Projektive Punktskalen und Summationsformeln für Tschebyscheff-Polynome

Zusammenfassung. Mit Hilfe der projektiven Geometrie werden neuartige Summationsformeln hergeleitet, die die Tschebyscheff-Polynome und modifizierte Polynome enthalten; z. B. gilt für alle : Daraus ergeben sich wiederum interessante Summationsformeln für die Hyperbelfunktionen; es gilt z. B. für positive reelle Zahlen t: Eingegangen am 28.02.96, revidierte Fassung am 25.08.97 / Angenommen am 09.10.97     

Modelling subsurface coupled chemical reactions and fluid flow over long time periods

Simulation of coupled chemical reactions and fluid flow in porous sedimentary basins over long time periods is a numerical challenge. Most models representing such a physical problem are solved as PDEs where efficient timestepping with controlled error is difficult. We use the differential algebraic equation system approach where robust adaptive timestepping algorithms are available in the solvers, e.g., RADAU5 and DASSL. Mathematical and numerical models for coupled chemical reactions and fluid flow are derived. The models have several interesting properties, e.g., strong nonlinearities and stiffness, which are discussed. We test the performance of our code. This revised version was published online in June 2006 with corrections to the Cover Date.     

Synchronized picosecond pulse generation with a synchronously pumped dye laser and a compressed Nd: YAG laser for non-degenerate transient grating experiments

We have set up a laser system which simultaneously provides synchronized picosecond pulses in the visible and at a wavelength of 1.06 m with a repetition rate of 76 MHz. The set-up consists of a dye laser synchronously pumped by the second harmonic of a cw mode-locked Nd:YAG laser and a fiber-grating compressor for the fundamental wavelength of the Nd;YAG laser. Crosscorrelation measurments reveal the time jitter between the two pulse trains to be less than 10 ps. As a first application we have performed non-degenerate transient grating experiments in semiconductors. The non-degenerate technique allows to use excitation energies well above the bandgap energy and to separate non-linear refractive index effects from photoinduced absorption or transmission changes.     

Syntheses of a Cp'Re=S derivative and more complex products

The reaction of Cp'ReCl(2)S(3) (Cp' = Me(4)EtC(5)) with slightly less than 2 equiv of a phosphine reagent results in the formation of [Cp'Re(Cl)(2)(mu-S)](2), 2, which has been characterized by an X-ray diffraction study. Reactions of 2 with nucleophiles did not lead to monomeric derivatives of the type Cp'ReS(Cl)(2)(Nuc). The reaction of Cp'ReCl(2)(SC(2)H(4)S) with (Me(3)Si)(2)S resulted in the formation of three new products: Cp'ReS(SC(2)H(4)S), 4; Cp'Re(S(3))(SC(2)H(4)S), 5; and a tetranuclear derivative, [(Cp'Re)(2)(mu-S)(mu,eta(2)-SC(2)H(4)S)(mu,eta(1)-SC(2)H(4)S](2)Cl(2), 6. Complexes 4 and 6 have been characterized by X-ray diffraction studies. The electrochemical properties of the mononuclear Re=S derivative, 4, are compared with those of Re=O and Re=NR analogues.     

Synthesis and structure of trigonal and tetragonal connectors for a "Tinkertoy" construction set

We describe a convergent synthesis of eight 1,3,5- and 1,2,4,5-substituted benzene derivatives with long rigid arms containing 4-pyridyl, 2,2'-bipyridyl, and 2,2'-bipyrimidyl termini, meant to be used as trigonal or tetragonal connectors for the construction of large molecular structures. The synthesis involved copper-free Pd-mediated coupling of terminal acetylenes to aryl halides. First, one of the termini of 1,3-diethynylbicyclo[1.1.1]pentane was coupled with a brominated aza heterocycle, and second, 3 equiv of the resulting extended arm were coupled with 1,3,5-triiodobenzene or 4 with 1,2,4,5-tetraiodobenzene. An improved large-scale synthesis for 1,3-diethynylbicyclo[1.1.1]pentane is described. The structures of two of the arms were determined by single-crystal X-ray analysis. Several long molecular rods with 4-pyridyl termini were obtained as byproducts, and a single-crystal X-ray structure is reported for the shortest of these.     

