Algorithms for deriving crystallographic space-group information

1999 ◽  
Vol 55 (2) ◽  
pp. 383-395 ◽  
Author(s):  
R. W. Grosse-Kunstleve
Keyword(s):  
Three Dimensional ◽  
Basis Matrix ◽  
Space Group ◽  
Space Groups ◽  
Group Type ◽  
Group Information ◽  
Commercial Applications ◽  
Crystallographic Space Groups ◽  
Symmetry Operations

Algorithms are presented for three-dimensional crystallographic space groups, handling tasks such as the generation of symmetry operations, the characterization of symmetry operations (determination of rotation-part type, axis direction, sense of rotation, screw or glide part and location part), the determination of space-group type [identified by the space-group number of the International Tables for Crystallography (Dordrecht: Kluwer Academic Publishers)] and the generation of structure-seminvariant vectors and moduli. The latter are an algebraic description of allowed origin shifts, which are important in crystal structure determination methods or for comparing crystal structures. The space-group type determination produces a change-of-basis matrix which transforms a given space-group representation to the standard one according to the International Tables for Crystallography. The algorithms were implemented and tested using the SgInfo library. The source code is free for non-commercial applications.

The crystallographic chameleon: when space groups change skin

2016 ◽  
Vol 72 (5) ◽  
pp. 523-538 ◽  
Author(s):  
Massimo Nespolo ◽  
Mois I. Aroyo
Keyword(s):  
Space Group ◽  
Practical Applications ◽  
Space Groups ◽  
Common Substructure ◽  
Group Information ◽  
Alternative Setting ◽  
The Common ◽  
Alternative Settings ◽  
Unnecessary Complication ◽  
Symmetry Operations

VolumeAofInternational Tables for Crystallographyis the reference for space-group information. However, the content is not exhaustive because for many space groups a variety of settings may be chosen but not all of them are described in detail or even fully listed. The use of alternative settings may seem an unnecessary complication when the purpose is just to describe a crystal structure; however, these are of the utmost importance for a number of tasks, such as the investigation of structure relations between polymorphs or derivative structures, the study of pseudo-symmetry and its potential consequences, and the analysis of the common substructure of twins. The aim of the article is twofold: (i) to present a guide to expressing the symmetry operations, the Hermann–Mauguin symbols and the Wyckoff positions of a space group in an alternative setting, and (ii) to point to alternative settings of space groups of possible practical applications and not listed in VolumeAofInternational Tables for Crystallography.

Extension of three-dimensional space-group generators to the (3+1) case

1999 ◽  
Vol 32 (3) ◽  
pp. 452-455
Author(s):  
Kazimierz Stróż
Keyword(s):  
Dimensional Space ◽  
Three Dimensional ◽  
Space Group ◽  
Space Groups ◽  
Symmetry Operations ◽  
Three Dimensional Space

A method of building up the generators of 775 (3+1)-dimensional superspace groups is proposed. The generators are based on the conventional space-group generators selected by Wondratschek and applied in theInternational Tables for Crystallography(1995, Vol. A). By the method, the generation of (3+1) space groups is found to be easier, the description of symmetry operations is closer to that used for the conventional space groups, and ambiguities in the (3+1) group notation are avoided.

SPGGEN: a computer program for retrieval of space-group information in several settings and generator-containing space-group symbols

2016 ◽  
Vol 49 (4) ◽  
pp. 1370-1376 ◽  
Author(s):  
Uri Shmueli
Keyword(s):  
Computer Program ◽  
Dimensional Space ◽  
Three Dimensional ◽  
General Setting ◽  
Detailed Comparison ◽  
Space Group ◽  
Space Groups ◽  
Group Number ◽  
Group Information ◽  
Three Dimensional Space

A brief outline of the algorithm for the derivation of a space group is followed by a detailed description of the explicit space-group symbols here employed. These space-group symbols are unique insofar as they contain explicitly the generators of the space group dealt with. Next, the implementation of the above in a computer program,SPGGEN, is briefly discussed and the options presented by the program are outlined. Briefly, these options are (i) conventional derivation of the space group from an explicit symbol, including a user-defined one; (ii) such derivation from the conventional space-group number only; (iii) introduction of a general setting into the derivation; (iv) introduction of a Cartesian setting into the derivation; and (v) treatment of some non-conventional settings of orthorhombic space groups. This is followed by a detailed comparison withInternational Tables for Crystallography, Vol. A, and by examples of the output ofSPGGEN. A complete tabulation of the explicit three-dimensional space-group symbols is readily accessed.

