Spherically inhomogeneous fluids. I. Percus–Yevick hard spheres: Osmotic coefficients and triplet correlations

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Abstract
For fluids in a spherically symmetric external field, it is shown that the Ornstein–Zernike convolution integral becomes a simple algebraic equation upon Legendre transformation. Applying the usual closure relations, the full inhomogeneous pair correlation functions become available. A discrete orthogonal transform pair is also derived, which, in conjunction with the Legendre factorization, makes computations feasable for these generic systems. The general method is here applied to a bulk uniform fluid of hard spheres, using the pair Percus–Yevick closure, but at the triplet level in the hierarchy of distribution functions. Consequently, the pair distribution function and the osmotic coefficient are better than the known analytic results. The method enables the accurate calculation of the triplet correlation function, and several examples are given.