From 9b6e724e1789e324ea1cd3f833c849c2dc62422f Mon Sep 17 00:00:00 2001 From: Pierre-Francois Loos Date: Sat, 24 Apr 2021 21:23:53 +0200 Subject: [PATCH] lectures 1 2021 --- 2021/Lecture_1/ISTPC_Loos_1.tex | 1229 +++++++++++++++++++++++++++ 2021/Lecture_1/fig/Be.pdf | Bin 0 -> 146508 bytes 2021/Lecture_1/fig/C.png | Bin 0 -> 76571 bytes 2021/Lecture_1/fig/CNRS.png | Bin 0 -> 123758 bytes 2021/Lecture_1/fig/Cramer.jpg | Bin 0 -> 31143 bytes 2021/Lecture_1/fig/Diagrams-CCD.png | Bin 0 -> 140827 bytes 2021/Lecture_1/fig/ERC.jpg | Bin 0 -> 57629 bytes 2021/Lecture_1/fig/Helgaker.jpg | Bin 0 -> 103839 bytes 2021/Lecture_1/fig/Jensen.jpg | Bin 0 -> 30315 bytes 2021/Lecture_1/fig/LCPQ.pdf | Bin 0 -> 15994 bytes 2021/Lecture_1/fig/MP2.png | Bin 0 -> 136968 bytes 2021/Lecture_1/fig/SBG.png | Bin 0 -> 128554 bytes 2021/Lecture_1/fig/Szabo.jpg | Bin 0 -> 3172373 bytes 2021/Lecture_1/fig/UPS.pdf | Bin 0 -> 235750 bytes 2021/Lecture_1/fig/basis.png | Bin 0 -> 527918 bytes 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b/2021/Lecture_1/ISTPC_Loos_1.tex @@ -0,0 +1,1229 @@ +\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{Pittsburgh} +\usecolortheme{seahorse} +\usepackage{mathpazo,libertine} + +\usepackage{algorithmicx,algorithm,algpseudocode} +\algnewcommand\algorithmicassert{\texttt{assert}} +\algnewcommand\Assert[1]{\State \algorithmicassert(#1)} +%\algrenewcommand{\algorithmiccomment}[1]{$\triangleright$ #1} + +%\usepackage[version=4]{mhchem} +\usepackage{amsmath,amsfonts,amssymb,bm,microtype,graphicx,wrapfig,geometry,physics,eurosym,multirow,pgfgantt} + +\usepackage{hyperref} +\hypersetup{ + colorlinks=true, + linkcolor=cyan, + filecolor=magenta, + urlcolor=blue, + citecolor=purple +} + +% operators +\newcommand{\hI}{\Hat{1}} +\newcommand{\hH}{\Hat{H}} +\newcommand{\hT}[2]{\Hat{T}_{#1}^{#2}} +\newcommand{\bH}{\mathbold{H}} +\newcommand{\br}{\mathbold{r}} +\newcommand{\la}{\lambda} +\newcommand{\si}{\sigma} + +\newcommand{\cJ}{\mathcal{J}} +\newcommand{\cK}{\mathcal{K}} + +% wave functions +\newcommand{\PsiO}{\Psi_0} +\newcommand{\PsiHF}{\Psi_\text{RHF}} +\newcommand{\PsiFCI}{\Psi_\text{FCI}} +\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{\EVCC}{E_\text{VCC}} +\newcommand{\ECCD}{E_\text{CCD}} + +\newcommand{\nEl}{n} +\newcommand{\nBas}{N} + +\newcommand{\ba}{\bm{a}} +\newcommand{\bb}{\bm{b}} +\newcommand{\bA}{\bm{A}} +\newcommand{\bB}{\bm{B}} +\newcommand{\bo}{\bm{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]{ + \purple{Hartree-Fock and post-Hartree-Fock methods: \\ + Computational aspects} + } + + \author[PF Loos]{Pierre-Fran\c{c}ois LOOS} + \date{Online ISTPC 2021 school --- April 27th, 2021} + \institute[CNRS@LCPQ]{ + Laboratoire de Chimie et Physique Quantiques (UMR 5626),\\ + Universit\'e de Toulouse, CNRS, UPS, Toulouse, France. + } + \titlegraphic{ + \includegraphics[width=0.3\textwidth]{fig/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 Computing the 2th-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}{One- and two-electron integrals} + \begin{block}{One-electron integrals: overlap \& core