Quantum effects on the rate constant and on the activation energy of the water dissociation reaction on (111) metal surfaces (Pt, Cu, Ru, and Rh) are described using an approach based on the approximate construction of an adiabatic potential energy surface (APES). Information about adsorption characteristics of the reactant and products is required to construct the APES. The Brønsted-Evans-Polanyi (BEP) relation between the calculated activation energies and the corresponding reaction energies, both accounting for quantum effects, is plotted. Tunneling is shown to lead to a decrease of the zero point energies (ZPE) corrected quantum activation energy, as compared to the ZPE-corrected classical one, by factors of 1.50, 1.64, and 1.99 for Cu, Rh, and Ru, while for platinum the correction is marginal (1.02-1.09, depending on the adsorption site for the hydrogen product). A possible deuterium kinetic isotope effect (KIE) in the system is considered. The value of this effect is estimated from 3.4 to 7.6 depending on metal. The reaction paths and mechanism of the water activation on a catalyst surface are interpreted on three-dimensional and two-dimensional pictures. Results of calculations are compared with published data.