A crystallographic and DFT study on a NHC complex of niobium oxide trifluoride

The complex [NbOF₃(Ipr)]₂, 1, was afforded in crystalline form by reaction of NbF₅ with the bulky NHC ligand 1,3-bis(2,6-diisopropylphenyl)imidazol-2-ylidene (IPr) in toluene by slow contact with moisture air. The molecular structure of 1 was ascertained by X-ray diffraction, providing the first example of a dinuclear NbOF₃ derivative and also a rare case of niobium compound with a monodentate NHC. A DFT investigation has shown that the Nb–C bond consists of a weak NHC to Nb σ donation, reinforced by an electrostatic contribution presumably favored by the presence of the ancillary fluoride ligands. The computed enthalpy for the dissociation of one Ipr from 1 is ca. 36 kcal mol^?1. The presence of bulky 2,6-diisopropylphenyl substituents on the carbene ligand has negligible influence on the Nb–C bond, as highlighted by DFT analyses on simplified models.     

Stationary focusing mean-field games

We consider stationary viscous mean-field games (MFG) systems in the case of local, decreasing and unbounded coupling. These systems arise in ergodic MFG theory and describe Nash equilibria of games with a large number of agents aiming at aggregation. We show how the dimension of the state space, the behavior of the coupling, and the Hamiltonian at infinity affect the existence and nonexistence of regular solutions. Our approach relies on the study of Sobolev regularity of the invariant measure and a blow-up procedure that is calibrated on the scaling properties of the system. In very special cases, we observe uniqueness of solutions. Finally, we apply our methods to obtain new existence results for MFG systems with competition, namely, when the coupling is local and increasing.     

Understanding Complex Electrochemical Impedance Spectroscopy in Corrosion Systems Using in‐situ Synchrotron Radiation Grazing Incidence X‐ray Diffraction

The difficulty in interpreting electrochemical impedance spectroscopy (EIS) data has limited the potential of the method despite its utility as an electrochemical method for studies of electrode reactions and interfacial structures. Taking the test case of the carbon dioxide corrosion of carbon steel, in‐situ synchrotron radiation grazing incidence X‐ray diffraction (SR‐GIXRD) has been demonstrated to be a powerful technique for validating and understanding the interfacial structures associated with complex EIS data. Carbon dioxide corrosion was found to occur in several steps, which may only be surmised by EIS, but the use of SR‐GIXRD in conjunction with EIS has enabled the establishment of a link between EIS time constants and structural changes associated with the evolution of corrosion products with time. By comparison to previous studies by the authors and others in this field, this communication provides the first direct experimental evidence linking SR‐GIXRD surface compositional data to otherwise indistinctive EIS time constants.     

Synthesis and Isolation of the Titanium–Scandium Endohedral Fullerenes—Sc2TiC@Ih‐C80, Sc2TiC@D5h‐C80 and Sc2TiC2@Ih‐C80: Metal Size Tuning of the TiIV/TiIII Redox Potentials

The formation of endohedral metallofullerenes (EMFs) in an electric arc is reported for the mixed‐metal Sc–Ti system utilizing methane as a reactive gas. Comparison of these results with those from the Sc/CH₄ and Ti/CH₄ systems as well as syntheses without methane revealed a strong mutual influence of all key components on the product distribution. Whereas a methane atmosphere alone suppresses the formation of empty cage fullerenes, the Ti/CH₄ system forms mainly empty cage fullerenes. In contrast, the main fullerene products in the Sc/CH₄ system are Sc₄C₂@C₈₀ (the most abundant EMF from this synthesis), Sc₃C₂@C₈₀, isomers of Sc₂C₂@C₈₂, and the family Sc₂C_{2 n} (2 n=74, 76, 82, 86, 90, etc.), as well as Sc₃CH@C₈₀. The Sc–Ti/CH₄ system produces the mixed‐metal Sc₂TiC@C_{2 n} (2 n=68, 78, 80) and Sc₂TiC₂@C_{2 n} (2 n=80) clusterfullerene families. The molecular structures of the new, transition‐metal‐containing endohedral fullerenes, Sc₂TiC@I_h‐C₈₀, Sc₂TiC@D_5h‐C₈₀, and Sc₂TiC₂@I_h‐C₈₀, were characterized by NMR spectroscopy. The structure of Sc₂TiC@I_h‐C₈₀ was also determined by single‐crystal X‐ray diffraction, which demonstrated the presence of a short Ti=C double bond. Both Sc₂TiC‐ and Sc₂TiC₂‐containing clusterfullerenes have Ti‐localized LUMOs. Encapsulation of the redox‐active Ti ion inside the fullerene cage enables analysis of the cluster–cage strain in the endohedral fullerenes through electrochemical measurements.     

