Showing papers in "Journal of Mathematical Chemistry in 2005"

Trigonometrically fitted Runge–Kutta methods for the numerical solution of the Schrödinger equation

Abstract: In this paper we construct two trigonometrically fitted methods based on a classical Runge–Kutta method of England with fifth algebraic order. The methods will be used for the integration of the radial Schrodinger equation and have high efficiency as the results show. The efficiency is higher when using higher energy and this can be explained by the error analysis of the methods. More specifically the new methods have lower powers of the energy in the local truncation error and that keeps the error at lower values.

A family of multiderivative methods for the numerical solution of the Schrödinger equation

Abstract: A family of multiderivative methods with minimal phase-lag are introduced in this paper, for the numerical solution of the Schrodinger equation. The methods are called multiderivative since uses derivatives of order two, four or six. Numerical application of the new obtained methods to the Schrodinger equation shows their efficiency compared with other similar well known methods of the literature.

Sixth algebraic order trigonometrically fitted predictor–corrector methods for the numerical solution of the radial Schrödinger equation

Abstract: Our new trigonometrically fitted predictor–corrector (P–C) schemes presented here are based on the well known Adams–Bashforth–Moulton methods: the predictor is based on the fifth order Adams–Bashforth scheme and the corrector on the sixth order Adams–Moulton scheme. We tested the efficiency of our newly developed schemes against well known methods, with excellent results. The numerical experiments showed that at least one of our schemes is noticeably more efficient compared to other methods, some of which are specially designed for this type of problem. It is also worth mentioning that this is the first time that sixth algebraic order trigonometrically fitted Adams–Bashforth–Moulton P–C schemes are used to efficiently solve the radial Schrodinger equation.

Exponentially fitted symplectic methods for the numerical integration of the Schrödinger equation

Abstract: The computation of the energy eigenvalues of the one-dimensional time-independent Schrodinger equation is considered. Exponentially fitted and trigonometrically fitted symplectic integrators are obtained, by modification of the first and second order Yoshida symplectic methods. Numerical results are obtained for the one-dimensional harmonic oscillator and Morse potential.

Properties of the Mittag-Leffler Relaxation Function

Abstract: The Mittag-Leffler relaxation function, E α (−x), with 0 ≤ α ≤ 1, which arises in the description of complex relaxation processes, is studied. A relation that gives the relaxation function in terms of two Mittag-Leffler functions with positive arguments is obtained, and from it a new form of the inverse Laplace transform of E α (−x) is derived and used to obtain a new integral representation of this function, its asymptotic behaviour and a new recurrence relation. It is also shown that the fastest initial decay of E α (−x) occurs for α =1/2, a result that displays the peculiar nature of the interpolation made by the Mittag-Leffler relaxation function between a pure exponential and a hyperbolic function.

Numerical solution of the two-dimensional time independent Schrödinger equation with Numerov-type methods

Abstract: The solution of the two-dimensional time-independent Schrodinger equation is considered by partial discretization. The discretized problem is treated as an ordinary differential equation problem and Numerov type methods are used to solve it. Specifically the classical Numerov method, the exponentially and trigonometrically fitting modified Numerov methods of Vanden Berghe et al. [Int. J. Comp. Math 32 (1990) 233–242], and the minimum phase-lag method of Rao et al. [Int. J. Comp. Math 37 (1990) 63–77] are applied to this problem. All methods are applied for the computation of the eigenvalues of the two-dimensional harmonic oscillator and the two-dimensional Henon–Heils potential. The results are compared with the results produced by full discterization.

On the Randić Index

Abstract: The Randic index of an organic molecule whose molecular graph is G is defined as the sum of (d(u)d(v))−1/2 over all pairs of adjacent vertices of G, where d(u) is the degree of the vertex u in G. In Discrete Mathematics 257, 29–38 by Delorme et al. gave a best-possible lower bound on the Randic index of a triangle-free graph G with given minimum degree δ(G). In the paper, we first point out a mistake in the proof of their result (Theorem 2 of [2002]), and then we will show that the result holds when δ(G) ≥ 2.

