Results from electronic structure studies and quantum scattering calculations are presented for the reaction of gas-phase H atoms with H atoms adsorbed onto a graphite surface to form H-2(g). H can chemisorb on graphite directly over a carbon atom, with the carbon puckering out of the surface plane by several tenths of an Angstrom. Using an ab initio approach based on the density functional theory, and treating the graphite substrate as a slab, we compute the potential energy surface for this reaction, for three cases. In the first case the adsorbed H is initially in the chemisorbed state and the lattice is held fixed in the puckered position during the reaction. In the second case the adsorbed H is initially in the chemisorbed state, but the lattice is allowed to fully relax for each configuration of the two H atoms. In the third case the H initially on the surface is in the physisorbed state. We use a fully quantum mechanical scattering approach to compute reaction cross sections and product H-2 translational, rotational, and vibrational state distributions for each case.