The Tight-Binding Method

As noted already, the starting point of the analysis is an approximation to the total electronic wave function for the solid. In particular, we seek a wave function of the form 

The sum over i in eqns (4.57) and (4.58) runs over the number of atoms N in the solid, while the sum on a is over the number of orbitals per site. The atomic-like orbital of type a centered on site i is written as i, a) = (/ Q.(r — R,). Indeed, the nature of these orbitals and the size of the parameter n strikes right to the heart of the minimalist character of the semiempirical tight-binding method. 

our expansion will center on the outer 3s and 3p electrons. In this case, the total wave function is built up out of the four such basis functions at each site which for the site we label as i, 3s , i, Px), i, 3py and i, 3p2 . Note that in keeping with our aim to shed unnecessary degrees of freedom, all reference to core electrons has been omitted they are assumed to be unchanged by the proximity of neighboring atoms. In addition, the basis itself is incomplete in the mathematical sense, since we have not included higher energy atomic states such as i, 4s , etc. in the basis. 

The nature of the matrices and 5 is subtle and as a result has been tackled from a number of different perspectives. Since our aim is to present ideas rather than to give an exhaustive description of each and every alternative approach to a given problem, the present discussion will try to highlight some of the distinguishing features of these quantities. The matrix provides a measure of the overlap of wave functions on adjacent sites, and is often described as the overlap matrix . For the purposes of the present discussion, we adopt the simplest 

For example, for two pyorbitals, the matrix element can be split up into a term involving Vppa and a second term involving Vppn. As a result of these 

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Formulation of the Tight Binding Method in the Notation of Second Quantization... 

Figure 5 Density of states of Ni V clusters with N — 5, 6, and 7, calculated by the tight binding method sp (dashed lines) and d (continuous lines). Positive and negative values correspond to up and down J, spins, respectively. The Fermi level is at the energy zero. Adapted with permission from Ref. 45.

One of the first ways used to calculate approximate solutions of (6) was the tight-binding method Here we consider an array of N potential wells, of which two are shown in Fig. 1.1, and suppose that in each well individually the electrons can have a number of bound states with energies WQiWl9...,Wn and with wave functions < 0, ,..., 4>n. When the electron is allowed to move from one well to another, a band containing N states is formed from each bound state of a single well. The approximate wave function describing this motion is... 

It should be emphasized that in metals the d-states, for which tight-binding functions may be used, lie above the zero of the muffin-tin potential The reason why the tight-binding method can still be used is the following. The radial part of the Schrodinger equation is... 

FIGURE 6.4 (a) The re-bands of a 2D graphene layer derived on the basis of the tight-binding method. The... 

The Tight-Binding Method. The tight-binding method starts from the Hamiltonian for the ionic (or molecular) core ... 

This chapter begins a series of chapters devoted to electronic structure and transport properties. In the present chapter, the foundation for understanding band structures of crystalline solids is laid. The presumption is, of course, that said electronic structures are more appropriately described from the standpoint of an MO (or Bloch)-type approach, rather than the Heitler-London valence-bond approach. This chapter will start with the many-body Schrodinger equation and the independent-electron (Hartree-Fock) approximation. This is followed with Bloch s theorem for wave functions in a periodic potential and an introduction to reciprocal space. Two general approaches are then described for solving the extended electronic structure problem, the free-electron model and the LCAO method, both of which rely on the independent-electron approximation. Finally, the consequences of the independent-electron approximation are examined. Chapter 5 studies the tight-binding method in detail. Chapter 6 focuses on electron and atomic dynamics (i.e. transport properties), and the metal-nonmetal transition is discussed in Chapter 7. 

In the present contribution, we will examine the fundamentals of such an approach. We first describe some basic notions of the tight-binding method to build the COs of an infinite periodic solid. Then we consider how to analyze the nature of these COs from the viewpoint of orbital interaction by using some one-dimensional (ID) examples. We then introduce the notion of density of states (DOS) and its chemical analysis, which is especially valuable in understanding the structure of complex 3D sohds or in studying surface related phenomena. Later, we introduce the concept of Fermi surface needed to examine the transport properties of metallic systems and consider the different electronic instabilities of metals. Finally, a brief consideration of the more frequently used computational approaches to the electronic structure of solids is presented. 

At the lowest level of sophistication of quantum treatments, the tight-binding method and the semi-empirical HF method reduce the complexity of the interacting electron system to the diagonalization of an effective one-electron Hamiltonian matrix, whose elements contain empirical parameters. The electronic wave functions are expanded on a minimal basis set of atomic or Slater orbitals centered on the atoms and usually restricted to valence orbitals. The matrix elements are self-consistently determined or not, depending upon the method. 

In the tight-binding method, the elements of the Hamiltonian matrix axe treated as adjustable parameters to be fitted to experimental or first-... 

The diffusion of individual Pb atoms on the Cu(llO) surface has been investigated by molecular dynamics simulations (section 3.4). In this case the interaction potential is derived from a phenomenological model similar to that used in the tight binding method [44, 45]. This potential satisfactorily describes bulk and surface properties of noble and transition metals except for the surface energies. In the Pb/Cu(l 10) studies the tight binding functional form is used to describe the Pb-Pb and Pb-Cu interactions. Parameters for both the pure metal and cross interaction potentials are obtained from fits to experimental values. 

In the tight-binding method, the wavefunctions are constructed of localized atomic levels, but an electron in such a level will be found with equal probability in any cell of a monatomic crystal [22, p. 185]. Nevertheless, these freely moving electrons do not necessarily contribute to electric conduction, since in the semiclassical picture of a filled band half the electrons move to the right and half to the left and this remains true even if an electric field is applied. For partially filled bands, the exact half/half-symmetry is broken by an electric field, and a current will flow. Note that half-filled bands can give an electric current but cannot... 

Because they are so computationally intensive, ab initio and semiempirical studies are limited to models that are about 10 rings or less. In order to study more reahstic carbon structures, approximations in the form of the Hamiltonian (i.e., Schrodinger equation) are necessary. The tight-binding method, in which the many-body wave function is expressed as a product of individual atomic orbitals, localized on the atomic centers, is one such approximation that has been successfully applied to amorphous and porous carbon systems [47]. 

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