Many-body effects in a water hexamer attached to a metal surface have been studied by quantum calculations. The metal component of the interface has been replaced by a set of geometrical constraints (virtual surface) [J. Phys. Chem. A 2000, 105, 4084-4095] which permits the analysis of the properties of the interface independently of the precise electronic structure of the metal, as a function of the surface lattice constant. Our calculations show that cooperative forces have a significant influence on the energy and geometry of the metal-water interface. The energy decomposition of the energy of formation of the water hexamer demonstrates how strong cooperative effects favor the growth of the water bilayer in the experimentally observed range of surface lattice constants, while a model based solely on two-center energies predicts the formation of the water bilayer at unphysically large values of surface lattice constant.