Trajectory-surface-hopping-algorithm

Jasper, A. W., Stechmann, S. N., Truhlar, D. G. (2002). Fewest-switches with time uncertainty A modified trajectory surface-hopping algorithm with better accuracy for classically forbidden electronic transitions. Journal of Chemical Physics, 116(13), 5424-5431. 

Figure 9. Snapshots of the phase space distribution (PSD) obtained from classical trajectory simulations based on the fewest-switches surface-hopping algorithm of a 50 K initial canonical ensemble [46], Na atoms are indicated by black circles, and F atoms are indicated by gray crosses. Dynamics on the hrst excited state starting at the Cj structure (t = 0 fs) over the structure with broken Na-Na bond t = 90 fs) and subsequently over broken ionic Na-F bond (t = 220 fs) toward the conical intersection region (t = 400 fs), Dynamics on the ground state after branching of the PSD from the hrst excited state leads to strong spatial delocalization (t = 600 fs). The C2v isomer can be identihed at 800 fs in the center-of-mass distribution. See color insert.

Fabiano, E., Groenhof, G., 8c Thiel, W. (2008a). Approximate switching algorithms for trajectory surface hopping. Chemical Physics, 351(1-3), 111-116. 

The simplest way to add a non-adiabatic correction to the classical BO dynamics method outlined above in Section n.B is to use what is known as surface hopping. First introduced on an intuitive basis by Bjerre and Nikitin [200] and Tully and Preston [201], a number of variations have been developed [202-205], and are reviewed in [28,206]. Reference [204] also includes technical details of practical algorithms. These methods all use standard classical trajectories that use the hopping procedure to sample the different states, and so add non-adiabatic effects. A different scheme was introduced by Miller and George [207] which, although based on the same ideas, uses complex coordinates and momenta. 

The chapter is organized as follows The quantum-classical Liouville dynamics scheme is first outlined and a rigorous surface hopping trajectory algorithm for its implementation is presented. The iterative linearized density matrix propagation approach is then described and an approach for its implementation is presented. In the Model Simulations section the comparable performance of the two methods is documented for the generalized spin-boson model and numerical convergence issues are mentioned. In the Conclusions we review the perspectives of this study. 

Various schemes have been proposed for the solution of the quantum-classical Liouville equation [13,21-24]. Here we describe the sequential short-time algorithm that represents the solution in an ensemble of surface-hopping trajectories [25,26]. 

Several algorithms have been constructed to simulate the solution of the QCLE. The simulation methods usually utilize particular representations of the quantum subsystem. Surface-hopping schemes that make use of the adiabatic basis have been constructed density matrix evolution has been carried out in the diabatic basis using trajectory-based methods, some of which make use of a mapping representation of the diabatic states.A representation of the dynamics in the force basis has been implemented to simulate the dynamics using the multithreads algorithm. ... 

This algorithm corresponds to a surface-hopping trajectory picture of the dynamics. The short-time segments involve evolution along the surface (aa ), which may be adiabatic (when a = a ) or the arithmetic mean of two adiabatic surfaces (when o, =ol ), governed by the propagator These trajec-... 

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