9.2 Molecular spectra

Molecular energy levels are more complicated than atomic energy levels because molecules can also vibrate and rotate. The energies associated with such motions lie in different ranges and can therefore be studied separately. Electronic transitions are of order 1 eV, vibrational transitions are of order ${10}^{\mathrm{-2}}\text{eV},$ and rotational transitions are of order ${10}^{\mathrm{-3}}\text{eV}\text{.}$ For complex molecules, these energy changes are difficult to characterize, so we begin with the simple case of a diatomic molecule.

According to classical mechanics, the energy of rotation of a diatomic molecule is given by

where I is the moment of inertia and L is the angular momentum. According to quantum mechanics, the rotational angular momentum is quantized:

where l is the orbital angular quantum number. The allowed rotational energy level    of a diatomic molecule is therefore

where the characteristic rotational energy of a molecule is defined as

For a diatomic molecule, the moment of inertia with reduced mass $\mu$ is

where $r_0$ is the total distance between the atoms. The energy difference between rotational levels is therefore

A detailed study of transitions between rotational energy levels brought about by the absorption or emission of radiation (a so-called electric dipole transition    ) requires that

This rule, known as a selection rule    , limits the possible transitions from one quantum state to another. [link] is the selection rule for rotational energy transitions. It applies only to diatomic molecules that have an electric dipole moment. For this reason, symmetric molecules such as ${\text{H}}_2$ and ${\text{N}}_2$ do not experience rotational energy transitions due to the absorption or emission of electromagnetic radiation.