The quantized Hamilton dynamics (QHD) method, which was introduced and developed in J. Chem. Phys. 113, 6557 (2000) to the second order, is extended to the third and fourth orders. The QHD formalism represents an extension of classical mechanics and allows for the derivation of a hierarchy of equations of motion which converge with the quantum-mechanical limit. Here, the second, third, and fourth order QHD approximations are applied to two model problems: the decay of a particle in a metastable cubic potential and the intermode energy exchange observed in the Henon-Heiles system. The QHD results exhibit good convergence with the quantum data with increasing order yet preserve the computational efficiency of classical calculations. The second order QHD approximation already does an excellent job in maintaining the zero-point energy in the Henon-Heiles system and describing moderate tunneling events in the metastable potential. Extensions to higher orders substantially improve the QHD results for deep tunneling and are capable of describing the finer details of energy exchange.