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Symmetry operationIn the context of molecules and crystals a symmetry operation may be defined as a permutation of atoms such that the molecule or crystal is transformed into a state indistinguishable from the starting state. Two basic facts follow from this definition, which emphasize its usefulness.
Additional recommended knowledge
MoleculesProper rotation operationsThese are denoted by C_{n}^{m} and are rotations of 360/n º, performed m times. The superscript m is omitted if it is equal to one. C_{1}, rotation by 360º, is called the Identity operation and is denoted by E or I. C_{n}^{n} , n rotations 360/n º is also an Identity operation. Improper rotation operationsThese are denoted by S_{n}^{m} and are rotations 360/n º followed by reflection in a plane perpendicular to the rotation axis. S_{1} is usually denoted as σ, a reflection operation about a mirror plane. S_{2} is usually denoted as i, an inversion operation about an inversion centre. When n is an even number S_{n}^{n} = E, but when n is odd S_{n}^{2n} = E. Rotation axes, mirror planes and inversion centres are symmetry elements, not operations. The rotation axis of highest order is known as the principal rotation axis. It is conventional to set the Cartesian z axis of the molecule to contain the principal rotation axis. ExamplesDichloromethane, CH_{2}Cl_{2}. There is a C_{2} rotation axis which passes through the carbon atom and the mid points between the two hydrogen atoms and the two chlorine atoms. Define the z axis as colinear with the C_{2} axis, the xz plane as containing CH_{2} and the yz plane as containing CCl_{2}. A C_{2} rotation operation permutes the two hydrogen atoms and the two chlorine atoms. Reflection in the yz plane permutes the hydrogen atoms while reflection in the xz plane permutes the chlorine atoms. The four symmetry operations E, C_{2}, σ(xz)and σ(yz) form the point group C_{2v}. Note that if any two operations are carried out in succession the result is the same as if a single operation of the group had been performed.
Methane, CH_{4}. In addition to the proper rotations of order 2 and 3 there are three mutually perpendicular S_{4} axes which pass halfway between the CH bonds and six mirror planes. Note that S_{4}^{2} = C_{2}.
. . . CrystalsIn crystals screw rotations and/or glide reflections are additionally possible. These are rotations or reflections together with partial translation. The Bravais lattices may be considered as representing translational symmetry operations. Combinations of operations of the crystallographic point groups with the addition symmetry operations produce the 230 crystallographic space groups.
ReferencesF.A. Cotton Chemical applications of group theory, Wiley, 1962, 1971 

This article is licensed under the GNU Free Documentation License. It uses material from the Wikipedia article "Symmetry_operation". A list of authors is available in Wikipedia. 