Congruence criteria for finite subsets of complex projective and complex hyperbolic spaces

We investigate congruence classes and direct congruence classes of m-tuples in the complex projective space ℂP ⁿ . For direct congruence one allows only isometries which are induced by linear (instead of semilinear) mappings. We establish a canonical bijection between the set of direct congruence classes of m-tuples of points in ℂP ⁿ and the set of equivalence classes of positive semidefinite Hermitean m×m-matrices of rank at most n+1 with 1's on the diagonal. As a corollary we get that the direct congruence class of an m-tuple is uniquely determined by the direct congruence classes of all of its triangles, provided that no pair of points of the m-tuple has distance π/2. Examples show that the situation changes drastically if one replaces direct congruence classes by congruence classes or if distances π/2 are allowed. Finally we do the same kind of investigation also for the complex hyperbolic space ℂH ⁿ . Most of the results are completely analogous, however, there are also some interesting differences. Received: 15 January 1996     

The shape invariant of triangles and trigonometry in two-point homogeneous spaces

We define a fourth basic invariant, which, besides the lengths of the three sides of a triangle, determines a triangle in the complex and quaternion projective spaces P ⁿ and P ⁿ (n2) uniquely up to isometry. We give inequalities describing the exact range of the four basic invariants. We express the angular invariants of a triangle with our basic invariants, giving a new completely elementary proof of the laws of trigonometry. As a corollary we derive a large number of congruence theorems. Finally we get, in exactly the same way, the corresponding results for triangles in the complex and quaternion hyperbolic spaces H ⁿ and H ⁿ (n2).     

A natural 15N approach to determine the biological fixation of atmospheric nitrogen by biological soil crusts of the Negev Desert

Biological soil crusts are important cryptogamic communities covering the sand dunes of the north-western Negev. The biological crusts contain cyanobacteria and other free-living N(2)-fixing bacteria and are hence able to fix atmospheric nitrogen (N). This is why they are considered to be one of the main N input pathways into the desert ecosystem. However, up to now, in situ determinations of the N(2) fixation in the field are not known to have been carried out. We examined the natural (15)N method to determine the biological N(2) fixation by these soil crusts under field conditions. This novel natural (15)N method uses the lichen Squamarina with symbiotic green algae--which are unable to fix N(2)--as a reference in order to determine N(2) fixation. Depending on the sampling location and year, the relative biological fixation of atmospheric nitrogen was estimated at 84-91% of the total N content of the biological soil crust. The cyanobacteria-containing soil lichen Collema had a fixation rate of about 88%. These fixation rates were used to derive an absolute atmospheric N input of 10-41 kg N ha(-1) year(-1). These values are reasonable results for the fixation of atmospheric N(2) by the biological crusts and cyanolichens and are in agreement with other comparable lab investigations. As far as we are aware, the results presented are the first to have been obtained from in situ field measurements, albeit only one location of the Negev with a small number of samples was investigated.     

Surface modification of highly dispersed rubber fillers and pigments by titanate proadhesive and hydrophobic compounds

A method of modifying silicate and carbonate fillers with titanate coupling agents and proadhesive agents was worked out. The modification aimed at hydrophobization of filler surface by introduction to the surface of functional groups with chemical affinity to polymers. Optimum amounts of modifying substances and appropriate conditions for performing the modification process were established. The obtained fillers showed a high degree of hydrophobicity.The modified fillers were applied in rubber mixtures based on butadienestyrene or natural rubber, in polyurethanes, in PVC, and as pigments in oil dyes of high corossive resistance.     

Binding N2, N2H2, N2H4, and NH3 to transition-metal sulfur sites: modeling potential intermediates of biological N2 fixation

In the quest for low-molecular-weight metal sulfur complexes that bind nitrogenase-relevant small molecules and can serve as model complexes for nitrogenase, compounds with the [Ru(PiPr(3))('N(2)Me(2)S(2)')] fragment were found ('N(2)Me(2)S(2)'(2-)=1,2-ethanediamine-N,N'-dimethyl-N,N'-bis(2-benzenethiolate)(2-)). This fragment enabled the synthesis of a first series of chiral metal sulfur complexes, [Ru(L)(PiPr(3))('N(2)Me(2)S(2)')] with L=N(2), N(2)H(2), N(2)H(4), and NH(3), that meet the biological constraint of forming under mild conditions. The reaction of [Ru(NCCH(3))(PiPr(3))('N(2)Me(2)S(2)')] (1) with NH(3) gave the ammonia complex [Ru(NH(3))(PiPr(3))('N(2)Me(2)S(2)')] (4), which readily exchanged NH(3) for N(2) to yield the mononuclear dinitrogen complex [Ru(N(2))(PiPr(3))('N(2)Me(2)S(2)')] (2) in almost quantitative yield. Complex 2, obtained by this new efficient synthesis, was the starting material for the synthesis of dinuclear (R,R)- and (S,S)-[micro-N(2)[Ru(PiPr(3))('N(2)Me(2)S(2)')](2)] ((R,R)-/(S,S)-3). (Both 2 and 3 have been reported previously.) The as-yet inexplicable behavior of complex 3 to form also the R,S isomer in solution has been revealed by DFT calculations and (2)D NMR spectroscopy studies. The reaction of 1 or 2 with anhydrous hydrazine yielded the hydrazine complex [Ru(N(2)H(4))(PiPr(3))('N(2)Me(2)S(2)')] (6), which is a highly reactive intermediate. Disproportionation of 6 resulted in the formation of mononuclear diazene complexes, the ammonia complex 4, and finally the dinuclear diazene complex [micro-N(2)H(2)[Ru(PiPr(3))('N(2)Me(2)S(2)')](2)] (5). Dinuclear complex 5 could also be obtained directly in an independent synthesis from 1 and N(2)H(2), which was generated in situ by acidolysis of K(2)N(2)(CO(2))(2). Treatment of 6 with CH(2)Cl(2), however, formed a chloromethylated diazene species [[Ru(PiPr(3))('N(2)Me(2)S(2)')]-micro-N(2)H(2)[Ru(Cl)('N(2)Me(2)S(2)CH(2)Cl')]] (9) ('N(2)Me(2)S(2)CH(2)Cl'(2-) =1,2-ethanediamine-N,N'-dimethyl-N-(2-benzenethiolate)(1-)-N'-(2-benzenechloromethylthioether)(1-)]. The molecular structures of 4, 5, and 9 were determined by X-ray crystal structure analysis, and the labile N(2)H(4) complex 6 was characterized by NMR spectroscopy.     

