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The validity of semiclassical expansions in the power of $\hbar$ for thequantum Green's function have been extensively tested for billiards systems,but in the case of chaotic dynamics with smooth potential, even if formula areexisting, a quantitative comparison is still missing. In this paper, extendingthe theory developed by Gaspard et al., Adv. Chem. Phys. XC 105 (1995), basedon the classical Green's functions, we present an efficient method allowing thecalculation of $\hbar$ corrections for the propagator, the quantum Green'sfunction, and their traces. Especially, we show that the previously publishedexpressions for $\hbar$ corrections to the traces are incomplete.Comment: 19 pages, 4 figures, to be published in Phys. Rev. E shortened  version, corrected typo

ħ corrections in semiclassical formulas for smooth chaotic dynamics