Morse potential energy levels

Figure 3.6 shows the Morse potential energy curves for two hypothetical electronic states of a diatomic molecule, the vibrational energy levels for each, and the shape of the vibrational wavefunctions (i//) within... 

A simple case is the portrayal of the ground state of the OH radical, see Figure 2.14. For simplicity, we show a part of the Morse potential energy curve with the highest bound and quasibound vibrational levels indicated by solid lines and dashed line, respectively, Figure 2.14a. The spectral density is... 

Figure 28-7 In force field calculations, different levels of approximations are used to reproduce the stretching and compression of chemical bonds The plot shows a Morse potential energy function siipenmposed with various power series approximations (quadratic, cubic, and quartic functions) Note that the bottoms of the curves, representing the bond length for most chemical bonds of interest to medicinal chemists, almost overlap exactly. This nearly perfect fit in the bonding region is the reason simple harmonic functions can be used to calculate tx>nd lengths for unstrained molecular structures in the force lield method.

The Morse potential energy curve may be seen as a function that may be approximated (as the energy increases) by wider and wider parabohc sections. No wonder, therefore, that the level separation decreases. The number of energy levels is finite. ... 

There is no theoretical basis for a Morse potential energy curve. Its form is empirical (that is, based on observation), but it is useful. First, it shows a dissociation limit, just as real diatomic molecules experience. The dissociation energy, D, appears in two places in the form of the Morse potential, as a premultiplicative term and as part of the definition of the constant a. It accurately predicts the observed trend of closer-spaced vibrational levels as the vibrational quantum number increases. Although diatomic molecules (and larger molecules also) do not behave as perfect Morse oscillators, the Morse potential is usually a better fit to the real vibrational behavior of molecules. 

Figure 7.34 Potential Morse, the energy levels and the dissociation energy D.

Two of the most severe limitations of the harmonie oseillator model, the laek of anharmonieity (i.e., non-uniform energy level spaeings) and laek of bond dissoeiation, result from the quadratie nature of its potential. By introdueing model potentials that allow for proper bond dissoeiation (i.e., that do not inerease without bound as x=>°o), the major shorteomings of the harmonie oseillator pieture ean be overeome. The so-ealled Morse potential (see the figure below)... 

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 ... 

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. 

The symmetric stretching surface behaves like the usual Morse potential with two CH bonds undergoing dissociation simultaneously. The potential surface widens from the harmonic profile and the vibrational levels come closer when the energy increases frequencies are shifted towards longer wavelengths. 

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.

Since all photochemical reactions require the absorption of a photon, the result is that the reactant molecule is raised to a higher energy level. The outcome of this process depends on the nature of the upper and lower electronic states of the molecule. Four types of absorption behaviour are possible and we will first illustrate these by referring to Morse curves for the simple, diatomic, molecules. Although the potential energy of a complex molecule as a function of its molecular geometry is not a simple two-dimensional curve but a complex multidimensional surface, the conclusions arrived at by the use of Morse curves are instructive. 

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... 

Figure 3.2 shows the experimental potential well for the H2 molecule, compared with the harmonic fit and the Morse potential. Horizontal lines represent quantized energy levels. Note that, as vibrational quantum number n increases, the energy gap between neighboring levels diminishes and the equilibrium distance increases, due to the anharmonicity of the potential well. The latter fact is responsible for the thermal expansion of the substance. 

Figure 1. Morse potential V(q), vibrational levels Ev, and wave functions < (q) for the model OH (adapted from Ref. 14). The arrows indicate various selective vibrational transitions as well as above-threshold dissociations (ATDs) induced by IR femtosecond/picosecond laser pulses, as discussed in Sections III.A-III.D see Figs. 3-5. Horizontal bars on the arrows mark multiple photon energies ha of the laser pulses cf. Table 1. The resulting ATD spectrum is illustrated by the insert above threshold.

The main problem with this equation is the description of the potential energy term (V). As we shall see, insertion of a restricted form of the potential allows one to express data on the ro-vibrational levels in terms of semi-empirical constants. If the Morse potential is used, the ro-vibrational energies are given by the expression... 

The true potential energy curve can be determined from experiment if sufficient data about the vibrational levels are obtained we will describe how this is accomplished later in this chapter. As we mentioned earlier, improvements over the Morse potential have been described, a particularly important one being due to Hulburt and Hirschfelder [64] ... 

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