Intergrowth polytypoids as modulated structures: a superspace description of the Sr n (Nb,Ti) n O3n + 2 compound series

2001 ◽  
Vol 57 (4) ◽  
pp. 471-484 ◽  
Author(s):  
L. Elcoro ◽  
J. M. Perez-Mato ◽  
R. L. Withers
Keyword(s):  
Dimensional Space ◽  
Three Dimensional ◽  
Separation Distance ◽  
Space Group ◽  
Space Groups ◽  
Cell Parameters ◽  
Average Structure ◽  
Layer Stacking ◽  
Compound Series ◽  
Three Dimensional Space

A new, unified superspace approach to the structural characterization of the perovskite-related Sr n (Nb,Ti) n O3n + 2 compound series, strontium niobium/titanium oxide, is presented. To a first approximation, the structure of any member of this compound series can be described in terms of the stacking of (110)-bounded perovskite slabs, the number of atomic layers in a single perovskite slab varying systematically with composition. The various composition-dependent layer-stacking sequences can be interpreted in terms of the structural modulation of a common underlying average structure. The average interlayer separation distance is directly related to the average structure periodicity along the layer stacking direction, while an inherent modulation thereof is produced by the presence of different types of layers (particularly vacant layers) along this stacking direction. The fundamental atomic modulation is therefore occupational and can be described by means of crenel (step-like) functions which define occupational atomic domains in the superspace, similarly to what occurs for quasicrystals. While in a standard crystallographic approach, one must describe each structure (in particular the space group and cell parameters) separately for each composition, the proposed superspace model is essentially common to the whole compound series. The superspace symmetry group is unique, while the primary modulation wavevector and the width of some occupation domains vary linearly with composition. For each rational composition, the corresponding conventional three-dimensional space group can be derived from the common superspace group. The resultant possible three-dimensional space groups are in agreement with all the symmetries reported for members of the series. The symmetry-breaking phase transitions with temperature observed in many compounds can be explained in terms of a change in superspace group, again in common for the whole compound series. Inclusion of the incommensurate phases, present in many compounds of the series, lifts the analysis into a five-dimensional superspace. The various four-dimensional superspace groups reported for this incommensurate phase at different compositions are shown to be predictable from a proposed five-dimensional superspace group apparently common to the whole compound series. A comparison with the scarce number of refined structures in this system and the homologous (Nb,Ca)6Ti6O20 compound demonstrates the suitability of the proposed formalism.

TABLES, a program to display space-group symmetry information in three dimensions

1987 ◽  
Vol 20 (6) ◽  
pp. 532-535 ◽  
Author(s):  
C. Abad-Zapatero ◽  
T. J. O'Donnell
Keyword(s):  
Computer Graphics ◽  
Computer Program ◽  
Three Dimensional ◽  
Spatial Location ◽  
Three Dimensions ◽  
Group Symmetry ◽  
Space Group ◽  
Interactive Display ◽  
Symmetry Operators ◽  
Group Information

TABLES is a computer program developed to display the crystal symmetry and the spatial location of the different symmetry operators for a given space group using interactive computer graphics. It allows the three-dimensional interactive display of the space-group information contained in International Tables for Crystallography [(1983), Vol. A. Dordrecht: Reidel]. Such a program is useful as a teaching aid in crystallography and is valuable for exploring molecular packing arrangements.

