Parabolic potential curves

Semiempirical Model of Radical Reaction as an Intersection of Two Parabolic Potential Curves... 

SEMIEMPIRICAL MODEL OF RADICAL REACTION AS AN INTERSECTION OF TWO PARABOLIC POTENTIAL CURVES... 

Figure 2.7 (a) Experimental (full line) and ideally parabolic (dashed line) electrocapillary curves and the corresponding (b) charge vs. potential, and (c) capacity vs. potential curves. 

Organic adsorption (cont.) lateral interactions, 983 and the maximum of the coverage-potential curve test, 971 naphty 1 compounds, 982 and the parabolic coverage-potential curve test. 970... 

Fig. 6.109. Coverage vs. potential curve for the adsorption of organic molecules on electrodes. The coverage due to the adsorption process of an organic molecule from solution follows a parabolic path, with a maximum close to the pzc.

The main advantage of the hat-curved potential is that it is possible to narrow the width Avor of the librational absorption band by decreasing the form factor /. Indeed, Avor attains its maximum value when/ = 1. Note that / = 1 is just the case of the hat flat or its simplified variant, the hybrid model, both of which were described in Section IV. The latter was often applied before (VIG) and is characterized by a rather wide absorption band, especially in the case of heavy water. In another extreme case, / — 0, the linewidth Avor becomes very low. When / = 0, we have the case of the parabolic potential well, whose dielectric response was described, for example, in GT and VIG. Thus, when the form factor/of the hat-curved well decreases from 1 to 0, the width Avor decreases from its maximum to some minimum value. 

The origin of this threshold could easily be explained. If p2 force constant Ka and the second term involves both constants Krt and k. At a very small (52 the first parabolic-potential term predominates, to which the nonzero threshold resonance frequency vstr corresponds to (50 —> 0. This frequency is determined completely by the force constant Ka. Since the second term is positive, the potential curve m([1) becomes steeper for larger p this leads to the increase of the RR frequency vstr. 

Figure 68. Dielectric loss calculated for the composite model (solid lines) experimental curves [17, 51, 54] (dashed lines) contribution to loss due to dipoles performing restricted rotation in the parabolic potential (dot-dashed lines).

Figure 1-6 Potential energy curve for a diatomic molecule. Solid line indicates a Morse potential that approximates the actual potential. Broken line is a parabolic potential for a harmonic oscillator. De and D0 are the theoretical and spectroscopic dissociation energies, respectively.

The capacitance-potential curves have parabolic shapes with a minimum at Eq = 0 (i.e., at the potential of zero charge). 

It was recently shown (Ratner and Levine, 1980) that the Marcus cross-relation (62) can be derived rigorously for the case that / = 1 by a thermodynamic treatment without postulating any microscopic model of the activation process. The only assumptions made were (1) the activation process for each species is independent of its reaction partner, and (2) the activated states of the participating species (A, [A-], B and [B ]+) are the same for the self-exchange reactions and for the cross reaction. Note that the following assumptions need not be made (3) applicability of the Franck-Condon principle, (4) validity of the transition-state theory, (5) parabolic potential energy curves, (6) solvent as a dielectric continuum and (7) electron transfer is... 

The potential curve for a diatomic molecule (blue) where / e represents the equilibrium bond distance and De is the bond dissociation energy. The parabolic curve (red) represents the behavior of a true harmonic oscillator. 

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