Time-reversible implementation of MASH for efficient nonadiabatic molecular dynamics

In this work, we describe various improved implementations of the mapping approach to surface hopping (MASH) for simulating nonadiabatic dynamics. These include time-reversible and piecewise-continuous integrators, which is only formally possible because of the deterministic nature of the underlying MASH equations of motion. The new algorithms allow for the use of either wave-function overlaps or nonadiabatic coupling vectors to propagate the spin, which encodes the electronic state. For a given time-step, $\Delta t$, it is demonstrated that the global error for these methods is $\mathcal{O}(\Delta t^2)$ compared to the $\mathcal{O}(\Delta t)$ error of standard implementations. This allows larger time-steps to be used for a desired error tolerance, or conversely, more accurate observables given a fixed value of $\Delta t$. The newly developed integrators thus provide further advantages for the MASH method, demonstrating that it can be implemented more efficiently than other surface-hopping approaches, which cannot construct time-reversible integrators due to their stochastic nature.