Dissipated tunneling model

In realistic systems, the separation of the modes according to their frequencies and subsequent reduction to one dimension is often impossible with the above-described methods. In this case an accurate multidimensional analysis is needed. Another case in which a multidimensional study is required and which obviously cannot be accounted for within the dissipative tunneling model is that of complex PES with several saddle points and therefore with several MEPs and tunneling paths. 

In realistic systems the separation of the modes by their frequencies and subsequent reduction to one dimension with the methods described above is often not possible. In this case an accurate multidimensional analysis is needed. Another case in which a multidimensional study is required and which obviously cannot be accounted for within the dissipative tunneling model is that of a complex PES with several saddle points and therefore several MEPs and tunneling paths. Whereas the goal of the previous models is to carry out analytical calculations and gain insight into the physical picture, the multidimensional calculations are expected to give a quantitative description of concrete chemical systems. However, at present we are just at the beginning of this process, and only a few examples of numerical multidimensional computations, mostly on rather idealized PES s, have been performed so far. Nonetheless, these... 

PMMA shows no thermally activated dissipation peak. PMMA shows a minimum in the internal friction at r 30 K where PS shows a maximum. Nevertheless, within the tunneling model and introducing a thermally activated relaxation rate as well as a Gaussian distributon density of states of tunneUng systems, it is possible to understand the acoustical properties. From the fits we may conclude that the density of states of tunneling systems for PMMA is restricted to smaller energies, i.e. a smaller width than for PS. 

We use the minimal transport model described in the previous sections. For convenience, we present the Hamiltonian here once more. The full Hamiltonian is the sum of the molecular Hamiltonian Hm, the Hamiltonians of the leads Hr(l)> the tunneling Hamiltonian Ht describing the molecule-to-lead coupling, the vibron Hamiltonian Hy including electron-vibron interaction and coupling of vibrations to the environment (describing dissipation of vibrons)... 

The resulting dissipation model, first introduced in Refs. 20-22, has later been popularized by A.O. Caldeira and A.J. Leggett, who applied it to the quantum mechanical tunneling of a macroscopic variable [23,24]. This model (also put forward at the same time in Refs. 25-27 for the description of quantum Brownian motion) is now referred to as the Caldeira-Leggett dissipation model. It is widely used for the description of the dynamics of classical as well as... 

In the second section the calculation of the rate constant was discussed from the classical mechanics viewpoint. Voth, Chandler, and Miller derived a quantum mechanical expression for the rate constant based on a path integral formalism. Using this expression as a starting point, Voth and O Gormani derived an effective barrier model to allow the calculation of the barrier tunneling contribution to the quantum mechanical rate constant for reactions in dissipative baths. The spirit of their derivation is quite similar to that which treats Grote-Hynes theory o as transition state theory for a parabolic barrier in a harmonic bath. 

The cause of this kind of decoherence is evidently thermal because of the alteration of the deBroglie wavelength of the wave packet associated with the quantum particle. Dissipation, on the other hand, arises from the exchange of energy between the system, which, in this case, comprise the tunneling particle and the environment of the electron cloud, or, for that matter, phonons that get excited because of the elastic distortion created by the particle (which are called interstitial sites). In either case, the coupling with the environment can be modeled as... 

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