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The authors formulate MASH-PLDM, a nonadiabatic dynamics method that integrates the partially linearized density matrix with the mapping approach to surface hopping via two coupled Bloch sphere spin vectors. This dual-vector representation explicitly captures electronic coherences by propagating trajectories on average adiabatic surfaces when vectors occupy opposing hemispheres, overcoming a critical failure mode of standard surface hopping. The framework significantly improves accuracy across spin-boson and scattering benchmarks while enabling the calculation of multi-time correlation functions required for time-resolved nonlinear spectroscopy.
Tracking dual spin vectors on the Bloch sphere resolves surface hopping's long-standing coherence breakdown, unlocking accurate multi-time correlation functions for nonlinear spectroscopy.
We derive a partially linearized version of the mapping approach to surface hopping (MASH) that provides a new method for simulating nonadiabatic dynamics in molecular systems. As in the original MASH formalism, trajectories travel on adiabatic surfaces with hops determined by the dynamics of a spin vector on the Bloch sphere. However, because the new approach is based on the partially linearized density matrix (PLDM), it has two spin vectors, which may be in the same or different hemispheres of the Bloch sphere. The former scenario describes an electronic population moving on a single adiabatic surface, whereas the latter describes a coherence moving on the average surface. We show that this new method, called MASH-PLDM, leads to improved results over the original MASH dynamics for a variety of scattering problems and spin-boson models, especially in cases where coherences play an important role. A further benefit of the new methodology is that it can treat multi-time correlation functions such as those required for nonlinear spectroscopy. We demonstrate this capability with a proof-of-principle simulation of time-resolved pump-probe spectroscopy.