The virial theorem in relativistic quantum mechanics

For a class of potentials including the Coulomb potential q = μr^?1 with ¦ μ ¦ < 1 (1) (i.e., atomic numbers Z ? 137), the virial theorem $\text{(u, α ·}\text{p}\text{u) = (u, r(}\text{?q}\text{?r}\text{)u)}$ is shown to hold, u being an eigenfunction of the operator

$\text{Hu = TU : = (α ·}\text{p}\text{+ β + q)u}$

,

$\text{D(H) = {u ¦ u ∈ [H}{}_{\mathrm{loc}}{}^1\text{(}\text{R}\text{+}{}^3\text{)]}{}^4\text{, r}{}^{\mathrm{<mtext>?</mtext><mtext>1</mtext><mtext>2</mtext>}}\text{u, TU ∈ [L}{}^2\text{(R)}{}^3\text{]}{}^4\text{}}$

($\text{R}$₊³ := $\text{R}$?{0}). The result implies in particular that H with (1) does not have any eigenvalues embedded in the continuum. The proof uses a scale transformation.     

The First Catalytic Synthesis of an Acrylate from CO2 and an Alkene—A Rational Approach

For more than three decades the catalytic synthesis of acrylates from the cheap and abundantly available C₁ building block carbon dioxide and alkenes has been an unsolved problem in catalysis research, both in academia and industry. Herein, we describe a homogeneous catalyst based on nickel that permits the catalytic synthesis of the industrially highly relevant acrylate sodium acrylate from CO₂, ethylene, and a base, as demonstrated, at this stage, by a turnover number of greater than 10 with respect to the metal.     

Explicit uniform bounds on integrals of Bessel functions and trace theorems for Fourier transforms

Explicit and partly sharp estimates are given of integrals over the square of Bessel functions with an integrable weight which can be singular at the origin. They are uniform with respect to the order of the Bessel functions and provide explicit bounds for some smoothing estimates as well as for the L² restrictions of Fourier transforms onto spheres in which are independent of the radius of the sphere. For more special weights these restrictions are shown to be Hölder continuous with a Hölder constant having this independence as well. To illustrate the use of these results a uniform resolvent estimate of the free Dirac operator with mass in dimensions is derived.     

A New Class of Lanthanide Complexes with Three Ligand Centered Radicals: NMR Evaluation of Ligand Field Energy Splitting and Magnetic Coupling

Combination of three radical anionic Ph-BIAN ligands (Ph-BIAN=bis-(phenylimino)-acenaphthenequinone) with lanthanoid ions leads to a series of homoleptic, six-coordinate complexes of the type Ln(Ph-BIAN)₃. Magnetic coupling data were measured by paramagnetic solution NMR spectroscopy. Combining ¹H NMR with ²H NMR of partially deuterated compounds allowed a detailed study of the magnetic susceptibility anisotropies over a large temperature range. The observed chemical shifts were separated into ligand- and metal-centered contributions by comparison with the Y analogue (diamagnetic at the metal). The metal-centered contributions of the complexes with the paramagnetic ions could then be separated into pseudocontact and Fermi contact shifts. The latter is large within the Ph-BIAN scaffold, which shows that magnetic coupling is significant between the lanthanide ion and the radical ligand. Pseudocontact shifts were further correlated to structural data obtained from X-ray diffraction experiments. Ligand-field parameters were determined by fitting the temperature dependence of the observed magnetic susceptibility anisotropies. The electronic structure determined by this approach shows, that the Er and Tm analogues are candidates for single molecule magnets (SMM). These results demonstrate the possibilities for the application of NMR spectroscopy in investigations of paramagnetic systems in general and single molecule magnets in particular.     

Ylide Chemistry: An Account of Structural,Conformational and Redox Investigations

Abstract

Structural data of phosphorus ylides are summarized and discussed. Conformational preferences in solution and in the solid state, as followed by NMR spectroscopy and X-ray diffraction, indicate a strong gauche effect for the lone pairs of electrons at the ylidic carbon atom. One- and two-electron oxydation of ylide carbanions is accomplished with copper(II) as the oxidant, leading to oxidative coupling or to halogenation, respectively.