Angle

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

$$\alpha = \arccos \frac{\vec{r}_{ij} \cdot \vec{r}_{kj}} {\vert\vec{r}_{ij}\vert \vert\vec{r}_{kj}\vert} \; .$$ (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:

$$\frac{\partial \alpha}{\partial \vec{r}_i} = \frac{\partial \alpha}{\partial \cos \alpha} \; \frac{\partial \c... ...( \frac{\vec{r}_{ij}}{r_{ij}} \cos \alpha - \frac{\vec{r}_{kj}}{r_{kj}} \right)$$ (7.19)

$$\frac{\partial \alpha}{\partial \vec{r}_k} = \frac{\partial \alpha}{\partial \cos \alpha} \; \frac{\partial \c... ...( \frac{\vec{r}_{kj}}{r_{kj}} \cos \alpha - \frac{\vec{r}_{ij}}{r_{ij}} \right)$$ (7.20)

$$\frac{\partial \alpha} {\partial \vec{r}_j} = -\frac{\partial d} {\partial \vec{r}_i} -\frac{\partial d} {\partial \vec{r}_k}$$ (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.