  Long-timescale simulations of the diffusion of a H$_2$O admolecule on the
(0001) basal plane of ice Ih were carried out over a temperature range of 100
to 200 K using the adaptive kinetic Monte Carlo method and TIP4P/2005f
interaction potential function. The arrangement of dangling H atoms was varied
from the proton-disordered surface to the perfectly ordered Fletcher surface. A
large variety of sites was found leading to a broad distribution in adsorption
energy at both types of surfaces. Up to 4 % of the sites on the
proton-disordered surface have an adsorption energy exceeding the cohesive
energy of ice Ih. The mean squared displacement of a simulated trajectory at
175 K for the proton-disordered surface gave a diffusion constant of
6$\cdot$10$^{-10}$ cm$^2$/s, consistent with an upper bound previously reported
from experimental measurements. During the simulation, dangling H atoms were
found to rearrange so as to reduce clustering, thereby approaching a linear
Fletcher type arrangement. Diffusion on the perfectly ordered Fletcher surface
was estimated to be significantly faster, especially in the direction along the
rows of dangling hydrogen atoms. From simulations over the range in
temperature, an effective activation energy of diffusion was estimated to be
0.16 eV and 0.22 eV for diffusion parallel and perpendicular to the rows,
respectively. Even a slight disruption of the rows of the Fletcher surface made
the diffusion isotropic.
