Morse potential potentials

Many other kinds of clusters differ sharply from this simple relationship. Clusters of rare-gas atoms such as argon tend to have structures based on icosahedral geometry. This geometry cannot be the basis for a lattice it simply does not have the translational symmetry necessary to build a lattice. The spacings between neighbors in shells distant from the central atom differ from those near the core. The cluster sizes for which complete, filled icosahedra can be made, namely 13, 55, 137,..., are called magic numbers and the structures are called Mackay icosahedra. Not all the most stable structures of such clusters have icosahedral structures, and the specific structure of the most stable form depends on the forces that bind the cluster together. ° 4i por example, a number of clusters bound by Lennard-Jones or Morse potentials, potentials V(R) that depend only on the distance R between pairs of particles. The first of these has the form... 

Heather R and Metiu H 1985 Some remarks concerning the propagation of a Gaussian wave packet trapped in a Morse potential Chem. Phys. Lett. 118 558-63... 

During initialization and final analysis of the QCT calculations, the numerical values of the Morse potential paiameters that we have used aie given as... 

The center of the wavepacket thus evolves along the trajectory defined by classical mechanics. This is in fact a general result for wavepackets in a hannonic potential, and follows from the Ehrenfest theorem [147] [see Eqs. (154,155) in Appendix C]. The equations of motion are straightforward to integrate, with the exception of the width matrix, Eq. (44). This equation is numerically unstable, and has been found to cause problems in practical applications using Morse potentials [148]. As a result, Heller inboduced the P-Z method as an alternative propagation method [24]. In this, the matrix A, is rewritten as a product of matrices... 

Next, we replace the stiff spring potential a(r — 1) /2 by the Morse potential... 

Compared with the Morse potential, Hooke s law performs reasonably well in the equilibrium area near If, where the shape of the Morse function is more or less quadratic (see Figure 7-9 in the minimum-energy region). To improve the performance of the harmonic potential for non-equilibrium bond lengths also, higher-order terms can be added to the potential according to Eq. (21). 

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. 

A Morse fun Cl ion best approxini ales a boiul polcn iial. One of ihc obvious clifl erciices between a Morse and harinoiiic poleiUial is lhal only (he Morse potential can describe a dissociating bond. 

Two of the most severe limitations of the harmonie oseillator model, the laek of anharmonieity (i.e., non-uniform energy level spaeings) and laek of bond dissoeiation, result from the quadratie nature of its potential. By introdueing model potentials that allow for proper bond dissoeiation (i.e., that do not inerease without bound as x=>°o), the major shorteomings of the harmonie oseillator pieture ean be overeome. The so-ealled Morse potential (see the figure below)... 

Here, Dg is the bond dissoeiation energy, rg is the equilibrium bond length, and a is a eonstant that eharaeterizes the steepness of the potential and determines the vibrational frequeneies. The advantage of using the Morse potential to improve upon harmonie-oseillator-level predietions is that its energy levels and wavefunetions are also known exaetly. The energies are given in terms of the parameters of the potential as follows ... 

Bond stretching is most often described by a harmonic oscillator equation. It is sometimes described by a Morse potential. In rare cases, bond stretching will be described by a Leonard-Jones or quartic potential. Cubic equations have been used for describing bond stretching, but suffer from becoming completely repulsive once the bond has been stretched past a certain point. 

FIGURE 6.2 Hannonic, cubic, and Morse potential curves used to describe the energy due to bond stretching in molecular mechanics force fields. 

Results using this technique are better for force helds made to describe geometries away from equilibrium. For example, it is better to use Morse potentials than harmonic potentials to describe bond stretching. Some researchers have created force helds for a specihc reaction. These are made by htting to the potential energy surface obtained from ah initio calculations. This is useful for examining dynamics on the surface, but it is much more work than simply using ah initio methods to hnd a transition structure. 

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

Although two-parameter models are rather restrictive, three-parameter models of the intermolecular potential have been developed which provide reasonable descriptions of the thermodynamic behavior of solids. Examples include the Morse potential, the exponential-six potential, and, more recently, a form proposed by Rose et al. (1984) for metals. 

The hot bond fracture stress Morse potential energy function by [1]... 

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. 

There are a number of three- to five-parameter potential functions in the literature, of which the Morse potential is the most popular a typical five-parameter potential is the Linnett function (Linnett, 1940, 1942) ... 

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