Chemistry 401

Symmetry and Point Groups

Symmetry is one way to indicate geometrical aspects of a molecule (or object)

Symmetry elements are geometric operations that leave the appearance of a molecule identical (in all respects) to the starting point:

 

Types of symmetry elements

1. Inversion i(x,y,z) → (–x,–y,–z)

 

2. Rotations C_nrotates the molecule by 360°/n; by convention, rotations are clockwise; successive rotations are denoted by superscripts:

C₆² = C₃

C₃² = 120° rotation counterclockwise

0° rotations are denoted by E.

Infinitely small rotations are denoted by C_∞

 

3. Mirror planes σ reflection through a plane; the plane is often denoted by a subscript

a mirror is a composite symmetry element : σ_xy = C_2zi

mirrors through bonds are usually denoted σ_v

mirrors between bonds are usually denoted σ_d

mirrors perpendicular to the highest n rotation axis are usually denoted σ_h

 

4. Improper rotations S_n a rotation followed by a mirror perpendicular to the rotation

S_n = C_nσ_h this can exist independent of the rotation or the mirror, e.g. S₄ in CH₄

note: S₂ = σC₂ = i

 

Point Group : the collection of all of the symmetry operations present in a molecule

Point Groups have mathematical significance, but we will take them simply as geometrical descriptions of molecules

To find point groups, we only need to identify certain key symmetry elements