A method for calculating quasiclassical trajectories on potential energy surfaces defined using a sequence of model quadratic surfaces (QCT/GH) is suggested, and tested for atom-diatom collisions against the traditional quasiclassical trajectory approach. A simple model is also suggested to constrain the classical energy of a bound vibrational mode to be greater than a specified amount, namely, its zero-point energy value. Essentially the model consists of assuming that the sum of the energies in the nonrelevant vibrational modes (typically unbound modes) of the supermolecular complex acts as a pool from which energy may be taken to compensate any leak of vibrational energy in the relevant bound modes, hence preventing the latter from falling below zero-point value. Extensive QCT/GH trajectory calculations carried out for the H+H-2 exchange reaction, which occurs over an energy barrier, as well as exploratory trajectories for the reaction O + OH --> O-2 + H, which occurs on a potential energy surface with a deep chemical well, have shown that the total energy and total angular momentum are conserved within a small numerical tolerance. Correcting for the leak of zero-point vibrational energy still leaves the total energy rigorously conserved but the total angular momentum is then only approximately kept constant. For H + H-2 (upsilon = O, j = O) --> H-2 (upsilon’, j’) + H, the calculated state-to-state QCT/GH cross sections show reasonably good agreement with those of converged quantum results reported in the literature for the same Hg potential energy surface. This agreement does not deteriorate after correction of zero-point energy leak. For both H-3 and HO2, accurate global analytical potential energy surfaces based on the double many-body expansion method have been utilized. Using these prototype systems, an assessment is made of the difficulties encountered on direct reaction dynamics using the novel QCT/GH method.