The quantization of Hamilton dynamics (QHD) [J. Chem. Phys. 113, 6557 (2000)] that efficiently generalizes classical mechanics to include quantum tunneling and zero-point energy effects is extended to a general position dependent potential. A Taylor series expansion of the potential is considered both around a fixed point and around the moving instantaneous value of the position variable. The equations-of-motion obtained for the moving frame are significantly simpler than for the fixed frame, while still satisfying the classical limit. The number of the QHD variables and the order of the Taylor expansion of the potential constitute two independent approximation parameters. Conservation of the total energy and the Heisenberg commutator relationship is established for the second-order QHD that includes linear and quadratic variables. The formal results are illustrated by examples, including the harmonic oscillator, tunneling in a doublewell potential, and energy exchange between coupled Morse oscillators representing the stretching vibrations of the SPC-F water molecule commonly used in molecular dynamics simulations. QHD provides an improvement over classical mechanics in all cases. The QHD data agree with the exact data in the large h quantum limit and in the (h) over bar =0 classical limit, with deviations observed for the intermediate h values, depending on the system and initial conditions. QHD provides a quantitative short time approximation. The second-order QHD becomes inappropriate when quantum wave packets split. Higher-order QHD approximations are required in such cases. The second-order QHD is particularly suitable for the description of zero-point energy effects and properly treats moderate tunneling events.