Model-free theory

This model can also be extended to further levels of librational motion, each with its own order parameter and correlation time, and has been underpinned more formally in the three-r case of two librational levels plus rotation [7]. It is clearly of a general nature. One may indeed choose to abandon relationship (4.23) and to re-interpret in (4.24) as a more general order parameter. One thereby achieves a model-free theory, which has advantages over alternative, more specific models for polymer motion (see chapter 6), when the motional details are not clear. 

Lipari G and Szabo A 1982 Model-free approach to the interpretation of nuclear magnetic resonance relaxation in macromolecules 1. Theory and range of validity J. Am. Chem. Soc. 104 4546-59... 

The integral equation method is free of the disadvantages of the continuum model and simulation techniques mentioned in the foregoing, and it gives a microscopic picture of the solvent effect within a reasonable computational time. Since details of the RISM-SCF/ MCSCF method are discussed in the following section we here briefly sketch the reference interaction site model (RISM) theory. 

The free theory for the quench models is provided by the potential (4), where A = 0 and m2(t) changes signs either instantaneously or for a finite period. In the Minkowski spacetime, we can apply the LvN method simply by letting R = 1. Before the phase transition (rrii = (mg + m2)1/2), all the modes are stable and oscillate around the true vacuum ... 

To indicate to the reader that the theory is based on the observable distribution of charge and its theoretical consequences. It is thus not only model-free but relates directly to the measurable properties of a system. 

We now direct our attention to the calculation of the a i parameters. The first and second derivatives, dE /dNk)° and d Ef /dNl)°, are most conveniently obtained from SCF-Xa theory [174], whieh offers the advantage of permitting calculations for any desired integer or fractional electron population. It is, indeed, important to account for the fact that these derivatives depend on N. The difficulty is that calculations of this sort cannot be performed directly for atoms that are actually part of a molecule. So one resorts to model free-atom calculations to mimic the behavior of atoms that are in a molecule but do not experience interactions with the other atoms in the host molecule (Table 10.3). 

The inconsistency between predictions made by MO REPEs and VB REPEs may be attributed to the different natures of simple MO and VB theories, i.e., the simple MO theory is a one-electron model, free of electron correlation, but VB theory is a many-electron, correlation model. This difference may result in the different topological dependence of -conjugation in MO and VB models, as illustrated by the [n]phenanthrene series. 

The parabolic dependence of energy band on k expected by the free electron model is not observed in Fig. 10. Different choices of orbital exponent giving less diffuse atomic orbitals promote such a behavior, but more work is needed to determine the criteria for choosing parameters in the infinite-model EH theory. Other choices of orbital exponents also predict cubic geometry to be more stable than linear. 

Clearly, one has to make an allowance for the electron in the Nernst-type theories— which are thermodynamically valid (model free) and therefore can be used today. This was done by a term Ecq (potential x the electronic charge = energy in electrostatics). The E was bought of by Nernst as the potential between the metal and the solution. Now, the thermodynamic equilibrium for Eq. (3.100) ... 

The present paper is concerned with mixtures composed of a highly nonideal solute and a multicomponent ideal solvent. A model-free methodology, based on the Kirkwood—Buff (KB) theory of solutions, was employed. The quaternary mixture was considered as an example, and the full set of expressions for the derivatives of the chemical potentials with respect to the number of particles, the partial molar volumes, and the isothermal compressibility were derived on the basis of the KB theory of solutions. Further, the expressions for the derivatives of the activity coefficients were applied to quaternary mixtures composed of a solute and an ideal ternary solvent. It was shown that the activity coefBcient of a solute at infinite dilution in an ideal ternary solvent can be predicted in terms of the activity coefBcients of the solute at infinite dilution in subsystems (solute + the individual three solvents, or solute + two binaries among the solvent species). The methodology could be extended to a system formed of a solute + a multicomponent ideal mixed solvent. The obtained equations were used to predict the gas solubilities and the solubilities of crystalline nonelectrolytes in multicomponent ideal mixed solvents. Good agreement between the predicted and experimental solubilities was obtained. 

In this paper, a previously developed expression for the activity coefficient of a solute at infinite dilution in multi-component solutions [22—24) will be applied to the solubility of environmentally significantcompounds in aqueous solvent mixtu res. The above expression for the activity coefficient of a solute at infinite dilution in multicomponent solutions [22— 24) is based on the fluctuation theory of solutions [25). This model-free thermodynamic expression can be applied to both binary and multicomponent solvents. 

The theoretical models discussed above are frequently employed in the description of the kinetics of gas-phase reactions, especially reactions of atoms and free radicals. This class of reactions is of interest in a broader scientific context, and a better understanding of their mechanism is of primary importance for the development of chemical modeling. Free atoms and radicals are very reactive species, which occur in and take part in many different reaction systems. Therefore, a radical reaction usually proceeds in competition with a few parallel or subsequent processes. The kinetic behavior of the reaction system may be very complicated and difficult for quantitative description. Theoretical investigations of the reaction kinetics provide information useful for a better understanding and correct interpretation of experimental findings. Results of ab initio calculations are employed to evaluate the rate constant in terms of the computational methods of the reaction rate theory. 

In view of the above comments, error estimates are usually made on the basis of overall reproducibility of, and matching between independent experimental or theoretical results, rather than on the basis of the precision reachable with a particular measurement and refinement model. There are several approaches that allow us to gain quantitative information on experimental reproducibility and uncertainties. These include the pseudoatom interpretation of error free, theoretical data [56, 57, 70-72], comparative analysis of experimental data sets in terms of different constrained models [73], theory versus experimental comparison of results obtained for the same system [74-76], systematic studies on a series of related compounds [77], and the simultaneous analysis of data collected at different temperatures [66]. 

Conventional models or theories consider only the segmental relaxation of amorphous polymers and the primary relaxation of nonpolymeric glass-formers in the change of molecular mobility with temperature and pressure (and concomitant changes in free volume and/or configurational entropy) leading to vitrification. Here we wish to recognise two different kinds of secondary relaxation processes. There... 

For diffusion of liquid through rubbery polymer composites, Fickian and non-Fickian diffusion theories are frequently used to describe the mechanism of transport, but for gas or vapour, other models have been developed to fit experimental data of diffusion profiles. The models of gas transport include Maxwell s model," free volume increase mechanism," solubility increase mechanism," nanogap hypothesis," Nielsen model, " " Bharadwaj model, ° Cussler model " " and Gusev and Lusti model, " etc. 

---