Surface-hopping scheme

When the quantum-classical Liouville equation is expressed in the adiabatic basis, the most difficult terms to simulate come from the off-diagonal force matrix elements, which give rise to the nonadiabatic coupling matrix elements. As described above, contributions coming from this term were computed using the momentum-jump approximation in the context of a surface-hopping scheme. 

The QCL approach discussed thus far in this chapter provides a good approximation to the quantum dynamics of condensed phase systems. Most often other approximate quantum-classical methods, such as mean field and surface-hopping schemes, have been commonly employed to treat the same class of problems as the QCLE. These methods are attractive due to their computational simplicity however, many important quantum features, such as quantum coherence and correlations, are not properly handled in these approaches. In this section we discuss these methods and show that starting from the QCLE, an approximate theory in its own right, further approximations lead to these other approaches. 

The surface-hopping trajectories obtained in the adiabatic representation of the QCLE contain nonadiabatic transitions between potential surfaces including both single adiabatic potential surfaces and the mean of two adiabatic surfaces. This picture is qualitatively different from surface-hopping schemes [2,56] which make the ansatz that classical coordinates follow some trajectory, R(t), while the quantum subsystem wave function, expanded in the adiabatic basis, is evolved according to the time dependent Schrodinger equation. The potential surfaces that the classical trajectories evolve along correspond to one of the adiabatic surfaces used in the expansion of the subsystem wavefunction, while the subsystem evolution is carried out coherently and may develop into linear combinations of these states. In such schemes, the environment does not experience the force associated with the true quantum state of the subsystem and decoherence by the environment is not automatically taken into account. Nonetheless, these methods have provided com-... 

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. 

With regard to example (c), Elstner et al. developed an efficient algorithm of nonadiabatic dynamics for the hole transfer dynamics of solvated DNA, with a combination of the fragment orbital DFT method and the surface hopping scheme in the spirit of a QM/MM approach [225]. 

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. ... 

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