Dihedral angle

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

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

where

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

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

$$\frac{ \; {d}\chi} { \; {d}\vec{r}} = \frac{ \; {d}\chi}{ \; {d}\cos \chi} \frac{ \; {d}\cos \chi}{ \; {d}\vec{r}}$$ (7.24)

where

$$\frac{ \; {d}\chi}{ \; {d}\cos \chi} = \left(\frac{ \; {d}\cos \chi}{ \; {d}\chi}\right)^{-1} = -\frac{1}{\sin \chi}$$ (7.25)

and

$$\frac{\partial \cos \chi}{\partial \vec{r}_i} = \vec{r}_{kj} \times \vec{a}$$ (7.26)

$$\frac{\partial \cos \chi}{\partial \vec{r}_j} = \vec{r}_{ik} \times \vec{a} - \vec{r}_{kl} \times \vec{b}$$ (7.27)

$$\frac{\partial \cos \chi}{\partial \vec{r}_k} = \vec{r}_{jl} \times \vec{b} - \vec{r}_{ij} \times \vec{a}$$ (7.28)

$$\frac{\partial \cos \chi}{\partial \vec{r}_l} = \vec{r}_{ij} \times \vec{b}$$ (7.29)

$$\vec{a} = \frac{1}{\vert\vec{r}_{ij} \times \vec{r}_{kj}\vert} \; \left(\fr... ...}_{ij} \times \vec{r}_{kj}} {\vert\vec{r}_{ij} \times \vec{r}_{kj}\vert}\right)$$ (7.30)

$$\vec{b} = \frac{1}{\vert\vec{r}_{kj} \times \vec{r}_{kl}\vert} \; \left(\fr... ...} \times \vec{r}_{kl}} {\vert\vec{r}_{kj} \times \vec{r}_{kl}\vert}\right) \; .$$ (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!#]:

$$\vec{r}_{mj} = \vec{r}_{ij} \times \vec{r}_{kj}$$ (7.32)

$$\vec{r}_{nk} = \vec{r}_{kj} \times \vec{r}_{kl}$$ (7.33)

$$\frac{\partial \chi}{\partial \vec{r}_i} = \frac{r_{kj}}{r^2_{mj}} \vec{r}_{mj}$$ (7.34)

$$\frac{\partial \chi}{\partial \vec{r}_l} = -\frac{r_{kj}}{r^2_{nk}} \vec{r}_{nk}$$ (7.35)

$$\frac{\partial \chi}{\partial \vec{r}_j} = \left(\frac{\vec{r}_{ij} \cdot \vec{r}_{kj}}{r^2_{kj}} - 1 \right... ...{r}_{kl} \cdot \vec{r}_{kj}}{r^2_{kj}} \frac{\partial \chi}{\partial \vec{r}_l}$$ (7.36)

$$\frac{\partial \chi}{\partial \vec{r}_k} = \left(\frac{\vec{r}_{kl} \cdot \vec{r}_{kj}}{r^2_{kj}} - 1 \right... ...{r}_{ij} \cdot \vec{r}_{kj}}{r^2_{kj}} \frac{\partial \chi}{\partial \vec{r}_i}$$ (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.