144
6 Equilibrium Structures from Spectroscopy
confirmed by the value of the equilibrium inertial defect, Δ e = −0.0014(6) uÅ
2 ,
which is different from zero although two orders of magnitude smaller than the
ground-state value, 0 = 0.163 uÅ
2 . The electronic correction is also not negligible. The centrifugal distortion correction is much larger than the uncertainty of
the ground-state rotational constants, but it has a small effect on the accuracy of
the structure: there is a tiny increase of 0.001 pm for the SO bond length. Thus,
except in very accurate works, it may be neglected. Neglecting the electronic correction, decreases the r(SO) bond length by 0.01 pm and increases the ∠(OSO) bond
angle by 0.007°. These deviations may be considered as negligible for most works.
This conclusion also applies to the smaller
γ /4 correction. Using the equilibrium moments of inertia derived from Table 6.8 allows us to obtain the equilibrium
structure: r e (SO) = 143.0782(15) pm and ∠(OSO) = 119.3297(30)°. The estimated
uncertainties are quite small, but they do not take into account the systematic errors.
The true uncertainties may be one order of magnitude larger. See also Sects. 6.10
and 9.11.
The high accuracy achieved for SO 2 is mainly because the structure of this
molecule can be determined from the moments of inertia of a single isotopologue.
Hence, the equations are well-conditioned; see Sect. 9.4.1. When more than one
isotopologue (the most frequent case) has to be used, the situation is less favorable
because it is difficult to avoid the problem of ill-conditioning. A typical example
is given by the structure of the formyl cation HCO+. Its structure was determined
using the experimental equilibrium rotational constants of H
12 C
16 O
+ , H
13 C
16 O
+ ,
and H
12 C
18 O
+ : r e (CH) = 109.72 pm and r e (CO) = 110.47 pm (Woods 1988). The
results disagree with several accurate ab initio calculations; in particular for the CH
bond, which seems much too long (Puzzarini et al. 1996a). An obvious explanation
is that the fit was not well-conditioned and the two bond lengths were fully correlated
[see Chap. 9 and Demaison et al. (1997)] because the rotational constant of DCO+ is
missing. The rotational constant for this species was measured later, and its inclusion
in the fit considerably improved the conditioning. However, statistical diagnostics
indicate that this constant is not compatible with the other three. A careful analysis
of the rovibrational spectra shows that the experimental rovibrational constant α 1
of D
12 C
16 O+ is heavily perturbed because the state υ 1 = 1 is in strong interaction
with a nearby state. The unperturbed value of α 1 (obtained from the ab initio force
field) is 339.79 MHz whereas the experimental value is more than 100 MHz smaller,
227.45 MHz (Puzzarini et al. 1996a).
The υ 1 state at 2585.91 cm
−1 may interact with two nearby states, either υ 2 = 4
0
at 2621.36 cm
−1 or the combination state 01
1 0 at 2574.66 cm
−1 . It was first assumed
that the resonance was with the υ 2 = 4
0 state. However, soon after, an analysis of the
rotational spectra of different υ 2 states indicated that the υ 2 = 4
0 is not significantly
perturbed (in MHz): α 2 (υ 2 = 1) = −98.540; α 2 (υ 2 = 2) = −98.665; α 2 (υ 2 = 3) =
−98.856; and α 2 (υ 2 = 4) = −99.092.
The correct explanation is that a Coriolis interaction couples the e component of
the 11
1 0 state to the 100 state. The solution is to determine α 1 from the combination B(10
0 0) + 2B(01
1 1) where the Coriolis contributions to each perturbed state
6 Equilibrium Structures from Spectroscopy
confirmed by the value of the equilibrium inertial defect, Δ e = −0.0014(6) uÅ
2 ,
which is different from zero although two orders of magnitude smaller than the
ground-state value, 0 = 0.163 uÅ
2 . The electronic correction is also not negligible. The centrifugal distortion correction is much larger than the uncertainty of
the ground-state rotational constants, but it has a small effect on the accuracy of
the structure: there is a tiny increase of 0.001 pm for the SO bond length. Thus,
except in very accurate works, it may be neglected. Neglecting the electronic correction, decreases the r(SO) bond length by 0.01 pm and increases the ∠(OSO) bond
angle by 0.007°. These deviations may be considered as negligible for most works.
This conclusion also applies to the smaller
γ /4 correction. Using the equilibrium moments of inertia derived from Table 6.8 allows us to obtain the equilibrium
structure: r e (SO) = 143.0782(15) pm and ∠(OSO) = 119.3297(30)°. The estimated
uncertainties are quite small, but they do not take into account the systematic errors.
The true uncertainties may be one order of magnitude larger. See also Sects. 6.10
and 9.11.
The high accuracy achieved for SO 2 is mainly because the structure of this
molecule can be determined from the moments of inertia of a single isotopologue.
Hence, the equations are well-conditioned; see Sect. 9.4.1. When more than one
isotopologue (the most frequent case) has to be used, the situation is less favorable
because it is difficult to avoid the problem of ill-conditioning. A typical example
is given by the structure of the formyl cation HCO+. Its structure was determined
using the experimental equilibrium rotational constants of H
12 C
16 O
+ , H
13 C
16 O
+ ,
and H
12 C
18 O
+ : r e (CH) = 109.72 pm and r e (CO) = 110.47 pm (Woods 1988). The
results disagree with several accurate ab initio calculations; in particular for the CH
bond, which seems much too long (Puzzarini et al. 1996a). An obvious explanation
is that the fit was not well-conditioned and the two bond lengths were fully correlated
[see Chap. 9 and Demaison et al. (1997)] because the rotational constant of DCO+ is
missing. The rotational constant for this species was measured later, and its inclusion
in the fit considerably improved the conditioning. However, statistical diagnostics
indicate that this constant is not compatible with the other three. A careful analysis
of the rovibrational spectra shows that the experimental rovibrational constant α 1
of D
12 C
16 O+ is heavily perturbed because the state υ 1 = 1 is in strong interaction
with a nearby state. The unperturbed value of α 1 (obtained from the ab initio force
field) is 339.79 MHz whereas the experimental value is more than 100 MHz smaller,
227.45 MHz (Puzzarini et al. 1996a).
The υ 1 state at 2585.91 cm
−1 may interact with two nearby states, either υ 2 = 4
0
at 2621.36 cm
−1 or the combination state 01
1 0 at 2574.66 cm
−1 . It was first assumed
that the resonance was with the υ 2 = 4
0 state. However, soon after, an analysis of the
rotational spectra of different υ 2 states indicated that the υ 2 = 4
0 is not significantly
perturbed (in MHz): α 2 (υ 2 = 1) = −98.540; α 2 (υ 2 = 2) = −98.665; α 2 (υ 2 = 3) =
−98.856; and α 2 (υ 2 = 4) = −99.092.
The correct explanation is that a Coriolis interaction couples the e component of
the 11
1 0 state to the 100 state. The solution is to determine α 1 from the combination B(10
0 0) + 2B(01
1 1) where the Coriolis contributions to each perturbed state