Journal of Mathematical Chemistry最新文献

This study proposes the application of the bond dual descriptor, computed using finite difference approximation, to investigate the electron density reorganization in π-conjugated compounds upon nucleophilic/electrophilic attack. Compared to the traditional bond Fukui function analysis, this descriptor offers a simpler approach, reducing the complexity of potential rearrangement scenarios in half and simplifying the interpretation. A series of ethylene derivatives have been tested allowing to explain the activation of the double bond in ethylene derivatives, the rearrangement of the electron density and its reduction in activation energies. In addition, it is shown that it is possible to explain the reactivity on Michael acceptors, the rearrangement of cis-1,3,5-hexatriene to form 1,3-cyclohexadiene and the preference of C₆₀ to react through its [6,6] over [5,6] bonds. The bond dual descriptor complements the atomic dual descriptor, enabling a comprehensive analysis of the chemical reactivity of π-conjugated compounds.

The theory of atomic structure is conceptually built on hydrogen-like orbitals and computed using Slater or Gaussian orbitals, owing to the relative difficulty of computing integrals concerning the hydrogenic orbitals. The optimal set of hydrogenic orbitals in an atom is obtained by minimizing the energy with respect to the orbitals. The Coulomb integral is difficult to compute due to the inverse distance relationship. In this paper, we evaluate the Coulomb integral and its derivative using two expressions for the inverse distance: the Laplace expression and the Legendre expression. The two expressions for inverse distance are similar and yield different integral forms. The Laplace expression yields the Coulomb integral as a sum of hypergeometric functions while the Legendre expression yields a compact polynomial form. The derivative of the Coulomb integral (computed using both forms) with respect to the decay constant is also provided.

The methodology for calculating the phase-lag and amplification-factor for both explicit and implicit multistep methods for first-order differential equations was recently developed by one of the authors. The objective of this study is to develop low-order Adams–Bashforth–Moulton predictor–corrector algorithms that eradicate phase-lag, amplification-factor. The stability regions of the newly established methodologies will also be highlighted. Furthermore, we will examine our results from numerical experiments employing the newly developed approaches.

The paper discusses a mathematical model for non-Michaelis–Menten kinetics, which involves a substrate forming a complex with the immobilized catalyst. A new Hosoya polynomial approximation method (HPAM) is applied for solving the reaction–diffusion equations. Analytical expressions are established to the nonlinear reaction–diffusion equation arising in electro catalytic thin film with an arbitrary shape models using the Hosoya polynomials. The main idea of the proposed research work is that the nonlinear reaction–diffusion problems are converted into a system of algebraic equations using the Hosoya polynomials. Analytical expressions for substrate concentration profiles are derived in closed and simplified forms for various geometries (planar, cylindrical, and spherical), along with the corresponding steady-state amperometric current response. The proposed results are validated with the other available results. Moreover, the utility of HPAM is investigated to be simple, straight forward, efficient and flexible. Also, the paper examines how different parameters influence the substrate concentration in the above models.

In this paper a numerical method on a posteriori mesh is presented to solve a second-order singularly perturbed convection-diffusion equation. A second-order three-point difference method is used to discretize the singularly perturbed convection-diffusion equation. A posteriori error analysis involving simple calculations with fewer proof techniques is developed for the second-order three-point difference method on an arbitrary mesh. A solution-adaptive algorithm based on a posteriori error analysis is designed to generate a posteriori mesh and the approximation solution. Numerical experiments verify the second-order uniform convergence of this method.

The Dűrer Solid has an art and mathematical history ever since it appeared in Dűrer’s 1514 copper engraving, Melencolia I. The associated truncated triangular trapezohedron is an intriguing structure owing to the symmetry, chirality concepts and combinatorial properties. The heptahedral version corresponds to the generalized Petersen’s graph G(7,2). It is shown that the heptagonal antiprism exhibits interesting combinatorial and chirality properties. The heptagonal bipyramid boron complex has been studied recently, and thus the developed techniques were applied to the multiple-quantum NMR of this complex. These graphs and solids serve as representations in dynamic stereochemistry of fluxional molecules as well as in material and molecular science. We consider symmetry-driven combinatorial-group theory through generating functions for the face, edge and vertex colorings of the title graphs and solids for all irreducible representations of the respective symmetry groups. We demonstrate of our computational-combinatorics techniques with tables of combinatorial numbers for the various colorings of these graphs for all irreducible representations. The chirality of the colorings can be directly inferred from the combinatorial enumerations. Applications to the ¹⁰B and ¹¹B MQ-NMR and nuclear spin statistics of the heptagonal bipyramid complex are considered.

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Complex dynamical systems, such as multicomponent chemical reactions, can exhibit chaotic behaviour, posing challenges for process control and optimisation. The pursuit of effective control methodologies to stabilise chaotic reaction systems constitutes a broad field of research and application. This study proposes the application of optimal linear control to mitigate chaotic behaviour in a model of four-component chemical reactions within a continuous stirred-tank reactor (CSTR). The methodology involves representing the reaction system through differential equations and minimising the Hamilton–Jacobi-Bellman functional equation via a linear feedback controller based on a Lyapunov function. Numerical simulations validate the methodology’s efficacy, demonstrating the controller’s capacity to transition the system of equations from a chaotic state to a stable periodic regime. The results highlight the potential of optimal linear control for optimising the model of complex chemical processes, thereby opening possibilities for technological applications in specific scenarios. Optimal linear control has proven effective in stabilising the model of the reaction system, presenting itself as a promising tool for the design of industrial processes involving continuous flow reactors. In these reactors, precise control of concentrations is crucial to ensure process quality and safety.

The electronic states arising from the 5g-row of the periodic table are far less explored compared to other lighter elements in the periodic table. Relativistic effects are extremely large for the 5g-row elements (E-121 to E-138) due to a large nuclear charge approaching the inverse of the fine structure constant. The large relativistic effects result in a molecular spinor nature of the electronic states leading to a substantial mixing of electronic states with different spin multiplicities and differing spatial symmetries. Hence we present the combinatorial enumeration of ω–ω states arising for the 5g-row dimers by employing multinomially-driven symmetric function methods. The combinatorial techniques enumerate all possible ω–ω states originating from the relativistic 2-component molecular spinors for all the 5g-row dimers. There is a combinatorial explosion of electronic states, for example, a 5g-mid-row dimer with 18 electrons exhibits (text{4,537,567,650}) terms in the S-functions. The S-functions for the 5g-row dimers are constructed, and from the S-function terms, those that comply with the Pauli exclusion principle result in a large number of relativistic ω–ω states for the 5g-row dimers. We have constructed the combinatorial tables for the relativistic ω–ω states of several 5g-row dimers subsequent to stipulating several conditions imposed on the level of electron excitations so that the combinatorics is manageable. Furthermore we invoke the electron–hole equivalence to mirror the ω–ω states for the remaining dimers of the 5g-row to enumerate them. Consequently, the developed technique enumerates the ω–ω states arising from up to 18 open-shell states without any restrictions making it applicable to numerous ω–ω states with varied quantum numbers.