Real Space Tight-Binding Methods

Earlier in this chapter, we introduced pair and cluster functionals as approximate effective theories in which explicit reference to the electronic degrees of freedom had been removed. In each case, we introduced these total energy schemes on 

As a preliminary to our discussion of the emergence of both pair functionals and angular forces from tight-binding theory, we first discuss the moments of the density of states which serve as a useful basis for these developments. We begin with the recognition that every distribution has its moments. In particular, the n moment of the density of states is defined by 

The moments of the density of states can be seen to be related to the electronic structure itself through the following argument. The shape of the electronic density of states determines the outcome of the one-electron sum embodied in eqn (4.66). The shape of the density of states is in turn determined by its moments. 

These are the two lowest-order moments beyond the second moment itself. In this expression, the quantities h R) characterize the strength of interaction between orbitals that are separated by a distance R. Yet higher-order moments can be obtained by recourse to the same basic ideas. 

The rectangular band model posits that the density of states is constant over some range of energies and zero otherwise. The constant value associated with the density of states is chosen such that 

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Electronic structure methods for studies of nanostructures can be divided broadly into supercell methods and real-space methods. Supercell methods use standard k-space electronic structure techniques separating periodically repeated nanostructures by distances large enough to neglect their interactions. Direct space methods do not need to use periodic boundary conditions. Various electronic structure methods are developed and applied using both approaches. In this section we will shortly discuss few popular but powerful electronic structure methods the pseudopotential method, linear muffin-tin orbital and related methods, and tight-binding methods. 

On the other side, semiempirical model Hamiltonians have been proposed to overcome the difficulties related to symmetry restriction These methods are formulated in the real space and use localized basis so that nonperiodic systems without symmetry are easy to deal with. In the particular case of magnetic systems the selfconsistent tight-binding model has been successfully applied for the study of free and supported clusters surfaces and over lay-... 

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