A second-order scheme for the “Brusselator” reaction–diffusion system

A second-order method is developed for the numerical solution of the initial-value problems , , and , , , in which the functions and , where A and B are positive real constants, are the reaction terms arising from the mathematical modelling of chemical systems such as in enzymatic reactions and plasma and laser physics in multiple coupling between modes. The method is based on three first-order methods for solving u and v, respectively. In addition to being second-order accurate in space and time, the method is seen to converge to the correct fixed point ( , V* = A/B) provided . The approach adopted is extended to solve a class of non-linear reaction–diffusion equations in two-space dimensions known as the “Brusselator” system. The algorithm is implemented in parallel using two processors, each solving a linear algebraic system as opposed to solving non-linear systems, which is often required when integrating non-linear partial differential equations (PDEs). This revised version was published online in July 2006 with corrections to the Cover Date.     

Two‐center p‐space integrals: diatomic molecular moments

Analytical expressions for the calculation of both orbital and total moments of diatomic molecules using Gaussian‐type orbitals are formulated. Moments with are computed for the ground state of 35 diatomic molecules at equilibrium bond length using 6‐311G(d,p) basis sets. In order to test our expressions, these expectation momentum values are compared with the values calculated using self‐consistent‐field wave functions of Hartree–Fock quality, which give Hartree–Fock limit energies. This revised version was published online in July 2006 with corrections to the Cover Date.     

NMR spectra of icosahedral (Ih and I) fullerenes

By analyzing the topological structures of the three types of icosahedral fullerenes: (1) , (2) and (3) k,\;h,k = 1,2,...} \right)$$ " align="middle" border="0"> , we have obtained theoretically the ¹³C NMR spectra with natural abundance for ¹³C of all the icosahedral (I_h and I) fullerenes.     

Three limit cycles for a first‐order exothermic chemical reaction in a continuous stirred‐tank reactor

We study a first‐order exothermic chemical reaction in a continuous stirred‐tank reactor modelled by a 3‐parameter family of vector fields in . We prove that there exist regions in which contain points that depend on parameters such that the chemical reaction has 0, 1, 2, or 3 small amplitude limit cycles that surround the origin. We conclude that this model can reach two stable small amplitude limit cycles. Finally, we show that one of these regions contains the point in the parameter space considered by Gurel and Lapidus [6] who proved numerically the existence of one stable limit cycle. This revised version was published online in July 2006 with corrections to the Cover Date.     

A perturbation theoretical method for determination of the dependence of the intramolecular X–H(D) potential on the hydrogen bond strength

A perturbation theoretical method is proposed that allows determination of the function describing the dependence of the intramolecular XH(D) potential on the hydrogen bond strength. Treating the XH(D) oscillators as mixed cubic–quartic oscillators in which the stretch–stretch couplings of the ν(X–H(D)) mode with the internal modes of the radical X may be neglected, and introducing the hydrogen bonding influence through the changes in the harmonic diagonal force constants (as proposed by Sceats and Rice [41]), it is possible to extract the analytical form of the function ) from the experimental versus correlations. The other parameters obtained by the empirical correlations, within this model, also have an exact physical meaning. The method is applicable to weak hydrogen bonds as well as to hydrogen bonds of intermediate strength, and, strictly speaking, to linear X– systems. However, extension to arbitrary geometry may be easily done. This revised version was published online in July 2006 with corrections to the Cover Date.     

A Class of Orthogonal Polynomials Suggested by a Trigonometric Hamiltonian: Symmetric States

A new subclass of the Jacobi polynomials arising in the exact analytical solution of the one-dimensional Schrödinger equation with a trigonometric potential has been introduced. The polynomials which consist of a free parameter are not ultraspherical polynomials and have been simply named the -polynomials since they are generated by a trigonometric Hamiltonian. In certain sense, it is shown that the -polynomials can be regarded as a generalisation of the airfoil polynomials or the Chebyshev polynomials of the third kind. This paper is intended to discuss the basic properties of the polynomials so defined.     

On the cycle index of point groups

In this brief work we express the cycle index of the molecular point groups as a function of a limited number of initial geometrical parameters. Such parameters are the number of elements composing the domain D of sites of substitutions in the molecule belonging to the point group G, and the numbers of sites of D lying on the symmetry elements for G.     

Harmonic oscillator tensors. III: Efficient algorithms for evaluating matrix elements of vibrational operators

Succinct expressions for the matrix elements of various vibrational operators have been derived in the basis of the nondegenerate harmonic oscillator. Among these are the matrix elements of and , which are found to be dependent upon two quantities and their derivatives. Furthermore, the derivative property of the commutator is used to obtain an explicit expression for the derivatives of an operator in terms of its nested commutator with the conjugate momentum. It may be applied to any of the above cases to obtain the matrix representatives of expressions such as the mixed products , for example. In addition, a simple expression for 1/q is given and its derivatives may be evaluated by this commutator technique. Also the matrix elements of a Gaussian-type operator has been evaluated. This revised version was published online in July 2006 with corrections to the Cover Date.     

