Quantum geometrical molecular dynamics.

We develop a unified and exact quantum geometric framework to understand and model molecular reactive quantum dynamics. The critical roles of quantum geometry of adiabatic electronic states in both adiabatic and nonadiabatic quantum dynamics are unveiled. A numerically exact geometric quantum molecular dynamics method is developed via discrete local trivialization of the projected electronic Hilbert space bundle over nuclear configuration space, eliminating all singularities from nonanalytic adiabatic electronic states. In it, the singular electronic quantum geometric tensor-Abelian for adiabatic dynamics and non-Abelian for nonadiabatic dynamics-is fully encoded in the global electronic overlap matrix. Numerical illustrations demonstrate that atomic motion, whether adiabatic or nonadiabatic, is governed not only by variations in electronic energies (potential energy surfaces) but also by variations in electronic states (electronic quantum geometry). Beyond quantum molecular dynamics, the strategy of discrete local trivialization can be extended to describe quantum dynamics, possibly non-Hermitian, on arbitrary, especially nondifferential fiber bundles.