Product Integration Rules at Clenshaw-Curtis and Related Points: A Robust Implementation

Product integration rules generalizing the Fej?r, Clenshaw-Curtis,and Filippi quadrature rules respectively are derived for integralswith trigonometric and hyperbolic weight factors. The Chebyshevmoments of the weight functions are found to be given by well-conditionedexpressions, in terms of hypergeometric functions ₀F₁. An a priori error estimator is discussed which is shown bothto avoid wasteful invocation of the integration rule and toincrease significantly the robustness of the automatic quadratureprocedure. Then, specializing to extended Clenshaw-Curtis (ECC) rules,three types of a posteriori error estimates are considered andthe existence of a great risk of their failure is demonstratedby large scale validation tests. An empirical error estimator,superseding them for slowly varying integrands, is found toresult in a spectacular increase in the output reliability. Finally, enhancements in the control of the interval subdivisionstrategy aiming at increasing code robustness is discussed.Comparison with the code DQAWO of QUADPACK, with about a hundredthousand solved integrals, is illustrative of the increasedrobustness and error estimate reliability of our computer codeimplementation of the ECC rules.     

C3-chiral tripodal amido complexes

A comprehensive study into the coordination chemistry of two C3-chiral tripodal amido ligands has been carried out. The amido ligands contain a trisilylmethane backbone and chiral peripheral substituents. The amine precursors. HC(SiMe2NH[(S)-1-phenylethyl]]3 (1) and HC[SiMe2NH[(R)-1-indanyl]]3 (2) were found to be in equilibrium in solution with the cyclic diamines HC[SiMe2N[(S)-1-phenylethyl]2](SiMe2NH-[(S)-1-phenylethyl]] (3) and HC[SiMe2NH[(R)-1-indanyl]][SiMe2NH[(R)-1-indanyl]) (4), which are generated upon ejection of one molecule of the chiral primary amine. Reaction of these equilibrium mixtures with three molar equivalents of butyllithium instantaneously gave the trilithium triamides HC[SiMe2N(Li)[(S)-1-phenylethyl]]3 (5) and HC[SiMe2N(Li)[(R)-1-indanyl]]3 (6), both of which were characterised by an X-ray diffraction study. Both lithium compounds possess a central heteroadamantane core, in which the two-coordinate Li atoms are additionally weakly solvated by the three aryl groups of the chiral peripheral substituents, the Li-C contacts being in the range of 2.65-2.73 A. Reaction of 5 and 6 with [TiCl4(thf)2] and ZrCl4 gave the corresponding amido complexes [TiCl-[HC[SiMe2N[(S)-1-phenylethyl]]3]] (7), [TiCl(HC[SiMe2N[(R)-1-indanyl]]3]] (8), [ZrCl[HC[SiMe2N[(S)-1-phenylethyl]]3]] (9) and [ZrCl[HC[SiMe2N[(R)-1-indanyl]]3]] (10), respectively. Of these, compound 7 was structurally characterised by X-ray structure analysis and was shown to possess a C3-symmetrical arrangement of the tripod ligand. The chiral anionic dinuclear complex [Li-(OEt2)4][Zr2Cl3[HC[SiMe2N[(S)-1-phenylethyl]]3]2] (11) was isolated from reaction mixtures leading to 9. An X-ray diffraction study established its dimeric structure, in which the chiral amido ligands cap the two metal centres, which are linked through three symmetrically arranged, bridging chloro ligands. Reaction of 9 and 10 with a series of alkyl Grignard and alkyllithium reagents yielded the corresponding alkylzirconium complexes. X-ray structure analyses of [Zr(CH3)[HC[SiMe2N[(S)-1-phenylethyl]]3]] (12) and [Zr(CH3)-[HC[SiMe2N)[(R)-1-indanyl]]3]] (20) established their detailed molecular arrangements. While the reaction of 12 with the aryl ketones PhC(O)R (R = CH = CHPh, iPr, Et) gave the corresponding C-O insertion products, which contain an additional chiral centre in the alkoxy group, with low stereoselectivity (0-40% de). The corresponding conversions with several aryl aldehydes yielded the alkoxo complexes with high stereoselectivity. Upon hydrolysis, the chiral alcohols were isolated and shown to have enantiomeric excesses between 68 and 82%. High stereodiscrimination was also observed in the insertion reactions of several chiral ketones and aldehydes. However, this was shown to originate primarily from the chirality of the substrate. In analogous experiments with carbonyl compounds, the ethyl- and butyl-zirconium analogues of 12 did not undergo CO insertion into the metal-alkyl bond. Instead, beta-elimination and formal insertion into the metal-hydride bond occurred. It was found that the elimination of the alkene was induced by     

