The Spin-MInt Algorithm: an Accurate and Symplectic Propagator for the Spin-Mapping Representation of Nonadiabatic Dynamics

Mapping methods, including the Meyer-Miller-Stock-Thoss (MMST) mapping and spin-mapping, are commonly utilised to simulate nonadiabatic dynamics by propagating classical mapping variable trajectories. Recent work confirmed the Momentum Integral (MInt) algorithm is the only known symplectic algorithm for the MMST Hamiltonian. To our knowledge, no symplectic algorithm has been published for the spin-mapping representation without obtaining Cartesian variables and utilising the MInt algorithm. Here, we present the Spin-MInt algorithm which directly propagates the spin-mapping variables. First, we consider a two-level system which maps onto a spin-vector on a Bloch sphere. Despite the spin-variables being non-canonical, we rigorously prove the Spin-MInt is symplectic using a canonical variable transformation. We determine that the Spin-MInt is a symmetrical, second-order, time-reversible, angle invariant and geometric structure preserving algorithm. Computationally, for a one-dimensional spin-boson model, the Spin-MInt and MInt algorithms are symplectic, satisfy Liouville's theorem, provide second-order energy conservation and are more accurate than a previously-published angle-based algorithm. We additionally present accurate correlation functions for a multi-dimensional spin-boson model. We also extend this methodology to a general number of electronic states and present accurate population results for a three-state Morse potential. The Spin-MInt is faster than the MInt algorithm for all tested models, particularly so for large nuclear degrees of freedom. We believe this to be the first known symplectic algorithm for propagating the nonadiabatic spin-mapping Hamiltonian and one of the first rigorously symplectic algorithms in the case of non-trivial coupling between canonical and spin systems. These results should guide and improve future simulations.