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

TL;DR

This work introduces Spin-MInt, the first symplectic propagator directly for the spin-mapping representation of nonadiabatic dynamics. By splitting the spin-mapping Hamiltonian analogously to MMST and applying a flow map that mirrors the MInt structure, Spin-MInt achieves symmetric, second-order, time-reversible, and structure-preserving propagation while preserving Liouville's theorem and energy to second order. The authors prove symplecticity through a canonical-variable transformation and demonstrate that Spin-MInt yields identical evolution to MInt for two-state systems and extends accurately to three-state Morse potentials, with notable speed advantages for two-state cases. These results provide a rigorous, efficient framework for spin-mapping simulations and offer a path toward scalable, symplectic nonadiabatic dynamics across multiple electronic states.

Abstract

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 converting to MMST 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 and determine that the Spin-MInt is a symplectic, symmetrical, second-order, time-reversible, angle invariant and geometric structure preserving algorithm. Despite spin-variables resulting in a non-invertible structure matrix, we rigorously prove the Spin-MInt is symplectic using a canonical variable transformation. Computationally, we find that 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. The Spin-MInt is significantly faster than the MInt algorithm for two electronic states. Secondly, we extend this methodology to more than two electronic states and present accurate population results for a three-state morse potential. 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.

Figures (7)

• Figure 1: The expectation values of the spin-operators form a basis of Cartesian coordinates where a Bloch sphere of radius, $r_s$, can be defined. The motion of the spin-vector (purple) on the surface is circular around the Hamiltonian vector. When the spin-vector is on either of the poles (orange circle), the population is entirely in one state.
• Figure 2: Propagation of $u_x$ for a single trajectory using Model 1 and the MInt (cyan dotted circles), Spin-MInt (purple dashed) and angle-based (orange) algorithms at three different timesteps: 0.1 (top), 0.01 (middle) and 0.001 (bottom). The Spin-MInt and MInt results are identical at all timesteps whilst the angle-based algorithm deviates for the 0.1 and 0.01 timesteps indicating lower accuracy.
• Figure 3: (a) The error matrix Frobenius norm and (b) the Liouville's theorem criterion as a function of time using Model 1 and $\Delta t = 0.01$, for a single trajectory using the Spin-MInt (purple), MInt (cyan) and angle-based (orange) algorithms. As the MInt and Spin-MInt algorithms measure floating point error in both plots, Liouville's theorem and symplecticity are satisfied. The angle-based algorithm does not satisfy symplecticity or Liouville's theorem.
• Figure 4: (a) The error matrix Frobenius norm and (b) the Liouville's theorem criterion as a function of time using Model 1 and $\Delta t = 0.1$, averaged over a million trajectories using the Spin-MInt (purple) and MInt (cyan) algorithms. In (a) the algorithms are symplectic and in (b) the algorithms satisfy Liouville's theorem as both plots measure floating point errors.
• Figure 5: Energy conservation of Model 1 with (a) a single trajectory and $\Delta t = 0.1$ (solid/circle) and (b) averaged using $\Delta t = 0.1$ (solid/circle), $\Delta t = 0.01$ (dotted/triangle) and $\Delta t = 1.0$ (dashed/square) with the Spin-MInt (purple) and MInt (cyan) algorithms. The energy is the same for both algorithms and is conserved by noting the y-axis scale on (a) and is second-order with respect to timestep in (b).