Empirical Potentials for Metals and Semiconductors

Perhaps the most important consideration when discussing the development and use of empirical potentials for studying atomic solids is that pairwise potential models are often not very suitable The performance of pairwise potential models can be bad for transition metals and even worse for semiconductors There are a number of reasons why this is so, many of which are due to the fundamental behaviour of pairwise potentials for certain experimental properties. The most oft-quoted properties are as follows  

The ratio between the cohesive energy and the melting temperature, EJk T. The cohesive energy is the energy cost of removing an atom from within the solid matrix. This ratio is observed to be approximately 30 in metals but about 10 in pairwise systems. 

The ratio between the elastic constants C12/C44. Elastic constants will be discussed in Section 5.10 for a cubic solid there are three distinct values, which are labelled Q], Cj2 and C44. For a two-body system the ratio is exactly 1 (this is known as the Cauchy relationship). For metals and oxides deviation from unity is common, gold has a particularly high value, which is indicative of its high malleability. 

The surface properties of metals are such that the surface tends to relax inwards but systems described by two-body interactions tend to relax outwards. 

The origins of the Finnis-Sinclair potential [Finnis and Sinclair 1984] lie in the density of states and the moments theorem. Recall that the density of states D E) (see Section 3.8.5) describes the distribution of electronic states in the system. D(E) gives the number of states between E and E + SE. Such a distribution can be described in terms of its moments. The moments are usually defined relative to the energy of the atomic orbital from which the molecular orbitals are formed. The mth moment, f/ , is given by  

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