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A triatomic molecule of great interest is the carbon dioxide molecule, which consists of one carbon atom and two oxygen atoms.  To a reasonable approximation we can assume that these three atoms, within the carbon dioxide molecule, lie in a line.  In order to investigate how these atoms vibrate within the molecule, we need a simple conceptualisation of the molecule that we can subject to mathematical analysis.

Such a conceptualisation is shown in the following figure.

This figure shows the carbon and oxygen atoms lying in a line.  The two oxygen atoms are assumed to be “connected” to the central carbon atom via springs with spring constant k.  The oxygen atoms are taken to be of mass m, and the carbon atom is taken to have mass M.

As in the previous sections, we can investigate the vibrations of the carbon dioxide molecule by writing down the equations of motion for the system.  To do this, we assume that each of the atoms is displaced from its equilibrium position, and hence evaluate the forces acting on the atoms due to the stretching or contraction of the springs that connect the atoms.

To simplify matters, we will consider vibrations only along the line of the molecule.  Let us assume, therefore, that each of the atoms is displaced from its equilibrium position, as shown in the above figure.  Consider the left-most oxygen atom.  The spring that connects it to the carbon atom is stretched by an amount x₂ – x₁ in the above figure.  Using this reasoning, the equations of motion for the three atoms are easily found to be:

Thus, in contrast to the simple systems considered in the previous sections, we now have three equations to consider.  In order to find oscillatory solutions of these equations, we will assume trial solutions of the form:

The A coefficients and the frequencies ω are to be determined.  It is now straightforward to substitute these solutions into the equations of motion and simplify the resulting equations by cancelling the common cosine terms.  Substituting in the trial solutions leads to the following three equations:

                                        (A)

Thus we are left with a system of three homogeneous equations for the amplitudes A.  Clearly, a trivial solution of these equations is

However, this is of little interest for our current problem.  In order for these equations to have a non-trivial solution, the determinant of the coefficients of the amplitudes A in equation (A) must equal zero.  This leads to the so-called secular equation for the system of equations (A).

Expanding this determinant leads to the following equation for the frequency ω.

This is a cubic equation from which we can determine the possible value of ω.  It is worth noting that there will be as many possible values of ω as there are “degrees of freedom” in the system.  In our simple system here, there are three – the possible movements of the three atoms along the line of the molecule.  Had we allowed motion of the atoms in more than one dimension (for example upwards and out of the plane of the figure above), then we would have additional equations of motion and additional allowed values of ω.

The equation above tells us that there are three allowed values of ω.  We can then substitute the allowed values of ω into the equations (A) to find the values of the amplitudes A.  The first solution is as follows:

This is a trivial solution.  There is no oscillatory behaviour, and this solution corresponds to a simple translation of the atoms along the line of the molecule.

The next two solutions correspond to oscillatory behaviour:

The first of these corresponds to a vibration of the molecule in which the two oxygen atoms vibrate in anti-phase to one another, while the carbon atom remains stationary.  In the second case, the oxygen atoms vibrate in phase, whereas the carbon atom vibrates in anti-phase, albeit with a different amplitude.

The three solutions are illustrated in the following figure.

Had we considered the possibility of movement of the atoms in the other two dimensions, then other modes of vibration would be possible.  Modes 2 and 3 in the above diagram are sometimes referred to as the “stretch mode” and the “asymmetric stretch mode” respectively.

THE END