256
9 Least-Squares Method
Actually, in many cases, a simple plot of the residuals r i of the fit as a function of
the calculated ˆ
y i is easier to use to point out an autocorrelation.
9.10.3 Effect of the Autocorrelation on the Value
of the Parameters
It is essential to check how much this autocorrelation affects the parameters. The
accuracy of the semiexperimental method has been analyzed in great detail, see
Sect. 6.7, and no significant effect of the autocorrelation was found on the values of
the parameters.
Example 11 A nice example strengthening this conclusion is the structure of HC 5 N
known to be quite sensitive to the accuracy of the rotational constants. Indeed, some
bond lengths in HC 5 N vary greatly when only one rotational constant is increased by
only 0.1 MHz, see Sect. 9.4.1, Example 5. On the other hand, when all moments of
inertia are decreased by 10% of the rovibrational contribution (i.e., about 0.11 MHz),
the final structure is hardly modified, see Table 9.7.
In this particular case, there is an easy way to explain this behavior. The difference
between the exact equilibrium structure, r e , and the optimized semiexperimental
structure, r
se
e , may be written for the ith internal coordinate
r e (i) − r
se
e (i) =
j
∂r (i)
∂ I j
δ I j
(9.70)
As the errors are mainly systematic, the δI j are almost equal. The ∂r(i)/∂I j are also
of the same order of magnitude, but some of them are negative, whereas the others are
positive. If all atoms are substituted once, the sum δ I j
∂r (i)/∂ I j nearly cancels.
Table 9.7 Influence of decrease of 10% of the rovibration correction on the semiexperimental
equilibrium structure of H–C 5 ≡C 4 –C 3 ≡C 2 –C 1 ≡N (in pm) (Bizzocchi et al. 2004)
r se
e
a
r se
e
*b
r se
e − r se
e
*
s c
r(H–C 5 )
106.220
106.169
−0.05
0.003
r(C 4 ≡C 5 )
120.949
120.940
−0.01
0.005
r(C 3 –C 4 )
136.518
136.483
−0.03
0.011
r(C 2 ≡C 3 )
121.284
121.354
0.07
0.016
r(C 1 –C 2 )
137.079
137.030
−0.05
0.011
r(C 1 ≡N)
116.209
116.196
−0.01
0.005
a Semiexperimental structure using the original equilibrium moments of inertia
b All moments of inertia are decreased by 10% of the rovibrational correction
c Standard deviation of the semiexperimental structure, column 2
9 Least-Squares Method
Actually, in many cases, a simple plot of the residuals r i of the fit as a function of
the calculated ˆ
y i is easier to use to point out an autocorrelation.
9.10.3 Effect of the Autocorrelation on the Value
of the Parameters
It is essential to check how much this autocorrelation affects the parameters. The
accuracy of the semiexperimental method has been analyzed in great detail, see
Sect. 6.7, and no significant effect of the autocorrelation was found on the values of
the parameters.
Example 11 A nice example strengthening this conclusion is the structure of HC 5 N
known to be quite sensitive to the accuracy of the rotational constants. Indeed, some
bond lengths in HC 5 N vary greatly when only one rotational constant is increased by
only 0.1 MHz, see Sect. 9.4.1, Example 5. On the other hand, when all moments of
inertia are decreased by 10% of the rovibrational contribution (i.e., about 0.11 MHz),
the final structure is hardly modified, see Table 9.7.
In this particular case, there is an easy way to explain this behavior. The difference
between the exact equilibrium structure, r e , and the optimized semiexperimental
structure, r
se
e , may be written for the ith internal coordinate
r e (i) − r
se
e (i) =
j
∂r (i)
∂ I j
δ I j
(9.70)
As the errors are mainly systematic, the δI j are almost equal. The ∂r(i)/∂I j are also
of the same order of magnitude, but some of them are negative, whereas the others are
positive. If all atoms are substituted once, the sum δ I j
∂r (i)/∂ I j nearly cancels.
Table 9.7 Influence of decrease of 10% of the rovibration correction on the semiexperimental
equilibrium structure of H–C 5 ≡C 4 –C 3 ≡C 2 –C 1 ≡N (in pm) (Bizzocchi et al. 2004)
r se
e
a
r se
e
*b
r se
e − r se
e
*
s c
r(H–C 5 )
106.220
106.169
−0.05
0.003
r(C 4 ≡C 5 )
120.949
120.940
−0.01
0.005
r(C 3 –C 4 )
136.518
136.483
−0.03
0.011
r(C 2 ≡C 3 )
121.284
121.354
0.07
0.016
r(C 1 –C 2 )
137.079
137.030
−0.05
0.011
r(C 1 ≡N)
116.209
116.196
−0.01
0.005
a Semiexperimental structure using the original equilibrium moments of inertia
b All moments of inertia are decreased by 10% of the rovibrational correction
c Standard deviation of the semiexperimental structure, column 2