Within the harmonic and rigid rotor approximations, the rotational-vibrational energy levels (in wavenumbers) of a diatomic molecule are given by

, where

,

are the vibrational and rotational quantum numbers, respectively,

is the harmonic vibrational constant, and

is the rotational constant. The right panel shows the ground and first excited vibrational states, labeled

and

, respectively, with their associated manifolds of rotational states labeled

and

. Allowed transitions must obey the selection rules

,

. The left graphic shows transitions where

,

(referred to as the

-branch and shown in blue), and

(referred to as the

-branch and shown in red). The arrows shown on the right indicate a transition between a pair of energy levels. The corresponding transition frequency is highlighted in the spectrum on the left.