Geometric Algebra in Linear Algebra and Geometry

This article explores the use of geometric algebra in linear and multilinear algebra, and in affine, projective and conformal geometries. Our principal objective is to show how the rich algebraic tools of geometric algebra are fully compatible with and augment the more traditional tools of matrix algebra. The novel concept of an h-twistor makes possible a simple new proof of the striking relationship between conformal transformations in a pseudo-Euclidean space to isometries in a pseudo-Euclidean space of two higher dimensions. The utility of the h-twistor concept, which is a generalization of the idea of a Penrose twistor to a pseudo-Euclidean space of arbitrary signature, is amply demonstrated in a new treatment of the Schwarzian derivative.     

Reactions of lithiated 2,5-dimethylazaferrocene with selected electrophiles

Lithiation of 2,5-dimethylazaferrocene 1 with sec-BuLi/TMEDA in THF at −78 °C proceeds (as shown by quenching with D₂O) to comparable extent on the methyl groups and the Cp ring. However, the outcome of the reaction of the lithiated 1 with an electrophile depends on the nature of this electrophile. In the reaction with 4-methoxybenzaldehyde only the product originated from the lateral lithiation is formed, whereas the reaction with 4-methoxyacetophenone and 4,4′-dimethoxybenzophenone afforded mixtures of the products resulting from lateral and ring-lithiation. Similar results were also obtained in the reaction of lithiated 1 with chlorodiphenylphosphine and diphenyl diselenide. On the other hand, the exclusive formation of the Cp-substituted product was observed in the reaction of lithiated 1 with N,N-dimethylformamide. The structures of selected products (oily compounds were transformed into the corresponding crystalline W(CO)₅-complexes) were confirmed by X-ray diffraction. The presented reactions open a novel entry to specifically substituted azaferrocenes (especially those containing heteroatom substituents) with potential applications as ligands for the homogenous catalysis.     

Structure and Vibrational Spectra of 1:1 Chloranilic Acid (CLA)—Tetramethylpyrazine (TMP) Complex

In searching for new molecular materials the crystals of 1:1 complexes of tetramethylpyrazine (TMP) with chloranilic acid (CLA) were grown. In the crystalline lattice of TMP.CLA infinite chains of hydrogen bonded components are formed. Very strong (d_(O·sN) = 2.692(2) {?}) hydrogen bonds without proton transfer are present. The components of the complex form stacks of parallel arrangement with the distance typical of van der Waals interaction. One can conclude that in packing, interactions via hydrogen bonds play a decisive role. In the IR spectrum one observes a broad, continuous absorption with well resolved trio typical of strong hydrogen bonds. A detailed analysis of IR and Raman spectra in the low frequency region related to deformation vibrations of CH₃ groups shows that strong interaction via hydrogen bonding only slightly affects the dynamics of these groups.     

Influence of redox properties on the activity of iron oxide catalysts in dehydrogenation of propane with CO2

Different Fe-containing catalysts (pure Fe₂O₃, Fe₂O₃ supported on active carbon or g-Al₂O₃, and hydrotalcite derived Mg-Fe oxides) were examined in the dehydrogenation of propane performed in an Ar or CO₂ atmosphere at 873 K. A promoting effect of carbon dioxide was found for the Fe₂O₃ and Fe₂O₃/AC samples. The catalytic results are discussed in terms of redox properties of the catalysts determined by temperature-programmed reduction (TPR). This revised version was published online in August 2006 with corrections to the Cover Date.     

Spectrum analysis of U-doped LaBr3 single crystals. Part 1: crystal-field analysis

Single crystals of U³⁺:LaBr₃ were grown by the Bridgman-Stockbarger technique. High-resolution polarized absorption spectra of the crystals were recorded at 4.2 K in the 4000-50,000 cm⁻¹ range. Sixty-four experimental crystal-field energy levels of the U³⁺ ion were fitted to a semiempirical Hamiltonian employing free-ion, one-electron crystal-field as well as two-particle correlation crystal-field (CCF) operators with an r.m.s. deviation of 28 cm⁻¹. The performed analysis of the spectra enabled the determination of crystal-field parameters and assignment of the observed 5f³→5f³ transitions. The effects of selected CCF operators on the splitting of some specific U³⁺ multiplets have been investigated and the obtained values of Hamiltonian parameters are discussed and compared with those reported in previous analyses.     

