Showing papers in "Journal of Mathematical Chemistry in 1999"

General foundations of high-dimensional model representations

Abstract: A family of multivariate representations is introduced to capture the input–output relationships of high‐dimensional physical systems with many input variables. A systematic mapping procedure between the inputs and outputs is prescribed to reveal the hierarchy of correlations amongst the input variables. It is argued that for most well‐defined physical systems, only relatively low‐order correlations of the input variables are expected to have an impact upon the output. The high‐dimensional model representations (HDMR) utilize this property to present an exact hierarchical representation of the physical system. At each new level of HDMR, higher‐order correlated effects of the input variables are introduced. Tests on several systems indicate that the few lowest‐order terms are often sufficient to represent the model in equivalent form to good accuracy. The input variables may be either finite‐dimensional (i.e., a vector of parameters chosen from the Euclidean space $$\mathcal{R}^n$$ ) or may be infinite‐dimensional as in the function space $${\text{C}}^n \left[ {0,1} \right]$$ . Each hierarchical level of HDMR is obtained by applying a suitable projection operator to the output function and each of these levels are orthogonal to each other with respect to an appropriately defined inner product. A family of HDMRs may be generated with each having distinct character by the use of different choices of projection operators. Two types of HDMRs are illustrated in the paper: ANOVA‐HDMR is the same as the analysis of variance (ANOVA) decomposition used in statistics. Another cut‐HDMR will be shown to be computationally more efficient than the ANOVA decomposition. Application of the HDMR tools can dramatically reduce the computational effort needed in representing the input–output relationships of a physical system. In addition, the hierarchy of identified correlation functions can provide valuable insight into the model structure. The notion of a model in the paper also encompasses input–output relationships developed with laboratory experiments, and the HDMR concepts are equally applicable in this domain. HDMRs can be classified as non‐regressive, non‐parametric learning networks. Selected applications of the HDMR concept are presented along with a discussion of its general utility.

A proof of the triangle inequality for the Tanimoto distance

Abstract: A distance, or dissimilarity measure, can be defined based on the Tanimoto coefficient, a similarity measure widely applied to chemical structures. A new, simple proof that this distance satisfies the triangle inequality is presented.

ISOEFF98. A program for studies of isotope effects using Hessian modifications

Abstract: A new program for calculations of isotope effects has been developed. It requires only force constants for substrate and transition state as the external data. All other calculational steps are integrated into the program. ISOEFF98 has features of Hessian modification and scale factor optimization. The first of these allows studies of isotope effect changes upon weakening or strengthening of internal coordinates. The second feature allows fitting of the calculated isotope effect to the experimental value by scaling of molecular frequencies.

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

Abstract: A second-order method is developed for the numerical solution of the initial-value problems \(u' \equiv du/dt = f_1 \left( {u,v} \right)\), \(t > 0\), \(u\left( 0 \right) = U^0 \) and \(v' \equiv dv/dt = f_2 \left( {u,v} \right)\), \(t > 0\), \(v\left( 0 \right) = V^0 \), in which the functions \(f_1 \left( {u,v} \right) = B + u^2 v - \left( {A + 1} \right)u\) and \(f_2 \left( {u,v} \right) = Au - u^2 v\), 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 (\(U^ * = B\), V* = A/B) provided \(1 - A + B^2 \geqslant 0\). 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).

Kinetic approach to chemical reactions and inelastic transitions in a rarefied gas

Abstract: A kinetic approach is presented for the analysis of a gas mixture with two kinds of nonconservative interactions. In a bimolecular chemical reaction, mass transfer and energy of chemical link arise, and in inelastic mechanical encounters, molecules get excited or de‐excited due to their quantized structure. Molecules undergo transitions between energy levels also by absorption and emission of photons of the self‐consistent radiation field. From the kinetic Boltzmann‐type equations, the problem of equilibria and of their stability is addressed. A detailed balance principle is proved and a Lyapunov functional is constructed; mass action law and Planck's law of radiation are recovered.

