I recently learnt that overtone absorption is

**classically forbidden,**i.e. it is intrinsically quantum mechanical (just like tunnelling, reflection above a barrier, interference, entanglement, ...). It does not occur in the limit that Planck's constant goes to zero.

Explicitly if you take an anharmonic oscillator and drive it with an external field of frequency 2 nu, you cannot get the oscillator to go at 2 nu.

Furthermore, this involves

**dynamical tunnelling,**i.e. there is no potential barrier in real space, but rather tunnelling occurs in phase space.

There is a nice article by Eric Heller where he shows that overtone excitation is like reflection above a potential barrier.

The figure below shows the classical phase space (and Poincare surfaces) for an anharmonic (Morse) oscillator coupled to a driving field with 4 times the harmonic frequency of the oscillator.

Heller states "the local phase space structure near the [resonance] islands is the same as the above-barrier problem". See the Figure below.

Works by Lehmann and by Medvedev, explicitly shows how the transition probability for overtone excitation (i.e. the relevant matrix element) is dominated by the semi-classical dynamics in the classically forbidden region of the potential, particularly the inner wall.

This is currently of interest to me because I am working on a paper about the intensity of overtone modes in hydrogen bonded systems. In the Condon approximation overtone excitation can only occur if the potential is anharmonic. Alternatively it can arise due to non-linear terms in the dipole surface (i.e. electrical anharmonicity) (equivalent to the break-down of the Condon approximation). It does seem that the matrix element for the overtone intensity is quite sense to the finer details of the potential and thus the nuclear wave functions.

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