R1 = 1.50 R2 = 1.10 R3 = 1.00 R4 = 1.20 A1 = 110.0 A2 = 90.0 A3 = 90.0 D1 = 120.0 D2 = 0.0 D3 = 180.0

A dummy atom is just a point in space and has no significance in the actual calculation. The above two Z-matrices give identical Cartesian coordinates. The R3 variable has arbitrarily been given a distance of 1.00, and the D2 torsional angle of 0.0° is also arbitrary - any other values may be substituted without affecting the coordinates of the real atoms. Similarly, the A2 and A3 angles should just add up to 180°; their individual values are not significant. The function of a dummy atom in this case is to break up the problematic 180° angle into two 90° angles. It should be noted that the introduction of dummy atoms does not increase the number of (non-redundant) parameters, although there are formally three more variables for each dummy atom. The dummy variables may be identified by excluding them from the symbolic variable list, or by explicitly forcing them to be non-optimizable parameters.

When a molecule is symmetric, it is often convenient to start the numbering with atoms lying on a rotation axis or in a symmetry plane. If there are no real atoms on a rotation axis or in a mirror plane, dummy atoms can be useful for defining the symmetry element. Consider for example the cyclopropenyl system, which has D3h symmetry. Without dummy atoms, one of the C—C bond lengths will be given in terms of the two other C—C distances and the C—C—C angle, and it will be complicated to force the three C—C bonds to be identical. By introducing two dummy atoms to define the C3 axis, this becomes easy.

Figure D.4 Atom numbering for the cyclopropyl system

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