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On Filtering Schemes in the Quantum-Classical Liouville Approach to Non-adiabatic Dynamics

We study a number of filtering schemes for the reduction of the statistical error in non-adiabatic calculations by means of the quantum-classical Liouville equation. In particular, we focus on a scheme based on setting a threshold value on the sampling weights, so that when the threshold is overcome the value of the weight is reset, and on another approach which prunes the ensemble of the allowed non-adiabatic transitions according to a generalised sampling probability. Both methods have advantages and drawbacks, however their combination drastically improves the performance of an algorithm known as the Sequential Short Time Step Propagation [D. MacKernan et al., J. Phys: Condens. Matter {\bf 14} 9069 (2002)], which is derived from a simple first order expansion of the quantum-classical propagator. Such an algorithm together with the combined filtering procedures produce results that compare very well with those obtained by means of numerically "exact" quantum calculations for the spin-boson model, even for intermediate and strong coupling regimes.