Galerkin-Laguerre Spectral Solution of Self-Similar Boundary Layer Problems

In this work the Laguerre basis for the biharmonic equation introduced by Jie Shen is employed in the spectral solution of self-similar problems of the boundary layer theory. An original Petrov-Galerkin formulation of the Falkner-Skan equation is presented which is based on a judiciously chosen special basis function to capture the asymptotic behaviour of the unknown. A spectral method of remarkable simplicity is obtained for computing Falkner-Skan-Cooke boundary layer flows. The accuracy and efficiency of the Laguerre spectral approximation is illustrated by determining the linear stability of nonseparated and separated flows according to the Orr-Sommerfeld equation. The pentadiagonal matrices representing the derivative operators are explicitly provided in an Appendix to aid an immediate implementation of the spectral solution algorithms

Dinuclear rhenium pyridazine complexes containing bridging chalcogenide anions : synthesis, characterization and computational study

The synthesis of a series of neutral dinuclear rhenium complexes of the general formula [Re2(m-ER)2(CO)6- (m-pydz)] (pydz = pyridazine; E = S, Se or Te; R = methyl or phenyl; the TeMe is not included) has been carried out via new, either one-pot or two-step, procedures. The one-pot synthesis consists of the oxidative addition of RE\u2013ER across the Re\u2013Re bond of [Re2(CO)10], in the presence of 1 equivalent of pyridazine, and affords the corresponding dinuclear complexes in high yields (ca. 85%). Furthermore, a general two-step procedure has been carried out, which involves the synthesis of heterocubane-like [Re4(m3-ER)4(CO)12] molecules and their reaction with pyridazine, quantitatively affording the corresponding dinuclear species through a symmetric [2+2] fragmentation pathway. The molecular structure of the complexes has been elucidated by single crystal XRD analysis, and TD-DFT calculations predicted the existence of conformers differing in the orientation of the chalcogen substituents with respect to the pyridazine ligand. The relative stabilities and the activation barriers for the interconversion have been calculated, observing a regular trend that has been rationalized depending on the hybridization of the chalcogen atom. Variable temperature NMR studies experimentally confirmed the theoretical prediction, showing, in solution, two conformers with different relative amounts and different interconversion rates between them, depending on the chalcogen nature. From the electrochemical point of view the S, Se and Te complexes display a bi-electronic reversible oxidation peak, differently from the two mono-electronic irreversible oxidation peaks previously observed for the O derivatives. Moreover, a progressive narrowing of the HOMO\u2013LUMO gap on going from O to Te, arising from the increase of the HOMO level, has been observed. This is in line with the decreasing electron-withdrawing strength of the chalcogenide bridging ligand, so that the energy gap for the telluride derivative is 1.64 eV, the smallest value in the whole family of the di-rhenium pyridazine complexes. The spectroscopic HOMO\u2013LUMO gap parallels this trend, with a significant red-shift of the metal-to-ligand charge transfer absorption, making the telluride complex highly promising as a photosensitizer in the field of solar energy conversion. In agreement with the narrow HOMO\u2013LUMO gap, no photoluminescence has been observed upon optical excitation

Electrochemical Characterization and CO2 Reduction Reaction of a Family of Pyridazine-Bridged Dinuclear Mn(I) Carbonyl Complexes

Three recently synthesized neutral dinuclear carbonyl manganese complexes with the pyridazine bridging ligand, of general formula [Mn2(μ-ER)2(CO)6(μ-pydz)] (pydz = pyridazine; E = O or S; R = methyl or phenyl), have been investigated by cyclic voltammetry in dimethylformamide and acetonitrile both under an inert argon atmosphere and in the presence of carbon dioxide. This family of Mn(I) compounds behaves interestingly at negative potentials in the presence of CO2. Based on this behavior, which is herein discussed, a rather efficient catalytic mechanism for the CO2 reduction reaction toward the generation of CO has been hypothesized

Dinuclear Re(I) Complexes as New Electrocatalytic Systems for CO2 Reduction

A family of dinuclear tricarbonyl rhenium (I) complexes containing bridging 1,2-diazine ligand and halide anions as ancillary ligands and able to catalyze CO2 reduction is presented. Electrochemical studies show that the highest catalytic efficiency is obtained for the complex containing the 4,5-bipenthyl-pyridazine and iodide as ancillary halogen ligands. This complex gives rise to TOF=15 s−1 that clearly outperforms the values reported for the benchmark mononuclear Re(CO)3Cl(bpy) (11.1 s−1). The role of the substituents on the pyridazine ligand and the nature of the bridging halide ligands on the catalytic activity have been deeply investigated through a systematic study on the structure-properties relationship to understand the improved catalytic efficiencies of this class of complexes

Spectral Matrix Conditioning in Cylindrical and Spherical Elliptic Equations

In the spectral solution of 3-D Poisson equations in cylindrical and spherical coordinates including the axis or the center, it is convenient to employ radial basis functions that depend on the Fourier wavenumber or on the latitudinal mode. This idea has been adopted by Matsushima and Marcus and by Verkley for planar problems and pursued by the present authors for spherical ones. For the Dirichlet boundary value problem in both geometries, original bases have been introduced built upon Jacobi polynomials which lead to a purely diagonal representation of the radial second-order differential operator of all spectral modes. This note details the origin of such a diagonalization which extends to cylindrical and spherical regions the properties of the Legendre basis introduced by Jie Shen for Cartesian domains. Closed form expressions are derived for the diagonal elements of the stiffness matrices as well as for the elements of the tridiagonal mass matrices occurring in evolutionary problems. Furthermore, the bound on the condition number of the spectral matrices associated with the Helmholtz equation are determined, proving in a rigorous way one of the main advantages of the proposed radial bases