Summary of the Theories

As an introduction to the peculiar properties of the spin Hamiltonians, we first give a short summary of the theory of spin relaxation in liquids where the problem is in fact a Brownian motion one. Then we consider the many-spin problem in solids and apply the general formalism of the theory of irreversible processes developed by Prigogine and his co-workers. We also analyse some aspects of the recent work of Caspers and Tjon on this subject. Finally, we indicate the special interest of spin-spin relaxation phenomena in connection with non-Markovian processes. 

Several monographs2-5 have detailed discussions dealing with heavy-atom and primary and secondary hydrogen-deuterium kinetic isotope effects. The monograph by Melander and Saunders5 covers the entire area particularly well. For this reason, only a brief summary of the theory of kinetic isotope effects as well as their important uses in the determination of reaction mechanism and transition-state geometry will be presented. 

This review is a brief summary of the theory and application of the CORCEMA-ST protocol in the quantitative analysis of STD-NMR data. The reader is referred to the various original articles for the details. 

This review summarizes our efforts up to this time. In the next section an overview of our formulation will be presented. We will start with a summary of the theory of MCD. We will then provide a short summary of TDDFT theory to establish our starting point. Bringing together MCD and TDDFT theory allows us to formulate our approach for calculating MCD spectra. To finish the theoretical section we provide some notes on the steps of a calculation and interpretation of the results. 

G. L. Richmond contributes an authoritative summary of the theory and applications of second harmonic generation by a laser beam reflected from an electrode surface. This powerful new in-situ technique yields information on the structural and electronic properties of electrode surfaces. 

The structure of this contribution is as follows. After a brief summary of the theory of optical activity, with particular emphasis on the computational challenges induced by the presence of the magnetic dipole operator, we will focus on theoretical studies of solvent effects on these properties, which to a large extent has been done using various polarizable dielectric continuum models. Our purpose is not to give an exhaustive review of all theoretical studies of solvent effects on natural optical activity but rather to focus on a few representative studies in order to illustrate the importance of the solvent effects and the accuracy that can be expected from different theoretical methods. 

This section discusses a selection of NMR results with an emphasis on powder studies, on experiments that describe the dynamics of water at the protein surface, and on lysozyme as a model protein. Methods and theory are not discussed. For review discussions see Kuntz and Kauz-mann (1974), Bryant (1978), Koenig (1980), and Fung (1986). A recent review by Bryant (1988) is an elegant summary of the theory and results for NMR measurements of protein hydration, in powders and in solution. 

The thermal diffusion factor a is proportional to the mass difference, (mi — mo)/(mi + m2). The thermal diffusion process depends on the transport of momentum in collisions between unlike molecules. The momentum transport vanishes for Maxwellian molecules, particles which repel one another with a force which falls off as the inverse fifth power of the distance between them. If the repulsive force between the molecules falls off more rapidly than the fifth power of the distance, then the light molecule will concentrate in the high temperature region of the space, while the heavy molecule concentrates in the cold temperature region. When the force law falls off less rapidly than the fifth power of the distance, then the thermal diffusion separation occurs in the opposite sense. The theory of the thermal diffusion factor a is as yet incomplete even for classical molecules. A summary of the theory has been given by Jones and Furry 15) and by Hirschfelder, Curtiss, and Bird 14), Since the thermal diffusion factor a for isotope mixtures is small, of the order of 10", it remained for Clusius and Dickel (8) to develop an elegant countercurrent system which could multiply the elementary effect. 

In his review, Cannon presents brief but useful summaries of the theories of Stepanov and of Batuev (345a). 

Two of the leading mathematicians of the nineteenth century, Riemann and Weierstrass, played crucial roles in the further development of minimal surface theory. Their most important result, for which Weierstrass is accredited, is the triplet of integrals known as the Weierstrass equations. These equations offer a useful route to periodic minimal surfaces. A summary of the theory follows. 

However, it should be noted that this is only a somewhat simplified summary of the theory of PEL brushes. For a more detailed discussion the reader is referred to Refs. [6-35]. 

A comprehensive summary of the theory and applications of simulated annealing has been published by van Laarhoven and Aarts [14]. 

The summary of the theory of kinetics of solid-state reactions given here is applicable to TG investigations and also equally to rate measurements by other methods of thermal analyses. For more extensive treatments and further background information see References 4,5,10,29,71,76,103. 

Most chapters start with a brief summary of the theory in order to put the spreadsheet exercises in perspective, and to define the nomenclature used. The standard versions of Excel 95 through Excel 2000 for Windows 95 or Windows 98 are used. Many exercises use the Solver and the Analysis ToolPak, both of which are available in the standard Excel packages but may have to be loaded separately, as add-ins, in case this was not done initially. When use of chapter 10 is contemplated, the VBA help file should also be loaded. 

