Question

What are the two modes of motion of a diatomic molecule about its centre of mass? Explain briefly the origin of the discrete energy level spectrum associated with one of these modes.

Answer 1

The two modes of motion of a diatomic molecule are (i) rotation and (ii) vibration. 

The first order rotational energy is ℏ² J (J + 1)/2I₀, where I₀ = M R²₀ is the moment of inertia of the molecule about an axis perpendicular to the line joining the nuclei; the energy being the same as for the rigid rotator. Clearly the spacing between successive levels is unequal; it progressively increases with the increasing value of J , where J = 0, 1, 2 ... The spectrum called band spectrum arises due to optical transitions between rotational levels. The band spectrum is actually a line spectrum, but is thus called because the lines are so closely spaced and unresolved with an ordinary spectrograph, and give the appearance of a band. 

The second mode consists of to and fro vibrations of the atoms about the equilibrium position. The motion is described as simple harmonic motion. The energy levels are given by E_n = ℏω (n + 1/2), where n = 0, 1, 2 ... and are equally spaced. However as J or n increases, the spacing between levels becomes smaller than that predicted from the simple rigid rotator and harmonic oscillator.