Symmetry operation
For example, a 1⁄3 turn rotation of a regular triangle about its center, a reflection of a square across its diagonal, a translation of the Euclidean plane, or a point reflection of a sphere through its center are all symmetry operations.
[1] 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 emphasizes its usefulness.
In the context of molecular symmetry, quantum 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.
Because every molecule is indistinguishable from itself if nothing is done to it, every object possesses at least the identity operation.
In the identity operation, no change can be observed for the molecule.
Even the most asymmetric molecule possesses the identity operation.
The need for such an identity operation arises from the mathematical requirements of group theory.
Its orientation relative to the principal axis of the molecule is indicated by a subscript.
Through the reflection of each mirror plane, the molecule must be able to produce an identical image of itself.
In an inversion through a centre of symmetry, i (the element), we imagine taking each point in a molecule and then moving it out the same distance on the other side.
As a result, all the cartesian coordinates of the atoms are inverted (i.e. x,y,z to –x,–y,–z).
The symbol used to represent inversion center is i.
When the inversion operation is carried out n times, it is denoted by in, where
Examples of molecules that have an inversion center include certain molecules with octahedral geometry (general formula AB6), square planar geometry (general formula AB4), and ethylene (H2C=CH2).
Examples of molecules without inversion centers are cyclopentadienide (C5H−5) and molecules with trigonal pyramidal geometry (general formula AB3).
Here the molecule can be rotated into equivalent positions around an axis.
If the H2O molecule is rotated by 180° about an axis passing through the oxygen atom, no detectable difference before and after the C2 operation is observed.
The improper rotation is represented by the symbol Sn where n is the order.
Since the improper rotation is the combination of a proper rotation and a reflection, Sn will always exist whenever Cn and a perpendicular plane exist separately.
[3] S1 is usually denoted as σ, a reflection operation about a mirror plane.
It is conventional to set the Cartesian z-axis of the molecule to contain the principal rotation axis.
Reflection in the yz plane permutes the hydrogen atoms while reflection in the xz plane permutes the chlorine atoms.
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.
In crystals, screw rotations and/or glide reflections are additionally possible.
These operations may change based on the dimensions of the crystal lattice.
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.
Molecular symmetry Crystal structure Crystallographic restriction theorem F. A.
Cotton Chemical applications of group theory, Wiley, 1962, 1971
