6.4 Empirical Structures
133
be compared with the equilibrium structure: r e (C = O) = 117.59(4) pm, r e (C–Cl)
= 173.75(2) pm and ∠ e (ClCCl) = 111.85(2) pm (Demaison and Császár 2012).
A comparison of 45 equilibrium and effective angles has been made (Rudolph and
Demaison 2011). The median absolute deviation (MAD) [for definition, see (9.35)]
is only 0.2°, corresponding to a standard deviation of 0.3° but with a maximum
deviation of 1.7°. This outcome is the reason that the effective angles sometimes
seem more reliable than the distances. The comparison of r 0 bond lengths to r e
values for 55 bonds indicates that the MAD is 0.3 pm with a maximum deviation of
1.4 pm (Rudolph and Demaison 2011).
When hydrogen atoms are present, the determined bond lengths are affected by
a mostly systematic error of about 0.5(2) pm but with r e – r 0 > 0 for the ≡C(sp)–
H bond and r e − r 0 < 0 when an sp
2 or sp
3 carbon is involved (Demaison and
Wlodarczak 1994). Finally, as the rovibrational contributions to the inertial moments
are predominantly positive, an r 0 structure may appear slightly expanded compared
to the equilibrium structure. Indeed, in many cases, r 0 > r e .
In conclusion, the r 0 method is useful for a first estimate of the structure, but it
cannot be considered as reliable, except in the cases when the least-squares equations
are well-conditioned.
6.4.3 Substitution-Like Structures
6.4.3.1 Kraitchman’s Equations
A better assumption than the neglect of the rovibrational correction is to assume that
it remains constant upon isotopic substitution. Indeed, in many cases, the range of the
ε
ξ values is small compared to the mean values. For instance, in the particular case of
OCS, the mean value of ε = I 0 − I e is 0.250 uÅ
2 for twelve isotopologues whereas
the range is only 0.018 uÅ
2 ; see Table 6.2. However, there are at least two cases where
this assumption is not valid: (i) when a hydrogen atom is substituted by a deuterium
atom because the change of mass is large; (ii) when there is a large rotation of axes
upon isotopic substitution. Following a suggestion by Costain (1958), analytical
equations derived by Kraitchman (1953) have been widely used to determine the
Cartesian coordinates of the substituted atoms; it is the so-called r s method. For a
linear molecule, the calculation is quite simple. If we assume that z is the axis of the
molecule, and if the Cartesian coordinates are measured from the center-of-mass of
the parent molecule, the equilibrium moment of inertia of the parent molecule is
I
x
e = I
y
e =
m i z
2
i
(6.6)
where the sum is over all atoms, m i is the mass of atom i and z i its Cartesian coordinate.
If one atom is substituted, its mass becomes m + m and the equilibrium moment
of inertia of the isotopic species is
133
be compared with the equilibrium structure: r e (C = O) = 117.59(4) pm, r e (C–Cl)
= 173.75(2) pm and ∠ e (ClCCl) = 111.85(2) pm (Demaison and Császár 2012).
A comparison of 45 equilibrium and effective angles has been made (Rudolph and
Demaison 2011). The median absolute deviation (MAD) [for definition, see (9.35)]
is only 0.2°, corresponding to a standard deviation of 0.3° but with a maximum
deviation of 1.7°. This outcome is the reason that the effective angles sometimes
seem more reliable than the distances. The comparison of r 0 bond lengths to r e
values for 55 bonds indicates that the MAD is 0.3 pm with a maximum deviation of
1.4 pm (Rudolph and Demaison 2011).
When hydrogen atoms are present, the determined bond lengths are affected by
a mostly systematic error of about 0.5(2) pm but with r e – r 0 > 0 for the ≡C(sp)–
H bond and r e − r 0 < 0 when an sp
2 or sp
3 carbon is involved (Demaison and
Wlodarczak 1994). Finally, as the rovibrational contributions to the inertial moments
are predominantly positive, an r 0 structure may appear slightly expanded compared
to the equilibrium structure. Indeed, in many cases, r 0 > r e .
In conclusion, the r 0 method is useful for a first estimate of the structure, but it
cannot be considered as reliable, except in the cases when the least-squares equations
are well-conditioned.
6.4.3 Substitution-Like Structures
6.4.3.1 Kraitchman’s Equations
A better assumption than the neglect of the rovibrational correction is to assume that
it remains constant upon isotopic substitution. Indeed, in many cases, the range of the
ε
ξ values is small compared to the mean values. For instance, in the particular case of
OCS, the mean value of ε = I 0 − I e is 0.250 uÅ
2 for twelve isotopologues whereas
the range is only 0.018 uÅ
2 ; see Table 6.2. However, there are at least two cases where
this assumption is not valid: (i) when a hydrogen atom is substituted by a deuterium
atom because the change of mass is large; (ii) when there is a large rotation of axes
upon isotopic substitution. Following a suggestion by Costain (1958), analytical
equations derived by Kraitchman (1953) have been widely used to determine the
Cartesian coordinates of the substituted atoms; it is the so-called r s method. For a
linear molecule, the calculation is quite simple. If we assume that z is the axis of the
molecule, and if the Cartesian coordinates are measured from the center-of-mass of
the parent molecule, the equilibrium moment of inertia of the parent molecule is
I
x
e = I
y
e =
m i z
2
i
(6.6)
where the sum is over all atoms, m i is the mass of atom i and z i its Cartesian coordinate.
If one atom is substituted, its mass becomes m + m and the equilibrium moment
of inertia of the isotopic species is