The Morse Potential

The harmonic potential is a good starting place for a discussion of vibrating molecules, but analysis of the vibrational spectrum shows that real diatomic 

This potential actually contains three parameters, and R, and so should 

In the case of a simple calculation shows that the dissociation energy 

This potential actually contains three parameters, De, fc and Re, and so should be capable of giving a better representation to the potential energy curve than the simple harmonic, which contains just the two parameters ks and Rc. 

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During initialization and final analysis of the QCT calculations, the numerical values of the Morse potential paiameters that we have used aie given as... 

Next, we replace the stiff spring potential a(r — 1) /2 by the Morse potential... 

Compared with the Morse potential, Hooke s law performs reasonably well in the equilibrium area near If, where the shape of the Morse function is more or less quadratic (see Figure 7-9 in the minimum-energy region). To improve the performance of the harmonic potential for non-equilibrium bond lengths also, higher-order terms can be added to the potential according to Eq. (21). 

Here, Dg is the bond dissoeiation energy, rg is the equilibrium bond length, and a is a eonstant that eharaeterizes the steepness of the potential and determines the vibrational frequeneies. The advantage of using the Morse potential to improve upon harmonie-oseillator-level predietions is that its energy levels and wavefunetions are also known exaetly. The energies are given in terms of the parameters of the potential as follows ... 

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. 

Although two-parameter models are rather restrictive, three-parameter models of the intermolecular potential have been developed which provide reasonable descriptions of the thermodynamic behavior of solids. Examples include the Morse potential, the exponential-six potential, and, more recently, a form proposed by Rose et al. (1984) for metals. 

The hot bond fracture stress Morse potential energy function by [1]... 

There are a number of three- to five-parameter potential functions in the literature, of which the Morse potential is the most popular a typical five-parameter potential is the Linnett function (Linnett, 1940, 1942) ... 

For each bond type, i.e. a bond between two atom types A and B, there are at least two parameters to be determined, and /fg . The higher-order expansions and the Morse potential have one additional parameter (a or D) to be determined. 

A third functional form, which has an exponential dependence and the correct general shape, is the Morse potential, eq. (2.5). It does not have the dependence at long range, but as mentioned above, in reality there are also R, /f etc. terms. The D and a parameters of a Morse function describing vdw will of course be much smaller than for str, and Ro will be longer. 

Fig. 18. The Morse potential energy of a bond under equilibrium ( — ) and in the presence of an applied force equal to 0.6 fb (-)...

The principle of action and counteraction impose the condition that the acting external force must be balanced by the internal molecular forces. For an isolated diatomic molecule, this internal force (also called the binding force) is given by the derivative of the Morse potential (Fig. 19) ... 

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 ... 

Fig. 20. Bond scission activation energy and lifetime (Tt) plotted as a function of applied force. The solid curve is derived from Eq. (65) based on the Morse potential, the other data are redrawn from Ref. [101]. The upper abscissa gives the overall elastic strain before failure. The numbers indicate the minimum chain lengths which will fail at a particular force...

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... 

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 10.2 The Morse potential for the ground state of a diatomic molecule.

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... 

During initialization and final analysis of the QCT calculations, the numerical values of the Morse potential parameters that we have used are given as De = 4.580 eV, re = 0.7416 A, and (3 = 1.974 A-1. Moreover, the potential energy as a function of internuclear distances obtained from the analytical expression (with the above parameters) and the LSTH [75,76] surface asymptotically agreed very well. 

The viscosity therefore replaces the restraint on diffusion arising from the interaction of atoms expressed by the Morse potential in Swalin s treatment. 

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.

Many other useful forms have been proposed (Steele and Lippincott, 1962) and their parameters were related to spectroscopic constants as will be given for the Morse potential by Eq. (1.14). Quite often, the potential V(r) is expanded as a power series in the displacement from equilibrium (force field method)... 

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