Correlation functions of classical fluids. III. The method of partition function variation applied to the chemical potential: Cases of PY and HNC2

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Abstract
By adopting the formalism of inhomogeneous systems, a variational expansion of the partition function as a functional of the singlet density is developed. Its successive derivatives are given in terms of correlation functions of increasing orders. A theorem is proven relating the inhomogeneous partition function caused by a source particle to that of the homogeneous medium, from which a general expression for the chemical potential μ is obtained as an infinite series in integrals involving correlation functions of all orders. This series is truncated or resummed by considering three approximate theories on correlation functions: the Percus‐Yevick approximation (PY), the hypernetted chain (HNC) approximation, and its extension HNC2. New expressions for μ in PY and HNC2 are given, while the results for HNC check with the formula given by Morita. To render our results on HNC2 open to the HNC2 distribution functions of Verlet and Levesque, we have generalized the Ornstein‐Zernike relation to third order correlation functions, called OZ3. Four alternative forms are obtained, two of which, first derived here, are of much simpler form. The convolution form of Wu‐Chien of the third order distribution function can be neatly summarized in light of OZ3 and is compared with other approximations. The μ formula for PY is found to be valid at low densities due to the convergence property. Examination of the hard sphere case shows that it is applicable to medium densities of ρd3 up to 0.5.