Coincidence hopping

More recently, D. Emin [24] developed an alternative analysis of activated hopping by introducing the concept of coincidence. The tunneling of an electron from one site to the next occurs when the energy state of the second site coincides with that of the first one. Such a coincidence is insured by the thermal deformations of the lattice. By comparing the lifetime of such a coincidence and the electron transit time, one can identify two classes of hopping processes. If the coincidence lime is much laigcr than the transit lime, the jump is adiabatic the electron has lime to follow the lattice deformations. In the reverse case, the jump is non-adia-batic. 

As discussed above, this discrepancy may be caused by classically forbidden electronic transitions—that is, cases in which a proposed hopping process is rejected due to a lack of nuclear kinetic energy. Figure 11c supports this idea by showing the absolute numbers of successful (thick fine) and rejected (thin line) surface hops. In accordance with the initial decay of the adiabatic population, the number of successful surface hops is largest during the first 20 fs. For larger times, the number of rejected hops exceeds the number of successful surface hops. This behavior clearly coincides with the onset of the deviations between the two classically evaluated curves Nk t) and P t). We therefore conclude that the observed breakdown of the consistency relation (42) is indeed caused by classically forbidden electronic transitions. 

Figure 65. Potential-energy surface for first excited singlet state of linear Hj1. Cut through surface at large R coincides with potential curve of at small R2 and with H2 curve at large R2 (see Fig. 63). There is a seam connecting this surface to that shown in Fig. 64, along which potential surface hopping can occur.2...

The main equation for the d-electron GF in PAM coincides with the equation for the Hubbard model if the hopping matrix elements t, ) in the Hubbard model are replaced by the effective ones Athat are V2 and depend on frequency. By iteration of this equation with respect to Aij(u>) one can construct a perturbation theory near the atomic limit. A singular term in the expansions, describing the interaction of d-electrons with spin fluctuations, was found. This term leads to a resonance peak near the Fermi-level with a width of the order of the Kondo temperature. The dynamical spin susceptibility in the paramagnetic phase in the hydrodynamic limit was also calculated. 

For t2= 0 the Hamiltonian H describes a one-dimensional Hubbard lattice (chain) with the alternating values of hopping integrals. Evidently, the spectrum of such a system must coincide with the set of singular values [Pg.706]

Figure 10-14. Time evolution of the nonadiabatic surface hopping parameter, P10 (Eq. 10-10), for a transition from the 5 excited state to the. V0 ground state for representative 7Me-keto (fast oscillating, small amplitude dark grey curve) and 9Me-keto (fast oscillating, large amplitude light grey curve) G trajectories. The steep increase of P10 at / 10 fs in the case of 9Me-keto coincides with the transition from a quasi-planar to an out-of-plane distorted structure. At / 40 fs the amino group starts rotating...

Gill (1972) was the first to suggest that charge transport in polymers occurred by polaron hopping. The application of polaron theory to transport in polymers was first described by Sahvun (1984). Schein et al. (1990), and Schein (1992). The models described by Sahvun and Schein and coworkers lead to a mobility that is a product of a Boltzmann probability of energy coincidence and the probability a carrier will hop to an adjacent site by thermal activation once... 

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