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

Sami Mustapha, Philippe D'Arco, Marco De La Pierre, Yves Noël, Matteo Ferrabone, Roberto Dovesi
2013 Journal of Physics: Condensed Matter  
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 symmetry can be used in the identification of symmetry independent configurations (SICs) is here discussed. First, Pólya's enumeration theory is
more » ... dopted to determine the number of SICs, in the case of both varying and fixed composition, for colors in number of two or higher. Then, De Bruijn's generalization is presented, which allows to analyze the case where 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 theories do help in solving the difficult problem of identifying the complete list of SICs. SICs representatives are here obtained by adopting an orderly generation approach, based on the lexicographic ordering, that 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, that permits to efficiently generate 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 the square with C 4v symmetry and the real cases of garnet and olivine mineral families.
doi:10.1088/0953-8984/25/10/105401 pmid:23388579 fatcat:eysnsnend5hndb7irxitd7bpym