Tight binding approximation

Previoiisly we had started with free electrons and then added interactions with the ion cores to modify the versus k relationships which provide the basis for the bands and energy gaps in conductors. Now we start with the bound states of the core electrons in the atoms making up the lattice and add the interactions that lead to similar bands in insulators and semiconductors. 

The tight-binding approximation uses the method of linear combination of atomic orbitals (LCAO) to approximate the wavefunction of one electron under the influence of 

Energy well of an individual atom in which A, B, and C are bound states. 

Illustration of a periodic potential with a band structure of bound electrons (A and B) and nearly free electrons in the Fermi region (C). 

We now compute the energy of the electrons imder the influence of the other atoms in the crystal. 

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Whereas the tight-binding approximation works well for certain types of solid, for other s. items it is often more useful to consider the valence electrons as free particles whose motion is modulated by the presence of the lattice. Our starting point here is the Schrodinger equation for a free particle in a one-dimensional, infinitely large box ... 

Fig. 4. Calculated density of states for two zigzag individual SWCNTs with (a) semiconducting (10, 0) and (b) metallic (9, 0) configurations. Tight-binding approximation was used for the calculation [6].

TOTAL ENERGY CALCULATIONS IN THE TIGHT-BINDING APPROXIMATION... 

We will limit ourselves here to transition metals. It is well known that in these metals, the cohesive properties are largely dominated by the valence d electrons, and consequently, sp electrons can be neglected save for the elements with an almost empty or filled d valence shelP. Since the valence d atomic orbitals are rather localized, the d electronic states in the solid are well described in the tight-binding approximation. In this approximation, the cohesive energy of a bulk crystal is usually written as ... 

Here, u is the displacement of the /ith molecule from its equilibrium position and M the reduced mass of each molecular site. Second, the electron is described within the frame of the tight-binding approximation, where it is assumed that the effect of the potential at a given site of the one-dimensional chain is limited to its nearest neighbors. In that case, the energy dispersion of the electron is given by... 

Within the tight binding approximation, it imphes a decrease in electron locahzation energy ... 

Fig. 3 (a) Crystal structure of (DMET)2FeBr4. The dotted and dashed lines denote the intermo-lecular anion—anion and donor-anion contacts, respectively, (b) Fermi surfaces obtained for a donor layer around z = 1/2 using the tight-binding approximation. The solid arrow represents the nesting vector Q (a b )/2... 

The Hiickel approximation (3.136) is equivalent to neglect of fi and p" (the tight-binding approximation), leading to the simpler Hiickel-type matrix h(HMO) ... 

Comparison with (3.156) shows that F(NBO) is intrinsically of significantly higher accuracy than h(HMO) for describing the actual pi interactions of benzene. Because F(nbo) js tjje fundamental starting point for localized NBO analysis of conjugafive interactions, we can conclude that the NBO donor-acceptor picture is inherently more accurate than that based on the Hiickel tight-binding approximation. 

ANG AO ATA BF CB CF CNDO CPA DBA DOS FL GF HFA LDOS LMTO MO NN TBA VB VCA WSL Anderson-Newns-Grimley atomic orbital average t-matrix approximation Bessel function conduction band continued fraction complete neglect of differential overlap coherent-potential approximation disordered binary alloy density of states Fermi level Green function Flartree-Fock approximation local density of states linear muffin-tin orbital molecular orbital nearest neighbour tight-binding approximation valence band virtual crystal approximation Wannier-Stark ladder... 

These moments can be calculated using the tight binding approximation. Introducing a complete set of atomic orbitals za) satisfying the equations... 

Rose and Benjamin (see also Halley and Hautman ) utilized molecular dynamic simulations to compute the free energy function for an electron transfer reaction, Fe (aq) + e Fe (aq) at an electrodesolution interface. In this treatment, Fe (aq) in water is considered to be fixed next to a metal electrode. In this tight-binding approximation, the electron transfer is viewed as a transition between two states, Y yand Pf. In Pj, the electron is at the Fermi level of the metal and the water is in equilibrium with the Fe ion. In Pf, the electron is localized on the ion, and the water is in equilibrium with the Fe" ions. The initial state Hamiltonian H, is expressed as... 

The simplest model of a solid is a linear chain of N atoms, with one end of the chain corresponding to the surface. If an atomic orbital r> (r = 1,..., N) is associated with the rth atom, then, in the tight-binding approximation, the matrix elements of the Hamiltonian for the solid, can... 

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. 

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... 

Let the chain atoms be numbered by an index m(—N m N), and let the two foreign atoms X and n be in interaction with the chain at positions - -n and —n. The wave functions for the system are either even or odd in the center of symmetry at the chain atom number 0. It is sufficient, therefore, to consider the range w 0. In the tight-binding approximation, we now have Equation (4) for all m n with the boundary conditions... 

To establish a quantitative relation between F and G for the entire tip and the entire sample, we have to consider all the states in the tip and the sample. A rigorous treatment is complicated. The following treatment is based on the approximate additivity of atomic force and tunneling conductance with respect to the atoms of the tip. In other words, the force between the entire tip and the sample can be approximated as the sum of the force between the individual atoms in the tip and the entire sample, so does tunneling conductivity. Because the tip is made of transition metals, for example, W, Pt, and Ir, the tight-binding approximation, and consequently, additivity, are reasonable assumptions. Under this approximation, the total force is... 

In order to understand the concept, it is customary to present a sort of paradox arising from the solution of the Hamiltonian (11) in the tight-binding approximation (Eq. (12)). 

In the case of narrow bands - and this will be the case of hybridized 5 f bands when 5 f electrons are itinerant - an approximate treatment has to be done. Kubo and Obata have studied the case of transition metals in the tight binding approximation. The narrow band susceptibility is the sum of 4 terms... 

In Eq. (3.106), A/2 is the period of the lattice potential, Xq is the x-coordinate of the bottom of the harmonic potential and f/ determines the tightness of the transverse confinement of a particle in the lattice. Using the tight-binding approximation, the condensate order parameter is [80, 81]... 

Consider a simple cubic lattice of p valent ajtoms which form nearest neighbour bonds only. Show that the bandstructure E(k, 0,0) is given within the tight binding approximation by... 

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