4.5 Symmetric Top
85
A =
h
8π 2 I y
= B =
h
8π 2 I x
> C =
h
8π 2 I z
(4.26)
and the corresponding rotational energy in frequency unit is
E J K =
E
J K
h
= (C − B)K
2
+ B J (J + 1)
(4.27)
Provided that the molecule has a permanent electric dipole moment, the selection
rules for absorption or emission of electromagnetic radiation are
J = 0, ±1 and K = 0 for K = 0
(4.28)
and
J = ±1 and K = 0 for K = 0
(4.29)
The absorption frequencies are obtained as
ν = 2B(J + 1)
(4.30)
It has to be noted that all transitions of same J but different K have the same
value. Actually, the inclusion of centrifugal distortion may resolve this degeneracy,
see next section.
4.5.2 Centrifugal Distortion
From the discussion of the diatomic molecule, Sect. 3.5, it appears that the first term
of the rotational energy is quadratic in J (or, more exactly, is a function of P
2 ),
whereas the next term, the centrifugal distortion term, is quartic in J (function of
P
4 ). We may anticipate that the same behavior will apply to the symmetric top. In
other words, we will have a term in J
2 (J + 1)
2 , J(J + 1)K
2 , and K
4 . The rotational
energy taking into account the centrifugal distortion may be written
E J K =
E
J K
h
= (A − B)K
2
+ B J (J + 1) − D J J
2
(J + 1)
2
− D J K J (J + 1)K
2
− D K K
4
(4.31)
This equation is valid for a prolate top. For an oblate top, it is enough to replace
A by C.
Taking into account the selection rules, the expression for the rotational frequencies is