Partition of <Emphasis Type="Italic">π</Emphasis>-electrons between faces of polyhedral carbon aggregates

We report for the trivalent regular and semiregular polyhedra (three Platonic and seven Archimedean carbon polyhedra) the π-electron partition between rings of various sizes based on considering all their resonance structures. It was found that small odd-membered (3-and 5-membered) faces are assigned a lower share of π-electrons than that corresponding to equipartition (i.e., 1/3 of an electron for carbon atoms shared between three rings). In contrast, 4-membered rings obtain a larger share of π-electrons than that corresponding to equipartition.     

介孔Co3O4多面体的制备及电化学性能

通过水热处理Co(NO₃)₂与(NH₄)₂S₂O₈合成了CoOOH多面体, 再经高温煅烧得到具有介孔结构的Co₃O₄多面体; 利用扫描电子显微镜(SEM)、 透射电子显微镜(TEM)、 X射线衍射(XRD)和N2吸附\|脱附实验等手段对其结构和组成进行了表征; 研究了反应时间、 反应温度及(NH₄)₂S₂O₈浓度等因素对CoOOH多面体形貌的影响, 分析了CoOOH多面体的形成机理. 性能测试结果表明, 该介孔Co₃O₄多面体具有良好的葡萄糖电化学检测活性, 检测线性范围为0.05~1.8 mmol/L, 响应灵敏度为148 μA·cm^-2·mmol·L^-1, 检出限为1 μmol/L.     

Truncating and chamfering diagrams of regular polyhedra

Truncating the vertex and chamfering the edges of a polyhedron is the classic way to obtain Archimedean polyhedra taking a Platonic solid as the starting point. We have considered the set of polyhedra obtained by this method and have found singular diagrams that show a special relation between these Archimedean polyhedra.     

An assembling strategy for DNA cages with minimum strands

DNA polyhedra are artificial cage-like architectures based on interlocked and interlinked DNA double-strands. Using fewer strands to construct DNA cages shows an important role in the design of single-stranded DNA molecules. However, construction methods for DNA polyhedra from topological perspective remains not well understood. In this study, we theoretically propose an assembling strategy for DNA polyhedra with minimum strands based on computer algorithm. The results show that this efficient method could search DNA polyhedra with fewer strands faster. Our research provides new insights into design and synthesis for DNA polyhedra with required topological structures.     

COORDINATION BINDING MODES FOR POLYMERIC CADMIUM DICARBOXYLATE HYDRATE COMPLEXES

Abstract

Single crystal X-ray crystal diffraction, IR and TGA-DTA data and searches in the Cambridge Structural Database (CSD) have been used to analyze coordination binding modes of Cd(II) in dicarboxylate complexes with anions of different length. The results, discussed in this paper, show that Cd(II) presents a range of coordination geometries. Six, seven and eight coordination with different degrees of departure from regular polyhedra are observed. The length of the ions influences the availability of donor oxygens and induces bidentate, six or four member chelating rings, and monodentate binding, which produces polymerization of different dimensionality.     

Chemical applications of topology and group theory

This paper characterizesforbidden polyhedra, which are polyhedra with fewer than 9 vertices which cannot be formed using only the 9s,p, andd atomic orbitals. In this connection polyhedra are of particular interest if their symmetry groups are direct product groups of the typeR × C′ _s in whichR is a group containing only proper rotations andC′ _s is eitherC _s orC _i in which the non-identity element is an inversion center or a reflection plane which is called theprimary plane of the groupR ×C′ _s . Using this terminology polyhedra of the following types are shown always to be forbidden polyhedra: (1) Polyhedra having 8 vertices, such direct product symmetry point groups, and either an inversion center or aprimary plane fixing either 0 or 6 vertices; (2) Polyhedra having a 6-fold or higherC _n rotation axis. However, these conditions are not necessary for a polyhedron to be forbidden since in addition to one 7-vertex polyhedron and ten 8-vertex polyhedra satisfying one or both of the above conditions there are two forbiddenC _3v 8-vertex polyhedra which satisfy neither of the above conditions. For part 15 of this series see reference 1.     

