The rigid rotor is a good starting point from which to construct a model of a rotating molecule. It is assumed that component atoms are point masses connected by rigid bonds. A linear molecule lies on a single axis and each atom moves on the surface of a sphere around the centre of mass. The two degrees of rotational freedom correspond to the spherical coordinates θ and φ which describe the direction of the molecular axis, and the quantum state is determined by two quantum numbers J and M. J defines the magnitude of the rotational angular momentum, and M its component about an axis fixed in space, such as an external electric or magnetic field. In the absence of external fields, the energy depends only on J. Under the rigid rotor model, the rotational energy levels, F(J), of the molecule can be expressed as,
where  is the rotational constant of the molecule and is related to the moment of inertia of the molecule. In a linear molecule the moment of inertia about an axis perpendicular to the molecular axis is unique, that is, , so 
For a diatomic molecule
where m and m are the masses of the atoms and d is the distance between them.
Selection rules dictate that during emission or absorption the rotational quantum number has to change by unity; i.e., . Thus, the locations of the lines in a rotational spectrum will be given by
where  denotes the lower level and  denotes the upper level involved in the transition.
The diagram illustrates rotational transitions that obey the =1 selection rule. The dashed lines show how these transitions map onto features that can be observed experimentally.  Adjacent  transitions are separated by 2B in the observed spectrum. Frequency or wavenumber units can also be used for the x axis of this plot.