1.4: Molecular Point Groups 1

The symmetry properties of molecules (i.e. the atoms of a molecule form a basis set) are described by point groups, since all the symmetry elements in a molecule will intersect at a common point, which is not shifted by any of the symmetry operations. There are also symmetry groups, called space groups, which contain operators involving translational motion.

The point groups are listed below along with their distinguishing symmetry elements

C₁ : E (h = 1) \(\Longrightarrow\) no symmetry

C_s : σ (h = 2) \(\Longrightarrow\) only a mirror plane

C_i : i (h = 2) \(\Longrightarrow\) only an inversion center (rare point group)

isomer of dichloro(difluoro)ethane

C_n : C_n and all powers up to C_n ⁿ = E (h = 2) \(\Longrightarrow\) a cyclic point group

C_nv : C_n and nσ_v (h = 2n) … by convention a σ_v contains C_n (as opposed to σ_h which is normal to C_n). For n even, there are \(\frac{n}{2} \sigma_{v}\) and \(\frac{n}{2} \sigma_{v_{v}}\) ' with the σ_v containing the most atoms and the σ_vs containing the least atoms

Consider a second example:

C_nh : C_n and σ_h (normal to C_n) are generators of S_n operations as well (h = 2n)

S_2n : S_2n and all powers up to S_2n ²ⁿ = E (h = 2n).

The F’s do not lie in the plane of the cyclopentane rings. If they did, then other symmetry operations arise; these are easiest to see by looking down the line indicated below:

Note S_n, where n is odd, is redundant with C_nh because S_n ⁿ = σ_h for n odd. As an example consider a S₃ point group. S₃ is the generator for S₃, S₃ ² (= C₃ ² ) S₃ ³ (= σ_h), S₃ ⁴ (= C₃), S₃ ⁵ , S₃ ⁶ (= E). The C₃’s and σ_h are the distinguishing elements of the C_3h point group.