Essentials of Computational Chemistry

362 10 THERMODYNAMIC PROPERTIES
(The superscript ‘o’ indicates that a ‘standard state’ is being referred to (see below).) Conventionally,
however, the last two terms in the final line of Eq. (10.9), i.e., those deriving from
Stirling’s approximation, are typically assigned to the translational partition function as well.
As they have no temperature dependence, this has no impact on Utrans; however, the entropy
of translation becomes
S o 2πMkBT trans = R ln
h2
3/2 o V
+ 5
(10.18)
2
Note that, because we are working under the assumption of an ideal gas, V o /NA in the term
in brackets can be replaced by kBT/P o .
A noteworthy aspect of Eqs. (10.16) and (10.18) is that they are altogether free of any
requirement to carry out an electronic structure calculation. Equation (10.16) is well known
for an ideal gas and is entirely independent of the molecule in question, and Eq. (10.18) can
be computed trivially as soon as the molecular weight is specified. Note, however, that the
units chosen for the various quantities must be such that the argument of the logarithm in
Eq. (10.18) (i.e., the partition function), is unitless.
The translational partition function is a function of both temperature and volume. However,
none of the other partition functions have a volume dependence. It is thus convenient to
eliminate the volume dependence of Strans by agreeing to report values that use exclusively
some volume that has been agreed upon by convention. The choices of the numerical value
of V and its associated units define a ‘standard state’ (or, more accurately, they contribute
to an overall definition that may be considerably more detailed, as described further below).
The most typical standard state used in theoretical calculations of entropies of translation
is the volume occupied by one mole of ideal gas at 298 K and 1 atm pressure, namely,
V o = 24.5 L.
10.3.5 Molecular Rotational Partition Function
In Section 9.3.1, the approach that is taken to solving the rigid-rotor nuclear Schrödinger
equation in order to compute rotational wave functions and energy levels was outlined,
and the particular cases of diatomic molecules and polyatomic prolate tops were explicitly
presented. The solution for the diatomic case is general for any linear molecule, so long as the
molecular moment of inertia I is computed in the appropriate fashion for more than 2 atoms
[Eq. (9.40)]. When the energy levels from Eq. (9.39) are used in the rotational partition
function with their appropriate degeneracies, as usual taking the lowest energy rotational
eigenvalue as the zero of energy, the sum can again be well approximated as an indefinite
integral at ‘normal’ temperatures, and solving that integral we find for a linear molecule that
q linear
rot (T ) = 8π 2 IkBT
σh 2
NA
(10.19)
where σ is 1 for asymmetric linear molecules and 2 for symmetric linear molecules (i.e.,
belonging to the C∞v and D∞h point groups, respectively). To be more precise, the validity