Two sub-series of Mayer's virial series are summed to infinite order. The resulting equations are a natural generalisation of the Bethe-Peierls-Guggenheim approximation. They are solved for three simple yet non-trivial cases to illustrate the physics involved. It seems that both vacancies and the strong correlation of neighbouring particles have been taken into account simultaneously.
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Shinn-Tyan Wu (1977) studied this question.
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