Lecture 8: Symmetry Point Groups

Read section 3.3, 3.4 from your textbook. We'll use this information to work through some symmetry problems in class. Your next quiz is due Sunday evening, 2/4/07.

Point groups are ways to categorize molecules based on their symmetry. We'll use information from the point groups in constructing molecular orbital diagrams and to interpret vibrational and electronic properties of molecules. The selection rules for UV-visible and IR absorption spectroscopy relate to point group symmetry.

    We'll use this flow chart to assign point groups to molecules. There is a related chart in your textbook.

Special Groups

  1. Linear Molecules
    All linear molecules are in either Dh or Cv point groups.
      Those linear molecules with a horizontal plane (), such as H2, are in the Dh group.
      Linear molecules without a horizontal plane, such as CO or NCS-, are in the Cv group.



  2. High Symmetry Molecules

    You should be able to pick out the high symmetry groups without even thinking about individual symmetry elements. Remember that the nature of the atoms as well as their arrangement is important. Methane is strickly tetrahedral but chloromethane is not!

      (a) Molecules that are tetrahedral, such as CH4, are in the Td point group.

      (b) Molecules that are octahedral, such as SF6, are in the Oh point group.

      (c) Molecules that are icosohedral, such as B12, are in the Ih point group.
  3. Low Symmetry Molecules
    Next, look for rotation axes. If there are no proper or improper rotation axes, the molecule must belong to one of the low symmetry groups.

      Cs : only element is a mirror plane

      C1 : no symmetry elements

      Ci : only element is a center of symmetry



  4. Molecules with Only Sn (n is even) Symmetry
    Molecules that have only an even numbered improper rotation axis and no other symmetry elements are very rare. This isomer of B4N4R8 is one example.

The C Groups

Molecules that have proper rotation axes belong to either the C or the D groups.

The first step in classifying these is to find the highest order rotation axis.

Are there C2 axes perpendicular to the highest order axis? If not, the molecule is in one of the C point groups.


    (a) Cn
    The molecule has a Cn axis only.

    (b) Cnv
    The molecule has a Cn axis and n (mirror planes that contain the axis).

    (c) Cnh
    The molecule has a Cn axis and one (mirror plane perpendicular to the axis). The Cn axis is also an Sn axis.

The D Groups

If there are n C2 axes perpendicular to the principle axis, the molecule belongs to one of the D point groups.

    (a) Dn
    There are only the principle rotation axis Cn and n C2 axes perpendicular to it.

    (b) Dnd
    There are also n planes and an S2n axis.

    (c) Dnh
    There is also a plane and n planes. The Cn axis is coincedent with a Sn axis.

Character Tables

Each point group has a character table associated with it. The character table describes how the molecule, its molecular orbitals, its bonds transform with the symmetry elements of the point group. Below is the character table for the Dh point group that we can use for the hydrogen molecule.

Dh E C2 C i
A1g 1 1 1 1 1 1
A1u 1 -1 1 -1 1 -1


The number 1 means that the object remains the same after the symmetry operation and -1 means that the object is inverted. Other numbers describe other changes after the symmetry operation.

Consider the molecular obitals of H2.
What happens to the bonding molecular orbital in each of the symmetry operations?
    E, unchanged, 1
    C2, unchanged, 1
    C, unchanged, 1
    i, unchanged, 1
    , unchanged, 1
    S, unchanged, 1
    This orbital transforms according to A1g.

Now what happens to the antibonding orbital?
    E, unchanged, 1
    C2, inverted, -1
    C, unchanged, 1
    i, inverted, -1
    , unchanged, 1
    S, inverted, -1
    This orbital transforms according to A1u.