minor corrections and additions
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@ -228,13 +228,18 @@ decoration={snake,
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\begin{equation}
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\begin{split}
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\braket{\ba_1\ba_2}{\bb_1\bb_2}
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& \equiv \mel{\ba_1\ba_2}{\alert{r_{12}^{-1}}}{\bb_1\bb_2}
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& \equiv \mel{\ba_1\ba_2}{\alert{r_{12}^{-1}}}{\bb_1\bb_2}
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\\
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& = \iint \phi_{\ba_1}^{\bA_1}(\br_1) \phi_{\ba_2}^{\bA_2}(\br_2) \,\alert{\frac{1}{r_{12}}} \,
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\phi_{\bb_1}^{\bB_1}(\br_1) \phi_{\bb_2}^{\bB_2}(\br_2) d\br_1 d\br_2
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\end{split}
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\end{equation}
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\end{block}
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\end{column}
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\begin{column}{0.3\textwidth}
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\includegraphics[width=\textwidth]{fig/SBG}
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\end{column}
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\end{columns}
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%
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\begin{block}{Gaussian-type orbital (GTO)}
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\small
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@ -246,17 +251,12 @@ decoration={snake,
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\text{\blue{Primitive} GTO} & = \sket{\ba}
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= (x-A_x)^{a_x} (y-A_y)^{a_y} (z-A_z)^{a_z} e^{-\alpha \abs{ \br -\bA }^2}
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\end{align*}
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\begin{itemize}
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\item \textbf{\purple{Exponent:}} $\alpha$
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\item \textbf{\purple{Center:}} $\bA = (A_x, A_y, A_z)$
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\item \textbf{\purple{Angular momentum:}} $\ba = (a_x, a_y, a_z)$ and total angular momentum $a=a_x + a_y + a_z$
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\end{itemize}
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\end{block}
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\end{column}
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\begin{column}{0.3\textwidth}
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\includegraphics[width=\textwidth]{fig/SBG}
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\end{column}
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\end{columns}
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\begin{itemize}
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\item \textbf{\purple{Exponent:}} $\alpha$
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\item \textbf{\purple{Center:}} $\bA = (A_x, A_y, A_z)$
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\item \textbf{\purple{Angular momentum:}} $\ba = (a_x, a_y, a_z)$ and total angular momentum $a=a_x + a_y + a_z$
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\end{itemize}
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%
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\end{frame}
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@ -268,7 +268,7 @@ decoration={snake,
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\item Same center $\bA$
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\item Same angular momentum $\ba$
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\item Different exponent $\violet{\alpha_k}$
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\item Contraction coefficient $\blue{D_k}$
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\item Contraction coefficient $\blue{D_k}$ and degree $K$
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\end{itemize}
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\begin{equation}
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\underbrace{\braket{\ba_1\ba_2}{\bb_1\bb_2}}_{\text{\green{contracted ERI}}}
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@ -339,12 +339,22 @@ decoration={snake,
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%
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\begin{frame}{Upper bounds for ERIs}
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\begin{block}{A ``good'' upper bound must be}
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\begin{itemize}
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\item tight (i.e., a good estimate)
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\item simple (i.e, cheap to compute)
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\end{itemize}
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\end{block}
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\begin{columns}
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\begin{column}{0.35\textwidth}
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\begin{block}{A ``good'' upper bound must be}
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\begin{itemize}
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\item tight (i.e., a good estimate)
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\item simple (i.e, cheap to compute)
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\end{itemize}
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\end{block}
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\end{column}
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\begin{column}{0.65\textwidth}
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\begin{equation}
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\boxed{\abs{(\bm{\red{a}} \bm{\blue{b}}|\bm{\orange{c}} \bm{\green{d}})} \le B}
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\end{equation}
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\end{column}
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\end{columns}
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\bigskip
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\begin{block}{Cauchy-Schwartz bound}
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\begin{equation}
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\abs{(\bm{\red{a}} \bm{\blue{b}}|\bm{\orange{c}} \bm{\green{d}})}
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@ -374,12 +384,16 @@ decoration={snake,
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\begin{frame}{Asymptotic scaling of two-electron integrals}
