The Euler characteristic as a basis for teaching topology concepts to crystallographers

The simple Euler polyhedral formula, expressed as an alternating count of the bounding faces, edges and vertices of any polyhedron, V − E + F = 2, is a fundamental concept in several branches of mathematics. Obviously, it is important in geometry, but it is also well known in topology, where a similar telescoping sum is known as the Euler characteristic χ of any finite space. The value of χ can also be computed for the unit polyhedra (such as the unit cell, the asymmetric unit or Dirichlet domain) which build, in a symmetric fashion, the infinite crystal lattices in all space groups. In this application χ has a modified form (χm) and value because the addends have to be weighted according to their symmetry. Although derived in geometry (in fact in crystallography), χm has an elegant topological interpretation through the concept of orbifolds. Alternatively, χm can be illustrated using the theorems of Harriot and Descartes, which predate the discovery made by Euler. Those historical theorems, which focus on angular defects of polyhedra, are beautifully expressed in the formula of de Gua de Malves. In a still more general interpretation, the theorem of Gauss–Bonnet links the Euler characteristic with the general curvature of any closed space. This article presents an overview of these interesting aspects of mathematics with Euler's formula as the leitmotif. Finally, a game is designed, allowing readers to absorb the concept of the Euler characteristic in an entertaining way.

Crystal structure and Hirshfeld surface analysis of the hydrated 2:1 adduct of piperazine-1,4-diium 3,5-dinitro-2-oxidobenzoate and piperazine

The crystal structure of the adduct piperazine-1,4-diium 3,5-dinitro-2-oxidobenzoate–piperazine–water (2/1/2) shows the existence of a 3,5-dinitrosalicylate dianion (DNSA2−) and a protonated piperazine-1,4-diium cation (PIP2+) along with a piperazine molecule. The formula of the title adduct in the asymmetric unit is 2C4H12N2 2+·2C7H2N2O7 2−·C4H10N2·2H2O with Z = 1. The piperazine ring in the piperazine-1,4-diium cation and in the neutral piperazine molecule adopt chair conformations. All O atoms in the DNSA2− moiety and the water molecule act as hydrogen-bonding acceptors for various intermolecular O—H...O, N—H...O and C—H...O interactions, which stabilize the crystal structure. Various supramolecular architectures formed by the different intermolecular interactions are discussed. The relative contribution of various intermolecular contacts is analysed with the aid of two-dimensional (full and decomposed) fingerprint plots, indicating that H...O/O...H (50.2%) and H...H (36.2%) contacts are the major contributors to the stabilization of the crystal structure.

Vibrational Dynamics in crystalline 4-(dimethylamino) benzaldehyde: Inelastic Neutron Scattering and Periodic DFT Study

The structure and dynamics of crystalline 4-(dimethylamino) benzaldehyde, 4DMAB, are assessed through INS spectroscopy combined with periodic DFT calculations. The excellent agreement between experimental and calculated spectra is the basis for a reliable assignment of INS bands. The external phonon modes of crystalline 4DMAB are quite well described by the simulated spectrum, as well as the modes involving low-frequency molecular vibrations. Crystal field splitting is predicted and observed for the modes assigned to the dimethylamino group. Concerning the torsional motion of methyl groups, four individual bands are identified and assigned to specific methyl groups in the asymmetric unit. The torsional frequencies of the four methyl groups in the asymmetric unit fall in a region of ca. 190 ± 20 cm−1, close to the range of values observed for methyl groups bonding to unsaturated carbon atoms. The hybridization state of the X atom in X-CH3 seems to play a key role in determining the methyl torsional frequency.

Crystal structure and Hirshfeld surface analysis of 3-methyl-4-oxo-N-phenyl-3,4-dihydroquinazoline-2-carbothioamide

The asymmetric unit of the title compound, C16H13N3OS, comprises two molecules (A and B) with similar conformations that differ mainly in the orientation of the phenyl group relative to the rest of the molecule, as expressed by the Cthioamide—Nthioamide—Cphenyl—Cphenyl torsion angle of 49.3 (3)° for molecule A and of 5.4 (3)° for molecule B. In the crystal, two intermolecular N—H...N hydrogen bonds lead to the formation of a dimer with R 2 2(10) graph-set notation. A Hirshfeld surface analysis revealed that H...H interactions are the most important intermolecular interactions, contributing 40.9% to the Hirshfeld surface.

