Potential energy surfaces surface-hopping method

The trajectory surface hopping method is an additional extension of the classical trajectory method. Potential energy surfaces are constructed for each electronic state involved in the collision. In addition, a function has to be obtained that defines the locus of points at which hops between surfaces can occur. Still another function is necessary which gives the probability of such jumps as a function of nuclear positions and velocities.28 Diatomics-in-molecules surfaces approximated the two lowest singlet potential surfaces of H3. The surfaces have been shown30 to be in good agreement with accurate ab initio calculations by Conroy.31... 

In those cases where the dynamics cannot be adequately described by motion on a single adiabatic potential-energy surface, additional dynamical assumptions must be made. The pioneering technique is the trajectory surface hopping (TSH) method (55,56) of Tully and co-workers. In these calculations, the classical trajectories are integrated on an adiabatic potential energy surface (typically generated by diatomics in molecules, DIM... 

Let us briefly review what MQC methods have so far been applied to the dynamics at conical intersections. Here the surface-hopping method has been the most popular approach, " in particular in combination with an on-the-fly ab initio evaluation of the potential-energy. Furthermore, various self-consistent-field methods have been employed to describe internal-conversion dynamics associated with a conical intersection, including the mean-field trajectory method, the classical electron... 

In what follows, we introduce the model Hamiltonian pertaining to the molecular systems under consideration. As has been discussed in detail in previous chapters of this book, a curve-crossing problem can be formulated in the adiabatic as well as in a diabatic electronic representation. Depending on the system under consideration and on the specific method used, both representations have been employed in mixed quantum-classical (MQC) approaches. While the diabatic representation is advantageous to model potential-energy surfaces in the vicinity of an intersection and has been used in mean-field type approaches, other MQC approaches such as the surface-hopping method usually employ the adiabatic representation. 

Genetic algorithm (GA) is a kind of widely used method to determine the global minimum structure of a cluster. It can sample the potential energy surface efficiently and hop from one region of the PES to another region rather easily. It is inspired by Darwinian evolution theory that only the fittest individuals can survive. The basic philosophy of GA is to mimic the natural selection and evolution processes in nature. The essential idea of GA procedure is to allow a population of a number of individual candidates to evolve under a given selection rule that maximizes the fitness function. 

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. 

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