C?H Bond dissociation of acetylene: Local density functional calculations

The C? H bond dissociation energy of acetylene was computed by both ab initio approaches and density functional theory in a local density approximation (DFT–LDA ). Structures and energies for acetylene and its dissociation products (the ethynyl and hydrogen radicals) are presented and compared. Using directly computed HCCH and HCC· energies and the exact H· value, the DFT–LDA calculations are found to yield C? H dissociation energies ranging from 129 to 131 kcal/mol, in good agreement with recent experimental and the highest level theoretical results. The DFT–LDA results show little dependence upon the computational procedure used to obtain geometries.     

Structures and synthesis of framework Rb and Cs uranyl arsenates and their relationships with their phosphate analogues

Two hydrated uranyl arsenates, Cs₂(UO₂)[(UO₂)(AsO₄)]₄(H₂O)₂ (CsUAs) and Rb₂(UO₂)[(UO₂)(AsO₄)]₄(H₂O)_4.5 (RbUAs), were synthesized by hydrothermal methods. Intensity data were collected at room temperature using MoKα radiation and a CCD-based area detector. The crystal structure of RbUAs was solved by direct methods, whereas the structure model of the phosphate Cs₂(UO₂)[(UO₂)(PO₄)]₄(H₂O)₂ was used for CsUAs; both were refined by full-matrix least-squares techniques on the basis of F² to agreement indices (CsUAs, RbUAs) wR₂=0.061,0.041, for all data, and R₁=0.032,0.021, calculated for 5098, 4991 unique observed reflections (|F_o|>4σ_F), respectively. The compound CsUAs is orthorhombic, space group Cmc2₁, Z=4, a=15.157(2), b=14.079(2), c=13.439(2) Å, V=2867.9(1) Å³. RbUAs is monoclinic, space group C2/m, Z=4, a=13.4619(4), b=15.8463(5), c=14.0068(4) Å, β=92.311(1)°, V=2985.52(2) Å³. The structures consist of sheets of arsenate tetrahedra and uranyl pentagonal bipyramids, with composition [(UO₂)(AsO₄)]⁻, that are topologically identical to the uranyl silicate sheets in uranophane-beta. These sheets are connected by a uranyl pentagonal bipyramid in the interlayer that shares corners with two arsenate tetrahedra on each of two adjacent sheets and whose fifth equatorial vertex is an H₂O group, resulting in an open framework with alkali metal cations in the larger cavities of the structures. CsUAs is isostructural with its phosphate analogue, and has two Cs atoms and a H₂O group in its structural cavities. RbUAs is not isostructural with its phosphate analogue, although it has a homeotypic framework. Its structural cavities are occupied by three Rb atoms and four H₂O groups; one Rb position and three of the interstitial H₂O groups are half-occupied. The partial occupancies of these positions probably result from the accommodation of the larger As atoms (relative to P) in the framework and resultant larger cavities.     

Pyrolysis of homoadamant-3-ene

Pyrolysis of homoadamant-3-ene ($\text{5}$), generated from 1-adamantylcarbene ($\text{7}$), leads to the same three olefins ($\text{2}$, $\text{3}$, and $\text{4}$) that are produced from pyrolysis of 3-homoadamantyl acetate ($\text{1}$).     

New Cyclophanes as Initiator Cores for the Construction of Dendritic Receptors: Host-guest complexation in aqueous solutions and structures of solid-state inclusion compounds

