Properties of α-monoclinic selenium

The growth of bulky and platelet shaped α-monoclinic crystals is discussed. A simple method is devised for identifying and orienting them.

The density, previously in disagreement with the value calculated from x-ray studies, is carefully redetermined, and found to be in good agreement with the latter.

The relative dielectric constant is determined, an effort being made to eliminate errors inherent in previous measurements, which have not been in agreement. A two parameter model is derived which explains the anisotropy in the relative dielectric constant of orthorhombic sulfur, which is also composed of 8-atom puckered ring molecules. The model works less well for α-monoclinic selenium. The relative dielectric constant anisotropy is quite noticeable, being 6.06 along the crystal b axis, and 8.52-8.93 normal to the axis.

Thin crystal platelets of α-monoclinic selenium (less than 1µ thick) are used to extend optical transmission measurements up to 4.5eV. Previously the measurements extended up to 2.1 eV, limited by the thickness of the available crystals. The absorption edge is at 2.20 eV, with changes in slope of the absorption coefficient occurring at 2.85 eV and 3.8 eV. Measurement of transmission through solutions of selenium in CS_2 and trichlorethylene yield an absorption edge of 2.75 eV. There is evidence the selenium exists in solution partly as Se_8 rings, the building block of monoclinic selenium. Transmission is measured at low temperatures (80°K and 10°K) using the platelets. The absorption edge is at 2.38 eV and 2.39 eV, respectively, for the two temperatures. Measurements at low temperatures with polarized and unpolarized light reveal interesting absorption anisotropy near 2.65 eV.

I. The crystal structure of trimesic acid. II. Topics in crystallographic calculations

I. Trimesic acid (1, 3, 5-benzenetricarboxylic acid) crystallizes with a monoclinic unit cell of dimensions a = 26.52 A, b = 16.42 A, c = 26.55 A, and β = 91.53° with 48 molecules /unit cell. Extinctions indicated a space group of Cc or C2/c; a satisfactory structure was obtained in the latter with 6 molecules/asymmetric unit - C₅₄O₃₆H₃₆ with a formula weight of 1261 g. Of approximately 12,000 independent reflections within the CuK_α sphere, intensities of 11,563 were recorded visually from equi-inclination Weissenberg photographs.

The structure was solved by packing considerations aided by molecular transforms and two- and three-dimensional Patterson functions. Hydrogen positions were found on difference maps. A total of 978 parameters were refined by least squares; these included hydrogen parameters and anisotropic temperature factors for the C and O atoms. The final R factor was 0.0675; the final "goodness of fit" was 1.49. All calculations were carried out on the Caltech IBM 7040-7094 computer using the CRYRM Crystallographic Computing System.

The six independent molecules fall into two groups of three nearly parallel molecules. All molecules are connected by carboxylto- carboxyl hydrogen bond pairs to form a continuous array of sixmolecule rings with a chicken-wire appearance. These arrays bend to assume two orientations, forming pleated sheets. Arrays in different orientations interpenetrate - three molecules in one orientation passing through the holes of three parallel arrays in the alternate orientation - to produce a completely interlocking network. One third of the carboxyl hydrogen atoms were found to be disordered.

II. Optical transforms as related to x-ray diffraction patterns are discussed with reference to the theory of Fraunhofer diffraction.

The use of a systems approach in crystallographic computing is discussed with special emphasis on the way in which this has been done at the California Institute of Technology.

An efficient manner of calculating Fourier and Patterson maps on a digital computer is presented. Expressions for the calculation of to-scale maps for standard sections and for general-plane sections are developed; space-group-specific expressions in a form suitable for computers are given for all space groups except the hexagonal ones.

Expressions for the calculation of settings for an Eulerian-cradle diffractometer are developed for both the general triclinic case and the orthogonal case.

Photographic materials on pp. 4, 6, 10, and 20 are essential and will not reproduce clearly on Xerox copies. Photographic copies should be ordered.

Strategies for the Stereoselective Synthesis of Carbon Quaternary Centers via Transition Metal-Catalyzed Alkylation of Enolate Compounds

Notwithstanding advances in modern chemical methods, the selective installation of sterically encumbered carbon stereocenters, in particular all-carbon quaternary centers, remains an unsolved problem in organic chemistry. The prevalence of all-carbon quaternary centers in biologically active natural products and pharmaceutical compounds provides a strong impetus to address current limitations in the state of the art of their generation. This thesis presents four related projects, all of which share in the goal of constructing highly-congested carbon centers in a stereoselective manner, and in the use of transition-metal catalyzed alkylation as a means to address that goal.

The first research described is an extension of allylic alkylation methodology previously developed in the Stoltz group to small, strained rings. This research constitutes the first transition metal-catalyzed enantioselective α-alkylation of cyclobutanones. Under Pd-catalysis, this chemistry affords all–carbon α-quaternary cyclobutanones in good to excellent yields and enantioselectivities.

Next is described our development of a (trimethylsilyl)ethyl β-ketoester class of enolate precursors, and their application in palladium–catalyzed asymmetric allylic alkylation to yield a variety of α-quaternary ketones and lactams. Independent coupling partner synthesis engenders enhanced allyl substrate scope relative to allyl β-ketoester substrates; highly functionalized α-quaternary ketones generated by the union of our fluoride-triggered β-ketoesters and sensitive allylic alkylation coupling partners serve to demonstrate the utility of this method for complex fragment coupling.

Lastly, our development of an Ir-catalyzed asymmetric allylic alkylation of cyclic β-ketoesters to afford highly congested, vicinal stereocenters comprised of tertiary and all-carbon quaternary centers with outstanding regio-, diastereo-, and enantiocontrol is detailed. Implementation of a subsequent Pd-catalyzed alkylation affords dialkylated products with pinpoint stereochemical control of both chiral centers. The chemistry is then extended to include acyclic β-ketoesters and similar levels of selective and functional group tolerance are observed. Critical to the successful development of this method was the employment of iridium catalysis in concert with N-aryl-phosphoramidite ligands.

Some constructions, related to noncommutative tori; Fredholm modules and the Beilinson-Bloch regulator

A noncommutative 2-torus is one of the main toy models of noncommutative geometry, and a noncommutative n-torus is a straightforward generalization of it. In 1980, Pimsner and Voiculescu in [17] described a 6-term exact sequence, which allows for the computation of the K-theory of noncommutative tori. It follows that both even and odd K-groups of n-dimensional noncommutative tori are free abelian groups on 2^n-1 generators. In 1981, the Powers-Rieffel projector was described [19], which, together with the class of identity, generates the even K-theory of noncommutative 2-tori. In 1984, Elliott [10] computed trace and Chern character on these K-groups. According to Rieffel [20], the odd K-theory of a noncommutative n-torus coincides with the group of connected components of the elements of the algebra. In particular, generators of K-theory can be chosen to be invertible elements of the algebra. In Chapter 1, we derive an explicit formula for the First nontrivial generator of the odd K-theory of noncommutative tori. This gives the full set of generators for the odd K-theory of noncommutative 3-tori and 4-tori.

In Chapter 2, we apply the graded-commutative framework of differential geometry to the polynomial subalgebra of the noncommutative torus algebra. We use the framework of differential geometry described in [27], [14], [25], [26]. In order to apply this framework to noncommutative torus, the notion of the graded-commutative algebra has to be generalized: the "signs" should be allowed to take values in U(1), rather than just {-1,1}. Such generalization is well-known (see, e.g., [8] in the context of linear algebra). We reformulate relevant results of [27], [14], [25], [26] using this extended notion of sign. We show how this framework can be used to construct differential operators, differential forms, and jet spaces on noncommutative tori. Then, we compare the constructed differential forms to the ones, obtained from the spectral triple of the noncommutative torus. Sections 2.1-2.3 recall the basic notions from [27], [14], [25], [26], with the required change of the notion of "sign". In Section 2.4, we apply these notions to the polynomial subalgebra of the noncommutative torus algebra. This polynomial subalgebra is similar to a free graded-commutative algebra. We show that, when restricted to the polynomial subalgebra, Connes construction of differential forms gives the same answer as the one obtained from the graded-commutative differential geometry. One may try to extend these notions to the smooth noncommutative torus algebra, but this was not done in this work.

