- Symmetry operation
In the context of
molecular symmetry , a symmetry operation may be defined as apermutation of atoms such that themolecule orcrystal is transformed into a state indistinguishable from the starting state. Two basic facts follow from this definition, which emphasize its usefulness.
# Physical properties and wave functions must beinvariant with respect to symmetry operations.
# Symmetry operations can be collected together in groups which are isomorphous topermutation groups .Molecules
Proper rotation operations
These are denoted by "Cnm" and are rotations of 360/"n" º, performed "m" times. The superscript "m" is omitted if it is equal to one.
"C1", rotation by 360º, is called the Identity operation and is denoted by "E" or "I".
"Cnn" , "n" rotations 360/"n" º is also an Identity operation.
Improper rotation operations
These are denoted by "Snm" and are rotations 360/"n" º followed by reflection in a plane perpendicular to the rotation axis.
"S1" is usually denoted as σ, a reflection operation about a mirror plane.
"S2" is usually denoted as "i", an inversion operation about an inversion centre.
When "n" is an even number "Snn" = "E", but when "n" is odd "Sn2n" = "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.
Examples
C2v. 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, CH4. In addition to the proper rotations of order 2 and 3 there are three mutually perpendicular "S4" axes which pass half-way between the C-H bonds and six mirror planes. Note that "S42" = "C2"..
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Crystals
In 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 thecrystallographic point groups with the addition symmetry operations produce the 230 crystallographicspace groups .References
F.A. Cotton "Chemical applications of group theory", Wiley, 1962, 1971
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