Recall the expression for the configurational partition function:
Suppose that the potential U can be written as a sum of two contributions
where 
is, in some sense, small compared to 
. An extra
bonus can be had if the partition function for can be
evaluated analytically.
Let
Then, we may express 
as
where 
means average with respect to
only. If is small, then the average can be expanded
in powers of :
The free energy is given by
Separating A into two contributions, we have
where 
is independent of and is given by
and
We wish to develop an expansion for 
of the general form
where 
are a set of expansion coefficients that are
determined by the condition that such an expansion be consistent
with 
.
Using the fact that
we have that
Equating this expansion to the proposed expansion for , we
obtain
This must be solved for each of the undetermined parameters ,
which can be done by equating like powers of 
on both
sides of the equation. Thus, from the 
term, we find,
from the right side:
and from the left side, the l=1 and k=1 term contributes:
from which it can be easily seen that
Likewise, from the 
term,
and from the left side, we see that the l=1,k=2 and l=2,k=1 terms contribute:
Thus,
For 
, the right sides gives:
the left side contributes the l=1,k=3, k=2,l=2 and l=3,k=1 terms:
Thus,
Now, the free energy, up to the third order term is given by
In order to evaluate 
, suppose that
is given by a pair potential
Then,
The free energy is therefore given by