Analytical inversion of the Laplace transform without contour integration: application to luminescence decay laws and other relaxation functions

Abstract: Laplace transforms find application in many fields, including time-resolved luminescence. In this work, relations that allow a direct (i.e., dispensing contour integration) analytical calculation of the original function from its transform are re-derived. The results are used for the determination of distributions of rate constants of several relaxation functions, including the stretched exponential and the compressed hyperbolic luminescence decay laws, and the asymptotic power law relaxation function.

On bicyclic graphs with minimal energies

Abstract: The energy of a graph is defined as the sum of the absolute values of all the eigenvalues of the graph. Let G(n) be the class of bicyclic graphs G on n vertices and containing no disjoint odd cycles of lengths k and l with k + l ≡ 2 (mod 4). In this paper, the graphs in G(n) with minimal, second-minimal and third-minimal energies are determined.

Partitioning of π -electrons in rings of polycyclic conjugated hydrocarbons: Part 6. Comparison with other methods for estimating the local aromaticity of rings in polycyclic benzenoids

Abstract: By adopting the convention that shared double bonds in polycyclic conjugated hydrocarbons contribute with one π-electron and unshared ones with two π-electrons, a partition of π-electrons in each ring (π-electron content, EC) can be obtained by averaging over all Kekule structures, which are assumed to have equal weights. This affords a simple measure of local aromaticity that is comparable with other such local aromaticity indices in polycyclic benzenoids.

Entropy/information bond indices of molecular fragments

Abstract: The communication channels of the disconnected (mutually non-bonded, closed) parts of the molecule are investigated. The entropy/information indices of such subsystems are proposed as measures of the intra-fragment (internal) information bond-order and its covalent/ionic components. The molecular fragment bond-order conservation and a competition between its ionic and covalent contributions are examined. An approximate scheme in the spirit of the grouping theorem of the Information Theory (IT), for combining the subsystem entropy/information data into the corresponding global quantities describing the system as a whole, is derived and tested. It uses the independent subsystem approximation to estimate the entropy/information indices of the inter-fragment (external) chemical interactions in the molecule. The applications to simple orbital models, including the three-orbital model of the transition state complex and the π bond systems (butadiene and benzene) in the Huckel theory approximation, are used to illustrate the concepts proposed.

Classical and quantum study of the motion of a particle in a gravitational field

Abstract: The classical and the quantum mechanical description of a one-dimensional motion of a particle in the presence of a gravitational field is thoroughly discussed. The attention is centered on the evolution of classical and quantum mechanical position probability distribution function. The classical case has been compared with three different quantum cases: (a) a quantum stationary case, (b) a quantum non-stationary zero approximation case, where the wave packet has the shape of the first eigenfunction, and (c) a quantum non-stationary general case, where the wave packet is a superposition of stationary states.

On the modelling of solid state reactions. Synthesis of YAG

Abstract: There is a model of yttrium aluminium garnet (YAG) synthesis presented in this article. The developed model is based on nonlinear reaction–diffusion partial differential equations. The solution was carried out numerically using finite difference techniques. We got dependability curves for diffusion and reaction rates and offered possible method to localize values of diffusion and reaction rate constants precisely enough.

A variable-step Numerov method for the numerical solution of the Schrödinger equation

Abstract: Numerov’s method is one of the most widely used algorithms for solving second-order ordinary differential equations of the form y’’ = f(x,y) The one-dimensional time-independent Schrodinger equation is a particular example of this type of equation In this article we present a variable-step Numerov method for the numerical solution of the Schrodinger equation

k-resonance in Toroidal Polyhexes*

Abstract: This paper considers the k -resonance of a toroidal polyhex (or toroidal graphitoid) with a string (p, q, t) of three integers (p ≥ 2, q ≥ 2, 0 ≤ t ≤ p − 1). A toroidal polyhex G is said to be k-resonant if, for 1≤ i ≤ k, any i disjoint hexagons are mutually resonant, that is, G has a Kekule structure (perfect matching) M such that these hexagons are M-alternating (in and off M). Characterizations for 1, 2 and 3-resonant toroidal polyhexes are given respectively in this paper.

