HF-LCAO Examples - Molecular Modelling - page 291

17.6 Thermodynamic Properties

In Chapter 8 I discussed the canonical ensemble and the canonical partition function

exp

E i
k B T

Q = − (17.7)

i

I added a * superscript to focus attention on the fact that E ∗ refers to the collection of
particles in each cell of the ensemble. There will be many possible allowed values for
E ∗ , and the N particles in each cell will contribute in some way to make up the total. So
far, I have made no assumptions whatever about these particles, neither have I made any
assumption about the way in which they might interact with each other.
Suppose now that the N particles are identical molecules, but that they form essentially
an ideal gas. One characteristic of an ideal gas is that there is no interaction between the
particles, so the total energy E ∗ of the N particles will be a simple sum of the particle
energies. If I label the molecules 1, 2, 3,..., N then we can write

E ∗ = ε (1) + ε (2) +···+ ε ( N ) (17.8)

Each molecular energy will contain a kinetic and a potential part, but there are no inter-
molecular interactions because of the ideal gas behaviour. So for each possible value of E ∗
we have

ε (1)
i

ε (2)
i

ε ( N )
i

E i = + +···+ (17.9)

Each allowed value of E ∗ will correspond to different values of the constituent ε's and a
simple rearrangement of Q shows that

exp exp

...
n

exp

ε (2)
j
k B T

ε (1)
i
k B T

ε ( N )
n
k B T

Q = − − − (17.10)

i j

I have used a different dummy index i , j ,..., n but the sums are all the same since they refer
to identical particles. At first sight we should therefore write

exp

N

ε (1)
i
k B T

= −

Q

i

This assumes that the N particles can somehow be distinguished one from another (we say
that such particles are distinguishable ). The laws of quantum mechanics tell us that seem-
ingly identical particles are truly identical, they are indistinguishable and to allow for this
we have to divide Q by the number of ways in which N things can be permuted:

N

exp

1
N

ε i
k B T

Q = − (17.11)

!

i

I have dropped all reference to particle '1', since all N are identical.
The summation

exp

ε i
k B T

q = − (17.12)

i