# 1.4: Molecular Point Groups 1

- Page ID
- 221672

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

*, which contain operators involving translational motion.*

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

**C _{1} **: 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

_{v}s 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 S_{3} point group. S_{3} is the generator for S_{3}, S_{3}^{ 2 }(= C_{3}^{ 2} ) S_{3}^{ 3 }(= σ_{h}), S_{3}^{ 4} (= C_{3}), S_{3}^{ 5} , S_{3}^{ 6} (= E). The C_{3}’s and σ_{h} are the distinguishing elements of the C_{3h} point group.