Cosine potential model

The complex susceptibility x( ) yielded by Eq. (9), combined with Eq. (22) when the small oscillation approximation is abandoned, may be calculated using the shift theorem for Fourier transforms combined with the matrix continued fraction solution for the fixed center of oscillation cosine potential model treated in detail in Ref. 25. Thus we shall merely outline that solution as far as it is needed here and refer the reader to Ref. 25 for the various matrix manipulations, and so on. On considering the orientational autocorrelation function of the surroundings ps(t) and expanding the double exponential, we have... 

The summation for the coefficient of the P2 term likewise includes phase insensitive contributions where I m = Im, but also now one has terms for which I = I 2, which introduce a partial dependence on relative phase shifts— specifically on the cosine of the relative phase shift. Again, this conclusion has long been recognized for example, by an explicit factor cos(ri j — ri i) in one term of the Cooper-Zare formula for the photoelectron (3 parameter in a central potential model [43]. 

A. Previous models of water (see 1-6 in Section V.A.l) and also the hat-curved model itself cannot describe properly the R-band arising in water and therefore cannot explain a small isotope shift of the center frequency vR. Indeed, in these models the R-band arises due to free rotors. Since the moment of inertia I of D20 molecule is about twice that of H20, the estimated center of the R-band for D20 would be placed at y/2 lower frequency than for H20. This result would contradict the recorded experimental data, since vR(D20) vR(H20) 200 cm-1. The first attempt to overcome this difficulty was made in GT, p. 549, where the cosine-squared (CS) potential model was formally (i.e., irrespective of a physical origin of such potential) applied for description of dielectric response of rotators moving above the CS well (in this work the librators were assumed to move in the rectangular well). The nonuniform CS potential yields a rather narrow absorption band this property agrees with the experimental data [17, 42, 54]. The absorption-peak position Vcs depends on the field parameter p of the model given by... 

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. 

Figure 37. Absorption-frequency dependence, water H20 at temperature 27°C. Calculation for the HC—HO model (solid line) and for the hybrid-cosine-squared potential model (dashed-and-dotted line). Dahsed curve Experimental data [42], (b) Same as in Fig. 34c but refers to T — 300 K.

Figure 43. Wideband FIR spectra calculated for the composite hat-curved-cosine-squared potential model (solid lines) dashed-and-dotted lines mark the contribution due to dipoles vibrating in the shallow CS well. Water H20 (a, c, e) and water D20 (b, d, f) at 22.2°C. Absorption coefficient (a-d) and dielectric loss (e, f) in Figs, a, b, e, f, dashed lines refer to the experiment [17, 51, 54]. In Figs, c, d dahsed lines mark the contribution to absorption due to dipoles reorienting in a deep hat-curved well.

Analysis by a dynamical model that incorporated a large-angle range on either side of the planar anti form gave good agreement with previous data at 225 °C [1]. A three-term cosine potential for internal rotation about the N-N bond was employed. The nozzle temperature was 225 °C. 

Many of the torsional terms in the AMBER force field contain just one term from the cosine series expansion, but for some bonds it was found necessary to include more than one term. For example, to correctly model the tendency of O-C—C-O bonds to adopt a gauche conformation, a torsional potential with two terms was used for the O—C—C—O contribution ... 

FIGURE 19.1 Cosine type angle dependence of the potential energy of the molecule or radical orientation in the model of anisotropic hard cage of the polymer matrix. 

The essential properties of incommensurate modulated structures can be studied within a simple one-dimensional model, the well-known Frenkel-Kontorova model . The competing interactions between the substrate potential and the lateral adatom interactions are modeled by a chain of adatoms, coupled with harmonic springs of force constant K, placed in a cosine substrate potential of amplitude V and periodicity b (see Fig. 27). The microscopic energy of this model is ... 

The primary aim of the staggered model approach is to describe the curvature of the potential function at the minima. Therefore, in order to use this approach for barrier determination, one has to choose a potential function with a curvature at the minima equal to that derived for the staggered model. The most obvious choice of potential function is again the one-term cosine function. It is therefore no surprise that the two approaches, the staggered model approach and the approach using Eqs. (13) and (14), lead to the same results if they use the same potential function. 

B. Hat-Curved-Cosine-Squared Potential Composite Model... 

Second, an alternative hat-curved-cosine-squared potential (HC-CS) model is also considered, which, as it seems, is more adeuate than the HC-HO model. The CS potential is assumed to govern angular deflections of H-bonded rigid dipole from equilibrium H-bond direction. The HC-CS model agrees very well with the experimental spectra of water. 

Inside a rectangular well a dipole rotates freely until it suffers instantaneous collision with a wall of the well and then is reflected, while in the field models a continuously acting static force tends to decrease the deflection of a dipole from the symmetry axis of the potential. Therefore, if a dipole has a sufficiently low energy, it would start backward motion at such a point inside the well, where its kinetic energy vanishes. Irrespective of the nature of forces governing the motion of a dipole in a liquid, we may formally regard the parabolic, cosine, or cosine squared potential wells as the simplest potential profiles useful for our studies. The linear dielectric response was found for this model, for example, in VIG (p. 359) and GT (p. 249). 

More success was gained in the calculations [32] based on application of the cosine squared (CS) potential applied to nonassociated liquids (CH3F and CHF3). However, the results obtained by using the CS model are poor if compared with those given by the hybrid model, since the CS model yields... 

Thus, evolution of semiphenomenological molecular models mentioned in Section V.A (items 1-6) have led to the hat-curved model as a model with a rounded potential well. This model combines useful properties of the rectangular potential well and those peculiar to the field models based on application of the parabolic, cosine, or cosine-squared potentials. Namely, the hat-curved model retains the main advantage of the rectangular-well model—its possibility to describe both the librational and the Debye-relaxation bands. 

B. In Ref. 7 an approach was presented, resembling that given in item A, but now a preliminary physical interpretation of the model was for the first time presented and the mathematical theory used was more rigorous. Each water molecule was assumed to participate, like a solid body, in the motions directed along different axes characterized by different projections pj of a molecule dipole moment p. In Ref. 7, two potential wells were introduced, rectangular and cosine squared. 

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