Hamiltonian (Appendix A)} + \begin{equation} + S_{\mu\nu} = \int \phi_\mu(\orange{\br}) \phi_\nu(\orange{\br}) d\orange{\br} + \end{equation} + \begin{equation} + H_{\mu\nu} = \int \phi_\mu(\orange{\br}) \hH^\text{c}(\orange{\br}) \phi_\nu(\orange{\br}) d\orange{\br} + \end{equation} + \end{block} + \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} + \end{block} + % + \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*} + \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{block} + \end{column} + \begin{column}{0.3\textwidth} + \includegraphics[width=\textwidth]{fig/SBG} + \end{column} + \end{columns} + % +\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 $\blue{D_k}$ + \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) = \green{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} + \green{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 + \\ + & = \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} + \end{block} +\end{frame} +% + +\begin{frame}{Upper bounds for ERIs} + \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} + \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 for polyenes} + \begin{equation} + (\bm{\red{a}} \bm{\blue{b}}|\bm{\orange{c}} \bm{\green{d}}) = (\bm{\red{a}} \bm{\blue{b}}| \mathcal{O}_2 | \bm{\orange{c}} \bm{\green{d}}) + \end{equation} + \bigskip + $$N_\text{sig} = c\,N^{\alpha}$$ + \center + \begin{tabular}{lcrccrc} + \hline + \hline + Molecule & $N$ & \mc{2}{c}{\red{$\mathcal{O}_2 = r_{12}^{-1}$}} && \mc{2}{c}{\orange{$\mathcal{O}_2 = e^{-\la r_{12}^2}$ ($\la = 1$)}} \\ + \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} +\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} + \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) = a fancy way of writing 1} + \begin{equation} + \boxed{\sum_{\green{A}=1}^{\red{\infty}} |\green{A}) (\green{A}| = 1 + \qqtext{and, in practice, } + \sum_{\green{A}=1}^{\red{K}} |\green{A}) (\green{A}| \approx 1} + \end{equation} + \end{block} + \begin{block}{Computing the Coulomb matrix with the RI} + \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}) I_{\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 Julien's lectures} + \begin{gather} + K_{\mu\nu}^\text{LDA} + = \int \phi_{\mu}(\br) \violet{v_\text{xc}}(\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}} + \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{xc}}(\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}{MP2 correlation energy} + \begin{block}{MP2 is the simplest way of catching a good chunk of correlation:} + \begin{equation} + \begin{split} + \green{E_\text{c}^\text{(2)}} + &= \sum_{\red{ij}}^{\text{occ}}\sum_{\blue{ab}}^{\text{virt}} + \frac{ \braket{\red{ij}}{\blue{ab}} (2 \braket{\red{ij}}{\blue{ab}} - \braket{\red{ij}}{\blue{ba}})} + {\epsilon_{\red{i}} + \epsilon_{\red{j}} - \epsilon_{\blue{a}} - \epsilon_{\blue{b}}} + \\ + & = \underbrace{ + 2 \sum_{\red{ij}}\sum_{\blue{ab}} + \frac{ \braket{\red{ij}}{\blue{ab}}^2} + {\epsilon_{\red{i}} + \epsilon_{\red{j}} - \epsilon_{\blue{a}} - \epsilon_{\blue{b}}} + }_{\text{direct