Surfaces hopping matrix elements

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. 

Yet, some theoretical problems are left to be discussed to seek for the ultimate and idealistic features as a nonadiabatic-transition theory Although a trajectory thus hopping plural times converges to run on an adiabatic potential surface asymptotically, the off-diagonal density matrix element Pij t) does not vanish practically, as in the original SET. This is ascribed to an incomplete treatment of the nuclear-electronic entanglement. This issue, often referred to as the problem of decoherence, is originated from the nuclear wavepacket bifurcation due to different slopes of potential surfaces, which will be discussed more precisely below. 

Recently, Hoffmann and Schatz(7ij have developed a new level of treatment of spin-orbit effects in bimolecular reactions which enables a more sophisticated treatment of intersystem crossing d3mamics than in the past. In this treatment high quality electronic structure n thods are used to determine global surfaces for the reaction and spin-orbit matrix elements, and then trajectory surface hopping (TSH) methods are used to determine properties of the bimolecular collisions such as reactive cross sections and state distribution information. In an application of this theory to the O + H2 reaction, the spin-orbit matrix elements were determined as a function of position, and then TSH calculations were done within a diabatic representation to determine cross sections. Intersystem crossing effects were found to be small for O + H2 due to... 

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