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\begin{block}{Number of significant two-electron integrals for polyenes}
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\begin{block}{Number of significant two-electron integrals}
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\begin{equation}
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(\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}})
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(\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}})
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\end{equation}
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\end{block}
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\bigskip
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\begin{block}{Long-range vs short-range operators}
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\begin{equation}
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N_\text{sig} = c\,N^{\alpha}
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\end{equation}
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\bigskip
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$$N_\text{sig} = c\,N^{\alpha}$$
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\center
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\begin{tabular}{lcrccrc}
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\hline
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@ -425,42 +439,48 @@ decoration={snake,
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\end{frame}
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\begin{frame}{Late-contraction path algorithm (Head-Gordon-Pople \& PRISM inspired)}
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\begin{tikzpicture}
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\begin{scope}[
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very thick,
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node distance=1.5cm,on grid,>=stealth',
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boxSP/.style={rectangle,draw,fill=purple!40},
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box0m/.style={rectangle,draw,fill=red!40},
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boxCm/.style={rectangle,draw,fill=gray!40},
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boxA/.style={rectangle,draw,fill=red!40},
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boxAA/.style={rectangle,draw,fill=red!40},
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boxAAA/.style={rectangle,draw,fill=red!40},
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boxC/.style={rectangle,draw,fill=gray!40},
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boxCC/.style={rectangle,draw,fill=gray!40},
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boxCCC/.style={rectangle,draw,fill=orange!40},
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boxCCCCCC/.style={rectangle,draw,fill=green!40},
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],
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\node [boxSP, align=center] (SP) {Shell-pair \\ data};
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\node [box0m, align=center] (0m) [right=of 1,xshift=1.25cm] {$\sbraket{00}{00}^{\bm{m}}$};
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\node [boxCm, align=center] (Cm) [right=of 0m,xshift=1.75cm] {$\braket{00}{00}^{\bm{m}}$};
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\node [boxA, align=center] (A) [below=of 0m] {$\sbraket{0 a_2}{00}^{\bm{m}}$};
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\node [boxC, align=center] (C) [right=of A,xshift=1.75cm] {$\braket{0 a_2}{00}^{\bm{m}}$};
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\node [boxAA, align=center] (AA) [below=of A] {$\sbraket{a_1 a_2}{00}$};
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\node [boxCC, align=center] (CC) [right=of AA,xshift=1.75cm] {$\braket{a_1 a_2}{00}$};
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\node [boxCCCCCC, align=center] (CCCC) [right=of CC,xshift=2cm] {$\braket{a_1 a_2}{b_1 b_2}$};
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\path
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(SP) edge[->] node[below,blue]{T$_0$} (0m)
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(0m) edge[->] node[left,orange]{T$_1$} node [right,red]{VRR$_1$} (A)
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(0m) edge[->,gray!70] (Cm)
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(A) edge[->] node[left,orange]{T$_2$} node [right,red]{VRR$_2$} (AA)
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(A) edge[->,gray!70] (C)
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(AA) edge[->] node [below,blue]{CC} (CC)
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(Cm) edge[->,gray!70] (C)
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(C) edge[->,gray!70] (CC)
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(CC) edge[->] node [above,orange]{T$_3$} node [below,red]{HRR} (CCCC)
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;
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\end{scope}
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\end{tikzpicture}
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\begin{tikzpicture}
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\begin{scope}[
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very thick,
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node distance=1.5cm,on grid,>=stealth',
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boxSP/.style={rectangle,draw,fill=purple!40},
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box0m/.style={rectangle,draw,fill=red!40},
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boxCm/.style={rectangle,draw,fill=gray!40},
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boxA/.style={rectangle,draw,fill=red!40},
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boxAA/.style={rectangle,draw,fill=red!40},
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boxAAA/.style={rectangle,draw,fill=red!40},
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boxC/.style={rectangle,draw,fill=gray!40},
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boxCC/.style={rectangle,draw,fill=gray!40},
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boxCCC/.style={rectangle,draw,fill=orange!40},
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boxCCCCCC/.style={rectangle,draw,fill=green!40},
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],
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\node [boxSP, align=center] (SP) {Shell-pair \\ data};
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\node [box0m, align=center] (0m) [right=of 1,xshift=1.25cm] {$\sbraket{00}{00}^{\bm{m}}$};
