8.5 Empirical Correlations (Legon and Demaison 2011)
219
constants. They also showed that a similar relationship is also valid for cubic and
quartic force constants.
It would be extremely interesting to be able to derive the bond length from
the stretching force constant. Unfortunately, with the exception of the diatomic
molecules, it is more difficult to obtain accurate force constants than bond lengths.
There is, however, an important exception: the bond X–H (with X=C, N, O, …).
The stretching frequency of such a bond has a much higher fundamental frequency
than the other modes due to the small mass of hydrogen (above 2900 cm
−1 ). If this
stretching mode is isolated, it is possible to treat it separately. Indeed, the secondorder perturbation calculation shows that the coupling terms will have a negligible
effect because the energy difference is large. In such a case, there is a relationship
between the bond length and the corresponding stretching vibrational frequency. It
was first pointed out by Bernstein (1962) and extensively developed by McKean
(1978; 1989) who used selective deuteriation, all the hydrogens being replaced by
deuterium, except for the CH bond of interest. Although this method gives good
results, it is sometimes thorny to use because the synthesis at a deuterated molecule
with only one hydrogen left may be extremely complicated. Furthermore, anharmonic resonances may still affect the stretching vibration. An alternative way is to
use overtone bands to determine isolated stretching frequencies. These bands result
from the multiexcitation of a single vibrational mode. The motions of the individual bonds become increasingly localized upon increasing excitation, and thus,
the internal couplings become smaller (Henry 1981, 1987). The band origin of as
many overtones of the CH stretch as possible needs to be measured and inserted into
a so-called Birge–Sponer plot (1926):
ν = υ ˜
ω − υ(υ + 1) ˜
ωx
(8.10)
where ˜
ω is the mechanical frequency and ˜
ωx the first anharmonic correction term.
This procedure thus also allows the frequency of the “unperturbed” fundamental
band, ν is , to be determined:
ν is = ˜
ω − 2 ˜
ωx
(8.11)
A least-squares fit of 60 equilibrium CH bond lengths gave the following
expression (Demaison and Rudolph 2008)
r e (CH)
pm
= 130.47(29) − 7.311(97) × 10
−3
ν is (CH)
cm
−1
(8.12)
with a correlation coefficient ρ = 0.990 and a standard deviation σ = 0.11 pm,
that corresponds to the expected accuracy of the bond lengths. See also Fig. 8.4.
Although this relationship is highly satisfactory, a careful analysis of the residuals
of the fit shows that they are large and all positive for the acetynyl H–C≡bonds. A
least-square’s fit of only the ethynyl data (9 points) gives a significantly different
correlation
219
constants. They also showed that a similar relationship is also valid for cubic and
quartic force constants.
It would be extremely interesting to be able to derive the bond length from
the stretching force constant. Unfortunately, with the exception of the diatomic
molecules, it is more difficult to obtain accurate force constants than bond lengths.
There is, however, an important exception: the bond X–H (with X=C, N, O, …).
The stretching frequency of such a bond has a much higher fundamental frequency
than the other modes due to the small mass of hydrogen (above 2900 cm
−1 ). If this
stretching mode is isolated, it is possible to treat it separately. Indeed, the secondorder perturbation calculation shows that the coupling terms will have a negligible
effect because the energy difference is large. In such a case, there is a relationship
between the bond length and the corresponding stretching vibrational frequency. It
was first pointed out by Bernstein (1962) and extensively developed by McKean
(1978; 1989) who used selective deuteriation, all the hydrogens being replaced by
deuterium, except for the CH bond of interest. Although this method gives good
results, it is sometimes thorny to use because the synthesis at a deuterated molecule
with only one hydrogen left may be extremely complicated. Furthermore, anharmonic resonances may still affect the stretching vibration. An alternative way is to
use overtone bands to determine isolated stretching frequencies. These bands result
from the multiexcitation of a single vibrational mode. The motions of the individual bonds become increasingly localized upon increasing excitation, and thus,
the internal couplings become smaller (Henry 1981, 1987). The band origin of as
many overtones of the CH stretch as possible needs to be measured and inserted into
a so-called Birge–Sponer plot (1926):
ν = υ ˜
ω − υ(υ + 1) ˜
ωx
(8.10)
where ˜
ω is the mechanical frequency and ˜
ωx the first anharmonic correction term.
This procedure thus also allows the frequency of the “unperturbed” fundamental
band, ν is , to be determined:
ν is = ˜
ω − 2 ˜
ωx
(8.11)
A least-squares fit of 60 equilibrium CH bond lengths gave the following
expression (Demaison and Rudolph 2008)
r e (CH)
pm
= 130.47(29) − 7.311(97) × 10
−3
ν is (CH)
cm
−1
(8.12)
with a correlation coefficient ρ = 0.990 and a standard deviation σ = 0.11 pm,
that corresponds to the expected accuracy of the bond lengths. See also Fig. 8.4.
Although this relationship is highly satisfactory, a careful analysis of the residuals
of the fit shows that they are large and all positive for the acetynyl H–C≡bonds. A
least-square’s fit of only the ethynyl data (9 points) gives a significantly different
correlation