Essential spectrum and -solutions of one-dimensional Schrödinger operators

In 1949, Hartman and Wintner showed that if the eigenvalue equations of a one-dimensional Schrödinger operator possess square integrable solutions, then the essential spectrum is nowhere dense. Furthermore, they conjectured that this statement could be improved and that under this condition the essential spectrum might always be void. This is shown to be false. It is proved that, on the contrary, every closed, nowhere dense set does occur as the essential spectrum of Schrödinger operators which satisfy the condition of existence of -solutions. The proof of this theorem is based on inverse spectral theory.

     

On the interpretation of random forces derived by projection operators

We study the derivation of a Langevin equation from a microscopic basis in order to elucidate the nature of the random force. We arrive at the conclusion that the consistent interpretation of the microscopic Langevin equation in terms of a stochastic differential equation (SDE) is according to I o rules. In addition, the random force is in general not Gaussian, and it is hence not completely characterized by its second moments.     

A new route to trinuclear carbocyanines

Amides or vinylogous amides react with tosyl chloride-pyridine to form activated intermediates which condense with Fischer's base or their vinylogs to give carbocyanine structures. Under the same conditions formylated Fischer's base reacts with vinylogous Fischer's bases to give trinuclear carbocyanines in good yields. Their structure and the limitations of this route are discussed.     

The role of radicals in metal-assisted oxygenation reactions

The oxo-functionalization of organic substrates with the aid of metal oxo moieties is of fundamental importance not only in nature but also in academic and industrial research. Nevertheless the corresponding reaction mechanisms remain among the most enigmatic in chemistry and few of them are understood in detail. Recent research efforts have resulted in significantly improved information: in the cases of many oxygenation reactions evidence has been provided for the occurrence radical intermediates, even though the high selectivity observed suggests to a different mechanism. Examples stem from various areas of chemistry and include processes involving molecular metal oxo complexes, gas-phase and matrix-isolated species, metalloenzymes, and solid-state oxide surfaces. This review treats this seemingly wide variety of systems with the aim of providing an overview of common reactivity patterns and principles, as well as open problems.     

Trennung von V,Cr und Mn aus Stahl und Aktivierungsanalyse der Legierungselemente mit 14 MeV-Neutronen

The elements are separated from iron by solvent-extraction. Cr, V and Mn are separated by ion-exchange chromatography with a strong acidic cation-exchanger. The elements are determined by 14 MeV-neutron activation analysis using γ-spectroscopy. The method is suitable for a concentration range of 0.05 to 5%.     

Joint QTAIM and Hirshfeld study of the sigma and pi charge distribution and electron delocalization in carbonyl compounds: a comparative study with the resonance model

Atomic charges and delocalization indexes (DIs) for a series of carbonyl compounds comprising dimethyl ketone, acetaldehyde, acetic acid, methyl acetate, acetamide, methyl vinyl ketone, divinyl ketone, and benzoic acid were studied using two different atomic partitionings: the QTAIM and the Hirshfeld (stockholder) scheme. The resonance model, traditionally employed to explain the reactivity of these compounds, is not in line with the total atomic charges and DIs calculated by both methodologies. However, the resonance model is supported to some extent by the pi charges and pi DIs calculated by both schemes, but the calculated values indicate that the pi population delocalizes only to a small degree. Although the absolute values of QTAIM and stockholder atomic charges are significantly different, the pi charges and the values of the DIs show similar trends for all the atoms and molecules of this study; this is especially the case for the pi DIs. A study of the electron density on the level of a single MO performed for CO, H2CO, F2CO, and H2CS reveals that the differences in the atomic sigma charges computed with both partitionings can be traced back to their different treatment of interatomic regions.     

Effective atom volumes for implicit solvent models: comparison between Voronoi volumes and minimum fluctuation volumes

An essential element of implicit solvent models, such as the generalized Born method, is a knowledge of the volume associated with the individual atoms of the solute. Two approaches for determining atomic volumes for the generalized Born model are described; one is based on Voronoi polyhedra and the other, on minimizing the fluctuations in the overall volume of the solute. Volumes to be used with various parameter sets for protein and nucleic acids in the CHARMM force field are determined from a large set of known structures. The volumes resulting from the two different approaches are compared with respect to various parameters, including the size and solvent accessibility of the structures from which they are determined. The question of whether to include hydrogens in the atomic representation of the solute volume is examined. Copyright 2001 John Wiley & Sons, Inc. J Comput Chem 22: 1857-1879, 2001