On the use of symmetry in configurational analysis for the simulation of disordered solids

The starting point for a quantum mechanical investigation of disordered systems usually implies calculations on a limited subset of configurations, generated by defining either the composition of interest or a set of compositions ranging from one end member to another, within an appropriate supercell of the primitive cell of the pure compound. The way in which symmetry can be used in the identification of symmetry independent configurations (SICs) is discussed here. First, Pólya's enumeration theory is adopted to determine the number of SICs, in the case of both varying and fixed composition, for colors numbering two or higher. Then, De Bruijn's generalization is presented, which allows analysis of the case where the colors are symmetry related, e.g. spin up and down in magnetic systems. In spite of their efficiency in counting SICs, neither Pólya's nor De Bruijn's theory helps in solving the difficult problem of identifying the complete list of SICs. Representative SICs are obtained by adopting an orderly generation approach, based on lexicographic ordering, which offers the advantage of avoiding the (computationally expensive) analysis and storage of all the possible configurations. When the number of colors increases, this strategy can be combined with the surjective resolution principle, which permits the efficient generation of SICs of a problem in |R| colors starting from the ones obtained for the (|R| − 1)-colors case. The whole scheme is documented by means of three examples: the abstract case of a square with C4v symmetry and the real cases of the garnet and olivine mineral families.