We present a detailed analysis of the fundamental atomic-scale processes that determine the surface smoothness of hydrogenated amorphous silicon (a-Si:H) thin films. The analysis is based on a synergistic combination of molecular-dynamics (MD) simulations of radical precursor migration on surfaces of a-Si:H films that are deposited computationally using MD simulation with first-principles density functional theory (DFT) calculations on the hydrogen-terminated Si(001)-(2x1) surface. The surfaces of the MD-grown a-Si:H films are remarkably smooth, as the mobile precursor, the SiH3 radical, diffuses fast and incorporates in surface valleys. Analysis of the MD simulations of SiH3 radical migration on a-Si:H surfaces yields an effective diffusion barrier of 0.16 eV. The low diffusion barrier on the a-Si:H surface is attributed to SiH3 migration through overcoordinated surface Si atoms, where the radical remains weakly bonded to the surface at all times and does not break any strong Si-Si bonds along its migration pathway. Furthermore, the diffusing SiH3 radical incorporates into the a-Si:H film only when it transfers an H atom and forms a second Si-Si backbond. On rough a-Si:H films, such H transfer from diffusing SiH3 radicals is more likely to occur in surface valleys, even when the dangling bond (DB) density is low and DBs are not present in surface valleys. In addition, this H-transfer process is thermally activated with activation energy barriers (E-a) over the range 0.29-0.65 eV; E-a is determined by the Si-Si interatomic distance between the Si of the SiH3 radical and the surface Si atom to which the H is transferred. The preferential incorporation in valleys is explained by both the increased residence time of the migrating precursor in valleys and the decreased activation barrier for incorporation reactions occurring in valleys. The mechanism and activation barrier for the H-transfer reaction on the a-Si:H surface were validated by performing first-principles DFT calculations on the crystalline Si surface.