Resonance tongues in the quasi-periodic Hill-Schrödinger equation with three frequencies

In this paper we investigate numerically the following Hill’s equation x″ + (a + bq(t))x = 0 where $ q(t) = \cos t + \cos \sqrt {2t} + \cos \sqrt {3t} $ is a quasi-periodic forcing with three rationally independent frequencies. It appears, also, as the eigenvalue equation of a Schrödinger operator with quasi-periodic potential. Massive numerical computations were performed for the rotation number and the Lyapunov exponent in order to detect open and collapsed gaps, resonance tongues. Our results show that the quasi-periodic case with three independent frequencies is very different not only from the periodic analogs, but also from the case of two frequencies. Indeed, for large values of b the spectrum contains open intervals at the bottom. From a dynamical point of view we numerically give evidence of the existence of open intervals of a, for large b, where the system is nonuniformly hyperbolic: the system does not have an exponential dichotomy but the Lyapunov exponent is positive. In contrast with the region with zero Lyapunov exponents, both the rotation number and the Lyapunov exponent do not seem to have square root behavior at endpoints of gaps. The rate of convergence to the rotation number and the Lyapunov exponent in the nonuniformly hyperbolic case is also seen to be different from the reducible case.     

A simple procedure to obtain the extreme core allocations of an assignment market

Given an assignment market, we introduce a set of vectors, one for each possible ordering on the player set, which we name the max-payoff vectors. Each one of these vectors is obtained recursively only making use of the assignment matrix. Those max-payoff vectors that are efficient turn out to give the extreme core allocations of the market. When the assignment game has a large core, all the max-payoff vectors are extreme core allocations.     

Comparison of density functionals for reactions of sulfur ylides with aldehydes and olefins

The reactions of a model sulfur ylide with formaldehyde and 1,1-dicianoethylene, leading to the formation of an epoxyde and a cyclopropane, respectively, have been studied using different computational methods, and the results have been compared to those obtained with the CBS-QB3 method. The second step of these reactions presents transition states similar to that of an SN2 reaction. Depending on the degree of electron delocalization at the transition state, a different amount of exact exchange is necessary in the exchange functional to obtain accurate energy barriers. This amount is larger for the reaction of formaldehyde, in which the transition state is more delocalized, than for the reaction of 1,1-dicianoethylene. Similar results have been obtained for symmetric and non-symmetric SN2 reactions. The calculation of the reaction path has shown that the error relative to CBS-QB3 tends to increase when approaching the transition state. Among the different computational methods, PBE1PBE is the one to provide the most accurate energy barriers and reaction energies, whereas BB1K leads to the best results for the reaction path before the transition state.     

Palladium-NHC complexes do catalyse the diboration of alkenes: mechanistic insights

Novel catalytic activation of the B-B bond by palladium(II)-NHC complexes in presence of a mild base (NaOAc) and an excess of diboron reagent enables chemoselective 1,2-diboration of alkenes, suggesting the heterolytic cleavage of diboron rather than oxidative addition of a B-B bond to the metal.     

Ester versus polyketone formation in the palladium-diphosphine catalyzed carbonylation of ethene

The origin of the chemoselectivity of palladium catalysts containing bidentate phosphine ligands toward either methoxycarbonylation of ethene or the copolymerization of ethene and carbon monoxide was investigated using density functional theory based calculations. For a palladium catalyst containing the electron-donating bis(dimethylphosphino)ethane (dmpe) ligand, the rate determining step for chain propagation is shown to be the insertion of ethene into the metal-acyl bond. The high barrier for chain propagation is attributed to the low stability of the ethene intermediate, (dmpe)Pd(ethene)(C(O)CH3). For the competing methanolysis process, the most likely pathway involves the formation of (dmpe)Pd(CH3OH)(C(O)CH3) via dissociative ligand exchange, followed by a solvent mediated proton-transfer/reductive- elimination process. The overall barrier for this process is higher than the barrier for ethene insertion into the palladium-acetyl bond, in line with the experimentally observed preference of this type of catalyst toward the formation of polyketone. Electronic bite angle effects on the rates of ethene insertion and ethanoyl methanolysis were evaluated using four electronically and sterically related ligands (Me)2P(CH2)nP(Me)2 (n = 1-4). Steric effects were studied for larger tert-butyl substituted ligands using a QM/MM methodology. The results show that ethene coordination to the metal center and subsequent insertion into the palladium-ethanoyl bond are disfavored by the addition of steric bulk around the metal center. Key intermediates in the methanolysis mechanism, on the other hand, are stabilized because of electronic effects caused by increasing the bite angle of the diphosphine ligand. The combined effects explain successfully which ligands give polymer and which ones give methyl propionate as the major products of the reaction.     

