Unitary group approach to the many-electron problem. I. Matrix element evaluation and shift operators

This is the first paper in a series of three directed toward the evaluation of spin-dependent Hamiltonians directly in the spin-orbit basis. In this paper we present a new and complete derivation of the matrix elements of the U(n) generators in the electronic Gel'fand basis. The approach employed differs from previous treatments in that the matrix elements of nonelementary generators are obtained directly. A general matrix element formula is derived which explicitly demonstrates the segment level formalism obtained previously by Shavitt using different methods. A simple relationship between the matrix elements of raising and lowering generators is determined which indicates that in CI calculations, only the matrix elements of raising generators need be calculated. Some results on the matrix elements of products of two generators are also presented.     

Transient absorption studies and numerical modeling of iodine photoreduction by nanocrystalline TiO2 films

We have used transient absorption spectroscopy to study the reaction between photogenerated electrons in a dye-free nanocrystalline titanium dioxide film and an iodine/iodide redox couple. Recombination kinetics was measured by recording the transient optical signal following band gap excitation by a UV laser pulse. In the presence of a methanol hole scavenger in the electrolyte, a long-lived (0.1-1 s) red/infrared absorbance is observed and assigned to photogenerated electrons forming Ti(3+) species. In the presence of iodine and excess iodide in the electrolyte, the signal decays on a millisecond-microsecond time scale, assigned to reduction of the redox couple by photogenerated electrons in the TiO(2). The electron lifetime decreases inversely with increasing iodine concentration, indicating that the back reaction is first order in [I(2)]. No evidence for I(2)(-) is observed, indicating that the reaction mechanism does not involve the formation of I(2)(-) as an intermediate. The shape of the kinetics evolves from monoexponential at low [I(2)] to stretched-exponential as [I(2)] increases. A Monte Carlo continuous-time random walk model is implemented to simulate the kinetics and its [I(2)] dependence and used to address the order of the recombination reaction with respect to electron density, n. The model incorporates the diffusion of oxidized species from the electrolyte toward the TiO(2) surface as well as electron trapping and transport in the TiO(2). In the limit of low [I(2)], the monoexponential kinetics is explained by the recombination reaction being rate limited by the diffusion of the oxidized species in the electrolyte. The stretched-exponential behavior at high [I(2)] can be explained by the reaction being rate limited by the transport of electrons through a distribution of trap states toward reactive sites at the TiO(2)-electrolyte interface, similar to the mechanism proposed previously for the kinetics of electron-dye cation recombination. Such trap-limited recombination can also explain the superlinear dependence of electron recombination rate on electron density, which has been reported elsewhere, without the need for a reaction mechanism that is second order in n. In contrast, a second-order reaction mechanism in a trap-free medium cannot explain the observed kinetics, although a second-order mechanism incorporating electron trapping cannot be conclusively ruled out by the data. We propose that the most likely reaction scheme, that is first order in both [I(2)] and n, is the dissociative reduction of I(2) onto the metal oxide surface, followed by a second electron reduction of the resulting adsorbed iodine radical, and that empirical second-order behavior of the electron lifetime is most likely explained by electron trapping rather than by a second-order recombination mechanism.     

Hydrophobic interactions and osmotic second virial coefficients for methanol in water

A recent theory of the hydrophobic effect together with a simple model for an alcohol molecule is used to calculate the osmotic (McMillan-Mayer) second virial coefficientB ₂ for methanol dissolved in water. We use this calculation to study the validity of common arguments that try to draw microscopic structural information from experimental virial coefficient data. In disagreement with many workers, we find that the hydrophobic interaction between hard spheres in water is attractive and that its strength diminishes as temperature is raised. Models that have come to the opposite conclusions have neglected complications inherent to real solutes such as the role of the hydroxy groups in affecting the correlations between the apolar portions of neighboring alcohols. The calculations reported here indicate that this neglect is a poor approximation for methanol. Our calculations also show that osmotic virial coefficients are sensitive to subtle details in the potentials of mean force. Therefore, slowly varying (e.g., dispersion) interactions may also contribute significantly to the values of these coefficients without significantly changing the solvent structure near the solute molecules.     

The role of solvent fluctuations in hydrophobic assembly

We use a coarse-grained solvent model to study the self-assembly of two nanoscale hydrophobic particles in water. We show how solvent degrees of freedom are involved in the process. By using tools of transition path sampling, we elucidate the reaction coordinates describing the assembly. In accord with earlier expectations, we find that fluctuations of the liquid-vapor-like interface surrounding the solutes are significant, in this case leading to the formation of a vapor tunnel between the two solute particles. This tunnel accelerates assembly. While considering this specific model system, the approach we use illustrates a methodology that is broadly applicable.     

Regularity of the powers of an ideal

For a homogeneous ideal I of S = A[X_o,... ,X_n] we compare the Castelnuovo-Mumford regularities of the powers of I to the regularity reg I of I itself.     

Tunable size and spectral properties of fluorescent nanoGUMBOS in modified sodium deoxycholate hydrogels

Microstructures of sodium deoxycholate hydrogels were altered considerably in the presence of variable tris(hydroxymethyl)aminomethane (TRIS) concentrations. These observations were confirmed by use of X-ray diffraction, polarized optical microscopy, rheology, and differential scanning calorimetry measurements. Our studies reveal enhanced gel crystallinity and rigidity with increasing TRIS concentrations. The tunable hydrogel microstructures obtained under various conditions have been successfully utilized as templates to synthesize cyanine-based fluorescent nanoGUMBOS (nanoparticles from a group of uniform materials based on organic salts). A systematic variation in size (70-200 nm), with relatively low polydispersity and tunable spectral properties of [HMT][AOT] nanoGUMBOS, was achieved by use of these modified hydrogels. The gel microstructures are observed to direct the size as well as molecular self-assembly of the nanomaterials, thereby tuning their spectral properties. These modified hydrogels were also found to possess other interesting properties such as variable morphologies ranging from fibrous to spherulitic, variable degrees of crystallinity, rigidity, optical activity, and release profiles which can be exploited for a multitude of applications. Hence, this study demonstrates a novel method for modification of sodium deoxycholate hydrogels, their applications as templates for nanomaterials synthesis, as well as their potential applications in biotechnology and drug delivery.     

Tuning supported catalyst reactivity with dendrimer-templated Pt-Cu nanoparticles

The effects of particle composition on heterogeneous catalysis were studied using dendrimer-encapsulated nanoparticles (DENs) as precursors to supported Pt-Cu catalysts. Bimetallic Pt-Cu DENs with varying Pt/Cu ratios were prepared in an anaerobic aqueous solution and deposited onto a high-purity commercial alumina support. The dendrimer template was then thermally removed to yield supported nanoparticle catalysts, which were studied with toluene hydrogenation and CO oxidation catalysis as well as infrared spectroscopy of adsorbed CO. Incorporating Cu into Pt nanoparticles had opposite effects on the two test reactions. Cu acted as a mild promoter for CO oxidation catalysis, and the promoting effect was independent of the amount of Cu present. Conversely, Cu acted as a strong poison for toluene hydrogenation catalysis, and the normalized rate tracked inversely with Cu content. Infrared spectroscopy of the supported nanoparticles indicated that electronic effects (electron donation from Cu to Pt) were minimal for these materials. Consequently, the catalysis results are interpreted in terms of potential structural differences as a function of Cu incorporation and reaction conditions.