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Symmetry is extremely important in chemistry, particularly in the areas of crystallography and spectroscopy. Consequently, point group symmetry is usually taught within the first few years of a graduate program of study. Consideration of symmetry elements and operations leads to point groups and then an understanding of character tables. The identification of all of the operations in a point group is a typical exercise at this stage. A common difficulty with this problem is that students will often miss an operation or group of operations, particularly the improper rotations which are more difficult to identify. However, knowledge of the total number of operations in a point group allows an easy check that all of the operations have been found. I developed the following simple method to determine the number of operations from only the point group name; the method does not appear in textbooks (at least those available to me) and so is either unknown to the chemistry community or has been “lost”. The method involves the following three steps: - Assign 1 for C or S, 2 for D, 12 for T, 24 for O, or 60 for I.
- Multiply by n for any numerical subscript.
- Multiply by 2 for any letter subscript label (s, v, d, h, i).
For example: D5h has 2 × 5 × 2 = 20 operations; C3v has 1 × 3 × 2 = 6 operations; and OH has 24 × 2 = 48 operations. Note that the method correctly predicts that the linear point groups C∞v and D∞h will each have an infinite number of operations. If you aren’t so concerned with the S, T, O and I point groups, the method is especially simple to remember. I should also add that it is not just the students who find this a useful tool!
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