Protonation microequilibrium treatment of polybasic compounds with any possible symmetry

Abstract: Microequilibrium treatment that has previously been limited to tridentate ligands is generalized to arbitrary number of functional groups in the molecule and the role of symmetry is also investigated. Cumulative microconstant, a new type of equilibrium parameter, is introduced, allowing an equivalent, but more compact mathematical treatment of large microequilibrium systems. The sufficient number of independent pieces of information for the unambiguous determination of all microconstants is deduced. It has been concluded that even if protonation mole fraction for all the basic sites is available, determinability of all the microconstants is rather the exception than the case, without a priori simplifying assumptions. It has been shown that all microconstants can only be determined from protonation mole fractions for molecules of up to three groups. For molecules of four groups and beyond, only specific symmetry and the concomitant simplification of the microequilibrium system make the strict, complete microspeciation feasible. As a case study, the protonation scheme and the complete microspeciation of a tetradentate ligand is analyzed in detail.

A family of P-stable exponentially‐fitted methods for the numerical solution of the Schrödinger equation

Abstract: A family of P‐stable exponentially‐fitted methods for the numerical solution of the Schrodinger equation is developed in this paper. An application to the resonance problem of the radial Schrodinger equation indicates that the new method is generally more efficient than the previously developed exponentially‐fitted methods of the same kind.

On minimal energy ordering of acyclic conjugated molecules

Abstract: The energy of a graph is defined as the sum of the absolute values of all the eigenvalues of the graph. Gutman (Acyclic conjugated molecules, trees and their energies, J. Math. Chem. 1 (1987) 123–143) proposes two conjectures about the minimum of the energy of conjugated trees (trees with a perfect matching), which are verified by Zhang and Li (On acyclic conjugated molecules with minimal energies, Discrete Appl. Math. 92 (1999) 71–84). This paper focuses on the trees of conjugated hydrocarbon\(n/2\)/EquationSource> trees in the class in the increasing order of their energies.

Embedded eighth order methods for the numerical solution of the Schrödinger equation

Abstract: A new method for the approximate numerical integration of the radial Schrodinger equation is developed in this paper. Phase-lag and stability analysis of the new method is included. The new method is called the embedded method because of a simple natural error control mechanism. Numerical results obtained for the phase-shift problem of the radial Schrodinger equation show the validity of the developed theory.

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

Abstract: The basic ideas of the Improved Bardeen–Cooper–Schrieffer (IBCS) approach to the first‐ and second‐order Reduced Density Matrices (1‐ and 2‐RDM) are briefly reviewed. The molecular orbital occupations \(\left\{ {\rho _q } \right\}\) are expressed by means of new quantities \(\left\{ {\gamma _q } \right\}\), which, satisfying a trigonometric relation, guarantee the non‐idempontent condition. Thus, a variational method is introduced to determine \(\left\{ {\rho _q } \right\}\), involving only an unconstrained minimization which may be performed using a conjugate gradient technique. A new effective Hamiltonian \(\hat V\) which is composed of the Coulomb, exchange and exchange‐time 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 self‐consistent‐field method is proposed: the iterative procedure is repeated until convergence is reached for the actual density matrix.

Bound states in the Kratzer plus polynomial potentials and the new form of perturbation theory

Abstract: The Schrodinger equation with potentials of the Kratzer plus polynomial type (say, quartic V(r) = Ar 4 + Br 3 + Cr 2 + Dr + F/r + G/r 2 etc.) is considered and a new method of exact construction of some of its bound states is presented. Our approach is made feasible via a combination of the traditional use of the infinite series ψ(r)(terminated rigorously after N + 1 terms at certain specific couplings and energies) with several new ideas. We proceed in two steps. Firstly, in the strong coupling regime with G → ∞, we find the exact, complete and compact unperturbed solution of our N + 1 coupled and nonlinear algebraic conditions of the termination. Secondly, we adapt the current Rayleigh–Schrodinger perturbation theory to our nonlinear equations and define the general G < ∞ bound states via an innovated, triple perturbation series. In its tests we show how all the corrections appear in integer arithmetics and remain, therefore, exact.

The heptakisoctahedral group and its relevance to carbon allotropes with negative curvature

Abstract: We introduce the pollakispolyhedral groups and describe in detail the representational structure of PSL(2,7) or 7 O, the automorphism group of the Klein graph composed of 56 trivalent vertices arranged in 24 heptagonal faces Leapfrog and quadruple transformations of the graph are described and their eigenvalue spectra derived Considered as carbon frameworks on the “plumber's nightmare” surface these chiral structures exhibit significant steric strain which prevents the molecular realisation of the Klein symmetry

A new transformation for the Lotka–Volterra problem

Abstract: The Lotka–Volterra dynamical system \(\left( {\dot x_1 = ax_1 - bx_1 x_2 ,\;\dot x_2 = - cx_2 + bx_1 x_2 } \right)\) is reduced to a single second‐order autonomous ordinary differential equation by means of a new variable transformation. Formal analytic solutions are presented for this latter differential equation.

