From a443a7865b03f526edfbd32746ff1b3dfe4d0f0c Mon Sep 17 00:00:00 2001 From: Pierre-Francois Loos Date: Mon, 26 Apr 2021 07:14:04 +0200 Subject: [PATCH] minor corrections and additions --- 2021/Lecture_1/ISTPC_Loos_1.tex | 147 ++++++++++++++++++-------------- 1 file changed, 85 insertions(+), 62 deletions(-) diff --git a/2021/Lecture_1/ISTPC_Loos_1.tex b/2021/Lecture_1/ISTPC_Loos_1.tex index e2e37c9..869310b 100644 --- a/2021/Lecture_1/ISTPC_Loos_1.tex +++ b/2021/Lecture_1/ISTPC_Loos_1.tex @@ -228,13 +228,18 @@ decoration={snake, \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} + & \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} + \end{column} + \begin{column}{0.3\textwidth} + \includegraphics[width=\textwidth]{fig/SBG} + \end{column} + \end{columns} % \begin{block}{Gaussian-type orbital (GTO)} \small @@ -246,17 +251,12 @@ decoration={snake, \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} + \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} @@ -268,7 +268,7 @@ decoration={snake, \item Same center $\bA$ \item Same angular momentum $\ba$ \item Different exponent $\violet{\alpha_k}$ - \item Contraction coefficient $\blue{D_k}$ + \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}}} @@ -339,12 +339,22 @@ decoration={snake, % \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{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}})} @@ -374,12 +384,16 @@ decoration={snake, \begin{frame}{Asymptotic scaling of two-electron integrals} - \begin{block}{Number of significant two-electron integrals for polyenes} + \begin{block}{Number of significant two-electron integrals} \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}}) + (\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} - \bigskip - $$N_\text{sig} = c\,N^{\alpha}$$ \center \begin{tabular}{lcrccrc} \hline @@ -425,42 +439,48 @@ decoration={snake, \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} + \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} @@ -538,6 +558,8 @@ decoration={snake, \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)} @@ -632,17 +654,17 @@ decoration={snake, \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 + = \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}} + \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{xc}}(\underbrace{\red{\br_k}}_{\text{\red{roots}}}) \phi_{\nu}(\red{\br_k}) + \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}) @@ -880,6 +902,7 @@ decoration={snake, \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 + \stackrel{\text{\blue{quad.}}}{\approx} \frac{1}{4} \blue{\sum_{k=1}^{N_\text{grid}}} \blue{w_k} \sum_{ij}\sum_{ab} \mel{i(\blue{t_k})j(\blue{t_k})}{}{a(\blue{t_k})b(\blue{t_k})}^2 \end{split} \end{equation} \begin{equation} @@ -895,7 +918,7 @@ decoration={snake, %%% SLIDE 2 %%% -\begin{frame}{Theory} +\begin{frame}{Coupled-Cluster Theory} \begin{block}{A few random thoughts about coupled cluster (CC)} \begin{itemize} \bigskip