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.     

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).     

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.     

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.     

Effect of solvent on the partial molal volumes and heat capacities of nonelectrolytes

Group contributions to in seven solvents and to in three solvents have been tabulated. The variation of group parameters is discussed in terms of the solvent compressibility coefficient, _T. The scaled particle theory (SPT) is used to calculate cavity contributions to and C _p2 ^o . Interaction contributions are obtained from the cavity terms and and values estimated through the additivity schemes. values are more sensitive to solute-solvent interactions than in water and less sensitive in methanol. The SPT results for heat capacities support the concept of structural promotion by hydrophobic solutes in water.     

Some analytical results forABC,ABCD, andXBCD coupled spin 1/2 systems. II

In the preceding paper, it was shown that the calculation of the density matrix(t) for multiply connectedABC, etc., spin 1/2 spin systems can be greatly simplified by subdividing the Hamiltonian into , where is a suitable linear combination of the constants of the motion. In this paper, a framework for the determination of the time evolution of high-order multipolar quantum states is presented and discussed. It is shown that the necessary mathematical labour is reduced to a minimum by (i) exploiting the fact that is a good quantum number, and (ii) using the theory of partitioned matrices. For example, it is shown that for a generaln-coupled spin 1/2 system, the spin dynamics of the multipolar states, whereK _max is the maximum tensorial rank, can be determined without the need to diagonalize the full 2 ⁿ × 2 ⁿ Hamiltonian matrix, wheren is the number of spins. In fact, to describe the time evolution of the multipolar states it is only necessary to diagonalize twon ×n matrices at most. Finally, some cautionary remarks are made concerning the use of the weak-coupling approximation.     

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.     

Some analytical results forABC,ABCD, andXBCD coupled spin 1/2 systems. I

By choosing a suitable linear combination of the constants of the motion , it is shown that the calculation of the density matrix(t) can be simplified by subdividing the Hamiltonian into . In particular, this technique can be used to obtain closed form solutions for the eigenfunctions and eigenvalues of spin 1/2ABC andXBCD spin systems, evolving in the presence of Zeeman offsets, scalar coupling and dipolar interactions. In general, the eigenvalues and eigenvalues of are very transparent, while those of require more effort. Nevertheless, simplifications can be made. Firstly, the effective size of the Hamiltonian matrix which needs to be considered, is reduced fromN ×N to at least (N – 2) × (N – 2), while forXBC ... systems it is reduced to (N – 4) × (N – 4). Secondly, the highest rank and highest/lowest order tensor operators available to the spin ensemble are constants of the motion under . Finally, by exploiting the fact that is a good quantum number, it is possible to block-diagonalize the matrix into no more than 3 × 3 matrices.     

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.     

Some conjectures on S n -derived nuclear permutational(or NMR) spin encodings

On existence of limiting ‐module decompositional sets for weak ‐branching at high n; on Voronoi polyhedral dual as geometric analogues to Cayley’s SU2 embedding theorem; and on SU dual group with retention of self‐associacy over subduced irrep set, as being the sufficient further condition to ensure the determinacy of SU embeddings. In the context of structures arising from nuclear permutation (NP) or NMR dual‐group spin algebras, the first conjecture sets out the high‐index n, and thus weak‐branching limit (WBL), aspects of module decompositions as giving rise to a set of numerical values for the associated Kostka coefficients which are invariant to further incrementation in the index; the existence of such combinatorial limit properties, implicit in sst tableaux enumerations, has not been addressed in the mathematics literature to date. Conjectures 2 and 3 are concerned with the questions of geometric and sufficient algebraic realisations of the determinacy of natural finite group embeddings in specific SU permutation groups. In conjecture 2, the Voronoi dual‐structures to the regular polyhedra for NP/NMR automorphic SU2 embedded spin symmetries hold the key to physical insight. Specifically, they provide a novel combinatorial geometric view of Cayley’s theorem; the mid‐face intersecting ‐axes of the initial NMR automorphic solids become (vertex) body‐diagonal axes of the specific‐dual Voronoi polyhedra, where a distinctness condition from the spin‐sites gives raise to a geometric statement of Cayley’s theorem. Conjecture 3 is concerned with SU embeddings for which the simple Cayley criterion alone is an insufficient condition to guarantee determinacy. The self‐associacy property and its retention over the subduced irrep‐subset(s) (i.e., comparable to studies of system‐invariants via Yamanouchi–Gel’fand subduction chains) is now seen as indicative of retention of determinacy for such SU group embeddings through the above sufficiency condition. This revised version was published online in July 2006 with corrections to the Cover Date.     

