Point Groups

What are point groups?

Point groups are a way of classifying molecules according to the symmetry operations they possess. They simplify the analysis of molecules, since once the point group has been determined, we can look up how many symmetry operations of each type the molecule must have, rather than having to find all of them.
There are several different point groups. We will begin by looking at the ones with the greatest symmetry:

Octohedral Molecules

The definition of an octohedral molecule is very strict in Group Theory. Only those molecules in which all the ligands are identical fit the definition, such as SF₆ and [Cr(CO)₆]. Molecules such as [Cr(CO)₅Cl] do not have have the same high degree of symmetry and do not fit this point group.

Molecules of this sort are given the point group O_h.

Tetrahedral Molecules

Again, only strictly tetrahedral molecules such as CH₄ fit this definition.

Tetrahedral molecules of the type ML₄ belong to point group T_d.

The Point Group D_nh

Many molecules belong to the point group D_nh. They all have a s_h mirror plane at right angles to the principal C_n rotation axis.

There are three major families of these compounds:

The first contains one central atom, with n ligands placed symmetrically around the central atom in a plane. There may be two additional ligands placed above and below the central atom (or not, e.g. in the case of a square planar compound.)

The second family has two eclipsed groups above and below the central atom. An extreme example of this is the D_¥h in molecules such as CO₂, i.e. O=C=O.

Note that this is only if the groups are eclipsed.

The third family is very similar to the second, but instead of having a central atom there are two or more atoms separating the outer groups:

Again, this is only if the groups are eclipsed

The Point Group D_nd

Molecules which belong to the point group D_nd all have a S_2n axis of improper rotation coincident with the principal C_n axis. There are two major groups of compounds belonging to this point group:

The first has two staggered groups above and below the central atom.

Note that the rings are staggered.

The second group is similar, but instead of having a central atom has two or more atoms separating the outer groups:

The Point Group D_n

Many molecules belond to the point group D_n. They all have n C₂ axes at right angles to the principal C_n axis, but do not have a mirror plane at right angles to the principal axis, or an S_2n improper rotation axis coaxial with it.
There are two major families of compounds. They are similar to the D_nh and D_nd point groups shown above, but the groups above and below the central atom(s) are neither eclipsed nor staggered, but in an intermediate position.

The first "family" has two groups above and below the central atom:

Note that the rings are neither eclipsed nor staggered,
but in an intermediate orientation.

The second group is very similar to the first, but instead of having a central atom has two or more separating the outer groups:

Ethane, neither eclipsed nor staggered.

The Point Group C_nv

Molecules in the point group C_nv all have a principal axis of rotation, C_n, but unlike the D_n, D_nh and D_nd groups, do not have nC₂ axes at right angles to it. A great many common molecules belong to this point group, including H₂O, NH₃.
Any heteronuclear diatomic, eg. H-Cl, has point group C_¥v.

Two molecules of point groups C_nv.

How to assign Point Groups

The easiest way to assign a point group to a molecule is by finding the principal axis - the rotational axis with the highest order of symmetry. Once this has been done, the molecule will fall into one of four simple groups:

1. Molecules which possess no axes of symmetry (other than C₁, i.e. the Identity Operation.)

2. Molecule possesses one axis of symmetry only (C_n)

3. Molecule possesses one principal axis of symmetry C_n with n C axes at right angles to it

4. Molecule possesses several principal axes of symmetry. These are the special groups.