text{HF} \rangle & | \text{S} \rangle & | \text{D} \rangle & | \text{T} \rangle & | \text{Q} \rangle & \cdots \\ + \langle \text{HF} | & \langle \text{HF} | \hH | \text{HF} \rangle & \langle \text{HF} | \hH | \text{S} \rangle & \langle \text{HF} | \hH | \text{D} \rangle & \langle \text{HF} | \hH | \text{T} \rangle & \langle \text{HF} | \hH | \text{Q} \rangle & \cdots \\ + \langle \text{S} | & \langle \text{S} | \hH | \text{HF} \rangle & \langle \text{S} | \hH | \text{S} \rangle & \langle \text{S} | \hH | \text{D} \rangle & \langle \text{S} | \hH | \text{T} \rangle & \langle \text{S} | \hH | \text{Q} \rangle & \cdots \\ + \langle \text{D} | & \langle \text{D} | \hH |\text{HF} \rangle & \langle \text{D} | \hH | \text{S} \rangle & \langle \text{D} | \hH | \text{D} \rangle & \langle \text{D} | \hH| \text{T} \rangle & \langle \text{D} | \hH | \text{Q} \rangle & \cdots \\ + \langle \text{T} | & \langle \text{T} | \hH |\text{HF} \rangle & \langle \text{T} | \hH | \text{S} \rangle & \langle \text{T} | \hH | \text{D} \rangle & \langle \text{T} |\hH | \text{T} \rangle & \langle \text{T} | \hH | \text{Q} \rangle & \cdots \\ + \langle \text{Q} | & \langle \text{Q} | \hH | \text{HF} \rangle & \langle \text{Q} | \hH | \text{S} \rangle & \langle \text{Q} | \hH | \text{D} \rangle & \langle \text{Q} | \hH | \text{T} \rangle & \langle \text{Q} | \hH | \text{Q} \rangle & \cdots \\ + \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots \\ + \end{array} + \end{equation} +\end{frame} +\begin{frame}{The FCI matrix: \green{after pruning}} + \begin{equation} + \boxed{ + \ket{\Phi_0} + = c_0 \ket{\text{HF}} + + c_\text{S} \ket{\text{S}} + + c_\text{D} \ket{\text{D}} + + c_\text{T} \ket{\text{T}} + + c_\text{Q} \ket{\text{Q}} + + \ldots + } + \end{equation} + \bigskip + \begin{equation} + \bH = + \begin{array}{ccccccc} + & | \text{HF} \rangle & | \text{S} \rangle & | \text{D} \rangle & | \text{T} \rangle & | \text{Q} \rangle & \cdots \\ + \langle \text{HF} | & \langle \text{HF} | \hH | \text{HF} \rangle & 0 & \langle \text{HF} | \hH | \text{D} \rangle & 0 & 0 & \cdots \\ + \langle \text{S} | & 0 & \langle \text{S} | \hH | \text{S} \rangle & \langle \text{S} | \hH | \text{D} \rangle & \langle \text{S} | \hH | \text{T} \rangle & 0 & \cdots \\ + \langle \text{D} | & \langle \text{D} | \hH | \text{HF} \rangle & \langle \text{D} | \hH | \text{S} \rangle & \langle \text{D} | \hH | \text{D} \rangle & \langle \text{D} | \hH | \text{T} \rangle & \langle \text{D} | \hH | \text{Q} \rangle & \cdots \\ + \langle \text{T} | & 0 & \langle \text{T} | \hH | \text{S} \rangle & \langle \text{T} | \hH | \text{D} \rangle & \langle \text{T} | \hH | \text{T} \rangle & \langle \text{T} | \hH | \text{Q} \rangle & \cdots \\ + \langle \text{Q} | & 0 & 0 & \langle \text{Q} | \hH | \text{D} \rangle & \langle \text{Q} | \hH | \text{T} \rangle & \langle \text{Q} | \hH | \text{Q} \rangle & \cdots \\ + \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots \\ + \end{array} + \end{equation} +\end{frame} + +\begin{frame}{Rules \& Observations} + \begin{enumerate} + \item No coupling between HF ground state $\ket{ \text{HF} }$ and single excitations $ \ket{ \text{S} }$\\ + \violet{$\Rightarrow$ Brillouin's theorem} + \begin{equation} + \violet{\mel{ \text{HF} }{ \hH }{ \text{S} } = 0} + \end{equation} + \item No coupling between $\ket{ \text{HF} }$ and triples $\ket{ \text{T} }$ , quadruples $ \ket{ \text{Q} }$ , etc. \\ + \alert{$\Rightarrow$ Slater-Condon rules} + \begin{gather} + \alert{\mel{ \text{HF} }{ \hH }{ \text{T} } = \mel{ \text{HF} }{ \hH }{ \text{Q} } = \ldots = 0} + \\ + \alert{\mel{ \text{S} }{ \hH }{ \text{Q} } = \ldots = 0} + \end{gather} + \item $ \ket{ \text{S} }$ have small effect but mix indirectly with $\ket{ \text{D} }$\\ + \orange{$\Rightarrow$ CID $\neq$ CISD} + \begin{equation} + \orange{\mel{ \text{HF} }{ \hH }{ \text{S} } = 0 \qq{but} \mel{ \text{S} }{ \hH }{ \text{D} } \neq 0} + \end{equation} + \item $ \ket{ \text{D} }$ have large effect and $ \ket{ \text{Q} }$ more important than $ \ket{ \text{T} }$\\ + \blue{$\Rightarrow$ CID gives most of the correlation energy} + \begin{equation} + \blue{\mel{ \text{HF} }{ \hH }{ \text{D} } \gg \mel{ \text{HF} }{ \hH }{ \text{Q} } \gg \mel{ \text{HF} }{ \hH }{ \text{T} }} + \end{equation} + \item \purple{Of course, this matrix is never explicitly built in practice (Davidson algorithm)...