A graph-theoretical approach to statistics and dynamics of tree-like molecules

Abstract: We present a new graph‐theoretical method for calculating the dynamical and statistical properties of a Gaussian chain with various molecular architecture. The characteristic polynomial for its line graph, which has the bonds of a molecular graph as its beads with adjacency of bonds as in the graph, makes it possible to provide us with the general equation for calculating the radius of gyration of Gaussian chains and their relaxation spectra.

Non‐linear transformations for rapid and efficient evaluation of multicenter bielectronic integrals over B functions

Abstract: This work presents an extremely efficient non‐linear transformation based on a certain Hankel type transform, originally due to A. Sidi. The approach is applied to evaluating Coulomb integrals in the molecular context. These integrals are bielectronic one‐, two‐, three‐ and four‐center terms arising from the interactions of electron distributions over a Slater type orbital basis. They occur in many millions of terms, even for small molecules, and require rapid and accurate evaluation. The present work shows how we can reduce the order of the linear differential equation required to be satisfied by the integrand considerably. Calculation times as short as 10-2 ms were obtained for four‐center terms (the least favorable case) on an IBM RS6000‐340 workstation. This method represents a considerable advance on previous work on Coulomb integrals.

Quantum algebraic-combinatoric study of the conformational properties of n-alkanes. I

Abstract: In a previous paper we solved a well-defined subcase of the whole conformational enumeration problem of n-alkanes This paper offers a general solution Most probably the rules determining the sequences of the conformers slightly depend on the force field applied Nevertheless, the methods presented here can be applied in every case

Modelling a biosensor based on the heterogeneous microreactor

Abstract: Modelling of the amperometric biosensors based on carbon paste electrodes encrusted with a single heterogeneous microreactor is analyzed. The microreactor was constructed from CPC‐silica carrier and was loaded with glucose oxidase. The model is based on non‐stationary diffusion–reaction equations containing a non‐linear term related to the enzymatic reaction. A homogenization process having an effective algorithm for the digital modelling of the operation of the microreactor is proposed. The influence of the size, geometrical form, and the position of a microreactor on the operation of biosensors are investigated.

A self-assembling neural network for modeling polymers

Abstract: A central problem in modeling protein and other polymer structures is the generation of self‐avoiding chains which obey a priori distance restraint information which could include a folding potential function. This problem is usually addressed with a lattice model or a torsion space model of the polymer. Exhaustive searches in these spaces are of necessity exponentially complex. A new computer algorithm for modeling polymers and polymeric systems is described. This algorithm is a randomized algorithm based on a self‐assembling or Kohonen neural network. Given a defined chain topology, a defined spatial extent, and a prior probability distribution, it finds a set of coordinates which reproduce these properties. The convergence rate of the algorithm is linear with respect to the number of distance terms included. Modifications to the standard Kohonen algorithm to include a defined spatial metric, and a modified update rule improve the convergence of the standard algorithm and result in a highly efficient algorithm for building polymer models which are self avoiding and consistent with prior probability information and interatomic distance restraints.

Another way to implement the Powell formula for updating Hessian matrices related to transition structures

Abstract: A way to update the Hessian matrix according to the Powell formula is given. With this formula one does not need to store the full Hessian matrix at any iteration. A method to find transition structures, which is a combination of the quasi‐Newton–Raphson augmented Hessian algorithm with the proposed Powell update scheme, is also given. The diagonalization of the augmented Hessian matrix is carried out by Lanczos‐like methods. In this way, during all the optimization process, one avoids to store full matrices.

Exchange and correlation in the Thomas-Fermi approximation

Abstract: We present the exchange and correlation potential calculated analytically as a function of the Hartree potential. We arrived at this expression by using the Thomas–Fermi approximation. This is an alternative way of calculating the exchange and correlation potential which can be very efficient.

