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Pierre-Francois Loos 2021-04-26 07:14:04 +02:00
parent 9b6e724e17
commit a443a7865b
1 changed files with 85 additions and 62 deletions
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2021/Lecture_1/ISTPC_Loos_1.tex
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@ -235,6 +235,11 @@ decoration={snake,
\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*}
\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{block}
\end{column}
\begin{column}{0.3\textwidth}
\includegraphics[width=\textwidth]{fig/SBG}
\end{column}
\end{columns}
%
\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{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
$$N_\text{sig} = c\,N^{\alpha}$$
\begin{block}{Long-range vs short-range operators}
\begin{equation}
N_\text{sig} = c\,N^{\alpha}
\end{equation}
\center
\begin{tabular}{lcrccrc}
\hline
@ -461,6 +475,12 @@ decoration={snake,
;
\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