Spherical Rotors

A spherical rotor is one for which the three moments of inertia (about mutually perpendicular axes) are equal. This implies that the molecule must be highly symmetric. (In fact, molecules that are spherical rotors must belong to a cubic or icosahedral point group – see the symmetry section, here, for a further explanation.)

Examples of spherical rotors are tetrahedral molecules such as methane, CH4, and octahedral molecules such as SF6:

We shall designate the moment of inertia around any of the axes in a spherical rotor as I, which means that the classical expression for the energy of such a body is given by:

where J is the angular momentum of the body. From the section on the quantum mechanics of rotation, we know that angular momentum can be written as a function of a quantum number, which we shall call J. The expression is:

Note italic J is used for the magnitude of the angular momentum, a normal J is used for the quantum number.

In the majority of uses that follow it will be the quantum number that is used, but the context should make it clear.
The energy of a spherical rotor is thus quantised as follows:

which indicates that the spacing of the energy levels increases with increasing quantum number:

It can be seen from the above expression that the heavier the molecule is (the larger I is), the closer the energy levels are.

It is common to express the energy in terms of the rotational constant, B, of the molecule, which is defined as:

which makes the expression for the energy:

The rotational constant has units of cm-1 , making it a wavenumber. The energy of a rotational state is commonly reported as the rotational term of that level, F(J), which is also a wavenumber. It is obtained by division of the energy by hc: