  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.
