Improvement of the Internal Consistency in Trajectory Surface Hopping

This paper addresses the issue of internal consistency in the molecular dynamics with quantum transitions (MDQT) surface hopping method. The MDQT method is based on Tully's fewest switches algorithm, which is designed to ensure that the fraction of trajectories on each surface is equivalent to the corresponding average quantum probability determined by coherent propagation of the quantum amplitudes. For many systems, however, this internal consistency is not maintained. Two reasons for this discrepancy are the existence of classically forbidden transitions and the divergence of the independent trajectories. This paper presents a modified MDQT method that improves the internal consistency. The classically forbidden switches are eliminated by utilizing modified velocities for the integration of the quantum amplitudes, and the difficulties due to divergent trajectories are alleviated by removing the coherence of the quantum amplitudes when each trajectory leaves a nonadiabatic coupling region. The standard and modified MDQT methods are compared to fully quantum calculations for a classic model for ultrafast electronic relaxation (i.e., a two-state three-mode model of the conically intersecting S₁ and S₂ excited states of pyrazine). The standard MDQT calculations exhibit significant discrepancies between the fraction of trajectories in each state and the corresponding average quantum probability. The modified MDQT method leads to remarkable internal consistency for this model system.