  We present a nonadiabatic classical-trajectory approach that offers the best
of both worlds between fewest-switches surface hopping (FSSH) and
quasiclassical mapping dynamics. This mapping approach to surface hopping
(MASH) propagates the nuclei on the active adiabatic potential-energy surface,
like in FSSH. However, unlike in FSSH, transitions between active surfaces are
deterministic and occur when the electronic mapping variables evolve between
specified regions of the electronic phase space. This guarantees internal
consistency between the active surface and the electronic degrees of freedom
throughout the dynamics. MASH is rigorously derivable from exact quantum
mechanics, as a limit of the quantum-classical Liouville equation (QCLE),
leading to a unique prescription for momentum rescaling and frustrated hops.
Hence, a quantum-jump procedure can in principle be used to systematically
converge the accuracy of the results to that of the QCLE. This jump procedure
also provides a rigorous framework for deriving approximate decoherence
corrections similar to those proposed for FSSH. We apply MASH to simulate the
nonadiabatic dynamics in various model systems and show that it consistently
produces more accurate results than FSSH, at a comparable computational cost.
