140
6 Equilibrium Structures from Spectroscopy
Table 6.5 Different structures of OCSe and SO 2 (distances in pm, angles in deg.)
OCSe a
SO b
2
r(OC)
r(CSe)
κ c
r(SO)
∠(OSO)
κ c
r 0
115.364(59)
171.299(44)
102
143.358(17)
119.420(30)
2.98
r
(1)
m
115.532(14)
170.816(22)
268
143.084(24)
119.442(32)
343
r
(2)
m
115.373(2)
170.950(2)
899
143.068(12)
119.340(20)
1486
r e
115.327(2)
170.981(2)
131
143.0782(15)
119.3297(30)
8
a Unit weighted fit of 27 isotopologues, except for r e where only 8 isotopologues were used
b Unit weighted fit of 17 isotopologues, except for r e where only one isotopologue was used
c Condition number
axis is exceptional (it mainly concerns oblate top molecules, which are much less
common than prolate top molecules).
The different structures of two small molecules, the linear OCSe and the triatomic
SO 2 , are compared in Table 6.5. The improvement, when going from r 0 to r
(2)
m via
r
(1)
m , is shown by the decrease of the standard deviation of the fitted parameters, but
there is a parallel increase of the condition number κ (a large condition number is
an indicator of ill-conditioning; see Sect. 9.4.1). The r
(2)
m structure is within a few
tenths of pm of the r e structure, which may be considered as satisfactory. However,
in the case of OCSe, the r
(1)
m (OC) bond length is still far from the r e value. Because
of the small value of the Cartesian coordinate of the central carbon atom, the r
(2)
m
approximation is required.
The weak points of this method are that there are more parameters to fit: c a , c b ,
c c , d a , d b , d c , and δ H and that (6.16) is only an approximation. When the leastsquares system is ill-conditioned, which is the rule when the number of parameters
to determine is large, the small remaining errors due to the approximate model are
considerably amplified and the derived parameters are inaccurate. For this reason,
the general r
(2)
m method was rarely used with success on a large molecule. However,
a trivial amendment improves it significantly: using the method of mixed estimation
(see Sect. 9.7), the structural parameters are fitted concurrently to predicate parameters and moments of inertia, associated with appropriate uncertainties. The predicate
parameters are usually obtained by high-level quantum-chemical computations; see
Chap. 2. With this modification, it was possible to determine accurate structures
for molecules as large as fructose, C 6 H 12 O 6 , (24 atoms, 66 independent structural
parameters) (Vogt et al. 2016). As an example, the r
(2)
m and the semiexperimental
equilibrium structures of the equatorial conformer of ethynylcyclohexane, C 8 H 12 ,
are compared in Table 6.6 (Vogt et al. 2018). The agreement is very satisfactory.
6 Equilibrium Structures from Spectroscopy
Table 6.5 Different structures of OCSe and SO 2 (distances in pm, angles in deg.)
OCSe a
SO b
2
r(OC)
r(CSe)
κ c
r(SO)
∠(OSO)
κ c
r 0
115.364(59)
171.299(44)
102
143.358(17)
119.420(30)
2.98
r
(1)
m
115.532(14)
170.816(22)
268
143.084(24)
119.442(32)
343
r
(2)
m
115.373(2)
170.950(2)
899
143.068(12)
119.340(20)
1486
r e
115.327(2)
170.981(2)
131
143.0782(15)
119.3297(30)
8
a Unit weighted fit of 27 isotopologues, except for r e where only 8 isotopologues were used
b Unit weighted fit of 17 isotopologues, except for r e where only one isotopologue was used
c Condition number
axis is exceptional (it mainly concerns oblate top molecules, which are much less
common than prolate top molecules).
The different structures of two small molecules, the linear OCSe and the triatomic
SO 2 , are compared in Table 6.5. The improvement, when going from r 0 to r
(2)
m via
r
(1)
m , is shown by the decrease of the standard deviation of the fitted parameters, but
there is a parallel increase of the condition number κ (a large condition number is
an indicator of ill-conditioning; see Sect. 9.4.1). The r
(2)
m structure is within a few
tenths of pm of the r e structure, which may be considered as satisfactory. However,
in the case of OCSe, the r
(1)
m (OC) bond length is still far from the r e value. Because
of the small value of the Cartesian coordinate of the central carbon atom, the r
(2)
m
approximation is required.
The weak points of this method are that there are more parameters to fit: c a , c b ,
c c , d a , d b , d c , and δ H and that (6.16) is only an approximation. When the leastsquares system is ill-conditioned, which is the rule when the number of parameters
to determine is large, the small remaining errors due to the approximate model are
considerably amplified and the derived parameters are inaccurate. For this reason,
the general r
(2)
m method was rarely used with success on a large molecule. However,
a trivial amendment improves it significantly: using the method of mixed estimation
(see Sect. 9.7), the structural parameters are fitted concurrently to predicate parameters and moments of inertia, associated with appropriate uncertainties. The predicate
parameters are usually obtained by high-level quantum-chemical computations; see
Chap. 2. With this modification, it was possible to determine accurate structures
for molecules as large as fructose, C 6 H 12 O 6 , (24 atoms, 66 independent structural
parameters) (Vogt et al. 2016). As an example, the r
(2)
m and the semiexperimental
equilibrium structures of the equatorial conformer of ethynylcyclohexane, C 8 H 12 ,
are compared in Table 6.6 (Vogt et al. 2018). The agreement is very satisfactory.