Morse equation, potential

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),... 

Morse equation phys chem An equation according to which the potential energy of a diatomic molecule in a given electronic state is given by a Morse potential. mors i kwa-zh3n ... 

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 ... 

The shapes of the absorption band associated with the intensities of vibrational transitions, are sensitive functions of the equilibrium bond length, about which approximately harmonic vibrational oscillations occur. Potential energy curves for a diatomic molecule (Figure 4.2), are commonly represented by Morse equation,... 

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). 

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... 

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... 

Fig. 5. Potential energy-distance relation for a simple charge transfer reaction using Morse equations to represent variation of energy with distance during stretching and forming of bonds (19).

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 ... 

Analytical solution of the Schrodinger equation for the Morse-function potential... 

Variational transition state theory (VTST) is useful when no TS can be explicitly identified (for Morse-like potential, e.g., direct bond dissociation), where there is no TS. It is based on the idea that there is a bottleneck in the phase space during the dissociatioa This can be explained by the fact that during the dissociation process the molecule needs to reach at a certain point a very specific conformation, without which it cannot go further to disassociate. The Arrhenius equation can be written in terms of exponential of Gibbs free energy and exponential of entropy, which characterize the nmnber of distinct states reachable with that amount of energy. 

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... 

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. 

The attractive energies 4D(cr/r)6 and ae2/2 r4 have two important effects on the vibrational energy transfer (a) they speed up the approaching collision partners so that the kinetic energy of the relative motion is increased, and (b) they modify the slope of the repulsive part of the interaction potential on which the transition probability depends. By letting m °°, we have completely ignored the second effect while we have over-emphasized the first. Note that Equation 12 is identical to an expression we could obtain when the interaction potential is assumed as U(r) = A [exp (— r/a)] — (ae2/2aA) — D. Similarly, if we assume a modified Morse potential of the form... 

In order to evaluate the above expression, solutions were found for the Schrodinger equation using the Morse potential for rotational quantum number i not equal to zero ... 

An independent estimate of the amount of p character of these bonds can be made with use of the assumption that a linear extrapolation of the low-lying vibrational energy levels (as indicated by the Morse potential function) will lead to the energy level of the atomic state involved in the bond. The equation... 

Numerical solutions of the Fleitler-London, or of density functional equations, show how energies depend on separation distance, but it is more instructive to consider semiempirical equations such as the Morse potential, or especially, the very simple Rydberg equation which has been shown to apply... 

A semi-classical treatment171-175 of the model depicted in Fig. 15, based on the Morse curve theory of thermal dissociative electron transfer described earlier, allows the prediction of the quantum yield as a function of the electronic matrix coupling element, H.54 The various states to be considered in the region where the zero-order potential energy curves cross each other are shown in the insert of Fig. 15. The treatment of the whole kinetics leads to the expression of the complete quenching fragmentation quantum yield, oc, given in equation (61)... 

---