part}} + - \underbrace{ + \sum_{\red{ij}}\sum_{\blue{ab}} + \frac{ \braket{\red{ij}}{\blue{ab}} \braket{\red{ij}}{\blue{ba}}} + {\epsilon_{\red{i}} + \epsilon_{\red{j}} - \epsilon_{\blue{a}} - \epsilon_{\blue{b}}} + }_{\text{exchange part}} + \end{split} + \end{equation} + \centering + \includegraphics[width=0.5\textwidth]{fig/MP2} + \end{block} +\end{frame} + + +\begin{frame}{Computing the MP2 correlation energy} + \begin{block}{How much does it cost to compute the MP2 correlation energy?} + \begin{algorithmic} + \Procedure{MP2 correlation energy}{} + \State $\green{E_\text{c}^\text{(2)}} = 0$ + \For{$\red{i}=1,O$} + \For{$\red{j}=1,O$} + \For{$\blue{a}=1,V$} + \For{$\blue{b}=1,V$} + \State $\purple{\Delta} = \epsilon_{\red{i}} + \epsilon_{\red{j}} - \epsilon_{\blue{a}} - \epsilon_{\blue{b}}$ + \State $\green{E_\text{c}^\text{(2)}} + = \green{E_\text{c}^\text{(2)}} + + (2 \braket{\red{ij}}{\blue{ab}}^2 - \braket{\red{ij}}{\blue{ab}}\braket{\red{ij}}{\blue{ba}})/\purple{\Delta} $ + \EndFor + \EndFor + \EndFor + \EndFor + \EndProcedure + \Comment{\bf \red{$\order{N^4}$ because there are four loops!}} + \end{algorithmic} + \end{block} +\end{frame} +% + +%%% SLIDE X %%% +\begin{frame}{AO to MO transformation (Take 1)} + \begin{block}{The naive way...} + \begin{equation} + \underbrace{\green{(pq|rs)}}_{\text{\purple{\bf MO integrals}}} + = \sum_{\red{\mu\nu\la\si}} c_{\red{\mu} \green{p}} c_{\red{\nu} \green{q}} c_{\red{\la} \green{r}} c_{\red{\si} \green{s}} + \underbrace{\red{(\mu\nu|\la\si)}}_{\text{\purple{\bf AO integrals}}} + \end{equation} + \end{block} + \begin{algorithmic} + \scriptsize + \Procedure{AO-to-MO Transformation}{} + \For{$\red{p}=1,N$} + \For{$\blue{q}=1,N$} + \For{$\orange{r}=1,N$} + \For{$\violet{s}=1,N$} + \State $(\red{p}\blue{q}|\orange{r}\violet{s}) = 0$ + \Comment{Initialization of the array} + \For{$\red{\mu}=1,N$} + \For{$\blue{\nu}=1,N$} + \For{$\orange{\la}=1,N$} + \For{$\violet{\si}=1,N$} + \State + $(\red{p}\blue{q}|\orange{r}\violet{s}) = (\red{p}\blue{q}|\orange{r}\violet{s}) + + c_{\red{\mu p}} c_{\blue{\nu q}} c_{\orange{\la r}} c_{\violet{\si s}} + (\red{\mu}\blue{\nu}|\orange{\la}\violet{\si})$ + \Comment{Accumulation step} + \EndFor + \EndFor + \EndFor + \EndFor + \EndFor + \EndFor + \EndFor + \EndFor + \Comment{\bf \red{This is a $\order{N^8}$ algorithm! You won't do much quantum chemistry with this...}} + \EndProcedure + \end{algorithmic} +\end{frame} +% + +%%% SLIDE X %%% +\begin{frame}{AO to MO transformation (Take 2)} + \begin{block}{Semi-direct algorithm...} + \begin{equation} + (\red{p}\blue{q}|\orange{r}\violet{s}) + = + \underbrace{ + \sum_{\red{\mu p}} c_{\red{\mu p}} + \underbrace{ + \qty{ \sum_{\blue{\nu q}} c_{\blue{\nu q}} + \underbrace{ + \qty[ \sum_{\orange{\la r}} c_{\orange{\la r}} + \underbrace{ + \qty( \sum_{\violet{\si s}} c_{\violet{\si s}} + (\red{\mu}\blue{\nu}|\orange{\la}\violet{\si}) + ) + }_{\text{\violet{Step \#1}}} + ] + }_{\text{\orange{Step \#2}}} + } + }_{\text{\blue{Step \#3}}} + }_{\text{\red{Step \#4}}} + \end{equation} + \end{block} +\end{frame} + +%%% SLIDE