Appendix E
Spherical/Symmetric Top Molecules
E.1 Linear Molecules: A Brief Review
We have seen (section “Heteronuclear Diatomic Molecules”) that the classical
expression for the rotational partition function for a linear molecule may be obtained
by approximating the sum over the rotational angular quantum number j of the
rotational energy states by treating j as a continuous variable, and integrating over
it. More generally, we have
z rot (T ) =
1
σ
j,m
e
−E rot (j,m)/(k B T ) ,
(E.1.1)
with E rot (j, m) = ¯
h 2 j (j + 1)/2I , and σ a symmetry number associated with the
molecular symmetry group. Upon defining the characteristic rotational temperature
rot by rot ≡ ¯
h 2 /(2I k B ), we obtain
z rot (T ) =
1
σ
j,m
e
−j (j+1)) rot /T
or
z rot (T ) =
1
σ
∞
j =0
(2j + 1)e
−j (j+1)) rot /T .
(E.1.2)
The classical approximation to z rot (T ) is thus given by
z rot (T ) =
1
σ
∞
0
(2j + 1)e
−j (j+1)) rot /T d j ,
© Springer Nature Switzerland AG 2021
F. R. W. McCourt, Statistical Thermodynamics for Pure and Applied Sciences,
https://doi.org/10.1007/978-3-030-52006-9
639