Tight-Binding Theory and Computational Materials Synthesis
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greater the computational time for a given number of atoms. Thus, to simulate the dynamics of 1,000 independent silicon atoms at the ab initio level, one needs a massively parallel supercomputer. With tight binding, a simulation of comparable accuracy of 1,000 Si atoms could be performed on a workstation. With an empirical potential, such as that due to Tersoff,8 one could perform a somewhat less accurate simulation of 1,000 Si atoms on a workstation requiring less than 1% of the CPU time of the tightbinding simulation. However, the reliability of a simulation depends not only on the accuracy of the adopted atomic interactions but also on whether a sufficient number of atoms has been treated in order to model the phenomenon or process of interest. For this reason, an atomistic simulator often has to make use of the full range of descriptions of atomic interactions up and down the hierarchy. Being in the middle of the hierarchy, the tight-binding approximation enjoys more of the accuracy of the ab initio simulations than empirical potentials, but is typically two to three orders of magnitude faster than ab initio techniques. This article is intended to serve as an elementary introduction to the theory of tight binding and to illustrate its application to a new area in materials modeling, namely the synthesis of materials from molecular precursors. This illustration highlights a particular strength of tight binding in being able to describe chemical processes at small computational cost compared with an ab initio simulation. However, tight binding is not without its limitations and a brief discussion of them is included. We close with a look forward and briefly introduce the O(N) ("order N") methods that are enabling simulations of thousands of atoms to be performed on current workstations.
Basic Theory In the wave-mechanical formulation of quantum mechanics, the wave function is of central importance. We will start with this familiar approach and show how it leads to the concept of the density matrix. The density matrix not only allows much greater insight into bonding and cohesion in materials, but it also enables very efficient computational schemes to be developed. The wave function of each electronic eigenstate of the system is expanded in an atomic-like basis set. By "atomic like," we mean that each basis function is anchored to an atom, and whereas its angular dependence resembles that of an atomic orbital, its radial dependence may be quite different. The basis functions are normally limited to those atomic states that take part in bonding—those in the outermost shell of the atom. Thus, if ipM is the wave function of the nth eigenstate of the system and , is the ;th basis function, we have (i)
Note that the i subscripts are composite labels, denoting the atomic site of the orbital and whether it has s, p, or d angular momentum. The expansion coefficients, c ,•'"', are determined by substituting the wave function i/f("' into the Schrodinger equation: lHj,w = E'-'V'"'
(2)
where H is the Hamiltonian oper
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