Universality of bridge functions and its relation to variational perturbation theory and additivity of equations of state

Featuring the modified hypernetted-chain (MHNC) scheme as a variational fitting procedure, we demonstrate that the accuracy of the variational perturbation theory (VPT) and of the method based on additivity of equations of state is determined by the excess entropy dependence of the bridge-function parameters [i.e., $\eta \mathrm{}\left(s\right)\mathrm{}$ when the Percus-Yevick hard-sphere bridge functions are employed]. It is found that $\eta \mathrm{}\left(s\right)\mathrm{}$ is nearly universal for all soft (i.e., "physical") potentials while it is distinctly different for the hard spheres, providing a graphical display of the "jump" in pair-potential space (with respect to accuracy of VPT) from "hard" to "soft" behavior. The universality of $\eta \mathrm{}\left(s\right)\mathrm{}$ provides a local criterion for the MHNC scheme that should be useful for inverting structure-factor data in order to obtain the potential. An alternative local MHNC criterion due to Lado is rederived and extended, and it is also analyzed in light of the plot of $\eta \mathrm{}\left(s\right)\mathrm{}$.