} + \end{enumerate} +\end{frame} + +\begin{frame}{Slater-Condon rules: One-electron operators} + \begin{equation} + \boxed{\cO_1 = \sum_i^N h(i)} + \end{equation} + \begin{block}{\green{Case 1 = differ by zero spinorbital}: $\ket{K} = \ket{\cdots ij \cdots}$} + \begin{equation} + \mel{K}{\cO_1}{K} = \sum_i^N \mel{i}{h}{i} + \end{equation} + \end{block} + \begin{block}{\orange{Case 2 = differ by one spinorbital}: $\ket{K} = \ket{\cdots ij \cdots}$ and $\ket{L} = \ket{\cdots aj \cdots}$} + \begin{equation} + \mel{K}{\cO_1}{L} = \mel{i}{h}{a} + \end{equation} + \end{block} + \begin{block}{\red{Case 3 = differ by two spinorbitals}: $\ket{K} = \ket{\cdots ij \cdots}$ and $\ket{L} = \ket{\cdots ab \cdots}$} + \begin{equation} + \mel{K}{\cO_1}{L} = 0 + \end{equation} + \end{block} +\end{frame} + +\begin{frame}{Slater-Condon rules: Two-electron operators} + \begin{equation} + \boxed{\cO_2 = \sum_{i \tau$} + \State Form linear array $\green{u_{ij}^{ab}}$ + \State Compute intermediate arrays $\mel*{kl}{X_1}{ij}$, $\mel*{b}{X_2}{c}$, $\mel*{k}{X_3}{j}$, and $\mel*{il}{X_4}{ad}$. + \State Form quadratic array $\orange{v_{ij}^{ab}}$ + \State Compute residues: $\blue{r_{ij}^{ab}} = \mel*{ij}{}{ab} + \Delta_{ij}^{ab} \amp{ij}{ab} + \green{u_{ij}^{ab}} + \orange{v_{ij}^{ab}}$ + \State Update amplitudes: $\red{t_{ij}^{ab}} \leftarrow \red{t_{ij}^{ab}} - \blue{r_{ij}^{ab}}/\Delta_{ij}^{ab}$ + \EndWhile + \State Compute CCD energy: $\ECCD = \EHF + \frac{1}{4} \sum_{ij} \sum_{ab} \red{t_{ij}^{ab}} \mel*{ij}{}{ab}$ + \EndProcedure + \end{algorithmic} + \end{block} + \end{column} + \begin{column}{0.3\textwidth} + \centering + \includegraphics[width=\textwidth]{fig/Diagrams-CCD} + \end{column} + \end{columns} +\end{frame} + +\begin{frame}{Illustration for the \ce{Be} atom} + \begin{block}{Correlation energy of \ce{Be} in a 4s2p basis set} + \bigskip + \begin{table} + \small + \begin{tabular}{llcclcclcc} + \hline \hline + Scaling & Level & $\Delta E_c$ & \% & Level & $\Delta E_c$ & \% & Level & $\Delta E_c$ & \% \\ + \hline + $N^5$ &MP2 & 0.053174 & 67.85 & & & & & & \\ + $N^6$ &MP3 & 0.067949 & 86.70 & CISD & 0.075277 & 96.05 & CCSD & 0.078176 & 99.75 \\ + $N^7$ &MP4 & 0.074121 & 94.58 & & & & CCSD(T) & 0.078361 & 99.99 \\ + $N^8$ &MP5 & 0.076918 & 98.15 & CISDT & 0.075465 & 96.29 & CCSDT & 0.078364 & 99.99 \\ + $N^9$ &MP6 & 0.078090 & 99.64 & & & & & & \\ + $N^{10}$ &MP7 & 0.078493 & 100.15 & CISDTQ & 0.078372 & 100 & CCSDTQ & 0.078372 & 100 \\ + \hline \hline + \end{tabular} + \end{table} + \bigskip + \alert{As a rule of thumb:}\\ + HF $\ll$ MP2 $<$ CISD $<$ MP4(SDQ) $\sim$ CCSD $<$ MP4 $<$ CCSD(T) + \end{block} +\end{frame} + + +%%% FINAL SLIDE %%% +%----------------------------------------------------- +\begin{frame}{Good books} + \begin{columns} + \begin{column}{0.7\textwidth} + \begin{itemize} + \item Introduction to Computational Chemistry (Jensen) + \\ + \vspace{1cm} + \item Essentials of Computational Chemistry (Cramer) + \\ + \vspace{1cm} + \item Modern Quantum Chemistry (Szabo \& Ostlund) + \\ + \vspace{1cm} + \item Molecular Electronic Structure Theory (Helgaker, Jorgensen \& Olsen) + \\ + \vspace{1cm} + \end{itemize} + \end{column} + \begin{column}{0.3\textwidth} + \centering + \includegraphics[height=0.3\textwidth]{fig/Jensen} + \\ + \bigskip + \includegraphics[height=0.3\textwidth]{fig/Cramer} + \\ + \bigskip + \includegraphics[height=0.3\textwidth]{fig/Szabo} + \\ + \bigskip + \includegraphics[height=0.3\textwidth]{fig/Helgaker} + \end{column} + \end{columns} +\end{frame} + + +\end{document} diff --git a/2024/postHF/fig/Be.pdf b/2024/postHF/fig/Be.pdf new file mode 100644 index 0000000..274260a Binary files /dev/null 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