Incorporation of stochastic chemistry on dust grains in the Meudon PDR code using moment equations. I. Application to the formation of H{2} and HD

Context: Unlike gas-phase reactions, chemical reactions taking place on interstellar dust grain surfaces cannot always be modeled by rate equations. Because of the small grain sizes and low flux, these reactions may exhibit large fluctuations and thus require stochastic methods such as the moment equations.
Aims: We evaluate the formation rates of H{2}, HD, and D{2} molecules on dust grain surfaces and their abundances in the gas phase under interstellar conditions.
Methods: We incorporate the moment equations into the Meudon PDR code and compare the results with those obtained from the rate equations.
Results: We find that within the experimental constraints on the energy barriers for both diffusion and desorption and the density of adsorption sites on the grain surface, H{2}, HD and D{2} molecules can be formed efficiently on dust grains.
Conclusions: In a wide range of conditions, the moment equation results agree with those obtained from the rate equations. However, for a range of relatively high grain temperatures, there are significant deviations: the rate equations fail, while the moment equations provide accurate results. The incorporation of the moment equations into the PDR code can be extended to other reactions taking place on grain surfaces.