Gaussian mean curvature flow

In the Euclidean Space \mathbb Rⁿ⁺¹{\mathbb {R}^{n+1}} with a density e^{e\frac12 n m2 |x|2}, (e = ±1){e^{\varepsilon \frac12 n \mu^2 |x|^2},} {(\varepsilon =\pm1}), we consider the flow of a hypersurface driven by its mean curvature associated to this density. We give a detailed account of the evolution of a convex hypersurface under this flow. In particular, when e = -1{ \varepsilon=-1} (Gaussian density), the hypersurface can expand to infinity or contract to a convex hypersurface (not necessarily a sphere) depending on the relation between the bound of its principal curvatures and μ.     

On the Convolution of the Generalized Finite Legendre Transform

The translation operator and the convolution for the finite Legendre transformation are investigated in the space ??(?1,1) of testing-functions and its dual through an approach that emphasizes the close similarity existing between this transform and the infinite Mehler - Fock transformation. The theory developed is used in solving some distributional boundary-value problems.     

Electronic Structure of Rh(2)(&mgr;-CO)(CO)(2)(H(2)PCH(2)PH(2))(2). An Example of a Non-A-Frame Structure

Calculations based on density functional theory (DFT) and Hartree-Fock configuration interaction (HF-CI) methodology have been carried out to investigate the rhodium-rhodium coupling in Rh(2)(CO)(2)(dppm)(2), 1 (dppm = Ph(2)PCH(2)PPh(2)) and in Rh(2)(&mgr;-CO)(CO)(2)(dppm)(2), 2. DFT geometries, obtained with the Dgauss program, are in good agreement with those determined from X-ray, but HF geometries, calculated using the same basis sets, yield bond distances systematically too long. Calculations indicate that the rhodium atoms in 1 are linked by a single bond. The insertion of a semibridging carbonyl between the two metal atoms leads to a shortening of the rhodium-rhodium distance and also to a noticeable weakening of the metal-metal interaction. Both effects, and also the stabilization of the HOMO of 2, are related to an observed change from square planar to tetrahedral of the ligand environment of the Rh atom proximal to the inserted CO. Both MO analysis and bond characterization from the topology of the charge density confirm the existence of a bonding interaction between the semibridging carbonyl and the distal rhodium atom. The electronic structures of the dicationic complex [Rh(2)(CO)(3)(dppm)(2)](2+) and of the A-frame-like, isoelectronic system Rh(2)Br(2)(&mgr;-CO) (dppm)(2) are also discussed. The electron deformation density is derived from 2 by means of several methodological approaches, namely, HF, HF-CI, DFT, and DFT + gradient corrections. The HF deformation density obtained in the plane containing the metals and the three CO ligands is discussed, as well as the "correlation density" obtained from the difference maps DFT - HF and CI - HF.     

ChemInform Abstract: X2Y2 Isomers: Tuning Structure and Relative Stability Through Electronegativity Differences (X: H,Li, Na,F, Cl,Br, I; Y: O,S, Se,Te).

Density functional calculations on XYYX and X₂YY isomers of the X₂Y₂ species (X: H, Li, Na, F, Cl, Br, I; Y: O, S, Se, Te) show that the XYYX isomers are more stable than the X₂YY forms except for X = F and Y = S and Te, for which the F₂SS and F₂TeTe isomers are slightly more stable.     

Observation and mechanistic study of facile C-O bond formation between a well-defined aryl-copper(III) complex and oxygen nucleophiles

