Accurate and stable numerical Hartree–Fock calculations for atoms. I. The 1s2 ground state of H−, He,Li+, and Be++

New techniques have been developed for atomic self-consistent-field calculations by numerical integration. For the origin and tail regions we present analytical expansions which can represent the solutions to high accuracy. For the numerical integration in the central region a five-point generalization of the Numerov formula is used; the error term is of the order h¹⁰. While this formula is unstable if used in the customary way, stability is achieved by using a Gaussian elimination technique. The new procedures are tested on the ground state of the helium isoelectronic series; with 251 integration points all quantities are calculated with an inherent accuracy of better than 10^–11.     

Cyclotrimerisierung von Ferrocenderivaten: Triferrocenyl-benzole

Zusammenfassung Cyclotrimerisierung von Ferrocenylacetylen und seinen Derivaten führt unter verschiedenen Bedingungen ausschließlich zu 1,2,4-Triferrocenyl-benzolen. Davon ist besonders die glatte Trimerisierung bei der rein thermischen Behandlung von Ferrocenyl-bzw. Methylferrocenylacetylenen bemerkenswert. Optisch aktives (-Methylferrocenyl)-acetylen liefert dabei ein stark aktives 1,2,4-Tri-(-methylferrocenyl)-benzol. Aus Ferrocenyl-methyl-bzw. Ferrocenyl-phenyl-acetylen erhält man bei Verwendung von Co₂(CO)₈ als Katalysator die sterisch stark überladenen (unsymm.) Trimethyl- bzw. Triphenyl-triferrocenyl-benzole.Die recht starren Konformationen der unsymm. Benzolderivate sowie des aus Acetylferrocen zugänglichen 1,3,5-Triferrocenyl-benzols lassen sich aus ihrenNMR- undUV-Spektren, sowie aus derORD des optisch aktiven Trimeren ableiten.Polycyclotrimerisierung von 1,1-Diacetylferrocen ergibt ein thermisch ziemlich stabiles, strukturell interessantes Polymeres.

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Cyclotrimerization of ferrocenyl acetylene and its derivatives under various conditions yields exclusively 1.2.4-triferrocenylbenzenes. Therefrom, the smooth trimerization upon purely thermal treatment of ferrocenyl- and methylferrocenyl-acetylenes seems especially remarkable. Optically active (-methylferrocenyl)-acetylene thereby affords a strongly active 1.2.4-tri-(-methylferrocenyl)-benzene. From ferrocenyl-methyl- and ferrocenyl-phenyl-acetylene, resp., employing Co₂(CO)₈ as a catalyst, sterically overcrowded (unsymmetrical) trimethyl- and triphenyl-triferrocenylbenzenes are obtained.The rather rigid conformations of the unsymmetrical benzene derivatives and of 1.3.5-triferrocenylbenzene (accessible from acetylferrocene) can be deduced from theirNMR- andUV-spectra, as well as from theORD of the optically active trimer.Polycyclotrimerization of 1.1-diacetylferrocene affords a thermically rather stable, structurally interesting polymer.

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Mit 3 Abbildungen

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Herrn Prof. Dr.J. W. Breitenbach zum 60. Geburtstag gewidmet.

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38. Mitt.:H. Falk undG. Haller, Mh. Chem.99, 279 (1968); die vorliegende Arbeit zählt unter dem Titel Cyclotrimerisierung von Ferrocenylacetylenen als 8. Mitt. über Ferrocen-Acetylene. 7. Mitt. s.G. Haller undK. Schlögl, Mh. Chem.98, 603 (1967).     

Aktivit?t und Selektivit?t der Palladiumkatalysatoren bei der Hydrierung isomerer n-Octine

Ohne Zusammenfassung     

A method for finding the roots of a polynomial

Summary Two variants of a method for finding the roots of a polynomial are described. A proof of the method for a general polynomial with complex coefficients is not based on neighbourhoods of saddle points which are sufficiently small. The speed of convergence is guaranteed since the method is a modification of the downhill method and since it can be used in combination with an arbitrary method which quickly converges in practice, but whose convergence cannot be proved.     

Comparison of four cholesterol‐based stationary phases for the separation of steroid hormones

The chromatographic behavior of steroid hormones on four cholesterol‐bonded stationary phases with different structures in binary methanol/water mobile phases was studied. Of the stationary phases tested, the commercially available stationary phases Cogent UDC cholesterol? and COSMOSIL cholester? provided better separations of steroid hormones in comparison to homemade aminocholesterol and diaminocholesterol stationary phases. The results show that the temperature has a significant influence on the retention and selectivity for steroid hormones separation. The temperature increase may cause changes in the elution order. From the dependences of the retention (ln k) on temperature (1/T), the standard partial molar enthalpy and standard partial molar entropy were calculated and their enthalpic and entropic contributions to the retention were compared. The enthalpic effects principally control the retention mechanism.     

Small-slope simulation of acoustic backscatter from a physical model of an elastic ocean bottom

An underwater acoustic experiment with a two-dimensional rough interface, milled from a slab of PVC, was performed at a tank facility. The purpose was to verify the predictions of numerical models of acoustic rough surface scattering, using a manufactured physical model of an ocean bottom that featured shear effects, nonhomogeneous roughness statistics, and root-mean-square roughness amplitude on the order of the acoustic wavelength. Predictions of the received time series and interface scattering strength in the 100-300 kHz band were obtained from the Bottom Reverberation from Inhomogeneities and Surfaces-Small-Slope Approximation (BORIS-SSA) numerical scattering model. The predictions were made using direct measurements of scattering model inputs-specifically, the geoacoustic properties from laboratory analysis of material samples and the grid of surface heights from a touch-trigger probe. BORIS-SSA predictions for the amplitude of the received time series were shown to be accurate with a root-mean-square residual error of about 1 dB, while errors for the scattering strength prediction were higher (2-3.5 dB). The work is part of an ongoing effort to use physical models to examine a variety of acoustic scattering and propagation phenomena involving the ocean bottom.     

Wide scattered spaces and morasses

We show that it is relatively consistent with ZFC that ω² is arbitrarily large and every sequence s=〈sα:α<ω₂〉 of infinite cardinals with sα?ω² is the cardinal sequence of some locally compact scattered space.