Continuous coupling of the binding energy

In section 3.4.1, we have seen that the chemical potential could be expressed in terms of g(R) provided that we also know the dependence of g(R) on either T or p. We now derive a third expression due to Kirkwood (1933), which employs the idea of a coupling parameter gj The ultimate expression for the chemical potential would be an integral over both R and involving the function g(R, ). We start by defining an auxiliary potential function as follows  

however, that within the assumption of pairwise additivity of the total potential energy, the quantity UN is unaffected by this coupling of the binding energy of the newly added particle. 

For each function U(0 we also define the corresponding configurational partition function by 

The expression (3.50) for the chemical potential can be rewritten using the above notation as 

It is instructive to go through the formal steps in (3.66). They are very similar to those in section 3.1. The only new feature in (3.66) is the appearance of the parameter in the pair distribution functions. 

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