Relativistic virial theorem for diatomic molecules. Application to H 2 +

Summary The virial theorem for a molecule in the relativistic clamped-nuclei approximation is derived. The individual energy contributionsA (momentum energy),B (mass energy),T=A+B (kinetic energy) andV (potential energy) are expressed in terms ofE, E/R (derivate w.r.t. the nuclear coordinates) and the relativistic correction E/² (derivative w.r.t. Sommerfield's fine-structure constant ). IfE and E/R are known as functions of , then all individual energy terms are also known as functions of . As an example, numerical results for H ₂ ⁺ are presented. The relativistic and nonrelativistic potential energy curves and the paradoxical behavior of their different contributions are analyzed and interpreted in both the largeR and shortR ranges.Dedicated to Professor W. Kutzelnigg on the occasion of his 60th birthday     

Theory-enforced Re-investigation of the Origin of the Large Metal–Carbon Bond Strength in PdCH2I+ and its reactions with unsaturated hydrocarbons

State-of-the-art ab initio studies demonstrate that the reaction Pd⁺ + CH₃I → PdCH₂I⁺ + H^. is endothermic by ca. 20 kcal/mol, which translates into a bond dissociation energy (BDE) of ca. 83 kcal/mol for the Pd⁺? CH₂I bond. This figure is in agreement with an experimental bracket of 68 kcal/mol < BDE(Pd⁺? CH₂I) < 92 kcal/mol. Based on these findings, the previously studied Pd⁺/CH₃I system was re-investigated, and double-resonance experiments demonstrate that the formation of PdCH₂I⁺ occurs stepwise via PdCH as a reactive intermediate. Further, ion/molecule reactions of PdCH₂I⁺ with unsaturated hydrocarbons are studied, which reveal the formation of carbon–carbon bonds in the gas phase.     

Polyelectrolyte adsorption on charged surfaces: study by electrokinetic measurements

In order to describe the influence of cationic polyelectrolytes on flocculation of disperse systems the adsorption of poly (diallyldimethylammonium chloride) (PDADMAC) onto silica, mica and acidic polymer latex was investigated. The plateau value of the adsorption isotherms grows with increasing surface charge density of the substrates and electrolyte concentration. The adsorbed layer of the polycation was characterized by zeta potential measurements with KCl solutions of constant ionic strength and varied pH. The zero point of the charge as well as the shape of the zeta potential–pH plot depends on the coverage of the surface with polycations. For fully covered substrates the zero point of the charge as well as the pK_A and pK_B values calculated by a stochastic search programme are independent of the substrate. Maximum flocculation was observed at about 30% of the plateau value of the adsorption isotherms.     

Relativistic perturbation theory of molecular structure

Summary The recently developed relativistic double perturbation theory is extended to handle relativistic changes of molecular structure more easily. This is achieved by simple coordinate scalings. Accurate higher order mixed perturbation energies for H ₂ ⁺ are calculated. The relativistic changes of bond energy,DE, of bond length,R _e , and especially of force constant,k, and of anharmonicity,a, are large, up to 100%·(Z/c)². The dominant contributions tok anda are due to the indirect change of the nonrelativistick anda connected with the relativistic change of bond length. Accordingly the relativistic changes obey Badger's and Gordy's rules (–RDEk).Dedicated to Prof. Klaus Ruedenberg in appreciation of his fundamental contributions to both formal theory and physical explanations in quantum chemistry     

Chemical Bonding and Pressure‐Induced Change of the Electron Configuration of Ytterbium in β‐YbAgGa2

Single‐phase polycrystalline samples of the intermetallic compound β‐YbAgGa₂ were synthesized by inductive heating and subsequent annealing for eight weeks at 670 K. Magnetic properties were characterized by susceptibility measurements and indicated intermediate valence of ytterbium at ambient pressure. Angle‐dispersive X‐ray powder diffraction data of orthorhombic β‐YbAgGa₂ indicate stability of the phase in the investigated pressure range from 0.1 MPa (ambient pressure) to 19 GPa. The pressure‐induced volume decrease is accompanied by an increase of the effective valence from 2.17 at ambient conditions to 2.71 at 16 GPa as evaluated by X‐ray absorption spectroscopy at the Yb L_III threshold. Analysis of the chemical bonding in β‐YbAgGa₂ by integrating the electron density of the polyanion in basins as defined by the electron localization function results in an electron count Yb^2.7+[(Ag^0.2—)(Ga1(3b)^1.0—)(Ga2(4b)^1.5—)]. This finding is close to the expected values calculated by means of the Zintl rules and fits well the results of magnetic susceptibility measurements and XAS investigations.     