A reconstruction of the Beilinson-Bloch regulator (for curves) via Fredholm modules was given by Eugene Ha in [12]. However, the proof in [12] contains a critical gap; in Chapter 3, we close this gap. More specifically, we do this by obtaining some technical results, and by proving Property 4 of Section 3.7 (see Theorem 3.9.4), which implies that such reformulation is, indeed, possible. The main motivation for this reformulation is the longer-term goal of finding possible analogs of the second K-group (in the context of algebraic geometry and K-theory of rings) and of the regulators for noncommutative spaces. This work should be seen as a necessary preliminary step for that purpose.

For the convenience of the reader, we also give a short description of the results from [12], as well as some background material on central extensions and Connes-Karoubi character.

Electrocatalytic reduction of dioxygen at chemically modified electrodes

The kinetics of the reduction of O₂ by Ru(NH₃)₆⁺² as catalyzed by cobalt(II) tetrakis(4-N-methylpyridyl)porphyrin are described both in homogeneous solution and when the reactants are confined to Nafion coatings on graphite electrodes. The catalytic mechanism is determined and the factors that can control the total reduction currents at Nafion-coated electrodes are specified. A kinetic zone diagram for analyzing the behavior of catalyst-mediator-substrate systems at polymer coated electrodes is presented and utilized in identifying the current-limiting processes. Good agreement is demonstrated between calculated and measured reduction currents at rotating disk electrodes. The experimental conditions that will yield the optimum performance of coated electrodes are discussed, and a relationship is derived for the optimal coating thickness.

The relation between the reduction potentials of adsorbed and unadsorbed cobalt(III) tetrakis(4-N-methylpyridyl)porphyrin and those where it catalyzes the electroreduction of dioxygen is described. There is an unusually large change in the formal potential of the Co(III) couple upon the adsorption of the porphyrin on the graphite electrode surface. The mechanism in which the (inevitably) adsorbed porphyrin catalyzes the reduction of O₂ is in accord with a general mechanistic scheme proposed for most monomeric cobalt porphyrins.

Four new dimeric metalloporphyrins (prepared in the laboratory of Professor C. K. Chang) have the two porphyrin rings linked by an anthracene bridge attached to meso positions. The electrocatalytic behavior of the diporphyrins towards the reduction of O₂ at graphite electrodes has been examined for the following combination of metal centers: Co-Cu, Co-Fe, Fe-Fe, Fe-H₂. The Co-Cu diporphyrin catalyzes the reduction of O₂ to H₂O₂ but no further. The other three catalysts all exhibit mixed reduction pathways leading to both H₂O₂ and H₂O. However, the pathways that lead to H₂O do not involve H₂O₂ as an intermediate. A possible mechanistic scheme is offered to account for the observed behavior.

Imidazolium-Based Organic Structure Directing Agents for the Synthesis of Microporous Materials

The central theme of this thesis is the use of imidazolium-based organic structure directing agents (OSDAs) in microporous materials synthesis. Imidazoliums are advantageous OSDAs as they are relatively inexpensive and simple to prepare, show robust stability under microporous material synthesis conditions, have led to a wide range of products, and have many permutations in structure that can be explored. The work I present involves the use of mono-, di-, and triquaternary imidazolium-based OSDAs in a wide variety of microporous material syntheses. Much of this work was motivated by successful computational predictions (Chapter 2) that led me to continue to explore these types of OSDAs. Some of the important discoveries with these OSDAs include the following: 1) Experimental evaluation and confirmation of a computational method that predicted a new OSDA for pure-silica STW, a desired framework containing helical pores that was previously very difficult to synthesize. 2) Discovery of a number of new imidazolium OSDAs to synthesize zeolite RTH, a zeolite desired for both the methanol-to-olefins reaction as well as NOX reduction in exhaust gases. This discovery enables the use of RTH for many additional investigations as the previous OSDA used to make this framework was difficult to synthesize, such that no large scale preparations would be practical. 3) The synthesis of pure-silica RTH by topotactic condensation from a layered precursor (denoted CIT-10), that can also be pillared to make a new framework material with an expanded pore system, denoted CIT-11, that can be calcined to form a new microporous material, denoted CIT-12. CIT-10 is also interesting since it is the first layered material to contain 8 membered rings through the layers, making it potentially useful in separations if delamination methods can be developed. 4) The synthesis of a new microporous material, denoted CIT-7 (framework code CSV) that contains a 2-dimensional system of 8 and 10 membered rings with a large cage at channel intersections. This material is especially important since it can be synthesized as a pure-silica framework under low-water, fluoride-mediated synthesis conditions, and as an aluminosilicate material under hydroxide mediated conditions. 5) The synthesis of high-silica heulandite (HEU) by topotactic condensation as well as direct synthesis, demonstrating new, more hydrothermally stable compositions of a previously known framework. 6) The synthesis of germanosilicate and aluminophosphate LTA using a triquaternary OSDA. All of these materials show the diverse range of products that can be formed from OSDAs that can be prepared by straightforward syntheses and have made many of these materials accessible for the first time under facile zeolite synthesis conditions.

