A model for the pseudorotation of cycloheptane

A very simple model for a seven-membered ring having fixed bond lengths and fixed ring bond angles is presented. The model pseudorotates in both the twist-chair/chair and the twist-boat/boat conformations and gives reasonably accurate detailed geometries.     

Origin and Use of Hydroxyl Group Tolerance in Cationic Molybdenum Imido Alkylidene N-Heterocyclic Carbene Catalysts

The origin of hydroxyl group tolerance in neutral and especially cationic molybdenum imido alkylidene N-heterocyclic carbene (NHC) complexes has been investigated. A wide range of catalysts was prepared and tested. Most cationic complexes can be handled in air without difficulty and display an unprecedented stability towards water and alcohols. NHC complexes were successfully used with substrates containing the hydroxyl functionality in acyclic diene metathesis polymerization, homo-, cross and ring-opening cross metathesis reactions. The catalysts remain active even in 2-PrOH and are applicable in ring-opening metathesis polymerization and alkene homometathesis using alcohols as solvent. The use of weakly basic bidentate, hemilabile anionic ligands such as triflate or pentafluorobenzoate and weakly basic aromatic imido ligands in combination with a sterically demanding 1,3-dimesitylimidazol-2-ylidene NHC ligand was found essential for reactive and yet robust catalysts.     

Entangling different degrees of freedom by quadrature squeezing cylindrically polarized modes

Quantum systems such as, for example, photons, atoms, or Bose-Einstein condensates, prepared in complex states where entanglement between distinct degrees of freedom is present, may display several intriguing features. In this Letter we introduce the concept of such complex quantum states for intense beams of light by exploiting the properties of cylindrically polarized modes. We show that already in a classical picture the spatial and polarization field variables of these modes cannot be factorized. Theoretically it is proven that by quadrature squeezing cylindrically polarized modes one generates entanglement between these two different degrees of freedom. Experimentally we demonstrate amplitude squeezing of an azimuthally polarized mode by exploiting the nonlinear Kerr effect in a specially tailored photonic crystal fiber. These results display that such novel continuous-variable entangled systems can, in principle, be realized.     

Upper Bound on the Packing Density of Regular Tetrahedra and Octahedra

Aristotle contended that (regular) tetrahedra tile space, an opinion that remained widespread until it was observed that non-overlapping tetrahedra cannot subtend a solid angle of 4π around a point if this point lies on a tetrahedron edge. From this 15th century argument, we can deduce that tetrahedra do not tile space but, more than 500 years later, we are unaware of any known non-trivial upper bound to the packing density of tetrahedra. In this article, we calculate such a bound. To this end, we show the existence, in any packing of regular tetrahedra, of a set of disjoint spheres centered on tetrahedron edges, so that each sphere is not fully covered by the packing. The bound on the amount of space that is not covered in each sphere is obtained in a recursive way by building on the solid angle argument. The argument can be readily modified to apply to other polyhedra. The resulting lower bound on the fraction of empty space in a packing of regular tetrahedra is 2.6…×10⁻²⁵ and reaches 1.4…×10⁻¹² for regular octahedra.     

Honig

Ohne Zusammenfassung