Anharmonic oscillator, matrix element

The corresponding level broadening equals half. In fact is the diagonal kinetic coefficient characterizing the rate of phonon-assisted escape from the ground state [Ambegaokar 1987]. In harmonic approximation for the well the only nonzero matrix element is that with /= 1,K0 Q /> = <5o, where is the zero-point spread of the harmonic oscillator. For an anharmonic potential, other matrix elements contribute to (2.52). 

It was shown above that the cubic term in the potential function for the anharmonic oscillator cannot, for reasons of symmetry, contribute to a first-order perturbation. However, if the matrix elements of = ax3 are evaluated, it is found that this term results in a second-order correction to the... 

Calculation of Matrix Elements for One-Dimensional Quantum-Mechanical Problems and the Application to Anharmonic Oscillators. 

Finally there are effects of electrical anharmonicity incorporating higher derivatives of the dipole moment to consider. The intensities of transitions between successive levels of the vibrator are not necessarily given accurately by harmonic or anharmonic matrix elements. For example, a ratio of (IM1212/2 /x0i 2) = (1 + (A/m)) is obtained for a Morse oscillator. 

We are now almost ready to convert into a rovibrational language both the algebraic branching rules discussed previously and the dynamical symmetry operator (4.23), complete with its matrix elements in the local basis (4.6). The first step is a straightforward one since it does not depend on the specific equilibrium shape of the molecule. We can consider uncoupled anharmonic bond stretching vibrations in a fashion similar to that in the single-oscillator problem. By recalling Eq. (2.67), we obtain... 

Note added in proof In view of the failure of the harmonic oscillator model to account for the observed rate of activation in unimolecular dissociation reactions (the dissociation lag problem) these calculations have been repeated for a Morse anharmonic oscillator with transition between nearest and next-nearest neighbor levels [S. K. Kim, /. Chem. Phys. (to be published)]. The numerical evaluation of the analytical results obtained by Kim has not yet been carried out. From the results obtained by us and our co-workers [Barley, Montroll, Rubin, and Shuler, /. Chem. Phys. in press)] on the relaxation of vibrational nonequilibrium distributions of a system of Morse anharmonic oscillators it seems clear, however, that the anharmonic oscillator model with weak interactions (i.e., adiabatic perturbation type matrix elements) does not constitute much of an improvement on the harmonic oscillator model in giving the observed rates of activation. The answer to tliis problem would seem to lie in a recalculation of the collisional matrix elements for translational-vibrational energy exchange which takes account of the strong interactions in highly energetic collisions which can lead to direct dissociation. 

Harris, D.O., Engerhohn, G.G., Gwitm, W.D. Calculation of matrix elements for one-dimensional quantum-mechanical problems and the appHcation to anharmonic oscillators, J. Chem. Phys. 1965, 43,1515-7. 

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