On a thermodynamic path proposed by Lutsko, Wolf, and Yip, a solid is transformed at fixed density into an ideal gas. The accuracy of molecular dynamics simulation results for the free energy of solids using this path is compromised by the presence of a singularity on the path and by the nonergodic behavior of the ideal gas. We present calculations of the third law free energies of the face-centered-cubic (fcc) and hexagonal-close-packed (hcp) solids of argon (modeled as a Lennard-Jones system) to illustrate how the singularity and the nonergodic dynamics introduce errors. Strategies for removing the singularity and the nonergodic dynamics are presented. Finally, an analytic expansion around the ideal state is developed to provide an alternate route to the free energy of those states for which simulation estimates give large statistical uncertainties. The quantitative errors in these new approaches to the simulation of the free energy of a solid are given.