Algorithms to search for crystal structures that optimize some extensive property(energy, volume, etc) typically make use of random particle reorganizations in thecontext of one or more numerical techniques such as simulated annealing, geneticalgorithms or biased random walks, applied to the coordinates of every particle in theunit cell, together with the cell angles and lengths. In this paper we describe therestriction of such searches to predefined isopointal sets, breaking the problem intocountable sub-problems which exploit crystal symmetries to reduce the dimensionalityof the search space. Applying this method to the search for maximally packedmixtures of hard spheres of two sizes, we demonstrate that the densest packedstructures can be identified by searches within a couple of isopointal sets. For theA2B system, the densest known packings over the entire tested range0.2 < rA/rB < 2.5, including some improvements on previous optima, can all be identified by searcheswithin a single isopointal set. In the case of the AB composition, searches of twoisopointal sets generate the densest packed structures over the radius ratio range0.2 < rA/rB < 5.0.
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