Symmetry breaking in HF wave functions of Fe(CH)2

HF and CAS calculations for linear geometry of Fe(CH)₂ with symmetry have been performed. The basis sets used were DZ and DZ + P with ECP on the iron atom. Two closedshell and one quintet RHF wave functions have been found, and . All of them are singlet and triplet unstable in the wide range of Fe–CH distances. Singlet instability leads to the Charge Density Wave (CDW) brokensymmetry wave function with two electrons on carbon or orbital in the dissociation limit. Triplet instabilities lead to two brokensymmetry HF wave functions of Axial Spin Density Wave (ASDW) type, ASDW₁ and ASDW₂. In the dissociation limit they give carbon atoms with two electrons on and orbitals coupled to singlet and triplet, respectively. The stability conditions for CDW, ASDW₁ and ASDW₂ instabilities have been derived. Other HF wave functions with spin symmetry unrestricted have been also found. CAS(8,8), CAS(10,10) and CAS(12,12) calculations for singlet, triplet and quintet states of Fe(CH)₂ have been carried out. In all CAS calculations the singlet state has the lowest energy. The Fe–CH equilibrium distances obtained from closedshell RHF wave functions are much shorter and from brokensymmetry wave functions are much longer than those obtained from CAS calculations.     

Handling multicomponent systems in \mathbb{R}^n. I: Theoretical results

The phases of multicomponent systems (mixtures, states, etc.) containing the compounds are , where and . For (quaternary or higher dimensional systems), the displaying methods and visual investigations in the dimensional Euclidean space are tangentially or not at all described in the literature. In this paper we first develop the theoretical (both mathematical and computational) background in any dimension in . We focus not only on the important points, lines, surfaces of these systems, and computing method of the states of some processes in such systems, but also on the approximating methods of the above mentioned lines and surfaces, and, finally, on the question which is the region where a state (a point) falls into. Using the above results a computer program for PC's was created for evaluating and displaying the approximated surfaces. This program is described in I. Szalkai, SALT3DIM.exe – A program for handling 4 component mixtures, Preprint No. 047, University of Veszprém (1996), and the computing results are planned to be published in a forthcoming paper (I. Szalkai, Handling multicomponent systems in . II: Computational results, J. Chem. Inf. Comput. Sci., submitted).     

Some analytical results for spin 1/2 spin systems. III: The benzene ring

In two earlier papers on multiply connectedABCD spin 1/2 spin systems, it was shown that it is possible to simplify the calculation of (i) the time dependent density matrix(t), and (ii) the time evolution of high-order multiple quantum operators, evolving in the presence of differing Zeeman offsets, scalar coupling and dipolar interactions, by subdividing the Hamiltonian into , where , is a suitable linear combination of the constants of the motion. In this paper, these techniques are applied to the benzene ring with particular emphasis on high-order multiple-quantum NMR experiments, and their interpretation in terms of specific wavefunctions. In particular, it is shown that excellent agreement between theory and experiment can be obtained for the energy splittings witnessed in the m = +4,+5 MQ-NMR transitions, with minimum effort.     

Two Algorithms for Computing the Randles–Sevcik Function from Electrochemistry

We derive two expansions of the Randles–Sevcik function : an asymptotic expansion of for x and its Taylor expansion at any x ₀ . These expansions are accompanied by error bounds for the remainder at any order of the approximation.     

Geometric and computational aspects of polymer reconfiguration

We examine a few computational geometric problems concerning the structures of polymers. We use a standard model of a polymer, a polygonal chain (path of line segments) in three dimensions. The chain can be reconfigured in any manner as long as the edge lengths and the angles between consecutive edges remain fixed, and no two edges cross during the motion. We discuss preliminary results on the following problems.Given a chain, select some interior edge , defining two subchains which are adjacent to . We keep the two subchains individually rigid and rotate one around while leaving the other fixed in space, while maintaining the vertex-angles at . We call this motion an edge spin at . An O(n ²) algorithm for this problem is given as well as an (nlogn) lower bound on the time complexity.In determining whether a chain can be reconfigured from one conformation to another, it is useful to consider reconfiguring through some canonical conformation. In our three-dimensional case, the most obvious choice is to flatten a chain into the plane. However, we demonstrate that determining if a given chain can be reconfigured into the plane without self-intersecting is NP-hard, even if the restriction that it must lie monotonically is added. We then provide an O(n) algorithm to decide if a chain has a non-crossing convex coil conformation (where all angles turn in the same direction), although we cannot yet decide if a sequence of motions to reconfigure a chain into a convex coil conformation exists.     

A generalized self‐consistent‐field procedure in the improved BCS theory

The basic ideas of the Improved Bardeen–Cooper–Schrieffer (IBCS) approach to the first and secondorder Reduced Density Matrices (1 and 2RDM) are briefly reviewed. The molecular orbital occupations are expressed by means of new quantities , which, satisfying a trigonometric relation, guarantee the nonidempontent condition. Thus, a variational method is introduced to determine , involving only an unconstrained minimization which may be performed using a conjugate gradient technique. A new effective Hamiltonian which is composed of the Coulomb, exchange and exchangetime inversion operators is also presented. It leads exactly to equations of Hartree–Fock type, however, the electronic field includes now an arbitrary number of orbitals and fractional occupation numbers. Accordingly, a generalized selfconsistentfield method is proposed: the iterative procedure is repeated until convergence is reached for the actual density matrix.     

