NMR studies of solid state—solvent and H/D isotope effects on hydrogen bond geometries of 1:1 complexes of collidine with carboxylic acids

¹H and ¹⁵N NMR spectra of 10 complexes exhibiting strong OHN hydrogen bonds formed by ¹⁵N-labeled collidine and different proton donors, partially deuterated in mobile proton sites, have been observed by low-temperature NMR spectroscopy using a low-freezing CDF₃/CDF₂Cl mixture as polar aprotic solvent. The following proton donors have been used: HCl, formic acid, acetic acid, various substituted benzoic acids and HBF₄. The slow hydrogen bond exchange regime could be reached below 140 K, which allowed us to resolve ¹⁵N signal splittings due to H/D isotopic substitution. The valence bond order model is used to link the observed NMR parameters to hydrogen bond geometries. The results are compared to those obtained previously [Magn. Reson. Chem. 39 (2001) S18] for the same complexes in the organic solids. The increase of the dielectric constant from the organic solids to the solution (30 at 130 K) leads to a change of the hydrogen bond geometries along the geometric correlation line towards the zwitterionic structures, where the proton is partially transferred from oxygen to nitrogen. Whereas the changes of spectroscopic and, hence, geometric parameters are small for the systems which are already zwitterionic in the solid state, large changes are observed for molecular complexes which exhibit almost a full proton transfer from oxygen to nitrogen in the polar liquid solvent.     

Ein Existenz- und Eindeutigkeitssatz für die Lösungen einer Funktionalgleichung aus der Nörlundschen Theorie der Differenzengleichungen

Ohne Zusammenfassung     

Bridging the gap between homogeneous and heterogeneous catalysis: ortho/para H(2) conversion, hydrogen isotope scrambling, and hydrogenation of olefins by Ir(CO)Cl(PPh(3))(2)

Some transition metal complexes are known to catalyze ortho/para hydrogen conversion, hydrogen isotope scrambling, and hydrogenation reactions in liquid solution. Using the example of Vaska's complex, we present here evidence by NMR that the solvent is not necessary for these reactions to occur. Thus, solid frozen solutions or polycrystalline powdered samples of homogeneous catalysts may become heterogeneous catalysts. Comparative liquid- and solid-state studies provide novel insight into the reaction mechanisms.     

Uniform convergence and a posteriori error estimators for the enhanced strain finite element method

Summary. Enhanced strain elements, frequently employed in practice, are known to improve the approximation of standard (non-enhanced) displacement-based elements in finite element computations. The first contribution in this work towards a complete theoretical explanation for this observation is a proof of robust convergence of enhanced element schemes: it is shown that such schemes are locking-free in the incompressible limit, in the sense that the error bound in the a priori estimate is independent of the relevant Lamé constant. The second contribution is a residual-based a posteriori error estimate; the L ² norm of the stress error is estimated by a reliable and efficient estimator that can be computed from the residuals. Mathematics Subject Classification (2000):65N30     

All first-order averaging techniques for a posteriori finite element error control on unstructured grids are efficient and reliable

All first-order averaging or gradient-recovery operators for lowest-order finite element methods are shown to allow for an efficient a posteriori error estimation in an isotropic, elliptic model problem in a bounded Lipschitz domain in . Given a piecewise constant discrete flux (that is the gradient of a discrete displacement) as an approximation to the unknown exact flux (that is the gradient of the exact displacement), recent results verify efficiency and reliability of

in the sense that is a lower and upper bound of the flux error up to multiplicative constants and higher-order terms. The averaging space consists of piecewise polynomial and globally continuous finite element functions in components with carefully designed boundary conditions. The minimal value is frequently replaced by some averaging operator applied within a simple post-processing to . The result provides a reliable error bound with .

This paper establishes and so equivalence of and . This implies efficiency of for a large class of patchwise averaging techniques which includes the ZZ-gradient-recovery technique. The bound established for tetrahedral finite elements appears striking in that the shape of the elements does not enter: The equivalence is robust with respect to anisotropic meshes. The main arguments in the proof are Ascoli's lemma, a strengthened Cauchy inequality, and elementary calculations with mass matrices.

     

A convergent adaptive finite element method for an optimal design problem

The optimal design problem for maximal torsion stiffness of an infinite bar of given geometry and unknown distribution of two materials of prescribed amounts is one model example in topology optimisation. It eventually leads to a degenerate convex minimisation problem. The numerical analysis is therefore delicate for possibly multiple primal variables u but unique derivatives σ : = DW(D u). Even fine a posteriori error estimates still suffer from the reliability-efficiency gap. However, it motivates a simple edge-based adaptive mesh-refining algorithm (AFEM) that is not a priori guaranteed to refine everywhere. Its convergence proof is therefore based on energy estimates and some refined convexity control. Numerical experiments illustrate even nearly optimal convergence rates of the proposed AFEM. Supported by the DFG Research Center MATHEON “Mathematics for key technologies” in Berlin.