next up previous contents index
Next: Atomic solvent accessibility Up: Features and their derivatives Previous: Angle   Contents   Index

Dihedral angle

Dihedral angle is defined by points $i$, $j$, $k$, and $l$ ($ijkl$):

\begin{displaymath}
\chi = \mbox{sign}(\chi) \arccos\frac{
(\vec{r}_{ij} \time...
...\vec{r}_{kj}\vert
\vert\vec{r}_{kj} \times \vec{r}_{kl}\vert}
\end{displaymath} (7.22)

where
\begin{displaymath}
\mbox{sign}(\chi) = \mbox{sign}[\vec{r}_{kj} \cdot
(\vec{...
...\vec{r}_{kj}) \times
(\vec{r}_{kj} \times \vec{r}_{kl})] \; .
\end{displaymath} (7.23)

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

\begin{displaymath}
\frac{ \; {d}\chi} { \; {d}\vec{r}} =
\frac{ \; {d}\chi}{ \; {d}\cos \chi} \frac{ \; {d}\cos \chi}{ \; {d}\vec{r}}
\end{displaymath} (7.24)

where
\begin{displaymath}
\frac{ \; {d}\chi}{ \; {d}\cos \chi} = \left(\frac{ \; {d}\cos \chi}{ \; {d}\chi}\right)^{-1} =
-\frac{1}{\sin \chi}
\end{displaymath} (7.25)

and
$\displaystyle \frac{\partial \cos \chi}{\partial \vec{r}_i}$ $\textstyle =$ $\displaystyle \vec{r}_{kj} \times \vec{a}$ (7.26)
$\displaystyle \frac{\partial \cos \chi}{\partial \vec{r}_j}$ $\textstyle =$ $\displaystyle \vec{r}_{ik} \times \vec{a} -
\vec{r}_{kl} \times \vec{b}$ (7.27)
$\displaystyle \frac{\partial \cos \chi}{\partial \vec{r}_k}$ $\textstyle =$ $\displaystyle \vec{r}_{jl} \times \vec{b} -
\vec{r}_{ij} \times \vec{a}$ (7.28)
$\displaystyle \frac{\partial \cos \chi}{\partial \vec{r}_l}$ $\textstyle =$ $\displaystyle \vec{r}_{ij} \times \vec{b}$ (7.29)
$\displaystyle \vec{a}$ $\textstyle =$ $\displaystyle \frac{1}{\vert\vec{r}_{ij} \times \vec{r}_{kj}\vert} \;
\left(\fr...
...}_{ij} \times \vec{r}_{kj}}
{\vert\vec{r}_{ij} \times \vec{r}_{kj}\vert}\right)$ (7.30)
$\displaystyle \vec{b}$ $\textstyle =$ $\displaystyle \frac{1}{\vert\vec{r}_{kj} \times \vec{r}_{kl}\vert} \;
\left(\fr...
...} \times \vec{r}_{kl}}
{\vert\vec{r}_{kj} \times \vec{r}_{kl}\vert}\right) \; .$ (7.31)

These equations for the derivatives have a numerical instability when the angle goes to 0. Thus, the following set of equations is used instead [#!Sch93!#]:


$\displaystyle \vec{r}_{mj}$ $\textstyle =$ $\displaystyle \vec{r}_{ij} \times \vec{r}_{kj}$ (7.32)
$\displaystyle \vec{r}_{nk}$ $\textstyle =$ $\displaystyle \vec{r}_{kj} \times \vec{r}_{kl}$ (7.33)
$\displaystyle \frac{\partial \chi}{\partial \vec{r}_i}$ $\textstyle =$ $\displaystyle \frac{r_{kj}}{r^2_{mj}} \vec{r}_{mj}$ (7.34)
$\displaystyle \frac{\partial \chi}{\partial \vec{r}_l}$ $\textstyle =$ $\displaystyle -\frac{r_{kj}}{r^2_{nk}} \vec{r}_{nk}$ (7.35)
$\displaystyle \frac{\partial \chi}{\partial \vec{r}_j}$ $\textstyle =$ $\displaystyle \left(\frac{\vec{r}_{ij} \cdot \vec{r}_{kj}}{r^2_{kj}} - 1 \right...
...{r}_{kl} \cdot \vec{r}_{kj}}{r^2_{kj}}
\frac{\partial \chi}{\partial \vec{r}_l}$ (7.36)
$\displaystyle \frac{\partial \chi}{\partial \vec{r}_k}$ $\textstyle =$ $\displaystyle \left(\frac{\vec{r}_{kl} \cdot \vec{r}_{kj}}{r^2_{kj}} - 1 \right...
...{r}_{ij} \cdot \vec{r}_{kj}}{r^2_{kj}}
\frac{\partial \chi}{\partial \vec{r}_i}$ (7.37)

The only possible instability in these equations is when the length of the central bond of the dihedral, $r_{kj}$, goes to 0. In such a case, which should not happen, the derivatives are set to 0. The expressions for an improper dihedral angle, as opposed to a dihedral or dihedral angle, are the same, except that indices $ijkl$ are permuted to $ikjl$. In both cases, covalent bonds $ij$, $jk$, and $kl$ are defining the angle.


next up previous contents index
Next: Atomic solvent accessibility Up: Features and their derivatives Previous: Angle   Contents   Index
Bozidar BJ Jerkovic 2001-12-21