  Nonadiabatic dynamical processes are one of the most important quantum
mechanical phenomena in chemical, materials, biological, and environmental
molecular systems, where the coupling between different electronic states is
either inherent in the molecular structure or induced by the (intense) external
field. The curse of dimensionality indicates the intractable exponential
scaling of calculation effort with system size and restricts the implementation
of numerically exact approaches for realistic large systems. The phase space
formulation of quantum mechanics offers an important theoretical framework for
constructing practical approximate trajectory-based methods for quantum
dynamics. This Account reviews our recent progress in phase space mapping
theory: a unified framework for constructing the mapping Hamiltonian on phase
space for coupled F-state systems where the renowned Meyer-Miller Hamiltonian
model is a special case, a general phase space formulation of quantum mechanics
for nonadiabatic systems where the electronic degrees of freedom are mapped
onto constraint space and the nuclear degrees of freedom are mapped onto
infinite space, and an isomorphism between the mapping phase space approach for
nonadiabatic systems and that for nonequilibrium electron transport processes.
