Introduction to Rigid Rotors
In a discussion of rotational energy levels, a very important property is the moment of inertia, I, of the molecule about any particular rotational axis. The moment of inertia of a molecule is generated by taking the mass of each atom in the molecule, multiplying it by the square of its perpendicular distance from the rotational axis, and summing these values together. i.e.
Note that the rotational axis must be one that passes through the centre of mass of the molecule.
The moment of inertia of a molecule is a measure of how difficult it is to rotationally accelerate the molecule – the larger the moment of inertia, the smaller the increase in angular momentum for a given applied torque.
The moment of inertia depends upon the masses of the atoms present and the molecular geometry, indicating that rotational spectroscopy will be able to give information about bond lengths and angles.
In general, the rotational properties of any molecule can be expressed using the moments of inertia about three mutually perpendicular axes. Conventionally, these axes are labelled Ia, Ib and Ic, choosing the axes in such a way that Ic ³ Ib ³ Ia . Note that for linear molecules, the moment of inertia around the molecular axis is zero, as all the atoms lie on the axis of rotation so are at zero distance from it.
For our purposes we shall make the supposition that molecules are rigid rotors, bodies that do not distort under the stress of rotation. Rigid rotors can be classified into four types:
Spherical rotors have three equal moments of inertia.
Symmetric rotors have two equal moments of inertia.
Linear rotors have one moment of inertia (that around the molecular axis) equal to zero.
Asymmetric rotors have three different moments of inertia.
The rotational energy levels of a rigid rotor may be obtained by construction and solution of the appropriate Schrodinger equation, but there is a much simpler approach that may be used. Classical mechanics gives expressions for the energy of a rotating body in terms of the angular momentum, and we may obtain the analogous quantum mechanical expressions by substitution of the quantum expressions for angular momentum.
The classical expression for a body rotating about a given axis with angular velocity ω is:
(Note the similarity to the classical expression for linear kinetic energy, E = mv2/2 – the moment of inertia is the rotational equivalent of the mass, and the angular velocity replaces the linear velocity.) A body free to rotate about three mutually perpendicular axes has an energy given by:
where the letters a, b and c distinguish the three rotational axes. Since the classical angular momentum is given by J = Iω, it follows that:
Now, in the discussion of the quantum mechanics of rotation in three dimensions, we stated that the magnitude of the angular momentum was given by a quantum number l, which was restricted to positive integral values (and zero). We may use this result generally in our discussion of rigid rotors.