The Role of Ligand Topology in the Decomplexation of Luminescent Lanthanide Complexes by Dipicolinic Acid

In this study, we present the aqueous solution behavior of two luminescent lanthanide antenna complexes (Eu³⁺? 1 , Dy³⁺? 9 ) with different ligand topologies in the presence of dipicolinic acid (DPA, pyridine‐2,6‐dicarboxylic acid). Macrocyclic (1,4,7,10‐tetraazacyclododecane‐1,4,7‐triacetic acid, DO3A, 9 ) and acyclic (1,4,7‐triazaheptane‐1,1,7,7‐tetraacetic acid, DTTA, 1 ) ligands have been selected to form a ratiometric pair in which Dy³⁺? 9 acts as a reference and Eu³⁺? 1 acts as a probe for the recognition of DPA. The pair of luminescent complexes in water reveals the capability to work as a DPA luminescent sensor. The change of emission intensity of Eu³⁺ indicates the occurrence of a new sensitization path for the lanthanide cation through excitation of DPA. NMR evidence implies the presence of free 1 and mass spectrometry shows the formation of emitting [EuDPA₂]^? as a result of a ligand exchange reaction.     

An Approach by Representation of Algebras for Decoherence-Free Subspaces

The aim of this paper is to present a general algebraic formulation for the decoherence-free subspaces (DFSs). In order to build the DFSs we consider the tensor product of Clifford algebras and left minimal ideals. States, error operators and projection operators are defined in a purely algebraic point of view. For this purpose, we initially generalize some results of Pauli and Artin about semisimple algebras. Then we derive orthogonality theorems for associative algebras analogous to theorems for finite groups. Some advantages and perspectives are also discussed.     

Asymptotic stability and bifurcations of 3D piecewise smooth vector fields

The paper deals with the analysis of the behavior of a nonsmooth three-dimensional vector field around a typal singularity. We focus on a class of generic one-parameter families \({Z_{\lambda}}\) of Filippov systems and address the persistence problem for the asymptotic stability when the parameter varies near the bifurcation value \({\lambda = 0}\).     

Free‐Standing Membranes to Study the Optical Properties of Anodic TiO2 Nanotube Layers

In the present work we investigate various optical properties (such as light absorption and reflectance) of anodic TiO₂ nanotube layers directly transferred as self‐standing membranes onto quartz substrates. This allows investigation in a transmission geometry which provides significantly more reliable data than measurements on the metallic Ti substrate. Light transmission and reflectance measurements were carried out for layers of thickness varying from 1.8 to 50 μm, and the layers were investigated in their amorphous and crystalline forms. A series of wavelength‐dependent light attenuation coefficients are extrapolated and found to match the photocurrent versus irradiation wavelength behavior. A feature specific to anodic nanotubes is that their intrinsic carbon contamination content causes a proportional sub‐bandgap response. Overall, the extracted data provide a valuable basis and understanding for the design of photo‐electrochemical devices based on TiO₂ nanotubes.     

The swap graph of the finite soluble groups

A graph is reflexive if the second largest eigenvalue of its adjacency matrix is less than or equal to 2. In this paper, we characterize trees whose line graphs are reflexive. It turns out that these trees can be of arbitrary order—they can have either a unique vertex of arbitrary degree or pendant paths of arbitrary lengths, or both. Since the reflexive line graphs are Salem graphs, we also relate some of our results to the Salem (graph) numbers.