ZurL 2-Bestapproximation eines konvexen Körpers durch einen bewegten konvexen Körper

LetK andL be convex bodies of then-dimensional Euclidean spaceE ⁿ . The -distance ofK andL is defined by , whereh _k andh _L denote the support functions ofK andL restricted to the unit sphereS ^n–1 E ⁿ . We consider the problem of best approximation ofL by the images k ofK under proper rigid motions :E ⁿ E ⁿ . A motion that minimizes ²(K L) is called optimal for (K, L) in the sense of the metric ². The results in the general case (n2 arbitrary) base on the fact that the Steiner points of K andL coincide if is optimal for (K, L). Forn=2 we obtain a relationship between convex convolution bodies and the underlying approximation problem.     

Coordination and ligand exchange dynamics of solvated metal ions

Recent developments in computer speed and capacity have opened the access to highly accurate molecular dynamics simulations based on quantum mechanically calculated forces for the chemically relevant region around ions in solution (QM/MM formalism). This accuracy, although still extremely consuming (30-300 days of CP time per simulation), is needed for reliable structural details and ligand exchange rates. A large number of main group and transition metal ions have been investigated by this approach, giving very detailed insight into the properties of these ions in solution and allowing to classify the ions by various characteristics. Most first-row transition metal ions have a very stable first hexa-coordinated solvation shell, whose vibrational distortions, however, strongly influence the dynamics of the second shell. The dynamical Jahn-Teller effect - shown to be a femto- and picosecond phenomenon - can strongly influence ligand coordination and exchange dynamics. A large number of ions with very labile solvation shell such as most main group ions, but also transition metal ions, e.g. Ag(I) and Hg(II), can change their coordination within the picosecond scale, leading to an almost simultaneous presence of several species hardly accessible by present experimental techniques. Among these ions, the structure breakers are of particular interest, and it could be shown that there are two types of them, one with a large and very labile first coordination shell such as Cs(I), the other characterised by a small first but an unusually large second solvation shell such as Au(I). Investigations of metal ions coordinated to ammonia ligands have shown that coordination to hetero-atoms can accelerate the ligand exchange reaction rates by several orders of magnitude, e.g. for Cu(II) and Ni(II). Simulations of ions in aqueous ammonia gave a very detailed picture of the complexity of species almost simultaneously present and illustrate the enormous difficulties encountered when trying to fit X-ray or neutron diffraction data for such systems. In general, ligand exchange rates situated in the picosecond range are far below the NMR scale, and as femtosecond laser pulse spectroscopy could not be applied so far to ionic solutions, accurate simulations have become a very important tool to access structure and dynamics of solvated ions. A number of VIDEO clips supplied on the Web as supporting material illustrates the processes occurring in solutions of the metal ions.     

An extended ab initio QM/MM MD approach to structure and dynamics of Zn(II) in aqueous solution

Structural and dynamical properties of Zn(II) in aqueous solution were investigated, based on an ab initio quantum mechanical/molecular mechanical (QM/MM) molecular dynamics simulation at double-zeta Hartree-Fock quantum mechanical level including the first and second hydration shells into the QM region. The inclusion of the second shell in the QM region resulted in significant changes in the properties of the hydrate. The first shell coordination number was found to be 6, the second shell consists of approximately 14 water molecules. The structural properties were determined in terms of RDF, ADF, tilt and theta angle distributions, while dynamics were characterized by mean ligand residence times, ion-ligand stretching frequencies and the vibrational and librational motions of water ligands.     

