Exact and approximate expressions for the three-particle direct correlation function and some related functions

An exact expression for the three-particle direct correlation function, $c_3$, of a general fluid is derived. Exact expressions for the three-particle total correlation function, $h_3$, &, &, and δ$h_2$/δ${\rho }_1$ are also found. The first terms of the black ${\rho }_1$-circle, $h_2$-bonds expansion of $c_3$ and δ$h_2$/δ${\rho }_1$ are explicitly evaluated and, in particular, it is found that the leading term of the $c_3$ expansion is $c_3$=${\prod }_i$,${\mathit{j}=1;\mathit{i}<\mathit{j}}^3$ $h_2$(i,j). The behavior of this term is discussed for several configurations of a Lennard-Jones fluid in the neighborhood of the triple point. A comparatively more complex structure than that of $c_2$ is found. The correction terms are analyzed and it is found that they are small but not negligible: they tend to enhance the excluded-volume effect due to the strong repulsion of the potential. A scheme to obtain the higher-order direct correlation functions is also outlined. The relevance of these results to the modified hypernetted-chain approximation and to the theory of inhomogeneous fluids is briefly discussed.