New implementation of semi-classical trajectory surface hopping dynamic simulation has been developed and applied to the photoisomerization of cis- and trans- isomers on the gas phase. This method not only uses the exponential model to the modification of the originally analytical non-adiabatic transition probability formula, but also involves the constrained Hamiltonian system into the constrained molecular dynamic simulation. Two-dimensional potential energy surfaces of ground S0 and excited S1 states are constructed analytically fitting to ab initio calculations in terms of torsion angle and one dihedral angle around the central ethylenic C＝C bond as variables, and the other internal coordinates are all fixed at configuration of one-bond flip conical intersection. The analytical PESs are quite accurate and the mean absolute error is less than 2.4 kcal·mol^-1, and much less than 1.0 kcal·mol^-1 around conical intersection region. A straight seam line is found on potential energy surfaces that simply separates the cis-area with the trans-area. The constrained Hamiltonian system is employed to run trajectories in the Cartesian coordinate system and surface hopping in terms of the two internal dihedral angles. Typical trajectories are found in which the torsion angle changes monotonically for both cis- to trans- and trans- to cisisomerizations. This is an exact picture of one-bond flip mechanism of photoisomerization around the conical intersection. Quantum yield for trans- to cis- isomerization is simulated as 60.45% in very good agreement with experimental value 55.0%, while quantum yield for cis- to trans- isomerization is simulated as 42.3% in comparison with experimental value 35.0%. As the S₁ energy in local minimum of cis-area is higher than that in trans-area, and thus cis- to trans- isomerization is quite possible to access to another Hula-Twist conical intersection. These simulation results demonstrate that the computed cumulative quantum yield and reaction mechanism are consistent with the previously experimental and theoretical results. This means that the present trajectory surface hopping method would be good at the dynamic simulation on the large system with or without constraint Hamiltonian in comparison with the quantum molecular dynamics.

Key words: new implementation of semi-classical dynamic simulation, constrained Hamiltonian system, Zhu-Nakamura theory, isomerization of cis- and trans-stilbene, two dimensional analytical potential energy surfaces