Theoretical physics and mathematics

Raymond Daudel, Treatise of Theoretical Physics and Mathematical Physics, No. XXIV Quantum Theory of Chemical Reactivity, Gauthier-Villars, Paris, 1967. 

Department of Theoretical Physics and Mathematical Methods, Gdatisk University of Technology, ul. Narutowicza 11/12, Pl-80952 Gdansk, Poland... 

The energy hypersurface describes the energy of chemical systems with a given set of atoms as a function of the spatial distribution of the atomic nuclei. If one could compute the complete energy hypersurface for any set of atoms and analyse the corresponding data in a suitable m.anner, one could predict most of the relevant properties of molecular systems with the methods of theoretical physics and mathematics. 

Hydrodynamic volume refers to the combined physical properties of size and shape. Molecules of larger volume have a limited ability to enter the pores and elute the fastest. A molecule larger than the stationary phase pore volume elutes first and defines the column s void volume (Vo). In contrast, intermediate and smaller volume molecules may enter the pores and therefore elute later. As a measure of hydrodynamic volume (size and shape), SE-HPLC provides an approximation of a molecule s apparent molecular weight. For further descriptions of theoretical models and mathematical equations relating to SE-HPLC, the reader is referred to Refs. 2-5. 

Perturbation theory is one of the oldest and most useful, general techniques in applied mathematics. Its initial applications to physics were in celestial mechanics, and its goal was to explain how the presence of bodies other than the sun perturbed the elliptical orbits of planets. Today, there is hardly a field of theoretical physics and chemistry in which perturbation theory is not used. Many beautiful, fundamental results have been obtained using this approach. Perturbation techniques are also used with great success in other fields of science, such as mathematics, engineering, and economics. 

In its disciplinary development at the end of the nineteenth century, physical chemistry served as a bridge, not a wedge, between the mathematical abstractions of theoretical physics and the metaphorical descriptions of organic chemistry. Not only through novel theories but also through control of new instrumentation, much of it electrical and optical in nature, physical chemistry was to revitalize and transform techniques in the chemical laboratory and theories of chemical explanation. Notably for our concerns, speculations about reaction mechanisms in hydrocarbon chemistry were to begin to proliferate in the early 1900s. 

The disciplinary titles of the practitioners of quantum chemistry and chemical physics varied. Pauling initially wanted his title at Caltech to be professor of theoretical chemistry and mathematical physics, but he accepted CalTech s dropping "mathematical physics" 115 and later preferred to be known as a chemist. 116 Slater always had his principal appointment in a physics department. Mulliken, who had taken his degree at Chicago in chemistry, returned in 1928 as associate professor of physics and retired in 1983 as professor of chemistry and physics. 117... 

Philosophical" or theoretical chemistry was wide-ranging during most of the nineteenth century. In contrast, late-nineteenth-century physical chemists and twentieth-century physicists tended to narrow the definition of theoretical chemistry, eliminating organic structure theory and making theoretical chemistry almost exclusively physical and mathematical. An early indicator of this trend is Noyes s deletion of structure theory from the course in theoretical chemistry at MIT. A later indicator is the special issue of Chemical Reviews in 1991 which carries the title, "Theoretical Chemistry," and begins with an introductory editorial entitled simply "Quantum Theory of Matter." 5... 

In practice, the primary objective of chemical thermodynamics is to estabhsh a criterion for determining the feasibility or spontaneity of a given physical or chemical transformation. For example, we may be interested in a criterion for determining the feasibility of a spontaneous transformation from one phase to another, such as the conversion of graphite to diamond, or the spontaneous direction of a metabohc reaction that occurs in a cell. On the basis of the first and second laws of thermod5m-amics, which are expressed in terms of Gibbs s functions, several additional theoretical concepts and mathematical functions have been developed that provide a powerful approach to the solution of these questions. 

As full theoretical discussion of this problem involves enormous difficulties, of both physical and mathematical nature, the practice nowadays is to make use of calculations based on an idealized picture of the phenomenon. These idealizations are based on the following assumptions ... 

Van Dilla MA, Dean PN, Laerum OD, Melamed MR, eds. (1985). Flow Cytometry Instrumentation and Data Analysis. Academic Press, London. A venerable, but still current book with an emphasis on the physics and mathematics of flow systems and data analysis. It has some excellent (and readable) articles on some theoretical subjects. 

Perhaps the most remarkable feature of modem chemical theory is the seamless transition it makes from a microscopic level (dealing directly with the properties of atoms) to describe the structure, reactivity and energetics of molecules as complicated as proteins and enzymes. The foundations of this theoretical structure are based on physics and mathematics at a somewhat higher level than is normally found in high school. In particular, calculus provides an indispensable tool for understanding how particles move and interact, except in somewhat artificial limits (such as perfectly constant velocity or acceleration). It also provides a direct connection between some observable quantities, such as force and energy. 

