Introduction

Point groups are a quick and easy way to gain knowledge of a molecule. They not only contain a molecule's  symmetry elements, but also give rise to a charcter table, which is a the complete set of irreducible representations for a point group. A molecule's point group can be determined by either elucidating each symmetry element contained in a molecule or by proper use of the Schreiber chart (see below).

Point groups usually consist (but are not limited to) the following elements:

E - The identity operator. This operation leaves a molecule completely unchanged and exists for mathematical purposes.

C_n - The C_n proper axis of rotation is a 360/n° rotation that when performed leaves a molecule the same. A proper  rotation with the highest value of n is known as the major axis of rotation.

σ - The mirror plane. The mirror plane can be described as a plane which produces a reflection of part of a molecule that is unnoticeable and can be labeled as either σ_h , σ_v , σ_d.

i - The inversion center. A molecule has a center of inversion if, when inverted, the molecule is unchanged.

See the section on symmetry elements for a more thorough explanation of each

Each point group is associated with a specific combination of symmetry elements

Each point group has it's own combination of symmetry elements. Listed below are some of the many point groups and their respective symmetry elements according to category followed by a representative example.

Non axial groups

C₁: E    C₁: E, i

C_n groups

C₂: E, C₂ (notice the major axis of rotation is the point group)    C₃: E, C₃, C₃²

H₂O₂    C₂

D_n groups

D₂: E C₂(z), C₂(y), C₂(x)    D3: E, 2C3, 3C2

C_nv groups

C_2v: E, C₂, σ_v(xz), σ_v'(yz)    C_3v: E, 2C3, 3σ_v

H₂O    C_2v

Cnh groups

C_2h: E, C₂, i, σ_h    C_3h: E, C₃, C₃², σ_h, S₃, S₃³

B(OH)₃

Dnh groups

D2h: E

C₂H₄    D_2h

How to determine a molecules point group

A molecule's point group can be determined by calculating all the symmetry elements of a molecule and matching them to it's respective point group. This process, however, is greatly simplified when the Schreiber chart is used:

References

1. Housecroft, Catherine E.; Sharpe, Alan G., Inorganic Chemistry, 3^rd Ed. Pearson Education Limited 2008, Chapter 4 
2. Koster, George F.; Wheeler, Robert G.; Dimmock, John O., The Properties of Thirty-Two Point Groups, The MIT Press 1963  
3. Bertolucci, Michael D.; Harris, Daniel C., Symmetry and Spectroscopy: An Introduction to Vibrational and Electronic Spectroscopy, Dover Publications 1989  

Outside Links

• Point groups in 3-D

Problems

1) Determine the point group of BH₃ by calculating all it's symmetry elements then use the chart and determine which method is faster.

2) Determine the point groups of BH₃ and NH₃. Why is there a difference?

3) What is the point group of PPh₃?

4) Determine the point groups of CO₂ and H₂O and then compare them.

5) Propose a molecule with no symmetry. What is it's point group?

Contributors

This page viewed 8094 times
The ChemWiki has 9249 Modules.

   UC Davis ChemWiki by University of California, Davis is licensed under a Creative Commons Attribution-Noncommercial-Share Alike 3.0 United States License.  Tweet

Permissions beyond the scope of this license may be available at copyright@ucdavis.edu. Terms of Use

Powered by Mindtouch Core 2010