N-Heterocyclic Olefins as Electron Donors in Combination with Triarylborane Acceptors: Synthesis,Optical and Electronic Properties of D–π–A Compounds

N-heterocyclic olefins (NHOs), relatives of N-heterocyclic carbenes (NHCs), exhibit high nucleophilicity and soft Lewis basic character. To investigate their π-electron donating ability, NHOs were attached to triarylborane π-acceptors (A) giving donor (D)–π–A compounds 1 – 3 . In addition, an enamine π-donor analogue ( 4 ) was synthesized for comparison. UV–visible absorption studies show a larger red shift for the NHO-containing boranes than for the enamine analogue, a relative of cyclic (alkyl)(amino) carbenes (CAACs). Solvent-dependent emission studies indicate that 1 – 4 have moderate intramolecular charge-transfer (ICT) behavior. Electrochemical investigations reveal that the NHO-containing boranes have extremely low reversible oxidation potentials (e.g., for 3 , =−0.40 V vs. ferrocene/ferrocenium, Fc/Fc⁺, in THF). Time-dependent (TD) DFT calculations show that the HOMOs of 1 – 3 are much more destabilized than that of the enamine-containing 4 , which confirms the stronger donating ability of NHOs.     

Targeted ferromagnetic coupling in a trinuclear copperII complex: analysis of the St = 3/2 spin ground state

The trinuclear Cu(II) complex [(talen)Cu(II)(3)] (1) using the new triplesalen ligand H(6)talen has been synthesized and structurally characterized. The three Cu(II) ions are bridged in a m-phenylene linkage by the phloroglucinol backbone of the ligand. This m-phenylene bridging mode results in ferromagnetic couplings with an S(t) = (3)/(2) spin ground state, which has been analyzed by means of EPR spectroscopy and DFT calculations. The EPR spectrum exhibits an unprecedented pattern of 10 hyperfine lines due to the coupling of three Cu(II) ions (I = (3)/(2)). Resonances around g = 4 in both perpendicular and parallel mode EPR spectra demonstrate a zero-field splitting of D approximately 74 x 10(-4) cm(-1) arising from anisotropic/antisymmetric exchange interactions. The DFT calculations show an alteration in the sign of the spin densities of the central benzene ring corroborating the spin-polarization mechanism as origin for the ferromagnetic coupling.     

Rearrangement reactions of the transient lewis acids (CF(3))(3)B and (CF(3))(3)BCF(2): an experimental and theoretical study

Short-lived (CF(3))(3)B and (CF(3))(3)BCF(2) are generated as intermediates by thermal dissociation of (CF(3))(3)BCO and F(-) abstraction from the weak coordinating anion [B(CF(3))(4)](-), respectively. Both Lewis acids cannot be detected because of their instability with respect to rearrangement reactions at the B-C-F moiety. A cascade of 1,2-fluorine shifts to boron followed by perfluoroalkyl group migrations and also difluorocarbene transfer reactions occur. In the gas phase, (CF(3))(3)B rearranges to a mixture of linear perfluoroalkyldifluoroboranes C(n)()F(2)(n)()(+1)BF(2) (n = 2-7), while the respective reactions of (CF(3))(3)BCF(2) result in a mixture of linear (n = 2-4) and branched monoperfluoroalkyldifluoroboranes, e.g., (C(2)F(5))(CF(3))FCBF(2). For comparison, the reactions of [CF(3)BF(3)](-) and [C(2)F(5)BF(3)](-) with AsF(5) are studied, and the products in the case of [CF(3)BF(3)](-) are BF(3) and C(2)F(5)BF(2) whereas in the case of [C(2)F(5)BF(3)](-), C(2)F(5)BF(2) is the sole product. In contrast to reports in the literature, it is found that CF(3)BF(2) is too unstable at room temperature to be detected. The decomposition of (CF(3))(3)BCO in anhydrous HF leads to a mixture of the new conjugate Br?nsted-Lewis acids [H(2)F][(CF(3))(3)BF] and [H(2)F][C(2)F(5)BF(3)]. All reactions are modeled by density functional calculations. The energy barriers of the transition states are low in agreement with the experimental results that (CF(3))(3)B and (CF(3))(3)BCF(2) are short-lived intermediates. Since CF(2) complexes are key intermediates in the rearrangement reactions of (CF(3))(3)B and (CF(3))(3)BCF(2), CF(2) affinities of some perfluoroalkylfluoroboranes are presented. CF(2) affinities are compared to CO and F(-) affinities of selected boranes showing a trend in Lewis acidity, and its influence on the stability of the complexes is discussed. Fluoride ion affinities are calculated for a variety of different fluoroboranes, including perfluorocarboranes, and compared to those of the title compounds.