Clathrate solvates of tetra­kis(4‐methoxy­carbonyl­phenyl)porphyrin and its zinc(II)–pyridine complex,in which the porphyrin host structures are stabilized by porphyrin–porphyrin stacking and C—H⋯O attractions

Tetra­kis(4‐methoxy­carbonyl­phenyl)porphyrin, or tetra­methyl 4,4′,4′′,4′′′‐porphyrin‐5,10,15,20‐tetra­benzoate, crystallizes as a nitro­benzene 1.9‐solvate, C₅₂H₃₈N₄O₈·1.9C₆H₅NO₂, (I). The solvent mol­ecules are contained in extended channels which propagate through the host lattice between parallel screw/glide‐related columns of offset‐stacked porphyrin entities. Side packing of these columns involves π–π inter­actions between the methoxy­carbonyl­phenyl residues. Mol­ecules of the porphyrin host lie on crystallographic inversion centres. The zinc(II)–pyridine derivative pyridine­(tetra­methyl 4,4′,4′′,4′′′‐porphyrin‐5,10,15,20‐tetra­benzoato)zinc(II), [Zn(C₅₂H₃₆N₄O₈)(C₅H₅N)], (II), is a square‐pyramidal five‐coordinate complex with pyridine as an apical ligand, which crystallizes as a chloro­form–pyridine solvate. The metallo­porphyrin–pyridine units form an open layered arrangement, occluding the non‐coordinated solvent moieties within the intra­layer inter­porphyrin voids. Within such arrays, the host porphyrin mol­ecules are in contact with one another through the peripheral methoxy­carbonyl substituents. The crystal packing consists of a bilayered arrangement of inversion‐related porphyrin layers, with the axial ligands mutually penetrating into the voids of neighbouring arrays and tight offset stacking of these bilayers.     

A Variational Approximation Scheme for¶Three-Dimensional Elastodynamics¶with Polyconvex Energy

We construct a variational approximation scheme for the equations of three-dimensional elastodynamics with polyconvex stored energy. The scheme is motivated by some recently discovered geometric identities (Qin [18]) for the null Lagrangians (the determinant and cofactor matrix), and by an associated embedding of the equations of elastodynamics into an enlarged system which is endowed with a convex entropy. The scheme decreases the energy, and its solvability is reduced to the solution of a constrained convex minimization problem. We prove that the approximating process admits regular weak solutions, which in the limit produce a measure-valued solution for polyconvex elastodynamics that satisfies the classical weak form of the geometric identities. This latter property is related to the weak continuity properties of minors of Jacobian matrices, here exploited in a time-dependent setting. Accepted November 18, 2000?Published online April 23, 2001     

Semiglobal results for on a complex space with arbitrary singularities

We obtain some -results for the operator on forms that vanish to high order on the singular set of a complex space.

     

Multiple solutions of constant sign for nonlinear nonsmooth eigenvalue problems near resonance

In this paper we study a class of nonlinear elliptic eigenvalue problems driven by the p-Laplacian and having a nonsmooth locally Lipschitz potential. We show that as the parameter approaches (= the principal eigenvalue of ) from the right, the problem has three nontrivial solutions of constant sign. Our approach is variational based on the nonsmooth critical point theory for locally Lipschitz functions. In the process of the proof we also establish a generalization of a recent result of Brezis and Nirenberg for C₀¹ versus W₀^1,p minimizers of a locally Lipschitz functional. In addition we prove a result of independent interest on the existence of an additional critical point in the presence of a local minimizer of constant sign. Finally by restricting further the asymptotic behavior of the potential at infinity, we show that for all the problem has two solutions one strictly positive and the other strictly negative.Received: 7 January 2003, Accepted: 12 May 2003, Published online: 4 September 2003Mathematics Subject Classification (2000): 35J20, 35J85, 35R70     

Variational Approach to Uniform Rigid Rotation of Ginzburg—Landau Vortices

Time periodic solutions for the hyperbolic gauged Ginzburg–Landau system, with spatial domain the unit disc, are shown to exist. Time periodic solutions representing bound states of vortices rotating about one another have been previously obtained in the near self-dual limit, using perturbative techniques. In contrast, we here take a variational approach, the solutions being obtained as critical points of an indefinite functional. We consider a special class of solutions which map out, uniformly in time, an orbit of the rotation group SO(2). It is shown that in the limit of large coupling constant the solutions have nontrivial time dependence or, as is shown to be equivalent, are not radially symmetric in any gauge.     

Large Isotope Effects in Organometallic Chemistry

The kinetic isotope effect (KIE) is key to understanding reaction mechanisms in many areas of chemistry and chemical biology, including organometallic chemistry. This ratio of rate constants, k_H/k_D, typically falls between 1–7. However, KIEs up to 105 have been reported, and can even be so large that reactivity with deuterium is unobserved. We collect here examples of large KIEs across organometallic chemistry, in catalytic and stoichiometric reactions, along with their mechanistic interpretations. Large KIEs occur in proton transfer reactions such as protonation of organometallic complexes and clusters, protonolysis of metal–carbon bonds, and dihydrogen reactivity. C−H activation reactions with large KIEs occur with late and early transition metals, photogenerated intermediates, and abstraction by metal-oxo complexes. We categorize the mechanistic interpretations of large KIEs into the following three types: (a) proton tunneling, (b) compound effects from multiple steps, and (c) semi-classical effects on a single step. This comprehensive collection of large KIEs in organometallics provides context for future mechanistic interpretation.     

Combining Nitridoborates,Nitrides and Hydrides—Synthesis and Characterization of the Multianionic Sr6N[BN2]2H3

Multianionic metal hydrides, which exhibit a wide variety of physical properties and complex structures, have recently attracted growing interest. Here we present Sr₆N[BN₂]₂H₃, prepared in a solid-state ampoule reaction at 800 °C, as the first combination of nitridoborate, nitride and hydride anions within a single compound. The crystal structure was solved from single-crystal X-ray and neutron powder diffraction data in space group P2₁/c (no. 14), revealing a three-dimensional network of undulated layers of nitridoborate units, strontium atoms and hydride together with nitride anions. Magic angle spinning (MAS) NMR and vibrational spectroscopy in combination with quantum chemical calculations further confirm the structure model. Electrochemical measurements suggest the existence of hydride ion conductivity, allowing the hydrides to migrate along the layers.     

Inhomogeneous multiscale dynamics in harmonic lattices

We use projection operators to address the coarse-grained multiscale problem in harmonic systems. Stochastic equations of motion for the coarse-grained variables, with an inhomogeneous level of coarse graining in both time and space, are presented. In contrast to previous approaches that typically start with thermodynamic averages, the key element of our approach is the use of a projection matrix chosen both for its physical appeal in analogy to mechanical stability theory and for its algebraic properties. We show that thermodynamic equilibrium can be recovered and obtain the fluctuation dissipation theorem a posteriori. All system-specific information can be computed from a series of feasible molecular dynamics simulations. We recover previous results in the literature and show how this approach can be used to extend the quasicontinuum approach and comment on implications for dissipative particle dynamics type of methods. Contrary to what is assumed in the latter models, the stochastic process of all coarse-grained variables is not necessarily Markovian, even though the variables are slow. Our approach is applicable to any system in which the coarse-grained regions are linear. As an example, we apply it to the dynamics of a single mesoscopic particle in the infinite one-dimensional harmonic chain.