Effective field model

Fig. 4.17 Effective field model for anisotropic hyperfine structure. +Ba and —Ba are the fields at the nucleus from the hyperfine couphng with an unpaired electron for mg = + /2 and = -Vi, respectively. Sn is in the direction of the applied field with a magnitude of Bjv = B+ and...

Onsager s reaction field model in its original fonn offers a description of major aspects of equilibrium solvation effects on reaction rates in solution that includes the basic physical ideas, but the inlierent simplifications seriously limit its practical use for quantitative predictions. It smce has been extended along several lines, some of which are briefly sunnnarized in the next section. 

Continuum models go one step frirtlier and drop the notion of particles altogether. Two classes of models shall be discussed field theoretical models that describe the equilibrium properties in temis of spatially varying fields of mesoscopic quantities (e.g., density or composition of a mixture) and effective interface models that describe the state of the system only in temis of the position of mterfaces. Sometimes these models can be derived from a mesoscopic model (e.g., the Edwards Hamiltonian for polymeric systems) but often the Hamiltonians are based on general symmetry considerations (e.g., Landau-Ginzburg models). These models are well suited to examine the generic universal features of mesoscopic behaviour. 

The simplest reaction field model is a spherical cavity, where only the net charge and dipole moment of the molecule are taken into account, and cavity/dispersion effects are neglected. For a net charge in a cavity of radius a, the difference in energy between vacuum and a medium with a dielectric constant of e is given by the Bom model. ... 

This would not be expected simply on the basis of a crystal-field model, for the d orbitals will contract with increasing positive charge and hence interact less well with the ligand point charges . The modest decreases in bond length as one traverses the series (Eq. 6.9) are unlikely to compensate for, let alone override, the effects of such orbital contraction. Finally, to add to the confusion, we also note from Eq. (6.7) that zio -t values increase as we go down the periodic table (Eq. 6.10). 

The presented scheme offers several extensions. For example, the model gives a clear route for an additional inclusion of entanglement constraints and packing effects [15]. Again, this can be realized with the successful mean field models based on the conformational tube picture [7,9] where the chains do not have free access to the total space between the cross-links but are trapped in a cage due to the additional topological restrictions, as visualized in the cartoon. 

Marriott and Topsom have recently developed theoretical scales of substituent field and resonance parameters. The former correspond to the traditional inductive parameters but these authors are firm believers in the field model of the so-called inductive effect and use the symbol The theoretical substituent field effect scale is based on ab initio molecular orbital calculations of energies or electron populations of simple molecular systems. The results of the calculations are well correlated with Op values for a small number of substituents whose Op values on the various experimental scales (gas-phase, non-polar solvents, polar solvents) are concordant, and the regression equations are the basis for theoretical Op values of about 50 substituents. These include SOMe and S02Me at 0.37 and 0.60 respectively, which agree well with inherent best values in the literature of 0.36 and 0.58. However, it should be noted that a, for SOMe is given as 0.50 by Ehrenson and coworkers . 

More recently, Saez et al. [27] have carried out numerical simulations to characterize the ultrasonic field propagation and to obtain the spatial distribution of the mechanical effects. The model is based on the assumption of linear wave propagation in a homogeneous media and the results are based on the solution of the... 

Flexible force field models require 30 or more beads in Cartesian path integral descriptions to obtain converged intramolecular energies. Rigid water models require only about five beads due to the weaker intermolecular quantum effects. 

TIP5P(PIMC) force field model. This implies that quantum effects are nonneg-ligible. 

The finite-temperature field theory has been the most popular approach to equilibrium phase transitions (L. Dolan et.al., 1974). The effective potential of quantum fluctuations around a classical background provides a convenient tool to describe phase transitions. The symmetry breaking or restoration mechanism can be illustrated by a scalar field model with broken symmetry... 

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