3.9 Higher-Order Effects
73
correction, (3.77), and the electronic correction, (3.55)], it gives the adiabatic bond
length r
ad
e which differs for different isotopologues.
As shown by Le Roy (1999), it is advantageous to choose one isotopologue as the
reference (α = 1) to which the Dunham parameters of all other species are related.
The Dunham parameters for (α = 1) are given by
Y
(α)
lk =
Y
(1)
lk +
m
(α)
A
m
(α)
A
δ
A
lk +
m
(α)
B
m
(α)
B
δ
B
lk
μ 1
μ α
k+l/2
(3.81)
where m
(α)
X is the mass of atom X(A, B) and m
(α)
X = m
(α)
X − m
(1)
X and μ α is the
reduced mass of isotopologue α. The relationship between δ
X
lk and
X
lk is
X
lk = −δ
X
lk
m
(1)
X
m e
Y
(1)
lk + δ
A
lk + δ
B
lk
−1
(3.82)
3.9.3 Effect of the Size of the Nuclei
In addition to the mass variation by isotopic substitution also, the nuclear size will
vary slightly giving rise to small changes in the Coulomb interaction between the
electrons and the nucleus (Tiemann et al. 1982b). When at least one of the atoms is
heavy, its nucleus can no longer be considered as a point charge and the charge is
distributed over a volume to produce the so-called field-shift contribution. This may
be pointed out by the large difference found for
A
01 and
B
01 , which are expected
to be approximately equal (Watson 1980). This isotope effect which is called field
shift in the theory of atomic spectra slightly modifies (3.79) where the mean square
nuclear charge radius
parameter V
A,B
01 is introduced
Y 01 =
¯
U 01
μ (l+2k)/2
1 + m e
A
01
m A
+
B
01
m B
+ V
A
01 δ
r
2
AA
+ V
B
01 δ
r
2
BB
(3.83)
with
¯
U 01 = U 01
1 + V
A
01
r
2
A
+ V
B
01
r
2
B
(3.84)
The field-shift parameter V
A,B
01 depends mainly on the electron density and its
derivatives with respect to the internuclear distance at the nucleus A or B. It is
isotopically independent. The difference δ
r
2
AA
is the mean square nuclear charge
radius on isotopic substitution A → A
. These differences are tabulated as well as
the values of
only significant when the accuracy of the measurements is extremely high and when
73
correction, (3.77), and the electronic correction, (3.55)], it gives the adiabatic bond
length r
ad
e which differs for different isotopologues.
As shown by Le Roy (1999), it is advantageous to choose one isotopologue as the
reference (α = 1) to which the Dunham parameters of all other species are related.
The Dunham parameters for (α = 1) are given by
Y
(α)
lk =
Y
(1)
lk +
m
(α)
A
m
(α)
A
δ
A
lk +
m
(α)
B
m
(α)
B
δ
B
lk
μ 1
μ α
k+l/2
(3.81)
where m
(α)
X is the mass of atom X(A, B) and m
(α)
X = m
(α)
X − m
(1)
X and μ α is the
reduced mass of isotopologue α. The relationship between δ
X
lk and
X
lk is
X
lk = −δ
X
lk
m
(1)
X
m e
Y
(1)
lk + δ
A
lk + δ
B
lk
−1
(3.82)
3.9.3 Effect of the Size of the Nuclei
In addition to the mass variation by isotopic substitution also, the nuclear size will
vary slightly giving rise to small changes in the Coulomb interaction between the
electrons and the nucleus (Tiemann et al. 1982b). When at least one of the atoms is
heavy, its nucleus can no longer be considered as a point charge and the charge is
distributed over a volume to produce the so-called field-shift contribution. This may
be pointed out by the large difference found for
A
01 and
B
01 , which are expected
to be approximately equal (Watson 1980). This isotope effect which is called field
shift in the theory of atomic spectra slightly modifies (3.79) where the mean square
nuclear charge radius
parameter V
A,B
01 is introduced
Y 01 =
¯
U 01
μ (l+2k)/2
1 + m e
A
01
m A
+
B
01
m B
+ V
A
01 δ
r
2
AA
+ V
B
01 δ
r
2
BB
(3.83)
with
¯
U 01 = U 01
1 + V
A
01
r
2
A
+ V
B
01
r
2
B
(3.84)
The field-shift parameter V
A,B
01 depends mainly on the electron density and its
derivatives with respect to the internuclear distance at the nucleus A or B. It is
isotopically independent. The difference δ
r
2
AA
is the mean square nuclear charge
radius on isotopic substitution A → A
. These differences are tabulated as well as
the values of
only significant when the accuracy of the measurements is extremely high and when