Non-adiabatic Dynamics using the Generators of the su(N) Lie Algebra

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. We use the generalized spin coherent states representation and the Stratonovich-Weyl transform to describe these electronic spin-mapping variables in the continuous variables space. Wigner representation is used to de- scribe 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 recently proposed spin-Linearized Semi-Classical (spin-LSC) dynamics. 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 mapping variables of the bosonic DOF (which are commonly referred to as the Meyer-Miller-Stock-Thoss mapping variables). The accuracy of this approach is tested with several challenging non-adiabatic model systems.