By Garrett P.
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Extra resources for Archimedean Zeta Integrals for Unitary Groups (2006)(en)(18s)
Each of the planes used to define the C2 axis directions is also a mirror plane for the molecule. 5c shows that these planes are between the C2 axes and so should be labelled σd . 5d). 5d it can be seen that the rotation and inversion are indeed different operations leading to different arrangements of the labelled atoms. The Newman projection shows that the C2 rotation leads to a reversal in the order of the hydrogen atoms within each methyl group from a clockwise arrangement of H1 , H2 and H3 to an anticlockwise ordering, whereas the inversion centre preserves the clockwise order.
A point group is a list of all symmetry operations that an object which belongs to the group can undergo and remain apparently unchanged. The set of operations that form a group must be complete, in the sense that if any two members of the group are applied in succession the result must also be a single operation which is a member of the group. e. it is not possible to generate a new symmetry operation by combining those in the group. This property of groups can be useful: ensuring that the group of operations is closed is one way of checking that all the operations that are possible have been identified.
These results can also be obtained by a simple rotation using the C3 axis, and so we write the equivalences S6 2 = C3 1 and S6 4 = C3 2 . Similarly, the S6 3 operation gives the same result as the inversion centre, and so only the latter is included in the list of unique operations for the molecule. In addition, the S6 6 operation is equivalent to the identity, and so the S6 element leads to only two unique operations: S6 1 and S6 5 . These observations on the S6 axis can be generalized to any improper rotation of even order.
Archimedean Zeta Integrals for Unitary Groups (2006)(en)(18s) by Garrett P.