The Chemical Bond Potential Energy Function

This section supplements the discussion in Section 19.4 on the Morse potential energy function. You can change the value of the variable Molecule to see the Morse function and its parameters for any of the following six diatomics: H₂, D₂, Li₂, F₂, HF, and LiF. The equilibrium bond lengths R_e and the two vibrational constants w_e and w_ex_e for these molecules (taken from Table 19.2) are built into this page, and the Morse function quantities D_e (the potential well depth), b, and v_D (the vibrational quantum number of the last—highest energy—vibrational level) are calculated from them. The vibrational levels for v = 0, 5, and 10 are also shown.

The Morse potential function itself is the red line, and the dotted orange line is the harmonic potential approximation to the Morse function. As you go from molecule to molecule, pay attention to the following changes:

1. H₂ and D₂ have the same potential parameters (nearly—that they don't agree exactly is a consequence of the approximations inherent in the Morse function itself), but note how the vibrational energy levels are lowered for D₂ in comparison to H₂. This is simply the effect of the change in reduced mass between H₂ and D₂.
2. Note that the heavier diatomics have many more vibrational levels than do the lighter ones, in general. (Look at the v_D values.)
3. Compare the harmonic approximation to the Morse curve. You can easily see that the harmonic approximation is not very good for highly excited vibrational levels of any chemical bond.

H₂ Morse vibrational wavefunctions

The QuickTime movie below shows the vibrational wavefunctions of the Morse potential for H₂. Note how the higher vibrational levels have wavefunctions that are largest in the region of the outer classical turning point and lack the symmetry of harmonic oscillator wavefunctions.

The QuickTime movie below follows the ground vibrational state (v = 0) wavefunction of H₂ as the rotational quantum number J is increased. Note how the total energy increases with rotational energy (increasing J). Note as well the change in appearance of the potential energy function. As J increases, the effect of the centrifugal potential becomes more and more important. Follow the location of the peak of the wavefunction as J increases. You will see that the peak moves to larger R as J increases, demonstrating centrifugal distortion, the general phenomenon of increasing bond length with increasing rotational motion. Finally, note that the energy of the highest J level, J = 33, is above the dissociation limit at V(R) =0. The molecule in this state is unstable and will spontaneously dissociate; it has a rotational energy greater than its bond energy.

Back to symmetry effects in vibration-rotation spectra. Ahead to triatomic normal modes of vibration. Back to the Index.   Copyright 1999, 2000 John S. Winn. All rights reserved. Last updated September 8, 2000.