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# Point Groups

Point groups are used to describe molecular symmetries and are a condesed representation of the symmetrry elements a molecule may posses. This includes both bond and orbital symmetry. Knowing molecular symmetry allows for a greater understanding of molecular structure and can help to predict many molecular properties.

### 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.

Cn - The Cn 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 σh , σv , σd.

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

Cn groups

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

H2O2    C2

Dn groups

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

Cnv groups

C2v: E, C2, σv(xz), σv'(yz)    C3v: E, 2C3, 3σv

H2O    C2v

Cnh groups

C2h: E, C2, i, σh    C3h: E, C3, C32, σh, S3, S33

B(OH)3

Dnh groups

D2h: E

C2H4    D2h

### 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, 3rd 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

### 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?

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