Tractability

The continuum treatment of dispersion forces due to Lifshitz [19,20] provides the appropriate analysis of retardation through quantum field theory. More recent analyses are more tractable and are described in some detail in several references [1,3,12,21,22],... 

This section has focused mainly on the internal dynamics of small molecules, where a coherent picture of the detailed mtemal motion has been emerging from intense efforts of many theoretical and experimental workers. A natural question is whether these kinds of issues will be important m the dynamics of larger molecules, and whether their investigation at the same level of detail will be profitable or tractable. 

The advantages of SIMS are its high sensitivity (ppm detection limit for certain elements), its ability to detect hydrogen and the emission of molecular fragments which often bear tractable relationships with the parent... 

The siipercell plane wave DFT approach is periodic in tliree dimensions, which has some disadvantages (i) thick vacuum layers are required so the slab does not interact with its images, (ii) for a tractably sized unit cell, only high adsorbate coverages are modelled readily and (iii) one is limited in accuracy by the fonn of the... 

The parameter /r tunes the stiffness of the potential. It is chosen such that the repulsive part of the Leimard-Jones potential makes a crossing of bonds highly improbable (e.g., k= 30). This off-lattice model has a rather realistic equation of state and reproduces many experimental features of polymer solutions. Due to the attractive interactions the model exhibits a liquid-vapour coexistence, and an isolated chain undergoes a transition from a self-avoiding walk at high temperatures to a collapsed globule at low temperatures. Since all interactions are continuous, the model is tractable by Monte Carlo simulations as well as by molecular dynamics. Generalizations of the Leimard-Jones potential to anisotropic pair interactions are available e.g., the Gay-Beme potential [29]. This latter potential has been employed to study non-spherical particles that possibly fomi liquid crystalline phases. 

Shakhnovich E 1998 Protein design a perspective from simple tractable models Folding Design 3 R45-R58... 

Understanding VER in condensed phases has proven difficult. The experiments are hard. The stmcturally simple systems (diatomic molecules) involve complicated relaxation mechanisms. The stmctures of polyatomic molecules are obviously more complex, but polyatomic systems are tractable because the VER mechanisms are somewhat simpler. 

At the present time there exist no flux relations wich a completely sound cheoretical basis, capable of describing transport in porous media over the whole range of pressures or pore sizes. All involve empiricism to a greater or less degree, or are based on a physically unrealistic representation of the structure of the porous medium. Existing models fall into two main classes in the first the medium is modeled as a network of interconnected capillaries, while in the second it is represented by an assembly of stationary obstacles dispersed in the gas on a molecular scale. The first type of model is closely related to the physical structure of the medium, but its development is hampered by the lack of a solution to the problem of transport in a capillary whose diameter is comparable to mean free path lengths in the gas mixture. The second type of model is more tenuously related to the real medium but more tractable theoretically. 

T-Jhile the stoichiometric relations have rendered the above problem tractable by permitting an explicit solution of the dusty gas model flux relations, it should be pointed out that they do not lead to equally radical simplifications with all flux models. In the case of the Feng and Stewart models [49- for example, Che total flux of species r is formed by in-... 

The primary problem with explicit solvent calculations is the significant amount of computer resources necessary. This may also require a significant amount of work for the researcher. One solution to this problem is to model the molecule of interest with quantum mechanics and the solvent with molecular mechanics as described in the previous chapter. Other ways to make the computational resource requirements tractable are to derive an analytic equation for the property of interest, use a group additivity method, or model the solvent as a continuum. 

Explicitly correlated wave functions have been shown to give very accurate results. Unfortunately, these calculations are only tractable for very small molecules. 

Many researchers have performed calculations that include the two large-magnitude components of the spinnors. This provides a balance between high accuracy and making the calculation tractable. Such calculations are often done on atoms in order to obtain the wave function description used to create relativistic core potentials. 

It is extremely difficult to know values for all of these parameters precisely. Therefore, absolute quantitation is almost never attempted. The determination of relative atomic ratios is an inherently more tractable approach, however. This method is best illustrated by consideration of a binary material composed exclusively of atoms A and B that is perfectiy homogeneous up to the surface. In this case, independent equations can be developed relating the number of atoms sampled to the xps intensity for each atom as follows ... 

Xps ndAes Instrumentation. The instmmentation required to perform xps and aes analyses is generally sophisticated and expensive (19). The need for UHV conditions in order to retain surface cleanliness for a tractable period of time was mentioned above. Beyond this requirement (and the hardware that accompanies it), the most important components of an electron spectrometer system are the source, the electron energy analyzer, and the electron detector. These will be discussed in turn below. 

Copoly(amide-imides) comprise an important class of copolyimides that have been developed into a commercial product. Incorporating the amide linkage into the PI makes the polymer more tractable than simple Pis, but involves a loss in thermal stabiUty. However, copoly(amide—imides) still possess quite good thermal stabiUties, intermediate between those of polyamides and Pis (12). They are relatively inexpensive to synthesize. 

