The primary selection rule governing rotational transitions in molecules, particularly for rigid rotors, is ΔJ = ±1.
Understanding which transitions are allowed and which are forbidden is fundamental to interpreting molecular rotational spectra. This rule dictates the changes in the rotational quantum number (J) that a molecule can undergo when absorbing or emitting electromagnetic radiation.
Understanding the Rotational Selection Rule
The selection rule for rotational transitions, as derived from the symmetries of the rotational wave functions in a rigid rotor, is ΔJ = ±1, where J is a rotational quantum number. This rule arises from the interaction of the molecule's electric dipole moment with the electric field of the light and the quantum mechanical requirement that the transition moment integral must be non-zero.
For a transition to occur, the molecule must possess a permanent electric dipole moment. Homonuclear diatomic molecules (like O₂, N₂) or symmetrical linear molecules (like CO₂) do not have a permanent dipole moment and therefore do not exhibit pure rotational spectra.
What ΔJ = ±1 Means
The rule ΔJ = ±1 specifies that during a rotational transition, the rotational quantum number J can only change by exactly one unit.
- ΔJ = +1: This corresponds to absorption of energy, where the molecule moves to a higher rotational energy level (J to J+1).
- ΔJ = -1: This corresponds to emission of energy, where the molecule moves to a lower rotational energy level (J to J-1).
Transitions where J changes by more than one unit (e.g., ΔJ = ±2, ±3, etc.) or where J does not change (ΔJ = 0) are forbidden by this selection rule in pure rotational spectroscopy.
Examples of Allowed and Forbidden Rotational Transitions
Based on the ΔJ = ±1 rule, we can easily identify which transitions are permitted:
-
Allowed Transitions:
- J = 0 to J = 1 (ΔJ = +1)
- J = 1 to J = 2 (ΔJ = +1)
- J = 2 to J = 1 (ΔJ = -1)
- J = 3 to J = 4 (ΔJ = +1)
- ... and so on for any initial J value.
-
Forbidden Transitions:
- J = 0 to J = 2 (ΔJ = +2)
- J = 1 to J = 3 (ΔJ = +2)
- J = 2 to J = 0 (ΔJ = -2)
- J = 1 to J = 1 (ΔJ = 0)
- ... and any transition where |ΔJ| ≠ 1.
This rule is crucial in determining the appearance of a rotational spectrum, where only lines corresponding to ΔJ = ±1 transitions are observed.
Summary Table: Rotational Selection Rules
| Change in J (ΔJ) | Status | Note |
|---|---|---|
| +1 | Allowed | Absorption (to higher J level) |
| -1 | Allowed | Emission (to lower J level) |
| 0 | Forbidden | No change in J |
| ±2, ±3, etc. | Forbidden | Changes by more than one unit |
In summary, for molecules with a permanent dipole moment undergoing pure rotational transitions, only those transitions resulting in a change of the rotational quantum number by exactly one unit (ΔJ = ±1) are spectroscopically active and observable.
