Page 310 - 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 827

2
whereσ is−1/b p = (2A − B − C)/(B − C), W(b p )isthe V = (1/2) fR , (58)
n
n
Wang reduced energy, and P is the average of P in the
Z Z
rigid asymmetric rotor basis |J , τ, M). These latter quan- where f is the stretching force constant and R represents
tities may be calculated, as mentioned previously, from the displacement coordinate, which measures the depar-
the eigenvectors obtained in diagonalization of the rigid- ture of the bond length from its equilibrium value. The
rotor energy matrix. For very slightly asymmetric tops, constant D J is defined by
n
P = K . The A, B, and C are effective rotational con-
n ∼
Z h 4
m 4B 3
stants that now contain a small contribution involving the D J = 3 = 2 e , (59)
distortion constants. The J , JK , and so on are the quar- h f (I e ) ω e
tic distortion coefficients. One may apply the above ex- with m the reduced mass; I e and B e are, respectively,
pressions to an oblate top (Z ↔ c) by interchanging A and the equilibrium moment of inertia and rotational constant
C and setting σ =−1/b o in the above energy expression. B e = h/8π I e . Here ω e = (1/2π)( f/m) 1/2 is the harmonic
2
To evaluate the rotational and distortion constants, dif- vibrational frequency. Thus from an analysis of the rota-
ferences between the observed and calculated rigid-rotor tional spectrum, precise values of D J can be obtained,
frequenciesareanalyzedviaEq.(55)bymeansoftheleast- which in turn yield, from the above expression, accurate
squares technique to determine adjustments in the original stretching force constants, or equivalently, ω e values.
rotational constants δA,δB, and δC as well as the distor- For other molecules, the details are more complicated,
tion constants J and so on. Quartic distortion constants but the principles are the same. The quadratic potential
for a few asymmetric tops are collected in Table V. function has the general form
The energy expression given in Eq. (57) can account
for a large number of asymmetric tops. However, for light 1
V = f ij R i R j . (60)
molecules with large rotational energies, such as H 2 O, 2
or when transitions from high J levels are studied, a
Infrared measurements yield the vibrational frequencies
first-order treatment does not suffice. Additional higher
associated with the various normal vibrational modes,
power terms in the angular momentum must be included
and these data, including isotopic frequency data, can be
in Eq. (57). Specifically, it is found that n + 1 distortion
used to evaluate the force constant matrix F = [ f ij ]. Since
contributions are added for each degree n in the angular these calculations are often ill conditioned and also since
6
momentum. Thus seven terms are added if P effects are
there are usually more force constants than vibrational
considered. The sextic distortion constants are denoted by
frequency data, both infrared data and the microwave dis-
J , JK , KJ , K ,φ J ,φ JK , and φ K and have been eval-
tortion constant data are often combined to help charac-
uated from a study of the rotational spectra for a large
terize the force constant matrix. Some examples are given
number of molecules. When such effects are important, a 6
in Table XIV. It may be noted that the P or sextic dis-
first-order treatment is not sufficient. In such cases, the en-
tortion constants depend on the cubic potential constants,
ergy matrix of r + d must be set up and diagonalized to
and these data have been employed to obtain information
obtain the general effects of centrifugal distortion. Proce-
on these anharmonic potential constants.
dures for effectively including such higher order distortion
effects are discussed elsewhere.
TABLE XIV Potential Constants Determined by
Combination of Infrared and Microwave Data
(mdyn/A) a
D. Information from Distortion Constants
Molecule f r f rr f αr 2 f rαr
The study of centrifugal distortion provides a number
of useful kinds of information. By including effects SO 2 10.006 0.024 0.793 0.189
of centrifugal distortion, one can obtain very accurate O 3 5.70 1.52 1.28 0.332
spectroscopic constants. These allow the prediction of OF 2 3.950 0.806 0.724 0.137
unmeasured transition frequencies with a high degree of ClO 2 7.018 −0.170 0.651 0.006
confidence over a wider range than provided by only NO 2 11.043 2.140 1.109 0.481
the rigid-rotor constants. Most importantly, however, the GeF 2 4.08 0.26 0.316 −0.01
centrifugal distortion constants provide information on SeO 2 6.91 0.03 0.488 0.009
the vibrational potential function, particularly for small
a The quadratic-valence force field potential function
molecules. This follows because the distortion constants
for bent triatomic molecules XY 2 is defined by
depend directly on the force constants, masses, and struc-
2
2 2 + f α δα + 2 f rα (δr 1 + δr 2 )δα
2V = f r δr + δr
ture of the molecule. This may be illustrated for a diatomic 1 2
molecule. The quadratic potential function is given by + 2 f rr δr 1 δr 2 .