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

830 Microwave Molecular Spectroscopy

for example, the 0 → 1 transition is missing in an excited ν = (1/2)q j (v j + 1)J(J + 1). (78)
bending mode with |l|= 1, and the 0 → 1 and 1 → 2 tran-
With sufficiently large q j or J, these transitions can
sitions are missing for |l|= 2. With the selection rules
fall into the microwave region. For HCN, v 2 = 1, q 2 =
J → J + 1,l →l, the rotational frequencies including
224.48 MHz, and a series of transitions is found in the
l-type doubling are
microwave region. In practice, it is found that q j varies
2
2
ν ± = 2B v (J + 1) − 4D v (J + 1)[(J + 1) − l ] slightly with J, and more elaborate expressions are re-
quired to fit accurately the l-type doublet spectra.
1
± q i (v j + 1)(J + 1). (77)
2 To define the rotation–vibration constants in terms of
more fundamental parameters or to understand the ori-
If l-type doubling represented by the last term is negligi-
gin of various nonrigidity effects in the rotational spec-
ble,therotationalspectrumislikethatforalinearmolecule
trum, the general rotation–vibration Hamiltonian must be
in a nondegenerate vibrational state except for the limita-
employed. This Hamiltonian contains pure rotation and
tions on the values of J and small effects in the distortion
vibration terms as well as interaction terms between ro-
correction because of the presence of the vibrational an-
tation and vibration. Perturbation treatments to various
gular momentum quantum number l.
orders are required to characterize the different rotation–
To distinguish the vibrational states, the value of |l| is
vibration effects. Space does not permit further discus-
added as a superscript to the vibrational quantum number
sion of this; however, we mention that such a perturbation
associated with the degenerate vibration. For a triatomic treatment shows that the α constants depend on the cubic
|l|
linear molecule, the states are specified by (v 1 ,v ,v 3 ).
2 potential energy constants of the molecule.
2
Thus, the notation (1, 2 , 0) corresponds to a state with the
v 1 stretching mode excited by one unit, the v 3 stretching
mode in its ground state, and the bending mode v 2 excited VII. INTERNAL ROTATION
by two units with l =± 2. The nonrigid rotor spectrum of
FC P is illustrated in Fig. 18. Excitation of the degenerate
Internal rotation involves the rotation of one part of a
vibrational mode produces a series of lines of decreasing
molecule relative to the other about a single bond. The ap-
intensity and l-type doubling is apparent when l =±1.
pearance of the rotational spectrum depends on the type of
It is also possible to observe direct transitions between
internal rotor and on the barrier height hindering internal
the closely spaced l-type doublet levels with J = 0. The
rotation. Rotation of a methyl group, hindered by a bar-
l-doublet transitions for |l|= 1 are given by
rier on the order of 3 kcal/mole, leads to a splitting of the

FIGURE 18 Moderate-resolution scan of the J = 2 → 3 transition of FC P observed with a 2800-V cm −1 Stark
modulation. Unassigned lines are labeled µ and the l-doublet Stark lobes by an asterisk. Excited-state lines from
the bending mode and the splitting of the l =±1 lines are shown. For the first excited bending state v 2 = 1, l-type
1
doubling gives two (0, 1 , 0) lines shifted to high frequency relative to the ground-state line (0, 0, 0). Similar, but much
0
2
weaker, lines are found for v 2 = 3. For the second excited state v 2 = 2, the lines (0, 2 , 0) and (0, 2 , 0) occur. [From
Krato, H. W., Nixon, J. F., and Simmons, N. P. C. (1980). J. Mol. Spectrosc. 82, 185.]