  The exact factorization approach has led to the development of new mixed
quantum-classical methods for simulating coupled electron-ion dynamics. We
compare their performance for dynamics when more than two electronic states are
occupied at a given time, and analyze: (1) the use of coupled versus auxiliary
trajectories in evaluating the electron-nuclear correlation terms, (2) the
approximation of using these terms within surface-hopping and Ehrenfest
frameworks, and (3) the relevance of the exact conditions of zero population
transfer away from nonadiabatic coupling regions and total energy conservation.
Dynamics through the three-state conical intersection in the uracil radical
cation as well as polaritonic models in one dimension are studied.
