Tétraèdre

Résumé

thumb|Un tétraèdre. thumb|Paul Sérusier, Tétraèdres, vers 1910. En géométrie, les tétraèdres (du grec tétra : quatre) sont des polyèdres de la famille des pyramides, composés de triangulaires, et . Propriétés combinatoires Le 3-simplexe est la représentation abstraite du tétraèdre ; dans ce modèle, les arêtes s'identifient aux 6 sous-ensembles à 2 éléments de l'ensemble des quatre sommets, et les faces aux 4 sous-ensembles à 3 éléments. Chaque sommet d'un tétraèdre est relié à tous les autres par une arête, et de même chaque face est reliée à toutes les autres par une arête. Ces caractéristiques sont rares : seulement deux polyèdres possédant la première propriété ont été découverts : le tétraèdre et le polyèdre de Császár, qui a , triangulaires et ; de même, seulement deux polyèdres possédant la seconde propriété ont été découverts, le tétraèdre et le polyèdre de Szilassi, qui a 14 sommets, 7 faces hexagonales et 21 arêtes ; les polyèdres de Császár et de Szilassi so

Source officielle

https://fr.wikipedia.org/wiki/T%C3%A9tra%C3%A8dre

À propos de ce résultat

Cette page est générée automatiquement et peut contenir des informations qui ne sont pas correctes, complètes, à jour ou pertinentes par rapport à votre recherche. Il en va de même pour toutes les autres pages de ce site. Veillez à vérifier les informations auprès des sources officielles de l'EPFL.

Publications associées (29)

Publications associées

Chargement

Personnes associées

Chargement

Unités associées

Chargement

Concepts associés (96)

Concepts associés

Chargement

Cours associés (25)

Cours associés

Chargement

Séances de cours associées (43)

Afficher plus

Séances de cours associées

Chargement

Properties of Ir4 clusters in the gas phase and on oxide substrates

Vladan Stevanovic

Results of a theoretical study on the properties of Ir4 clusters in the gas–phase and on oxide surfaces are presented. The work is based on density functional theory (DFT) within the generalized gradient approximation (GGA) and ultrasoft pseudopotentials. Properties of a small particle such as Ir4 cluster are entirely determined by its geometry. The already known result that the most stable form of Ir4 in the gas–phase is the square structure which is significantly more stable than the butterfly and tetrahedron is confirmed. This result is in contradiction with experiments which indicate that the oxide supported Ir4 adopts a tetrahedral configuration. It is shown in this thesis that the chemical environment has a strong influence on the relative stability of Ir4 clusters. On MgO(100) surface, the square isomer remains the most stable Ir4 structure, well separated in energy from the other two. Moreover, the tetrahedron is heavily distorted by the interaction with the surface oxygen. Presence of point defects (neutral and charged O vacancies) affects the energy ordering making tetrahedron and square very close in energy, but the structural distortion of the tetrahedron is even bigger and the predicted data do not correspond to experiments. On TiO2(110) the tetrahedron and square structures become degenerate and the butterfly becomes the least stable isomer. Moreover, structural distortions are very small, in agreement with experimental data. It is shown that the TiO2 surface influences the relative stability of the three isomers through a particularly strong electrostatic field. Interactions of Ir4 with H, C and O atoms as well as with CO molecules have been studied. Adsorption of a single C atom strongly influences the relative stability of the three isomers. Upon C adsorption, the butterfly becomes the most stable gas–phase isomer while on both surfaces the tetrahedron is the most probable structure. Adsorption of a single H or O atom does not produce the same effect. The interaction with CO molecules is also important given the experimental procedure used for producing supported Ir4 clusters. It is shown that on MgO(100), CO dissociation is as probable as the competing process CO desorption justifying the presence of carbon adatoms on Ir4 clusters which brings theoretical predictions in better agreement with experimental data.

EPFL2009

Chargement

Described are the synthesis and characterization of two, potentially tetradentate, N2S2 Schiff-base ligands, containing a disulfide bond, N,N'-bis(3-phenylprop-2-en-l-ylidene)-2,2'-disulfanediyldianiline (L-1) and N,N'-bis(3,3-diphenylprop-2-en-1-ylidene)-2,2'-disulfanediyldianiline (L-2), and their reaction with Zn2+. Surprisingly, both L-1 and L-2 undergo reductive disulfide bond scission upon reaction with Zn2+ in alcoholic media to give, under alcohol oxidation, the respective Zn(NS)2 complexes Zn(L-3)(2) (1) and Zn(L-4)(2) (2), where the L-3 and L-4 are the respective bidentate thiolate-imine anions. The ligands L-1 and L-2 and the complexes I and 2 have been characterized spectroscopically, and the crystal and molecular structures of the two complexes have been determined by single crystal X-ray diffraction. The coordination geometry around Zn(II) centers in both complexes is a distorted tetrahedron. In addition, DFT calculations (B3LYP/LANL2DZ/6-311++G(d,p)) support the structure of I. Cyclic voltammetric studies demonstrate that Zn(II) shifts the reduction potential of the disulfide ligands L-1 and L-2 to less negative values thus making them more susceptible to reductive cleavage of the disulfide bond. The results of semi-empirical PM6 calculations offer key insight into the nature of the transition state for this reaction. (C) 2014 Elsevier Ltd. All rights reserved.

Elsevier2014

Chargement

Structural elucidation and antibacterial activity of new dialkylstannyl- and chlorodialkylstannyl(IV) 3,4-dihydroisoquinoline-2-(1H)-carbodithioate

In extension with growing concern to develop new antimicrobial organometallic drugs, an attempt has been made to synthesize five new antimicrobial organotin(IV) carbodithioates with general formula, Me2SnClL (1), Me2SnL2 (2), Et2SnClL (3), Et2SnL2 (4), Bu2SnClL (5), Bu2SnL2 (6), (where L = Sodium 3,4-dihydroisoquinoline-2-(1H)-carbodithioate). Compounds have been synthesized by refluxing organotin(IV) chlorides with ligand in dry toluene for 7-8 h. Compound 1, 3 and 5 exhibited Penta coordination, while 2, 4 and 6 showed hexacoordination in solid state, as evident from the difference obtained between symmetric and asymmetric CSS stretches in IR spectra. Solid-state structures of 1 was further attested by single crystal analysis as distorted trigonal bipyramidal geometry. H-1-, C-13- and Sn-119- NMR spectroscopy was used to assess the geometry of the compounds in solution state. It was observed that 1, 3, 4 and 5 completely dissociated in solution and showed tetrahedral geometry, whereas 2 and 6 exhibit pentacoordinate geometry after partial dissociation in solution. The pronounced activity was observed for all the compounds against five different strains of bacteria. (C) 2021 Elsevier B.V. All rights reserved.

ELSEVIER2021

Chargement

Properties of Ir4 clusters in the gas phase and on oxide substrates

Vladan Stevanovic

EPFL2009