  We have used the Path Integral Monte Carlo method to simulate a monolayer of
molecular hydrogen on graphite above 1/3 submonolayer coverage. We find that at
low temperature and as the coverage increases the system undergoes a series of
transformations starting from the $\sqrt{3} \times \sqrt{3}$ commensurate solid
near 1/3 coverage. First, a phase is formed which is characterized by a
uniaxially compressed incommensurate solid with additional mass density
modulations along the same direction which can be viewed as an ordered
domain-wall solid with a characteristic domain-wall distance which depends on
the surface coverage. At low temperature and higher coverage there is a
transition to an incommensurate solid which is rotated relative to the
substrate commensurate lattice. As a function of temperature the domain-wall
ordering first melts into a fluid of domain-walls and at higher temperature the
solid melts into a uniform fluid. Regardless of the large amplitude of quantum
fluctuations, these phase transitions are analogous to those in classical
monolayer films. Our calculated values for the surface coverage and temperature
where these transitions occur, the calculated structure factors and specific
heat are in general agreement with the available experimental results with no
adjustable parameteters.
