Morse interaction

Zullig, J. J., and J. W. Morse, Interaction of organic acids with carbonate mineral surfaces in seawater and related solutions. I. Fatty acid adsorption , Geochim. Cosmochim. Acta, 52, 1667-1678 (1988). 

The distance dependence of chemical bonding forces can be modeled by a Morse interaction defined by... 

Berend and Benson s classical treatment of V-V transfer [81] employs a two-dimensional collision model of a pair of diatomics with identical Morse interaction potentials between each pair of atoms. The Morse range parameter a was determined from experimental data for the N2-N2 T-V process. In-all, six functions are employed, one between each pair of atoms (Figure 3.5). Molecule CD, oriented at an angle / relative to its velocity vector, collides with molecule AB, with impact parameter b. Molecule AB is taken to be oriented parallel to the velocity vector of CD. The instantaneous angle between the molecular axis of AB and the line joining the centers of mass is denoted t). Cross sections for the reactions... 

Relatively few measurements have been reported of vibrational energy transfer at low temperatures. Aside from the NO studies of Billingsley and Callear, already discussed, Miller and Millikan [190] have reported relaxation times for mixtures of CO with helium and hydrogen down to 100°K. A very clear deviation from the Landau-Teller linear extrapolation [191] was observed below 300°K for both mixtures. Shin [192] has examined the CO + He case in more detail, using the WKB approach and a Morse interaction potential. At low temperatures, he concludes, there are three important considerations ... 

The suppression-enhaneement probability for the VR event with the Morse interaction potential reads [6] ... 

Figure 1. Weakening of the Morse interaction by the bond-order term, which varies from isolated molecule towards 0 for a bond that is screened by intervening neighbor atoms.

The a in the denominator is a range parameter from the Morse interaction potential. It is typically 2 A . The q, term is a dimensionless quantity which characterizes the translational wave function and is related to the number of nodes in the wave function. It is through this function that we obtain the translational quantum number because the quantum number is also related to the number of nodes. In the vicinity of the Morse potential well, there will be approximately qJ2 nodes so that the effective translational quantum number change is approximated by... 

One potential for which the above prescription can be carried out analytically is the Morse interaction... 

To calculate the bonded interaction of two atoms, a Morse function is often used. It has the form described by Eq, (19). 

Figure 7-9. Variation of the potential energy of the bonded interaction of two atoms with the distance between them. The solid line comes close to the experimental situation by using a Morse function the broken line represents the approximation by a harmonic potential.

For each pair of interacting atoms (/r is their reduced mass), three parameters are needed D, (depth of the potential energy minimum, k (force constant of the par-tictilar bond), and l(, (reference bond length). The Morse ftinction will correctly allow the bond to dissociate, but has the disadvantage that it is computationally very expensive. Moreover, force fields arc normally not parameterized to handle bond dissociation. To circumvent these disadvantages, the Morse function is replaced by a simple harmonic potential, which describes bond stretching by Hooke s law (Eq. (20)). 

Additionally to and a third adjustable parameter a was introduced. For a-values between 14 and 15, a form very similar to the Lennard-Jones [12-6] potential can be obtained. The Buckingham type of potential has the disadvantage that it becomes attractive for very short interatomic distances. A Morse potential may also be used to model van der Waals interactions in a PEF, assuming that an adapted parameter set is available. 

The viscosity therefore replaces the restraint on diffusion arising from the interaction of atoms expressed by tire Morse potential in Swalin s treatment. 

FIG. 4 Normalized oxygen density profile perpendicular to the surface from simulations of pure water with adsorption energies of 12, 24, 36, and 48 kJ/mol (from bottom to top). The lower curves are shifted downwards by 0.5, 1.0, and 1.5 units. The inset shows the height of the first (diamonds) and second peak (crosses) as a function of adsorption energy. Water interacts with the surface through a Morse potential. (From Ref. 98.)... 

Morse potential). The constant C is defined such that t/ (r) = 0 at the cutoff distance R. The interaction range is determined by the parameter a, which Viduna et al. choose very large, a = 24. Hence the cutoff distance can be made small (R = 1.25cr in [144]). This model was first used by Gerroff et al. [147] and is discussed in some detail in Chapter 12 of this book. 

For small systems, where accurate interaction energy profiles are available, it has been shown that the Morse function actually gives a slightly better description than an Exp.-6, which again performs significantly better than a Lennard-Jones 12-6 potential. This is illustrated for the H2-He interaction in Figure 2.9. 

Figure 2.10. Part of the better description of the Morse and Exp.-6 potentials may be due to the fact that they have three parameters, while the Lennard-Jones potential only employs two. Since the equilibrium distance and the well depth fix two constants, there is no additional flexibility in the Lennard-Jones function to fit the form of the repulsive interaction.

It is difficult to point to the basic reason why the average-potential model is not better applicable to metallic solutions. Shimoji60 believes that a Lennard-Jones 6-12 potential is not adequate for metals and that a Morse potential would give better results when incorporated in the same kind of model. On the other hand, it is possible that the main trouble is that metal solutions do not obey a theorem of corresponding states. More specifically, the interaction eAB(r) may not be expressible by the same function as for the pure components because the solute is so strongly modified by the solvent. This point of view is supported by considerations of the electronic models of metal solutions.46 The idea that the solvent strongly modifies the solute metal is reached also through a consideration of the quasi-chemical theory applied to dilute solutions. This is the topic that we consider next. 

The attractive energies 4D(cr/r)6 and ae2/2 r4 have two important effects on the vibrational energy transfer (a) they speed up the approaching collision partners so that the kinetic energy of the relative motion is increased, and (b) they modify the slope of the repulsive part of the interaction potential on which the transition probability depends. By letting m °°, we have completely ignored the second effect while we have over-emphasized the first. Note that Equation 12 is identical to an expression we could obtain when the interaction potential is assumed as U(r) = A [exp (— r/a)] — (ae2/2aA) — D. Similarly, if we assume a modified Morse potential of the form... 

The hard-core limiting forms of U(r) do not lead to physically acceptable results. We conclude that this is caused by a complete neglect of the effect of the attractive forces on the slope of the repulsive part in U(r). If the interaction energy is assumed as the sum of a Morse exponential function and the polarization energy evaluated at r = r°, the resulting transition probabilities appear useful for analyzing ion-molecule collisions. 

Brouwer, A., Morse, D.C., and Lans, M.C. et al. (1998). Interactions of persistent enviromnen-tal organohalogens with the thyroid hormone system Mechanisms and possible conse-qnences for animal and hnman health. Toxicology and Industrial Health 14, 59-84. 

Early experimental spectroscopic investigations on Rg- XY complexes resulted in contradictory information regarding the interactions within them and their preferred geometries. Rovibronic absorption and LIF spectra revealed T-shaped excited- and ground-state configurations, wherein the Rg atom is confined to a plane perpendicular to the X—Y bond [10, 19, 28-30]. While these results were supported by the prediction of T-shaped structures based on pairwise additive Lennard-Jones or Morse atom-atom potentials, they seemed to be at odds with results from microwave spectroscopy experiments that were consistent with linear ground-state geometries [31, 32]. Some attempts were made to justify the contradictory results of the microwave and optical spectroscopic studies, and... 

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