9.3 Constrained Optimization

9.3.4 Dummy Atom Placement in Dihedral Constraints

Bond and dihedral angles cannot be constrained in Cartesian optimizations to exactly 0 or ±180. This is because the corresponding constraint normals are zero vectors. Also, dihedral constraints near these two limiting values (within, say 20) 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 can be redefined to two constraints of 90 with respect to a suitably positioned dummy atom. The same thing can be done with a 180 bond angle (long a familiar use in Z-matrix construction).

Typical usage is as shown in Table 9.2. Note that the order of atoms is important to obtain the correct signature on the dihedral angles. For a 0 dihedral constraint, atoms J and K should be switched in the definition of the second torsion constraint in Cartesian coordinates.

Internal Coordinates Cartesian Coordinates
$opt $opt
   tors I J K L 180.0    M 2 I J K
   tors I J K M 90
   tors M J K L 90
Table 9.2: Comparison of dihedral angle constraint method for adopted 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.