Fewest switch surface hopping

A final study that must be mentioned is a study by Haitmann et al. [249] on the ultrafast spechoscopy of the Na3p2 cluster. They derived an expression for the calculation of a pump-probe signal using a Wigner-type density mahix approach, which requires a time-dependent ensemble to be calculated after the initial excitation. This ensemble was obtained using fewest switches surface hopping, with trajectories inibally sampled from the thermalized vibronic Wigner function vertically excited onto the upper surface. 

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.

Surface Hopping Model (SHM) first proposed by Tully and Preston [444] is a practical method to cope with nonadiabatic transition. It is actually not a theory but an intuitive prescription to take account of quantum coherent jump by replacing with a classical hop from one potential energy surface to another with a transition probability that is borrowed from other theories of semiclassical (or full quantum mechanical) nonadiabatic transitions state theory such as Zhu-Nakamura method. The fewest switch surface hopping method [445] and the theory of natural decay of mixing [197, 452, 509, 515] are among the most advanced methodologies so far proposed to practically resolve the critical difficulty of SET and the primitive version of SHM. 

Tully s fewest switch surface hopping method and its variants... 

Recently, Shenvi [368] proposed to use the scheme of the fewest switching surface hopping algorithm along with the so-called phase-space adiabatic basis (Sec. 2.4), which diagonalize the Hamiltonian... 

How well the PSANB method works with such densely quasi-degenerate nonadiabatic systems was assessed. Prom the practical aspect, the PSANB scheme in fact reduces the number of initial points needed for qualitatively (but not quantitatively) converged result compared to the fewest switches surface hopping (FSSH) method, one of the most reliable and flexible methods to date. [496] This feature is preferable for tracing branches of electron wavepackets. 

Because of the stochastic nature of the fewest switches surface hopping approach, trajectories starting with the same initial conditions will give rise to different time development. Moreover, the initial conditions should reflect the initial phase space distribution. Therefore, the averages that define the state occupation should in principle be performed over this double ensemble of trajectories starting in different points of the phase space, several times in each one. Because of computational limitations, this procedure is usually reduced to a single ensemble of trajectories starting only once in different points of the phase space. 

Granucci, G., 8c Persico, M. (2007). Critical appraisal of the fewest switches algorithm for surface hopping. Journal of Chemical Physics, 126(13), 134114-134111. 

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. 

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