The spatial distribution and evolution of gas in the inner 10 au of protoplanetary disks form the basis for estimating the initial conditions of planet formation. Among the most important constraints derived from spectroscopic observations of the inner disk are the radial distributions of the major gas phase constituents, how the properties of the gas change with inner disk dust evolution, and how chemical abundances and excitation conditions are influenced by the high-energy radiation from the central star. We present a survey of the radial distribution, excitation, and evolution of inner disk molecular hydrogen (H$_{2}$) obtained as part of the $HST$/ULLYSES program. We analyze far-ultraviolet spectroscopy of 71 (63 accreting) pre-main sequence systems in the ULLYSES DR5 release to characterize the H$_{2}$ emission lines, H$_{2}$ dissociation continuum emission, and major photochemical/disk evolution driving UV emissions (Ly$\alpha$, UV continuum, and C IV). We use the widths of the H$_{2}$ emission lines to show that most fluorescent H$_{2}$ arises between 0.1 - 1.4 au from the parent star, and show positive correlations of the average emitting radius with the accretion luminosity and with the dust disk mass. We find a strong correlation between H$_{2}$ dissociation emission and both the accretion-dominated Ly$\alpha$ luminosity and the inner disk dust clearing, painting a picture where water molecules in the inner 3 au are exposed to and dissociated by strong Ly$\alpha$ emission as the opacity of the inner disk declines with time.
Comment: 19 pages, 11 figures. Accepted to the Astronomical Journal