Structure and ultrafast dynamics of liquid water: a quantum mechanics/molecular mechanics molecular dynamics simulations study

A quantum mechanics/molecular mechanics molecular dynamics simulation was performed for liquid water to investigate structural and dynamical properties of this peculiar liquid. The most important region containing a central reference molecule and all nearest surrounding molecules (first coordination shell) was treated by Hartree-Fock (HF), post-Hartree-Fock [second-order Moller-Plesset perturbation theory (MP2)], and hybrid density functional B3LYP [Becke's three parameter functional (B3) with the correlation functional of Lee, Yang, and Parr (LYP)] methods. In addition, another HF-level simulation (2HF) included the full second coordination shell. Site to site interactions between oxygen-oxygen, oxygen-hydrogen, and hydrogen-hydrogen atoms of all ab initio methods were compared to experimental data. The absence of a second peak and the appearance of a shoulder instead in the gO-O graph obtained from the 2HF simulation is notable, as this feature has been observed so far only for pressurized or heated water. Dynamical data show that the 2HF procedure compensates some of the deficiency of the HF one-shell simulation, reducing the difference between correlated (MP2) and HF results. B3LYP apparently leads to too rigid structures and thus to an artificial slow down of the dynamics.     

Zur Theorie der Erzeugung der dritten optischen Harmonischen bei optimaler Fokussierung

The optical third harmonic generation of focused Gaussian light beams is studied theoretically in uniaxial crystals with third order non-linearity. The numerical integration of the parabolic equation for optimum focusing condition shows that, for strong focusing, the third harmonic power is independent of the focusing parameter=I/b (I=crystal length,b=confocal parameter). Therefore, the third harmonic generation differs from the second harmonic generation. We neglect in our calculations pump depletion and the intensity dependence of the refractive index.

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Herrn V. Russu danke ich für Diskussionen. Frl. J. Bohn bin ich für die Durchführung der numerischen Rechnungen zu Dank verpflichtet.     

Bis(triazacyclohexane) sandwich complexes of (Cu(I))(2), Cu(II) and Zn(II): complexes with cuprophilic attraction between two cationic copper(I) leading to unusual reactivity with dioxygen

As the first 1st-row transition metal complexes having six tertiary amine donor groups, bis(triazacyclohexane) sandwich complexes [L2M](BF4)2 (L = benzyl- or p-fluorobenzyl-triazacyclohexane, M = Cu or Zn) have been obtained by the protonolysis of Et2Zn in the presence of L or by reaction of [Cu(MeCN)4](BF4) with L in CH2Cl2 and subsequent air oxidation via an unprecedented Cu(I)(2) sandwich complex containing a short Cu-Cu contact.     

Temperature dependence of structure and dynamics of the hydrated Ca2+ ion according to ab initio quantum mechanical charge field and classical molecular dynamics

Simulations using ab initio quantum mechanical charge field molecular dynamics (QMCF MD) and classical molecular dynamics using two‐body and three‐body potentials were performed to investigate the hydration of the Ca²⁺ ion at different temperatures. Results from the simulations demonstrate significant effects of temperature on solution dynamics and the corresponding composition and structure of hydrated Ca²⁺. Substantial increase in ligand exchange events was observed in going from 273.15 K to 368.15 K, resulting in a redistribution of coordination numbers to lower values. The effect of temperature is also visible in a red‐shift of the ion‐oxygen stretching frequencies, reflecting weakened ligand binding. Even the moderate increase from ambient to body temperature leads to significant changes in the properties of Ca²⁺ in aqueous environment. © 2009 Wiley Periodicals, Inc. J Comput Chem, 2010     

Improved measurement of the hydrogen 1S-2S transition frequency

We have measured the 1S-2S transition frequency in atomic hydrogen via two-photon spectroscopy on a 5.8 K atomic beam. We obtain f(1S-2S) = 2,466,061,413,187,035 (10) Hz for the hyperfine centroid, in agreement with, but 3.3 times better than the previous result [M. Fischer et al., Phys. Rev. Lett. 92, 230802 (2004)]. The improvement to a fractional frequency uncertainty of 4.2 × 10(-15) arises mainly from an improved stability of the spectroscopy laser, and a better determination of the main systematic uncertainties, namely, the second order Doppler and ac and dc Stark shifts. The probe laser frequency was phase coherently linked to the mobile cesium fountain clock FOM via a frequency comb.