Page 318 - 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 835

to give two equations in the two unknowns z Y , z Z . From TABLE XX Effective and Substitution Struc-
the coordinates the bond lengths are then given by tures of OCS
d ij =|z i − z j |. (91) Effective structures
Bond length ( ˚ A)
With equilibrium moments of inertia, the important point
isthattheparticularmethodofcalculationisnotimportant. Isotopic species used C O C S
12 34
16 O C S, 16 O C S 1.1647 1.5576
12 32
B. Average Structure
13 32
12 32
16 O C S, 16 O C S 1.1629 1.5591
13 34
12 34
Like the equilibrium structure, the average structure has 16 O C S, 16 O C S 1.1625 1.5594
12 32
12 32
a well-defined physical meaning. The vibrational effects 16 O C S, 18 O C S 1.1552 1.5653
contained in the moments of inertia may be divided into Average 1.1613 1.5604
α
α
harmonic ε (har) and anharmonic ε (anhar) contribu- Range 0.0095 0.0077
s s
tions, which depend, respectively, on the quadratic and
Substitution structures
cubic part of the potential energy function. To evaluate the
average structures, the moments of inertia for the average Bond length ( ˚ A)
configuration, denoted by I (α = a, b, c), are required. Parent molecule C O C S
∗
α
These may be obtained from the effective moments of
12 32
α
inertia by correcting for the ε (har) effects: 16 O C S 1.16012 1.56020
s
12 32
18 O C S 1.15979 1.56063

∗ v α
I = I − (v s + d s /2)ε (har). (92) 16 13 32
α α s O C S 1.16017 1.56008
s 16 O C S 1.16075 1.55963
12 34
Only a knowledge of the harmonic force constants Average 1.16021 1.56014
is required to make this correction. For a diatomic Range 0.00096 0.00100
2
b
molecule, ε (har) =−6h/8π ω e ; and for the ground state,
0
b
∗
I = I − ε (har)/2. The average bond length is then cal- 0
b
b
b
culated from with I the effective ground-state moment of inertia. As
apparent from Table XIX, the effective bond distance r 0
1/2
" #
m X + m Y varies with isotopic substitution, and r 0 >r e . In general,
r = I b ∗ . (93)
m X m Y structural parameters obtained by fitting effective mo-
In Table XIX, it is clear that r parameters change with ments of inertia are termed r 0 structures. For a linear XYZ
isotopicsubstitution,butr e parametersdonot,asexpected. molecule, two isotopic forms are required to determine
The anharmonic part of the potential function has the ef- the structure and also the assumption that the bond dis-
fect of displacing the average configuration of a molecule tances are unaffected by isotopic substitution. As we have
from its equilibrium configuration, and r >r e . Usually seen, this assumption is only approximately true; hence,
r >r 0 >r e , and replacement of H by D, which signifi- the structural parameters derived are less reliable particu-
cantly decreases the amplitude of vibration, causes a large larly for parameters involving H atoms. When more than a
shortening in r . Average structures for excited vibra- minimum number of isotopic species is available, different
tional states have also been evaluated; these clearly in- r 0 structurescanbeevaluated,andanestimateoftheuncer-
dicate the variation expected for a given vibrational ex- tainty in the structure can be obtained (see Table XX). This
citation. This measure of the molecular structure is most analysis, however, cannot in general indicate the closeness
meaningful for simple molecules. Two drawbacks are that of the r 0 to the r e parameters.
a knowledge of the harmonic force constants is required,
1. Inertial Defect
and if isotopic data are needed to evaluate the average
structure, then the isotopic shrinkage effects just men- The effects of vibration are readily apparent from a quan-
tioned must be ignored or estimated. This introduces some tity called the inertial defect , which is useful to charac-
ambiguity in the derived parameters. terize a planar molecule. The inertial defect is defined by
v
v
v
C. Effective Structure I − I − I = , (95)
a
b
c
The effective bond distance for a diatomic molecule is where c is the principal axis perpendicular to the molecu-
given by lar plane. Actually, for a planar molecule, it follows from
the definitions of the moments of inertia that should be
1/2
" #
m X + m Y 0
r 0 = I b , (94) identically zero. In reality this is true only if equilibrium
m X m Y moments of inertia are employed in the above relations.