Pair Morse

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)). 

The semiempirical methods combine experimental data with theory as a way to circumvent the calculational difficulties of pure theory. The first of these methods leads to what are called London-Eyring-Polanyi (LEP) potential energy surfaces. Consider the triatomic ABC system. For any pair of atoms the energy as a function of intermolecular distance r is represented by the Morse equation, Eq. (5-16),... 

In an effort to understand silicon surface diffusion, NoorBatcha, Raff and Thompson have employed molecular dynamics to model the motion of single silicon atoms on the Si(001) and Si(lll)surfaces. Morse functions are used for the pair forces, with the parameters being determined by the heat of sublimation. Because different forces were used for the diffusing and substrate atoms, the incorporation of gas-phase species into the crystal could not be directly modeled. Nonetheless, they were able to explore the characteristics of adsorption and diffusion for single atoms. 

The conclusions of Ciraci et al. (1990a) can be understood with a simple independent-atom model, that is, by considering the tip as a single atom, and the total force is the sum of the forces on every atom on the sample surface. Assuming that the force between individual pairs of atoms can be represented as a Morse curve, the z component of the force to the nth atom at a point in space, r, is ... 

An example of an Al(lll) surface is shown in Fig. 8.4. From the measured work function, cj)=3.5 eV, we find k = 0.96A", The atomic distance is a = 2.88 A, which also equals the parameter m in the Morse formula. An estimation of the parameter Uq in the Morse formula, or the binding energy per pair of A1 atoms, can be made as follows The evaporation heat of aluminum is 293 kJ/mol, which is 3.0 eV per atom. Aluminum is an fee crystal, where each atom has 12 nearest neighbors. Therefore, the binding energy per pair of A1 atoms is about 0.5 eV. Substitute these numbers into Eq. (8.16) and Eq. (8.17), we find the forces at the T site and the H sites which reproduce the result of first-principle calculations by Ciraci et al. (1990a). 

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. 

Eq. (9.12) does not require any specification of bonding - all atoms electrically interact with all other atoms. Sefcik et al. (2002) have combined QEq electrostatics with Morse potentials for non-electrostatic non-bonded interactions between all atom pairs to create a... 

Morse and Feshbach [499] have discussed the variation approach to a description of equations of motion for diffusion. Their approach is straightforward and is generalised here to consider the cases where there is an energy of interaction, U, between the pair of particles, separated by a distance r at time t. It is relatively easy to extend this to a many-body situation. The usual Euler form of the equation of motion is the Debye— Smoluchowski equation, which has been discussed in much detail before, viz. 

A major complication exists for constructing the Lagrangian density of a pair of particles diffusing relative to each other. The diffusion (Euler) equation is dissipative and the density of the diffusing species is not conserved. The Euler density, p, would lead to a space—time invariant, Sfr, which would not be constant. This difficulty requires the same approach as that used to handle the Schrodinger equation. Morse and Feshbach [499] define a reverse or backward diffusion equation where time goes backwards compared with that in eqn. (254)... 

Fig. 55. The probability that a pair of iodine atoms remain unreacted at a time f after they were formed with an initial separation of 0.7 nm. The encounter distance is 0.37 nm. A Morse potential energy getween the iodine atoms is imposed and the temperature was chosen as 300 K. The diffusion coefficient is 5 X 10 9 m2 s1 and iodine atom has a mass of 0.127 kg mol . Monte Carlo techniques were used to calculate the survival probability, with-----, toe — 2xl013s 1, cjc = 1013s l --------,...

Equation 2 is not a poor representation of the energy function because it can be shown that a Morse potential in real distances assumes a simple quadratic form if one uses a dimensionless bond order coordinate.1263 In any event, minimization of this function leads to the conclusion that the minimum energy is attained when all bonds have the same length. Furthermore, a bond alternating distortion that lengthens and shortens a pair of adjacent bonds by Ar can be shown to raise the cr-energy as in eq 3a. 

In Table II are given the three fundamental constants for a number of atom pairs which are of interest in chemical kinetics. From these constants Morse curves can be constructed as just shown for bromine, and these, in turn, are necessary for the calculation of activation energies of chemical reactions. 

Fundamental Constants for Calculating Morse Curves which Give the Energy of Atom Pairs as a Function of the Distance Separating Them... 

Exact calculations of AE have been carried out by Kelley and Wolfsberg [19] for colinear collisions between an atom and a diatomic molecule. The oscillator potential was considered to be both harmonic and Morse-type, and the interaction between the colliding pair was taken both as an exponential repulsion and as a Lennard-Jones 6 12 potential. Two important conclusions were reached First, when the initial energy of the oscillator increases, the total energy transferred from translation to vibration, AE, decreases. Second, the effect of using a Morse-oscillator potential in place of the harmonic oscillator was generally to decrease AE, often by more than a factor of 10. 

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... 

Figure 3.5 The two dimensional collision model for V-V transfer utilized by Berend and Benson [81] an atomic Morse-type interaction is employed between each pair of atoms.

Morse MD, Rice SA (1982) Tfests of effective pair potentials for water predicted ice structures. J Chem Phys 76 650- 660... 

Covalent Solids. Interatomic potentials are the most difficult to derive for covalent solids. The potential must predict the directional nature to the bonding (i.e. the bond angles). Most covalent solids have rather open crystal stmctures, not close packed ones. Pair potentials used with diatomic molecules, such as the Lennard-Jones and Morse potentials, are simply not adequate for solids because atoms interacting via only radial forces prefer to have as many neighbors as possible. Hence, qualitatively wrong covalent crystal stmctures are predicted. 

The principles of photoluminescence applied to solid oxide surfaces can be most easily understood by assuming some simplifications. For example, we can start by considering the Morse potential energy curves (Fig. 1) related to an ion pair such as M-+0-, taken as a harmonic oscillator to represent an oxide, typically an alkaline earth oxide. The absorption of light close to the fundamental absorption edge of this oxide leads to the excitation of an electron in the oxide ion followed by a charge-transfer process to create an exciton (an electron-hole pair), which is essentially... 

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