Cosine squared potential

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

The rectangular form of the well, being cmde, nevertheless allows us to obtain a rather adequate description of local-order intermolecular forces arising in a liquid, which generally presents a state intermediate between solid body and gas. We emphasize that popular [13, 14] parabolic, cosine, or cosine-squared potential wells generally give poor description of the wideband... 

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). 

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. 

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 42. Form of the cosine-squared potential, in which a dipole vibrates (solid line). The horizontal dashed line marks the mean angular amplitude ( 45°) on the potential curve C/(0). Dotted curve denotes dimensionless intermolecular static electric field. For the chosen p value (p = 0.8) about 46% and 56% of the dipoles perform complete rotation, respectively, in the case of ordinary and heavy water. Upper horizontal line marks the value of the potential near edge of well. In the center of regions A the potential undergoes minimum (zero value) and the absolute value of static field-maximum.

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.

Fitted (A) and Estimated (B) Parameters of the Hat-Curved-Cosine-Squared Potential Model11... 

Both hat-curved-harmonic oscillator and hat-curved-cosine-squared potential composite models considered in this section give excellent description of wideband spectra of water H20 and D20 in the range from 0 to 1000 cm-1. However, it appears that the physical picture of fast vibrations of the H-bonded molecules differ for these two approaches. In the first one, where... 

Figure 57. Forms of the potential well pertaining to the scheme shown in Fig. 56 for pure librations (a) and pure transverse translations (b). Solid line refers to the H-bond length L = 1.0 A, and dashed line refers to L = 1.42 A. Calculation for water H20 at 27°C. In Fig. (a), dashed-and-dotted curve refers to the cosine-squared potential.

Figure 65. Frequency dependence of dielectric loss in ice Ih. (a) Recorded [171] spectrum in the translational band, T = 100 K. (b) The main part of the loss line shown in Fig. (a), calculated for the constant field (solid line) and for the cosine-squared potential (dashed line) points represent experimental data from Ref. 171.

The first and second terms in the right-hand part are, respectively, the transverse and longitudinal components of the spectral function. In other words, these terms are stipulated by reorientation of the projections of a dipole moment, which are, respectively, normal and collinear to the potential symmetry axis. The potential under consideration comprises two wells with oppositely directed symmetry axes. Such is the cosine-squared potential... 

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