Abstract
We perform deep variational free energy calculations to investigate the dense hydrogen system at 1200 K and high pressures. In this computational framework, neural networks are used to model the free energy through the proton Boltzmann distribution and the electron wavefunction. By directly minimizing the free energy, our results reveal the emergence of a crystalline order associated with the center of mass of hydrogen molecules at approximately 180 GPa. This transition from atomic liquid to a molecular solid is marked by discontinuities in both the pressure and thermal entropy. Additionally, we discuss the broader implications and limitations of these findings in the context of recent studies of dense hydrogen under similar conditions.