5. Molecular rotation
5.1 Moments of inertia
The key molecular parameter needed to describe molecular
rotations is the moment of inertia, I, of the molecule defined as
I   mi ri
2
i
Definition of moment of inertia.
In this molecule there are three
identical atoms attached to the
B atom and three different but
mutually identical atoms
attached to the C atom. In this
example, the center of mass
lies on an axis passing through
the B and C atom, and the
perpendicular distances are
measured from the axis.
5.2 Rotational properties
The rotational properties of any molecule can be expressed in
terms of the moment of inertia about three perpendicular axes set
in the molecule. For linear molecules the moment of inertia around
the internuclear axis is zero. (why??)
An asymmetric rotor
has three different
moments of inertia;
all three rotation
axes coincide at the
centre of mass of the
molecule.
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5.3 Rigid rotors
We shall suppose initially that molecules are rigid rotors (i.e., do
not distort under the stress of rotation). There are four types then:
have one moment of inertia
equal to zero (CO2, HCl, HCCH)
(diatomics: I=mR2)
have three equal moments of
inertia (CH4, SF6)
have two equal moments of
inertia (NH3, CH3Cl)
have three different moments of
inertia (H2O, CH3OH)
5.4 Rotational energy levels
The rotational energy levels of a rigid rotor may be obtained by
solving the appropriate SE. However, there is a ‘short cut’ to the
exact expressions: The classical energy of a body rotating about
an axis a is
E a
1
2
I a a
2
A body free to rotate about three axes has an energy
E
1
1
1
2
2
2
I a a  I bb  I cc
2
2
2
Angular momentum about the axis a is
2
2
J a I aa , therefore
2
J
J
J
E a  b  c
2I a 2I b 2I c
2
5.5 Spherical rotors
When all three moments of inertia are equal to some value, I,
(CH4, SF6), the classical expression for energy is
J  Jb  Jc
J2
E a

2I
2I
2
2
2
where J is the magnitude of the angular momentum. The quantum
expression is obtained by
therefore
J 2  J ( J  1) 2
J=0, 1, 2, …
2
E J  J ( J  1)
2I
J=0, 1, 2, …
5.6 The rotational constant B
Definition of the rotational constant B
B

4cI
EJ  J ( J  1)hcB
J=0, 1, 2, …
rotational term (a wavenumber): F ( J )  BJ ( J  1)
The rotational energy levels of a linear or
spherical rotor. Note that the energy separation
between neighboring levels increases as J
increases.
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5.7 Degeneracies
The angular momentum of the molecule has a component on an
external, laboratory-fixed axis, M J  which is quantized:
M J  0,1,..., J
The significance of the quantum number MJ.
(a) When MJ is close to its maximum value, J,
most of the molecular rotation is around the
laboratory z-axis.
(b) an intermediate value of MJ.
(c) When MJ=0 the molecule has no angular
momentum about the z-axis.
5.8 Centrifugal distortion
The effect of rotation on a
molecule. The centrifugal force
arising from rotation distorts the
molecule, opening out bond
angles and stretching bonds
slightly. The effect is to increase
the moment of inertia of the
molecules and hence to
decrease its rotational constant.
The effect is taken into account empirically:
F ( J )  BJ ( J  1)  DJ J 2 ( J  1) 2
DJ: centrifugal distortion constant
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5.9 Rotational transitions
Typical values of B for ‘small’ molecules: 0.1 – 10 cm -1. Therefore,
rotational transitions lie in the microwave region of the spectrum!
A molecule must have a
permanent electric dipole
moment (it must be polar) for
the observation of a pure
rotational spectrum.
Diatomic homonuclear and
symmetric linear molecules as
well as spherical rotors are
normally inactive.
5.9.1 Rotational selection rules
For a linear molecule the transition moment vanishes unless:
J  1
M J  0,1
When a photon is absorbed by a
molecule, the angular momentum of
the combined system is conserved. If
the molecule is rotating in the same
sense as the spin of the incoming
photon, then J increases by 1
(absorption).
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5.10 The appearance of rotational spectra
When selection rules are applied for a rigid symmetric or linear rotor
allowed wavenumbers for J+1  J absorptions are:
 ( J  1  J )  2 B( J  1)
J=0, 1, 2, …
The rotational energy levels of a
linear rotor, the transitions allowed
by the selection rule DJ=±1, and a
typical pure rotational absorption
spectrum.
Measurement of n’s gives B and
hence bond lengths in case of
diatomic molecules.
5.11 The intensities of spectral lines
The Boltzmann distribution implies that the population decays
exponentially with J:
N J  Ng J e
 EJ
kT
and if the degeneracy gJ of level J is 2J+1 (linear rotor):
N J  (2 J  1)e
 hcBJ ( J 1)
kT
For a ‘typical’ molecule (OCS, B=0.2 cm -1) at room
temperature, Jmax is 22.
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5.12 Rotational Raman spectra
An electric field applied to a molecule
results in its distortion, and the
distorted molecule acquires a
contribution to its dipole moment
(even if it is nonpolar initially). The
polarizability may be different when
the field is applied (a) parallel or (b)
perpendicular to the molecular axis
(or, in general, in different directions
relative to the molecule); if that is so,
then the molecule has an anisotropic
polarizability.
5.12.1 Rotational Raman spectra-selection rules
The distortions induced in a
molecule by an applied electric
field returns to its initial value
after a rotation of only 180o (that
is, twice a revolution). This is the
origin of the DJ=±2 selection rule
in rotational Raman
spectroscopy.
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5.12.1 Rotational Raman spectra-selection rules
All linear molecules and diatomics (whether homonuclear or
heteronuclear) have anisotropic polarizabilities, and so are
rotationally Raman active.
Note that spherical rotors (e.g. CH4, SF6) are rotationally Raman
inactive as well as microwave inactive!
Rotational Raman selection rules are:
Linear rotors:
J  0,2
5.12.2 Rotational Raman spectra – energy levels
The rotational energy levels of a
linear rotor and the transitions
allowed by the DJ=±2 Raman
selection rules.
The form of a typical rotational
Raman spectrum is also shown.
The Rayleigh line is much
stronger than depicted in the
figure, it is shown as a weaker
line to improve visualization of the
Raman lines.
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