Structural analysis of certain linear operators representing chemical network systems via the existence and uniqueness theorems of spectral resolution. II

The methodology and theoretical framework of Part I of this series of articles have been further developed to the setting of the Banach algebra B (??), of all bounded operators acting on a Banach space ??. Using the above setting B (??), certain dynamical systems of chemical kinetic equations are illustrated in conjunction with Part I and with the analysis of more general systems, some of which will be made in Part III of this series of articles. The main theorem and its auxiliary theorem in the present article elucidate in a unifying manner the structure and the underlying pattern of the spectral symmetry of linear operators (acting on Banach spaces and Hilbert spaces) that are investigated in each of the research fields of dynamical systems and quantum chemistry involving the spectral symmetry of alternant hydrocarbons. © 1995 John Wiley & Sons, Inc.     

Structural analysis of certain linear operators representing chemical network systems via the existence and uniqueness theorems of spectral resolution. IV

Part IV of this series consists of two complementary subparts devoted to attain the following two goals: (i) By shifting from the previous setting of the Banach algebra B (ℬ︁)= B (ℬ︁, ℬ︁) to a broader setting of the space B (X, ℬ︁) of all bounded linear operators from a normed space X to a Banach space ℬ︁, we extend our previous theoretical framework to incorporate part of the theory of additive correlation involving the Asymptotic Linearity Theorems, which have been developed for a study of correlation between structure and properties in molecules having many identical moieties, especially in macromolecules having repeating units. (ii) By reverting our focus to the special algebra B (ℋ︁) with ℋ︁ being a Hilbert space, we develop a theorem which is useful for a structural analysis of spectral symmetry of linear operators representing physico-chemical network systems. © 1998 John Wiley & Sons, Inc. Int J Quant Chem 67 : 57–69, 1998     

Structural analysis of certain linear operators representing chemical network systems via the existence and uniqueness theorems of spectral resolution. I

The present article develops a methodology and a unifying theorem to treat, on an equal footing, mathematical phenomena that were hitherto studied separately in each of the research fields of dynamical systems and quantum chemistry involving the spectral symmetry of alternant hydrocarbons. This article also serves as a foundation of a theoretical framework for the analysis of certain dynamical systems of chemical kinetic equations, which shall be made in the context of operator algebra in Parts II and III of this series of papers. © 1995 John Wiley & Sons, Inc.     

Finding complex balanced and detailed balanced realizations of chemical reaction networks

Reversibility, weak reversibility and deficiency, detailed and complex balancing are generally not “encoded” in the kinetic differential equations but they are realization properties that may imply local or even global asymptotic stability of the underlying reaction kinetic system when further conditions are also fulfilled. In this paper, efficient numerical procedures are given for finding complex balanced or detailed balanced realizations of mass action type chemical reaction networks or kinetic dynamical systems in the framework of linear programming. The procedures are illustrated on numerical examples.     

Scalar nonlinearities in physics and chemistry

Scalar nonlinearities are operators of the form N(u) = 〈Au, u〉Bu, where A, B are linear operators, and 〈·,·〉 is the inner product in a Hilbert space ??. This paper reviews applications of scalar nonlinearities. We show that operators of the form N(u) are found in equations that describe phenomena of classical mechanics, open systems of quantum mechanics, and chemical physics.     

General criteria for assessing the accuracy of approximate wave functions and their densities

Master equations for propagators in quantum open systems and their spectral resolutions are derived. The Zwanzig partitioning scheme along the superoperator algebra are used to derive equations of motion for partitioned operators in a Liouville space. The reservoir influence on the dynamical evolution of operators is shown to lead explicitly to dissipative effects arising from memory terms in the evolution equations of such operators. It is also shown that spectral representations may be written in a self-consistent analytic way by means of the self-energy fields for transition energies of the system by taking into account the lack of the complete knowledge about the reservoir. A kinematic fluid interpretation of the resultant equations is given and an explicit form of the “collision” superoperator is obtained. Finally, a simple example to illustrate the determination of self-energy fields for the system–reservoir interaction corrections is given. © 1995 John Wiley & Sons, Inc.     

