Rotational Raman Spectroscopy

Raman spectroscopy can also be used to measure rotational spectra. The requirement for rotational Raman spectroscopy is that a molecule must have an anisotropic polarizability. The form of the polarizability tensor of a molecule has the same symmetry as the inertial tensor.

Spherical tops have isotropic polarizability a_xx = a_yy = a_zz.

Symmetric tops have axially symmetric polarizability a_xx = a_yy ¹ a_zz.

Asymmetric tops have a_xx ¹ a_yy ¹ a_zz.

The selection rules for rotational Raman can be derived from the Kramers-Heisenberg-Dirac expression for the polarizability tensor

The initial, intermediate, and final states can be specified as follows

|iñ º |gvJKMñ |nñ º |ev'J'K'M'ñ |fñ º |gvJ''K''M''ñ

The sum over intermediate states includes all possibile electronic (g for ground and e for excited), vibrational (v'), and rotational (J'K'M') levels. The transition dipoles then have the form

(m_r)_ge is the electronic transition moment. The transition moment is evaluated at the equilibrium nuclear configuration (the Condon approximation). Normalization requires that v = v' and so there are now vibrational transitions here. The electronic transition moment is projected onto the lab frame to get the components (m_X)_ge,(m_Y)_geand (m_Z)_ge. The matrix elements connecting initial rotational states with intermediate rotational states or intermediate rotational states with final rotational states are the same ones considered in the analysis of microwave selection rules.

Diatomics

For diatomics this means:

where r,s can equal cosq, sinqcosf or sinqsinf. Since the selection rule for each of the rotational transition moments

is DJ = ±1 there is an overall selection rule of DJ = 0, ±2.

Symmetric tops

For symmetric tops an allowed rotational transition, JMK à J''M''K'' requires:

Where the angle brackets represent integration over Euler angles, and N and N' can be 0, +1 or -1 depending on which components of the lab frame polarizability are under consideration. Since DJ = 0, ±1 for each of these matrix elements, the net selection rule for Raman scattering is DJ = 0, ±1, ±2.

Given the range of possible values for the selection rule DJ there are the following definitions for the branches of the rotational Raman spectrum.

O à DJ = -2

P à DJ = -1

Q à DJ = 0

R à DJ = 1

S à DJ = 2

We have not explicity considered nuclear spin. It should be recalled that the overall wavefunction must be anti-symmetric with respect to nuclear exchange for half-integral nuclear spins and symmetric with respect nuclear exchange for integral spin. We do not consider this further in the course.