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As is well known, approximate integral equations for liquids, such as thehypernetted chain (HNC) and Percus--Yevick (PY) theories, are in generalthermodynamically inconsistent in the sense that the macroscopic propertiesobtained from the spatial correlation functions depend on the route followed.In particular, the values of the fourth virial coefficient $B_4$ predicted bythe HNC and PY approximations via the virial route differ from those obtainedvia the compressibility route. Despite this, it is shown in this paper that thevalue of $B_4$ obtained from the virial route in the HNC theory is exactlythree halves the value obtained from the compressibility route in the PYtheory, irrespective of the interaction potential (whether isotropic or not),the number of components, and the dimensionality of the system. This simplerelationship is confirmed in one-component systems by analytical results forthe one-dimensional penetrable-square-well model and the three-dimensionalpenetrable-sphere model, as well as by numerical results for theone-dimensional Lennard--Jones model, the one-dimensional Gaussian core model,and the three-dimensional square-well model.

Simple relationship between the virial-route hypernetted-chain and the compressibility-route Percus--Yevick values of the fourth virial coefficient