Metallderivate von Molekülverbindungen. III. Molekül- und Kristallstruktur des Lithium-bis (trimethylsilyl)-phosphids · DME und des Lithium-dihydrogenphosphids · DME

Metal Derivatives of Molecular Compounds. III. Molecular and Crystal Structure of Lithium bis(trimethylsilyl)phosphide · DME and of Lithium dihydrogenphosphide · DME Lithium bis(trimethylsilyl)phosphide · DME 1 prepared from tris(trimethylsilyl)-phosphine and lithium methanide [2, 4] in 1,2-dimethoxyethane

1 1,2-Dimethoxyethan (DME); Tetrahydrofuran (THF); Bis[2-(dimethylamino)ethyl]methyl-amin (PMDETA).

, crystallizes in the orthorhombic space group Pnnn {a = 881.1(9); b = 1308.5(9); c = 1563.4(9) pm at ?120 ± 3°C; Z = 4 formula units}, lithium dihydrogenphosphide · DME 2 [10] prepared from phosphine and lithium- n -butanide in the same solvent, in P2 ₁ 2 ₁ 2 ₁ {a = 671.8(1); b = 878.6(1); c = 1332.2(2) pm at ?120 ± 3°C; Z = 4 formula units}. X-ray structure determinations (R _w = 0.036/0.045) show the bis(trimethylsilyl) derivative 1 to be dimeric with a planar P? Li? P? Li ring (P? Li 256 pm; Li? P? Li 76°; P? Li? P 104°), and the dihydrogenphosphide 2 to be polymeric with a linear Li? P? Li fragment (P? Li 254 to 260 pm; Li? P? Li 177°; P? Li? P 118°). The shortened P? Si distance (221 pm) of compound 1 and the structure of the PH ₂ group in 2 are discussed in detail. Lithium obtains its preferred coordination number 4 by a chelation with one molecule of 1,2-dimethoxyethane (Li? O 202 to 204 pm).     

Photoredox-enabled 1,2-dialkylation of α-substituted acrylates via Ireland–Claisen rearrangement

Herein, we report the 1,2-dialkylation of simple feedstock acrylates for the synthesis of valuable tertiary carboxylic acids by merging Giese-type radical addition with an Ireland–Claisen rearrangement. Key to success is the utilization of the reductive radical-polar crossover concept under photocatalytic reaction conditions to force the [3,3]-sigmatropic rearrangement after alkyl radical addition to allyl acrylates. Using readily available alkyl boronic acids as radical progenitors, this redox-neutral, transition-metal-free protocol allows the mild formation of two C(sp³)–C(sp³) bonds, thus providing rapid access to complex tertiary carboxylic acids in a single step. Moreover, this strategy enables the efficient synthesis of highly attractive α,α-dialkylated γ-amino butyric acids (GABAs) when α-silyl amines are used as radical precursors – a structural motif that was still inaccessible in related transformations. Depending on the nature of the radical precursors and their inherent oxidation potentials, either a photoredox-induced radical chain or a solely photoredox mechanism is proposed to be operative.

A photocatalytic 1,2-dialkylation of α-substituted acrylates is enabled by a reaction cascade combining reductive radical-polar crossover with the established Ireland–Claisen rearrangement for the synthesis of valuable tertiary carboxylic acids.     

Periodic,multiplicative number-theoretical functions

D. Leitmann andD. Wolke [3] proved that multiplicative functions, which are restrictions of continuous, periodic functionsF: with irrational period, are constant equal to 0 or 1. In this note a simpler proof for this result is given.     

Convergence of the Finite Difference Scheme for a Fast Diffusion Equation in Porous Media

We are concerned with an implicit scheme for the finite difference solution to a nonlinear parabolic equation with a multivalued coefficient that describes the fast diffusion in a porous medium. The boundary conditions contain the multivalued function as well. We prove the stability and the convergence of the scheme, emphasizing the precise nature of convergence in this specific case, and compute the error level of the approximating solution. The method is aimed to simplify the numerical computations for the solutions to equations of this type, without performing an approximation of the multivalued function. The theory is illustrated by numerical results.