Reaction graphs and a construction of reaction networks

Summary A general definition of reaction graphs is presented. For a pair of isomeric molecular graphs and , related by a chemical transformation , the reaction graph is determined using a maximal common subgraph defined for vertex mapping . A binary operation defined for graphs constructed over the same vertex set enables us to decompose the reaction graph into the sum of prototype reaction graphs. A decomposition of an overall reaction graph can be advantageously used for the construction of a reaction network. An oriented path in this network beginning at and ending at corresponds to a breakdown of the transformation into a sequence of intermediates.     

Dimension‐dependent two‐electron Hamiltonian matrix elements

Since the birth of quantum mechanics the ground state electronic energy of the twoelectron atom has received special attention. This is because the twoelectron system is the simplest atom to include electron–electron interactions. These interactions are key to understanding manyelectron systems. This paper adds to the knowledge of twoelectron atoms by presenting closed form solutions for Hamiltonian matrix elements at arbitrary spatial dimension, . The basis functions are the dependent hydrogenic wavefunctions: . The electron–electron repulsion integrals are solved by the Fourier integral transform.     

Studies on a new dinuclear Co<Superscript>II</Superscript>–pterin complex exhibiting reactivity towards phenylalanine and bromobenzene

2-Pivaloylamino-6-acetonyl-isoxanthopterin (1, ) has been reacted with under suitable conditions for synthesizing the new compound ] (2). It has been characterized by elemental analysis, electrospray ionization mass spectrometry, magnetic susceptibility measurement, different spectroscopic techniques, and cyclic voltammetry. Molecular mechanics (MM2) method provided with its optimized geometry (having lowest steric energy), consistent with the above data; the optimized bond lengths and bond angles data tally with the literature X-ray structural data. Reactivity of (2) towards phenylalanine in the presence of in methanol has been followed both kinetically and stoichiometrically; a reasonable amount of tyrosine could be recovered from the reaction medium. The negative value (−274.0 J mol⁻¹ indicates an associative pathway for this process. (2) is also able to react with bromobenzene as indicated by time-dependent absorption spectra as well as product identification. Efficacy of the pterin ligand residue of (2) in rendering the latter reactive towards the above-mentioned organic compounds, has been discussed on the basis of experimental evidence.     

Surface Fluorination of Carboxylated Nitrile Butadiene Rubber: An XPS Study

Because of their exceptional reactivity, fluorine and fluorinated gases are of primary importance for the modification of the surface properties of materials. This study is devoted to surface treatment of thin nitrile gloves, made of carboxylated nitrile butadiene rubber latex, using either direct fluorination (10% F₂gas diluted in N₂) or plasma-enhanced fluorination in radio-frequency cold plasmas using fluorinated gases (CF₄, CHF₃). Mechanisms of fluorination of these co-elastomers have been proposed on the basis of the assignment of the different components of the XPS spectra. Several mechanisms have been observed depending on the fluorination conditions. Although the modification of nitrile gloves is already effective for fluorination reactions at room temperature, an important activation is observed for experiments carried out at 90°C. When the treatments are carried out at room temperature, a gradual fluorination occurs: in the case of 10% diluted F₂ gas, monofluorinated C—F groups are the species most found at the surface and perfluoro groups CF _n are present in lower amount. An addition reaction takes place at the CH=CH double bonds of the polybutadiene entities, leading to CHF=CHF units. Whatever the fluorination method, thermal activation yields a more massive fluorination of the surface that finally leads to perfluorinated CF₂ groups and terminal —CF₃ groups.     

Preferential solvation in three component systems: Evaluation of Kirkwood-Buff parameters

The extent of local excess or deficiency of a component solvent near the solute in a mixed binary solvent has been calculated using the Hall formalism for the Kirkwood-Buff equation. The possibility of calculation of the two solute-solvent Kirkwood-Buff parameters using the values is discussed. A model calculation using literature data for preferential solvation in mixed binary solvents is presented. The solute-solvent and solvent-solvent interactions and the relative size of the solvents are also shown to be relevant factors in determining the values.     

Ionic Mobility, Structure, Phase Transition, and Conductivity in (NH4)3Sb4F15

Fluoride and ammonium ion dynamics in (NH₄)₃Sb₄F₁₅ was studied by ¹H and ¹⁹F NMR spectroscopy in the range 180-440 K. Types of ionic motion were determined, and their activation energies were estimated. The crystal structure of a single crystal of (NH₄)₃Sb₄F₁₅ (space group ) was solved. In the range K, a reversible phase transition has been found. Based on the experimental values of conductivity of (NH₄)₃Sb₄F₁₅ ( S/cm at T = 440 K), this antimony(III) fluoride is classified as a superionic conductor.