  Ring Polymer Surface-Hopping (RPSH) has been recently introduced as a
well-tailored method for incorporating nuclear quantum effects (NQEs), such as
zero-point energy and tunneling, into non-adiabatic molecular dynamics
simulations. The practical widespread usage of RPSH demands a comprehensive
benchmarking of different reaction regimes and conditions with equal emphasis
on demonstrating both the cons and pros of the method. Here, we investigate the
fundamental questions related to the conservation of energy and detailed
balance in the context of RPSH. Using Tully's avoided crossing model as well as
a 2-level system coupled to a classical bath undergoing Langevin dynamics, we
probe the critical problem of the proper treatment of the classically forbidden
transitions stemming from the surface hopping algorithm. We show that proper
treatment of these frustrated hops is key to the accurate description of
real-time dynamics as well as reproducing the exact quantum Boltzmann
population.
