Constructing the Effective Energy Cluster Expansions

On the practical side, the difficulties involved in constructing the alloy free energy are related to the fact that somehow the effective Hamiltonian must be informed on the basis of a relatively small set of calculations. For concreteness, we consider a binary alloy in which there are two different chemical constituents. For a lattice built up of N sites, the freedom to distribute the two constituents on these N sites results in 2 configurations (i.e. each site may be assigned one of 

In a landmark paper, Sanchez et al. (1984), examined the extent to which the types of cluster functions introduced above can serve as a basis for representing any function of configuration. For our present purposes, the key result to take away from their analysis is that this set of cluster functions is complete in the mathematical sense, implying that any function of configuration can be written as a linear combination of these cluster functions. On the other hand, for the whole exercise to be useful, we seek a minimal basis, that is to say, we truncate at some order with the expectation that a low-order expansion in such cluster 

What this equation tells us is to find all N clusters of type a and to evaluate the product CT1CT2 a for each such cluster and to add up the contributions from them all. In the particular case written above, our notation has been chosen such that there are n sites involved in the a cluster type. The entirety of these averaged correlation functions provides a measure of the configurational order associated with the structure of interest. What these equations tell us physically is that the energetics is expanded in terms of various correlation functions that characterize the extent to which adjacent atoms are like or unlike. 

In the approach which has come to be known as the Connolly-Williams method (Connolly and Williams, 1983), a systematic inversion of the cluster expansion is effected with the result that the parameters in the effective Hamiltonian are determined explicitly. In this case, the number of energies in the database is the same as the number of undetermined parameters in the cluster expansion. Other strategies include the use of least-squares analysis (Lu et al. 1991) and linear programming methods (Garbulsky and Ceder, 1995) in order to obtain some best choice of parameters. For example, in the least-squares analyses, the number of energies determined in the database exceeds the number of undetermined parameters in the effective Hamiltonian, which are then determined by minimizing a cost function of the form 

The subscript j refers to the j member of the database, and thus Ej is the energy of that member as obtained from the relevant atomistic calculation and 4 Q ( cr )7 

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