To investigate the fundamental principles of H2 formation in a giant molecular cloud, we derive the H I and H2 surface density (?H I and ?H2) images of the Perseus molecular cloud on sub-pc scales (~0.4?pc). We use the far-infrared data from the Improved Reprocessing of the IRAS Survey and the V-band extinction image provided by the COMPLETE Survey to estimate the dust column density image of Perseus. In combination with the H I data from the Galactic Arecibo L-band Feed Array H I Survey and an estimate of the local dust-to-gas ratio, we then derive the ?H2 distribution across Perseus. We find a relatively uniform ?H I ~ 6-8 M ??pc?2 for both dark and star-forming regions, suggesting a minimum H I surface density required to shield H2 against photodissociation. As a result, a remarkably tight and consistent relation is found between ?H2/?H I and ?H I + ?H2. The transition between the H I- and H2-dominated regions occurs at N(H I) + 2N(H2) ~ (8-14)?? 1020?cm?2. Our findings are consistent with predictions for H2 formation in equilibrium, suggesting that turbulence may not be of primary importance for H2 formation. However, the importance of a warm neutral medium for H2 shielding, an internal radiation field, and the timescale of H2 formation still remain as open questions. We also compare H2 and CO distributions and estimate the fraction of CO-dark gas, f DG ~ 0.3. While significant spatial variations of f DG are found, we do not find a clear correlation with the mean V-band extinction.

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