Algorithms for crystallographic groups

Eick, Bettina; Souvignier, B.

doi:10.1002/qua.20747

Cited by 8 publications

(21 citation statements)

References 19 publications

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“…[ (1,14,20) (2,15,16) (3,11,17)(4,12,18)(5,13,19) (32,45,46) (33,41,47) (34,42,48) (35,43,49), (1,2,3,4,5)(6,11,16,21,26) (7,12,17,22,27)(8,13,18,23,28)(9,14,19,24,29) (10,15,20,25,30)(31,32,33,34,35)(36,41,46,51,56) (37,42,47,52,57) (38,43,48,53,58) (39,44,49,54,59) (40,45,50,55,60) (9,52,49) (10,51,48) (11,35,28) (12,34,29) (13,…”

Section: Gap> Auall1;mentioning

confidence: 99%

“…It happens that the permutation r should act on d[j] as its transposition, or its inverse. And the product of the operations is given by ( gap> G:=Group(g1,g2,g3); gap> f1:=(1,2,3)(4,6,11)(5,10,12) (7,8,9); gap> f2:= (2,3,4,5,6)(8,9,10,11,12); gap> f3:=(1,7)(2,8) (3,9)(4,10) (5,11)(6,12); gap> F:=Group(f1,f2,f3); gap> r1:= ( 1,14,20) ( 2,15,16) ( 3, 11, 17)( 4, 12, 18)( 5, 13, 19) ( 6, 58, 22)( 7, 57, 23)( 8, 56, 24)( 9, 60, 25)( 10, 59, 21) ( 31, 44, 50) ( 32,45,46) ( 33,41,47) ( 34,42,48) ( 35,43,49) ( 36,28,52) ( 37,27,53) ( 38,26,54) ( 39,30,55) ( 40,29,51); gap> r2:= ( 1, 2, 3, 4, 5)( 6, 11, 16, 21, 26)( 7, 12, 17, 22, 27) ( 8, 13, 18, 23, 28)( 9, 14, 19, 24, 29) ( 10,15,20,25,30)…”

Section: Analysis Of Vibrational Mode In C 60mentioning

confidence: 99%

“…To decompose them into separated irreducible spaces, the following function is pre-# The atoms will be indexed as (i,j) (i=3,..,62, and j=1,2) # by the duplication of single C60. ( 1,14,20) ( 2,15,16) ( 3,11,17) ( 4,12,18) ( 5, 13, 19) ( 6, 58, 22) ( 7,57,23) ( 8,56,24) ( 9,60,25) ( 10, 59, 21) ( 31, 44, 50) ( 32,45,46) ( 33,41,47) ( 34,42,48) ( 35,43,49) ( 36,28,52) ( 37,27,53) ( 38,26,54) ( 39,30,55) ( 40,29,51); g2:= ( 1,…”

Section: Appendix E: the Computation Of Wyckoff Positions By Gapmentioning

confidence: 99%

“…And they analyze the energy spectrum of the molecule by the representation theory, with the hope that their results will find applications in the physical properties of the molecule. Eick and Souvignier give us a survey of the algorithms for space groups and crystallographic groups available in the computer algebra system GAP and in the software packages Carat and Cryst [28].…”

Section: Introductionmentioning

confidence: 99%

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Computer Algebra and Materials Physics

Kikuchi¹

2018

Springer Series in Materials Science

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This article is intended to an introductory lecture in material physics, in which the modern computational group theory and the electronic structure calculation are in collaboration. The effort of mathematicians in the field of group theory have ripened as a new trend, called "computer algebra", outcomes of which now can be available as handy computational packages, and would also be useful to physicists with practical purposes. This article, in the former part, explains how to use the computer algebra for the applications in the solid-state simulation, by means of one of the computer algebra package, the GAP system. The computer algebra enables us to obtain various group theoretical properties with ease, such as the representations, the character tables, the subgroups, etc. Furthermore, it would grant us a new perspective of material design, which could be executed in mathematically rigorous and systematic way. Some technical details and some computations which require the knowledge of a little higher mathematics (but computable easily by the computer algebra) are also given. The selected topics will provide the reader with some insights toward the dominating role of the symmetry in crystal, or, the "mathematical first principles" in it. In the latter part of the article, we analyze the relation between the structural symmetry and the electronic structure in C 60 (as an example to the system without periodicity). The principal object of the study is to illustrate the hierarchical change of the quantum-physical properties of the molecule, in accordance with the reduction of the symmetry (as it descends down in the ladder of subgroups). As an application, this article also presents the computation of the vibrational modes of the C 60 by means of the computer algebra. In order to serve the common interest of the researchers, the details of the computations (the required initial data and the small programs developed for the purpose) are explained as minutely as possible. * akihito kikuchi@gakushikai.jp (The corresponding author)

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Section: Gap> Auall1;mentioning

confidence: 99%

Section: Analysis Of Vibrational Mode In C 60mentioning

confidence: 99%

Section: Appendix E: the Computation Of Wyckoff Positions By Gapmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Computer Algebra and Materials Physics

Kikuchi¹

2018

Springer Series in Materials Science

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“…In order to improve readability, we have collected all the proofs in Appendix A, and we have devoted the last section to a detailed discussion of two specific examples in three dimensions: the derivation of all inequivalent hexagonal 2-lattices (Fadda & Zanzotto, 2001b) and all inequivalent cubic 3-lattices (Hosoya, 1987). Such results could also be obtained using the Wyckoff positions of the relevant space groups, which can, in turn, be determined in any dimension (Fuksa & Engel, 1994;Eick & Souvignier, 2006), but our approach has the advantage of not requiring the computation of high-dimensional space groups, and taking into account arithmetical equivalence and site symmetry by design.…”

Section: Introductionmentioning

confidence: 99%

An algorithm for the arithmetic classification of multilattices

Indelicato

2012

Acta Cryst Sect A

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A procedure for the construction and the classification of monoatomic multilattices in arbitrary dimension is developed. The algorithm allows one to determine the location of the points of all monoatomic multilattices with a given symmetry, or to determine whether two assigned multilattices are arithmetically equivalent. This approach is based on ideas from integral matrix theory, in particular the reduction to the Smith normal form, and can be coded to provide a classification software package.

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Factor groups, semidirect product and quantum chemistry

Trindade

2014

Int J of Quantum Chemistry

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In this article, we prove some general theorems about representations of finite groups arising from the inner semidirect product of groups. We show how these results can be used for standard applications of group theory in quantum chemistry through the orthogonality relations for the characters of irreducible representations. In this context, conditions for transitions between energy levels, projection operators, and basis functions were determined. This approach applies to composite systems and it is illustrated by the dihedral group related to glycolate oxidase enzyme. © 2014 Wiley Periodicals, Inc.

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Algorithms for crystallographic groups

Abstract: This article contains a survey on the algorithms for space groups and crystallographic groups available in the computer algebra system Gap [9] and in the software packages Carat [25] and Cryst [5].

Cited by 8 publications

References 19 publications

Computer Algebra and Materials Physics

Computer Algebra and Materials Physics

An algorithm for the arithmetic classification of multilattices

Factor groups, semidirect product and quantum chemistry

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