Harmonic Oscillator
The diatomic harmonic oscillator is a simple model for the vibration of chemical bonds.
The model consists of two masses, m1 and m2, connected by a Hooke's Law spring with force constant k.
The separation of lowest potential energy is denoted by l0.
For a different separation, the restoring force is -k(l-l0).
Therefore the potential energy is (1/2)k(l-l0)2.
The exact quantum mechanical description of this model, obtained by inserting the above expression for the potential energy into the one-dimensional Schroedinger equation, yields an infinite set of allowed energies, with equal spacing of successive energy levels.
En = (n+ 1/2)hf
n is a quantum number with discrete values of 0, 1, ... Planck's constant is denoted by h, and f is the frequency, which is determined by k and the two masses, present as their reduced mass, mu.
f = (1/2pi)(k/mu)1/2
mu = m1 m2/(m1 + m2)
This oscillator has several properties
- It is always in motion, because the lowest allowable energy is greater than 0
- All energy levels have the same frequency (but the amplitude increases as the energy level increases)
- An increase in k, which corresponds to a stiffer spring (or stiffer bond, for molecules) will increase the frequency.
Double bonds tend to have higher frequencies than single bonds.
- An increase in mass will lower the frequency. Thus substitution of deuterium for hydrogen results in a decrease in frequency of the stretching of C-H bonds, and bonds to hydrogen tend to have higher frequencies than bonds of carbon to heavier atoms.
- For k and reduced masses of the size found in real molecules, the separation of adjacent energy levels, hf, is larger than thermal energy (kT) at the usual temperatures.
Therefore nearly all of the oscillators are in their ground state, with energy (1/2)hf.
The transition observed in an infrared spectrum is the promotion of this vibration to its first excited state, with energy (3/2)hf.
This process requires energy hf, with f in the infrared region of the spectrum.
- For real covalent bonds, the magnitude of the oscillation in the vibrational ground state is only a small fraction (a few percent) of l0.
- For real covalent bonds, the time for one vibration is on the fs scale (9-21 fs for water, 14-50 fs for carbon dioxide).
The fastest vibrations are those involving the stretching of bonds to hydrogen.
The harmonic oscillator is a useful model because its expression for the potential energy (as a parabola) is an adequate approximation to the true potential energy for many vibrations in real molecules, as long as one confines attention to that part of the potential energy surface close to the minimum.
Important problems will arise if we probe the parts of this potential energy surface at much higher energies relative to the minimum.
For example, the harmonic oscillator does not dissociate, no matter how high its energy. But real chemical bonds can undergo dissociation, if the vibration is promoted to a sufficiently high state.
This fact implies that the parabola becomes a poor approximation to the true potential energy when we move far above the minimum.
The Morse potential is an improvement over the parabola in this high energy region.
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July 1, 1999
Wayne L. Mattice: wlm@polymer.uakron.edu