Orientation of the Abelian Point Groups C2v and D2h

Unlike all other groups, two of the Abelian groups, C2v and D2h, present novel problems in assigning the irreducible representations. For most groups, the symmetry axis is obvious, or if there are several axes, the principal axis is obvious. For C2v and D2h an ambiguity exists. Consider, for example, ethylene, a system of point-group D2h. Should the z axis be perpendicular to the plane of the molecule--that is the unique direction, or should it go through the two carbon atoms--that is also a unique direction, but for a different reason, or should it be the third orthogonal direction--which is also unique. The choice of z axis is important in order to correctly assign the B1g and B1u of point-group D2h. For both C2v and D2h the x and y axes must also be unambiguously defined in o

The convention used in MOPAC is the following:

If there are three C2 axes, the one with the largest number of atoms unmoved by a C2 operation is z. If there is only one C2 axis, that is z.

Once z is defined, the y axis is defined as the axis of the remaining two axes which has the larger number of atoms unmoved by the Σ symmetry operations.

The x axis is the remaining axis.

To see how this works, consider ethylene, with the C-C axis being along the x direction, and the plane of the system being xy. Under the eight operations of D2h, E, C2z, C2y, C2x, σxy, σxz, σyz, and i, the number of atoms unmoved are 6, 0, 0, 2, 6, 2, 0, and 0 respectively.

From this it follows that the old x axis is now re-defined as the z axis. The new y axis has to be chosen based on the number of atoms unmoved under the σxyand σxoperations (6 and 2). The new y axis is defined as being the old y axis. The remaining new axis x therefore is the old z axis.

The overall result is that the symmetry axes in ethylene are defined as: z - along the C-C bond; y - in the molecular plane, perpendicular to the C-C bond, and x - out-of-plane.

In MOPAC the orientation of the molecule is defined by the user, therefore the assignment of the symmetry axes might be confusing. If the irreducible representations of ethylene are assigned, and the atoms are defined using internal coordinates in the order C, C, H, H, H, H, then the p orbitals will reflect the orientation used in the previous discussion, but the representations will be correct according to the conventions just defined.