Vibrations of molecules

The following discussion of the vibrations of molecules uses the binomial theorem. If you aren’t familiar with this theorem, you should read about it in the appropriate section of a math textbook.

When two atoms are separated by a few atomic diameters, they can exert attractive forces on each other. But if the atoms are so close that their electron shells overlap, the atoms repel each other. Between these limits, there can be an equilibrium separation distance at which two atoms form a molecule. If these atoms are displaced slightly from equilibrium, they will oscillate

Let’s consider one type of interaction between atoms called the van der Waals interaction. Our immediate task here is to study oscillations, so we won’t go into the details of how this interaction arises. Let the center of one atom be at the origin and let the center of the other atom be a distance r away (Fig. 14.20a); the equilibrium distance between centers is r = R_0. Experiment shows that the van der Waals interaction can be described by the potential-energy function.

where U0 is a positive constant with units of joules. When the two atoms are very far apart, U = 0; when they are separated by the equilibrium distance r = R₀, U = -U₀. From Section 7.4, the force on the second atom is the negative derivative of Eq. (14.25):

Figures 14.20b and 14.20c plot the potential energy and force, respectively. The force is positive for r < R₀ and negative for r > R₀, so it is a restoring force. Let’s examine the restoring force Fr in Eq. (14.26). We let x represent the displacement from equilibrium:

In terms of x, the force Fr in Eq. (14.26) becomes

This looks nothing like Hooke’s law, Fx = -kx, so we might be tempted to conclude that molecular oscillations cannot be SHM. But let us restrict ourselves to small-amplitude oscillations so that the absolute value of the displacement x is small in comparison to R0 and the absolute value of the ratio x/R0 is much less than 1. We can then simplify Eq. (14.27) by using the binomial theorem:

If | u | is much less than 1, each successive term in Eq. (14.28) is much smaller than the one it follows, and we can safely approximate (1 + u)ⁿ by just the first two terms. In Eq. (14.27), u is replaced by x/R0 and n equals -13 or -7, so

This is just Hooke’s law, with force constant k = 72U₀/R0 ². (Note that k has the correct units, J/m2 or N/m.) So oscillations of molecules bound by the van der Waals interaction can be simple harmonic motion, provided that the amplitude is small in comparison to R0 so that the approximation | x/R₀ | <<1 used in the derivation of Eq. (14.29) is valid.

You can also use the binomial theorem to show that the potential energy U in Eq. (14.25) can be written as U ≈ 1 2 kx² + C, where C = -U₀ and k is again equal to 72U0/R0 2 . Adding a constant to the potential-energy function has no effect on the physics, so the system of two atoms is fundamentally no different from a mass attached to a horizontal spring for which U = 1/2 kx² .

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