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\begin{document}
$\mathrm{CH}_3\mathrm{NH}_2$
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$i$
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$0 \leq i < n$
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$2 (i-1) + 1$
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$2 i$
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$(i-1)\times 2 + 1$
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$i\times 2$
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$B$
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$\kappa_2$
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$C_{2v}$
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$C_1$
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$a$
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$b$
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$\bar{r}_a$
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$\bar{r}_b$
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$r$
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\[ r = \| \bar{r}_a - \bar{r}_b \| \]
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$c$
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$\bar{r}_c$
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$\theta$
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\[ \bar{u}_{ab} = \frac{\bar{r}_a - \bar{r}_b}{\| \bar{r}_a - \bar{r}_b \|}\]
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\[ \bar{u}_{cb} = \frac{\bar{r}_c - \bar{r}_b}{\| \bar{r}_c - \bar{r}_b \|}\]
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\[ \theta = \arccos ( \bar{u}_{ab} \cdot \bar{u}_{cb} ) \]
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$d$
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$\bar{r}_d$
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$\tau$
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\[ \bar{u}_{cd} = \frac{\bar{r}_c - \bar{r}_d}{\| \bar{r}_c - \bar{r}_b \|}\]
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\[ \bar{n}_{abc}= \frac{\bar{u}_{ab} \times \bar{u}_{cb}} {\| \bar{u}_{ab} \times \bar{u}_{cb} \|} \]
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\[ \bar{n}_{bcd}= \frac{\bar{u}_{cd} \times \bar{u}_{bc}} {\| \bar{u}_{cd} \times \bar{u}_{bc} \|} \]
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\[ s = \left\{ \begin{array}{ll} 1 & \mbox{if $(\bar{n}_{abc}\times\bar{n}_{bcd}) \cdot \bar{u}_{cb} > 0;$} \\ -1 & \mbox{otherwise} \end{array} \right. \]
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\[ \tau = s \arccos ( - \bar{n}_{abc} \cdot \bar{n}_{bcd} ) \]
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$\tau_s$
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\[ \bar{n}_{abc}= \frac{\bar{u}_{ab} \times \bar{u}_{cb}} {\| \bar{u}_{ab} \times \bar{u}_{cb} \|}\]
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\[ \bar{n}_{bcd}= \frac{\bar{u}_{cd} \times \bar{u}_{cb}} {\| \bar{u}_{cd} \times \bar{u}_{cb} \|}\]
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\[ s = \left\{ \begin{array}{ll} -1 & \mbox{if $(\bar{n}_{abc}\times\bar{n}_{bcd}) \cdot \bar{u}_{cb} > 0$} \\ 1 & \mbox{otherwise} \end{array} \right. \]
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\[ \tau_s = s \sqrt{\left(1-(\bar{u}_{ab} \cdot \bar{u}_{cb}\right)^2) \left(1-(\bar{u}_{cb} \cdot \bar{u}_{cd}\right)^2)} \arccos ( - \bar{n}_{abc} \cdot \bar{n}_{bcd} )\]
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$\bar{u}$
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$\bar{r}_a - \bar{r}_b$
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$\bar{r}_b - \bar{r}_c$
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$\theta_i$
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\[ \bar{u}_{cb} = \frac{\bar{r}_b - \bar{r}_c}{\| \bar{r}_c - \bar{r}_b \|}\]
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\[ \theta_i = \pi - \arccos ( \bar{u}_{ab} \cdot \bar{u} ) - \arccos ( \bar{u}_{cb} \cdot \bar{u} )\]
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$\theta_o$
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\[ \bar{n} = \frac{\bar{u} \times \bar{u}_{ab}} {\| \bar{u} \times \bar{u}_{ab} \|}\]
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\[ \theta_o = \pi - \arccos ( \bar{u}_{ab} \cdot \bar{n} ) - \arccos ( \bar{u}_{cb} \cdot \bar{n} )\]
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$n_0$
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$n_1$
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$n_2$
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$O$
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$X O X^T$
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$X$
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$x^d$
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$x$
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$O(n_\mathrm{basis}^5)$
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$n_\mathrm{socc}$
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$m$
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$n_\mathrm{socc} = m - 1$
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$n_\mathrm{docc}$
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$n_\mathrm{docc} = (c - n_\mathrm{socc})/2$
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$\Delta D$
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$D$
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$G$
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\[ \bar{x}' = f(\bar{x}) \]
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$m_{ij}$
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$a_i$
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$b_j$
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$T_d$
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$I_h$
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$n$
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\end{document}