Finite basis representations with nondirect product basis functions having structure similar to that of spherical harmonics

The currently most efficient finite basis representation (FBR) method [Corey et al., in Numerical Grid Methods and Their Applications to Schrodinger Equation, NATO ASI Series C, edited by C. Cerjan (Kluwer Academic, New York, 1993), Vol. 412, p. 1; Bramley et al., J. Chem. Phys. 100, 6175 (1994)] designed specifically to deal with nondirect product bases of structures phi(n) (l)(s)f(l)(u), chi(m) (l)(t)phi(n) (l)(s)f(l)(u), etc., employs very special l-independent grids and results in a symmetric FBR. While highly efficient, this method is not general enough. For instance, it cannot deal with nondirect product bases of the above structure efficiently if the functions phi(n) (l)(s) [and/or chi(m) (l)(t)] are discrete variable representation (DVR) functions of the infinite type. The optimal-generalized FBR(DVR) method [V. Szalay, J. Chem. Phys. 105, 6940 (1996)] is designed to deal with general, i.e., direct and/or nondirect product, bases and grids. This robust method, however, is too general, and its direct application can result in inefficient computer codes [Czako et al., J. Chem. Phys. 122, 024101 (2005)]. It is shown here how the optimal-generalized FBR method can be simplified in the case of nondirect product bases of structures phi(n) (l)(s)f(l)(u), chi(m) (l)(t)phi(n) (l)(s)f(l)(u), etc. As a result the commonly used symmetric FBR is recovered and simplified nonsymmetric FBRs utilizing very special l-dependent grids are obtained. The nonsymmetric FBRs are more general than the symmetric FBR in that they can be employed efficiently even when the functions phi(n) (l)(s) [and/or chi(m) (l)(t)] are DVR functions of the infinite type. Arithmetic operation counts and a simple numerical example presented show unambiguously that setting up the Hamiltonian matrix requires significantly less computer time when using one of the proposed nonsymmetric FBRs than that in the symmetric FBR. Therefore, application of this nonsymmetric FBR is more efficient than that of the symmetric FBR when one wants to diagonalize the Hamiltonian matrix either by a direct or via a basis-set contraction method. Enormous decrease of computer time can be achieved, with respect to a direct application of the optimal-generalized FBR, by employing one of the simplified nonsymmetric FBRs as is demonstrated in noniterative calculations of the low-lying vibrational energy levels of the H3+ molecular ion. The arithmetic operation counts of the Hamiltonian matrix vector products and the properties of a recently developed diagonalization method [Andreozzi et al., J. Phys. A Math. Gen. 35, L61 (2002)] suggest that the nonsymmetric FBR applied along with this particular diagonalization method is suitable to large scale iterative calculations. Whether or not the nonsymmetric FBR is competitive with the symmetric FBR in large-scale iterative calculations still has to be investigated numerically.     

Enantioselective Michael Addition of the 2‐(1‐Ethylpropoxy)acetaldehyde to N‐[(1Z)‐2‐Nitroethenyl]acetamide – Optimization of the Key Step in the Organocatalytic Oseltamivir Synthesis

Organocatalytic Michael addition of alkoxyacetaldehyde 1 to N‐protected 2‐nitroethene‐1‐amine 2 (Scheme 2) is a key step in the synthesis of an important antiviral agent, oseltamivir. Screening of a large array of structurally diverse acids as potential promoters led to the identification of several useful acidic additives for this reaction (Tables 1–4). Also other reaction parameters were investigated with the aim of improving the diastereoselectivity of the Michael addition, while maintaining high enantiomer purity and yield (Tables 5 and 6).     

“First‐Principles” kinetic monte carlo simulations revisited: CO oxidation over RuO2(110)

First principles‐based kinetic Monte Carlo (kMC) simulations are performed for the CO oxidation on RuO₂(110) under steady‐state reaction conditions. The simulations include a set of elementary reaction steps with activation energies taken from three different ab initio density functional theory studies. Critical comparison of the simulation results reveals that already small variations in the activation energies lead to distinctly different reaction scenarios on the surface, even to the point where the dominating elementary reaction step is substituted by another one. For a critical assessment of the chosen energy parameters, it is not sufficient to compare kMC simulations only to experimental turnover frequency (TOF) as a function of the reactant feed ratio. More appropriate benchmarks for kMC simulations are the actual distribution of reactants on the catalyst's surface during steady‐state reaction, as determined by in situ infrared spectroscopy and in situ scanning tunneling microscopy, and the temperature dependence of TOF in the from of Arrhenius plots. © 2012 Wiley Periodicals, Inc.     

