Dependent Tunneling in the Born-Oppenheimer Approximation

Distance-Dependent Tunneling in the Born-Oppenheimer Approximation 

First calculations of the optimum distance between the reactants, R, taking into account the dependence of the probability of proton transfer between the unexcited vibrational energy levels on the transfer distance have been performed in Ref. 42 assuming classical character of the reactant motion. Effects of this type were considered also in Ref. 43 in another model. It was shown that R depends on the temperature and this dependence leads to a distortion of the Arrhenius temperature dependence of the transition probability. 

A general method for the calculation of the transition probability in the harmonic approximation developed in Ref. 44 enabled us to take into account, in a rigorous way, both the dependence of the tunneling of the quantum particles on the coordinates of other degrees of freedom of the system and the effects of the inertia and nonadiabaticity of the tunneling particle, taking into account the mixing of the normal coordinates of the system in the initial and 

The dependence of the proton resonance integral J for the unexcited vibrational states on the vibrations of the crystal lattice was taken into account recently in Ref. 47 for proton transfer reactions in solids. The dependence of J on the nuclear coordinates was chosen phenomenologically as an exponential Gaussian function. 

Below we will use Eq. (16), which, in certain models in the Born-Oppenheimer approximation, enables us to take into account both the dependence of the proton tunneling between fixed vibrational states on the coordinates of other nuclei and the contribution to the transition probability arising from the excited vibrational states of the proton. Taking into account that the proton is the easiest nucleus and that proton transfer reactions occur often between heavy donor and acceptor molecules we will not consider here the effects of the inertia, nonadiabaticity, and mixing of the normal coordinates. These effects will be considered in Section V in the discussion of the processes of the transfer of heavier atoms. 

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