First, lets define the dipole moment of a molecule to be:
In order for a vibrational transition, the following must hold:
An expectation value of some component of the dipole moment
must not be equal to zero. If we expand the dipole moment as
a Taylor series in terms of the normal coordinates:
This can be truncated after the first two terms in the electrical
harmonicity approximation.
Substitution yields:
Since vibrational states one and two are orthogonal, the first term is zero.
Likewise, only those vibrational states belonging to the k normal coordinate
will be non-zero in the integral. Furthermore, by assuming that the only
accessable states are 
(mechanical harmonicity, the
equation reduces to:
In the double harmonic approximation combination
bands and overtones will not be determined.
Subsituting this into a definition for intensity yields:
