Fourier series criteria for operator decomposability

Let U be an invertible operator on a Banach space Y. U is said to betrigonometrically well-bounded provided the sequence {Uⁿ} _n =– is the Fourier-Stieltjes transform of a suitable projection-valued function E(·): [0, 2](Y). This class of operators is known to apply naturally to a variety of classical phenomena which exclude the presence of spectral measures. In the case Y reflexive we use the Cesáro means _n(U, t) of the trigonometric series _k0 k^–e^iktU^k, whichformally transfers the discrete Hilbert transform to Y, in order to give three separate necessary and sufficient conditions for U to be trigonometrically well-bounded. One of these conditions is sup {_n(U,t): n 1, t [0,2]} <      

A novel thermal rearrangement of the 2-thiabicyclo[3,1,0]hex-3-ene system. The crystal and molecular structure of ethyl-2,4-dichloro-5-hydroxy-6-methylbenzoate

Thermal rearrangement of 1,3-dichloro-6-ethoxycarbonyl-2-thiabicyclo[3, 1, 0]hex-3-ene does not follow the anticipated course, but leads instead to ethyl-2,4-dichloro-5-hydroxy-6-methylbenzoate, whose structure has been confirmed by an X-ray crystal structure determination.     

Thermolysis of fluorinated cycloalkylidene fulgides yields a new class of photochromic compounds

A series of fluorinated cycloalkylidene indolylfulgides has been designed, synthesized and characterized; most of the thermolysis products of these fulgides maintain photochromicity and display outstanding thermal and photochemical stability.     

Chemical bonding in hypervalent molecules: is the octet rule relevant?

The bonding in a large number of hypervalent molecules of P, As, S, Se, Te, Cl, and Br with the ligands F, Cl, O, CH(3), and CH(2) has been studied using the topological analysis of the electron localization function ELF. This function partitions the electron density of a molecule into core and valence basins and further classifies valence basins according to the number of core basins with which they have a contact. The number and geometry of these basins is generally in accord with the VSEPR model. The population of each basin can be obtained by integration, and so, the total population of the valence shell of an atom can be obtained as the sum of the populations of all the valence basins which share a boundary with its core basin. It was found that the population of the V(A, X) disynaptic basin corresponding to the bond, where A is the central atom and X the ligand, varies with the electronegativity of the ligand from approximately 2.0 for a weakly electronegative ligand such as CH(3) to less than 1.0 for a ligand such as F. We find that the total population of the valence shell of a hypervalent atom may vary from close to 10 for a period 15 element and close to 12 for a group 16 element to considerably less than 8 for an electronegative ligand such as F. For example, the phosphorus atom in PF(5) has a population of 5.37 electrons in its valence shell, whereas the arsenic atom in AsMe5 has a population of 9.68 electrons in its valence shell. By definition, hypervalent atoms do not obey the Lewis octet rule. They may or may not obey a modified octet rule that has taken the place of the Lewis octet rule in many recent discussions and according to which an atom in a molecule always has fewer than 8 electrons in its valence shell. We show that the bonds in hypervalent molecules are very similar to those in corresponding nonhypervalent (Lewis octet) molecules. They are all polar bonds ranging from weakly to strongly polar depending on the electronegativity of the ligands. The term hypervalent therefore has little significance except to indicate that an atom in a molecule is forming more than four electron pair bonds.     

Models of molecular geometry

Although the structure of almost any molecule can now be obtained by ab initio calculations chemists still look for simple answers to the question "What determines the geometry of a given molecule?" For this purpose they make use of various models such as the VSEPR model and qualitative quantum mechanical models such as those based on the valence bond theory. The present state of such models, and the support for them provided by recently developed methods for analyzing calculated electron densities, are reviewed and discussed in this tutorial review.     

The numerical stability of leaping methods for stochastic simulation of chemically reacting systems

Tau-leaping methods have recently been proposed for the acceleration of discrete stochastic simulation of chemically reacting systems. This paper considers the numerical stability of these methods. The concept of stochastic absolute stability is defined, discussed, and applied to the following leaping methods: the explicit tau, implicit tau, and trapezoidal tau.     

The geometry of nonmetal hydrides and the ligand radius of hydrogen

The aim of this paper was to investigate why the geometries of nonmetal hydrides are often not in accordance with the VSEPR model. From a consideration of interligand distances in a variety of BX(4), CX(4), and NX(4) molecules where X is a ligand or a lone pair and in which there are at least two H ligands we have shown that the hydrogen ligands are essentially close-packed. For each of the central atoms we have obtained a value for the ligand radius of hydrogen. These radii decrease with decreasing negative charge and increasing positive charge of the hydrogen ligand as the electronegativity of the central atom increases, as has been found previously for other ligands such as F and Cl. We show that ligand-ligand intractions are an important factor in determining bond angles in hydrides and that the ligand close-packing (LCP) model gives a better explanation of bond angles than the VSEPR model according to which bond angles depend on the electronegativity of the ligand rather than on its size. For example, although the very small angles in PH(3) and SH(2) are not in accord with the VSEPR model, they are consistent with the LCP model in that they are a consequence of the small size of hydrogen ligands which are pushed together by the lone pairs until they are almost close-packed.     

Diffusion of water vapor through a hydrophilic polymer film

Multifilm techniques have been used to measure the diffusion coefficient of water vapor in cellophane, and the data have been compared with the integral diffusion coefficient obtained in previous work with single films. The multifilm techniques lead to a much sharper resolution of the effect of concentration on diffusion, and the maximum integral diffusion coefficient. The diffusion coefficient for water vapor in cellophane peaks at a moisture content corresponding to about 70% R. H., which is presumably the “critical concentration” discussed in previous work on the thermodynamics of water sorption by cellophane.     

Characterization and photochemistry of 13-desmethyl bacteriorhodopsin

The photochemistry of the 13-desmethyl (DM) analogue of bacteriorhodopsin (BR) is examined by using spectroscopy, molecular orbital theory, and chromophore extraction followed by conformational analysis. The removal of the 13-methyl group permits the direct photochemical formation of a thermally stable, photochemically reversible state, P1(DM) (lambda(max) = 525 nm), which can be generated efficiently by exciting the resting state, bR(DM) with yellow or red light (lambda > 590 nm). Chromophore extraction analysis reveals that the retinal configuration in P1(DM) is 9-cis, identical to that of the retinal configuration in the native BR P1 state. Fourier transform infrared and Raman experiments on P1(DM) indicate an anti configuration around the C15=N bond, as would be expected of an O-state photoproduct. However, low-temperature spectroscopy and ambient, time-resolved studies indicate that the P1(DM) state forms primarily via thermal relaxation from the L(D)(DM) state. Theoretical studies on the BR binding site show that 13-dm retinal is capable of isomerizing into a 9-cis configuration with minimal steric hindrance from surrounding residues, in contrast to the native chromophore in which surrounding residues significantly obstruct the corresponding motion. Analysis of the photokinetic experiments indicates that the Arrhenius activation energy of the bR(DM) --> P1(DM) transition in 13-dm-BR is less than 0.6 kcal/mol (vs 22 +/-5 kcal/mol measured for the bR --> P (P1 and P2) reaction in 85:15 glycerol:water suspensions of wild type). Consequently, the P1(DM) state in 13-dm-BR can form directly from all-trans, 15-anti intermediates (bR(DM) and O(DM)) or all-trans, 15-syn (K(D)(DM)/L(D)(DM)) intermediates. This study demonstrates that the 13-methyl group, and its interactions with nearby binding site residues, is primarily responsible for channeling one-photon photochemical and thermal reactions and is limited to the all-trans and 13-cis species interconversions in the native protein.