Hopping term

The resulting expression is especially simple in the weak coupling case. In this case, the two propagators in Eq. (12) can be approximated by their first order (i.e, single hop) terms. (The zeroth order term makes no contribution of Kjf as long as i 5 f) In this weak coupling limit, the expression for Pif (t) can be expressed as ... 

We describe here that the redox oligomer wires fabricated with the stepwise coordination method show characteristic electron transport behavior distinct from conventional redox polymers. Redox polymers are representative electron-conducting substances in which redox species are connected to form a polymer wire.21-25 The electron transport was treated according to the concept of redox conduction, based on the dilfusional motion of collective electron transfer pathways, composed of electron hopping terms and/or physical diffusion.17,18,26-30 In the characterization of redox conduction, the Cottrell equation can be applied to the initial current—time curve after the potential step in potential step chronoamperometry (PSCA), which causes the redox reaction of the redox polymer film ... 

It is known that for a set of ID and 2D quantum spin models the exact ground state of which can be represented in the RVB form [15, 6, 36, 13, 31]. It is natural to try to find electronic models with an exact ground state at half-filling formed by SB functions in the same manner as for above mentioned spin models. The electronic models of these types include the correlated hopping of electrons as well as the spin interactions and pair hopping terms. 

Figure 49. Susceptibility spectra for propylene carbonate (Tg — 160 K) as measured by depolarized light scattering (top, data from Ref. 372) and dielectric spectroscopy (bottom, data from Ref. 9), each normalized by a temperature-independent static susceptibility. The full lines are fits from solutions of a two-component schematic MCT model. The dashed fines indicate a white noise spectrum. The dash—dotted line in the upper panel exhibits the asymptote of the critical spectrum. The dotted line shows the solution of the model at T — Tc with hopping terms being neglected. (From Ref. 380.)...

At sufficiently high temperatures, y becomes large, so that I (yi-1 can be replaced by its asympotic value e /(2ify), while r=Ay. The hopping term in Eq. (4.7) then dominates, and can be written as... 

Using new annihilation and creation operators, the hopping terms are written as... 

In the sums above, each sum is prime with respect to (i.e. excludes) the index of the previous sum, except for the inner-most sum, which is also prime with respect to i. For example, in the 4-hop term, k can equal i, but not j. Similarly, 1 can equal j, but not k or i. 

In the next step, to analyze the resonant processes associated with charge exchange between He+ and He , we consider a spin-less Newns-Anderson Hamiltonian [18] where the level and the hopping terms Tis /a that have been neglected to calculate ElHe" "] and [He , are intioduced. This Hamiltonian reads... 

Hubbard initiated a treatment where some electron interaction is included in a localized description with the aim to deal with magnetic features. His model has become a popular and widely used vehicle for the study of electron correlation. It is based on the reduction of the electron repulsion terms to intraatomic ones and to use only the form with the largest pair parameter, (Hint)Ao gAO gAO, for each atom A. Interatomic coupling is represented by a one-particle operator which moves particles from one atom to another, so-called hopping terms. The Hamiltonian is then... 

The charge carrier hopping term via the i-th potential barrier of height AE is defined by the concentration of carriers with the significant potential energy that corresponds to AE ksT and can be written as ... 

It is clearly seen that the adiabatic surfaces exhibit avoided crossing at the angles for which the hopping terms vanish, i.e. cp = t[/2 for the mono-dimensional case and and the four points (, = Irr / 2, = nH) for the bidimensional case. 

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