Analytic solutions to a family of Lotka–Volterra related differential equations

Abstract: An initial formal analysis of the analytic solution (C.M. Evans and G.L. Findley, J. Math. Chem. 25 (1999) 105–110.) to the Lotka–Volterra (LV) dynamical system is presented. A family of first‐order autonomous ordinary differential equations related to the LV system is derived, and the analytic solutions to these systems are given. Invariants for the latter systems are introduced, and a simple transformation which allows these systems to be reduced to Hamiltonian form is provided.

A naive look on the Hohenberg–Kohn theorem

Abstract: A generalised Hohenberg-Kohn theorem is described in terms of the sign of the second-order energy variation. Independently, it is also corroborated within the perturbation theoretical framework. An alternative formulation of the Hohenberg-Kohn theorem, based on the relationships involving the matrix representations of density functions and the Hamiltonian operator variations, is shown to extend the validity of the theorem to the excited states of the Hamiltonian operators possessing non-degenerate spectra. Finally, a connection with Brillouin's theorem when energy variation becomes stationary is also outlined.

A class of exactly solvable matrix models

Abstract: Some exactly solvable matrix models are discussed. Possible applications to problems in physical chemistry are pointed out, in particular the Huckel problem, the problem of torsional vibrations of polyatomic molecules, and of vibrations of finite polymer chains.

Chemical clock reactions: The effect of precursor consumption

Abstract: During a clock reaction an initial induction period is observed before a significant change in concentration of one of the chemical species occurs. In this study we develop the results of Billingham and Needham (1993) who studied a particular class of inhibited autocatalytic clock reactions. We obtain modified expressions for the length of the induction period and show that characteristic clock reaction behaviour is only observed within certain parameter limits.

Algebraic derivation of Franck–Condon overlap integrals for diatomic molecules

Abstract: We derive a new formula for Franck–Condon harmonic‐oscillator overlap integrals using an algebraic procedure based on a Bogoliubov transformation. We discuss how the formulation may be generalized to SU(2)‐based descriptions of anharmonic oscillator wave functions.

Continuous symmetry measures: A note in proof of the folding/unfolding method

Abstract: The measurement of the degree of symmetry proved to be a useful tool in the prediction of quantitative structural–physical correlations. These measurements have been based, in the most general form, on the folding/unfolding algorithm, for which we provide here a new and simpler proof. We generalize this proof to the case of objects composed of more than one (full) orbit. An important practical issue we consider is the division of the graph into symmetry orbits and the mapping of the symmetry group elements onto the points of the graph. The logical constraints imposed by the edges of the graph are reviewed and used for the successful resolution of the coupling between different orbits.

Two alternative derivations of Bridgman's theorem

Abstract: One of the fundamental results of Dimensional Analysis is the so‐called Bridgman' s theorem. This theorem states that the only functions that may have dimensional arguments are products of powers of the base quantities of a given system of units. In this work, Bridgman's theorem is discussed and rederived in two different ways, one not involving calculus, and a second one based on a Taylor series expansion analysis.

Multistationarity is neither necessary nor sufficient to oscillations

Abstract: The existence of more than one stationary point for certain parameter values is generally considered to be a necessary condition of the emergence of oscillatory solutions for some parameter values in reaction kinetics. Examples taken from both physics and chemical kinetics show that the condition is neither necessary nor sufficient.

New N -representability results with symmetry in molecular quantum chemistry

Abstract: We prove a new type of N-representability result: given a totally symmetric density function ρ, we construct a wavefunction Ψ such that the totally symmetric part of $$\rho \Psi $$ (its projection over the totally symmetric functions) be equal to ρ, and, furthermore, such that Ψ belongs to a given class of symmetry associated to the symmetry group of a molecule. Our proof uses deformations of density functions and which are solutions of a “Jacobian problem”. This allows us to formalize rigorously an idea of A. Gorling (Phys. Rev. A 47 (1993) 2783), for Density-Functional Theory in molecular quantum chemistry, by defining a density functional that takes into account the symmetry of the molecule under study.

Scalar wave diffraction from zero-range scatterers

Abstract: The zero‐range approximation for wave scattering is popularized. The general idea of the approximation is outlined and the theory is applied to solve an illustrative example: diffraction on a system of four identical scatterers forming a d-symmetry structure. The group theoretical methods are used to decompose a scattered wave into four partial waves adequate to the symmetry group of the target. Concepts of partial scattering amplitudes, associated phase shifts and partial cross sections for the nonspherical target considered are introduced and utilized. It is shown that under some conditions a phenomenon of resonant scattering may occur.