Polaron Marcus model

In the 2-level limit a perturbative approach has been used in two famous problems the Marcus model in chemistry and the small polaron model in physics. Both models describe hopping of an electron that drags the polarization cloud that it is formed because of its electrostatic coupling to the enviromnent. This enviromnent is the solvent in the Marcus model and the crystal vibrations (phonons) in the small polaron problem. The details of the coupling and of the polarization are different in these problems, but the Hamiltonian formulation is very similar. ... 

The Holstein model is a model for structural reorganization just as the Marcus model. The Holstein model applies to a polaron localized in a lattice. The polaron is an electron moving with the lattice deformation caused by the electron. The polaron moves through the lattice with a certain effective mass. There are some similarities to the Marcus model, but also some differences. The Marcus model is, in principle, a many-electron model, since it assumes the existence of parabolic total energy surfaces. The Holstein model is a one-electron model that includes... 

Several models can explain the carrier transport in organic semiconductors. However, none of them can be independently employed to explain the carrier transport phenomena and the mechanism at the same time. Among the theoretical models, the most often used models are the band transport model (Warta and Karl, 1985 Pemstich et al., 2008 Karl et al., 1991), polaron transport model (Holstein, 1959 Emin and Holstein, 1969 Marcus, 1960), hopping transport model (Vissenberg and Matters, 1998), and multiple trapping and release model (Horowitz et al., 1995 Le Comber and Spear, 1970). 

Most polaron models consider only electron transfer steps parallel and antiparallel to the applied field. Van der Auweraer et al. (1994) derived an expression for the mobility that takes into account isotropic hopping in three dimensions. The treatment is based on the Marcus theoiy (Marcus, 1964, 1968, 1984 Kester et al., 1974 Jortner, 1976 Sumi and Marcus, 1986 Jortner and Bixon, 1988) and assumes that energetic and positional disorder can be neglected. 

A connection can be established between the Marcus theory and the polaron model by noting that the mobility is connected to the electron transfer rate by [26]... 

A generalization of the Marcus theory establishes an important criterion for activationless ( band-like ) or localized transport namely, the former occurs when 2t>X, while the latter dominates when 2t molecular polaron model developed before can be derived from the fact that the reorganization energy is linked to the the molecular polarization time, and the transfer integral to the residence time, so the first inequality can also write and the... 

The first two terms denote the reactant and the metal, the last term affects electron exchange between the metal and the reactant c denotes a creation and c an annihilation operator. Just like in Marcus (and polaron) theory, the solvent modes are divided into a fast part, which is supposed to follow the electron transfer instantly, and a slow part. The latter is modeled as a phonon bath after transformation to a single, normalized reaction coordinate q, with corresponding momentum p, the corresponding part of the Hamiltonian is... 

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