Recent experiments by Lauhon and Ho using scanning tunneling microscopy (STM) observed the direct hopping of H and D on Cu(001) as a function of temperature, They found nearly temperature independent tunneling for H below 60 K, but could not detect the tunneling threshold for D (it is at least 1000 times lower than for H). The availability of such direct and accurate measurement provides the opportunity for validating the level of theory required to predict the diffusion of adsorbates on surfaces. Thus, we carried out density functional theory (DFT) using the generalized gradient approximation (GGA-II) on periodic slabs. The calculated tunneling rate of 4.74 X 10(-4) s(-1) for H is in close agreement with the experimental value of 4.4 X 10(-4) s(-1). We predict 4.66 X 10(-9) s(-1) for the tunneling rate of D (one hop every 83 months!). Between 60 and 80 K, the calculated thermally activated diffusion rate of H is 10(12.88) exp(-0.181 eV/kT) s(-1) in close agreement with the STM value: 10(12.9+/-0.3) exp(-0.197 eV/kT). For deuterium, between 50 and 80 K, the calculated rate is 10(12.70) exp(-0.175eV/kT) s(-1) in close agreement with the STM value: 10(12.7+/-0.2) exp(-0.194 eV/k-T) s(-1). These results validate that such first principle theory can be used to predict the diffusion (including tunneling) for adsorbates on surfaces, providing important data needed to unravel surface processes in catalysis and crystal growth.