How to determine symmetry elements from their point groups?
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Answer:
Rotation axis. A rotation by 360˚/n that brings a three-dimensional body into an equivalent configuration comprises a C ^ n symmetry operation. If this operation is performed a second time, the product C ^ nC ^ n equals a rotation by 2(360˚/n), which may be written as C ^ n2. If n is even, n/2 is integral and the rotation reduces to C ^ n/2. In general, a C ^ nm operation is reduced by dividing m and n by their least common divisor (e.g., C ^ 96 = C ^ 32). Continued rotation by 360˚/n generates the set of operations: C ^ n, C ^ n2, C ^ n3, C ^ n4, ... C ^ nn where C ^ nn = rotation by a full 360˚ = E ^ the identity. Therefore C ^ nn+m = C ^ nm. Operations resulting from a Cn symmetry axis comprise a group that is isomorphic to the cyclic group of order n.
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One special type of C ^ n operation exists only for linear molecules (e.g., HCl). Rotation by any angle around the internuclear axis defines a symmetry operation. This element is called C• axis and an infinite number of operations C ^ •f are associated with the element where f denotes rotation in decimal degrees. We already saw that molecules may contain more than one rotation axis. In BF3, depicted below, a three-fold axis emerges from the plane of the paper intersecting the center of the
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