Page 311 - 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

828 Microwave Molecular Spectroscopy

VI. ROTATION–VIBRATION INTERACTIONS arate rotational spectrum is obtained for each vibrational
state. These excited-state lines, or satellite spectra, may
In addition to centrifugal distortion effects, other nonrigid- be shifted only a few megahertz or many hundreds of
ity effects also alter the rotational spectra. The vibrational megahertz away from the ground-state line. Because of
motions of a polyatomic molecule may be described in the Boltzmann factor e −E v /kT , the line intensity decreases
terms of n normal modes of vibration, where n = 3N − 6 with increasing vibrational excitation, and only low-lying
(or 3N − 5 for a linear molecule) with N the number of vibrational levels give rise to lines with sufficient inten-
atoms. For linear and symmetric tops, degenerate vibra- sity to be observed. Except for diatomic molecules and
tions are present, and not all of these modes have different a few relatively simple polyatomic molecules, it is not
vibrational frequencies ω i . The modes with the same ω i possible to obtain sufficient data to determine all the α i
are usually grouped together and their number specified in Eqs. (61)–(63). Hence, it is not possible to correct the
by d i . As a molecule rotates, it vibrates rapidly, even in observed ground-state rotational constants to obtain the
the ground vibrational state, and the moments of inertia equilibrium constants, for instance,
are averaged in a complicated way over the molecular vi- a
A e = A 0 + α d i 2. (65)
i
brations. The rotational constants as well as the distortion
constants, and in fact almost all the molecular parame- Because of this, effective constants must be used to evalu-
ters derived from an analysis of rotational spectra, must ate the molecular structure, which introduces uncertainties
be considered as effective constants, that is, constants as- in the derived structural parameter. This is discussed fur-
sociated with a particular vibrational state. The depen- ther in Section VIII.
dence of the effective rotational constants on the vibra- Unless there is an accidental near-vibrational degener-
tional state v may, to a good approximation, be expressed acy, the rotational spectrum of an asymmetric top in an
by excited vibrational state is similar to that obtained in the

ground state, except that the spectrum is characterized
a d i
A v = A e − α i v i + , (61) by a slightly different set of rotation and distortion con-
2
stants. Other nonrigid effects are often more important

b d i for asymmetric tops, such as internal rotation, and these
B v = B e − α i v i + , (62)
2 are considered in Section VII. Similar statements apply to
linear and symmetric-top molecules in excited nondegen-

c d i
C v = C e − α i v i + , (63) erate vibrational states. For example, the rotational fre-
2 quencies for symmetric tops in nondegenerate vibrational
where A e andsoonaretheequilibriumrotationalconstants states are given by Eq. (54) with the rotation and distortion
(v) (v)
associated with the vibrationless state, v is specified by the constants replaced by effective constants B v , D , D .
J JK
vibrational quantum numbers (v 1 ,v 2 ,...,v i ,...), where On the other hand, when degenerate bending modes are
v i is the quantum number of the ith vibration, and d i is present, as with linear and symmetric tops, the spectrum
c
b
a
the corresponding degeneracy. The α ,α , and α are the in these excited states can be altered markedly. This effect
i i i
rotation–vibration constants for the ith mode and the A, B, is called l-type doubling and will be discussed for linear
and C rotational constants, respectively. The sum is over molecules after the general expression for the rotation–
the various vibrations, with degenerate vibrations counted vibration energy levels is given for a diatomic molecule.
only once. For diatomic and asymmetric tops, d i = 1. The
dependence on vibrational state of the effective distortion A. Diatomic Molecules
constants is similar to the above expressions, that is, for
D J of a symmetric top, we may write For diatomic molecules, since there is only one vibrational
(v) mode, enough excited states can be studied to enable the
D = D e + β i (v i + d i /2), (64)
J
evaluation of a number of rotation–vibration constants.
where β i is a small rotation–vibration constant. In many With the assumption of a Morse potential, the eigen-
cases, particularly in the literature, the rotation or distor- value equation, ψ = Eψ, for a diatomic molecule can
tion constants are designated simply A or D J , and the be solved directly, and the energy levels are specified by
symbol v is omitted. However, it is to be understood that
1 1 2

such parameters are, in general, dependent on the vibra- E v,J = ω e v + 2 − ω e χ e v + 2 + B v J(J + 1)
tional state. 2 2 3 3
− D v J (J + 1) + H v J (J + 1) + ··· . (66)
It is apparent from the above expressions that the ef-
fective rotational constants as well as the distortion con- The first two terms represent the vibrational energy
stants are different for each vibrational state, and a sep- and the last terms the effective rotational energy. The