  Likely candidates for the global potential energy minima of (H$_{2}$O)$_{n}$
clusters with $n\leq21$ on the (0001)-surface of graphite are found using
basin-hopping global optimization. The potential energy surfaces are
constructed using the TIP4P intermolecular potentials for the water molecules
(the TIP3P is also explored as a secondary choice), a Lennard-Jones
water-graphite potential, and a water-graphite polarization potential that is
built from classical electrostatic image methods and takes into account both
the perpendicular and parallel electric polarizations of graphite. This
potential energy surface produces a rather hydrophobic water-graphite
interaction. As a consequence, the water component of the lowest
graphite-(H$_{2}$O)$_{n}$ minima is quite closely related to low-lying minima
of the corresponding TIP4P (H$_{2}$O)$_{n}$ clusters. In about half of the
cases the geometrical substructure of the water molecules in the
graphite-(H$_{2}$O)$_{n}$ global minimum coincides with that of the
corresponding free water cluster. Exceptions occur when the interaction with
graphite induces a change in geometry. A comparison of our results with
available theoretical and experimental data is performed.
