Essentials of Computational Chemistry

358 10 THERMODYNAMIC PROPERTIES
where i runs over all possible energy states of the system having energy Ei
Boltzmann’s constant (1.3806 × 10
and kB is
−23 JK−1 ).
Within the canonical ensemble, and using established thermodynamic definitions, all of
the following are true
U = kBT 2
∂ ln Q
∂T
(10.3)
N,V
H = U + PV (10.4)
∂ ln Q
S = kB ln Q + kBT
(10.5)
∂T
N,V
G = H − TS (10.6)
where the notation associated with the partial derivatives in Eqs. (10.3) and (10.5) implies N
and V held constant during differentiation with respect to T , H is enthalpy, P is pressure,
S is entropy, and G is (Gibbs) free energy.
Of course, the elegance of Eqs. (10.3)–(10.6) is somewhat muted by the daunting prospect
of finding an explicit representation of Q that permits the necessary partial differentiations in
Eqs. (10.3) and (10.5) to be carried out. For a true ensemble, Q must be some fantastically
complex many-body function involving a staggeringly enormous number of energy levels.
So, in order to make progress, we indulge in a number of simplifying assumptions.
10.3.1 Ideal Gas Assumption
We begin by assuming that our ensemble is an ideal gas. The first consequence of this
assumption, since ideal gas molecules do not interact with one another, is that we may
rewrite the partition function as
Q(N,V,T)= 1
N!
= 1
N!
i
⎡
⎣
e −[ε1(V )+ε2(V )+···+εN (V )]i/kBT
⎤ ⎡
⎦ ⎣
e
j(1)
−εj(1)(V )/kBT
e
j(2)
−εj(2)(V )/kBT
= 1
levels
gke
N!
k
−εk(V
N
)/kBT
= [q(V,T )]N
N!
⎤ ⎡
⎦ ··· ⎣
e
j(N)
−εj(N)(V )/kBT
⎤
⎦
(10.7)
where the factor of 1/N! derives from the quantum mechanical indistinguishability of the
particles, ε is the total energy of an individual molecule, and the change on going from
the first to the second line derives from expressing the exponential of all possible sums of
energies as a product of all possible sums of exponentials of individual energies. On going