Simulating quantum dynamics in classical environments

Methods for simulating the dynamics of composite systems, where part of the system is treated quantum mechanically and its environment is treated classically, are discussed. Such quantum–classical systems arise in many physical contexts where certain degrees of freedom have an essential quantum character while the other degrees of freedom to which they are coupled may be treated classically to a good approximation. The dynamics of these composite systems are governed by a quantum–classical Liouville equation for either the density matrix or the dynamical variables which are operators in the Hilbert space of the quantum subsystem and functions of the classical phase space variables of the classical environment. Solutions of the evolution equations may be formulated in terms of surface-hopping dynamics involving ensembles of trajectory segments interspersed with quantum transitions. The surface-hopping schemes incorporate quantum coherence and account for energy exchanges between the quantum and classical degrees of freedom. Various simulation algorithms are discussed and illustrated with calculations on simple spin-boson models but the methods described here are applicable to realistic many-body environments.     

Kinetic study of atomization in atomic absorption analysis of solid samples using flame-furnace atomizer

Direct solid sampling analysis in atomic absorption spectrometry using flame-furnace atomizer allows to significantly decrease analysis duration, to avoid sample pollution and to exclude toxic reagents. The choice of chemical modifiers that decreases the detection limit and improves the repeatability of results is based on the mechanism of analyte-free atom formation. The kinetic approach developed here allows determination of pre-exponential factors k₀ and apparent activation energies E_a of the atomization processes for Pb(II) and Cd(II) compounds and enables use of CaCO₃ and KHF₂ as effective chemical modifiers for soil samples. A fast and precise technique for lead and cadmium determination in soils using the proposed chemical modifiers was developed.     

HOA (Heaviside Operational Ansatz) revisited: recent remarks on novel exact solution methodologies in wavefunction analysis

A viable methodology for the exact analytical solution of the multiparticle Schrodinger and Dirac equations has long been considered a holy grail of theoretical chemistry. Since a benchmark work by Torres-Vega and Frederick in the 1990s, the QPSR (Quantum Phase Space Representation) has been explored as an alternate method for solving various physical systems. Recently, the present author has developed an exact analytical symbolic solution scheme for broad classes of differential equations utilizing the HOA (Heaviside Operational Ansatz). An application of the scheme to chemical systems was initially presented in Journal of Mathematical Chemistry (Toward chemical applications of Heaviside Operational Ansatz: exact solution of radial Schrodinger equation for nonrelativistic N-particle system with pairwise 1/r(I) radial potential in quantum phase space. Journal of Mathematical Chemistry, 2009; 45(1):129–140). It is believed that the coupling of HOA with QPSR represents not only a fundamental breakthrough in theoretical physical chemistry, but it is promising as a basis for exact solution algorithms that would have tremendous impact on the capabilities of computational chemistry/physics. The novel methods allow the exact determination of the momentum [and configuration] space wavefunction from the QPSR wavefunction by way of a Fourier transform. In this note some remarks, examples and further directions, concerning HOA as a tool to solve and provide analytical insight into solutions of dynamical systems occurring in, but not limited to Mathematical Chemistry, are also posited.     

Isochronicity and limit cycle oscillation in chemical systems

Chemical oscillation is an interesting nonlinear dynamical phenomenon which arises due to complex stability condition of the steady state of a reaction far away from equilibrium which is usually characterised by a periodic attractor or a limit cycle around an interior stationary point. In this context Lienard equation is specifically used in the study of nonlinear dynamical properties of an open system which can be utilized to obtain the condition of limit cycle. In conjunction with the property of limit cycle oscillation, here we have shown the condition for isochronicity for different chemical oscillators with the help of renormalisation group method with multiple time scale analysis from a Lienard system. When two variable open system of equations are transformed into a Lienard system of equation the condition for limit cycle and isochronicity can be stated in a unified way. For any such nonlinear oscillator we have shown the route of a dynamical transformation of a limit cycle oscillation to a periodic orbit of centre type depending on the parameters of the system.     

