Morse potential approximation

Anharmonicity is conveniently treated using the Morse-potential approximation. The sine term of Eq. (8) is then to a first approximation replaced by uu2 2... 

Figure 5. Vibrational levels of HBr in the Morse potential approximation.

Here k and K are the harmonic constants of the longitudinal and hehcoidal springs, respectively. The last term in the Hamiltonian represents a Morse potential approximating the potential of the hydrogen bonds. Eventually, D and a are the depth and the inverse width of the Morse potential well, respectively. 

AMorse function best approximates a bond potential. One of the obvious differences between a Morse and harmonic potential is that only the Morse potential can describe a dissociating bond. 

In 1936, de Boer formulated his theory of a stressed bond which, despite its simplicity, still constitutes the basis for most models of chemical reactivity under stress [92], In order to fracture an unstressed bond which, in the absence of any vibration, is approximated by the Morse potential of Fig. 18, an energy D must be supplied. If, however, the bond is under tension due to a constant force feitt pulling on either end, the bond rupture activation energy will be decreased by an amount equivalent to the work performed by the mechanical force over the stretching distance from the equilibrium position. The bond potential energy in the presence of stress is given by ... 

With the approximated functions for e1 and e2 and with a Morse potential, M, that describes the observed properties of Eg we can solve eq. (1.56) and obtain... 

The harmonic approximation is only valid for small deviations of the atoms from their equilibrium positions. The most obvious shortcoming of the harmonic potential is that the bond between two atoms can not break. With physically more realistic potentials, such as the Lennard-Jones or the Morse potential, the energy levels are no longer equally spaced and vibrational transitions with An > 1 are no longer forbidden. Such transitions are called overtones. The overtone of gaseous CO at 4260 cm (slightly less than 2 x 2143 = 4286 cm ) is an example. 

An approximation to the potential U(R) for a diatomic molecule is the Morse potential... 

Figure 8.1 The harmonic potential and the Morse potential, together with vibrational energy levels. The harmonic potential is an acceptable approximation for molecular separations close to the equilibrium distance and vibrations up to the first excited level, but fails for higher excitations. The Morse potential is more realistic. Note that the separation between the vibrational levels decreases with increasing quantum number, implying, for example, that the second overtone occurs at a frequency slightly less than twice that of the fundamental vibration.

Figure 1.3 Potential curve of a molecule (ground state of HC1). The full curve is the Morse potential of Eq. (1.6). The dashed curve is the harmonic approximation. De is the dissociation energy, and re is the equilibrium separation.

For typical values of p, re and V, encountered in molecules, Eq. (l.ll) is an excellent approximation to the exact solution (better than l part in 109). The Morse potential is the simplest member of a family of potentials that give rise to a vibrational spectrum of the functional form E(v) = coc(v +1/2) -a>exe(v +1/2)2. This is quite realistic at lower levels of excitation. The vibrational spectrum does not however suffice, by itself, to specify the potential uniquely. The dependence of the eigenvalues on the rotational state is therefore important. For / 0 (as well as for the / = 0) the energy eigenvalues are given by... 

The expressions of the Sections 1.5 and 1.6 are general and apply to any solution of the Schrodinger equation. In the special case of a Morse potential, the radial integrals in Eq. (1.34) can be evaluated, with some approximations, in closed form. The approximation consists in replacing the lower limit of integration by -oo. This approximation is similar to that used in Section 1.3 when obtaining the wave functions. Thus... 

The typical behavior of M0 v is shown in Figure 1.6. One should note that, for the Morse potential, and in lowest approximation, the radial wave functions and thus v are independent of /. This is no longer the case for more general potentials and for the exact solution of the Morse problem. 

Dunham obtained these eigenvalues using the semiclassical approximation for the potential (1.8) which is an expansion in powers of (r - re)/re. The results for the Morse potential [Eq. (1.14)] can also be written in this form, as can results for other potentials. One therefore often uses Eq. (1.71) as a convenient empirical form. A slightly different form of (1.71) is... 

For these vibrations, the quantization scheme of Section 4.2 can be carried over without any modification (Iachello and Oss, 1991a). The potentials in each stretching coordinate 5 are in an anharmonic force field approximation represented by Morse potentials. The boson operators (Ot,xt) correspond to the quantization of anharmonic Morse oscillators, with classical Hamiltonian... 

Morse potential phys chem An approximate potential associated with the distance r between the nuclei of a diatomic molecule in a given electronic state it is V(r) = D 1 - expl - a(r - rj where q is the equilibrium distance, D is the dissociation energy, and n is a constant. mors po.ten-chol mosaic gold See stannic sulfide. mo za-ik. gold ... 

The corresponding EDFs for both potentials calculated for the same test set of the Fe(II) complexes are plotted in Figs. 5, 8. The systematic errors in all cases are close to zero, as can be seen from the parameters of the EDFs for the EHCF/MM with the Morse potential — r = —0.005, (7=0.056 with the NR potential — /r =0.002, (7=0.052. However, although somewhat improved the stiffness of the Morse potential again manifests itself the systematic errors for the separate LS and HS sets do exist and only approximately cancel each other in the total set, whereas for the NR... 

There have been several studies of the iodine-atom recombination reaction which have used numerical techniques, normally based on the Langevin equation. Bunker and Jacobson [534] made a Monte Carlo trajectory study to two iodine atoms in a cubical box of dimension 1.6 nm containing 26 carbon tetrachloride molecules (approximated as spheres). The iodine atom and carbon tetrachloride molecules interact with a Lennard—Jones potential and the iodine atoms can recombine on a Morse potential energy surface. The trajectives were followed for several picoseconds. When the atoms were formed about 0.5—0.7 nm apart initially, they took only a few picoseconds to migrate together and react. They noted that the motion of both iodine atoms never had time to develop a characteristic diffusive form before reaction occurred. The dominance of the cage effect over such short times was considerable. 

Thus this model predicts an infinite sequence of evenly spaced levels with no allowance for dissociation. An approximation to the effective potential for a diatomic molecule proposed by Morse has proved to be extremely useful. The Morse potential... 

In order to elucidate this concept, let us consider perhaps the simplest molecular system, the hydrogen molecular ion H2+. Within the Born-Oppenheimer approximation, the potential in the ground electronic state of this molecular ion is very well represented by the following Morse potential [48] ... 

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