Integral equations and closure relations for the bridge function and for the triplet correlation function

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Abstract
The first term in a systematic expansion of the two particle potential of mean force yields the hypernetted chain closure approximation. Here is it shown that the second term (a bridge function) consists of a convolution of two ternary correlation functions, and that these can be related by an integral equation analagous to that of Ornstein and Zernike. An expansion for the three particle potential of mean force which also retains the leading two terms completes a closed set of equations. Also discussed are methods for the numerical resolution of the equations, the asymptotic forms of the triplet total and direct correlation functions, and relations between various quaternary functions.