Application of coset representations to the construction of symmetry adapted functions

Summary A coset representation (G(/G _i )), which is defined algebraically by a coset decomposition of a finite groupG by its subgroupG _i , is shown to be a method for the decomposition of a regular body into its point group orbits. This proof also shows that each member of theG(/G _i ) orbit belongs to theG _i site-symmetry. In addition, a general equation concerning the multiplicities of such coset representations is derived and shown to involve Brester's equations and thek-value equations of framework groups as special cases. The relationship of the coset representation and the site-symmetry affords a general procedure for obtaining symmetry adapted functions.     

On the Bose-Einstein condensation of free relativistic bosons with or without mass

The Bose-Einstein condensation of free relativistic particles [=(M ² c ⁴ +c ² p ² ) ^1/2 –Mc ² ] is studied rigorously. For massless bosons (=cp), the condensation transition of third (second) order occurs in2 (3) dimensions (D). The molar heat capacity follows the T ² (T ³ ) law below the condensation temperature T_c [k ^B T_c=(2 ² c ² n/1.645) ^1/2 [( ^{2 3} c ³ n/1.202) ^1/3 ], reaches4.38 (10.8) R at T=T_c, and approaches the high-temperature-limit value2 (3) R with no jump (a jump equal to6.75R) in2 (3)D. For finite-mass (M) bosons, the phase transition occurs only in3D with the condensation temperature T_c always smaller than that of the corresponding nonrelativistic bosons [=(2M) ^–1 p ² ]. If the mass M is reduced to zero, the condensation temperature T_c grows monotonically and reaches eventually that of massless relativistic bosons. This mass-dependence of T_c is therefore distinct from the case of nonrelativistic bosons, where T_c grows to infinity as M 0. A brief discussion is given for a possible connection with the normal-to-super transition of the independently moving Cooper pairs (bosons).     

Tetrakis(2-hydroxyphenyl)ethene and derivatives. A structurally preorganized tetradentate ligand system for polymetallic coordination chemistry and catalysis

A topologically unique, conformationally constrained tetradentate ligand system for polymetallic coordination chemistry has been developed: tetrakis(2-hydroxyphenyl)ethene (1a) and substituted derivatives. The design exploits the planarity of the tetraphenylethylene core to impart rigidity to the roughly square oxygen binding array, while maintaining a degree of conformational mobility associated with rotation about the aryl-ethylene carbon-carbon bonds. Tetrakis(2-hydroxyphenyl)ethene derivatives are designed to promote multiple metal bridging over chelating coordination modes. The ligand is synthesized from anisole or 4-tert-butylanisole in four steps via the 2,2'-dimethoxybenzophenone hydrazones 4a,b. The sterically hindered ortho-substituted tetraphenylethylene core is produced in high yield by acid-catalyzed decomposition of the corresponding diaryl diazomethane prepared in situ by oxidation of the hydrazone using nickel peroxide. Deprotection of the methyl ethers using boron tribromide gives tetrakis(2-hydroxyphenyl)ethene (1a), characterized by X-ray crystallography, and tetrakis(5-tert-butyl-2-hydroxyphenyl)ethene (1b). Sterically isolating substituents in the 3-position can be installed via Claisen rearrangement/hydrogenation, providing tetrakis(3-n-propyl-2-hydroxyphenyl)ethene (6) efficiently. To illustrate potential applications of this unprecedented ligand class, two coordination complexes are reported, including tetrakis(2-diethylaluminoxyphenyl)ethene (8), a structurally robust eight-membered-ring aluminum/oxygen crown complex characterized both in solution and in the solid state.     

Mechanism of Mukaiyama-Michael Reaction of Ketene Silyl Acetal: Electron Transfer or Nucleophilic Addition?

Mechanism of Mukaiyama-Michael reaction of ketene silyl acetal has been discussed. The competition reaction employing various types of ketene silyl acetals reveals that those bearing more substituents at the beta-position react preferentially over less substituted ones. However, when ketene silyl acetals involve bulky siloxy and/or alkoxy group(s), less substituted compounds react preferentially. The Lewis acids play an important role in these reactions. Enhanced preference for the more sterically demanding Michael adducts is obtained with Bu(2)Sn(OTf)(2), SnCl(4), and Et(3)SiClO(4) in the former reaction while TiCl(4) gives the highest selectivity for the less sterically demanding products in the latter case. These results are interpreted in terms of alternative reaction mechanisms. The reaction of less bulky ketene silyl acetals are initiated by electron transfer from these compounds to a Lewis acid. On the other hand, bulkier ketene silyl acetals undergo a ubiquitous nucleophilic reaction. Such a mechanistic change is discussed based on a variety of experimental results as well as the semiempirical PM3 MO calculations.     

Sphericity governs both stereochemistry in a molecule and stereoisomerism among molecules

The concept of sphericity and relevant fundamental concepts that we have proposed have produced a systematized format for comprehending stereochemical phenomena. Permutability of ligands in conventional approaches is discussed from a stereochemical point of view. After the introduction of orbits governed by coset representations, the concepts of subduction and sphericity are proposed to characterize desymmetrization processes, with a tetrahedral skeleton as an example. The stereochemistry and stereoisomerism of the resulting promolecules (molecules formulated abstractly) are discussed in terms of the concept of sphericity as a common mathematical and logical framework. Thus, these promolecules are characterized by point group and permutation group symmetry. Prochirality, stereogenicity, prostereogenicity, and relevant topics are described in terms of the concept of sphericity.     

Selective and sensitive fluorometric determinations of cobalt(II) and hydrogen peroxide with fluorescein-hydrazide

Fluorophotometric determinations of cobalt(II) and hydrogen peroxide were investigated by using the fluorescence reaction between fluorescein-hydrazide (fl-NHNH(2)), and/or hydrogen peroxide, cobalt(II), respectively. The calibration graphs were liner in the range of 0-6.0 ng cobalt(II) and 0-1000 ng hydrogen peroxide per 10 ml at an emission wavelength (E(m)) of 530 nm with an exicitation wavelength (E(x)) of 508 nm, respectively. These proposed methods were selective and simple, and the effect of foreign ions was negligible in comparison with conventional reported methods such as nitroso R,4-(2-pyridylazo)resorcinol(PAR), alizarin, pyridine-2-aldehyde-2-pyridinehydrazone, stilbazo-cobalt(II), etc.