The one-dimensional tight binding model

This model consists of a row of identical atoms arranged so that their centers lie on a one-dimensional lattice (along the x-axis, say) with lattice-spacing a. Denote the electronic Hamiltonian for atom j by hj and the corresponding atomic orbitals by pjn 

The Hamiltonian of the full system is.H = Tijhj - - 7, where V is the interatomic interaction. We focus on the low-energy regime where the atomic orbitals are well localized about their corresponding atomic centers, and use this set of electronic states as a basis for the representation of the full problem. When a ckj, that is, the atoms are infinitely far from each other, V - 0 and we have 

For finite a both diagonal and non-diagonal elements of H change, and in particular 0. 

Consider then the Hamiltonian matrix in this atomic orbital representation. We denote 

By symmetry, these diagonal elements do not depend on j. We see that in the submatrix of H associated with the same atomic level n defined on each atom, all diagonal elements are the same Non-diagonal elements of H result from the interatomic coupling, and if the atomic centers are not too close to each other these elements will be small relative to the spacings between different s, that is, 

In this case our problem is reduced to diagonalizing each Hamiltonian sub-matrix associated with the same atomic level n (or with a group of degenerate atomic levels) and with the different atomic centers. 

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The electronic states for a given lattice configuration are given by the solution of fleiect- In the limit of equal bond lengths, we can use the result already obtained for the one-dimensional, tight binding model, Equation (4.22), to write down the result for the dispersion of the electron band as ... 

For the discussion of the physicochemical properties of conductive conjugated polymers, it is most important to set up the appropriate model and to employ the proper method of calculations. In this article most of the analyses have been based on the one-dimensional tight-binding CO methodology for polymers with periodical unit cells. This approach is useful because it gives not only the band structure but also information reflecting the nature of each atomic orbital in the unit cell. 

Strong intersite coupiing leads to the formation of uncorrelated electron-hole pairs, in which the optical transition is described appropriately using a band description [103]. The electronic structure of conjugated polymers was described by Su et al. [2,3] (SSH model) in terms of a quasi-one-dimensional tight-binding model in which the tt electrons are coupled to distortions in the polymer backbone by the electron-phonon interaction. Photon absorption makes an electron jump from the HOMO to the LUMO band (n—tt transition). This transition creates free carriers, which subsequently self-localize, thereby forming nonlinear excitations of... 

The most orthodox model involving a quasi-one-dimensional tight-binding band with electron scattering by acoustic phonons and molecular vibrations (one-phonon processes) has been analyzed carefully and in great detail [43,44]. Good agreement with experimental data is claimed by the proponents of this model. 

The strict generalization of the one-dimensional model treated in Sec. III,A leads to a crystal with its surface completely covered by adsorbed atoms. For this system, the tight-binding approximation gives a difference equation and boundary conditions which can be solved directly. The results show an important new feature. For a simple cubic lattice, the energy levels are given by the usual equation... 

The one-dimensional system has a lot of similarities with the planar n-systems treated using the Hiickel model in Chapter 3. In the case of solids, the Hiickel model becomes the tight-binding model. Let us first assume that we have a onedimensional metal containing a great number of atoms. Each atom provides one loosely bound electron, as in a linear tt-system. Our experience tells us that instead of a linear system, we could use a cyclic system. The wave functions are different just at the end points, and our system is assumed to be so large that the end points do not matter. 

If one assumes a one-dimensional open-chain model for the adatom-metal system, with one atomic orbital per atom and all orbital energies equivalent, within tight binding theory, analytical solutions for the molecular orbital energies can be found l. 

We begin by considering a one-dimensional model in which the crystal is represented by a straight chain of similar atoms and a foreign atom is in interaction with one end of the chain. This is the simplest model of the chemisorption process which may be expected to yield useful results (9). If the normal electronic structure of the chain consists of just one band, this one-dimensional model is easily treated in the tight-binding approximation. 

In this paper we have tried to describe the most important steps on the way from the simple tight-binding approximation to the simple fluctuation model for quasi-one-dimensional conductors. A major crossing on this way is certainly traversed in earlier sub-sections, where are given the arguments in favour of the single-order parameter phase transition theory for high-temperature quasi-one-dimensional conductors. 

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