Selection rules specify the possible transitions among quantum levels due to absorption or emission of electromagnetic radiation. Incident electromagnetic radiation presents an oscillating electric field E0cos(wt) that interacts with a transition dipole. The dipole operator is m = e · r where r is a vector pointing in a direction of space.
A dipole moment of a given state is
A transition dipole moment is a transient dipolar polarization created by an interaction of electromagnetic radiation with a molecule
In an experiment we present an electric field along the z axis (in the laboratory frame) and we may consider specifically the interaction between the transition dipole along the x, y, or z axis of the molecule with this radiation. If mz is zero then a transition is forbidden. The selection rule is a statement of when mz is non-zero.
We can consider selection rules for electronic, rotational, and vibrational transitions.
We consider a hydrogen atom. In order to observe emission of radiation from two statesmz must be non-zero. That is
For example, is the transition from Y1s to Y2s allowed?
Using the fact that z = r cosq in spherical polar coordinates we have
We can consider each of the three integrals separately.
If any one of these is non-zero the transition is not allowed. We can see specifically that we should consider the q integral. We make the substitution x = cosq, dx = -sinqdq and the integral becomes
which is zero. The result is an even function evaluated over odd limits. In a similar fashion we can show that transitions along the x or y axes are not allowed either. This presents a selection rule that transitions are forbidden forDl = 0. For electronic transitions the selection rules turn out to be Dl = ± 1 and Dm = 0. These result from the integrals over spherical harmonics which are the same for rigid rotator wavefunctions. We will prove the selection rules for rotational transitions keeping in mind that they are also valid for electronic transitions.
We can use the definition of the transition moment and the spherical harmonics to derive selection rules for a rigid rotator. Once again we assume that radiation is along the z axis.
Notice thatm must be non-zero in order for the transition moment to be non-zero. This proves that a molecule must have a permanent dipole moment in order to have a rotational spectrum.
The spherical harmonics can be written as
where NJM is a normalization constant. Using the standard substitution of
x = cosq we can express the rotational transition moment as
The integral overf is zero unless M = M' so DM = 0 is part of the rigid rotator selection rule. Integration over f for M = M' gives 2p so we have
We can evaluate this integral using the identity
Substituting into the integral one obtains an integral which will vanish unless J' = J + 1 or J' = J - 1.
This leads to the selection ruleDJ = ± 1 for absorptive rotational transitions. Keep in mind the physical interpretation of the quantum numbers J and M as the total angular momentum and z-component of angular momentum, respectively. As stated above in the section on electronic transitions, these selection rules also apply to the orbital angular momentum
(Dl = ±1, Dm = 0).
The harmonic oscillator wavefunctions are
where Hv(a1/2q) is a Hermite polynomial and a = (km/á2)1/2.
The transition dipole moment for electromagnetic radiation polarized along the z axis is
Note that we continue to use the general coordinate q although this can be z if the dipole moment of the molecule is aligned along the z axis. The transition moment can be expanded about the equilibrium nuclear separation.
wherem0 is the dipole moment at the equilibrium bond length and q is the displacement from that equilibrium state. From the first two terms in the expansion we have for the first term
This term is zero unless v = v’ and in that case there is no transition since the quantum number has not changed.
This integral can be evaluated using the Hermite polynomial identity known as a recursion relation
where x =Öaq. If we now substitute the recursion relation into the integral we find
which will be non-zero if v’ = v – 1 or v’ = v + 1. Thus, we see the origin of the vibrational transition selection rule that v =± 1. We also see that vibrational transitions will only occur if the dipole moment changes as a function nuclear motion.