Physical Chemistry Third Edition

986 23 Optical Spectroscopy and Photochemistry

and if the molecules lose energy to the radiation,

hν′−hνEupper−Elower (23.7-2)

The difference between the scattered and incident frequencies or reciprocal wavelengths

is called theRaman shift. Spectral lines corresponding to transitions from a lower

to a higher molecular energy as in Eq. (23.7-1) are calledStokes lines, and those

corresponding to Eq. (23.7-2) are calledanti-Stokes lines.

The selection rules for Raman transitions are different from those of absorption or

emission, and this makes it possible to observe transitions that are forbidden in emission

or absorption spectroscopy. The Raman selection rules for rotational and vibrational

transitions are:

∆J0,±2 (linear molecules) (23.7-3a)

∆J0,±1,±2 (nonlinear molecules) (23.7-3b)

∆v0,± 1 (23.7-3c)

The nuclear motion must modulate the polarizability of the molecule. (23.7-3d)

Thepolarizabilityis a measure of the tendency of a molecule to acquire an electric

dipole in the presence of an electric field (see Problem 19.44). For a molecule with the

same properties in all directions (an isotropic molecule), the induced momentμindis

proportional to the electric fieldEEEand in the same direction as the electric field:

μindαEEE (isotropic molecule) (23.7-4)

whereαis the polarizability (a scalar quantity), and whereEEEis the electric field

(a vector). A symmetric top molecule such as methane or sulfur hexafluoride obeys

Eq. (23.7-4). For an anisotropic molecule (with different properties in different direc-

tions) thexcomponent of the induced moment is given by

μx,indαxxEEEx+αxyEEEy+αxzEEEz (23.7-5)

with similar equations for theyandzcomponents. The polarizability is now a matrix

with nine components (atensor) with components that have two subscripts. Equation

(23.7-5) and its analogues become the same as Eq. (23.7-4) if

αxxαyyαzz (23.7-6)

and if the other components vanish.

Just as principal axes for rotation of a molecule could be found, principal axes for

the polarizability of a molecule can be found such that the polarizability “cross-terms”

with unequal indexes vanish. The components of the induced dipole are then given by

μx′,indαx′x′EEEx′ (23.7-7a)

μy′,indαy′y′EEEy′ (23.7-7b)

μz′,indαz′z′EEEz′ (23.7-7c)

where we label the principal axes byx′,y′, andz′. The principal axes will generally lie

in the symmetry elements of the molecule.