The notes above are not intended as a summary of the theory, but merely to give the reader some feeling for the use of Feynman graphs and their physical content. For a more complete account, see, e.g., Kelly [242]. 

This chapter describes the use of the CARS technique for gas temperature measurements in flames. The description includes the optical arrangement, a brief summary of the theory of CARS signal generation, data acquisition, and analysis. The establishment of a CARS measurement system for practical flames can be challenging and it is hoped that the practical information included in this chapter will be of assistance to a novice attempting to get measurable and accurate CARS temperatures in large flames. Since the establishment of CARS as a practical measurement technique for temperature measurements in flames, literature shows that a lot has been learned about this non-linear interaction resulting... 

For geochemical purposes, the dependence of isotope fractionation factors on temperature is the most important property. In principle, fractionation factors for isotope exchange reactions are also slightly pressure-dependent, but experimental studies have shown the pressure dependence to be of no importance within the outer earth environments (Hoefs 2004). Occasionally, the fractionation factors can be calculated by means of partition functions derivable from statistical mechanics. However, the interpretation of observed variations of the isotope distribution in nature is largely empirical and relies on observations in natural environments or experimental results obtained in laboratory studies. A brief summary of the theory of isotope exchange reactions is given by Hoefs (2004). 

A general ah initio VB wavefunction consisting of single and double excitations out of the SCF-MI determinant, are used to determine an ab initio interaction potential apt to the dynamics simulation of liquid water. A short summary of the theory involved will be presented. For a more detailed account see (Gianinetti et ai, 1996 Famulari, 1997 Specchio, 1997). The chosen scheme permits to eliminate BSSE in an a priori fashion, both at the SCF and at the correlated level. 

We presented a summary of the theory and of applications of an ab initio constant pressure variable cell shape molecular dynamics (VCS-MD) algorithm to the study of Earth forming mineral phases. The applications discussed here demonstrated the use of the technique as an efficient structure... 

The operating principles of multi-stage Isotope separating processes, such as the gaseous diffusion process, were first developed by Karl Cohen (19), and further exposed in a form more directly useful to chemical engineers by Benedict and Plgford ( ). Blgelelsen has presented a concise summary of the theory In his review ( ). ... 

For the main part this book gives an.account of our own work on the interaction of colloidal particles. It has not been written, however, on the assumption that the whole of the relevant literature is known to the reader. Accordingly we have added a number of chapters which may serve as an introduction to the main problem of the book. These chapters contain, for instance, a brief survey of some aspects of lyophobic colloids, a concise summary of the theory of the double layer, and a short introduction to the subject of Van der Waals— London forces. 

During his stay at NHL Mason was concerned with cument meters, a basic instrument in hydraulic research, with a paper published with Galen B. Schubauer (1904-1992), who later would greatly contribute to turbulence measurements. Mason then published two papers on the salt velocity method as proposed by Charles M. Allen (1871-1950), which was often applied in prototype experiments, to determine the discharge of turbines or that in mountainous rivers. Once with the Beach Erosion Board, Mason started with the publication of a series of papers, including for instance A study of progressive oscillatory waves in water, which was followed by A summary of the theory of oscillatory waves, co-authored by Morrough P. O Brien (1902-1988). 

Mulliken s interest in the electronic levels in molecules in the 1920s was stimulated by suggestions that their molecular spectra bore similarities to atomic spectra and definite relationships could be discerned for isosteric molecules. He found that the spectroscopic analogy between isosteric molecules could be extended to atoms with the same number of electrons, and this relatirmship was to lead subsequently to the united atom approach. He and Birge classified the electronic states in diatomic molecules using the same Russell-Saunders classification used previously for atomic states. Himd s theoretical analysis [ 149,162-166] of the nature of electronic states in molecules therefore proved to be timely for Mulliken and led him to publish [167-171] a summary of the theory and provide extra experimental evidence supporting it. In the molecular orbital theory Hund showed how the concept of atomic orbitals and the mathematical procedures developed to define them could... 

The volume of literature on this subject is so enormous that it is possible to make reference here only to more recent, reviews. Some summaries of the theories regarding silicosis which appeared since 1955 are as follows. 

The discussion that follows is not by any means a summary of the theories that have appeared on colloidal chemistry up to present time. Such a comprehensive presentation would be entirely beyond the scope of this book. On the contrary the author, with due consideration of the work of other investigators, has endeavored to present a point of view that has been prompted by years of personal observation and thought, and which at the same time gives a simple and comprehensive explanation of the largest number of facts. For purposes of illustration constitution formulas have been made use of, such as the author has previously employed in his monograph, Zur Erkenntnis der Kolloide. ... 

---