Oscillator, Morse

The Morse oscillator model is often used to go beyond the harmonic oscillator approximation. In this model, the potential Ej(R) is expressed in terms of the bond dissociation energy Dg and a parameter a related to the second derivative k of Ej(R) at Rg k = ( d2Ej/dR2) = 2a2Dg as follows ... 

An interesting improvement from the classical treatment of the bond under stress was proposed by Crist et al, [101], Considering the chain as a set of N-coupled Morse oscillators, these authors determined the elongation and time to failure as a function of the axial stress. The results, reported in Fig. 20, show a decreasing correlation between the total elastic strain before failure and the level of applied force with the chain length. To break a chain within some reasonable time interval (for example <10-3s) requires, however, the same level of stress (a0.7 fb) as found from the simpler de Boer s model. 

Longuet-Higgins phase-based treatment, three-particle reactive system, 157-168 MORBID Hamiltonian, Renner-Teller effect, triatomic molecules, benchmark handling, 621-623 Morse oscillator ... 

Designed to Calculate Energy Levels in Selected Range and Application to a (One-Dimensional) Morse Oscillator and (Three-Dimensional) HCN/HNC. 

Evaluation of diagonal and off-diagonal matrix elements for the one-dimensional and also for the rotating (Chapter 2) Morse oscillator has been discussed by many authors. The results quoted in the text are from Matsumoto (1988). 

This is precisely the spectrum of the one-dimensional Morse oscillator discussed in Section 1.9. 

The familiar consistency relation V0 = (ae/4xe of the Morse oscillator is seen to be satisfied. Another manifestation of this relation is xe = l/(/V + 1). 

As mentioned in the previous section this equation represents the energy eigenvalues of the Morse oscillator. In general one can write... 

This is a Dunham-like expansion but done around the anharmonic solution. It converges very quickly to the exact solution if the potential is not too different from that of a Morse oscillator (Figure 2.3). This will not, however, be the case for the highest-lying vibrational states just below the dissociation threshold. The inverse power dependence of the potential suggests that fractional powers of n must be included (LeRoy and Bernstein, 1970). 

The Morse oscillator was discussed by Levine and Wulfman (1979) and by Berrondo and Palma (1980). Levine (1982) is a review of algebraic work on onedimensional anharmonic vibrations. Additional work on one dimensional motion is found in Alhassid, Gtirsey, and Iachello (1983a, 1983b) and Levine (1988). 

Analytic expressions for matrix elements of the rotating Morse oscillator can be found in Heaps and Herzberg (1952) Elsum and Gordon (1982), Huffaker and Tran (1982), Requena et al. (1983), and Nagaoka and Yamabe (1988). 

For these vibrations, the quantization scheme of Section 4.2 can be carried over without any modification (Iachello and Oss, 1991a). The potentials in each stretching coordinate 5 are in an anharmonic force field approximation represented by Morse potentials. The boson operators (Ot,xt) correspond to the quantization of anharmonic Morse oscillators, with classical Hamiltonian... 

The potential function V(rx,r2) describes two coupled Morse oscillators with couplings involving the square root of the product of Morse oscillators. 

One can compare this expression with the so-called Morse oscillator-rigid bender Hamiltonian of Jensen (1988), where powers of products of Morse potentials appear. 

Billing, G. D., and Jolicard, G. (1983), The Linearly Forced Morse Oscillator, Chem. Phys. Lett., 102, 491. 

Elsum, I. R., and Gordon, R. G. (1982), Accurate Analytic Approximations for the Rotating Morse Oscillator Energies, Wave Functions, and Matrix Elements, J. Chem. Phys. 76, 5452. 

Huffaker, J. N., and Tran, L. B. (1982), Morse-Oscillator Matrix Elements Appropriate for Vibration-Rotation Intensities of Diatomic Molecules, 7. Chem. Phys. 76, 3838. 

Jensen, P. (1988), A New Morse Oscillator-Rigid Bender Internal Dynamics (MORBID) Hamiltonian for Triatomic Molecules, J. Mol. Sped. 128,478. 

Palma, A., Rivas-Silva, J. F., Durand, J. S., and Sandoval, L. (1992), Algebraic Approximation to the Franck-Condon Factors for the Morse Oscillator, Inf l J. Quant. Chem. 41,811. 

Recamier, J., and Berrondo, M. (1991), Vibration-Translation Energy Transfer in a Collision Between an Atom and a Morse Oscillator, Mol. Phys. 73,831. 

In the LM model, molecular vibrations are treated as motions of individual anharmonic bonds [38] (usually Morse oscillators). They therefore include anharmonicity, but not coupling between bonds, thus requiring inclusion of interbond coupling for obtaining a better description. For the case of t identical Morse oscillators, the energy levels related to the LM Hamiltonian are given by... 

This modification requires calculating the derivative of the Morse Oscillator... 

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