Notes on the Asymptotic Linearity Theorems

The present article provides three lemmas that initiate the generalization of the theory of additive correlation involving the Asymptotic Linearity Theorems, which were constructed for a study of the correlation between structure and properties in molecules having many identical moieties. The new tools provided here also help to pave the way to linking the above theory with a theoretical framework developed for the asymptotic analysis of certain chemical kinetic network systems.On leave from: Institute for Fundamental Chemistry, 34-4 Takano-Nishihiraki-cho, Sakyo-ku, Kyoto 606, Japan.     

The Topological Analysis of the Electron Localization Function. A Key for a Position Space Representation of Chemical Bonds

Summary. The purpose of the topological theory of Chemical Bonding is to provide a mathematical bridge between chemical concepts, such as bonds and lone pair, and rigorous Quantum Mechanics. The theory of dynamical systems enables to achieve a partition of the geometrical molecular space into basins of attractors bearing a chemical significance. A one to one correspondence is established between these basins and the chemical objects used to describe the bonding. The definition of population operators gives access to quantitative information and provides a firm basis for the concept of mesomery.     

Effects of Reversible Chemical Reaction on Morphology and Domain Growth of Phase Separating Binary Mixtures with Viscosity Difference

Summary: The effects of a reversible chemical reaction on morphology and dynamics of phase separating binary mixtures with viscosity difference are studied by numerically solving modified time‐dependent Ginzburg‐Landau and Navier‐Stokes equations. Much more interesting morphologies are observed in the system due to the coupling of reversible chemical reaction and viscosity difference between two components. When the chemical reaction rate is relatively low, the impact of viscosity difference on morphologies is prominent, so that the resulting patterns are affected by both reversible chemical reaction and viscosity difference. However, increasing the chemical reaction rate weakens the impact of viscosity difference on morphologies. Similarly, increasing the chemical reaction rate also suppresses the effects of viscosity difference on domain growth dynamics, which is prominent at the early stage of phase separation when the chemical reaction rate is relatively low. For both cases with relatively low and high chemical reaction rates, the average domain size eventually attains an equilibrium value due to the competition between the mixing of reversible chemical reaction and demixing of phase separation.

Domain patterns of a critical system with ϕ_ini = 0, and Γ₁ = Γ₂ = 0.001.     

The Topological Analysis of the Electron Localization Function. A Key for a Position Space Representation of Chemical Bonds

The purpose of the topological theory of Chemical Bonding is to provide a mathematical bridge between chemical concepts, such as bonds and lone pair, and rigorous Quantum Mechanics. The theory of dynamical systems enables to achieve a partition of the geometrical molecular space into basins of attractors bearing a chemical significance. A one to one correspondence is established between these basins and the chemical objects used to describe the bonding. The definition of population operators gives access to quantitative information and provides a firm basis for the concept of mesomery.     

Dynamical Approach to Multi-Equilibria Problems Considering the Debye–Hückel Theory of Electrolyte Solutions: Concentration Quotients as a Function of Ionic Strength

We recently described a dynamical approach to the equilibrium problem that involves the formulation of the kinetic rate equations for each species. The equilibrium concentrations are determined by evolving the initial concentrations via this dynamical system to their steady state values. This dynamical approach is particularly attractive because it can be extended easily to very large multi-equilibria systems and the effects of ionic strength also are easily included. Here we describe mathematical methods for the determination of steady state concentrations of all species with the consideration of their activities using several approximations of Debye–Hückel theory of electrolyte solutions. We describe the equations for a system that consists of a triprotic acid H₃A and its conjugate bases. With these equations, two types of multi-equilibria systems were studied and compared to experimental data. The first system is exemplified by case studies of solutions of acetate-buffered acetic acid and the second system is exemplified by the hydroxide titration of citric acid. The discussion focuses on the effect of ionic strength on pH and on the amplification of acidity by ionic strength. Ionic strength effects are shown to cause significant deviations from the widely used Henderson–Hasselbalch equation.     