Cyclophanes 3 and 4 were prepared as initiator cores for the construction of dendrophanes (dendritic cydophanes) 1 and 2 , respectively, which mimic recognition sites buried in globular proteins. The tetra-oxy[6.1.6.1]paracyclophane 3 was prepared by a short three-step route (Scheme 1) and possesses a cavity binding site shaped by two diphenylmethane units suitable for the inclusion of flat aromatic substrates such as benzene and naphthalene derivatives as was shown by ¹H-NMR binding titrations in basic D₂O phosphate buffer (Table 1). The larger cyclophane 4 , shaped by two wider naphthyl(phenyl)methane spacers, was prepared in a longer, ten-step synthesis (Scheme 2) which included as a key intermediate the tetrabromocyclophane 5 . ¹H-NMR Binding studies in basic borate buffer in D₂O/CD₃OD demonstrated that 4 is an efficient steroid receptor. In a series of steroids (Table 1), complexation strength decreased with increasing substrate polarity and increasing number of polar substituents; in addition, electrostatic repulsion between carboxylate residues of host and guest also affected the binding affinity strongly. The conformationally flexible tetrabromocyclophane 5 displayed a pronounced tendency to form solid-state inclusion compounds of defined stoichiometry, which were analyzed by X-ray crystallography (Fig. 2). 1,2-Dichloroethane formed a cavity inclusion complex 5a with 1:1 stoichiometry, while in the 1:3 inclusion compound 5b with benzene, one guest is fully buried in the macrocyclic cavity and two others are positioned in channels between the Cyclophanes in the crystal lattice. In the 1:2 inclusion compound 5c , two toluene molecules penetrate with their aromatic rings the macrocyclic cavity from opposite sides in an antiparallel fashion. On the other hand, p-xylene (= 1,4-dimethylbenzene) in the 1:1 compound 5d is sandwiched between the cyclophane molecules with its two Me groups penetrating the cavities of the two macrocycles. In the 1:2 inclusion compound 5e with tetralin (= 1,2,3,4-tetrahydronaphthalene), both host and guest are statically disordered. The shape of the macrocycle in 5a – e depends strongly on the nature of the guest (Fig. 4). Characteristic for these compounds is the pronounced tendency of 5 to undergo regular stacking and to form channels for guest inclusion; these channels can infinitely extend across the macrocyclic cavities (Fig. 6) or in the crystal lattice between neighboring cyclophane stacks (Fig. 5). Also, the crystal lattice of 5c displays a remarkable zig-zag pattern of short Br…?O contacts between neighboring macrocycles (Fig. 7).     

Some estimates of theH ∞-uniform distribution

The sequences (y _n )=(a+blogn+ _n ) are uniformly distributed in the sense of the summation methodH . In the present paper the speed of convergence of this procedure is estimated for these, sequences and for some other sequences. For the sequences (a+blogn) theH -means converge considerably faster than logarithmic means.     

On the structure of a random sum-free set

Summary There is a natural probability measure on the set of all sum-free sets of natural numbers. If we represent such a set by its characteristic functions, then the zero-one random variabless(i) are far from independent, and we cannot expect a law of large numbers to hold for them. In this paper I conjecture a decomposition of into countably many more tractible pieces (up to a null set). I prove that each piece has positive measure, and show that, within each piece, a random set almost surely has a density which is a fixed rational number depending only on the piece. For example, the first such piece is made up of sets consisting entirely of odd numbers; it has probability 0.218 ..., and its members almost surely have density 1/4.     

On quadratic and sesquilinear functionals

LetX be a leftA-module, whereA is either a complex Banach *-algebra with an identity element or the field of quaternions. The main result of this note is that forQ, anA-quadratic functional defined onX, there exists a sesquilinear functionalB such thatB(x,x)=Q(x) holds for allxX.     

Local limits and harmonic functions for nonisotropic random walks on free groups

Summary Nearest neighbour random walks on the homogeneous tree representing a free group withs generators (2s) are investigated. By use of generating functions and their analytic properties a local limit theorem is derived. A study of the harmonic functions corresponding to the random walk leads to properties that characterize ther-harmonic function connected with the local limits.     

Coloring of universal graphs

LetG be a graph andr a cardinal number. Extending the theorem of J. Folkman we show that if eitherr or clG are finite then there existsH with clH = clG andH (G) _r ¹ . Answering a question of A. Hajnal we show that countably universal graphU ₃ satisfiesU ₃ (U₃) _r ¹ for every finiter.