Forced Morse oscillator

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

Ballhausen CJ (1995) Time-dependent raising and lowering operators applied to the forced morse oscillator. J Phys Chem 99 2530-2533... 

Robertsand in the dissociation of a forced Morse oscillator by Hunt and Sridharan. In each case the number of bumps in the dissociation curve was found to be equal to the initial vibrational nimaber of the oscillator. 

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. 

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... 

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 ... 

The first applications of wavelet transforms to analyse time series in the field of chemical dynamics were those of Permann and Hamilton [47,48]. Their interest lay in modelling diatomic molecules, close to dissociation, perturbed by a photon. They modelled the reaction using the equation of motion for a forced and damped Morse oscillator, given by ... 

D. Permann and 1. Hamilton, Wavelet Analysis of Time Series for the Weakly Forced and Weakly Damped Morse Oscillator, Journal of Chemical Physics, 100 (1994), 379-386. 

The second derivative of V(x) calculated at the minimum of the well represents the force constant k of the Morse oscillator ... 

The parabola —D + kx best approximates V (x) close to x = 0 and represents the harmonic oscillator potential energy (with the force constant k). The Morse oscillator is hard to... 

As we can see, the intermolecular interactions have made the intramolecular vibration frequencies decrease, while the intermolecular frequencies have very low frequencies. The last effect is nothing strange, of course, because a change of intermolecular distances does require a small expenditure of energy (which means small force constants). Note that the simple Morse oscillator model considered in Chapter 4 (p. 198) gave the correct order of magnitude of the intermolecular frequency of two water molecules (235 cm , compared to the above, much more accurate, result of 183 cm ). 

Porter RN, Raff LM, Miller WH (1975) Quasiclassical selection of initial coordinates and momenta for a rotating morse oscillator. J Chem Phys 63 2214-2218 Canongia Lopes JN, Deschamps J, Padua AAH (2004) Modelling ionic liquids using a systematic all-atom force field. J Phys Chem B 108 2038-2047... 

KJ/mol and a gauche-trms energy difference of 2.5 kJ/mol). The constants D, and y [see Eq. (2)] represent the usual Morse oscillator parameters. The nonbonded terms e and a [Eq. (3a)] represent the Lennard-Jones parameters, b and C [Eq. 3b)] are related to the overlap and dispersion of the atoms i and j, and A is a parameter related to the position and well-depth of the interaction. The bending force constant is, and 6o [Eq. (4)] indicates the equilibrium value of the angle formed by the three atoms of interest. The above potential energy functions [Eqs. (2-5)] have been demonstrated to yield good spectroscopic, thermodynamic, and kinetic data, as well as to provide the atomistic details of temperature-dependent phase transitions for crystalline polymers. 

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, ) ... 

Finally, some papers which carry the theory of vibrational averaging to higher levels of approximation should be mentioned. Bartell, in his 1955 paper " on the Morse oscillator probability distribution, considers the effect of an increase of temperature on r. Bartell s theory is extended by Bartell and Kuchitsu and by Kuchitsu these papers show in particular that the effective mean amplitude obtained by refinement on the molecular intensity curve is not quite equal to the harmonic mean amplitude calculated from the harmonic force field. Bonham and co-workers have calculated the effect of temperature on both a Morse oscillator and an oscillator in an RKR potential energy curve. In their final paper an informative series of diagrams shows how the quantum-mechanical average passes into the classical average at high temperature. 

Here, the HC bond is taken to be a Morse oscillator (r is the stretch coordinate) while the HCC bending potential ( 6 is the HCC wag coordinate) is a harmonic oscillator with r-dependent force constant. The CC bond stretch coordinate R and the CCC bond angle are frozen at their equilibrium values. The stretch-bend G-matrix elements are ... 

The energy halfway between the bottom of the Morse oscillator well and the dissociation limit is —U, /2, or Djl when measured from the bottom of the well. Call the difference between minimum and maximum classically allowed R values at this energy 8R (by classically allowed we mean that timneUng is neglected). At the same energy from the bottom of the well, D, /2, a harmonic oscillator with the same force constant k, wiU predict a different value for 8R, where the Morse oscillator parameter... 

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...

These potential energy terms and their attendant empirical parameters define the force field (FF). More complicated FFs which use different and/or more complex functional forms are also possible. For example, the simple harmonic oscillator expression for bond stretching can be replaced by a Morse function, Euorse (3), or additional FF terms may be added such as the stretch-bend cross terms, Estb, (4) used in the Merck molecular force field (MMFF) (7-10) which may be useful for better describing vibrations and conformational energies. 

The harmonic oscillator model does not take into account the real nature of chemical bonds, which are not perfect springs. The force constant k decreases if the atoms are pulled apart and increases significantly if they are pushed close together. The vibrational levels, instead of being represented by a parabolic function as in equation (10.3), are contained in an envelope. This envelope can be described by the Morse equation (Fig. 10.5) ... 

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. 

Fig. 2.27. (a) The experimental potential curve for H. (full curve) the dotted curve represents a Morse fit to the data, and the dashed curve shows a harmonic-oscillator function with the force constant taken at r = (0.7414 A). The first... 

Diatomic molecules differ from harmonic oscillators mainly in that they may dissociate. If we pull a diatomic molecule with intemuclear distance R equal to the equilibrium distance Rg, then at the beginning, displacement x = R — Rg is indeed proportional to the force applied. However, afterward the pulling becomes easier and easier. Finally, the molecule dissociates i.e., we separate the two parts without any effort at aU. This fundamental difference with respect to the hannonic oscillator is qualitatively captured by the potential proposed by Morse (parameter a > 0) ... 

Before we proceed, we should remind ourselves on the Morse potential, which is the last term in Equation (27.10). This function, shown in Figure 27.22, represents the potential of the hydrogen bonds between base pairs of different strands. Of course, a curve a corresponds to the larger force than a curve b. As can be seen, smaller force means smaller value of the parameter a, as if a is the inverse width of the potential well. Of course, if the force between the nucleotides of different strands is larger than the amplitude of the oscillations will be smaller and vice versa. 

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