Classical sampling problem description

We conclude that the QCL description represents a promising approach to the treatment of multidimensional curve-crossing problems. The density-matrix formulation yields a consistent treatment of electronic populations and coherences, and the momentum changes associated with an electronic transition can be directly derived from the formalism without the need of ad hoc assumptions. Employing a Monte-Carlo sampling scheme of local classical trajectories, however, we have to face two major complications, that is, the representation of nonlocal phase-space operators and the sampling problem caused by rapidly varying phases. At the present time, the... 

The main practical problem in the implementation of the mixed quantum-classical dynamics method described in Section 4.2.4 is the nonlocal nature of the force in the equation of motion for the stationary-phase trajectories (Equation 4.29). Surface hopping methods provide an approximate, intuitive, stochastic alternative approach that uses the average dynamics of swarm of trajectories over the coupled surfaces to approximate the behavior of the nonlocal stationary-phase trajectory. The siu--face hopping method of Tully and Preston and Tully describes nonadiabatic dynamics even for systems with many particles. Commonly, the nuclei are treated classically, but it is important to consider a large niunber of trajectories in order to sample the quantum probability distribution in the phase space and, if necessary, a statistical distribution over states. In each of the many independent trajectories, the system evolves from the initial configuration for the time necessary for the description of the event of interest. The integration of a trajec-... 

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