Quasi-chemical theory for the statistical thermodynamics of the hard-sphere fluid

We develop a quasi-chemical theory for the study of packing thermodynamics in dense liquids. The situation of hard-core interactions is addressed by considering the binding of solvent molecules to a precisely defined cavity in order to assess the probability that the cavity is entirely evacuated. The primitive quasi-chemical approximation corresponds to an extension of the Poisson distribution used as a default model in an information theory approach. This primitive quasi-chemical theory is in good qualitative agreement with the observations for the hard-sphere fluid of occupancy distributions that are central to quasi-chemical theories but begins to be quantitatively erroneous for the equation of state in the dense liquid regime of pd(3) > 0.6. How the quasichemical approach can be iterated to treat correlation effects is addressed. Consideration of neglected correlation effects leads to a simple model for the form of those contributions neglected by the primitive quasi-chemical approximation. These considerations, supported by simulation observations, identify a "break away" phenomena that requires special thermodynamic consideration for the zero (0) occupancy case as distinct from the rest of the distribution. An empirical treatment leads to a one-parameter model occupancy distribution that accurately fits the hard-sphere equation of state and observed distributions.