Essentials of Computational Chemistry

10.3 ENSEMBLE PROPERTIES AND BASIC STATISTICAL MECHANICS 359
from the second to the third line, the sum has been changed so that it goes over discrete
energy levels, rather than individual states, and gk is the degeneracy of level k. The quantity
in brackets on the third line defines the molecular partition function q.
A second consequence of the ideal gas assumption is that PV in Eq. (10.4) may be
replaced by NkBT . In the special case where we are working with one mole of molecules,
in which case N = NA (Avogadro’s number), we may replace PV with RT ,whereR is the
universal gas constant (8.3145 J mol −1 K −1 ).
10.3.2 Separability of Energy Components
We have thus reduced the problem from finding the ensemble partition function Q to finding
the molecular partition function q. In order to make further progress, we assume that the
molecular energy ε can be expressed as a separable sum of electronic, translational, rotational,
and vibrational terms, i.e.,
q(V,T) =
=
=
levels
k
gke −εk(V )/kBT
levels
gke
k
−[εelec+εtrans(V )+εrot+εvib]k/kBT
elec
gie
i
−εi/kBT
⎡
trans
⎣ gje
j
−εj (V )/kBT
⎤
⎦
rot
gke −εk/kBT
k
vib
gle −εl/kBT
= qelec(T )qtrans(V, T )qrot(T )qvib(T ) (10.8)
where again advantage is taken of the ability to express an exponential of sums as a product
of sums of exponentials, and the separate lines make clear that the degeneracy of a total
molecular energy level is simply the product of the degeneracies of each of its contributing
components.
Note in Eqs. (10.3) and (10.5), Q always appears as the argument of the natural logarithm
function. Using Eqs. (10.7) and (10.8), our assumptions to this point allow us to write
[qelec(T )qtrans(V, T )qrot(T )qvib(T )]
ln[Q(N,V,T)] = ln
N
N!
= N{ln[qelec(T )] + ln[qtrans(V, T )] + ln[qelec(T )]
+ ln[qvib(T )]}−ln(N!)
≈ N{ln[qelec(T )] + ln[qtrans(V, T )]
+ ln[qelec(T )] + ln[qvib(T )]}−N ln N + N (10.9)
where going from the second to the third equality makes use of Stirling’s approximation for
ln(N!) when N is large. This separation of terms by the logarithm function makes evident
l