In this series of papers we study the structure of the atomic to molecular
transition in the giant atomic-molecular complexes that are the repositories of
most molecular gas in galaxies, with the ultimate goal of attaining a better
understanding of what determines galaxies' molecular content. Here we derive an
approximate analytic solution for the structure of a photodissociation region
(PDR) in a cloud of finite size that is bathed in an external dissociating
radiation field. Our solution extends previous work, which with few exceptions
has been restricted to a one-dimensional treatment of the radiation field. We
show that our analytic results compare favorably to exact numerical
calculations in the one-dimensional limit. However, our more general geometry
provides a more realistic representation than a semi-infinite slab for
atomic-molecular complexes exposed to the interstellar radiation field,
particularly in environments such as low-metallicity dwarf galaxies where the
curvature and finite size of the atomic envelope cannot be neglected. For
clouds that are at least 20% molecular we obtain analytic expressions for the
molecular fraction in terms of properties of the gas and radiation field that
are accurate to tens of percent, while for clouds of lower molecular content we
obtain upper limits. As a side benefit, our analysis helps clarify when
self-shielding is the dominant process in H_2 formation, and under what
circumstances shielding by dust makes a significant contribution.Comment: 19 pages, 11 figures, emulateapj style, accepted to ApJ. Discussion
slightly changed from previous version, and some new analytic approximations
added. Underlying results unchange