Symmetry Elements and Their Notation

There are five kinds of symmetry operations that one can utilize to move an object through a maximum number of indistinguishable configurations. One is the trivial identity operation E. Each of the other kinds of symmetry operation has an associated symmetry element in the object. For example, our ammonia model has three reflection operations, each of which has an associated reflection plane as its symmetry element. It also has two rotation operations and these are associated with a common rotation axis as symmetry element. The axis is said to be three-fold in this case because the associated rotations are each one-third of a complete cycle. In general, rotation by iTt/n radians is said to occur about an -fold axis. Another kind of operation—one we have encountered before is inversion, and it has a point of inversion as its symmetry element. Finally, there is an operation known as improper rotation. In this operation, we first rotate the object by some fraction of a cycle about an axis, and then reflect it through a plane perpendicular to the rotation axis. The axis is the symmetry element and is called an improper axis. 

The following two examples should help clarify the nature of these symmetry elements. Consider first the ethane molecule in its eclipsed conformation (I). 

The following symmetry elements can be identified. (It is important that you satisfy yourself that you see these elements in the sketch.) 

One three-fold axis coincident with the C-C bond. 

Three two-fold axes perpendicular to the C-C bond and intersecting its midpoint. 

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