Page 312 - 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 829

v (=0, 1, 2,...) and J (=0, 1, 2,...) are, respectively, the Furthermore, the effects of vibration on B v and D v are
vibrational and rotational quantum numbers, ω e is the har- given by Eqs. (62) and (64). However, the rotational spec-
monic vibrational frequency, and ω e χ e is the anharmonic- trum is complicated by the presence of degenerate bending
ity constant. The effective rotation and distortion constants modes of vibration. The bending mode is twofold degen-
are defined by erate since the linear molecule may bend in either of two
orthogonal planes. In the case of OCS, for example, there
1 2
1

B v = B e − α e v + + γ e v + + ··· , (67)
2 2 are 3 · 3 − 5 = 4 vibrational modes, labeled v 1 ,v 2 , and v 3
1 with the bending mode twofold degenerate, d 2 = 2. With

D v = D e + β e v + +· · · , (68)
2
excitation of a single degenerate bending mode v j , an an-
H v = H e +· · · . (69) gular momentum p =lh is generated along the molecular
The α e ,γ e , and β e are the rotation–vibration interaction axis analogous to a symmetric top with l similar to K. The
constants representing corrections for the effect of vibra- possible values of l are
tion. The selection rules for pure rotational transitions are
l = v j ,v j−2 ,v j−4 ,..., −v j . (74)
J → J + I,v → v, and the rotational frequencies are eas-
ily shown to be Hence, for v j = 1,l =±1; v j = 2,l = 0, ±2; and so on. In
addition, a Coriolis interaction betweenrotation andvibra-
3
3
ν = 2B v (J + 1) − 4D v (J + 1) + H v (J + 1)
tion exists that can remove the ±l degeneracy when l = 0.
3 3
× [(J + 2) − J ]. (70) The linear molecule behaves in an excited bending state
as if it were slightly bent; and, like a slightly asymmetric
To evaluate all of the constants, measurements of rota-
rotor, where the ±K degeneracy is lifted, the ±l degen-
tional transitions in at least three vibrational states (e.g.,
eracy is lifted. This is called l-type doubling. A detailed
v = 0, 1, and 2) must be made. Each vibrational state is
treatment for the energies including the l-type splitting of
analyzed via the above equation. For example, from the
the levels gives
data B 0 , B 1 , and B 2 , Eq. (67) yields
2
2 2
E = B v [J(J + 1) − l ] − D v [J(J + 1) − l ]
1
B e = (15B 0 − 10B 1 + 3B 2 ). (71)
8
1
Some spectroscopic constants obtained for a few selected ± q j (v j + 1)J(J + 1), (75)
4
diatomic molecules are collected in Table XV. Information where q j is the coupling constant characterizing, the split-
on the vibrational constants can also be obtained from the ting for the bending mode v j . This constant is usually sig-
rotational constants, for example, nificant only for the case |l|= 1. Here B v and D v have
4B 3 their usual meaning. If the splitting term is omitted, it is
2 e
ω = , (72) apparent that the levels are doubly degenerate since they
e
D e 2
depend on l , except when l = 0. However, J represents
2

α e ω e the total angular momentum quantum number including
ω e χ e = B e 2 + 1 . (73)
6B e the vibrational angular momentum. Hence,
J =|l|, |l|+ 1, |l|+ 2,..., (76)
B. I-Type Doubling in Linear Molecules
and depending on l, certain values of J are missing. In
For linear molecules, the rotational frequencies in excited particular, for |l|= 1, J = 1 is the lowest value of J, while
nondegenerate vibrational states are specified by Eq. (51). for |l|= 2, J = 2 is the lowest value. As a result of this,
TABLE XV Selected Molecular Constants of Some Diatomic Molecules
Diatomic
molecule B e (MHz) − −α e (MHz) D e (kHz) ω e (cm − −1 ) ω e χ e (cm − −1 ) r e ( ˚ A)
28 Si O 21,787.453 151.026 29.38 1252 5.96 1.50973
16
32
74 Ge S 5,593.1019 22.4569 2.41 569 1.723 2.0120772
74 130
Ge Te 1,958.7903 5.1702 0.353 308 0.62 2.3401556
120 16
Sn O 10,664.189 64.243 7.98 882 3.93 1.832198
120 32
Sn S 4,103.0013 15.1585 1.272 491.6 1.412 2.2090172
208 32
Pb S 3,487.1435 13.0373 1.012 431.8 1.277 2.2868535
208 80
Pb Se 1,516.9358 3.8952 0.210 272.3 0.552 2.402223