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Figure 5.4 Rotational levels of H2 for the first two vibrational states. Within the v = 0 state, the J = 2 ^ 0 transition at 28.2 |m is displayed. Also shown is the transition giving the 1-0 S(1) rovibrational line at 2.12 |m. Note that two different energy scales are used.

The moment of inertia of H2 is the smallest of any diatomic molecule, so equation (5.6) shows why its energy levels are widely spaced. Figure 5.4 displays the ladder of rotational levels. Here the energy E is given both as an equivalent temperature, E/kB, and in wave numbers, E/hc. Note that the two scales are similar numerically, since kB/hc = 0.70 deg-1 cm-1. The second measure is convenient in that the energy difference between two states yields directly the inverse of the wavelength of the emitted photon.1 As already noted, the rotational levels of H2 decay principally through electric quadrupole transitions, in which J decreases by 2. The lowest possible transition, J =2 ^ 0, has an associated energy change of 510 K and the relatively low Einstein A-value of 3.0 x 10-11 s-1. Each decay produces a photon of wavelength 28.2 |m. This far-infrared line has been detected through spaceborne observations.

The total energy of the hydrogen molecule is the sum of its rotational, vibrational, and electronic contributions:

In quantum mechanics, the energy of a simple harmonic oscillator of natural frequency va is

where w0 = 2nva, and where the vibrational quantum number v can be 0, 1, 2, etc. The J = 0 rotational level of the v = 1 state has an energy above ground equivalent to 6.6 x 103 K and an A-value of 8.5 x 10-7 s-1. In the hot environments where vibrational states are excited, the molecule relaxes through rovibrational transitions, in which both J and v change. (See Figure 5.4.) Here, the change in v is unrestricted, while A J can be 0 or ±2. Suppose v' and v"

1 In this chapter only, we follow the spectroscopic definition of wave number, k =