Multiplication in Riesz spaces

A.G. Vulih has shown how an essentially unique intrinsic multiplication can be defined in certain types of Riesz spaces (vector lattices) L. In general, the multiplication is not universally defined in L, but L can always be imbedded in a large space L^# in which multiplication is universally defined.

If ф is a normal integral in L, then ф can be extended to a normal integral on a large space L₁(ф) in L^#, and L₁(ф) may be regarded as an abstract integral space. A very general form of the Radon-Nikodym theorem can be proved in L₁(ф), and this can be used to give a relatively simple proof of a theorem of Segal giving a necessary and sufficient condition that the Radon-Nikodym theorem hold in a measure space.

In another application, the multiplication is used to give a representation of certain Riesz spaces as rings of operators on a Hilbert space.

The binding and catalytic properties of lysozyme

The binding and catalytic properties of hen's egg white lysozyme have been studied by a variety of techniques. These studies show that the enzyme has three contiguous binding subsites, A, B, and C. The application of nuclear magnetic resonance (NMR) spectroscopy to probe the binding environment of several saccharides to lysozyme has demonstrated that the reducing end sugar rings of chitotriose, chitobiose and the β-form of N-acetylglucosamine all bind in subsite C. The central sugar ring of chitotriose and the sugar ring at the nonreducing end of chitobiose were found to bind in subsite B, while the nonreducing end sugar residue of chitotriose occupied subsite A. The dynamics of the binding process has also been investigated by NMR. The formation rate constant of chitobiose--and chitotriose-enzyme complexes were found to be about 4 X 10^-6 M^-1 sec^-1 with small activation energies.

The stereochemical path of the lysozyme catalyzed hydrolysis of glycosidic bonds has been shown to proceed with at least 99.7% retention of configuration at C-1 of the sugar. The lysozyme catalyzed hydrolysis of glucosidic bonds has been shown to be largely carbonium ion in character by virtue of the α-deuterium kinetic isotope effect (k_H/k_D = 1.11) observed for the reaction. It is probable that the mechanism of action of the enzyme involves a carbonium ion intermediate which is stereospecifically quenched by solvent. However, acetamido group participation cannot be ruled out for natural substrates.

Fast Lattice Green's Function Methods for Viscous Incompressible Flows on Unbounded Domains

In this thesis, a collection of novel numerical techniques culminating in a fast, parallel method for the direct numerical simulation of incompressible viscous flows around surfaces immersed in unbounded fluid domains is presented. At the core of all these techniques is the use of the fundamental solutions, or lattice Green’s functions, of discrete operators to solve inhomogeneous elliptic difference equations arising in the discretization of the three-dimensional incompressible Navier-Stokes equations on unbounded regular grids. In addition to automatically enforcing the natural free-space boundary conditions, these new lattice Green’s function techniques facilitate the implementation of robust staggered-Cartesian-grid flow solvers with efficient nodal distributions and fast multipole methods. The provable conservation and stability properties of the appropriately combined discretization and solution techniques ensure robust numerical solutions. Numerical experiments on thin vortex rings, low-aspect-ratio flat plates, and spheres are used verify the accuracy, physical fidelity, and computational efficiency of the present formulations.