Monomers. A wide variety of monomers can be used, and they are chosen on the basis of cost and abiUty to impart specific properties to the final product. Water solubiUties of iadustriaHy important monomers are shown ia Table 1 (38). The solubiUty of the monomer ia water affects the physical chemistry of the polymerization. Functional monomers like methacrylic and acryUc acid, infinitely soluble ia water, are also used. These monomers impart long-term shelf stabiUty to latices by acting as emulsifiers. The polymerization behavior of some monomers, such as methacrylic acid, as well as the final latex properties are iafiuenced by pH. For optimum results with these acids, polymerization is best performed at a pH of ca 2. After polymerization, the latex is neutralized to give adequate shelf stabiUty at tractable viscosities. 

An all aromatic polyetherimide is made by Du Pont from reaction of pyromelUtic dianhydride and 4,4 -oxydianiline and is sold as Kapton. It possesses excellent thermal stabiUty, mechanical characteristics, and electrical properties, as indicated in Table 3. The high heat-deflection temperature of the resin limits its processibiUty. Kapton is available as general-purpose film and used in appHcations such as washers and gaskets. Often the resin is not used directly rather, the more tractable polyamide acid intermediate is appHed in solution to a surface and then is thermally imidi2ed as the solvent evaporates. 

The majority of polyetherimides are tractable and their polymerization can be performed in solution or in the melt. High molecular weight polyetherimides have been synthesized via one-step imide—amine exchange reaction between bis(etherimide)s and diamine (67) according to the following ... 

Numerical simulations are designed to solve, for the material body in question, the system of equations expressing the fundamental laws of physics to which the dynamic response of the body must conform. The detail provided by such first-principles solutions can often be used to develop simplified methods for predicting the outcome of physical processes. These simplified analytic techniques have the virtue of calculational efficiency and are, therefore, preferable to numerical simulations for parameter sensitivity studies. Typically, rather restrictive assumptions are made on the bounds of material response in order to simplify the problem and make it tractable to analytic methods of solution. Thus, analytic methods lack the generality of numerical simulations and care must be taken to apply them only to problems where the assumptions on which they are based will be valid. 

Computational methods have played an exceedingly important role in understanding the fundamental aspects of shock compression and in solving complex shock-wave problems. Major advances in the numerical algorithms used for solving dynamic problems, coupled with the tremendous increase in computational capabilities, have made many problems tractable that only a few years ago could not have been solved. It is now possible to perform two-dimensional molecular dynamics simulations with a high degree of accuracy, and three-dimensional problems can also be solved with moderate accuracy. 

A tractable example is the pmf between two particular particles in a macromolecule as a function of their separation q. The free energy to increase q by 8q becomes... 

Having demonstrated that our simulation reproduces the neutron data reasonably well, we may critically evaluate the models used to interpret the data. For the models to be analytically tractable, it is generally assumed that the center-of-mass and internal motions are decoupled so that the total intermediate scattering function can be written as a product of the expression for the center-of-mass motion and that for the internal motions. We have confirmed the validity of the decoupling assumption over a wide range of Q (data not shown). In the next two sections we take a closer look at our simulation to see to what extent the dynamics is consistent with models used to describe the dynamics. We discuss the motion of the center of mass in the next section and the internal dynamics of the hydrocarbon chains in Section IV.F. 

Although the correlation function formalism provides formally exact expressions for the rate constant, only the parabolic barrier has proven to be analytically tractable in this way. It is difficult to consistently follow up the relationship between the flux-flux correlation function expression and the semiclassical Im F formulae atoo. So far, the correlation function approach has mostly been used for fairly high temperatures in order to accurately study the quantum corrections to CLST, while the behavior of the functions Cf, Cf, and C, far below has not been studied. A number of papers have appeared (see, e.g., Tromp and Miller [1986], Makri [1991]) implementing the correlation function formalism for two-dimensional PES. 

The limited tractability of the polymer makes processing in conventional plastics form very difficult. Nevertheless the materials have been used in the manufacture of seals, gaskets and piston rings (Vespel-Du Pont) and also as the binder resin for diamond grinding wheels. 

The successful introduction of the polyimides stimulated attempts to produce somewhat more tractable materials without too serious a loss of heat resistance. This led to the availability of a polyamide-imides, polyester-imides and the polybismaleinimides, and in 1982 the polyether-imides. 

However, momomers based on these units alone have extremely high melting points and are intractable. This has led to the use of a variety of techniques to produce more tractable materials which may be processed without degradation whilst retaining many of the features of liquid crystalline materials. These techniques include ... 

At the present time, doped ICPs are not normally capable of being processed like normal thermoplastics. Processes usually Involve high-pressure moulding of finely powdered polymers under vacuum or an inert gas. However, modification of some ICPs with, for example, alkyl or alkoxy side groups may produce soluble, and thus more tractable, polymers. 

In order to make the mathematics tractable, approximations must be made. The choice of approximations has produced a variety of MO methods, the judicious application of which can provide valuable insight into questions of bonding, structure, dynamics, and reactivity. The discussion that follows will not be sufficiently detailed or complete for the reader to understand how the calculations are performed or the details of the approximations. Instead, the nature of the information that is obtained will be described, and the ways in which organic chemists have applied the results of MO theoiy will be illustrated. Several excellent books are available which provide detailed treatment of various aspects of MO methods. 

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