Stereoselective peterson alkoxycarbonylmethylenation reaction of substituted cyclohexanones

In contrast to previous results the Peterson olefination reaction of substituted cyclohexanones with metallotrimethylsilylacetates (1) affords moderate to high stereoselectivity.     

The first structurally characterized nonorganometallic titanium(III) alkoxo-bridged dinuclear complexes

The reaction of [Ti4(OMe)14Cl2] (1) with an excess of AlMe3 gave the cocrystallite [Ti2(mu-OMe)2(mu-Cl)Cl3(thf)3].[Ti2(mu-OMe)3Cl3(thf)3] (2.3) species in a 1:1 ratio. Similar to 2, [Ti2(mu-OEt)2(mu-Cl)Cl3-(thf)3] (4) was obtained in the reaction of an equimolar mixture of TiCl4 and Ti(OEt)4 with Al/AlMe3. The short distance [2.543(1)av A in 2.3 and 2.599(1) A in 4] between "Ti(+3)" atoms, their diamagnetism, and ELF analysis indicate the presence of a Ti-Ti bond.     

Phlogophites intercalated with Al2O3 pillars and modified with transition metals as catalysts of the DeNOx process

Natural phlogophite, pre-treated with acids and intercalated with alumina pillars, was used as catalytic support. Transition metals (Fe, Cu) were deposited on the surface of the modified clay materials by an ion-exchange method. The obtained samples were characterized with respect to structure (XRD), texture (BET), composition (EPMA) and chemical nature of the deposited transition metals species (UV-vis-DRS). The phlogophite based materials have been found to be active and selective catalysts of the DeNOx process. The Fe-containing samples were catalytically active at lower temperatures than the clays modified with copper. A competitive ammonia oxidation by oxygen decreased the effectiveness of the DeNOx process in the high temperature range.     

Direct calculation of lattice sums. A method to account for the crystal field effects

A method of direct calculation of lattice sums in three-dimensional crystals is reported. The method is based on annihilation of some lowest multipole moments of the unit cell by a redefinition of the unit cell content. As a result, properties of the infinite crystal can be calculated as usual by taking a finite cluster of unit cells, but surrounded by an additional surface layer of a charge density (e.g., a layer of point charges). This charge density distribution produces the electric field approximating that one of the rest of the infinite crystal. The method proposed is easily applicable in the SCFLCAO procedure as well as in any method using a cluster representation for an infinite crystal. The validity of the infinite crystal model for a finite crystal is also discussed.     

Matrix-isolation FT-IR and DFT theoretical studies of the intramolecular hydrogen bonding in Mannich bases

FT-IR Ar-matrix isolated spectra were studied for dichloro- (Cl₂-MB) and tetrachloroderivatives (Cl₄-MB) of the ortho Mannich base. The spectra were analyzed based on the DFT calculated frequencies and intensities and compared with those recorded in CCl₄ solution in the region of the ν(OH) and ν(OD) vibrations. The matrix-isolated spectra are characterized by narrower ν(OH) and ν(OD) bands with much better resolved fine structure than in solution. The fine structure originates from the anharmonic coupling with the low frequency modes as well as from Fermi resonance. The ν(OD) band shapes can be reproduced exclusively by assuming the Fermi resonance with overtones and summation of the frequencies of modes into which the bridge atoms are involved. The frequency isotopic ratio (ISR) is for both compounds 1.33 while the half-width ratios are equal to 1.82 and 1.94, for Cl₂-MB and Cl₄-MB, respectively.