Page 309 - 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

826 Microwave Molecular Spectroscopy

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spectrum. If distortion is taken into account, terms in the ν = 2B(J + 1) − 4D J (J + 1) − 2D JK (J + 1)K . (54)
angular momentum components of the fourth power, sixth
power, and so forth are introduced into the Hamiltonian. Note that neither A nor D K affects the rotational spec-
Although the centrifugal distortion constant are very small trum. The first correction term, involving D J , alters the
relative to the rotational constants, they produce signif- even spacing between successive J → J + 1 transitions,
icant effects in the rotational spectrum, particularly for while the last term also separates the superposed lines of
asymmetric tops. different K values into J +1 closely spaced lines with the
2
separation increasing as K . This is illustrated in Fig. 7.
A. Diatomic and Linear Rotors Typical values of D J and D JK are collected in Table IV.
D J is always positive, whereas D JK and D K may be pos-
For diatomic and linear rotors, distortion terms of the type itive or negative.
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4
DP and HP are added to the rigid-rotor Hamilitonian. Higher order effects introduce additional distortion
4
Considering only the major P effect and noting (J, terms and also distortion terms that can give rise to
4
4
2
2
M|P |J, M) = h J (J + 1) , we have for the rotational
splittings of certain K-levels. The effects of centrifugal
energy of a nonrigid diatomic or linear molecule
distortion on the observation of forbidden K =±3 tran-
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E J = BJ(J + 1) − D J J (J + 1) 2 (50) sitions have already been mentioned. Induced dipole mo-
ments also allow the observation of pure rotational spectra
and the frequencies are given by of spherical tops which, because they have no permanent
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ν = 2B(J + 1) − 4D J (J + 1) , (51) dipole moment, would otherwise have no rotational spec-
tra. For CH 4 , the distortion moment is on the order of
where D J is the distortion constant, and both B and D J 5 × 10 −6 D. Both J → J + 1 and J → J transitions have
are in frequency units. The physical picture of distortion been observed. The leading terms in the frequency equa-
in a diatomic molecule is quite simple. As the molecular tion for the J → J + 1 transitions are like those for a linear
bond stretches, I increases and B decreases, leading to molecule, Eq. (51); however, the molecular distortion in
a decrease in the rotational energy and a shift to lower such molecules is more complicated, and additional terms
frequency relative to the rigid rotor frequency. If P 6 are required to adequately characterize the rotational spec-
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effects are considered, then a term H J J (J + 1) must trum. Such observations have provided the rotation and
be added to Eq. (50). Distortion effects are small and distortion constants. Some examples of nonpolar molecu-
primarily important only for high J values. Some typical lar studies via microwave spectroscopy are spherical tops
values of distortion constants are listed in Table III. with T d symmetry like CH 4 , SiH 4 , and GeH 4 and those
with D 3h symmetry like BF 3 and SO 3 . For SO 3 , a planar
B. Symmetric-Top Molecules molecule, the centrifugally induced rotational spectrum
˚
provides r e = 1.4175 A.
A first-order treatment of centrifugal distortion yields for
the distortion Hamiltonian
4 2 2 4
d =−D J P − D JK P P − D K P , (52) C. Asymmetric-Top Molecules
Z Z
which is diagonal in the symmetric-top basis since the The evaluation of centrifugal distortion in asymmetric ro-
only nonvanishing matrix elements (in units of h) are tors is considerably more complex than for linear or sym-
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2
2
2
4
(J, K, M|P |J, K, M) = J (J + 1) , (J, K, M|P P |J, metric tops, and because of the nature of the spectrum, par-
Z
4
2
4
K, M) = K J(J + 1), and (J, K, M|P |J, K, M) = K . ticularly large distortion shifts (say 1000 MHz or larger)
Z
4
The energy of a nonrigid prolate symmetric top is can be observed. A first-order treatment of the P distor-
tion effects gives, for the energy of a semirigid prolate
2
2
2
E J,K = BJ(J + 1) + (A − B)K − D J J (J + 1)
asymmetric rotor (Z ↔ a),
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− D JK J(J + 1)K − D K K . (53)
E = E r + E d (55)
For an oblate symmetric top, the unique axis is designated
c, and the energy expression may be obtained from the E r = (1/2)(B + C)J(J + 1)
above expression by replacement of A by C. The con-
+ [A − (1/2)(B + C)]W(b p ) (56)
stants D J , D K , and D JK essentially represent the distor- 2 2 2 !
tion effects of end-over-end rotation, rotation about the E d =− J J (J + 1) − JK J(J + 1) P Z
symmetry axis, and the interaction between these motions, 4 ! 2 !
− K P − 2δ J σ J(J + 1) W(b p ) − P Z
Z
respectively. With the selection rules J → J + 1, K → K,
2 ! 4 !
the rotational frequencies are found to be − 2δ K σ W(b p ) P − P , (57)
Z Z