6.4 Empirical Structures
137
Nevertheless, the r s method has the advantage of permitting a partial structure
determination when only a few isotopologues have been studied. However, a comparison of r 0 and r s bond lengths to r e values for 55 bonds indicates that the maximum
deviation of the residuals is much larger for r s − r e than for r 0 − r e , 2.1 pm instead
of 1.4 pm, respectively (Rudolph and Demaison 2011).
When possible (i.e., when the rotational constants of a sufficiently large set of
isotopologues is available), it is better to use the least-squares fitting method where,
in addition to the internal coordinates, the three rovibrational corrections ε
ξ (ξ =
a, b, c) are fitted. This method is sometimes called r Iε , (Rudolph 1991). It avoids
the problem of imaginary coordinates, and the first-moment equations (m i z i = 0)
are automatically satisfied. Its obvious inconvenience is that this method introduces
three supplementary parameters, which is often enough to undermine the quality of
the fit. For this reason, the method described in Sect. 6.5 is preferred.
6.4.3.3 r c Method
For molecules with few atoms, it is possible to considerably improve the accuracy
of the substitution method. It can be shown that (Watson 1973; Smith and Watson
1978)
2I
s
ξ − I
0
ξ = I
e
ξ +
1
M
n
i=1
m i m i
∂
2
(Mε
ξ
)
∂m
2
i
+ · · ·
(6.14)
where ξ = a, b, c and n is the number of atoms. The substitution moment of inertia, I
s
ξ ,
must be calculated using the squares of all atomic position coordinates as individually
obtained by the r s method, even if the square is negative. Nakata et al. (1980, 1981)
suggested the use of complementary sets of isotopologues in order to eliminate the
first term in the summation of (6.14). In other words, substitutions with positive and
negative m i have to be used. This method gives accurate results; see for instance
Table 8 of Demaison et al. (1997). The large number of isotopologues required limits
its application to very small molecules.
6.5 Mass-Dependent Structures
It was observed that the rovibrational correction ε is approximately proportional to
the square root of the moment of inertia. It was first found empirically (Demaison
and Nemes 1979) and, then, justified theoretically by Watson (1973) who confirmed
that ε is a homogeneous function of atomic masses of degree ½.
Therefore, the ground-state moment of inertia may be approximated by
I
ξ
0 = I
ξ
m + c ξ
I
ξ
m
(6.15)
137
Nevertheless, the r s method has the advantage of permitting a partial structure
determination when only a few isotopologues have been studied. However, a comparison of r 0 and r s bond lengths to r e values for 55 bonds indicates that the maximum
deviation of the residuals is much larger for r s − r e than for r 0 − r e , 2.1 pm instead
of 1.4 pm, respectively (Rudolph and Demaison 2011).
When possible (i.e., when the rotational constants of a sufficiently large set of
isotopologues is available), it is better to use the least-squares fitting method where,
in addition to the internal coordinates, the three rovibrational corrections ε
ξ (ξ =
a, b, c) are fitted. This method is sometimes called r Iε , (Rudolph 1991). It avoids
the problem of imaginary coordinates, and the first-moment equations (m i z i = 0)
are automatically satisfied. Its obvious inconvenience is that this method introduces
three supplementary parameters, which is often enough to undermine the quality of
the fit. For this reason, the method described in Sect. 6.5 is preferred.
6.4.3.3 r c Method
For molecules with few atoms, it is possible to considerably improve the accuracy
of the substitution method. It can be shown that (Watson 1973; Smith and Watson
1978)
2I
s
ξ − I
0
ξ = I
e
ξ +
1
M
n
i=1
m i m i
∂
2
(Mε
ξ
)
∂m
2
i
+ · · ·
(6.14)
where ξ = a, b, c and n is the number of atoms. The substitution moment of inertia, I
s
ξ ,
must be calculated using the squares of all atomic position coordinates as individually
obtained by the r s method, even if the square is negative. Nakata et al. (1980, 1981)
suggested the use of complementary sets of isotopologues in order to eliminate the
first term in the summation of (6.14). In other words, substitutions with positive and
negative m i have to be used. This method gives accurate results; see for instance
Table 8 of Demaison et al. (1997). The large number of isotopologues required limits
its application to very small molecules.
6.5 Mass-Dependent Structures
It was observed that the rovibrational correction ε is approximately proportional to
the square root of the moment of inertia. It was first found empirically (Demaison
and Nemes 1979) and, then, justified theoretically by Watson (1973) who confirmed
that ε is a homogeneous function of atomic masses of degree ½.
Therefore, the ground-state moment of inertia may be approximated by
I
ξ
0 = I
ξ
m + c ξ
I
ξ
m
(6.15)