We study molecular para-hydrogen (p-${rm H_{2}}$) and ortho-deuterium (o-${rm D_{2}}$) in two dimensions and in the limit of zero temperature by means of the diffusion Monte Carlo method. We report energetic and structural properties of both systems like the total and kinetic energy per particle, radial pair distribution function, and Lindemanns ratio in the low pressure regime. By comparing the total energy per particle as a function of the density in liquid and solid p-${rm H_{2}}$, we show that molecular para-hydrogen, and also ortho-deuterium, remain solid at zero temperature. Interestingly, we assess the quality of three different symmetrized trial wave functions, based on the Nosanow-Jastrow model, in the p-${rm H_{2}}$ solid film at the variational level. In particular, we analyze a new type of symmetrized trial wave function which has been used very recently to describe solid $^{4}$He and found that also characterizes hydrogen satisfactorily. With this wave function, we show that the one-body density matrix $varrho_{1} (r)$ of solid p-${rm H_{2}}$ possesses off-diagonal long range order, with a condensate fraction that increases sizably in the negative pressure regime.
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