A quantum mechanical nonadiabatic theory of dissociative adsorption of diatomic molecules X2 on metal surface is presented. The following reaction coordinates are used to construct crossing diabatic potential energy surfaces (PES): the distance y between the atoms of the X2 molecule, the distance x of the X2 molecular axis from the surface, the set of coordinates describing possible displacements of metal atoms under adsorption. Expression for the rate constant is derived using the model potentials describing vibrations along these coordinates. The calculated dependency of the rate constant W on the reaction heat ΔE is compared with that in classical approximation. It is shown that quantum effects lead to a weaker dependence of W on ΔE as compared to that for classical one.