We have applied variational transition-state theory with multidimensional semiclassical transmission coefficients to calculate the surface diffusion rate constants of an H atom on the (100) face of Cu. The solid surface is modeled by using the embedded-cluster approach with movable cluster sizes of 8, 14, 24, and 28 Cu atoms embedded in fixed substrates containing up to 228 additional Cu atoms. We used the empirical pairwise interaction potential of Gregory, Gelb, and Silbey for the H-Cu potential and that of Halicioglu and Pound for Cu-Cu. For temperatures above 200 K, the surface diffusion rate constants are well-converged for a 6-atom cluster, but below 160 K, the rate constants converge very slowly even for a 28-atom cluster. We found that below 160 K the tunneling mechanism dominates the surface diffusion of H on Cu, and this is a phonon-assisted process in which it is essential for quantitative accuracy to include movable metal atom coordinates to specify the tunneling path.