Mathematical aspects of molecular replacement. IV. Measure-theoretic decompositions of motion spaces

2017 ◽  
Vol 73 (5) ◽  
pp. 387-402 ◽  
Author(s):  
Gregory S. Chirikjian ◽  
Sajdeh Sajjadi ◽  
Bernard Shiffman ◽  
Steven M. Zucker
Keyword(s):  
General Theory ◽  
Rigid Body ◽  
Three Dimensional ◽  
Molecular Replacement ◽  
Translation Group ◽  
Common Occurrence ◽  
Space Group ◽  
Space Groups ◽  
Relevant Case ◽  
The Subject

In molecular-replacement (MR) searches, spaces of motions are explored for determining the appropriate placement of rigid-body models of macromolecules in crystallographic asymmetric units. The properties of the space of non-redundant motions in an MR search, called a `motion space', are the subject of this series of papers. This paper, the fourth in the series, builds on the others by showing that when the space group of a macromolecular crystal can be decomposed into a product of two space subgroups that share only the lattice translation group, the decomposition of the group provides different decompositions of the corresponding motion spaces. Then an MR search can be implemented by trading off between regions of the translation and rotation subspaces. The results of this paper constrain the allowable shapes and sizes of these subspaces. Special choices result when the space group is decomposed into a product of a normal Bieberbach subgroup and a symmorphic subgroup (which is a common occurrence in the space groups encountered in protein crystallography). Examples of Sohncke space groups are used to illustrate the general theory in the three-dimensional case (which is the relevant case for MR), but the general theory in this paper applies to any dimension.

scholarly journals From space to superspace and back: Superspace Group Finder

2008 ◽  
Vol 41 (6) ◽  
pp. 1182-1186 ◽  
Author(s):  
Ivan Orlov ◽  
Lukas Palatinus ◽  
Gervais Chapuis
Keyword(s):  
Crystal Structure ◽  
Dimensional Space ◽  
Flexible Structures ◽  
Three Dimensional ◽  
Real Space ◽  
Space Group ◽  
Space Groups ◽  
Group A ◽  
Modular Composition ◽  
Three Dimensional Space

The symmetry of a commensurately modulated crystal structure can be described in two different ways: in terms of a conventional three-dimensional space group or using the superspace concept in (3 +d) dimensions. The three-dimensional space group is obtained as a real-space section of the (3 +d) superspace group. A complete network was constructed linking (3 + 1) superspace groups and the corresponding three-dimensional space groups derived from rational sections. A database has been established and is available at http://superspace.epfl.ch/finder/. It is particularly useful for finding common superspace groups for various series of modular (`composition-flexible') structures and phase transitions. The use of the database is illustrated with examples from various fields of crystal chemistry.

scholarly journals Isomers and polymorphs of (E,E)-1,4-bis(nitrophenyl)-2,3-diaza-1,3-butadienes

2006 ◽  
Vol 62 (4) ◽  
pp. 666-675 ◽  
Author(s):  
Christopher Glidewell ◽  
John N. Low ◽  
Janet M. S. Skakle ◽  
James L. Wardell
Keyword(s):  
Hydrogen Bond ◽  
Hydrogen Bonds ◽  
Three Dimensional ◽  
Stacking Interactions ◽  
Space Group ◽  
Stacking Interaction ◽  
Space Groups ◽  
Π Stacking ◽  
Π Stacking Interaction ◽  
Polymorphic Forms

The structures of five of the possible six isomers of (E,E)-1,4-bis(nitrophenyl)-2,3-diaza-1,3-butadiene are reported, including two polymorphs of one of the isomers. (E,E)-1,4-Bis(2-nitrophenyl)-2,3-diaza-1,3-butadiene, C14H10N4O4 (I), crystallizes in two polymorphic forms (Ia) and (Ib) in which the molecules lie across centres of inversion in space groups P21/n and P21/c, respectively: the molecules in (Ia) and (Ib) are linked into chains by aromatic π...π stacking interactions and C—H...π(arene) hydrogen bonds, respectively. Molecules of (E,E)-1-(2-nitrophenyl)-4-(3-nitrophenyl)-2,3-diaza-1,3-butadiene (II) are linked into sheets by two independent C—H...O hydrogen bonds. The molecules of (E,E)-1,4-bis(3-nitrophenyl)-2,3-diaza-1,3-butadiene (III) lie across inversion centres in the space group P21/n, and a combination of a C—H...O hydrogen bond and a π...π stacking interaction links the molecules into sheets. A total of four independent C—H...O hydrogen bonds link the molecules of (E,E)-1-(3-nitrophenyl)-4-(4-nitrophenyl)-2,3-diaza-1,3-butadiene (IV) into sheets. In (E,E)-1,4-bis(4-nitrophenyl)-2,3-diaza-1,3-butadiene (V) the molecules, which lie across centres of inversion in the space group P21/n, are linked by just two independent C—H...O hydrogen bonds into a three-dimensional framework.