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\node [boxCm, align=center] (Cm) [right=of 0m,xshift=1.75cm] {$\braket{00}{00}^{\bm{m}}$};
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\node [boxA, align=center] (A) [below=of 0m] {$\sbraket{0 a_2}{00}^{\bm{m}}$};
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\node [boxC, align=center] (C) [right=of A,xshift=1.75cm] {$\braket{0 a_2}{00}^{\bm{m}}$};
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\node [boxAA, align=center] (AA) [below=of A] {$\sbraket{a_1 a_2}{00}$};
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\node [boxCC, align=center] (CC) [right=of AA,xshift=1.75cm] {$\braket{a_1 a_2}{00}$};
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\node [boxCCCCCC, align=center] (CCCC) [right=of CC,xshift=2cm] {$\braket{a_1 a_2}{b_1 b_2}$};
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\path
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(SP) edge[->] node[below,blue]{T$_0$} (0m)
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(0m) edge[->] node[left,orange]{T$_1$} node [right,red]{VRR$_1$} (A)
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(0m) edge[->,gray!70] (Cm)
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(A) edge[->] node[left,orange]{T$_2$} node [right,red]{VRR$_2$} (AA)
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(A) edge[->,gray!70] (C)
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(AA) edge[->] node [below,blue]{CC} (CC)
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(Cm) edge[->,gray!70] (C)
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(C) edge[->,gray!70] (CC)
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(CC) edge[->] node [above,orange]{T$_3$} node [below,red]{HRR} (CCCC)
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;
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\end{scope}
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\end{tikzpicture}
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\bigskip
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\begin{itemize}
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\item \red{HRR} = horizontal recurrence relation [Obara-Saika]
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\item \red{VRR} = vertical recurrence relation
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\item \blue{CC} = bra contraction
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\end{itemize}
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\end{frame}
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%\begin{frame}{Screening algorithm for two-electron integrals}
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@ -538,6 +558,8 @@ decoration={snake,
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\begin{block}{Density matrix (closed-shell system)}
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\begin{equation}
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P_{\red{\mu \nu}} = 2 \sum_{i}^\text{occ} C_{\red{\mu} i} C_{\red{\nu} i}
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\qqtext{or}
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\boxed{\bm{P} = \bm{C} \cdot \bm{C}^{\dag}}
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\end{equation}
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\end{block}
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\begin{block}{Fock matrix in the AO basis (closed-shell system)}
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@ -632,17 +654,17 @@ decoration={snake,
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\begin{block}{LDA exchange (in theory) = cf Julien's lectures}
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\begin{gather}
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K_{\mu\nu}^\text{LDA}
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= \int \phi_{\mu}(\br) \violet{v_\text{xc}}(\br) \phi_{\nu}(\br) d\br
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= \int \phi_{\mu}(\br) \violet{v_\text{x}^\text{LDA}}(\br) \phi_{\nu}(\br) d\br
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= \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}}}
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\\
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\blue{\rho}(\br) = \sum_{\mu \nu} \phi_{\mu}(\br) \blue{P_{\mu \nu}} \phi_{\nu}(\br)
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\end{gather}
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\end{block}
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\begin{block}{LDA exchange (in practice) = \alert{numerical integration via quadrature}}
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\begin{block}{LDA exchange (in practice) = \alert{numerical integration via quadrature} = $\int f(x) dx \approx \sum_k w_k f(x_k)$}
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\begin{gather}
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\underbrace{K_{\mu\nu}^\text{LDA}}_{\green{\order{N_\text{grid} N^2}}}
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\approx \sum_{k=1}^{\purple{N_\text{grid}}}
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\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})
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\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})
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= \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})
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\\
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\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})
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@ -880,6 +902,7 @@ decoration={snake,
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\mel{ij}{}{ab}^2 \exp[-(\purple{\epsilon_{i} + \epsilon_{j} - \epsilon_{a} - \epsilon_{b}}) \blue{t}] d\blue{t}
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\\
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& = \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
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\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
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\end{split}
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\end{equation}
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\begin{equation}
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@ -895,7 +918,7 @@ decoration={snake,
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%%% SLIDE 2 %%%
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\begin{frame}{Theory}
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\begin{frame}{Coupled-Cluster Theory}
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\begin{block}{A few random thoughts about coupled cluster (CC)}
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\begin{itemize}
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\bigskip
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