Surface-hopping models

3 Surface hopping scheme and beyond 4.3.1 Surface hopping model 

To overcome such severe limitation of the naive surface-hopping scheme, many improved versions have been proposed, including the important one due to Tully himself [445], which will be discussed in the next subsection. 

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By using this approach, it is possible to calculate vibrational state-selected cross-sections from minimal END trajectories obtained with a classical description of the nuclei. We have studied vibrationally excited H2(v) molecules produced in collisions with 30-eV protons [42,43]. The relevant experiments were performed by Toennies et al. [46] with comparisons to theoretical studies using the trajectory surface hopping model [11,47] fTSHM). This system has also stimulated a quantum mechanical study [48] using diatomics-in-molecule (DIM) surfaces [49] and invoicing the infinite-onler sudden approximation (lOSA). 

The trajectory surface hopping model of ion-molecule reaction dynamics has realized an impressive agreement between theory and experiment in this reaction, i.e. H+ + H2, and it provides the experimentalist with a realistic and workable theory to use in the comparison with and interpretation of experimental results. As reliable potential energy surfaces become available for other ion-molecule systems, we can expect further tests of this theory and its applicability to more complicated reactions. 

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. 

Surface hopping can be caused not only by the standard nonadiabatic interactions like that in Eq. (4.42) but others such as the spin-orbit interactions. Likewise one can apply the spirit of the surface hopping model to other coupled-state systems. For instance, one can incorporate the electron dynamics such as that in Eq. (4.41) to calculate the transition probability, with which a path can jump from one adiabatic state to another. 

The greatest advantage of the surface hopping model is its simplicity, which can be applied to any adiabatic representations. For instance. 

As explained above, the naive surface hopping model lets a classical trajectory running on an adiabatic potential siu face jumps to another at a specific point with a transition probability borrowed from other theories. Although... 

Muller, U., 8c Stock, G. (1997). Surface-hopping modeling of photoinduced relaxation dynamics on coupled potential-energy surfaces. Journal of Chemical Physics, 107(16), 6230-6245. 

Another topic in the classical treatment of reactive collisions which has advanced considerably in recent years concerns the treatment of electronically nonadiabatic processes. Early work on this topic followed either the semiclassical complex trajectory method of George and Miller,or the more approximate surface hopping model of Tully and Preston.Recent work in this field by McCurdy, Meyer, and Miller " has attempted to develop a purely classical description of the electronic degrees of freedom, thereby replacing the many-surface aspect of the dynamics with extra classical degrees of freedom (one for each surface beyond the first) which represent the collective electronic motions to which the nuclear motions can couple to cause transitions. This means that a multiple-surface problem can now be treated by standard" trajectory methods, which is a considerable computational simplification. Applications to the f ( Pi/2) 2... 

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