Anharmonic Oscillator

A parabola is a useful approximation to the shape of the potential energy function for stretching of real chemical bonds at small distortions, where the potential energy is only slightly larger than the minimum. But a parabola becomes a poor approximation at larger distortions (and larger energies) because

• The parabola is symmetric, but the true potential energy function is asymmetric for large distortions. The true potential energy rises faster for compression than for extension.
• Real bonds can experience dissociation at high temperatures, but the parabola prohibits dissociation, because the potential energy rises without limit as the stretching increases.

When these anharmonic aspects of the oscillation become important, the energy levels obtained from the Schroedinger equation upon substitution of the Morse potential for the parabola of the harmonic oscillator provide a better description of the vibration of the system. The Morse potential is

V = D_e{1 - exp[-a(l-l₀)]}²

It approaches a constant limiting value, D_e, as l goes to infinity, but becomes very large as l becomes much smaller than l₀.

Dissociation energy

The dissociation energy is slightly smaller than D_e, because we need to subtract from D_e the vibrational energy in the ground state, (1/2)hf. Therefore the dissociation energy specified by the Morse potential is D_e - (1/2)hf.

Energy levels

With insertion of the Morse potential, the one-dimensional Schroedinger equation gives a set of energy levels with energies that are no longer equally spaced (as was the case with the harmonic oscillator), but instead become crowded closer and closer together as the energy level increases.

E = (n + 1/2)hf - (n+ 1/2)² x_e hf

Here the quantum number n has integer values 0, 1, ..., just as in the harmonic oscillator. The energy levels of the harmonic oscillator are recovered if we retain only the first term on the right-hand side (if x_e = 0). x_e is called the anharmonicity constant. It is related to the terms in the Morse potential as

x_e = hf/4D_e

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