Increasing the Computational Efficiency of Combinatoric Searches

• Main Author: Morgan, Wiley Spencer
• Format: Others
• Published: BYU ScholarsArchive 2016
• Subjects: enumeration; algorithm; combinatorics; Astrophysics and Astronomy
• Online Access: https://scholarsarchive.byu.edu/etd/6528; https://scholarsarchive.byu.edu/cgi/viewcontent.cgi?article=7528&context=etd

A new algorithm for the enumeration of derivative superstructures of a crystal is presented. The algorithm will help increase the efficiency of computational material design methods such as cluster expansion by increasing the size and diversity of the types of systems that can be modeled. Modeling potential alloys requires the exploration of all possible configurations of atoms. Additionally, modeling the thermal properties of materials requires knowledge of the possible ways of displacing the atoms. One solution to finding all symmetrically unique configurations and displacements is to generate the complete list of possible configurations and remove those that are symmetrically equivalent. This approach, however, suffers from the combinatoric explosion that happens when the supercell size is large, when there are more than two atom types, or when atomic displacements are included in the system. The combinatoric explosion is a problem because the large number of possible arrangements makes finding the relatively small number of unique arrangements for these systems impractical. The algorithm presented here is an extension of an existing algorithm [Hart & Forcade (2008a), Hart & Forcade (2009a), Hart et al. (2012a) Hart, Nelson, & Forcade] to include the extra configurational degree of freedom from the inclusion of displacement directions. The algorithm makes use of another recently developed algorithm for the Pólya [Pólya & Read (1987), Pólya (1937), Rosenbrock et al.(2015) Rosenbrock, Morgan, Hart, Curtarolo, & Forcade] counting theorem to inform the user of the total number of unique arrangements before performing the enumeration and to ensure that the list of unique arrangements will fit in system memory. The algorithm also uses group theory to eliminate large classes of arrangements rather than eliminating arrangements one by one. The three major topics of this paper will be presented in this order, first the Pólya algorithm, second the new algorithm for eliminating duplicate structures, and third the algorithms extension to include displacement directions. With these tools, it is possible to avoid the combinatoric explosion and enumerate previously inaccessible systems, including those that contain displaced atoms.