320
6 Molecular Systems
z rot−nuc (T ) =
(2I a + 1) 2
σ
πB
A
e
α(4−m)/12 α
−
3
2
1 +
α 2
90
(1 − m)
2
+ · · ·
(1 + δ
) ,
(6.3.25)
in which σ = 3 is the symmetry number for C 3V molecules, α ≡ rot /T , m ≡ B/A
for prolate symmetric top molecules (B/C for oblate tops), and δ is the nuclear spin
symmetry correction factor given by
δ
2
(2I a + 1) 2 e
π 2 m(αm−6)/(54α) .
(6.3.26)
Centrifugal distortion effects for symmetric top molecules are significantly more
complicated than those for linear and spherical top molecules. We recall that the
energy levels for a rigid-rotor symmetric top are given by
E(j, k) = Bj (j + 1) + (A − B)k
2 ,
for prolate symmetric tops (A is replaced by C for oblate symmetric tops), with j
the usual rotational quantum number and k the quantum number associated with the
projection of the rotational angular momentum onto the body-fixed z-axis (which
is taken to lie along the molecular symmetry axis). When centrifugal distortion
effects are considered, the rather simple energy expression for a prolate symmetric
top molecule is replaced at the level of quartic distortions by [28]
E(j, k) = Bj (j + 1) + (A − B)k
2
− D j [j (j + 1)]
2
− D jk j (j + 1)k
2
− D k k
4
+ · · ·
(6.3.27)
with three quartic centrifugal distortion terms, given in terms of three constants D j ,
D k , D jk , rather than only one as found for linear and spherical top molecules. At
the next level of approximation, there will be four additional (sextic) centrifugal
distortion terms, rather than only the single sextic term found for linear and
spherical top molecules. The quartic centrifugal distortion correction terms have
been available for a long time [29], and can be represented in terms of a centrifugal
distortion correction factor to lowest order in the quartic distortion constants. In its
simplest representation, z rot−nuc (T ) has been given by McDowell [25] as
z rot−nuc (T ) =
(2I a + 1) 3
σ
πB
A
e
α(4−m)/12 α
−
3
2
1 +
α 2
90
(1−m)
2
+ · · ·
(1+δ
)f cd .
(6.3.28)
The centrifugal distortion correction factor, f cd , is given by
f cd = 1 + ρ 0 + ρ 1 α
−1
+ · · · ,
(6.3.29)
in which ρ 0 and ρ 1 are given in terms of the quartic centrifugal distortion constants
[26]
6 Molecular Systems
z rot−nuc (T ) =
(2I a + 1) 2
σ
πB
A
e
α(4−m)/12 α
−
3
2
1 +
α 2
90
(1 − m)
2
+ · · ·
(1 + δ
) ,
(6.3.25)
in which σ = 3 is the symmetry number for C 3V molecules, α ≡ rot /T , m ≡ B/A
for prolate symmetric top molecules (B/C for oblate tops), and δ is the nuclear spin
symmetry correction factor given by
δ
2
(2I a + 1) 2 e
π 2 m(αm−6)/(54α) .
(6.3.26)
Centrifugal distortion effects for symmetric top molecules are significantly more
complicated than those for linear and spherical top molecules. We recall that the
energy levels for a rigid-rotor symmetric top are given by
E(j, k) = Bj (j + 1) + (A − B)k
2 ,
for prolate symmetric tops (A is replaced by C for oblate symmetric tops), with j
the usual rotational quantum number and k the quantum number associated with the
projection of the rotational angular momentum onto the body-fixed z-axis (which
is taken to lie along the molecular symmetry axis). When centrifugal distortion
effects are considered, the rather simple energy expression for a prolate symmetric
top molecule is replaced at the level of quartic distortions by [28]
E(j, k) = Bj (j + 1) + (A − B)k
2
− D j [j (j + 1)]
2
− D jk j (j + 1)k
2
− D k k
4
+ · · ·
(6.3.27)
with three quartic centrifugal distortion terms, given in terms of three constants D j ,
D k , D jk , rather than only one as found for linear and spherical top molecules. At
the next level of approximation, there will be four additional (sextic) centrifugal
distortion terms, rather than only the single sextic term found for linear and
spherical top molecules. The quartic centrifugal distortion correction terms have
been available for a long time [29], and can be represented in terms of a centrifugal
distortion correction factor to lowest order in the quartic distortion constants. In its
simplest representation, z rot−nuc (T ) has been given by McDowell [25] as
z rot−nuc (T ) =
(2I a + 1) 3
σ
πB
A
e
α(4−m)/12 α
−
3
2
1 +
α 2
90
(1−m)
2
+ · · ·
(1+δ
)f cd .
(6.3.28)
The centrifugal distortion correction factor, f cd , is given by
f cd = 1 + ρ 0 + ρ 1 α
−1
+ · · · ,
(6.3.29)
in which ρ 0 and ρ 1 are given in terms of the quartic centrifugal distortion constants
[26]