Iodine determination in biological materials Options for sample preparation and final determination

This paper describes the development of various new acid sample decomposition methods, as well as an extraction (leaching) method and compares them with the “Sch?niger Combustion” technique. The methods have been developed as sample preparation techniques for iodine determination in biological materials, especially in solid samples. ICP-MS (inductively coupled plasma mass spectrometry) and a catalytic technique are employed and discussed for the final determination of iodine concentrations. Accuracy and reliability of the different analytical methods are shown in the examples of different CRMs (certified reference materials) available for iodine. The results of an interlaboratory comparison are specifically presented for the extraction (leaching) method. Received: 11 May 1998 / Revised: 14 June 1998 / Accepted: 16 June 1998     

Thermodynamic Complexation of Dopamine with Zinc(II) in Media with Different Dielectric Constants

The complexation of zinc(II) with dopamine has been investigated by spectrophotometric measurements in mixed solvent system at an ionic strength of 0.2 mol•dm－3 sodium chloride, employed at at (15±0.1), (25±0.1), (35±0.1) ℃ at inin a pH range of ca. 6 to ca. 7 with a high ratio of ligand to metal. The effect of solvent systems on protonation and complexation are was was discussed. Linear relationships are werewere observed by plotting lg K versus 1/D, where K and D show stability and dielectric constants, respectively.     

Theoretical study of new acceptor and donor molecules based on polycyclic aromatic hydrocarbons

Functionalized polycyclic aromatic hydrocarbons (PAHs) are an interesting class of molecules in which the electronic state of the graphene-like hydrocarbon part is tuned by the functional group. Searching for new types of donor and acceptor molecules, a set of new PAHs has recently been investigated experimentally using ultraviolet photoelectron spectroscopy (UPS). In this work, the electronic structure of the PAHs is studied theoretically with the help of B3LYP hybrid density functionals. Using the ΔSCF method, electron binding energies have been determined which affirm, specify and complement the UPS data. Symmetry properties of molecular orbitals are analyzed for a categorization and an estimate of the related signal strength. While σ-like orbitals are difficult to detect in UPS spectra of condensed film, calculation provides a detailed insight into the hidden parts of the electronic structure of donor and acceptor molecules. In addition, a diffuse basis set (6-311++G**) was used to calculate electron affinity and LUMO eigenvalues. The calculated electron affinity (EA) provides a classification of the donor/acceptor properties of the studied molecules. Coronene-hexaone shows a high EA, comparable to TCNQ, which is a well-known classical acceptor. Calculated HOMO-LUMO gaps using the related eigenvalues have a good agreement with the experimental lowest excitation energies. TD-DFT also accurately predicts the measured optical gap.     

Lattice Boltzmann simulation of natural convection in nanofluid-filled 2D long enclosures at presence of magnetic field

In this paper, the effects of a magnetic field on natural convection flow in filled long enclosures with Cu/water nanofluid have been analyzed by lattice Boltzmann method. This study has been carried out for the pertinent parameters in the following ranges: the Rayleigh number of base fluid, Ra = 10³–10⁵, the volumetric fraction of nanoparticles between 0 and 6 %, the aspect ratio of the enclosure between A = 0.5 and 2. The Hartmann number has been varied from Ha = 0 to 90 with interval 30 while the magnetic field is considered at inclination angles of θ = 0°, 30°, 60° and 90°. Results show that the heat transfer decreases by the increment of Hartmann number for various Rayleigh numbers and the aspect ratios. Heat transfer decreases with the growth of the aspect ratio but this growth causes the effect of the nanoparticles to increase. The magnetic field augments the effect of the nanoparticles at high Rayleigh numbers (Ra = 10⁵). The effect of the nanoparticles rises for high Hartmann numbers when the aspect ratio increases. The rise in the magnetic field inclination improves heat transfer at aspect ratio of A = 0.5.