X %%% +\begin{frame}{Semi-direct algorithm} + \begin{block}{Semi-direct algorithm... \violet{Step \#1}} + \begin{algorithmic} + \Procedure{Semi-Direct Algorithm (\violet{Step \#1})}{} + \State Allocate temporary array $I$ of size $N^4$ + \For{$\red{\mu}=1,N$} + \For{$\blue{\nu}=1,N$} + \For{$\orange{\la}=1,N$} + \For{$\violet{\si}=1,N$} + \For{$\violet{s}=1,N$} + \State + $I_{\red{\mu}\blue{\nu}\orange{\la}\violet{s}} = I_{\red{\mu}\blue{\nu}\orange{\la}\violet{s}} + + c_{\violet{\si s}} (\red{\mu}\blue{\nu}|\orange{\la}\violet{\si})$ + \EndFor + \EndFor + \EndFor + \EndFor + \EndFor + \Comment{\violet{Step \#1} costs $\order{N^5}$ and $\order{N^4}$ storage} + \EndProcedure + \end{algorithmic} + \end{block} +\end{frame} +% +\begin{frame}{Semi-direct algorithm} + \begin{block}{Semi-direct algorithm... \orange{Step \#2}} + \begin{algorithmic} + \Procedure{Semi-Direct Algorithm (\orange{Step \#2})}{} + \State Allocate temporary array $J$ of size $N^4$ + \For{$\red{\mu}=1,N$} + \For{$\blue{\nu}=1,N$} + \For{$\orange{\la}=1,N$} + \For{$\orange{r}=1,N$} + \For{$\violet{s}=1,N$} + \State + $J_{\red{\mu}\blue{\nu}\orange{r}\violet{s}} = J_{\red{\mu}\blue{\nu}\orange{r}\violet{s}} + + c_{\orange{\la r}} I_{\red{\mu}\blue{\nu}\orange{\la}\violet{s}}$ + \EndFor + \EndFor + \EndFor + \EndFor + \EndFor + \Comment{\orange{Step \#2} costs $\order{N^5}$ and $\order{N^4}$ storage} + \EndProcedure + \end{algorithmic} + \end{block} +\end{frame} +% +\begin{frame}{Semi-direct algorithm} + \begin{block}{Semi-direct algorithm... \blue{Step \#3}} + \begin{algorithmic} + \Procedure{Semi-Direct Algorithm (\blue{Step \#3})}{} + \For{$\red{\mu}=1,N$} + \For{$\blue{\nu}=1,N$} + \For{$\blue{q}=1,N$} + \For{$\orange{r}=1,N$} + \For{$\violet{s}=1,N$} + \State + $I_{\red{\mu}\blue{q}\orange{r}\violet{s}} = I_{\red{\mu}\blue{q}\orange{r}\violet{s}} + + c_{\blue{\nu q}} J_{\red{\mu}\blue{\nu}\orange{r}\violet{s}}$ + \EndFor + \EndFor + \EndFor + \EndFor + \EndFor + \Comment{\blue{Step \#3} costs $\order{N^5}$ and no new storage} + \EndProcedure + \end{algorithmic} + \end{block} +\end{frame} +% +\begin{frame}{Semi-direct algorithm} + \begin{block}{Semi-direct algorithm... \red{Step \#4}} + \begin{algorithmic} + \Procedure{Semi-Direct Algorithm (\red{Step \#4})}{} + \For{$\red{\mu}=1,N$} + \For{$\red{p}=1,N$} + \For{$\blue{q}=1,N$} + \For{$\orange{r}=1,N$} + \For{$\violet{s}=1,N$} + \State + $(\red{p}\blue{q}|\orange{r}\violet{s}) = (\red{p}\blue{q}|\orange{r}\violet{s}) + + c_{\red{\mu p}} I_{\red{\mu}\blue{q}\orange{r}\violet{s}}$ + \EndFor + \EndFor + \EndFor + \EndFor + \EndFor + \Comment{\red{Step \#4} costs $\order{N^5}$ and no new storage} + \EndProcedure + \end{algorithmic} + \end{block} +\end{frame} + +\begin{frame}{Laplace transform} + \begin{block}{Alml{\"o}f's trick} + \begin{equation} + \boxed{\frac{1}{\purple{\Delta}} = \blue{\int_0^\infty} \exp(-\purple{\Delta} \blue{t}) d\blue{t}} + \end{equation} + \begin{equation} + \begin{split} + \green{E_\text{c}^\text{(2)}} + & = \frac{1}{4} \sum_{ij}\sum_{ab} + \frac{\mel{ij}{}{ab}^2} + {\purple{\epsilon_{i} + \epsilon_{j} - \epsilon_{a} - \epsilon_{b}}} + \\ + & = \frac{1}{4} \blue{\int_0^{\infty}} \sum_{ij}\sum_{ab} + \mel{ij}{}{ab}^2 \exp[-(\purple{\epsilon_{i} + \epsilon_{j} - \epsilon_{a} - \epsilon_{b}}) \blue{t}] d\blue{t} + \\ + & = \frac{1}{4} \blue{\int_0^{\infty}} \sum_{ij}\sum_{ab} \mel{i(\blue{t})j(\blue{t})}{}{a(\blue{t})b(\blue{t})}^2 + \end{split} + \end{equation} + \begin{equation} + \ket*{p} \equiv \ket*{\varphi_p(0)} + \qqtext{and} + \ket*{p(\blue{t})} \equiv \ket*{\varphi_p(\blue{t})} = \ket*{\varphi_p(0) e^{\pm\frac{1}{2} \purple{\epsilon_p} \blue{t}}} + \end{equation} + \\ + \bigskip + At this stage, one can play more games (e.g., localized orbitals, RI, stochastic sampling, quadrature, etc) + \end{block} +\end{frame} + + +%%% SLIDE 2 %%% +\begin{frame}{Theory} + \begin{block}{A few random thoughts about coupled cluster (CC)} + \begin{itemize} + \bigskip + \item CC theory comes from \alert{nuclear physics} (Thank you Physics!) + \bigskip + \item The idea behind CC is to include \alert{all corrections} of a given type to \alert{infinite order} + \bigskip + \item The CC wave function is an \alert{exponential \textit{ansatz}} + \bigskip + \item The CC energy is \alert{size-extensive}, but \alert{non-variational} + \bigskip + \item CC is considered as the \alert{gold standard} for weakly correlated systems + \end{itemize} + \end{block} +\end{frame} + +%%% SLIDE 3 %%% +\begin{frame}{Theory} + \begin{itemize} + \item CC wave function + \begin{equation} + \PsiCC = \alert{e^{\hT{}{}}} \PsiO% \qquad \text{where $\PsiO$ is a reference wave function} + \end{equation} + \item Excitation operator + \begin{equation} + \hT{}{} = \hT{1}{} + \hT{2}{} + \ldots + \hT{\nEl}{}% \qquad \text{where $\nEl$ is the number of electrons} + \end{equation} + \item Exponential \textit{ansatz} + \begin{equation} + \begin{split} + e^{\hT{}{}} + & = \hI + \hT{}{} + \frac{1}{2!} \hT{}{2} + \frac{1}{3!} \hT{}{3} + \ldots + \\ + & = \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{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}}} ) + + \ldots + \end{split} + \end{equation} + \end{itemize} +\end{frame} + +%%% SLIDE 4 %%% +\begin{frame}{Excitation operators} + \begin{itemize} + \item Singles + \begin{equation} + \hT{1}{} \PsiO = \sum_{i} \sum_{a} \underbrace{\amp{i}{a}}_{\text{\alert{amplitudes}}} \Det{i}{a} + \end{equation} + \item Doubles + \begin{equation} + \hT{2}{} \PsiO = \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} + +%%%% FINAL SLIDE %%% +%\begin{frame}{Further reading for CC methods} +% \begin{itemize} +% \item Pople et al. IJQC 14 (1978) 545 +% \bigskip +% \item Scuseria et al, JCP 86 (1987) 2881 +% \bigskip +% \item Jensen, \textit{``Introduction to Computational Chemistry''} (Wiley) +% \bigskip +% \item Crawford \& Schaefer, \textit{``An Introduction to Coupled Cluster Theory for Computational Chemists''}, Reviews in Computational Chemistry, Vol.