9.3 Weighted Least Squares Method
239
computed at the B3LYP/6-311 + G(3df, 2pd) level of theory, the standard deviation of these 15 semiexperimental rotational constants is 58 kHz in good agreement
with the corresponding standard deviation of the fit, 53 kHz. On the other hand,
the comparison of the rovibrational correction calculated with different force fields
indicates that the systematic error is probably larger than one MHz.
Example 2 Ethynylcyclohexane (Vogt et al. 2018). It is possible to estimate σ rand
because the equilibrium planar moment of inertia P b = (I a + I c – I b )/2 should be
constant for the parent species and the isotopologues
13 C1,
13 C4,
13 C7, and
13 C8,
due to the C s symmetry of the molecule. The range of P b may be used to estimate
the standard deviation (see (9.73)), which may be written
σ
2
(P b ) =
σ
2
(I a ) + σ
2
(I b ) + σ
2
(I c )
/4
(9.18)
For this molecule, it may be assumed that σ (I b ) = σ (I c ) (both I c and I b have
similar values and almost same rovibrational corrections). Furthermore, I a is three
times smaller than I b and its rovibrational correction about 1.7 times smaller, thus,
σ (I a ) ≈ σ (I b )/2 and σ (P b ) ≈ 3σ (I b )/4. This allows us to estimate the random
error of the moments of inertia.
Example 3 Semiexperimental equilibrium structure of PH 2 Cl which shows the
importance of choosing correct weights. The results of fits of nine moments of inertia
are given in Table 9.1. In the first fit, it is assumed that the A rotational constant is
twice less accurate than B and C. The weights are optimized in the second fit, see
Sect. 9.5. Note that the standard deviations attached to the parameters only reflect the
quality of the fit; they are not a reliable estimate of the accuracy of the parameters.
Table 9.1 Semiexperimental
equilibrium structure of
PH 2 Cl (distances in pm and
angles in degree) (Dréan et al.
1996b)
r a
e
Fit 1
Fit 2
r(PCl)
206.3(2)
205.90(47)
206.1215(38)
r(PH)
141.1(2)
141.79(66)
141.386(23)
∠(HPCl)
96.4(2)
97.5(19)
96.622(18)
∠(HPH)/2
46.4(1)
46.0(10)
46.616(10)
s(A) b
0.20
108
s(B) b
0.10
0.25
s(C) b
0.10
0.29
κ c
86.9
7.3
a Previous results
b Uncertainty of the rotational constant used for the weighting (in
MHz)
c Condition number, see Sect. 9.4.1
239
computed at the B3LYP/6-311 + G(3df, 2pd) level of theory, the standard deviation of these 15 semiexperimental rotational constants is 58 kHz in good agreement
with the corresponding standard deviation of the fit, 53 kHz. On the other hand,
the comparison of the rovibrational correction calculated with different force fields
indicates that the systematic error is probably larger than one MHz.
Example 2 Ethynylcyclohexane (Vogt et al. 2018). It is possible to estimate σ rand
because the equilibrium planar moment of inertia P b = (I a + I c – I b )/2 should be
constant for the parent species and the isotopologues
13 C1,
13 C4,
13 C7, and
13 C8,
due to the C s symmetry of the molecule. The range of P b may be used to estimate
the standard deviation (see (9.73)), which may be written
σ
2
(P b ) =
σ
2
(I a ) + σ
2
(I b ) + σ
2
(I c )
/4
(9.18)
For this molecule, it may be assumed that σ (I b ) = σ (I c ) (both I c and I b have
similar values and almost same rovibrational corrections). Furthermore, I a is three
times smaller than I b and its rovibrational correction about 1.7 times smaller, thus,
σ (I a ) ≈ σ (I b )/2 and σ (P b ) ≈ 3σ (I b )/4. This allows us to estimate the random
error of the moments of inertia.
Example 3 Semiexperimental equilibrium structure of PH 2 Cl which shows the
importance of choosing correct weights. The results of fits of nine moments of inertia
are given in Table 9.1. In the first fit, it is assumed that the A rotational constant is
twice less accurate than B and C. The weights are optimized in the second fit, see
Sect. 9.5. Note that the standard deviations attached to the parameters only reflect the
quality of the fit; they are not a reliable estimate of the accuracy of the parameters.
Table 9.1 Semiexperimental
equilibrium structure of
PH 2 Cl (distances in pm and
angles in degree) (Dréan et al.
1996b)
r a
e
Fit 1
Fit 2
r(PCl)
206.3(2)
205.90(47)
206.1215(38)
r(PH)
141.1(2)
141.79(66)
141.386(23)
∠(HPCl)
96.4(2)
97.5(19)
96.622(18)
∠(HPH)/2
46.4(1)
46.0(10)
46.616(10)
s(A) b
0.20
108
s(B) b
0.10
0.25
s(C) b
0.10
0.29
κ c
86.9
7.3
a Previous results
b Uncertainty of the rotational constant used for the weighting (in
MHz)
c Condition number, see Sect. 9.4.1