Synthesis of enantiomeric bridgehead substituted bisnoradamantane derivatives

The preparation of (+)- and (−)-12 by resolution of (±)-12 with (R)-N-phenylpantolactam, (R)-13, is described. From (+)- and (−)-12 a series of chiral bisnoradamantane derivatives, whose chirality stems from substitution at the bridgehead positions, have been obtained in both enantiomeric forms.     

Highly Active Aluminium Catalysts for the Formation of Organic Carbonates from CO2 and Oxiranes

Al^III complexes of amino‐tris(phenolate) ligand scaffolds have been prepared to attain highly Lewis acidic catalysts. Combination of the aforementioned systems with ammonium halides provides highly active catalysts for the synthesis of organic carbonates through addition of carbon dioxide to oxiranes with initial turnover frequencies among the highest reported to date within the context of cyclic carbonate formation. Density functional theory (DFT) studies combined with kinetic data provides a rational for the relative high activity found for these Al^III complexes, and the data are consistent with a monometallic mechanism. The activity and versatility of these Al^III complexes has also been evaluated against some state‐of‐the‐art catalysts and the combined results compare favorably in terms of catalyst construction, stability, activity, and applicability.     

The Energy Landscape of Uranyl–Peroxide Species

Nanoscale uranyl peroxide clusters containing UO₂²⁺ groups bonded through peroxide bridges to form polynuclear molecular species (polyoxometalates) exist both in solution and in the solid state. There is an extensive family of clusters containing 28 uranium atoms (U₂₈ clusters), with an encapsulated anion in the center, for example, [UO₂(O₂)_3?x(OH)_x^4?], [Nb(O₂)₄^3?], or [Ta(O₂)₄^3?]. The negative charge of these clusters is balanced by alkali ions, both encapsulated, and located exterior to the cluster. The present study reports measurement of enthalpy of formation for two such U₂₈ compounds, one of which is uranyl centered and the other is peroxotantalate centered. The [(Ta(O₂)₄]‐centered U₂₈ capsule is energetically more stable than the [(UO₂)(O₂)₃]‐centered capsule. These data, along with our prior studies on other uranyl–peroxide solids, are used to explore the energy landscape and define thermochemical trends in alkali–uranyl–peroxide systems. It was suggested that the energetic role of charge‐balancing alkali ions and their electrostatic interactions with the negatively charged uranyl–peroxide species is the dominant factor in defining energetic stability. These experimental data were supported by DFT calculations, which agree that the [(Ta(O₂)₄]‐centered U₂₈ capsule is more stable than the uranyl‐centered capsule. Moreover, the relative stability is controlled by the interactions of the encapsulated alkalis with the encapsulated anion. Thus, the role of alkali‐anion interactions was shown to be important at all length scales of uranyl–peroxide species: in both comparing clusters to clusters; and clusters to monomers or extended solids.     

Methylated Azoles, 2‐Tetrazenes,and Hydrazines

This report is a summary of our work on energetic materials. Herein, triazole‐, tetrazole‐, 2‐tetrazene‐ and hydrazine‐based energetic compounds are described. An overview of the synthetic methods and of the problematic around the quest of new explosives is given. Hydrogen‐bonding formation and alkylation (methylation) reactions were studied as a mean to decrease the sensitivities towards classical stimuli and increase their thermal and chemical stability. ¹⁵N NMR spectroscopy is a valuable tool for the assignment of the methylation site in keeping with the results of the crystal structure analysis. Lastly, the thermal/chemical stability, sensitivity data and energetic performance of the compounds is described.