Cluster Models of Co2+ at the Cation Sites of Zeolite ZSM-5

The local structure of Co₂₊ at the -, , and -cation sites of zeolite ZSM-5 was calculated in terms of density functional theory using the cluster approach. The local geometry of the oxygen environment of Co₂₊ is characterized; it is found that the ion stabilization energy increases in the series .     

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.     

Volumetric Properties of Binary Mixtures of Acetonitrile and Chloroalkanes at 25°C and Atmospheric Pressure

Excess molar volumes for binary mixtures of acetonitrile + dichloromethane, acetonitrile + trichloromethane, and acetonitrile + tetracloromethane at 25°C have been used to calculate partial molar volumes , excess partial molar volumes , and apparent molar volumes of each component as a function of composition. The V _m ^Evalues are negative over the entire composition range for the systems studied. The applicability of the Prigogine–Flory–Patterson theory was explored. The agreement between theoretical and experimental results is satisfactory for the systems with dichloromethane and tetrachloromethane. For the unsymmetrical behavior of the system with trichloromethane, however, the agreement is poor.     

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.     

On the Permanental Polynomials of Some Graphs

Let G be a simple graph with adjacency matrix A(G) and (G,x) the permanental polynomial of G. Let G × H denotes the Cartesian product of graphs G and H. Inspired by Kleins idea to compute the permanent of some matrices (Mol. Phy. 31 (3) (1976) 811–823), in this paper in terms of some orientation of graphs we study the permanental polynomial of a type of graphs. Here are some of our main results.1.If G is a bipartite graph containing no subgraph which is an even subdivision of K _2,3, then G has an orientation G ^e such that (G,x) = det (xI-A(G ^e )), where A(G ^e ) denotes the skew adjacency matrix of G ^e.2.Let G be a 2-connected outerplanar bipartite graph with n vertices. Then there exists a 2-connected outerplanar bipartite graph with 2n+2 vertices such that (G,x) is a factor of .3.Let T be an arbitrary tree with n vertices. Then , where ₁ , ₂ , ..., _n are the eigenvalues of T.     

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.     

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.     

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.     

Mechanism of the Reaction of Xenon Tetraoxide with Hydrogen: A Photochemical Reaction

The dependence of the rate of a photochemical reaction of XeO₄with H₂on the concentrations of H₂and the diluent gases He and CO₂is studied at room temperature. The results are compared with the data for the dark reaction. A mechanism with the elementary energetic branched-chain reaction OH (v) + XeO₄OH + O + XeO₃is proposed to explain the results of the study.     

The thermodynamics of dilution ofn-dodecyltrimethylammonium bromide micelles in aqueous solutions

Summary The relative partial molal thermodynamic functions ₁ and have been derived from reported experimental light scattering and calorimetric measurements on micellar solutions ofn-dodecyltrimethylammonium bromide. It is shown that the magnitudes of the thermodynamic functions are considerably larger than can be accounted for by the interaction of the electrical double layers surrounding the micelles. Possible enthalpy changes arising from a change in micelle shape, size and degree of dissociation on dilution are discussed.

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Zusammenfassung Die relativen molaren thermodynamischen Partialfunktionen , andS ₁wurden abgeleitet aus Literaturwerten von Lichtstreuungsmessungen und kalorimetrischen Untersuchungen an mizellaren Lösungen vonn-Dodecyltrimethylammoniumbromid. Es wird gezeigt, daß die Zahlenwerte der thermodynamischen Funktionen erheblich größer sind als von der Wechselwirkung der elektrischen Doppelschichten zu erwarten ist, die die Mizellen umgeben. Mögliche Enthalpieänderungen, die sich aus Änderungen von Größe, Gestalt und Dissoziationsgrad bei Verdünnung herleiten, werden diskutiert.