642
E Spherical/Symmetric Top Molecules
Fig. E.1 Symmetric top
molecule and body-fixed axis
system
The energy of the rigid rotational motion of a symmetric top molecule is given
by
E rot =
¯
h 2
2I B
(J
2
x + J
2
y ) +
¯
h 2
2I A
J
2
z
=
¯
h 2
2I B
(J
2
x + J
2
y + J
2
z ) −
¯
h 2
2I B
J
2
z +
¯
h 2
2I A
J
2
z .
(E.3.1)
The rotational Hamiltonian corresponding to this energy is then given by
H rot =
¯
h 2
2I B
J 2 +
¯
h 2
2I A
−
¯
h 2
2I B
J
2
z .
(E.3.2)
The eigenstates of this rigid body rotational Hamiltonian can be designated by
|jkm, and correspond to eigenenergies E jkm :
H rot |jkm = E jkm |jkm .
(E.3.3)
The quantum states |jkm are also simultaneous eigenstates of the set of
(mutually commuting) operators
J 2 ,
J z ,
J Z , so that the energy eigenvalues for this
molecule are
E jkm =
¯
h 2
2I B
j (j + 1) +
¯
h 2
2I A
−
¯
h 2
2I B
k
2 .
(E.3.4)
The quantity that appears in the exponent of the rotational partition function
expression is E jkm /(k B T ). For convenience this will be written as
E jkm
k B T
= αj (j + 1) + βk
2 ,
(E.3.5)