Several properties of the KSA nondiagrammatic cluster χ′(r)

A general integral expression, valid for all intermolecular potentials, for the nondiagrammatic cluster χ'(r) obtained in the Kirkwood superposition approximation (KSA), is found by integration in bipolar coordinates of the differential expression for this cluster as given by Nijboer and Van Hove. The volume integral of χ'(r) is shown to be minus four times the product of the second and third virial coefficients. An integral relationship establishing a correspondence between the virial and compressibility equations of state in the Kirkwood superposition approximation is derived.