  Recent very large molecular dynamics simulations of homogeneous nucleation
with $(1-8) \cdot 10^9$ Lennard-Jones atoms [Diemand et al. J. Chem. Phys. {\bf
139}, 074309 (2013)] allow us to accurately determine the formation free energy
of clusters over a wide range of cluster sizes. This is now possible because
such large simulations allow for very precise measurements of the cluster size
distribution in the steady state nucleation regime. The peaks of the free
energy curves give critical cluster sizes, which agree well with independent
estimates based on the nucleation theorem. Using these results, we derive an
analytical formula and a new scaling relation for nucleation rates: $\ln J' /
\eta$ is scaled by $\ln S / \eta$, where the supersaturation ratio is $S$,
$\eta$ is the dimensionless surface energy, and $J'$ is a dimensionless
nucleation rate. This relation can be derived using the free energy of cluster
formation at equilibrium which corresponds to the surface energy required to
form the vapor-liquid interface. At low temperatures (below the triple point),
we find that the surface energy divided by that of the classical nucleation
theory does not depend on temperature, which leads to the scaling relation and
implies a constant, positive Tolman length equal to half of the mean
inter-particle separation in the liquid phase.
