Non-adiabatic Mapping Dynamics in the Phase Space of the SU(N) Lie Group

We present the rigorous theoretical framework of the generalized spin mapping representation for non- adiabatic dynamics. This formalism is based on the generators of the su(N) Lie algebra to represent N discrete electronic states, thus preserving the size of the original Hilbert space in the state representation. The Stratonovich-Weyl transform is then used to map an operator in the Hilbert space to a continuous func- tion on the SU(N) Lie Group manifold which is a phase space of continuous variables. Wigner representation is used to describe the nuclear degrees of freedom. Using the above representations, we derived an exact expression of the time-correlation function as well as the exact quantum Liouvillian. Making the linearization approximation, this exact Liouvillian is reduced to the Liouvillian of the several recently proposed meth- ods. These expressions lead to a self-consistent trajectory-based method to simulate non-adiabatic dynamics, which is based entirely on the generalized spin mapping formalism to treat the electronic states without the necessity of converting back to the cartesian Meyer-Miller-Stock-Thoss mapping variables. We envision that the theoretical work presented in this work provides a rigorous and unified framework to formally derive non-adiabatic quantum dynamics approaches with continuous variables.