Page 331 - Instant notes

Rotational spectroscopy 317

rotational transitions of a polar linear
molecule.

The moment of inertia is the rotational equivalent of linear momentum. A body with a
large moment of inertia requires a greater twisting force to reach a certain rate of rotation
and possesses greater energy for a given rate of rotation than a body with a small moment
of inertia. Since B is inversely proportional to I, the larger the moment of inertia
(corresponding to a longer bond and heavier atoms) the smaller the rotational constant
and the more closely spaced the rotational energy levels.
The energies of rotational levels are usually quoted in units of reciprocal centimeter,
−1
or wavenumber, cm , instead of joules. The rotational constant in units of wavenumber
is:

Polyatomic molecules
The rotational energy levels of a linear polyatomic molecule are the same as for a
diatomic molecule and characterized by one rotational constant and one rotational
quantum number.
Non-linear molecules in which the moments of inertia about two axes are the same but
different from the third are called symmetric top rotors since they have one axis of
rotational symmetry. Example molecules are NH 3, CH 3Cl and PCl 5. The energy levels of
a symmetric rotor are characterized by two rotational quantum numbers J and K:
2
E J,K=BJ(J+1)+(A−B)K J=0, 1, 2,… K=J, J−1,…, −J
The rotational constants A and B correspond to the moments of inertia parallel and
perpendicular to the rotational symmetry axis of the molecule, respectively, and are given
(in wavenumber units) by:

The quantum number K indicates the extent of rotation about the symmetry axis. When
K=0 the molecule is rotating end-over-end only. Energy levels with K>0 are doubly
degenerate (K appears as a squared term in the energy level expression) and correspond
to states of clockwise or anticlockwise rotation about the symmetry axis.
In the most general case, a polyatomic molecule has three different moments of inertia
(an asymmetric rotor) and the expressions for the rotational energy levels of these
molecules are complex.

Microwave rotational spectroscopy: selection rules and transitions

For a molecule to interact with an electromagnetic field, and undergo a transition between
two energy levels, requires an electric dipole to be associated with the motion giving rise