Morse anharmonic oscillators

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. 

FIGURE 3.2 (a) Vibration of diatomic molecule, HC1, (b) potential energy of an ideal harmonic oscillator, and (c) an anharmonic oscillator described by the Morse function. 

Figure 2. Internuclear separation (top panel) and velocity (bottom panel) as a function of time for a Morse (dashed line) and harmonic (solid line) oscillator having the same total energy, ca. 50% of the dissociation energy of the Morse potential. Note the rapid change in velocity at the inner turning point and slow change in velocity at the outer turning point for an anharmonic oscillator.

Describe the correlation that may exist between the Morse curve and the energy potential curve of an anharmonic oscillator. 

Rl. Within a molecule, the stretching of the molecular bond lets the atoms come closer and move away one from each other. The Morse curve describes the potential energy induced by this stretching. At short distances, the repulsive forces are dominant, while at long distances, the attractive forces are dominant. However, when the distance that separates the two atoms within the bond is too important, the attractive forces are no longer efficient, and the Morse curve will look like an anharmonic oscillator. 

Figure 6.23. Potential curve for an anharmonic oscillator, the potential being represented by the Morse function, equation (6.187).

The correction for stretching anharmonicity can be made in a similar way using the differences in the mean square amplitudes parallel to the direction of the X-H bond using the leading term in the correction for a Morse-curve oscillation [205-207]... 

The author examines with success the efficiency of the methods by their applications to bound states for the one-dimensional harmonic oscillator, anharmonic oscillators, the Morse potential, and the modified Poschl-Teller potential in quantum mechanics. 

The authors in this paper present an explicit symplectic method for the numerical solution of the Schrodinger equation. A modified symplectic integrator with the trigonometrically fitted property which is based on this method is also produced. Our new methods are tested on the computation of the eigenvalues of the one-dimensional harmonic oscillator, the doubly anharmonic oscillator and the Morse potential. 

Parameters of the effective Hamiltonian for YCX3 are calibrated on the observed C-Y and C-X stretching fundamentals, overtones and combinations at 2 quanta of excitation. At the highest excitations, describing the energy of anharmonic oscillator by the Morse potential can lead to significant error [49]. 

Solution The above figure is an example of a potential-energy curve for an anharmonic oscillator, or a Morse curve (see Fig. 1). ... 

The heterogeneous nature of polymer melts at Tgtwinkling fractal theory (TFT) [Wool, 2008a,b]. Wool considers Tg to result from the molecular cooperativity that leads to dynamic percolating fractal structures below Tc. He assumes Boltzmann distribution of diatomic oscillators interacting via the Morse anharmonic potential. Integrating the latter from zero to the inflection point, he expresses the T dependence of solidified polymer fraction as... 

Examine the difference in the computed rate constants caused by using the harmonic or anharmonic oscillator models for the same reactant molecule also, examine the differences found by using the approximate and the exact Morse partition functions, but beware that the use of the exact partition functions will be almost an order of magnitude more time consuming. 

In the standard approximation, an anharmonic oscillator model is used to describe the vibrational motion. The vibrational amplitude and energy are largely dictated by the potential energy. From a descriptive point of view the simplest form of a potential is a so-called Morse potential, which requires only three mathematical parameters ... 

Nonlinear oscillators (NLOs) have been extensively used as realistic models for chemical bonds (Merzbacher 1970 Morse 1929), especially for describing the bondbreaking process (dissociation), simulating vibrational spectra of molecules (Child and Lawton 1981 Lehmann 1992), and modeling the nonlinear optical responses of several classes of molecules (Kirtman 1992 Takahashi and Mukamel 1994), catalytic bond activation, and dissociation processes (McCoy 1984). The nonlinear (anharmonic) oscillator, defined by the equation... 

Fig.7.n. The ground-state vibrational wave function i/r of the anharmonic oscillator (of potential energy Vi, taken here as the Morse oscillator potential energy t is asymmetric and shifted toward positive values of the displacement when compared to the wave function i/ro for the harmonic oscillator with the same force constant (the potential energy V2, ) ... 