On topological form in structures

This paper begins with a review of the Euler relation for the polyhedra and presents the corresponding Schläfli relation in n, the polygonality, and p, the connectivity of the polyhedra. The use of ordered pairs as given by (n, p), the Schläfli symbols, to organize the mapping of the polyhedra and its extension into the two-dimensional (2D) and three-dimensional (3D) networks is described. The topological form index, represented by l, is introduced and is defined as the ratio of the polygonality, n, to the connectivity, p, in a structure, it is given by l = n/p. Next a discussion is given of establishing a conventional metric of length in order to compare topological properties of the polyhedra and networks in 2D and 3D. A fundamental structural metric is assumed for the polyhedra. The metric for the polyhedra is, in turn, used to establish a metric for tilings in the Euclidean plane. The metrics for the polyhedra and 2D plane are used to establish a metric for networks in 3D. Once the metrics have been established, a conjecture is introduced, based upon the metrics assumed, that the area of the elementary polygonal circuit in the polyhedra and 2D and 3D networks is proportional to a function of the topological form index, l, for these structures. Data of the form indexes and the corresponding elementary polygonal circuit areas, for a selection of polyhedra and 2D and 3D networks is tabulated, and the results of a least squares regression analysis of the data plotted in a Cartesian space are reported. From the regression analysis it is seen that a quadratic in l, the form index, successfully correlates with the corresponding elementary polygonal circuit area data of the polyhedra and 2D and 3D networks. A brief discussion of the evident rigorousness of the Schläfli indexes (n, p) over all the polyhedra and 2D and 3D networks, based upon the correlation of the topological form index with elementary polygonal circuit area in these structures, and the suggestion that an Euler–Schläfli relation for the 2D and 3D networks, is possible, in terms of the Schläfli indexes, concludes the paper.     

极性晶体结晶习性的形成机理

将负离子配位多面体生长基元模型应用于对极性晶体结晶习性的研究。从结晶化学角度探讨了晶体中负离子配位多面体的结晶方位与晶体各族晶面显露规律，提出负离子配位多面体在晶体各族晶面上联结的稳定性决定了晶面的生长速率。在不同的生长温度和溶液碱浓度下，负离子配住多面体相互联结构成不同维度的生长基元，而不同维度的生长基元往晶体各族晶面上叠合的速率比例是在变化的，这是导致晶体结晶形态多变性的主要原因。同时提出：如果把PBC模型中的化学键链设定为配位多面体相联结的键链，使得极性晶体结晶习性中难以解释的问题就会迎刃而解，从而使PBC理论模型的应用会得到更进一步的拓宽。     

Reticular chemistry of metal-organic polyhedra

Metal-organic polyhedra (MOPs), are discrete metal-organic molecular assemblies. They are useful as host molecules that can provide tailorable internal volume in terms of metrics, functionality, and active metal sites. As a result, these materials are potentially useful for a variety of applications, such as highly selective guest inclusion and gas storage, and as nanoscale reaction vessels. This review identifies the nine most important polyhedra, and describes the design principles for the five polyhedra most likely to result from the assembly of secondary building units, and provides examples of these shapes that are known as metal-organic crystals.     

Symmetry factoring of the characteristic equations of graphs corresponding to polyhedra

A systematic procedure is described which uses two-and three-fold symmetry elements in graphs to reduce their adjacency matrices to lead to corresponding factorings of their characteristic polynomials. A graph splitting algorithm based on this matrix reduction procedure is described. Applications of these methods to the factoring of the characteristic polynomials of 28 polyhedra with nine or less vertices are given. General expressions for the eigenvalues of prisms, pyramids, and bipyramids in terms of the eigenvalues of their basal or equatorial regular polygons are calculated by closely related matrix methods.     

Multicomponent self-assembly of a nested co(24) @co(48) metal-organic polyhedral framework

A tale of two polyhedra: Two nested Archimedean metal-organic polyhedra, a rhombicuboctahedron (Co(48) cage) and a cuboctahedron (Co(24) cage), have been assembled from two types of cobalt dimers and two complementary ligands. Within the 3D covalent cubic array of outer Co(48) cages and framework lie encapsulated inner Co(24) cages that are linked into a separate "hidden" 3D framework.     