Chaoticity of some chemical attractors: a computer assisted proof

Abstract: In this paper we study dynamics of two chemical attractors. By means of computer assisted proof, we show that these chemical attractors are chaotic in terms of positive entropy. We prove that the fourth power of the Poincare map derived from one chemical attractor and the second power of the Poincare map derived from the other chemical attractor are semi-conjugate to the 2-shift map, therefore the entropies of the two Poincare maps are not less than \({{1}\over {4}}\)log 2 and \({{1}\over {2}}\)log 2, respectively. The positivity of entropies of these two maps shows that the corresponding attractors are chaotic.

Exact Solution of Schrodinger equation with deformed Ring-Shaped Potential

Abstract: Exact solution of the Schrodinger equation with deformed ring-shaped potential is obtained in the parabolic and spherical coordinates. The Nikiforov–Uvarov method is used in the solution. Eigenfunctions and corresponding energy eigenvalues are calculated analytically. The agreement of our results is good.

Stability of Conjugated Carbon Nanocones

Abstract: We use interlacing techniques to prove that carbon nanocones who have a Fries Kekule structure have closed Huckel shells, and that this result can be extended to all conjugated cones where each edge belongs to a hexagonal face and the configuration of the non-hexagonal faces are consistent with a Fries Kekule structure. Cones with Fries Kekule structure or substructure are topical—not only from a valence bond theoretical point of view—since a previous ab initioanalysis favored cones where the pentagons at the tip are configured as in a Fries Kekule structure. The question of interdependence will therefore be addressed.

A three-dimensional chemostat with quadratic yields

Abstract: A three dimensional chemostat with two microorganisms which are both with quadratic yields is studied. The stability of the equlibrium points, the existence of limit cycles, the Hopf bifurcation, and the positive invariant set for the system are discussed. We also prove the conditions that guarantee two limit cycles in the model.

Relation between the inverse Laplace transforms of I (t β ) and I (t): Application to the Mittag-Leffler and asymptotic inverse power law relaxation functions

Abstract: The relation between H(k), inverse Laplace transform of a relaxation function I(t), and Hβ(k), inverse Laplace transform of I(tβ), is obtained. It is shown that for β < 1 the function Hβ(k) can be expressed in terms of H(k) and of the Levy one-sided distribution Lβ(k). The obtained results are applied to the Mittag-Leffler and asymptotic inverse power law relaxation functions. A simple integral representation for the Levy one-sided density function L1/4(k) is also obtained.

Influence of the composition on the electronegativity and on the oxygen charge distribution in a binary hydrotalcite-like by modified Sanderson method

Abstract: An intuitive and computationally non-intensive model for the classification of Hydrotalcite-like compounds (HTLCs) based simply on the chemical composition using the Sanderson Method led to good prediction of basicity and different basic sites (oxygen atoms with different charge). That model was evaluated at different M3+ /(M2++M3+) ratio and with different divalent and trivalent metallic cations.

Analysis of a chemostat model with variable yield coefficient

Abstract: We investigate a chemostat model in which the growth rate is given by a Monod expression with a variable yield coefficient. This model has been investigated by previous researchers using numerical integration. We combine analytical results with path-following methods. The conditions for washout to occur are found. When washout does not occur we establish the conditions under which the reactor performance is maximised at either a finite or infinite residence time. We also determine the parameter region in which oscillations may be generated in the reactor, which was the primary feature of interest to earlier workers on this problem.

Flow analysis deconvolution for kinetic information reconstruction

Abstract: In this work it is proposed a methodology for the kinetic information reconstruction based on the definition of a macrotransport transfer function and a numerical regularisation method. In continuous flowing systems there could be a discrepancy, for fast enough processes, between the read measure in the detector and what actually happens in the chemical reactor. This difference is a consequence of the solute dispersion along the tube. To solve this problem we define a macrotransport transfer function from the Aris–Taylor dispersion theory which enables us a direct interpretation of the experimental data (convolution) or signal reconstruction (deconvolution). The methodology proposed consists in data processing using Tikhonov regularisation method in combination with a specific experimental procedure which allows to characterize the dispersion of the solute along the flowing system. Preliminary results for the determination of the specific area of a gas–liquid reactor are shown analysing the reaction data between the ozone and the Blue Indigotrisulfonate.

