β−type fractional Sturm‐Liouville Coulomb operator and applied results

In this article, β‐type fractional Sturm‐Liouville Coulomb operator is considered by Hilfer fractional derivative. Fundamental spectral theory is investigated for the aforementioned problem. In this context, it is shown that the operator is self‐adjoint, eigenfunctions correspond to the distinct eigenfunctions are orthogonal, and eigenvalues are real. Furthermore, applications of this problem are given by the Adomian decomposition method and the results are shown with visual graphs.     

Is it homogeneous or heterogeneous catalysis derived from [RhCp*Cl2]2? In operando XAFS, kinetic, and crucial kinetic poisoning evidence for subnanometer Rh4 cluster-based benzene hydrogenation catalysis

Determining the true, kinetically dominant catalytically active species, in the classic benzene hydrogenation system pioneered by Maitlis and co-workers 34 years ago starting with [RhCp*Cl(2)](2) (Cp* = [η(5)-C(5)(CH(3))(5)]), has proven to be one of the most challenging case studies in the quest to distinguish single-metal-based "homogeneous" from polymetallic, "heterogeneous" catalysis. The reason, this study will show, is the previous failure to use the proper combination of: (i) in operando spectroscopy to determine the dominant form(s) of the precatalyst's mass under catalysis (i.e., operating) conditions, and then crucially also (ii) the previous lack of the necessary kinetic studies, catalysis being a "wholly kinetic phenomenon" as J. Halpern long ago noted. An important contribution from this study will be to reveal the power of quantitiative kinetic poisoning experiments for distinguishing single-metal, or in the present case subnanometer Rh(4) cluster-based catalysis, from larger, polymetallic Rh(0)(n) nanoparticle catalysis, at least under favorable conditions. The combined in operando X-ray absorption fine structure (XAFS) spectroscopy and kinetic evidence provide a compelling case for Rh(4)-based, with average stoichiometry "Rh(4)Cp*(2.4)Cl(4)H(c)", benzene hydrogenation catalysis in 2-propanol with added Et(3)N and at 100 °C and 50 atm initial H(2) pressure. The results also reveal, however, that if even ca. 1.4% of the total soluble Rh(0)(n) had formed nanoparticles, then those Rh(0)(n) nanoparticles would have been able to account for all the observed benzene hydrogenation catalytic rate (using commercial, ca. 2 nm, polyethyleneglycol-dodecylether hydrosol stabilized Rh(0)(n) nanoparticles as a model system). The results--especially the poisoning methodology developed and employed--are of significant, broader interest since determining the nature of the true catalyst continues to be a central, often vexing issue in any and all catalytic reactions. The results are also of fundamental interest in that they add to a growing body of evidence indicating that certain, appropriately ligated, coordinatively unsaturated, subnanometer M(4) transition-metal clusters can be relatively robust catalysts. Also demonstrated herein is that Rh(4) clusters are poisoned by Hg(0), demonstrating for the first time that the classic Hg(0) poisoning test of "homogeneous" vs "heterogeneous" catalysts cannot distinguish Rh(4)-based subnanometer catalysts from Rh(0)(n) nanoparticle catalysts, at least for the present examples of these two specific, Rh-based catalysts.     

The Influence of Hughes Type Action

We call the action of an automorphism α of a finite group G a Hughes type action if it is described by conditions on the orders of elements of G ? α ? ? G. In the present paper we study the structure of finite group G admitting an automorphism α of prime order p so that the orders of elements in G ? α ? ? G are not divisible by p ².     

An electron paramagnetic resonance and density-functional theory study on the methyl isotropic hyperfine coupling constants in gamma-irradiated 2,6-di-tert-butyl-4-methylphenol

Single crystal of gammairradiated 2,6-di-tert-butyl-4-methylphenol (BHT) was investigated using an electron paramagnetic resonance (EPR) technique at different orientations in the magnetic field at room temperatures. Taking into consideration the chemical structure and the experimental spectra of the irradiated single crystal of BHT, we assumed that one phenoxyltype paramagnetic species was produced having an unpaired electron localized at the methyl fragment side of the phenyl ring. Depending on this assumption, one possible radical was modeled using the B3LYP/6-311+G(d) level of density-functional theory. EPR parameters were calculated for these modeled radical using the B3LYP/TZVP and B3LYP/EPR-III level. The averaged value of isotropic hydrogen hyperfine coupling constants of rotating methyl functional group of phenoxyl radical is calculated for the first time. Theoretically calculated values of the modeled radical are in reasonably good agreement with the experimental data determined from the spectra (differences in averaged coupling constant values smaller than 5%, and differences in isotropic g values fall into 1 ppt).