Reactions of dithiolate ligands in mononuclear complexes of rhenium(V)

The thermal reactions of the Re(V) dithiolate complex Cp'ReCl2(SCH2CH2S), 1 (where Cp' = EtMe4C5), and related derivatives have been studied. When 1 is heated in toluene in a sealed evacuated tube at 100 degrees C, a dehydrogenation reaction occurs to form a new rhenium complex with a dithiolene ligand, Cp'ReCl2(SCHCHS), 6, in ca. 40% yield. The structure of 6 has been confirmed by an X-ray diffraction study. Under the thermal conditions studied, 1 also undergoes an olefin extrusion reaction. Free ethene is detected in the NMR spectrum of the products, and insoluble rhenium products are also formed. When 1 is reacted with excess ethene under mild conditions, a new organic product, 1,4-dithiane, is formed. Complex 1 is also found to react with oxidants, such as O2 and S8, under mild conditions to form the dehydrogenation product 6. Kinetic studies of the thermal reaction of 1 and related derivatives have been completed, and possible mechanisms for the thermally induced dehydrogenation reaction are discussed.     

Covalent chemistry and conformational dynamics of topologically chiral amide-based molecular knots

The readily available in gram quantities tris(allyloxy)knot of the amide-type 5 (knotane) can be completely and partially deprotected with nBu(3)SnH in the presence of a palladium catalyst resulting in hydroxyknotanes 7-9. These, in turn, react with diethylchlorophosphate giving rise to knotanes equipped with between one and three phosphoryl groups. Sulfonylation of bis(allyloxy)monohydroxyknotane 8 with p-toluenesulfonyl chloride and, following removal of one or two allyl groups from the intermediate monosulfonate 13, give rise to sulfonyloxy-allyloxy-hydroxy- and sulfonyloxy-dihydroxy-knotanes 15 and 14, respectively. This provides a convenient method for the preparation of knotanes with any substitution pattern. All new knotanes have been isolated in preparative amounts and as highly pure substances with an exception of allyloxy-dihydroxyknotane 9. This compound could only be obtained as a mixture with the corresponding monohydroxy-derivative 8. The structures of all synthesized compounds were established by means of FAB and MALDI TOF mass spectrometry, (1)H and (31)P NMR spectroscopy. The triphosphorylated knotane 10 exhibits high solubility in alcohols, allowing its complete enantiomeric resolution with a commercially available chiral HPLC column. (1)H,(1)H DQF-COSY correlation spectroscopy along with H/D exchange experiments and ab initio calculations provided the first detailed (1)H NMR signal assignments of knotanes in [D(6)]DMSO solution. The combination of variable temperature (1)H and (31)P NMR spectroscopy and molecular modeling has been applied to study the conformational behavior of the new knotanes in different solvents. It has been shown that in DMSO solution at room temperature knotanes exist in a relatively rigid nonsymmetrical conformation similar to that found in the solid state while faster conformational exchange leading to the average D(3) symmetrical structure was detected in a number of other solvents.     

Determination of dissociation constants of 99mTechnetium radiopharmaceuticals by capillary electrophoresis

Capillary electrophoresis was applied to investigate pKa values of 99mTc radiotracers used in nuclear medicine. Therefore, the protonation equilibria of the carboxyl groups of 99mTc-mercaptoacetylglycylglycylglycine (99mTc-MAG3) and 99mTc-ethylenecysteine dimer (99mTc-EC) were studied by pH-dependent determination of electrophoretical velocities. 99mTc-ethylenecysteine dimer diethyl ester (99mTc-ECD) was used as a non-protonable standard. The capillary electrophoresis system was equipped with a radioactivity detector. Measurements were performed using a pressure-driven capillary zone electrophoresis which allowed runs even in the low pH range. For the determination of pKa values, the electrophoretical velocities of the analytes were referred to the electrophoretical velocities of tetraphenyle arsonium chloride as a positively charged marker. Calculation of pKa values was accomplished by non-linear curve fitting of both structure-based equilibria equations and sigmoidal decay functions to the experimental data. 99mTc-MAG3 was shown to have a carboxyl group pKa value of 4.22. The value for the carboxyl groups of 99mTc-EC is 2.90 (determined by structure-based equilibria equations), which represents a common value for both carboxyl groups. By the use of sigmoidal functions, similar values were elucidated. As expected, 99mTc-ECD shows no protonation step.