Symmetry operation

In the context of molecular symmetry, a symmetry operation is 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.

1. Physical properties must be invariant with respect to symmetry operations.
2. Symmetry operations can be collected together in groups which are isomorphous to permutation groups.

Wavefunctions need not be invariant, because the operation can multiply them by a phase or mix states within a degenerate representation, without affecting any physical property.

Molecules

Proper rotation operations

These are denoted by C_n^m and are rotations of 360°/n, performed m times. The superscript m is omitted if it is equal to one.

C₁, rotation by 360°, is called the Identity operation and is denoted by E or I.

C_nⁿ, n rotations 360°/n is also an Identity operation.

Improper rotation operations

These are denoted by S_n^m and are rotations of 360°/n 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 S_nⁿ = E, but when n is odd S_n²ⁿ = 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

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Dichloromethane, CH₂Cl₂. There is a C₂ rotation axis which passes through the carbon atom and the midpoints between the two hydrogen atoms and the two chlorine atoms. Define the z axis as co-linear with the C₂ axis, the xz plane as containing CH₂ and the yz plane as containing CCl₂. A C₂ 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, C₂, 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, 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.

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 the crystallographic point groups with the addition symmetry operations produce the 230 crystallographic space groups.

References

F. A. Cotton Chemical applications of group theory, Wiley, 1962, 1971