Here we give an overview of the current status and perspectives of theoretical treatments of solvent effects based on continuum solvation models where the solute is treated quantum mechanically. It is worth noting that our aim is not to give a detailed description of the physical and mathematical formalisms that underlie the different quantum mechanical self-consistent reaction field (QM-SCRF) models, since these issues have been covered in other contributions to the book. Rather, our goal is to illustrate the features that have contributed to make QM-SCRF continuum methods successful and to discuss their reliability for the study of chemical reactivity in solution. 

This book brings together the essential ideas and methods behind current applications of variational theory in theoretical physics and chemistry. The emphasis is on understanding physical and computational applications of variational methodology rather than on rigorous mathematical formalism. 

A programme to develop a theory of chemistry, not dictated by theoretical physics and free of unnecessary mathematical complications, is not supposed to be a paradigm in isolation. It should respect the discoveries of related disciplines, but not necessarily all of their interpretations. The implications of relativity and quantum theory are as important for the understanding of chemical phenomena as for physics, particularly in so far as these theories elucidate the structure of matter. This aspect is of vital importance to chemistry, but only a philosophical curiosity in physics. In the orthodox view of physics it is the outcome of experimental measurements which has theoretical significance - the chemist needs insight into the nature of elementary substances to understand and manipulate their systems of interest. With-... 

We have tried to make plausible and physically acceptable a result of fundamental importance. In no way should the adoption of the earlier heuristic arguments diminish in the eyes of nonexpert readers the importance of these results for physics and mathematics. Levy processes seem to be ubiquitous, and it is a challenge for theoretical physicists to find a way to establish if the traditional approaches of nonequilibrium statistical mechanics can satisfactorily account for them. 

Mathematics was incorporated into chemistry from physics, and for a long time it was used mainly as a tool for calculations. There is no doubt that such an approach led to important discoveries, and numerical computations are of great importance. Quantum chemistry may be seen as a good example of advances in theoretical physics, and numerical methods which together with the development of new computer hardware allowed us to calculate many properties of molecules with great accuracy. However, quantum chemistry is, in essence, the theoretical physics of molecules, and may be classified as the physical metatheory of chemistry which interprets phenomenological notions and concepts of chemistry and provides its physical background. 

In this chapter these phenomena will be systematically discussed. Related phenomena such as surface conduction and dielectric relaxation of sols will be included. In view of the fact that the rigorous theory is both physically and mathematically extremely involved, we shall discuss the phenomena in two steps. First, elementary theory will be presented. For electrophoresis this leads to the Huckel-Onsager and Helmholtz-Smoluchowskl equations. It gives the leading term for a number of simple conditions regarding potential and particle size and shape. Thereafter, more advanced theoretical treatments will follow. 

It has been of considerable interest to develop a theoretical model for predicting the behavior of fire. Excellent articles by Martin and others reflect the strides made in this direction through a number of investigations. Except for Martin s work, which is briefly reviewed, most of these studies (involving the disciplines of physics and mathematics) are beyond the scope of the present article. However, it should be noted that some of the formulas and correlations developed are based on the chemical kinetics, as well as on physical principles. Thus, the lack of sufiBcient knowledge regarding the nature of the combustion process and the reactions involved has led to serious limitations that have been handled by various forms of approximation. For instance, the pioneering work of Bamford, Crank, and Malan was based on the assumption that thermal decomposition. 

The plate models assume that the column is divided into a series of an arbitrary number of identical equilibrium stages, or theoretical plates, and that the mobile and the stationary phases in each of these successive plates are in equihbrimn. The plate models are in essence approximate, empirical models because they depict a continuous column of length I by a discrete number of well-mixed cells. Although any mixing mechanism is dearly absent from the actual physical system, plate models have been used successfully to characterize the column operation physically and mathematically. Therefore, by nature, plate models are empirical ones, which cannot be related to first principles. 

S. M. Blinder is Professor Emeritus of Chemistry and Physics at the University of Michigan, Ann Arbor. Born in New York City, he completed his PhD in Chemical Physics from Harvard in 1958 under the direction of W. E. Moffitt and J. H. Van Vleck (Nobel Laureate in Physics, 1977). Professor Blinder has over 100 research publications in several areas of theoretical chemistry and mathematical physics. He was the first to derive the exact Coulomb (hydrogen atom) propagator in Feynman s path-integral formulation of quantum mechanics. He is the author of three earlier books Advanced Physical Chemistry (Macmillan, 1969), Foundations of Quantum Dynamics (Academic Press, 1974), md Introduction to Quantum Mechanics in Chemistry, Materials Science and Biology (Elsevier, 2004). 

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