Fourth virial coefficient of additive hard-sphere mixtures in the   Percus-Yevick and hypernetted-chain approximations
  The fourth virial coefficient of additive hard-sphere mixtures, as predicted by the Percus-Yevick (PY) and hypernetted-chain (HNC) theories, is derived via the compressibility, virial, and chemical-potential routes, the outcomes being compared with exact results. Except in the case of the HNC compressibility route, the other five expressions exhibit a common structure involving the first three moments of the size distribution. In both theories the chemical-potential route is slightly better than the virial one and the best behavior is generally presented by the compressibility route. Moreover, the PY results with any of the three routes are more accurate than any of the HNC results. 
Elena Beltr\'an-Heredia and Andr\'es Santos
cond-mat.stat-mech cond-mat.soft physics.chem-ph