scholarly journals Normally supportive sublattices of crystallographic space groups

2020 ◽  
Vol 76 (1) ◽  
pp. 7-23
Author(s):  
Miles A. Clemens ◽  
Branton J. Campbell ◽  
Stephen P. Humphries
Keyword(s):  
Normal Subgroup ◽  
Point Group ◽  
Group Extension ◽  
Normal Subgroups ◽  
Symbolic Form ◽  
Space Group ◽  
Space Groups ◽  
Significant Step ◽  
Crystal Class ◽  
Crystallographic Space Groups

The tabulation of normal subgroups of 3D crystallographic space groups that are themselves 3D crystallographic space groups (csg's) is an ambitious goal, but would have a variety of applications. For convenience, such subgroups are referred to as `csg-normal' while normal subgroups of the crystallographic point group (cpg) of a crystallographic space group are referred to as `cpg-normal'. The point group of a csg-normal subgroup must be a cpg-normal subgroup. The present work takes a significant step towards that goal by tabulating the translational subgroups (a.k.a. sublattices) that are capable of supporting csg-normal subgroups. Two necessary conditions are identified on the relative sublattice basis that must be met in order for the sublattice to support csg-normal subgroups: one depends on the operations of the point group of the space group, while the other depends on the operations of the cpg-normal subgroup. Sublattices that meet these conditions are referred to as `normally supportive'. For each cpg-normal subgroup (excluding the identity subgroup 1) of each of the arithmetic crystal classes of 3D space groups, all of the normally supportive sublattices have been tabulated in symbolic form, such that most of the entries in the table contain one or more integer variables of infinite range; thus it could be more accurately described as a table of the infinite families of normally supportive sublattices. For a given pair of cpg-normal subgroup and normally supportive sublattice, csg-normal subgroups of the space groups of the parent arithmetic crystal class can be constructed via group extension, though in general such a pair does not guarantee the existence of a corresponding csg-normal subgroup.

The three-dimensional structures of two isoforms of nucleoside diphosphate kinase from bovine retina

1999 ◽  
Vol 55 (6) ◽  
pp. 1127-1135 ◽  
Author(s):  
Jane E. Ladner ◽  
Najmoutin G. Abdulaev ◽  
Dmitri L. Kakuev ◽  
Mária Tordová ◽  
Kevin D. Ridge ◽  
...  
Keyword(s):  
Active Site ◽  
Three Dimensional ◽  
Nucleoside Diphosphate Kinase ◽  
Asymmetric Unit ◽  
Ndp Kinase ◽  
Space Group ◽  
Space Groups ◽  
Bovine Retina ◽  
Biological Sources ◽  
Polypeptide Chains

The crystal structures of two isoforms of nucleoside diphosphate kinase from bovine retina overexpressed in Escherischia coli have been determined to 2.4 Å resolution. Both the isoforms, NBR-A and NBR-B, are hexameric and the fold of the monomer is in agreement with NDP-kinase structures from other biological sources. Although the polypeptide chains of the two isoforms differ by only two residues, they crystallize in different space groups. NBR-A crystallizes in space group P212121 with an entire hexamer in the asymmetric unit, while NBR-B crystallizes in space group P43212 with a trimer in the asymmetric unit. The highly conserved nucleotide-binding site observed in other nucleoside diphosphate kinase structures is also observed here. Both NBR-A and NBR-B were crystallized in the presence of cGMP. The nucleotide is bound with the base in the anti conformation. The NBR-A active site contained both cGMP and GDP each bound at half occupancy. Presumably, NBR-A had retained GDP (or GTP) from the purification process. The NBR-B active site contained only cGMP.

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