Crystal structure of a trigonal polymorph of aquadioxidobis(pentane-2,4-dionato-κ2 O,O′)uranium(VI)

The title compound, [UO2(acac)2(H2O)] consists of a uranyl(VI) unit ([O=U=O]2+) coordinated to two monoanionic acetylacetonate (acac, C5H7O2) ligands and one water molecule. The asymmetric unit includes a one-half of a uranium atom, one oxido ion, one-half of a water molecule and one acac ligand. The coordination about the uranium atom is distorted pentagonal–bipyramidal. The acac ligands and Ow atom comprise the equatorial plane, while the uranyl O atoms occupy the axial positions. Intermolecular hydrogen bonding between complexes results in the formation of two-dimensional hexagonal void channels along the c-axis direction with a diameter of 6.7 Å. The monoclinic (P21/c space group) polymorph was reported by Alcock & Flanders [(1987). Acta Cryst. C43, 1480–1483].

Synthesis and structural analysis of cis-bis(1,10-phenanthroline)dicarbonyl ruthenium(II) 1.72-trifluoromethanesulfonate 0.28-hexafluoridophosphate

Ruthenium(II) complexes containing both 1,10-phenanthroline (Phen) and carbonyl (CO) ligands are important molecules for various applications including catalysis. In this work, the molecular structure of [Ru(Phen)2(CO)2]2+ was determined via X-ray diffraction analysis for the first time. The complex exhibits substitutional disorder of one of counter-anions in the asymmetric unit, with different occupancies for CF3SO3- (0.72) and PF6- (0.28). The ruthenium atom is coordinated in a distorted octahedral environment by two carbonyl carbon atoms and four nitrogen atoms from bis-Phen ligands. The cation displays a cis configuration of the carbonyl ligands. Several hydrogen bonds and π-π interactions are present in the crystal. In addition to structural characterization, IR spectral data for the complex is compared with calculated values. These results provide fundamental data for understanding various properties of related ruthenium complexes.

Syntheses, Characterization, and X-ray Crystal Structure of Binuclear Lanthanide Complexes Assembled with Schiff Base and Acetate

Herein we reported the binuclear complexes of the 1-(pyridin-2-yl)-2-(pyridin-2-ylmethylene)hydrazine ligand (HL) [Ln2(HL)2(CH3COO)6].n(H2O) (Ln = Y, Pr, Gd and Er). The binuclear complexes are characterized by IR and physical measurement. Spectroscopic evidence indicated that the Schiff base HL behave an N3 coordination tridentate ligand. The complexes are formulated as [{Ln(1-(pyridin-2-ylmethylidene-kN)-2-(pyridin-2-yl-kN)hydrazine-kN1)(h2-OOCH3)2}{h1:h2:m2-OOCH3}2{Ln(1-(pyridin-2-ylmethylidene-kN)-2-(pyridin-2-yl-kN)hydrazine-kN1))(h2-OOCH3)2}].n(H2O). The structure of the praseodymium complex was elucidated by X-ray diffraction analysis. Suitable crystals were grown by slow evaporation of methanol solution. The asymmetric unit of the compound contains two neutral ligand molecules, two Pr3+ ions, four acetate anions acting in h2-OOCH3 mode, two acetate anions acting in h1:h2:m2-OOCH3 mode, and three uncoordinated water molecules. The praseodymium atom is ten coordinated and the coordination sphere is best described as a distorted bicapped square antiprism. The PrIII···PrIII distance is 4.2777(6) Å and the bridging angle Pr—O—Pr and O—Pr—O are respectively 115.8(3)° and 64.2(3)°. The structure is consolidated by intra and intermolecular hydrogen bond.

