Simplified harmonic models

In order to give a more quantitative view of the relative perturbation of each vibrator (C=C stretching and CH2 scissoring), the authors performed calculations for this complex based on simplified harmonic models involving mechanical couplings between the C=C... 

Fig. 6.31 Normalised intermediate scattering function from C-phycocyanin (CPC) obtained by spin-echo [335] compared to a full MD simulation (solid line) exhibiting a good quantitative matching. In contrast the MD results from simplified treatments as from protein without solvent (long dash-short dash /me), with point-like residues (Cpt-atoms) (dashed line) or coarse grained harmonic model (dash-dotted line) show similar slopes but deviate in particular in terms of the amplitude of initial decay. The latter deviation are (partly) explained by the employed technique of Fourier transformation. (Reprinted with permission from [348]. Copyright 2002 Elsevier)...

In finite boundary conditions the solute molecule is surrounded by a finite layer of explicit solvent. The missing bulk solvent is modeled by some form of boundary potential at the vacuum/solvent interface. A host of such potentials have been proposed, from the simple spherical half-harmonic potential, which models a hydrophobic container [22], to stochastic boundary conditions [23], which surround the finite system with shells of particles obeying simplified dynamics, and finally to the Beglov and Roux spherical solvent boundary potential [24], which approximates the exact potential of mean force due to the bulk solvent by a superposition of physically motivated tenns. 

The model presented here is simplified in several ways (harmonic approximation, purely classical treatment of inner-sphere reorganization), and it says little about the pre-exponential factor A. But it does... 

Figure 3.1 shows a simplified picture of an interface. It consists of a multilayer geometry where the surface layer of thickness d lies between two centrosymmetric media (1 and 2) which have two different linear dielectric constants e, and e2, respectively. When a monochromatic plane wave at frequency co is incident from medium 1, it induces a nonlinear source polarization in the surface layer and in the bulk of medium 2. This source polarization then radiates, and harmonic waves at 2 to emanate from the boundary in both the reflected and transmitted directions. In this model, medium 1 is assumed to be linear. 

A simplified model of the stretching frequency of 2 atoms and their connecting bond is offered by the harmonic oscillator of two masses connected by a spring (Herzberg 1950). 

The cosine-squared potential model was simplified in terms of the so-called stratified approximation, for which the spectral function Tcs(Z) is given in GT, p. 300 and in VIG, p. 462. We remark that the dielectric spectra calculated rigorously for the CS model agree with this approximation, while simpler quasi-harmonic approximation (GT, p. 285 VIG, p. 451) used in item A yields for p > la too narrow theoretical absorption band. 

FIG. 17. Comparison between Monte Carlo simulation (symbols) with decamers and the mean-field solution (lines) of the simplified model with the harmonic grafting potential. The net osmotic pressure is given as a function of separation for various value of surface o and y. solid line and circles y —1.0, one charge per 50.77 A2. Dotted line and squares y = 1.0, one charge per 101.54 A2. Dashed line and diamonds y = 0.875, one charge per 50.77 A2 [61]. 

The first usable results were obtained by Rice and Ramsperger" and Kassel,who were able to deduce from a simplified model of a molecule, consisting of a set of harmonic oscillators, that... 

The assignment of appropriate models to a given system has not always been strongly foimded. The number of observable rotational transitions is few and the presence of any translational vibrations merely complicates the picture. Often several similar solutions can be found for widely different a and h values [11]. Perhaps the greatest possibility for confusion occurs between 1-D and 2-DP type spectra. Early INS work avoided a full description of the potential in terms of spherical harmonics and usually worked within simplifying approximations specific to individual cases. The relationship between those models and the forms we develop here is not necessarily straightforward. 

The situation changes drastically if the field mode is allowed to interact with some detector placed inside the cavity. Following other findings [188,189] we demonstrate the effect in the framework of a simplified model, when a harmonic oscillator tuned to the frequency of the resonant mode is placed at the point of maximum of the amplitude mode function v /mn(x,y L ) in the 3D rectangular cavity. 

In simple model calculations we can mimic this effect by writing a Hamiltonian like Equation (10) in which Q(t) appears as an independent oscillator, but it must be understood that this Hamiltonian is a simplified model designed to produce a fluctuating PES, and Q(t) is a quantity that parametrizes this fluctuation. However, it is true that one cannot assume beforehand that the distance Q(t) is harmonic, one has to calculate it, Fourier transform it and check whether it is peaked at some frequency, as we will do in the examples in the next section. 

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