Fig. 1. Diabatic potential energy surfaces (blue and red solid lines) to construct the model base with two quantum levels and one classical degree of freedom, and adiabatic potential energy surfaces (red and blue solid lines) and the corresponding nonadiabatic couplings (green dotted lines) of Tully's (B) simple avoided crossing, (C) dual avoided crossing, and (D) extended coupling with reflection models.

Fig. 2. Error distribution for phase-corrected FSSH with (A) linear (Eq.(16)), (B) exponential (Eq.(17)), and (C) the traditional (Eq.(15)) decoherence time formulas in the training set with 120 chosen calculations. The Roman numbers Ⅰ, Ⅱ, and Ⅲ indicate the parameter sets with the smallest errors in (A), while the Roman numbers IV and V indicate the parameter sets with the smallest errors in (B). The minimum error in (C) is highlighted by the black arrow.

Fig. 3. (A) Average population error as a function of the initial nuclear momentum by FSSH and phase-corrected FSSH with Eq.(15), Eq.(16), and Eq.(17) for the decoherence time. The parameters \begin{document}$ A $\end{document} and \begin{document}$ B $\end{document} are shown as indicated. The 200 models in the model base with 26 different initial momenta and 4 transmission/reflection channels per model are considered. (B) Distribution of the corresponding population errors for surface hopping calculations with the chosen decoherence correction approaches as indicated.

Fig. 4. Transmission populations on (A) the lower and (B) the upper surfaces, reflection populations on (C) the lower and (D) the upper surfaces for the simple avoided crossing model of Tully. Open circles are exact quantum solutions by DVR. The results of FSSH are shown as green solid squares. The phase-corrected FSSH with Eq.(15), Eq.(16) (\begin{document}$ A $\end{document} = 0.5 and \begin{document}$ B $\end{document} = 1200), and Eq.(17) (\begin{document}$ A $\end{document} = 25 and \begin{document}$ B $\end{document} = 42) for the energy-based decoherence correction are shown as blue solid triangles, light blue diamonds, and red solid circles, respectively.

Fig. 5. Transmission populations on (A) the lower and (B) the upper surfaces, reflection populations on (C) the lower and (D) the upper surfaces for the dual avoided crossing model of Tully. The symbols are the same as those in FIG. 4.

Fig. 6. Transmission populations on (A) the lower and (B) the upper surfaces, reflection populations on (C) the lower and (D) the upper surfaces for the extended coupling with reflection model of Tully. The symbols are the same as those in FIG. 4.