Bond and dihedral angles cannot be constrained in Cartesian optimizations to exactly or . This is because the corresponding constraint normals are zero vectors. Also, dihedral constraints near these two limiting values (within, say ) 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 can be redefined to two constraints of with respect to a suitably positioned dummy atom. The same thing can be done with a 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 dihedral constraint, atoms J and K should be switched in the definition of the second torsion constraint in Cartesian coordinates.
|Internal Coordinates||Cartesian Coordinates|
|tors I J K L 180.0||M 2 I J K|
|tors I J K M 90|
|tors M J K L 90|
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.