10.3.4 Dummy Atom Placement in Dihedral Constraints

Bond and dihedral angles cannot be constrained in Cartesian optimizations to exactly $0^{\circ }$ or $\pm {180}^{\circ }$. This is because the corresponding constraint normals are zero vectors. Also, dihedral constraints near these two limiting values (within, say ${20}^{\circ }$) tend to oscillate and are difficult to converge.

These difficulties can be overcome by defining dummy atoms and redefining the constraints with respect to the dummy atoms. For example, a dihedral constraint of ${180}^{\circ }$ can be redefined to two constraints of ${90}^{\circ }$ with respect to a suitably positioned dummy atom. The same thing can be done with a ${180}^{\circ }$ bond angle (long a familiar use in Z-matrix construction).

Typical usage is as shown in Table 10.2. Note that the order of atoms is important to obtain the correct signature on the dihedral angles. For a $0^{\circ }$ dihedral constraint, atoms J and K should be switched in the definition of the second torsion constraint in Cartesian coordinates.

Note:  In almost all cases the above discussion is somewhat academic, as internal coordinates are now best imposed using delocalized internal coordinates and there is no restriction on the constraint values.