(μ-2,2′,2′′,2′′′-{[Pyrazine-2,3,5,6-tetrayltetrakis(methylene)]tetrakis(sulfanediyl)}tetraacetato)bis[aquanickel(II)] heptahydrate

Reaction of the ligand 2,2′,2′′,2′′′-{[pyrazine-2,3,5,6-tetrayltetrakis(methylene)]tetrakis(sulfanediyl)}tetraacetic acid (H4L1), with NiCl2 leads to the formation of a binuclear complex, (μ-2,2′,2′′,2′′′-{[pyrazine-2,3,5,6-tetrayltetrakis(methylene)]tetrakis(sulfanediyl)}tetraacetato-κ5 O,S,N 1,S′,O′:κ5 O′′,S′′,N 4,S′′′,O′′′)bis[aquanickel(II)] heptahydrate, {[Ni2(C16H16N2O8S4)(H2O)2]·7H2O} (I). It crystallizes with two half molecules in the asymmetric unit. The complete molecules are generated by inversion symmetry, with the center of the pyrazine rings being located at crystallographic centres of inversion. The ligand coordinates two NiII ions in a bis-pentadentate manner and the sixfold coordination sphere of each nickel(II) atom (NiS2O3N) is completed by a water molecule. The complex crystallized as a hepta-hydrate. The binuclear complexes are linked by Owater—H...Ocarbonyl hydrogen bonds, forming layers parallel to the (101) plane. This layered structure is additionally stabilized by weak C—H...O hydrogen bonds. Further O—H...O hydrogen bonds involving binuclear complexes and solvent water molecules, together with weak C—H...S hydrogen bonds, link the layers to form a supramolecular framework.

Dichlorido(η6-p-cymene)[tris(4-methoxyphenyl)phosphane]ruthenium(II)

The title compound, [RuCl2(C10H14)(C21H21O3P)], crystallizes with two complex molecules in the asymmetric unit. The RuII atom has a classical three-legged piano-stool environment being coordinated by a cymene ligand [Ru—centroid = 1.707 (2)/1.704 (2) Å], a tris(4-methoxyphenyl)phosphane ligand [Ru—P = 2.3629 (15)/2.3665 (15) Å] and two chloride atoms with the Ru—Cl bonds adopting two distinct values of 2.4068 (16)/2.4167 (16) and 2.4016 (15)/2.4244 (16) Å. The effective cone and solid angles for the phosphane ligands were calculated to be 149.5/150.2° and 25.3/25.6°, respectively. In the crystal, weak C—H...Cl/O/π interactions are observed. The crystal was refined as a two-component twin.

Structural Analysis of Saccharomyces cerevisiae Dihydroorotase Reveals Molecular Insights into the Tetramerization Mechanism

Dihydroorotase (DHOase), a dimetalloenzyme containing a carbamylated lysine within the active site, is a member of the cyclic amidohydrolase family, which also includes allantoinase (ALLase), dihydropyrimidinase (DHPase), hydantoinase, and imidase. Unlike most known cyclic amidohydrolases, which are tetrameric, DHOase exists as a monomer or dimer. Here, we report and analyze two crystal structures of the eukaryotic Saccharomyces cerevisiae DHOase (ScDHOase) complexed with malate. The structures of different DHOases were also compared. An asymmetric unit of these crystals contained four crystallographically independent ScDHOase monomers. ScDHOase shares structural similarity with Escherichia coli DHOase (EcDHOase). Unlike EcDHOase, ScDHOase can form tetramers, both in the crystalline state and in solution. In addition, the subunit-interacting residues of ScDHOase for dimerization and tetramerization are significantly different from those of other DHOases. The tetramerization pattern of ScDHOase is also different from those of DHPase and ALLase. Based on sequence analysis and structural evidence, we identify two unique helices (α6 and α10) and a loop (loop 7) for tetramerization, and discuss why the residues for tetramerization in ScDHOase are not necessarily conserved among DHOases.