Multi-State Trajectory Approach to Non-Adiabatic Dynamics: General Formalism and the Active State Trajectory Approximation

Abstract

A general theoretical framework is derived for the recently developed multi-state trajectory (MST) approach from the time dependent Schr\"odinger equation, resulting in equations of motion for coupled nuclear-electronic dynamics equivalent to Hamilton dynamics or Heisenberg equation based on a new multistate Meyer-Miller (MM) model. The derived MST formalism incorporates both diabatic and adiabatic representations as limiting cases, and reduces to Ehrenfest or Born-Oppenheimer dynamics in the mean field or the single state limits, respectively. By quantizing nuclear dynamics to a particular active state, the MST algorithm does not suffer from the instability caused by the negative instant electronic population variables unlike the standard MM dynamics. Furthermore the multistate representation for electron coupled nuclear dynamics with each state associated with one individual trajectory presumably captures single state dynamics better than the mean field description. The coupled electronic-nuclear coherence is incorporated consistently in the MST framework with no ad-hoc state switch and the associated momentum adjustment or parameters for artificial decoherence, unlike the original or modified surface hopping treatments. The implementation of the MST approach to benchmark problems shows reasonably good agreement with exact quantum calculations, and the results in both representations are similar in accuracy. The active state trajectory (AST) approximation of the MST approach provides a consistent interpretation to trajectory surface hopping, which predicts the transition probabilities reasonably well for multiple nonadiabatic transitions and conical intersection problems.