~14 (2000) +% \bigskip +% \item Shavitt \& Bartlett, \textit{``Many-Body Methods in Chemistry and Physics: MBPT and Coupled-Cluster Theory''} (Cambridge University Press) +% \end{itemize} +%\end{frame} + +%%% FINAL SLIDE %%% +%----------------------------------------------------- +\section{Books} +%----------------------------------------------------- +\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/2021/Lecture_1/fig/Be.pdf b/2021/Lecture_1/fig/Be.pdf new file mode 100644 index 0000000000000000000000000000000000000000..274260a5bd8ee0d206f9047f20940942889e0f18 GIT binary patch literal 146508 zcmeEu^;eW_*Y+)lC{!%o%lpflc?ao(Z9lL$ zDBe)m8d_lS^HZ?Mnb?>*nNe^8msBWNB+RXxOdNn$tM^VOVkSnm#wHYkf>@4D4kqud zv0U#ksNGk4tu}}!#U!J^^qNk-SNV1KU94xsNl6UP{>dm0#7I!+S06m-cP0KB_>@a@xnzjEGwcKeR)zkZUMo1F;-i_&}3-~T$8*f>$J|M__; zCXTkw4n`)96kLBUh}zmX0iQck+%5_D{qiQp=I=#pT`9nCfD5dwoZJ-b9B*~8fHnO4 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+\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{Pittsburgh} +\usecolortheme{seahorse} +\usepackage{mathpazo,libertine} + +\usepackage{algorithmicx,algorithm,algpseudocode} +\algnewcommand\algorithmicassert{\texttt{assert}} +\algnewcommand\Assert[1]{\State \algorithmicassert(#1)} +%\algrenewcommand{\algorithmiccomment}[1]{$\triangleright$ #1} + +%\usepackage[version=4]{mhchem} +\usepackage{amsmath,amsfonts,amssymb,bm,microtype,graphicx,wrapfig,geometry,physics,eurosym,multirow,pgfgantt} + +\usepackage{hyperref} +\hypersetup{ + colorlinks=true, + linkcolor=cyan, + filecolor=magenta, + urlcolor=blue, + citecolor=purple +} + +% operators +\newcommand{\hI}{\Hat{1}} +\newcommand{\hH}{\Hat{H}} +\newcommand{\hT}[2]{\Hat{T}_{#1}^{#2}} +\newcommand{\bH}{\mathbold{H}} +\newcommand{\br}{\mathbold{r}} +\newcommand{\la}{\lambda} +\newcommand{\si}{\sigma} + +\newcommand{\cJ}{\mathcal{J}} +\newcommand{\cK}{\mathcal{K}} + +% wave functions +\newcommand{\PsiO}{\Psi_0} +\newcommand{\PsiHF}{\Psi_\text{RHF}} +\newcommand{\PsiFCI}{\Psi_\text{FCI}} +\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{\EVCC}{E_\text{VCC}} +\newcommand{\ECCD}{E_\text{CCD}} + +\newcommand{\nEl}{n} +\newcommand{\nBas}{N} + +\newcommand{\ba}{\bm{a}} +\newcommand{\bb}{\bm{b}} +\newcommand{\bA}{\bm{A}} +\newcommand{\bB}{\bm{B}} +\newcommand{\bo}{\bm{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[$GW$/BSE methods in chemistry]{ + \purple{$GW$/BSE methods in chemistry: \\ + Computational aspects} + } + \author[PF Loos]{Pierre-Fran\c{c}ois LOOS} + \date{Online ISTPC 2021 school --- April 27th, 2021} + \institute[CNRS@LCPQ]{ + Laboratoire de Chimie et Physique Quantiques (UMR 5626),\\ + Universit\'e de Toulouse, CNRS, UPS, Toulouse, France. + } + \titlegraphic{ + \includegraphics[width=0.4\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} + +%%% SLIDE 1 %%% +\begin{frame} + \titlepage +\end{frame} +% + +%%% SLIDE 2 %%% +\begin{frame}{Today's program} + \begin{itemize} + \item + \end{itemize} +\end{frame} +% + + +%%% FINAL SLIDE %%% +%----------------------------------------------------- +\section{Books} +%----------------------------------------------------- +\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/2021/Lecture_2/fig/CNRS.png 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