A well-defined macrocyclic aryl–Cu(III) complex (2) reacts readily with a variety of oxygen nucleophiles, including carboxylic acids, phenols and alcohols, under mild conditions to form the corresponding aryl esters, biaryl ethers and alkyl aryl ethers. The relationship between these reactions and catalytic C-O coupling methods is demonstrated by the reaction of the macrocyclic aryl–Br species with acetic acid and p-fluorophenol in the presence of 10 mol% Cu(I). An aryl-Cu(III)-Br species 2(Br) was observed as an intermediate in the catalytic reaction. Investigation of the stoichiometric C-O bond-forming reactions revealed nucleophile-dependent changes in the mechanism. The reaction of 2 with carboxylic acids revealed a positive correlation between the log(k(obs)) and the pK(a) of the nucleophile (less-acidic nucleophiles react more rapidly), whereas a negative correlation was observed with most phenols (more-acidic phenols react more rapidly). The latter trend resembles previous observations with nitrogen nucleophiles. With carboxylic acids and acidic phenols, UV-visible spectroscopic data support the formation of a ground-state adduct between 2 and the oxygen nucleophile. Collectively, kinetic and spectroscopic data support a unified mechanism for aryl-O coupling from the Cu(III) complex, consisting of nucleophile coordination to the Cu(III) center, deprotonation of the coordinated nucleophile, and C-O (or C-N) reductive elimination from Cu(III).     

Single and double metallic layer-containing ruthenium dendrimers. Synthesis and catalytic properties

The reaction of a series of phosphanyl-terminated carbosilane dendrimers displaying only one phosphorus ligand per arm with [RuCl(2)(p-cymene)](2) resulted in the grafting of RuCl(p-cymene) moieties on the periphery of the dendrimer. In these species, the chloride ligand is easily displaced by the organic bases pyridine, 4-cyanopyridine and 4,4[prime or minute]-bipyridine to afford new cationic metallodendrimers. NMR studies have confirmed the chirality of the ruthenium centre. The species containing 4,4[prime or minute]-bipyridine reacts through the uncoordinated pyridyl nitrogen with a new equivalent of [RuCl(2)(p-cymene)](2) or [RhCl(CO)(2)](2) to lead to homo- or hetero-bimetallic layer-containing dendrimeric systems. The ruthenodendrimers were tested as catalysts in the transfer hydrogenation of cyclohexanone by propan-2-ol and their activity compared with that of some analogous mononuclear ruthenium(ii) complexes.     

Complexation of Hg2+ with α‐Lipoic and Dihydrolipoic Acids: Study by Differential Pulse Voltammetry on Rotating Au‐Disk Electrode and ESI‐MS

The complexation of the natural antioxidants α‐lipoic acid (ALA) and its reduced form dihydrolipoic acid (DHLA) with Hg²⁺ was investigated by a recently proposed differential pulse voltammetric (DPV) method using the rotating Au‐disk electrode. Complexation processes are proposed from the multivariate curve resolution by alternating least squares (MCR‐ALS) analysis of DPV titration data. Main complexes were both 1 : 1 Hg : ALA and Hg : DHLA, although the formation of 1 : 2 complexes can be also deduced. ALA and DHLA show different Hg²⁺‐binding patterns at different pH. Voltammetric findings are completed with the data obtained by electrospray ionization mass‐spectrometry (ESI‐MS), especially in negative mode.     

Theoretical and experimental study of the vertical excitation energies in the ionic and tautomeric forms of 4-aminomethylpyridine

4-Aminomethylpyridine (4-PAM) has been widely used as a model compound to elucidate the mechanisms of biological and biomedical action of the amino derivatives of vitamin B₆. By virtue of the presence of two ionizable groups (viz. a pyridine nitrogen and an amino function) in its structure, 4-PAM in solution occurs as various ionic and tautomeric forms in equilibrium. In this work, we optimized the geometries of such forms and found the protonation status of the ionizable groups in 4-PAM to affect the molecular geometry and frontier orbitals. In addition, we determined the experimental electronic excitation energies for each molecular species of 4-PAM from deconvoluted UV–vis spectra. The results thus obtained were compared with their theoretical counterparts as determined from TD-DFT calculations. Based on the outcome, the theoretical methodology used affords correct simulation of electronic excitation energies. The theoretical and experimental results showed that the deprotonation of the pyridine nitrogen has no effect on the energy of the first electronic transition, however it affects its intensity. Additionally, the deprotonation of both pyridine nitrogen and methylamino group increases the number of bands, by increasing the n–π^* transitions.