We have performed calculations on the dissociative chemisorption of H on un-reconstructed and reconstructed Au(111) with density functional theory, and dynamics calculations on this process on un-reconstructed Au(111). Due to a very late barrier for dissociation, H + Au(111) is a candidate H-metal system for which the dissociative chemisorption could be considerably affected by the energy transfer to electron-hole pairs. Minimum barrier geometries and potential energy surfaces were computed for six density functionals. The functionals tested yield minimum barrier heights in the range of 1.15-1.6 eV, and barriers that are even later than found for the similar H + Cu(111) system. The potential energy surfaces have been used in quasi-classical trajectory calculations of the initial (v,J) state resolved reaction probability for several vibrational states v and rotational states J of H and D. Our calculations may serve as predictions for state-resolved associative desorption experiments, from which initial state-resolved dissociative chemisorption probabilities can be extracted by invoking detailed balance. The vibrational efficacy η reported for D dissociating on un-reconstructed Au(111) (about 0.9) is similar to that found in earlier quantum dynamics calculations on H + Ag(111), but larger than found for D + Cu(111). With the two functionals tested most extensively, the reactivity of H and D exhibits an almost monotonic increase with increasing rotational quantum number J. Test calculations suggest that, for chemical accuracy (1 kcal/mol), the herringbone reconstruction of Au(111) should be modeled.