Magnetic fields play such essential roles in star formation as transporting angular momentum and driving outflows from a star-forming cloud, thereby controlling the formation efficiency of a circumstellar disc and also multiple stellar systems. The coupling of magnetic fields to the gas depends on its ionization degree. We calculate the temperature evolution and ionization degree of a cloud for various metallicities of Z/Zsun = 1e-6, 1e-5, 1e-4, 1e-3, 1e-2, 1e-1, and 1. We update the chemical network by reversing all the gas-phase processes and by considering grain-surface chemistry, including grain evaporation, thermal ionization of alkali metals, and thermionic emission from grains. The ionization degree at nH ~ 1e15-1e19 /cm^3 becomes up to eight orders of magnitude higher than that obtained in the previous model, owing to the thermionic emission and thermal ionization of K and Na, which have been neglected so far. Although magnetic fields dissipate owing to ambipolar diffusion or Ohmic loss at nH < 1e15 /cm^3, the fields recover strong coupling to the gas at nH ~ 1e15 /cm^3, which is lower by a few orders of magnitude compared to the previous work. We develop a reduced chemical network by choosing processes relevant to major coolants and charged species. The reduced network consists of 104 (161) reactions among 28 (38) species in the absence (presence, respectively) of ionization sources. The reduced model includes H2 and HD formation on grain surfaces as well as the depletion of O, C, OH, CO, and H2O on grain surfaces.
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