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The minimum mass that a virialized gas cloud must have in order to be able to cool in a Hubble time is computed, using a detailed treatment of the chemistry of molecular hydrogen. With a simple model for halo profiles, we reduce the problem to that of numerically integrating a system of chemical equations. The results agree well with numerically expensive three-dimensional simulations, and our approach has the advantage of being able to explore large regions of parameter space rapidly. The minimum baryonic mass M b is found to be strongly redshift dependent, dropping from 10 6 M &sun; at z∼ 15 to 5× 10 3 M &sun; at z∼ 100 as molecular cooling becomes effective. For z ≫ 100, M b rises again, as cosmic microwave background photons inhibit H 2 formation through the H-channel. Finally, for z ≫ 200, the H channel for H 2 formation becomes effective, driving M b down toward M b∼ 10 3 M &sun;. With a …