Tetrahedral Fulleroids

Fulleroids are cubic convex polyhedra with faces of size 5 or greater. They are suitable models of carbon molecules. In this paper sufficient and necessary conditions for existence of fulleroids of tetrahedral symmetry types and with pentagonal and n-gonal faces only depending on number n are presented. Either infinite series of examples are found to prove existence or nonexistence is proved using symmetry invariants.     

Stabilizing Metal–Organic Polyhedra (MOP): Issues and Strategies

Metal–organic polyhedra (MOPs) are discrete, metal–organic molecular entities composed of edge‐sharing molecular polygons or connected molecular vertices. Unlike the infinite metal–organic coordination networks popularized by metal–organic frameworks (MOFs), spherical MOPs, also known as nanocages, nanospheres, nanocapsules, or nanoballs, are obtained through the self‐organization of metal–carboxylate or metal–pyridine/pyrimidine links to afford cage‐like nanoarchitectures. MOPs offer much promise as porous materials owing to their well‐defined structures and solution processability. However, these advantages become moot if their poor aqueous stability and/or guest‐removal‐induced aggregation handicaps remain unaddressed. The concise premise of this contribution limits our discussion to the design principles in action behind recent developments in stable carboxylate MOPs. To highlight the structure–property relationships between the structural and compositional features of these metal carboxylate polyhedra, related scientific challenges and state‐of‐the‐art research directions for further exploration are presented in brief.     

A Phosphine Functionalized β-Diketimine Ligand for the Synthesis of Manifold Metal Complexes

A bis(diphenyl)-phosphine functionalized β-diketimine (PNac-H) was synthesized as a flexible ligand for transition metal complexes. The newly designed ligand features symmetrically placed phosphine moieties around a β-diketimine unit, forming a PNNP-type pocket. Due to the hard and soft donor atoms (N vs. P) the ligand can stabilize various coordination polyhedra. A complete series ranging from coordination numbers 2 to 6 was realized. Linear, trigonal planar, square planar, tetrahedral, square pyramidal, and octahedral coordination arrangements containing the PNac-ligand around the metal center were observed by using suitable metal sources. Hereby, PNac-H or its anion PNac⁻ acts as mono-, bi- and tetradendate ligand. Such a broad flexibility is unusual for a rigid tetradentate system. The structural motifs were realized by treatment of PNac-H with a series of late transition metal precursors, for example, silver, gold, nickel, copper, platinum, and rhodium. The new complexes have been fully characterized by single crystal X-ray diffraction, NMR, IR, UV/Vis spectroscopy, mass spectrometry as well as elemental analysis. Additionally, selected complexes were investigated regarding their photophysical properties. Thus, PNac-H proved to be an ideal ligand platform for the selective coordination and stabilization of various metal ions in diverse polyhedra and oxidation states.     

Hierarchical Modulation of Optical Anisotropy Driven by Metal Cation Polyhedra in Fluorooxoborates MIIB4O6F2 (MII=Be,Mg, Pb,Zn, Cd)

The enhancement mechanism of birefringence is very important to modulate optical anisotropy and materials design. Herein, the different cations extending from alkaline-earth to alkaline-earth, d¹⁰ electron configuration, and 6s² lone pair cations are highlighted to explore the influence on the birefringence. A flexible fluorooxoborate framework from AEB₄O₆F₂ (AE=Ca, Sr) is adopted for UV/deep-UV birefringent structures, namely, M^IIB₄O₆F₂ (M^II=Be, Mg, Pb, Zn, Cd). The maximal enhancement on birefringence can reach 46.6 % with the cation substitution from Ca, Sr to Be, Mg (route-I), Pb (route-II), and Zn, Cd (route-III). The influence of the cation size, the stereochemically active lone pair, and the binding capability of metal cation polyhedra is investigated for the hierarchical improvement on birefringence. Significantly, the BeB₄O₆F₂ structure features the shortest UV cutoff edge 146 nm among the available anhydrous beryllium borates with birefringence over 0.1 at 1064 nm, and the PbB₄O₆F₂ structure has the shortest UV cutoff edge 194 nm within the reported anhydrous lead borates that hold birefringence larger than 0.1 at 1064 nm. This work sheds light on how metal cation polyhedra modulate birefringence, which suggests a credible design strategy to obtain desirable birefringent structures by cation control.     

