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

Microwave Molecular Spectroscopy 837

zero-point vibrational effects tend to cancel, and more this method is primarily based on the observation that
consistent structural parameters can be obtained from ρ is constant for different choices of parent species to
4
s
s
ρ
0
0 ∼
different combinations of isotopic data. This is illustrated ca. 1/10 ([I ] g /[I ] g = [I ] 1 /[I ] 1 ), and the I com-
in Table XX by the different r s structures for OCS. puted from the above prescription give values very close
e
Note that the spread in the r s structures is 10 times to I .
ρ
smaller than the spread in the r 0 structures. One of the Once the set of L scaled moments of inertia I have
best approximations to the equilibrium structure is the been evaluated, the molecular structural parameters are
substitution structure, and numerous such structures have derived by means of a standard least-squares fitting of the
ρ
been determined, with typical structural uncertainties of I ’s. This is found to provide the best averaging of small
˚
◦
◦
±0.002 to ±0.005 A and ±0.2 to ±0.5 . residualvibrationaleffects.Foralineartriatomicmolecule
XYZ, the four moments of inertia would be analyzed for
E. Mass-Dependence Structure the parameters d YX and d YZ (see Table II). Importantly,
the method employs a minimal set of isotopic substitution
To evaluate the mass-dependence structure, various iso-
data compared to the mass-dependence method. It is, how-
topic species are employed to calculate the substitution
ever, necessary to select the parent such that all isotopic
coordinates of each nonequivalent atom for a given parent.
substitutions satisfy either m i > 0or m i < 0 for all
These substitution coordinates are then used to evaluate
atoms i. This minimizes residual vibrational effects. For
the moments of inertia, Eq. (5), which are called substitu- the general case, there are moments I and I (α = a, b, c)
0
s
α
s
α
tion moments of inertia I . The mass-dependence moment associated with each axis, and these are used to calculate
α
m
of inertia I is calculated from the relation the corresponding ρ a , ρ b , ρ c and the I a , I , I c . The mo-
ρ
ρ
ρ
α
b
ρ
s
I m = 2I − I 0 (α = a, b, c). (98) ments of inertia I are then analyzed by least squares for
α α α α
the structural parameters. Table XXIII compares several
m
To first order, the I moments are equal to the equilibrium
α structures for OCS. Results for SO 2 are summarized in
e
moments of inertia I . The above procedure is repeated Table XVIII. It is apparent that the r ρ structures compare
α
for another parent species. Once a sufficient number of most favorably with the r e structures. Similar results are
I m for different parent isotopic species have been deter-
found for other molecules.
mined, the moment-of-inertia equations may be solved to
This method based on the use of a set of scaled mometns
give the r m structure. The r m structure for SO 2 is given in
of inertia provides a molecular structure which is a bet-
Table XVIII. This measure of the molecular structure has
ter approximation to the r e structure than the conventional
limited applicability because of the large amount of pre- r s structure, particularly for heavy-atom molecules. For
cise isotopic moment-of-inertia data required and because molecules which contain hydrogen atoms, additional con-
e
the first-order approximation I = I is not sufficient es-
m ∼
siderations apply because of larger vibrational effects.
pecially for light atoms. Thus, hydrogen bond lengths can-
The quantity ρ is no longer virtually constant, but varies
not be determined by this method.
significantly when hydrogen atom substitution species
ρ
(H ↔ D, T ) are considered, and the I values do not
F. Scaled Structure
give reliable structures. Corrections now have to be ap-
A method has been proposed which employs a set of mo- plied to obtain near-equilibrium structures, and an empir-
ρ
ments [I ] g defined for L isotopic species and computed ical method has been developed. In particular, the mo-
from ments of inertia for the deuterated species are corrected
via
ρ
0
[I ] g = (2ρ − 1)[I ] g , g = 1, 2,..., L, (99)
ρ D
ρ D
(I ) = (I ) + . (101)
with the scaling factor obtained from corr
s
0

ρ = [I ] 1 [I ] 1 . (100)
TABLE XXIII Structural Calculations for Car-
0
Here [I ] 1 (g = 1) is the ground-state moment of the par- bonyl Sulfide (OCS) a
s
ent isotopic species and [I ] 1 is the substitution moment
r s r 0 r m r ρ r e
of inertia calculated from the set of substitution coordi-
nates of the parent species. The dataset, the L isotopic CO 1.1605 1.1568 1.1587 1.1551 1.1543
species, is that needed for evaluation of a complete sub- CS 1.5596 1.5645 1.5593 1.5621 1.5628
stitution structure. For a linear XYZ molecule, four iso-
a All distances in angstroms. [From Harmony, M. D.,
topic species (including the parent) are needed for the
and Taylor, W. H. (1988). J. Mol. Spectrosc. 118, 163;
substitution structure. From Eqs. (99) and (98), it follows see also Berry, R. J., and Harmony, M. D. (1988). Struct.
ρ
m
for the parent, g = 1, that [I ] 1 ≡ I . The rationale for Chem. 1, 49.]