Synthesis and anion recognition studies of novel 5,5-dioxidophenothiazine-1,9-diamides

Novel 5,5-dioxidophenothiazine-1,9-diacetamide and -dibenzamide receptor molecules were prepared starting from commercially available and relatively cheap chemicals. The anion recognition properties of these two diamides were investigated using UV–vis spectroscopy. Chloride formed a simple hydrogen-bonded complex with the diacetamide receptor, while fluoride, acetate and dihydrogen phosphate deprotonated both sensor molecules. Upon titration with fluoride deprotonation occurred via the formation of [HF₂]^?, and in the case of the diacetamide receptor complexation took place alongside deprotonation.     

Effect of grain orientation on chemical etching

The effect of grain orientation on the effectiveness of pre-etching before color etching was investigated by the help of electron back scattering diffraction and atomic force microscopy in case of cast iron. Strong correlation was found between the angle between the specimen normal and the [001] orientation of the ferrite grains and the depth of the etching. If the angle between the specimen normal and the [001] direction of the ferritic grain is small, then the speed of the etching is low, but the lateral variation of the etching speed within the grain is larger.     

Comparison of chemometric methods in the analysis of pharmaceuticals with hyperspectral Raman imaging

Chemical imaging method of vibrational spectroscopy, which provides both spectral and spatial information, creates a three‐dimensional (3D) dataset with a huge amount of data. When the components of the sample are unknown or their reference spectra are not available, the classical least squares (CLS) method cannot be applied to create visualized distribution maps. Raman image datasets can be evaluated even in such cases using multivariate (chemometric) methods for extracting the needed hidden information. The capability of chemometrics‐assisted Raman mapping is evaluated through the analysis of pharmaceutical tablets (considered as unknown) with the aim of estimating the pure component spectra based on the collected Raman image. Six chemometric methods, namely, principal component analysis (PCA), maximum autocorrelation factors (MAF), sample–sample 2D correlation spectroscopy (SS2D), self‐modeling mixture analysis (SMMA), multivariate curve resolution–alternating least squares (MCR‐ALS), and positive matrix factorization (PMF), were compared. SMMA was found to be the best choice to determine the number of components. MCR‐ALS and PMF provided the pure component spectra with the highest quality. MCR‐ALS was found to be superior to PMF in the estimation of Raman scores (which correspond to the concentrations) and yielded almost the same results as CLS (using the real reference spectra). Thus, the combination of Raman mapping and chemometrics could be successfully used to characterize unknown pharmaceuticals, identify their ingredients, and obtain information about their structures. This may be useful in the struggles against illegal and counterfeit products and also in the field of pharmaceutical industry when contaminants are to be identified. Copyright © 2011 John Wiley & Sons, Ltd.     

On the Wang–Uhlenbeck Problem in Discrete Velocity Space

The arguably simplest model for dynamics in phase space is the one where the velocity can jump between only two discrete values, ±v with rate constant k. For this model, which is the continuous-space version of a persistent random walk, analytic expressions are found for the first passage time distributions to the origin. Since the evolution equation of this model can be regarded as the two-state finite-difference approximation in velocity space of the Kramers–Klein equation, this work constitutes a solution of the simplest version of the Wang–Uhlenbeck problem. Formal solution (in Laplace space) of generalizations where the velocity can assume an arbitrary number of discrete states that mimic the Maxwell distribution is also provided.     

Localized partial traps in diffusion processes and random walks

Reaction-diffusion equations, in which the reaction is described by a sink term consisting of a sum of delta functions, are studied. It is shown that the Laplace transform of the reactive Green's function can be analytically expressed in terms of the Green's function for diffusion in the absence of reaction. Moreover, a simple relation between the Green's functions satisfying the radiation boundary condition and the reflecting boundary condition is obtained. Several applications are presented and the formalism is used to establish the relationship between the time-dependent geminate recombination yield and the bimolecular reaction rate for diffusion-influenced reactions. Finally, an analogous development for lattice random walks is presented.     

Friedrichs extensions of Schrödinger operators with singular potentials

The Friedrichs extension for the generalized spiked harmonic oscillator given by the singular differential operator −d²/dx²+Bx²+Ax⁻²+λx^−α (B>0, A?0) in L₂(0,∞) is studied. We look at two different domains of definition for each of these differential operators in L₂(0,∞), namely C₀^∞(0,∞) and D(T_2,F)∩D(M_λ,α), where the latter is a subspace of the Sobolev space W_2,2(0,∞). Adjoints of these differential operators on C₀^∞(0,∞) exist as result of the null-space properties of functionals. For the other domain, convolutions and Jensen and Minkowski integral inequalities, density of C₀^∞(0,∞) in D(T_2,F)∩D(M_λ,α) in L₂(0,∞) lead to the other adjoints. Further density properties C₀^∞(0,∞) in D(T_2,F)∩D(M_λ,α) yield the Friedrichs extension of these differential operators with domains of definition D(T_2,F)∩D(M_λ,α).