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Angle

Angle is defined by points $i$, $j$, and $k$, and spanned by vectors $ij$ and $kj$:

\begin{displaymath}
\alpha = \arccos \frac{\vec{r}_{ij} \cdot \vec{r}_{kj}}
{\vert\vec{r}_{ij}\vert \vert\vec{r}_{kj}\vert} \; .
\end{displaymath} (7.18)

It lies in the interval from 0 to 180${}^{o}$. Internal MODELLER units are radians.

The first derivatives of $\alpha$ with respect to Cartesian coordinates are:

$\displaystyle \frac{\partial \alpha}{\partial \vec{r}_i}$ $\textstyle =$ $\displaystyle \frac{\partial \alpha}{\partial \cos \alpha} \;
\frac{\partial \c...
...( \frac{\vec{r}_{ij}}{r_{ij}} \cos \alpha -
\frac{\vec{r}_{kj}}{r_{kj}} \right)$ (7.19)
$\displaystyle \frac{\partial \alpha}{\partial \vec{r}_k}$ $\textstyle =$ $\displaystyle \frac{\partial \alpha}{\partial \cos \alpha} \;
\frac{\partial \c...
...( \frac{\vec{r}_{kj}}{r_{kj}} \cos \alpha -
\frac{\vec{r}_{ij}}{r_{ij}} \right)$ (7.20)
$\displaystyle \frac{\partial \alpha} {\partial \vec{r}_j}$ $\textstyle =$ $\displaystyle -\frac{\partial d} {\partial \vec{r}_i}
-\frac{\partial d} {\partial \vec{r}_k}$ (7.21)

These equations for the derivatives have a numerical instability when the angle goes to 0 or to 180${}^{o}$. Presently, the problem is `solved' by testing for the size of the angle; if it is too small, the derivatives are set to 0 in the hope that other restraints will eventually pull the angle towards well behaved regions. Thus, angle restraints of 0 or 180${}^{o}$ should not be used in the conjugate gradients or molecular dynamics optimizations.


next up previous contents index
Next: Dihedral angle Up: Features and their derivatives Previous: Distance   Contents   Index
Bozidar BJ Jerkovic 2001-12-21