Infrared spectroscopy: What is the Condon approximation?

How do you calculate the absorption intensity associated with a molecular vibration?
First, why might you care?
This is not just a basic scientific issue that is only of interest to people working in molecular spectroscopy.
It actually lies at the heart of global warming. For example, why is methane a much worse greenhouse gas than carbon dioxide? It is because it has a much larger infrared (IR) absorption intensity in the relevant frequency range.

In the electronic ground state consider a transition from a vibrational level with quantum number j to one with i. The absorption intensity is given by

where the dipole matrix element between the two vibrational states is

I  use r to denote all the nuclear co-ordinates.

mu_g (r)  is the dipole moment of the molecule in the electronic ground state.

For notational simplicity I neglect the vector character of the dipole moment.

One can now make an approximation to greatly simplify evaluation of this matrix element and to provide some physical insight. One approximates the dipole moment by its first derivative term in a Taylor expansion.

This is known as the Condon approximation.

Aside: I can't find the original reference. Please let know if you know it.

This is a very useful approximation. First, it give some insight.

a.
It tells us that the IR intensity is dominated by the variation in the dipole moment of the electronic ground state with nuclear co-ordinates.

b.
If the nuclear wave functions are harmonic, then the only non-zero IR transition is that of the fundamental (i.e. from the ground state i=0 to the first vibrational excited state, i=1). There are no overtones, i.e. higher harmonics. This is known as the double harmonic approximation. (The first is the Condon approximation).
In reality, all potential energy surfaces are slightly anharmonic and so this leads to the presence of weak overtones in IR spectra. Their intensity can be used to estimate the amount of anharmonicity, both in the potential and the dipole moment surface (i.e. deviations from Condon).

Second, the Condon approximation makes calculations of intensities a lot easier. One does not need to calculate the full dipole surface, mu_g(r) just its first derivative at the equilibrium geometry. This is what almost all computational quantum chemistry codes do.

How reliable is the Condon approximation?
It seems to be very good for most molecules. Corrections are often only a few percent.
Here is one study by Juana Vazquez and John Stanton.
One can measure vibrational frequencies extremely accurately (especially in the gas phase), e.g. to within 0.01 per cent. In contrast, one can usually only measure vibrational intensities to within about 10 per cent. This provides less motivation to worry about corrections to Condon.

However, there are exceptions. Jim Skinner and collaborators have shown how for the OH stretch in liquid water one needs to take into account the dependence of the dipole moment on the nuclear co-ordinates of the surrounding water molecules.