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Our description of the reference contribution to the second virial coefficient follows from an exact mapping onto the second virial coefficient of hard spheres; we propose a simple algebraic equation for the corresponding effective diameter of the hard spheres, which correctly recovers the low- and high-temperature scaling and limits of the reference fluid's second virial coefficient.We test the theoretical free energy surface (FES) for two-step nucleation (TSN) proposed by Iwamatsu [J. Chem. Phys. 134, 164508 (2011)] by comparing the p