where Jrefers to the change in the Jquantum number for the rotational
transition.* This is for an ideal process. Under real conditions, this selec-
tion rule is not strictly adhered to, but variances from equation 14.18 are
rare and such transitions are only weakly absorbing or emitting (and so
practically unnoticeable). Equation 14.18 therefore defines allowedrota-
tional transitions. Other transitions where Jis different from 1 are
forbidden.
Linear molecules also have the zcomponent of their total angular momen-
tum quantized, and if the total angular momentum is changing by 1 unit, then
that unit may be in the z-axis. But then again, it may not, so the MJquantum
number might either stay the same or change by 1 unit. Therefore, the selec-
tion rule for MJis written
MJ0,1 for linear molecules (14.19)
We have already seen that symmetric top molecules have a second quantum
number,K, used to define the quantized figure-axis component of the angular
momentum. Interaction of the light field with the molecule’s dipole, which
must lie on the figure axis, is such that the electromagnetic field cannot change
the molecule’s rotation with respect to the figure axis. Therefore, the quantum
number Kis used to define the selection rule
K 0 (14.20)14.6 Rotational Spectroscopy
Now that the selection rules have been established, the energies of transition
can be determined. We want to determine E, the change in energy between
two rotational states involved in a transition. For now, we will express Ein
terms of units of energy (usually J).
For a diatomic or linear molecule,Ein a rotational transition is the dif-
ference in energy of two adjacent states. For absorption, the transition can be
labeled as E(J) →E(J 1), and the difference in energy is given by the differ-
ence between final and initial states:
EE(J)→E(J+1)E(J 1) E(J)(J)(J
2 I
1)^2
EE(J)→E(J+1)
(2J
2 I
2)^2
(J
I
1)^2
(14.21)
In an equation such as this that contains a quantum number, it is important
to remember which state the quantum number Jstands for. In this case, we are
considering absorption, and Jstands for the quantum number of the lower-
energy state. To be consistent in this definition ofJ, for emission the transition
will be written as E(J 1) →E(J), and the energy difference will have the same
magnitude but with a negative sign, indicating that energy is lost.Jstill repre-
sents the quantum number of the lower-energy state. It is typical to use equa-
tion 14.21 for absorption andemission spectroscopy, and just keep track of
whether light is absorbed or emitted.
(J 1)(J 1 1)^2
2 I14.6 Rotational Spectroscopy 473*In some circumstances, such as in odd-electron molecules like NO,J0 is allowed,
but we will not consider these here.