On the number of Kekulé structures in capped zigzag nanotubes

Abstract: Two theoretical formulae for the number of Kekule structures in general capped zigzag nanotubes are established: one of which is by using the techniques of the transfer matrices, the other involves the eigenvalues of the transfer matrix which reveals the asymptotic behaviour of this index. In effective, according to the symmetric aspect of the tubule, the order of the transfer matrix could be notably decreased. As an application, the closed expressions for four types are given out and the relevant numerical results for those of length up to 50 are listed.

A quantum theory of chemical processes and reaction rates based on diabatic electronic functions coupled in an external field

Abstract: The method discussed in this work provides a theoretical framework where simple chemical reactions resemble any other standard quantum process, i.e., a transition in quantum state mediated by the electromagnetic field. In our approach, quantum states are represented as a superposition of electronic diabatic basis functions, whose amplitudes can be modulated by the field and by the external control of nuclear configurations. Using a one-dimensional three-state model system, we show how chemical structure and dynamics can be represented in terms of these control parameters, and propose an algorithm to compute the reaction probabilities. Our analysis of effective energy barriers generalizes previous ideas on structural similarity between reactant, and product, and transition states using the geometry of conventional reaction paths. In the present context, exceptions to empirical rules such as the Hammond postulate appear as effects induced by the environment that supplies the external field acting on the quantum system.

Use of analytical relations in evaluation of exponential integral functions

Abstract: Analytical formulas through the initial values suitable for numerical computation are developed for the exponential integral functions En (x). The relationships obtained are numerically stable for all values of n and for x < 1. Numerical results are also given.

The total π-electron energy as a problem of moments: application of the Backus–Gilbert method

Abstract: The total π-electron energy problem can be formulated as a classical problem of moments. This observation allows us to apply general methodologies developed in the field of moment’s theory to solve the total π-electron energy problem. In the present article, we apply the Backus–Gilbert method to obtain analytical expressions for the total π-electron energy in terms of its spectral moments.

Relative positions of limit cycles in the continuous culture vessel with variable yield

Abstract: To estimate the relative position of limit cycles for a continuous culture vessel is always useful in the qualitative study of the system. In this paper, we construct an annular region containing all the limit cycles for the chemostat with variable yield model that was studied by Huang (J. Math. Chem. 5, 151–166. 1990), and by Pilyugin and Waltman (Math. Biosci. 182, 151–166. 2003).

Central limit theorem for chemical kinetics in complex systems

Abstract: The prevalence of apparently first-order kinetics of reactant disappearance in complex systems with many possible reaction pathways is usually attributed to the dominance of a single rate limiting step. Here, we investigate another possible explanation: that apparently first-order kinetics might arise because the aggregate behavior of many processes, with varying order of reaction and rate constant, approaches a “central limit” that is indistinguishable from first-order behavior. This hypothesis was investigated by simulating systems of increasing complexity and deriving relationships between the apparent reaction order of such systems and various measures of their complexity. Transformation of a chemical species by parallel irreversible reactions that are zero-, first-, or second-order is found to converge to a central limit as the number of parallel reactions becomes large. When all three reaction orders are represented, on average, in equal proportions, this central limit is experimentally indistinguishable from first-order. A measure of apparent reaction order was used to investigate the nature of the convergence both stochastically and by deriving theoretical limits. The range of systems that exhibit a central limit that is approximately first-order is found to be broad. First-order like behavior is also found to be favored when the distribution of material among the parallel processes (due to differences in rate constants for the individual reactions) is more complex. Our results show that a first-order central limit exists for the kinetics of chemical systems and that the variable controlling the convergence is the physical complexity of reaction systems.