Morse bond order

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. 

It is not necessary to restrict ourselves to bonds that are described by Morse potentials. We can regard eqn. (56) as a quadratic equation in x, use any form of the potential energy V(R) with the usual shape (i.e., a minimum, a repulsive barrier at short distances, and a monotonical increase at large distances), and determine x to get another definition of the bond order. This is called the unity bond index quadratic exponential potential (UBI QEP) method by Shustor-ovich and Sellers. ... 

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. 

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. 

For any process where only one elementary reaction coordinate changes, the energy will be a quadratic function of the corresponding reaction coordinate. This requires that some sort of bond order coordinate be used for any bond making/breaking process so that the Morse curve is transformed into a quadratic, and more generally that it may be necessary to carry out some coordinate transformation to obtain quadratic behavior. 

Heterolytic bond formation)cleavage can be treated as a simple reaction dimension. This assumes that with only bond cleavage, energy as a function of r will be described by a Morse curve, or in terms of a bond order-related coordinate energy will be described by a quadratic. 

Here the simplest way to proceed is to keep the Morse parameters (Q0, / (), and a) the same for both the isolated M-A bond and each additive M-A contribution within M -A so that these contributions would differ by their bond orders x,- (i = 1,2,..., n) only. Experimentally, Q(n) increases less than linearly with n... 

TABLE 9.6 Morse Parameters and Bond Order for Homonuclear Diatomic Group IA and IB Neutral and Anions... 

Before leaving this section, we should mention the application of simple bond-order conservation models (Shustorovich 1985, 1986, 1987, 1988) to molecule-surface PESs. In this model, one assumes that each gas atom-surface atom bond is a simple Morse potential, yielding the total interaction between A-S as ... 

It is well known [6, 14a, 27] that the Marcus relation may be derived from a model of intersecting parabolas for the reactant and product energy curves (Fig. 19.2). A parabola provides an unrealistic description of the energy profile for extension of an A-H bond far from its equilibrium length and towards complete dissociation a Morse curve offers a better description. It is perhaps not too widely appreciated [28, 29] that transformation of a Morse function (Eq. (19.7)) from bond length (r) to bond order (n) coordinates (Eq. (19.8)) yields a parabola (Eq. (19.9)). 

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.

Results of the estimation of bond energy with a metal surface by a phenomenological Bond Order Correlation - Morse Potential method [2]. 

In addition to reaction energies, activation barriers are also important in interpreting reaction processes. It is, of course, difBcult to locate transition states for surface reactions. In order to circumvent the problem, Shustorovich [78] and Shustorovich and Sellers [82] have developed the bond order conservation-Morse potential (BOC-MP) and UBI-QEP methods, respectively, to predict accurately activation energies for dissociations or combinations of adsorbates on metal surfaces. For the dissociation reaction of CHxOHads or H2O, the activation energy is given by... 

The problem is to formulate these relationships quantitatively. One approach, called the interacting state model (ISM) uses the Morse curves for the C-H and H-X bonds as the starting point and describes the TSs in terms of the length of the C-H and C-X bonds at the The total bond order is taken to be 1.0 unless one of the... 

Bond Order Conservation-Morse Potential (BOC-MP) formalism is used to study the change in activation energy for different elementary steps involving all possible chemical species on any metallic smrface in order to identify optimum catalysts for the production of methane, methanol or ethylene from synthesis gas. All possible steps are analyzed for the methanation reaction. It is assumed that the reaction occurs on pure metallic surfaces, thus, the influence of the support or promoters is not taken into account. The method suggests that Ni is a more selective for methanation reaction, in agreement with the fact that Ni/SiOz is the real catalyst used industrially. On the other hand on the surface of Fe or W the reaction may residt in the formation of adsorbed C, also in agreement with experiment. 

If the two center interaction that comprises the chemical bond is described by a Morse potential then a simple relation follows between the potential Q x) and the bond order ... 

A second approach which may be attractive for more complex surface systems involves the application of the Bond Order Conservation model that was developed by Shustorovich and co-workers . The BOC model treats the interaction between the adsorbate and the surface atom through the use of a Morse potential. The total heat of adsorption is then described by summing all interactions. The BOC model is based on the concept that the bonding potential for every atom in the system is conserved. The heat of adsorption for an atomic species A is described by the following expression ... 

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