In mixed quantum-classical molecular dynamics few but important degrees of freedom of a dynamical system are modeled quantum-mechanically while the remaining ones are treated within the classical approximation. Rothe methods established in the theory of partial differential equations are used to control both temporal and spatial discretization errors on grounds of a global tolerance criterion. The trapezoidal rule for adaptive integration of Liouville dynamics (TRAIL) [I. Horenko and M. Weiser, J. Comput. Chem. 24, 1921 (2003)] has been extended to account for non-adiabatic effects in molecular dynamics described by the quantum-classical Liouville equation. In the context of particle methods, the quality of the spatial approximation of the phase-space distributions is maximized while the numerical condition of the least-squares problem for the parameters of particles is minimized. The resulting dynamical scheme is based on a simultaneous propagation of moving particles (Gaussian and Dirac delta-like trajectories) in phase space employing a fully adaptive strategy to upgrade Dirac to Gaussian particles and, vice versa, downgrading Gaussians to Dirac-like trajectories. This allows for the combination of Monte-Carlo-based strategies for the sampling of densities and coherences in multi-dimensional problems with deterministic treatment of non-adiabatic effects.

Numerical examples demonstrate the application of the method to spin-boson systems in different dimensionality. Non-adiabatic effects occuring at conical intersections are treated in the diabatic representation. By decreasing the global tolerance, the numerical solution obtained from the TRAIL scheme are shown to converge towards exact results.