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 9.2. 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.

Internal Coordinates | Cartesian Coordinates |
---|---|

$opt | $opt |

CONSTRAINT | DUMMY |

tors I J K L 180.0 | M 2 I J K |

ENDCONSTRAINT | ENDDUMMY |

$end | CONSTRAINT |

tors I J K M 90 | |

tors M J K L 90 | |

ENDCONSTRAINT | |

$end |

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.