Page 314 - Academic Press Encyclopedia of Physical Science and Technology 3rd Analytical Chemistry

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Encyclopedia of Physical Science and Technology EN009N-447 July 19, 2001 23:3

Microwave Molecular Spectroscopy 831

spectral lines into doublets for asymmetric rotors. With a the two groups. If the barrier is very low, V 3 → 0, the form
very low barrier, such as found in CH 3 NO 2 , a quite com- of the above waveequation is that of a spatial rotor with
plex spectrum can be obtained. In cases where asymmetric a fixed axis of rotation, and solution gives for the energy
groups are connected by a single bond, internal rotation levels
cangiverisetodistinctrotationalisomers.Theserotational 2
E = Fm , (81)
isomers are often stable enough to give separate rotational
spectra even though they cannot be chemically separated. with m = 0, ±1, ±2,.... The internal motion is essen-
As illustrated in Fig. 10, the different rotational isomers tiallyfreerotationabouttheC Cbond,andtheinternalro-
canbereadilydistinguishedbecauseoftheveryhighsensi- tation states are specified by the quantum number m. This
tivity of the moments of inertia to the molecular geometry. is the case for molecules such as CH 3 NO 2 , where the bar-
rier height is 6.03 cal/mole and the rotation is effectively
A. Single Top with a Threefold Barrier ∼
free (note that at room temperature, RT = 600 cal/mole).
For CH 3 CH 3 or CH 3 CHO, there are three equivalent Actually, for this and other molecules of similar symme-
configurations for a complete rotation of the methyl group try, there are six equivalent configurations for a complete
about the C C bond. The potential function possesses internal revolution. The internal potential thus has sixfold
three potential energy minima and maxima as illustrated symmetry, and the leading term in V (α)isa V 6 term:
in Fig. 19. Since the methyl group is symmetric, the mo-
V (α) = (V 6 /2)(1 − cos 6α), (82)
ments of inertia of the molecule do not depend on the
orientation of the methyl group. The effects of internal ro- where we expect V 6 V 3 . For the low barrier (V 6 ), the
tation are transmitted to the rotational spectrum somewhat m =±3 levels are particularly sensitive to the barrier.
indirectly. If the barrier is very high, V 3 →∞, the internal mo-
The periodic potential function that describes the inter- tion of the methyl group corresponds to simple harmonic
nal rotation is expressed by torsional oscillation in each well. The cosine function in
2
Eq. (79) may be expanded, giving V (α) = (9V 3 /4)α , and
V (α) = (V 3 /2)(1 − cos 3α), (79)
the form of Eq. (80) is like that for a simple harmonic
where V 3 is the threefold barrier hindering internal rota- oscillator. Solution gives for the energy
tion and α the angle of internal rotation. The eigenvalue 1/2 1
E = 3(V 3 F) v + , (83)
equation for this internal motion is given by 2
2
δ U(α) with v = 0, 1, 2,.... The frequency of torsional oscilla-
−F + [V (α) − E]U(α) = 0, (80) tion is
δα 2
2
1/2
where V (α) is given by Eq. (79), and F = h /2I r with I r 3 V 3
ν = . (84)
the reduced moment of inertia for the relative rotation of 2π 2I r
For high barriers, the rotational spectrum exhibits transi-
tions in the ground and excited torsional states. Relative
intensity measurements can thus enable the determination
of the barrier. In particular, the intensity ratio between the
ground and first excited states is given by the Boltzmann
distribution law,
l v=0 /l v=1 = e −hν/kT . (85)
The above provides a measure of ν, and V 3 may be eval-
uated from Eq. (84).
For an infinite barrier, each torsional state v is threefold
degenerate corresponding to oscillations in any one of the
three equivalent potential wells. When the barrier is finite,
FIGURE 19 Schematic representation of the potential function the molecule may pass from one configuration to another
and torsional energy levels for a threefold barrier. A simple co- by tunneling through the barrier. This quantum mechan-
sine potential is depicted with three identical minima and max- ical tunneling effect leads to a splitting of the threefold
ima. The corresponding eclipsed and staggered configurations
for ethane are also indicated. Each torsional energy level is la- degeneracy into a nondegenerate level (designated by A)
beled by the torsional quantum number v. The torsional sublevels and a doubly degenerate level (designated by E). This
are denoted by A or E. torsional level splitting is illustrated in Fig. 19. Note that