  A surface-hopping algorithm recently derived from the exact factorization
approach, SHXF, [Ha, Lee, Min, J. Phys. Chem. Lett. 9, 1097 (2018)] introduces
an additional term in the electronic equation of surface-hopping, which couples
electronic states through the quantum momentum. This term not only provides a
first-principles description of decoherence but here we show it is crucial to
accurately capture non-adiabatic dynamics when more than two states are
occupied at any given time. Using a vibronic coupling model of the uracil
cation, we show that the lack of this term in traditional surface-hopping
methods, including those with decoherence-corrections, leads to failure to
predict the dynamics through a three-state intersection, while SHXF performs
similarly to the multi-configuration time-dependent Hartree quantum dynamics
benchmark.
