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In this series of papers we study the structure of the atomic to moleculartransition in the giant atomic-molecular complexes that are the repositories ofmost molecular gas in galaxies, with the ultimate goal of attaining a betterunderstanding of what determines galaxies' molecular content. Here we derive anapproximate analytic solution for the structure of a photodissociation region(PDR) in a cloud of finite size that is bathed in an external dissociatingradiation field. Our solution extends previous work, which with few exceptionshas been restricted to a one-dimensional treatment of the radiation field. Weshow that our analytic results compare favorably to exact numericalcalculations in the one-dimensional limit. However, our more general geometryprovides a more realistic representation than a semi-infinite slab foratomic-molecular complexes exposed to the interstellar radiation field,particularly in environments such as low-metallicity dwarf galaxies where thecurvature and finite size of the atomic envelope cannot be neglected. Forclouds that are at least 20% molecular we obtain analytic expressions for themolecular fraction in terms of properties of the gas and radiation field thatare accurate to tens of percent, while for clouds of lower molecular content weobtain upper limits. As a side benefit, our analysis helps clarify whenself-shielding is the dominant process in H_2 formation, and under whatcircumstances shielding by dust makes a significant contribution.

The Atomic to Molecular Transition in Galaxies. I: An Analytic Approximation for Photodissociation Fronts in Finite Clouds