Harmonic vs. anharmonic motion

(Classical Dynamics) J. Chem. Phys. 101, 8768 (1994): In this paper M. Ben-Nun and R.D. Levine show one possible way to describe the harmonic motion of the Morse oscillator. One of the key steps was actually done years before, in 1978, in the paper by W.C. DeMarcus: Am. J. Phys. 46, 733 (1978).

(Quantum Dynamics) In the above mentioned paper it is shown that one can transform the Morse potential to the quadratic potential, however such a transformation will affect the commutation relations between the newly defined coordinate and momentum operators. The possible way to have closed algebra for the Morse potential was described by Wulfman and Kumei using the time-dilation technique. See the details in the recent book by Wulfman, Dynamical Symmetry (ch. 10.4) The derivation can be found in a Master thesis of Kumei.

Coherent states of the anharmonic oscillator were described earlier, by S. Kais and R.D. Levine, Phys. Rev. A 41, 2301. An interesting twist for the expression of these coherent states in energy domain can be found for the so-called directed states. See for example here: J. Phys. Chem. 1987, 91, 21, 5462-5465. It is important to remember that the number of bound states is finite, therefore infinite sum over the states will necessarily include the states of the dissociative continuum.