Page 319 - 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

836 Microwave Molecular Spectroscopy

TABLE XXI Inertial Defects for Some Planar Molecules a where ν t is the low-frequency out-of-plane vibration and
I c is the out-of-plane principal moment of inertia. The first
2
2
Molecule (amu ∆ ˚ A ) Molecule (amu ∆ ˚ A )
term follows directly from theory neglecting other molec-
H 2 O 0.0486 H 2 S 0.0660 ular vibrations, but it overestimates the magnitude of ,
H 2 Se 0.0595 SO 2 0.1348 indicating the positive contributions from other vibrations
H 2 CO 0.0574 ClNO 2 0.2079 are not negligible. Analysis of a number of molecules
Furan 0.046 Fluorobenzene 0.033 with low-frequency out-of-plane vibrations yields the
Pyrrole 0.076 Benzonitrile 0.084 above empirical formula with α = 0.0186 and 0.00803
for aliphate and aromatic molecules, respectively. This
a = I c − I a − I b .
expression closely approximates observed inertial defects
for molecules with a low-frequency out-of-plane vibration
When effective moments are used, the residual defined
and is hence useful in judging the planarity of molecules
above is small but does not vanish. This arises because the
with small, negative inertial defects. It also follows from
vibrational effects associated with the different principal
the above relation that ν t can be estimated from an ob-
moments are slightly different. Nonetheless, a small
served inertial object.
inertial defect is usually good evidence of the planarity
of a particular molecule. This is illustrated in Table XXI.
As observed, a small, positive inertial defect provides D. Substitution Structure
evidence for molecular planarity. On the other hand, the
Substitution structures involve the use of Kraitchman’s
presence of a very low out-of-plane vibration can result
equations, which provide the position of an atom in a
in a negative inertial defects for planar molecules as il-
molecule utilizing the changes in moments of inertia from
lustrated in Table XXII. A simple relation has been de-
isotopic substitution. One isotopic form is selected as the
veloped to explain negative inertial defects observed for
parent molecule, and Kraitchman’s equations give coor-
planar molecules in the ground vibrational state,
dinates of the isotopically substituted atom in the center-
√
−1
˚ 2
=−(33.715/ν t ) (amu A cm ) + α I c , of-mass principal axis system of the parent molecule. For
diatomic or linear molecules, Kraitchman’s equation has
TABLE XXII Inertial Defects for Some Pla-
the form
nar Molecules with Low-Frequency Out-of-Plane
Vibrations (ν t ) a " M + m # 1/2
|z|= I x , (96)
2
Molecule ∆ (amu ˚ A ) ν t cm − −1 M m

CHO CHO −0.1286 108 where I x = I − I x is the difference in the moment of
x
CHO O CHO −0.1909 85 inertia of the isotopically substituted molecule of mass
S −0.164 90 M + m and the parent molecule of mass M. The m is,
hence, the mass change due to isotopic substitution. This
NO 2
relation can also be used to find the position of an atom lo-
−0.128 111 cated on the symmetry axis of a symmetric-top molecule.
CHO
Only absolute values of the coordinates are obtained from
relations such as Eq. (96). The sign of the coordinate must
−0.775 30
F CHCH 2 be assigned from other considerations such as the arrange-
ment of the atoms and an approximate location of the
CH 2 CH NO 2 −0.0665 100 center of mass. In general, to evaluate the bond distance
CHO CHS −0.070 104 b between two atoms, the effective moments of inertia must
−0.146 100 be obtained for the parent molecule and two singly substi-
NO
tuted species. Likewise, from the moment-of-inertia data
for a parent and three singly substituted species, an inter-
−0.127 57
NCO atomic angle can be evaluated. For a diatomic molecule,
the parent X—Y and two isotopic forms X —Y and X—
∗
−0.131 124 ∗
N CHO Y are required to give the internuclear distance
d X−Y =|z X − z Y |. (97)
a
= I c − I a − I b . For an extended compilation see
Since differences in the effective moments of inertia
Oka, T. (1995). J. Mol. Struct. 352, 225.
b
Estimated. are employed in calculating substitution structures, the