On Structural Features Necessary for Near‐IR‐Light Photocatalysts

In the search for photocatalysts that can directly utilize near‐IR (NIR) light, we investigated three oxides Cu₃(OH)₄SO₄ (antlerite), Cu₄(OH)₆SO₄, and Cu₂(OH)₃Cl by photodecomposing 2,4‐dichlorophenol over them under NIR irradiation and by comparing their electronic structures with that of the known NIR photocatalyst Cu₂(OH)PO₄. Both Cu₃(OH)₄SO₄ and Cu₄(OH)₆SO₄ are NIR photocatalysts, but Cu₂(OH)₃Cl is not. Thus, in addition to the presence of two different CuO_m and Cu′O_n polyhedra linked with Cu?O?Cu′ bridges, the presence of acceptor groups (e.g., SO₄, PO₄) linked to the metal oxygen polyhedra is necessary for NIR photocatalysts.     

次级键与晶体中锑(III)螯合物配位多面体研究

考虑立体活性孤对电子附近次级键配位原子的贡献, 对文献报道的三十个氨基多羧酸锑(III)螯合物的晶体结构中配位多面体描述进行了全面的修正. 配位多面体的几何构型指定采用了单位球内截多面体的两面角判据及其相关的ANVPDA程序. 所有配位多面体几何构型的修正均得到了键价计算的有力支持.     

次级键与晶体中锑(III)螯合物配位多面体研究

考虑立体活性孤对电子附近次级键配位原子的贡献,对文献报道的三十个氨基多羧酸锑(III)螯合物的晶体结构中配位多面体描述进行了全面的修正.配位多面体的几何构型指定采用了单位球内截多面体的两面角判据及其相关的ANVPDA程序.所有配位多面体几何构型的修正均得到了键价计算的有力支持.     

Cooperative Crystallization of Heterometallic Indium–Chromium Metal–Organic Polyhedra and Their Fast Proton Conductivity

Metal–organic polyhedra (MOPs) or frameworks (MOFs) based on Cr³⁺ are notoriously difficult to synthesize, especially as crystals large enough to be suitable for characterization of the structure or properties. It is now shown that the co‐existence of In³⁺ and Cr³⁺ induces a rapid crystal growth of large single crystals of heterometallic In‐Cr‐MOPs with the [M₈L₁₂] (M=In/Cr, L=dinegative 4,5‐imidazole‐dicarboxylate) cubane‐like structure. With a high concentration of protons from 12 carboxyl groups decorating every edge of the cube and an extensive H‐bonded network between cubes and surrounding H₂O molecules, the newly synthesized In‐Cr‐MOPs exhibit an exceptionally high proton conductivity (up to 5.8×10^?2 S cm^?1 at 22.5 °C and 98 % relative humidity, single crystal).     

Structural Principles of Polyhedral Allotropes of Phosphorus

We derive the structural principles of polyhedral allotropes of phosphorus, introducing three distinct families of black phosphorus nanostructures. The predicted tetrahedral, octahedral, and icosahedral phosphorus cages can also be considered as phosphorus fullerenes. Phosphorus cages up to P₈₈₈ are systematically investigated by quantum chemical methods, and their thermodynamic stabilities are compared with the experimentally known allotropic forms of phosphorus. The tetrahedral cages are thermodynamically favored over the octahedral and icosahedral structures, although large octahedral structures become nearly as stable as the tetrahedral ones. The stability trends of the studied polyhedral families can be rationalized on the basis of their structural characteristics. The phosphorus polyhedra can be further stabilized by fitting smaller structures inside larger ones, resulting in multilayered, bulk‐like cages. The synthesis of the predicted black phosphorus nanostructures is suggested to be viable from the thermodynamic point of view, and several approaches for their experimental preparation can be envisaged.