 Reduction formula

The reduction can be achieved using the reduction formula. It is a mathematical way of reducing that will always work when the answer cannot be spotted by eye. It is particularly useful when there are large numbers of bonds involved.

The vibrational modes of the molecule are reduced to produce a reducible representation into the irreducible representations. This method uses the following formula reduction formula:

 where ni is the number of times a symmetry species occurs in the reducible representation,
h is the ‘order of the group’: simply the total number of symmetry operations.
The summation is over all of the symmetry operations. For each symmetry operation, three numbers are multiplied together. These are:

• gc is the number of symmetry operations of that type

• ci is the character of the irreducible representation and

• cr is the character of the reducible representation.

The number should come out as zero or as a positive integer:

if it does not either the formula has been used incorrectly
or the reducible representation has been generated incorrectly...
or both.

This may sound very complicated, however as the examples show, it is actually just tedious and time consuming to use.

Click the character tables to view in a new page.

Examples:
 ammonia NH3 PF5 <Back to Predicting Vibrational spectra>