Chemistry plays an important role in the interstellar medium (ISM), regulating heating and cooling of the gas, and determining abundances of molecular species that trace gas properties in observations. Although solving the time-dependent equations is necessary for accurate abundances and temperature in the dynamic ISM, a full chemical network is too computationally expensive to incorporate in numerical simulations. In this paper, we propose a new simplified chemical network for hydrogen and carbon chemistry in the atomic and molecular ISM. We compare results from our chemical network in detail with results from a full photo-dissociation region (PDR) code, and also with the Nelson & Langer (1999) (NL99) network previously adopted in the simulation literature. We show that our chemical network gives similar results to the PDR code in the equilibrium abundances of all species over a wide range of densities, temperature, and metallicities, whereas the NL99 network shows significant disagreement. Applying our network in 1D models, we find that the $mathrm{CO}$-dominated regime delimits the coldest gas and that the corresponding temperature tracks the cosmic ray ionization rate in molecular clouds. We provide a simple fit for the locus of $mathrm{CO}$ dominated regions as a function of gas density and column. We also compare with observations of diffuse and translucent clouds. We find that the $mathrm{CO}$, $mathrm{CHx}$ and $mathrm{OHx}$ abundances are consistent with equilibrium predictions for densities $n=100-1000~mathrm{cm^{-3}}$, but the predicted equilibrium $mathrm{C}$ abundance is higher than observations, signaling the potential importance of non-equilibrium/dynamical effects.
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