Morse fitting function

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... 

A molecular dynamics calculation was performed for thorium mononitride ThN(cr) in the temperature range from 300 to 2800 K to evaluate the thermophysical properties, viz. the lattice parameter, linear thermal expansion coefficient, compressibility, heat capacity (C° ), and thermal conductivity. A Morse-type function added to the Busing-Ida type potential was employed as the potential function for interatomic interactions. The interatomic potential parameters were semi-empirically determined by fitting to the experimental variation of the lattice parameter with temperature. 

The a constant is the same as that appearing in the Morse function, but is usually taken as a fitting parameter. 

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.

FIGURE 3.2. Variation of the rate constants of dissociative electron transfer from aromatic anion radicals to butyl and benzyl halides as a function of steric hindrance. Data points from reference 10. Solid lines, best-fit parabola dashed lines, prediction of the Morse curve model, logAf-1 s-1). Adapted from Figure 3 of reference 6b, with permission from the American Chemical Society. 

The three parameters in the Morse function D, B, re are positive and are usually chosen to fit the bond dissociation energy, the harmonic vibrational frequency and the equilibrium bond length. At r = re, the Morse function V = 0. As r — D, V approaches D. For r re, V is large and positive, corresponding to short range repulsion. Although the Morse function has been used extensively, its representation of the potential away from re is not satisfactory. Several modifications have been proposed in Morse function. 

A number of other analytic potential functions such as Rotated-Morse-Curve-Spline (RMCS), Bond-Energy-Bond-Order (BEBO) have been used for fitting of ab initio surfaces. 

Solubilities of the Mg-caldte as a function of MaCOn constant. The solubility is expressed in line with Eq. (8.11) as lAP g-calcite = (Ca2 1 ) (Mg2+) CO 2). The solid curves represent the general trend of results on dissolution of biogenic and synthetic Mg-calcites. The curve fitting the data of Plummer and Mackenzie (1974) is dashed. The various points refer to the results of different researches. (For the origin of the data see Morse and Mackenzie, 1990.) (IAP = ion activity product.)... 

The substrate in these studies was restricted to be rigid, and Morse functions were used for the hydrogen-surface and two-body interactions. The parameters in the Morse functions were determined for single hydrogen atoms adsorbed on the tungsten surface by fitting to extended Huckel molecular orbital (EHMO) results, and the H2 Morse parameters were fit to gas-phase data. The Sato parameter, which enters the many-body LEPS prescription, was varied to produce a potential barrier for the desorption of H2 from the surface which matched experimental results. 

Apparently there are no experimental data on BeNe. If we fit a Morse function to the parameters we obtain for the dissociation curve, it is estimated that there would be 14-15 bound vibrational states for this Van der Waals molecule. Thus, VB theory predicts the existence of stable gaseous BeNe, if it is cold enough, since Dg is only 2kT for room temperature. 

IUMorse — Urkr /being 3.68 percent. Thus the Morse function generally gives a rather poor fit to U(R), but because of its simplicity and early invention, it has been widely used. 

The potential maximum could not be fitted by a parabola as in the singlet case a rA functional dependence was found to be more adequate. The deep chemical well at 1.045 A was represented by a Morse function, which reproduced Ginter s spectroscopic results.160... 

The frequency-time correlation function is dependent on the frequency and the force constants of the vibrational mode whose dephasing is being considered. They are determined by fitting the vibrational bond energies to a Morse potential of the following form ... 

Lippincott and Schroeder [1955, 1957] introduced a semiempirical two-dimensional PES and fitted their parameters from experimental data. Further studies in this direction were carried out by Savel ev and Sokolov [1975] and Sokolov and Savel ev [1977], Lautie and Novak [1980], Saitoh et al. [1981], and Emsley [1984]. These studies have shown that an adequate two-dimensional PES can be constructed from Morse functions of diatomic fragments XH and HY and repulsive functions representing the XY interaction. The values of rXH ar,d wXH and isotope effects as a function of R are in agreement with the experimental ones for OH O, OH-N, and NH-- N fragments. The dependencies rXH(/ ) and a>XH(R) collected by Novak [1974] are shown in Figure 6.1. The method of Lippincott and Schroeder [1957] is one of the versions of the general semiempirical method of bond energy-bond order (BEBO) developed by Johnston and Parr [1963] to construct a two-dimensional PES. 

Figure 3.14. Stabilities of calcite, and synthetic (closed squares) and biogenic (closed circles) magnesian calcites as a function of composition. Stabilities are expressed as -log IAPMg-Calcite- The curve is a hand-drawn "best" fit to the synthetic data. Also plotted are the results of precipitation experiments by Mucci and Morse (1984, open squares) and biogenic dissolution experiments by Walter and Morse (1984a, open circles). (After Bischoff et al., 1987.)...

Kupperman et a/.223-228 compared transition-state, classical, and quantum mechanical thermal rate constants using the SSMK surface.112 In ab initio potential energy surface calculations, the potential energy is known only as a list of values at selected geometries of the system. It becomes necessary, then, if trajectory calculations are to be made, to fit the calculated points to a smooth and continuous map . In their calculations, Kupperman et al. fit the collinear SSMK surface by the rotating Morse function procedure of Wall and Porter.227... 

More recently, Yates and Lester230 fitted Liu s surface with a slightly modified form of the Porter-Karplus formulas after first fitting Liu s H2 potential to a simple Morse function. They then use the resulting surface to calculate the three-dimensional classical trajectory of the system. Their empirical fit very closely duplicates Liu s saddle-point properties. Reaction probabilities on this surface are compared with those on the PK surface. 

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