Microtheme: Morse Potentials and Harmonic Oscillators
Physical Chemistry, CHM341 Fall 2016
Ossama Abu-Halawa
Bonds: Harmonic and Aharmonic Oscillation
When characterizing the motion of atoms, one can model their interaction using Hooke's law, where they interact with eachother in a spring like motion. The atoms interact in a series of attractive and repulsive forces. They can be modeled by measuring their potential, V(x), as a function of x. Thinking of Hooke's law and the spring model, x is the distance the molecule is displaced from the point of equilibrium. Using this model, a negative or positive displacement - a negative or positive x - gives rise to a potential that is symmetric on both sides of the equilibrium point of the spring. This potential energy arises due to the force acting on the masses to counter their motion; when the spring is compressed, the force of the spring acts in a such a way as to separate the masses, and when stretched, it acts in such a way as to bring them closer together. Figure 1 (adapted from figure 18.4 of Engel and Taylor's "Physical Chemistry") shows the spring model bond between two atoms of masses m1 and m2. When a graph of position, x, is plotted vs. potential, V(x), is plotted, the pattern of the relationship is a parabola. This is the classical harmonic model of the oscillation of motion between two atoms in a bond, which translates into the potential energy present between atoms in a bond.
One can then specificy distinct energy levels on the potential energy diagram and plot the wavefunctions and probability densities of the atoms on each energy level. When this is done, however, this model is in disagreement with the quantum mechanically derived probability densities of the electrons at each energy level. In other words, the harmonic oscillation model results in areas where the probability densities (eigenfunctions squared) go outside the potential energy bounds of the classical model. This is displayed in Figure 2. below (adapted from figure 18.9 of Engel and Taylor's "Physical Chemistry").
This, then, suggests that the classical harmonic model - the spring model - is not sufficient to describe the motion of atoms in a bond. In 1929, Morse proposed an aharmonic model to account for the vibrational motion of the atoms in a bond (equation 19.4 in Engel and Taylor's "Physical Chemsitry"):
In this equation, De is the dissociation energy of the bond, re is the equilibrium length of the bond, and a is a value related to the force constant and the dissociation energy. There are key differences in the characterization of the motion of bonds and their dissociation energies between the classical harmonic model and the aharmonic Morse potential model presented:
In the aharmonic model, the potential energy based on the displacement from the equilibrium position is not symmetrical as in the classical model. Rather, when the atoms are seperated by a small internuclear distance, r, the potential energy is very high (as the repulsion is extreme at such distance). When r is greater, the potential energy decreases to a point where is "levels off", as at this point the bonds has dissociated due to the great internuclear distance.
In the classical model, the average position of the electrons (derived from the probability densities) does not change as the energy level, n, increases, because the potential energies are perfectally symmetrical. In the aharmonic model, the potential energy is not symmetrical in regards to the displacement of the atoms from the equilibrium position, and therefore the average positions of the electrons changes with increasing energy levels.
In the classical model, the separation between energy levels is equal due to the symmetical nature of the potential energy. In the aharmonic model, however, the separation between energy levels decreases as n increases due to the shifting average electron positions and the corrollary decrease in potential energy at the shifting positions.
Real Life Examples: HF and HCl
The differences between the classical harmonic potential and Morse's aharmonic model are shown in the figures of HF and HCl, below:
The orange line is the parabolic relationship of the classical harmonic model, where the potential energy increases with internuclear distance. The blue line is the Morse potential of HF. The difference between the classical and aharmonic model accounts for the observable phenomenon of bonds breaking when the distance between the atoms is great enough; when the bonds are at a length of 2 angstroms, for example, the energy to overcome the bond is 1e-18 J in the aharmonic model vs. 2.5e-18 J in the classical model. The phenomena can also be observed with respect to the energy level diagram of the HF bond in the Morse model (below). What is shown is the decreasing distance between energy levels with increasing energy. This is substantiated to the point of the dissociation energy, where no difference between energy levels is seen as the energy levels off with the dissociation energy; this is the point where the bond breaks.
The same patterns displayed in the HF plot above are evident in the HCl plot as well. The green line is the classical harmonic model and the magenta line is the aharmonic Morse model.
Uses of the Morse Potential
The Morse potential can be used to compare chemical characteristics of models. Most notably, the Morse potential is used to compare the bond dissociation energies of the molecules. "HCl vs. HF Morse Potential" figure is an example of how the Morse Potential can be used to compare molecules; the blue line is HF and the magenta line is HCl. What is evident from the graph is that HF has a smaller bond length than HCl, evidenced by the location of the lowest energy state along the x-axis. Furthermore, HF has a higher dissociation energy than HCl, evidenced by the larger energy value of the plateau, where the bond dissociation takes place. Although the intial bond length is different between the two molecules, the dissociation occurs around the same length, 2.5 angstroms. This is determined by locating where the energy levels plateau.
