The Morse Equation

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),... 

Spectroscopic energy of dissociation of a diatomic molecule in the Morse equation... 

In an assembly of molecules such as a molecular crystal, there are minima, that is, pits, in the voids between the molecules, and the Poincare-Hopf relation is replaced by the Morse equation (Johnson 1992)... 

It can be seen at once that no simple relation (in particular not Ux = kip, a simple harmonic relation) can represent these potential energy-distance relations. As known since the 1930s, from gas phase spectroscopy, curves with the appearances of those shown in Figs. 9.15 and 9.16 can be represented in form by an empirical relation, the Morse equation ... 

However, it is of interest to use the Taylor-MacClaurin expansion procedure on the Morse equation and see what happens if one takes a(r- re) as 1. One obtains at once ... 

The harmonic oscillator model does not take into account the real nature of chemical bonds, which are not perfect springs. The force constant k decreases if the atoms are pulled apart and increases significantly if they are pushed close together. The vibrational levels, instead of being represented by a parabolic function as in equation (10.3), are contained in an envelope. This envelope can be described by the Morse equation (Fig. 10.5) ... 

The Morse equation describes the potential energy curve very satisfactorily. Its main disadvantage lies in the fact that when r = o, a finite, although large, value of the potential energy is obtained instead of an infinite value. Various modifications to this formula have been proposed by Rosen and Morse and by Poschl and Teller. ... 

In this model the ionization step is pictured as a tunneling of jmto-tons between two potential energy minima symmetrically located. Conway and co-workers used the Morse equation, adthough Lippincott and Schroeder s might be more suitable (see Section 8.2.5). 

The first term on the right-hand side in Equation (5.10) is the same as — AE in Equation (5.9), but explicitly showing that (3 depends on R. The dependence shown follows from the repulsion term, depending on the first power of the overlap integral, rather than on the square of it. Equation (5.10) is the solid-state version of the Morse equation. 

What is perhaps the most remarkable feature of these examples is the sharp sensitivity of the activation energy of bond-breaking processes to quite small changes in ground-state bond distances. An analogous observation applies to equilibrium distances and dissociation energies Dq of many classes of chemical bonds [62]. It can be accounted for by a simple modification of the Morse equation... 

A different model was proposed by Biirgi and Dunitz [53], who observed that the properties of fractional bonds can be described using simple modifications of interatomic potential functions, such as the Morse equation or the general inverse... 

The energy for a small displacement along the reaction pathway must actually depend on the distance changes for all bonds affected each of these changes can be taken as a linear function of the reaction coordinate Ar, the change in the exocyclic C-0 distance. Thus, in the harmonic approximation, one obtains 2V= k(AArif, and in order to yield this expression for small Arj the Morse equation is adapted in the following way ... 

Due to the nonlinearity of the potential energy-distance relations, one may expect the symmetry factor to depend on potential. Its effect will be examined in this section using Morse cmrves for the stretching of the bonds A—B+ and M—A (19). In Fig. 5, cimve D-D represents the potential energy-distance relation for stretching of the A—B+ bond, which is expressed by the Morse equation... 

The bond energy of a diatomic molecule varies with the bond length as shown in Figure 4.3. The energy is most favorable at the bottom of the potential well which corresponds to the equilibrium bond length. One equation that models the kind of relationship shown in Figure 4.3 is the Morse equation. 

We first find the potential energy as a function of internuclear distance for all the possible diatomic molecules that can be made from the atoms X, Y, and Z. This can be conveniently done by using spectroscopic data to obtain the constants in the Morse equation for the potential of a diatomic molecule ... 

Batsanov SS (1998) Estimation of the vtm der Watils radii of elements with the use of the Morse equation. Russ J Gen Chem 68 495-500... 

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