On the Choice of a Maximal Cluster in the Cluster Variational Method
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ON THE CHOICE OF A MAXIMAL CLUSTER IN THE CLUSTER VARIATIONAL METHOD DAVID A. VUL* AND DIDIER DE FONTAINE*t *Department of Materials Science and Mineral Engineering, University of California, Berkeley, CA 94720 tMaterials Sciences Division, Lawrence Berkeley Laboratory, Berkeley, CA 94720 ABSTRACT An explanation of the occurrence of unphysical solutions in the cluster variational method is given. A simple algorithm for the construction of an optimal maximal cluster, i.e., a cluster that guarantees correct results for any set of interatomic interactions, is suggested. Examples of optimal maximal clusters for various two- and three-dimensional lattices are presented. INTRODUCTION With the possible exception of simulation techniques, the cluster variational method (CVM), originally proposed by Kikuchi [1], is the most flexible and accurate one available for the calculation of thermodynamic properties of crystals. Despite recent advances by a number of authors [2-4], the choice of appropriate cluster approximations remains problematic. In particular, no simple rules exist for selection the choice of the largest cluster of lattice points to be retained in the formulation, i.e., the maximal cluster, compatible with a given range of effective interactions. In the CVM approximation, the maximal cluster (MC) is the largest set of atoms for which correlations are correctly taken into account. One might expect the accuracy of the method to increase monotonically with the number of lattice points in the cluster, but that is not the case: certain clusters of large "size" give completely unacceptable solutions so that a simple rule is required for ruling out "bad" clusters, and constructing "good" ones. Finel [4] has given a criteria for cluster selection based on the factorization of the density matrix. In the present work, we give an explanation for the appearance of inphysical solutions in the CVM and suggest a criterion which must be fulfilled to guarantee that a given MC is optimal, i.e., provides correct results for any set of interatomic interactions. We also suggest a simple rule for the construction of increasingly large optimal MC's for arbitrary lattices and present corresponding examples for various two- and three-dimensional lattices. CLUSTER EXPANSION FOR ENTROPY Consider a binary alloy with lattice sites i=1,2,... N and "spin" variables ri, si=l or -1, depending on occupation of the site i by an A or a B atom. The density of states p(N) as a function of atomic configurations ({i)can be written as statistical average over variables s or as an expansion over cluster functions Ga of all possible clusters in a lattice [3]. p(N)= 2-N<
s = 2-N(I +I
•C;) a
,
Gi
=
• E,
= ,(1)
iEQt
where E will be called the correlation parameter of cluster cc, and the angle brackets represent a suitable ensemble average. Expression (1) for p(N) can be represented conveniently in another form by introducing the "leveled" exponential function expLx = I+x which allows one to write the result of averaging as an expansion o
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