Molecular Symmetry and Group Theory

Lecture 1 — Symmetry elements and operations

Lecture Summary

The five symmetry elements necessary to specify completely the symmetry of all possible molecules are:

E Identity
Cn Proper Rotation Axis of order n
σ Mirror Plane
i Inversion Centre (centre of symmetry)
Sn Improper Rotation (rotation-reflection) axis of order n

A molecule is said to possess a symmetry element if the application of a symmetry operation generated by the element leaves the molecule indistinguishable from its starting position.

The elements E, σ and i each generate one operation:

E the ‘operation’ of doing nothing (all molecules possess the identity element!)
σ reflect in the plane
i invert through the centre of the molecule

The elements Cn and Sn can each generate a number of operations. This is because the effect of applying the basic operation more than once often counts as a separate operation.

Cn rotate clockwise through 360°/n
Sn rotate clockwise through 360°/n and reflect in the plane perpendicular to the rotation axis