Group Theory Solutions: #11. Identify the point group of each of the following pictures. Which are chiral? Which have dipole moments? A. fivepointed star There is a 5fold rotation axis perpendicular to the plane of the page. There are 5 C_{2} axes perpendicular to the principal rotation axis, so this is a D group. There is a mirror plane in the plane of the page (perpendicular to the principle rotation axis), so this group is D_{5h}, which is not chiral or polar. Notice that using the simplified search criteria, we didn't need to list out all of the symmetry elements that the picture has in order to decide on a point group (clearly there are improper rotations, etc.). B. Star of David There is a 6fold rotation axis perpendicular to the plane of the page. There are 6 C_{2} axes perpendicular to the principal rotation axis, so this is a D group. There is a mirror plane in the plane of the page (perpendicular to the principle rotation axis), so this group is D_{6h}. Notice that unlike the picture in part a, this picture has an inversion center. An interesting trivia fact is that all the D_{nh} groups where n is even have an inversion center; the D_{nh} groups where n is odd do not. In any case, no D_{nh} group molecule is polar or chiral. C. baseball Given this baseball (and assuming it looks the same from behind), we have the following symmetry elements: We also have mirror planes containing all three axes. Therefore the point group of this picture is D_{2h}, which is neither chiral or polar. D. pencil This linear picture has a Cinfinity rotation axis along the axis of the pencil and no inversion center. Therefore the pencil is Cinfinityv. This is not a chiral group, but it is polar.
