The Morse Oscillator

It is useful and instructive to treat the local bond stretch oscillators (but not the normal mode oscillators) as Morse oscillators (Jaffe and Brumer, 1980 Sibert, et al, 1982a and b Child, 1991 Mills and Robiette, 1985), 

The Morse oscillator has the convenient quantum mechanical property that the vibrational energy levels of a Morse oscillator have the simple form of 

Although the symmetric stretch can dissociate (by simultaneously breaking both bonds) similarly to a single bond local stretch, the antisymmetric stretch cannot dissociate to a knowable dissociation asymptote, thus there is no physical basis for treating an antisymmetric stretch (or any bend) as a Morse oscillator. 

An exact solution of the Morse oscillator Schrodinger equation yields 

Exactly the same relationship between (cum,xm) and (a,De) is derived by perturbation theory, 

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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 ... 

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... 

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). 

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. 

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

The initial exploration in this unit requires the students to compare the trajectories calculated for several different energies for both Morse oscillator and harmonic oscillator approximations of a specific diatomic molecule. Each pair of students is given parameters for a different molecule. The students explore the influence of initial conditions and of the parameters of the potential on the vibrational motion. The differences are visualized in several ways. The velocity and position as a function of time are plotted in Figure 2 for an energy approximately 50% of the Morse Oscillator dissociation energy. The potential, kinetic and total energy as a function of time are plotted for the same parameters in Figure 3. 

They solved the equation of motion for the two reactants and their respective nearest neighbour shells with a potential energy of interaction between the two iodine atoms of the Morse oscillator type. A Monte Carlo technique was used and reaction occurred immediately the iodine atoms came into contact. The results of these simulations ate shown in Fig. 55. As the frequency of oscillation coe, of the iodine atom in the... 

Fig. 12.4. ln(r/n) plotted versus [m(en - en i)]x 2 for the vibrational predissociation of HeCl2. m is the reduced mass of the van der Waals molecule and en — en i is the energy spacing between adjacent levels of the Morse oscillator. Note that n increases from the right- to the left-hand side Adapted from Cline, Evard, Thommen, and Janda (1986). 

Use Mathcad or some other symbolic algebra program to solve the Ai secular determinant of Table 41-2 in a manner similar to that shown in Fig. 5 for the Bi determinant. Modify the Mathematica commands shown in Fig. 6 to see the square of the I2 harmonic oscillator and Morse wavefunctions and their overlap product for v" = 2 and v = 0, 5, 10, 15, 20, and 25. Obtain plots of these results and discuss the trends that you see. Repeat the exercise for v = 40 and v" = 0, 1, 2, 3, 4, and 5 and note the dramatic intensity variations for the Morse oscillator. Emission from this state, which can be populated by the 520.8-nm krypton ion laser line, is strong to even v" levels but is very weak to odd v" levels (up to about v" = 30). 

Jensen, R Bunko-, P. R. The potential surface and stretching frequencies of X Bj methylene (CHj) detomined from expoiment using the morse oscillator-rigid bender intoBal dynamics hamioltonian, J. ChertL Phys. 1988, 89, 1327-1332. 

Now, the quantity u(s,Q) can be related to the mean square energy transfer for the Morse oscillator per one excursion into the repulsive region of the potential ... 

The quantity X(e) = 0(1 - e) / [l - (1 - e) / ] isjust the mean kinetic energy of the Morse oscillator. For collinear diatom-atom collisions, Nesbitt and Hynesderived a phase-averaged hard-sphere model in which... 

For K 2kT and /Km, the collinear result of Eq. (5.13) differs from the three-dimensional model of this work by a factor of 3. This difference is interpreted in terms of the projection of forces along the internuclear axis. The slightly different kinematic factors arise, in part, from the definition of the collision frequency that is used to derive, Eq. (5.11). The hard-sphere model gave excellent agreement with simulations for a very steep exponential repulsive potential with exponent 2a = 256h, where b is that of the Morse oscillator. It is to be remembered that Eq. (5.12) was derived from a stochastic model with three major assumptions ... 

In all cases the agreement is excellent when the empirical correction factor Si of Eq. (5.9) is used. This factor is 1.03, 1.37, and 2.86 for He, Ar, and Xe, resp>ectively. The important point is that this correction was deduced from computationsof the Morse oscillator with e = 0. The results of this work show that the correction applies over the entire range of oscillator excitation. This will be further demonstrated in the next section near dissociation. 

VI). The harmonic oscillator of model VI is replaced by the Morse oscillator. 

Thus, exploiting the built-in v-dependence of the Morse oscillator eigen-energies (Eq. (9.4.107)),... 

These results have been explained in terms of two models in which account is taken of a non-equilibrium distribution over vibrational states the truncated harmonic oscillator and the Morse oscillator with all transitions allowed [83]. The dissociation may take place from any vibrational level. It is shown that as the temperature is increased, the contribution to the decomposition process from the high vibrational levels is severely diminished and it is the lower states that make the major contribution. It is the reduction in the number of reactive states that is... 

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