Vibrational motion in isotopomers of the HeH+ molecular ion: An application of the electron nuclear dynamics method

The importance of isotopic substitution as a tool for elucidation of chemical reaction events originates in the fact that the Coulombic Hamiltonian is isotopically invariant except for the nuclear kinetic energy term. Thus, in theories of isotope effects based on the Born-Oppenheimer scheme, the basic presumption is the invariance of the potential energy surface (PES). We use, however, a fully dynamic approach, called Electron Nuclear Dynamics (END), which does not require a preconstructed PES. Since the END formalism is rather different from commonly used procedures, we study the anharmonic nuclear vibration in isotopic species of the HeH⁺ molecular ion as a model problem. A single time-dependent complex parametrized determinantal wave function is used for the electrons and the nuclei are treated classically. The time evolution of the nuclear and electronic dynamical variables obtained by integration of equations of motion are reported as bond length, nuclear kinetic energy, and Mulliken populations. The molecule vibrates as a classical object. The product of the reduced mass and the square of the vibrational frequency is isotopomer invariant for any common total energy. The difference between the total energy and the nuclear kinetic energy as a function of the internuclear distance is interpreted as the average dynamic potential. © 1997 John Wiley & Sons, Inc.     

Nonclassical critical indices in the statistical theory of liquids and equations of state with regular and scaling components

Nonclassical critical indices γ = 9/8 and β = 3/8 obtained in works using the statistical theory of liquids are used to approximate data on the isothermal compressibility of ⁴He and SF₆ on critical isochores and binodals, and on the shape of a binodal. The ratio of asymptotic compressibility coefficients Γ ₀ ⁺ /Γ ₀ ^? on critical isochores and binodals is determined from experiments using statistical theory and it is shown that Γ ₀ ⁺ /Γ ₀ ^? coincides with the magnitude of this ratio for linear scaling model with γ = 9/8 and β = 3/8 indices. Highly accurate P-ρ-T data on ⁴He and SF₆ are approximated at these values of the indices using new combined equations of state that include regular and scaling components. The advantage of describing of P-ρ-T data in this manner, compared to describing them with the same equations of state using the critical indices of the three-dimensional Ising model, is demonstrated.     

Mathematical basis of the integral formalism of chemical kinetics. Compact representation of the general solution of the first-order linear differential equation

The standard approach in chemical and photochemical kinetics is to proceed from the kinetic scheme to the corresponding system of first-order differential equations, and then to integrate it, analytically or numerically. An equivalent integral formulation circumventing such system was recently developed on the basis of physical arguments. The mathematical basis of this ansatz is discussed here. A compact representation of the general solution of the linear first-order differential equation is also obtained.     

On the molecular theory of relaxation processes and the viscoelastic properties of magnetic liquids

The kinetic equations for one-and two-particle distribution functions including the contributions of spatial correlation of density and correlation of velocities were used to obtain equations of generalized hydrodynamics of magnetic liquids, whose transfer coefficients microscopically depended on the spatial and time scales. (The relaxing flows in these equations comprise kinetic and potential parts, which ensures the inclusion of translational and structural relaxation processes.) The system of generalized hydrodynamics equations obtained can be used to study transfer phenomena in magnetic liquids. Smoluchowski equations for binary density n ₂(q ₁, q ₂, t) and binary flow of particles I ₂(q ₁, q ₂, t) were obtained and their general solutions found to construct a closure of the initial kinetic equation for the one-particle distribution function. The asymptotic behavior of these solutions at low frequencies (ω → 0) was analyzed and found to coincide with the long-time asymptotes of autocorrelation functions. The viscoelastic properties of magnetic liquids were studied over a wide range of frequencies in the presence of an external magnetic field H. Microscopic equations for viscosity coefficients and elastic moduli were found and their asymptotic behavior in slow and fast processes considered.