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A common strategy to simulate mixed quantum-classical dynamics is bypropagating classical trajectories with mapping variables, often using theMeyer-Miller-Stock-Thoss (MMST) Hamiltonian or the related spin-mappingapproach. When mapping the quantum subsystem, the coupled dynamics reduce to aset of equations of motion to integrate. Several numerical algorithms have beenproposed, but a thorough performance comparison appears to be lacking. Here, wecompare three time-propagation algorithms for the MMST Hamiltonian: theMomentum Integral (MInt) (arXiv:1709.07474), the Split-Liouvillian (SL)(arXiv:1609.00644), and the algorithm in arXiv:1201.1042 that we refer to asthe Degenerate Eigenvalue (DE) algorithm due to the approximation requiredduring derivation. We analyse the accuracy of individual trajectories,correlation functions, energy conservation, symplecticity, Liouville's theoremand the computational cost. We find that the MInt algorithm is the onlyrigorously symplectic algorithm. However, comparable accuracy at a lowercomputational cost can be obtained with the SL algorithm. The approximationimplicitly made within the DE algorithm conserves energy poorly, even for smalltimesteps, and thus leads to slightly different results. These results shouldguide future mapping-variable simulations.

Which Algorithm Best Propagates the Meyer-Miller-Stock-Thoss Mapping Hamiltonian for Non-Adiabatic Dynamics?