  The validity of semiclassical expansions in the power of $\hbar$ for the
quantum Green's function have been extensively tested for billiards systems,
but in the case of chaotic dynamics with smooth potential, even if formula are
existing, a quantitative comparison is still missing. In this paper, extending
the theory developed by Gaspard et al., Adv. Chem. Phys. XC 105 (1995), based
on the classical Green's functions, we present an efficient method allowing the
calculation of $\hbar$ corrections for the propagator, the quantum Green's
function, and their traces. Especially, we show that the previously published
expressions for $\hbar$ corrections to the traces are incomplete.
