132
6 Equilibrium Structures from Spectroscopy
6.4.2 Effective Structures (r 0 )
Historically, the effective structure method (r 0 ) is the oldest method. The structural
parameters are directly fitted to the ground-state moments of inertia. In other words,
the rovibrational corrections ε
ξ
= I
ξ
0 − I
ξ
e are neglected. These corrections are indeed
quite small compared to the values of the moments of inertia. They are less than 1%
in most cases (Demaison and Nemes 1979); see Table 6.2. Thus, neglecting these
corrections may seem to be a good approximation. However, the least-squares system
is often ill-conditioned; see Sects. 9.4.1 and 9.7. As a consequence, the solution may
be quite inaccurate. The accuracy is about 1% for diatomic molecules; see Sect. 3.8.2.
In favorable cases, the bond lengths in polyatomic molecules are determined with
an accuracy of about 0.2 pm and, for the angles, it is about 0.2–0.3°. For instance,
in the case of SO 2 , which is a very favorable case, the r 0 structure derived from the
I a and I b moments of inertia is: r 0 (SO) = 143.22 pm and ∠ 0 (SOS) = 119.535°.
These values are rather close to the equilibrium structure: r e (SO) = 143.0793(4)
pm and ∠ e (SOS) = 119.3290(2)° (Lafferty et al. 2009). This satisfactory agreement
is obtained for two reasons: i) the structure can be obtained from only one isotopic
species, which avoids the problem of ill-conditioning; ii) the molecule is planar, hence
the equilibrium moments of inertia are not independent (I
e
c = I
e
a + I
e
b ). However,
because of the inertial defect, the relation does not hold for the ground-state moments
of inertia. To determine the structure, it is possible to use either I a and I b , or I b and
I c , or I c and I a . The best results are obtained with the two smallest moments of
inertia (I a and I b ) because the rovibrational correction increases with the value of
the moment of inertia.
A striking counterexample is the r 0 structure of the planar molecule phosgene,
OCCl 2 . In this case, the rotational constants of at least two isotopologues are required
because there are three geometrical parameters to determine. A fit of the ground-state
moments of inertia of a full set of isotopologues gives r 0 (C = O) = 116.6(10) pm;
r 0 (C–Cl) = 174.6(10) pm and ∠ 0 (ClCCl) = 111.3(10) pm (Demaison et al. 1997) to
Table 6.2 Variation of the vibrational correction ε = I 0 − I e as a function of the moments of inertia
I 0 (uÅ 2 )
Molecule
Number of isotopologues
I 0
ε
ε/I 0 (%)
Mean
Range
HCN
11
11.707
0.049(1)
0.007
0.4
N 2 O
12
40.232
0.206(4)
0.015
0.5
OCS
12
83.101
0.250(6)
0.018
0.3
OCSe
27
125.019
0.349(9)
0.026
0.3
Experimental, semiexperimental and ab initio equilibrium structures; Demaison J; Molecular
Physics; 105:3109-3138; Dec 10, 2007, reprinted by permission of the publisher Taylor&Francis
Ltd, http://www.tandfonline.com
a Demaison (2007)
6 Equilibrium Structures from Spectroscopy
6.4.2 Effective Structures (r 0 )
Historically, the effective structure method (r 0 ) is the oldest method. The structural
parameters are directly fitted to the ground-state moments of inertia. In other words,
the rovibrational corrections ε
ξ
= I
ξ
0 − I
ξ
e are neglected. These corrections are indeed
quite small compared to the values of the moments of inertia. They are less than 1%
in most cases (Demaison and Nemes 1979); see Table 6.2. Thus, neglecting these
corrections may seem to be a good approximation. However, the least-squares system
is often ill-conditioned; see Sects. 9.4.1 and 9.7. As a consequence, the solution may
be quite inaccurate. The accuracy is about 1% for diatomic molecules; see Sect. 3.8.2.
In favorable cases, the bond lengths in polyatomic molecules are determined with
an accuracy of about 0.2 pm and, for the angles, it is about 0.2–0.3°. For instance,
in the case of SO 2 , which is a very favorable case, the r 0 structure derived from the
I a and I b moments of inertia is: r 0 (SO) = 143.22 pm and ∠ 0 (SOS) = 119.535°.
These values are rather close to the equilibrium structure: r e (SO) = 143.0793(4)
pm and ∠ e (SOS) = 119.3290(2)° (Lafferty et al. 2009). This satisfactory agreement
is obtained for two reasons: i) the structure can be obtained from only one isotopic
species, which avoids the problem of ill-conditioning; ii) the molecule is planar, hence
the equilibrium moments of inertia are not independent (I
e
c = I
e
a + I
e
b ). However,
because of the inertial defect, the relation does not hold for the ground-state moments
of inertia. To determine the structure, it is possible to use either I a and I b , or I b and
I c , or I c and I a . The best results are obtained with the two smallest moments of
inertia (I a and I b ) because the rovibrational correction increases with the value of
the moment of inertia.
A striking counterexample is the r 0 structure of the planar molecule phosgene,
OCCl 2 . In this case, the rotational constants of at least two isotopologues are required
because there are three geometrical parameters to determine. A fit of the ground-state
moments of inertia of a full set of isotopologues gives r 0 (C = O) = 116.6(10) pm;
r 0 (C–Cl) = 174.6(10) pm and ∠ 0 (ClCCl) = 111.3(10) pm (Demaison et al. 1997) to
Table 6.2 Variation of the vibrational correction ε = I 0 − I e as a function of the moments of inertia
I 0 (uÅ 2 )
Molecule
Number of isotopologues
I 0
ε
ε/I 0 (%)
Mean
Range
HCN
11
11.707
0.049(1)
0.007
0.4
N 2 O
12
40.232
0.206(4)
0.015
0.5
OCS
12
83.101
0.250(6)
0.018
0.3
OCSe
27
125.019
0.349(9)
0.026
0.3
Experimental, semiexperimental and ab initio equilibrium structures; Demaison J; Molecular
Physics; 105:3109-3138; Dec 10, 2007, reprinted by permission of the publisher Taylor&Francis
Ltd, http://www.tandfonline.com
a Demaison (2007)