As already stated, the Morse potential is our first example of a potential surface that describes a particular motion. The bond vibrates within the constraints imposed by this potential. One may ask, "At any given moment, what is the probability of having a particular bond length " This is similar to questions related to the probability of finding electrons at particular coordinates in space, which we will show in Chapter 14 is related to the square of the wave-function that describes the electron motion. The exact same procedure is used for bond vibrations. We square the wavefunction that describes the wave-like nature of the bond vibration. Let s explore this using the potential surface for a harmonic oscillator (such as with a normal spring), instead of an anharmonic oscillator (Morse potential). For the low energy vibrational states, the harmonic oscillator nicely mimics the anharmonic oscillator. 

Tbus the anharmonicity in the vibration is captured by the coefficient x, which adopts values typically less than 1 % of ft>e for bond stretches, but may be up to 5% for those involving hydrogen. It can be shown that any function capable of representing the variation of potential energy with displacement will lead to energy levels given by a power series in (v + 1/2), so the simple harmonic and Morse oscillators are particular cases of this general anharmonic oscillator, with the power series truncated after the first and second terms, respectively. 

The energy levels for a Morse function arc illustrated in Figure 2.14, alongside those for the harmonic potential. For the anharmonic oscillator, it is clear that the vibrational energy levels associated with a particular vibration have diminishing separation with increasing quantum number, but there is a finite number of them below the bond dissociation limit. In contrast, the harmonic model has equally spaced vibrational levels that in principle go on for ever. Around the equilibrium stmeture, however, the two functions are broadly similar. This is the validation for the harmonic approximation, which is routinely used in the calculation of vibrational frequencies by quantum mechanical methods (Section 3.3). 

The local mode description with only one bond such as X-H being in motion at a time leads us to consider the polyatomic molecule as a single diatomic molecule M-H where H is hydrogen and M denotes the full rest of the molecule. The main advantage of this description is the dramatic reduction of the number of coupling constants. The anharmonic oscillator potential adopted in the local mode model is the Morse potential, which takes account of the dissociation energy De (53). 

Here r, r, and are the atomic distance, equilibrium bond length, and potential-well depth, respectively. Further, h is Planck s constant and c is the light velocity in vacuum. The potential-curve shapes depend on the parameters and D. By analytically solving the Schrodinger equation with the anharmonic potential (Equation 2.3), one can obtain the vibrational energy levels ofthe Morse molecular oscillator ... 

The Morse constant a can be expressed through force constant and anharmonici-ty constant x of the oscillator formed by the bond atoms. The Morse function yields the energy eigenvalues of an anharmonic oscillator which are correct within a sizeable range of amplitudes (in the case of H2 for 0.4 < r/ro < 1.6). For large r, however, V becomes progressively too small [8b]. 

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). 

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... 

Fig. 3.1 Born-Oppenheimer vibrational potentials for a diatomic molecule corresponding to the CH fragment. The experimentally realistic anharmonic potential (solid line) is accurately described by the Morse function Vmorse = De[l — exp(a(r — r0)]2 with De = 397kJ/mol, a = 2A and ro = 1.086 A (A = Angstrom = 10 10m). Near the origin the BO potential is adequately approximated by the harmonic oscillator (Hooke s Law) function (dashed line), Vharm osc = f(r — ro)2/2. The harmonic oscillator force constant f = 2a2De...

Two of the most severe limitations of the harmonic oscillator model, the lack of anharmonicity (i.e., non-uniform energy level spacings) and lack of bond dissociation, result from the quadratic nature of its potential. By introducing model potentials that allow for proper bond dissociation (i.e., that do not increase without bound as x=>°°), the major shortcomings of the harmonic oscillator picture can be overcome. The so-called Morse potential (see the figure below)... 

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... 

---