Calculation of the dynamics of nonadiabatic transitions with multiple vibrational modes

We propose a new calculation method of the dynamics of nonadiabatic transition which is applicable to study coherent control of photoisomerizations. We show that we can obtain a good approximation of the real dynamics of the system with less computational cost by switching fully quantum mechanical (QM) calculation and classical (CM) calculation at every time step by evaluating a ``switching factor'' defined by \begin{equation} \eta = \max_{\phi_i,z_i} |\langle z_1 e^{i\phi_1},z_2 e^{i\phi_2},...,z_N e^{i\phi_N}|U|\Phi \rangle|^2, \end{equation} where $N$ is the number of the vibrational modes relevant to the photoisomerization, and $|\Phi \rangle$ shows the quantum mechanical state each wavepacket in the system. $|z_1e^{i\phi_1},z_2 e^{i\phi_2},...,z_N e^{i\phi_N} \rangle$ and $U$ denote the coherent state in $N$-dimensional space and a translation operator, respectively. The switching rule is:\\ ``When $\eta$ exceeds a threshold value $\eta_c$, we perform a QM calculation, and {\it vice versa}.''\\ By choosing an appropriate value of $\eta_c$ we obtain a approximated wavefunction at each time step which well-reproduces that derived by QM calculation.